Slurry density influence on ball mill behavior...SLURRY DENSITY INFLUENCE ON BALL MILL BEHAVIOR by'...
83
Slurry density influence on ball mill behavior Item Type text; Thesis-Reproduction (electronic) Authors Carson, Harry Benjamin, 1943- Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 09/08/2021 14:14:47 Link to Item http://hdl.handle.net/10150/318784
Transcript of Slurry density influence on ball mill behavior...SLURRY DENSITY INFLUENCE ON BALL MILL BEHAVIOR by'...
Slurry density influence on ball mill behavior
Item Type text; Thesis-Reproduction (electronic)
Authors Carson, Harry Benjamin, 1943-
Publisher The University of Arizona.
Rights Copyright © is held by the author. Digital access to this materialis made possible by the University Libraries, University of Arizona.Further transmission, reproduction or presentation (such aspublic display or performance) of protected items is prohibitedexcept with permission of the author.
Download date 09/08/2021 14:14:47
Link to Item http://hdl.handle.net/10150/318784
SLURRY DENSITY INFLUENCE ON BALL MILL BEHAVIOR
by'
• H a r r y Benj am.in.. Car s e n
A Thesis .. S u b m i t t e d to t h e F a c u l t y o f t h e
DEPARTMENT OF METALLURGICAL ENGINEERING
I n . P a r t i a l F u l f i l l m e n t o f the-.. R e q u i r e m e n t s For t h e . D e g re e o f
MASTER OF SCIENCE
I n t h e G r a d u a t e . G o l i e g e
THE UNIVERSITY OF ARIZONA
1 .9 6 9
(
STATEMENT BY AUTHOR
T h is t h e s i s h a s b e e n ■s u b m i t t e d i n p a r t i a l f u l f i l l m ent o f r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e . a t : The U n i v e r s i t y o f A r i z o n a and. i s d e p o s i t e d i n t h e U n i v e r s i t y L i b r a r y t o . be "made a v a i l a b l e t o ■b o r r o w e r s u n d e r ; r u l e s o f t h e . L i b r a r y . ..................
B r i e f q u o t a t i o n s f rom t h f s t h e s i s a r e a l l o w a b l e w i t h o u t , s p e c i a l p e r m i s s i o n , p r o v i d e d tha t" a c c u r a t e >.acknowledgment o f . s o u r c e ■ i s made’. - R e q u e s t s f o r permisTs.i.o..n f o r e x t e n d e d quo t a t i o n frumr . o r x e p r o d u c t f o n .o f " t h i s ' m a n u s c r i p t . i n w hole o r i n ' p a r t .may be g r a n t e d by ■ the".head": of" t h e m a jo r d e p a r t m e n t . o r - the" B e a n of ' the": G r a d u a t e -'Go'!lege when i n h i s j u d g m e n t t h e p r o p o s e d u s e . o f " t h e ’" m a t e r i a l : i s i n : t h e ; i n t e r e s t s o f s c h o l a r s h i p . . I n a l l o t h e r . i n s t a n c e s , h o w e v e r , p e r m i s s i o n m u s t be o b t a i n e d f r o m T t h e " a u t h o r ' . ............. ' A
' S I G N E P g
APPROVAL BY THESIS DIRECTOR
T h is t h e s i s h a s b e e n a p p r o v e d . o n t h e d a t e . shown b e lo w :
'nORST— -----A s s o c i a t e P r o f e s s o r o f
M e t a l l u r g i c a l .E n g i n e e r i n g
a t e
ACKNOWLEDGMENTS
The e x t r a o r d i n a r y e n c o u r a g e ment o f my a d v i s o r ,
Dr„ We E„- H o r s t , was i n v a l u a b l e d u r i n g t h i s w ork .
S p e c i a l t h a n k s a r e due my w i f e Sandy f a r h e r
c o o p e r a t i o n d u r i n g t h e p r e p a r a t i o n o f t h e m a n u s c r i p t 0
The D e p a r tm e n t o f M e t a l l u r g i c a l E n g i n e e r i n g ,
C o l l e g e o f M i n e s , p r o v i d e d f i n a n c i a l s u p p o r t t h r o u g h a
r e s e a r c h a s s i s t a n t s h i p e
TABLE OF CONTENTS
Page
T* 3̂ i?' l-i ElS ® © i t i © © a i i i @ @ @ ® © i 3 @ o ® ® © < B
LIST OF IL L U S T R A T IO N S........................ J . . . . . . . . v i i i
S TRACT Q © o o © e Q © © ® © © ® © o ® © ® o © o © 15C
INTRODUCTION . 1
EXPERIMENTAL APPARATUS . . . . . . . . . . . . . . . . 5
. F e e d . M a t e r i a l . © © » © © . . . © « © © © . © © 5c311 ™ Mi 1,1. U a 11 © © © o © © © © ® © © © © © © ® © 7
E x p e r i m e n t a l P r o c e d u r e 7
EXPERIMENTAL RESULTS . . 11
COMPUTATIONAL TECHNIQUES . . . . . . . . . . . . . . . 19
SLURRY INFLUENCE IN WELL-STIRRED-TANKCOMMINUTION M O D E L .......................................................................... 22
...De v e lo p m e n t - o f t h e Model © © © © . © » © © . . © 22. E v a l u a t i o n o f .P e r C e n t - S o l i d s . . . . . . . . . 26E v a l u a t i o n o f .V o lum e F r a c t i o n o f . S o l i d s . . . . 34E v a l u a t i o n o f A p p a r e n t . . S l u r r y V i s c o s i t y © © © . 36
AXIAL-DIFFUSION M O D E L ...................................................................... 43
A p p l i c a t i o n o f t h e Model . . . . . . . . . . . . 43
CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . 54
APPENDIX As NOMENCLATURE............................................................ 56
APPENDIX B s EXAMPLE OF DATA COLLECTED DURINGAN EXPERIMENTAL RUN . . . . . . . . . 58
APPENDIX C: FEED MATERIAL . . . © © . . . . . . . . 59
APPENDIX Ds EXPERIMENTAL AND COMPUTED DISCHARGEPARTICLE-SIZE DISTRIBUTIONS . . . . . 60
i v
V
TABLE OF CONTENTS--C o n t in u e d
Page
APPENDIX E% DIGITAL COMPUTER PROGRAMS . . . . . . . 64
SELECTED BIBLIOGRAPHY . . . . . . . . . . . . . . . 71
LIST OF TABLES
T ab le Page
1. E x p e r i m e n t a l R e s u l t s , Type X M a t e r i a l . . . . 12
2. E x p e r i m e n t a l R e s u l t s , Type 'Y M a t e r i a l . . . . 13
3 . C a l c u l a t e d R e s u l t s f o r Type X M a t e r i a l ' . . . . 14
4 . - C a l c u l a t e d R e s u l t s f o r Type Y M a t e r i a l . . . . 15
5 . F i r s t - O r d e r L e a s t - S q u a r e s E q u a t i o n s f o rI n v e n t o r y - I n t e n s i t y P r o f i l e s . . . . . . . . 17
6. C a l c u l a t e d A p p a r e n t S l u r r y V i s c o s i t yf r o m E q . 1 . . . . . . . . . . . . . . . . . 18
7. Comminut ion C o e f f i c i e n t s (TO . 27
8 . Summary o f R e s u l t s f o r (3 = a - a-,SEq. 8 (9 P a r a m e t e r s ) . . . 0 . . . . . . . . . 31
9 . Summary o f R e s u l t s f o r {3 = ( a 0 - a , S )Eq. 8 ( 8 P a r a m e t e r s ) ( a Q « 1 . 0 0 ) . . . . . . 32
10. Summary o f R e s u l t s f o r (3 = 1 - SEq. 8 ( 7 P a r a m e t e r s ) ......................................... 33
11. Summary o f R e s u l t s f o r (3 = a - a . X - a 9XEq. 9 ( 1 2 P a r a m e t e r s ) . . ° ......................................... 35
12. Summary o f R e s u l t s f o r (3 = 1 .0 /CBased on Eq. 1 ( 7 P a r a m e t e r s ) . . . . . . . 39
13. Summary o f R e s u l t s f o r (3 = 1 .0 /C w i t hA d j u s t e d P a r a m e t e r s E q c 1 (1 5P a r a m e t e r s ) ........................................................... 40
14. Summary o f R e s u l t s f o r A x i a l D i f f u s i o nModel w i t h (3 = a 0 - a^S - aoS Eq. 18and Two D^ s ( 1 2 P a r a m e t e r s ; . . . . . . . . 50
15. Summary o f R e s u l t s f o r A x i a l D i f f u s i o n1 Model w i t h (3 = a 0 - a^S Eq. 8 and Two
D ^ ’s (11 P a r a m e t e r s ) . . . . . . . . . . . . 51
v i
v i i
LIST OF TABLES--C o n t in u e d
T a b le Page
16. Summary o f R e s u l t s f o r A x i a l D i f f u s i o nModel f o r (3 = a 0 - a^X Eq. 19 and TwoD ^ ’ s (1 1 P a r a m e t e r s ) . . . . . . . . . . . . 52
17. Feed S i z e D i s t r i b u t i o n ......................... ..... . 59
18. E x p e r i m e n t a l and C a l c u l a t e d S i z eD i s t r i b u t i o n . . 61
LIST OF ILLUSTRATIONS
F ig u r e Page
1 o . Average p a r t i c l e s i z e d i s t r i b u t i o n o f-f e e ® © e @ © e ® ® o @ ® @ @ e © © © ® ®
2« - S c h e m a t i c d ra w in g - o f . e q u ip m e n t ® © © © © © . . 8
3© . S c h e m a t i c r e p r e s e n t a t i o n o f mass f l o wt h r o u g h t h e b a l l m i l l . © © © © © © © . . . » - 24
•4. • Com minut ion c o e f f i c i e n t ( k ) vs© p e r c e n ts o l i d s i n . s 1 u r r y © © © © © © . © © © © © © © 2 8
5 © , E le m e n t o f . s l u r r y a n d . g r i n d i n g m ed ia a l o n g t h e . a x i a l d i r e c t i o n o f . t h eb slI 1 . mi 11 © » © © © © © © © ©® . 0 ® © © ©
6 © I n v e n t o r y i n t e n s i t y f o r T e s t 10( m o d i f i e d g r a t e d i s c h a r g e ) © © © © . © . © © 47
7. FORTRAN p r o g r a m f o r s t e e p e s t a s c e n t m a i n l i n e . 65
8 © FORTRAN p r o g r a m f o r s u b r o u t i n e RSTAR © © . © © 6 6
9 . FORTRAN p r o g r a m . f o r n o n l i n e a r e s t i m a t i o n. ma 11). 11 ne © © © © © © © . @ © © © . © © @ @ ® 6 7
v i i i
ABSTRACT
The i n f l u e n c e o f s l u r r y s o lid s on . t h e . r a t e ;of
c o m m in u t io n was. i n c o r p o r a t e d i n t o t h e d i s t r i b u t e d - p a r a m . e t e r
- a x i a l - d i f f u s i o n a n d . lumped ̂ parame t e r . we. IT - s t i r r ed - 1 a n k ...
.m odels , f o r - w e t b a l l -mi 11 . g r i n d i n g ' . . ■ T h e s e m a t h e m a t i c a l
. 'models u s e . a . f i r s t s o r d e r ‘ com m inu t i on r r a t e to . p r e d i c t . t h e
p r o d u c t s i z e d i s t r i h u t i o n v e c t o r . The modele ,• a r e demon
s t r a t e d u s i n g . s e v e r a l i m p l e m e n t a t i o n s f o r t h e i n f l u e n c e o f
t h e s l u r r y , s o l i d s , on t h e c-omminution r a t e .
- The m o d e ls d i s c u s s e d above have b e e n . e x t e n d e d so
t h a t they , are . :now a p p l i c a b l e t o w ide v a r i a t i o n s i n s l u r r y
. s o l i d s . . - I t was a l s o d e m o n s t r a t e d t h a t t h e a x i a l - d i f f u s i o n .
■Mode 1 i s c a p a b l e . o f . a c c o m m o d a t i n g :wide: v a r i a t i o n s i n t h e
: f e e d p a r t i c l e , s i z e d i s t r i b u t i o n «
INTRODUCTION
' T h e . d e v e l e p r a e n t o f a m a t h e r a a t l c a l mode1 w h ich
d e s c r i b e s t h e com m in u t io n p r o c e s s h a s b e en t h e g o a l o f many
r e s e a r c h e r s .. T h e re have b een , s e v e r a l . r e a s o n s f o r s e e k i n g
t o d e v e l o p su c h a m o d e l , b u t t h e p r i m a r y r e a s o n i s f o r t h e
a p p l i c a t i o n of a u t o m a t i c p r o c e s s , c o n t r o l t o ■t h e com m in u t io n
p r o c e s s . ■ E x p e r i e n c e i n t h e p r o c e s s i n d u s t r i e s h a s shown
t h a t a p r o c e s s m u s t be d e s c r i b e d m a t h e m a t i c a l l y b e f o r e
a u t o m a t i c p r o c e s s c o n t r o l c a n be a p p l i e d .
S i z e - r e d u c t i o n p r o c e s s e s h ave be en h i s t o r i c a l l y
• r a t h e r i n e f f i c i e n t u n i t o p e r a t i o n s , * I n a d d ! t i o n , t h e
c o m m in u t io n p r o c e s s i s o f t e n ,an e x p e n s i v e p a r t : o f t h e
o v e r a l l o re t r e a t m e n t .c o s t . A t t i m e s t h i s u n i t o p e r a t i o n
r e p r e s e n t s up t o f i f t y p e r c e n t o f t h e o v e r a l l t r e a t m e n t
c o s t . - From t h i s i t c an be s e e n t h a t i t i s d e s i r a b l e t o
■ o p e r a t e t h e p r o c e s s a t maximum c a p a c i t y w h i l e . s t i l l m a i n
t a i n i n g t h e d e s i r e d g r i n d . A m a t h e m a t i c a l m ode l w h ich
d e s c r i b e s t h e c o m m in u t io n p r o c e s s would, a l l o w d e t e r m i n a t i o n
o f v optimum, c o n d i t i o n s t o a c h i e v e ■ t h e - g o a l s o u t l i n e d above .
T he re h ave b e e n s e v e r a l a t t e m p t s t o d e v e l o p m a t h e
m a t i c a l models, d e s c r i b i n g t h e co m m in u t io n p r o c e s s , and a l l
o f t h e s e have c o n t r i b u t e d t o t h e u n d e r s t a n d i n g o f t h e
p r o c e s s * The u se of. p r o b a b i l i t y t h e o r y t o a c c o u n t f o r . t h e
b r e a k a g e i n ;a b a l l m i l l h a s b e e n u s e d b y ■E p s t e i n ( 1 3 ) ,
2
Brown and a s s o c i a t e s (10.) . , ; A u s t i n and Gardner ; ( 3 ) , and B ass
/ ( 5 ) . To d e s c r i b e t h e . s i z e r e d u c t i o n o c c u r r i n g i n t h e - b a l l
m i l l t h e s e w o r k e r s u s e d a . s e l e c t i o n f u n c t i o n t o d e s c r i b e
th e . f r a c t i o n a l amount o f any g i v e n . s i z e t h a t w i l l be= b ro k e n
d u r i n g . g r in d in g . . . A l s o , m o s t p r o b a b i l i t y m odels , i n c o r
p o r a t e d a d i s t r i b u t i o n f u n c t i o n t o d e s c r i b e t h e p a r t i c l e
s i z e d i s t r i b u t i o n o f . t h e b r o k e n m a t e r i a l . - A l a r g e . amount
o f r e s e a r c h . e f f o r t h a s gone i n t o t h e d e v e lo p m e n t o f
c o m m in u t io n m o d e ls b a sed , on p r o b a b i l i t y t h e o r y . However,
n o ■s u c c e s s f u l a p p l i c a t i o n of t h i s t y p e o f m ode l f o r p r o c e s s
c o n t r o l h a s b e e n r e p o r t e d . A l s o , a s shown b y /L y n c h - (2 4 )
t h e p r o b a b i l i t y a p p r o a c h p r o v i d e s v e r y l i t t l e . i n f o r m a t i o n
• on how t h e ■breakage- o c c u r s •w i t h i n t h e g r i n d i n g mi 11,
A n o t h e r a p p r o a c h t o t h e p r o b l e m of d e s c r i b i n g a
- g r i n d i n g u n i t . .m a t h e m a t i c a l ly i s t h e k i n e t i c a p p r o a c h . ■ -
S e v e r a l . i n v e s t i g a t o r s h ave d e v e l o p e d m o d e ls b a s e d on
k i n e t i c t h e o r y and i t a p p e a r s t h a t t h e s e s h o u l d be e a s i e r
t o - a p p l y ' t o a u t o m a t i c p r o c e s s c o n t r o l t h a n t h e m o d e l s ba sed
o n p r o b a b i l i t y - t h e o r y . R o b e r t s • (2 8 ) and Bowdish. ( S ) :have
. c o r r e l a t e d t h e r a t e o f . change o f . m ate r i a l l a r g e r •t h a n ,a
g i v e n s i z e w i t h t h e m ass f r a c t i o n ;o f t h e o v e r - s i z e and t h e
e n e r g y i n p u t . - Use of a. f i r s t - o r d e r r a t e e q u a t i o n t o
d e s c r i b e t h e g r i n d i n g p r o c e s s h a s b e e n d e m o n s t r a t e d by
s e v e r a l i n v e s t i g a t o r s . A r b i t e r : and. Bhrany. ( 2 ) , ajad: F r e e h ,
H o r s t , . and. K e l l n e r . ( 1 5 ) .. have shown t h a t a f i r s t - o r d e r r a t e
e q u a t i o n i s a p p l i c a b l e t o b a t c h g r i n d i n g . o f q u a r t z .
-3
K e l s a l l .(21) u s e d a f i r s t - u r d e r . r a t e . e q u a t i o n t o d e s c r i b e
•• t h e b r e a k a g e in .a . - s m a l l c o n t i n u o u s wet b a l l m i l l .
• The a d e q u a c y •o f the- k i n e t i c a p p r o a c h h a s c l e a r l y
• be en :demons t r a t e d ' by7 Ke l i n e r .(20.) . H;orst . ( 1 9 ) , and
. P i z z u t o - Zamani 11.o ( 2 7 ) .. - P i z z u t o - Zamani 11 o (27 ) u s e d an
a x i a l ^ d i f f u s i o n , ( d i s t r i b u t e d - p a r a m e t e r ) m odel t o d e s c r i b e
t h e ' r a t e o f g r i n d i n g a s a f u n c t i o n o f t h e p r o c e s s v a r i a b l e s .
• K e l l n e r .(20) . and. H o r s t ( 1 9 ) u s e d the., a x i a l - d i f f u s i o n and
w e l l - s t i r r e d - t a n k ( lum ped-param e t e r ) . m odels , t o d e s c r i b e , t h e
g r i n d i n g . a c t i o n , i n a . c o n t i n u o u s wet b a l l m i l l .
■ I t h a s b e en d e m o n s t r a t e d by H o r s t . (1 9 ) . , - P i z z u t o -
Z a m a n i l l o (2-7) , and K e l s a l l .and. Reid ( 2 2 ) t h a t t h e c o n t e n t s
o f a d f e t b a l l m i l l a r e w e l l m ixed . T h e r e f o r e t h e we l i
s t ! r r e d r t a n k .mode1 c a n be u s e d t o a p p r o x i m a t e t h e b a l l m i l l
b e h a v io r , . However , t h e a x i a l - d i f f u s i o n m odel d e l i n e a t e s
more c l e a r l y w ha t o c c u r s w i th in , t h e b a l l m i l l . T h i s i s
p a r t i c u l a r l y -. true i n r e g a r d s t o c h a n g e s i n p a r t i c l e - s i z e
d i s t r i b u t i o n . a l o n g t h e a x i a l l e n g t h o f t h e b a l l m i l l .
- However , t h e e q u a t i o n s f b r t h e a x i a l - d i f f u s i o n m odel a r e
of . a d i f f e r e n t i a l f o rm .and more com plex t o m a n i p u l a t e i n
: c o m p ar i son t o • t h e a l g e b r a ! c e q u a t i o n s •• t h a t . r e s u i t - f o r • t h e
we 1 1 - s t i r r e d - t a n k ;mode 1. I t was c l ' e a r l y d e m o n s t r a t e d by
H o r s t . ( 1 9 ) t h a t t h e w e l l - s t i r r e d - t a n k m odel c a n be u s e d t o
d e s c r i b e t h e s i z e r e d u c t i o n . i n . a wet b a l l m i l l .
- The o b j e c t i v e o f t h i s : i n v e s t i g a t i o n i s t o i n c o r
p o r a t e i n t o t h e a x i a l - d i f f u s i o n m odel t h e i n f l u e n c e o f t h e
- s l u r r y ■ c h a r a c t e r i s t i c s . ' ©n t h e g r i n d i n g : ra te , . , T h i s w i l l be
a c c o m p l i s h e d by u t i l i z i n g t h e we 1 1 - s b i r r e d - t a n k mode1 t o
d e v e lo p . ' r e l a t i o n s h i p s - f o r - t h e com m in u t io n i r a t e , v s . s l u r r y
s o l i d s , co m m in u t io n ■ r a t e - v s . volume f r a c t i o n of, s l u r r y , , and
. c o m m in u t io n r a p e vs.. a p p a r e n t - s l u r r y - v i s c o s i t y , . The , r e l a - ■
t i o n s h i p s d e v e l o p e d w i l l t h e n be i n c o r p o r a t e d , i n t o t h e
. a x i a l - d i f f u s i o n model , t o •i s o l a t e t h e s l u r r y 1s i n f l u e n c e
f ro m t h e c o m m in u t io n ■ c o e f f i c i e n t s , . - T h i s s t u d y w i l l d e a l
.. o n ly -■ w i t h t h e s y s t e m * s • r e s p o n s e a t - s t e a d y • s t a t e and w i l l
n o t a t t e m p t t o d e s c r i b e t h e d y n am ic r e s p o n s e . . However , i t
would be e x p e c t e d t h a t t h e . b a s i c m ode l c o n s i d e r a t i o n s would
b e , a p p l i c a b l e - t o t h e f o r m u l a t i o n . o f a. dynam ic m o d e l .
EXPERIMENTAL APPARATUS
, . Feed M a t e r i a l
T h e . f e e d m a t e r i a l u sed - f e r t h e - g r i n d i n g t e s t s
p e r f o r m e d d u r i n g t h i s i n v e s t i g a t i o n was a p o r p h y r y c o p p e r
o r e •f ro m t h e M i s s i o n - U n i t of the- A m e r ic a n S m e l t i n g :and
•• R e f i n i n g - Company.* The o re u s e d . i n the.se t e s t s , h a s a work
: index- o f ■ 19- t o - 20 k w -h r p e r - t o n and. a s p e c i f i c g r a v i t y of
2 .7 7 0 . 0 9 a t 95 p e r c e n t c o n f i d e n c e l im i t - s X20),.. The o re
was p r e p a r e d by s t a g e c r u s h i n g t h r o u g h a r o l l . c r u s h e r w h ich
was i n c l o s e d c i r c u i t w i t h a v i b r a t i n g : s c r e e n . . I n t h i s
m a n n e r , a b a l l m i l l f e e d w i t h . a s m a l l amount o f f i n e
m a t e r i a l , was p r e p a r e d .
• F o r t h i s i n v e s t i g a t i o n two. feed.: s i z e , d i s t r i b u t i o n s
were used. . Type X m a t e r i a l , was. 100 p e r . c e n t minus. 3 mesh
.and .Bad t h e . a v e r a g e p a r t i c l e s i z e - d i s t r i b u t i o n - shown ■ i n
- F i g . 1. Type Y m a t e r i a l was 100 p f e r - c e n t - m in u s 6 mesh and
. h a d t h e . a v e r a g e p a r t i c l e s i z e d i s t r i b u t i o n : s h o w n ,i n - F i g . 1 .
The c o m p le t e p a r t i c l e =s i z e d i s t r i b u t i o n f o r t h e i n d i v i d u a l
o re l o t s u s e d i s g i v e n i n .A p p p n d ix . C , -- T a b le 17 . - As shown
• • in : ! F ig , 1, the . p a r t i c l e s i z e d i s t r i b u t i o n of t h e two t y p e s
■ o f . o r e i’id : q u i t e , d i f f e r e n t .
Mas
s F
ract
ion
Hel
d
Type X Type Y
S i z e
S i z e Mesh
1 3 X 62 6 X 103 10 X 204 20 X 355 35 X 656 65 X 1507 150 X 2708 -270
F i g . 1. A ve rag e p a r t i c l e s i z e d i s t r i b u t i o n o f f e e d .
B a l l - M i l l U n i t
The b a l l - m i 11 u n i t u s e d i n t h i s i n v e s t i g a t i o n i s a
D env er E qu ip m en t Company 16 iS x, 16" p i l o t s c a l e u n i t « The
m i l l i s . e q u ip p e d w i t h a g r a t e d i s c h a r g e and o p e r a t e s a t
5 3 . 8 . rpm; w h ich i s 7 8 .5 p e r c e n t o f t h e c r i t i c a l sp e ed f o r
t h e u n i t . . The b a l l l o a d was m a i n t a i n e d . a t 230 pounds
. d u r i n g t h e t e s t p e r i o d and. t h e b a l l d i s t r i b u t i o n - w a s t h e
s a m e . a s t h a t r e p o r t e d b y "K e l i n e r ( 2 0 ) . - The maximum s i z e
b a l l was 2 . 5 i n c h e s i n d i a m e t e r and t h e minimum s i z e b a l l
was. 0 . 7 5 i n c h e s i n d i a m e t e r .
The o re was f e d t o t h e b a l l m i l l b y ; a D e n v e r E q u i p
m ent . Company b e l t f e e d e r and t h e f e e d r a t e was a d j u s t e d by
c o n t r o l l i n g t h e b e l t s p e e d and s l i d e - g a t e o p e n in g s - W a t e r
- was s u p p l i e d , f ro m a c o n s t a n t - h e a d t a n k and t h e f l o w was
m o n i t o r e d by a - F i s c h e r - P o r t e r f l o w - m e t e r . . ■ A d e t a i l e d
. d e s c r i p t i o n o f t h e e x p e r i m e n t a l a p p a r a t u s h a s p r e v i o u s l y
' b e e n , r e p o r t e d ( 1 9 ) . . : A s c h e m a t i c o f t h i s a p p a r a t u s i s shown
. i n ' F i g . 2 .
E x p e r i m e n t a l P r o c e d u r e
S i x t e s t s were made u s i n g t h e •Type Y f e e d . and a
n o m in a l t h r e e p ou n ds p e r m i n u t e . f e e d r a t e . F o r t h e s e t e s t s
t h e p e r c e n t s o l i d s in t h e s l u r r y . w a s v a r i e d b e tw e e n 58 and
72 p e r c e n t . A t o t a l , o f . f o u r t e s t s were made u s i n g t h e
Type X f e e d , m a t e r i a l and t h e - same f e e d . r a t e . . F o r t h e s e
Feed
FeedHopperG r a te
D i s c h a r g e __ FeedGate
B e l t F e e d e r
Di s c h a r g eF low m ete r
B a l l M i l l C o n s t a n t - h e a d W ate r
F i g . 2 . S c h e m a t i c d raw in g of e q u ip m e n t .
00
l a t t e r t e s t s t h e p e r c e n t - . s o l i d s i n t h e s l u r r y was a l s o
v a r i e d h i tw e e n ■ 58 . an d .72 p e r c e n t .
■ The e x p e r i m e n t r a l ' p r o c e d u r e u s e d d u r i n g t h i s . • i n v e s -
t i g a t i o n : c l o s e l y p a r a l l e l e d t h a t o u t l i n e d . . b y P i z z u t o -
Z a m a n i l l o . ( 2 7 ) . B a l l - m i l l g r i n d i n g t e s t s were made t o
e v a l u a t e t h e p a r t i c l e - s i z e d i s t r i b u t i o n .o f . t h e g ro u n d
. p r o d u c t and t h e p a r t i c l e - s i z e d i s t r i b u t i o n a l o n g t h e a x i a l
l e n g t h o f t h e m i l l . •• A t o t a l - o f t e n t e s t s were made u n d e r
v a r i o u s p r o c e s s c o n d i t i o n s .
The e x p e r i m e n t a l p r o c e d u r e u s e d i s sum m arized
b e low . The o re t o be u s e d f o r t h e t e s t was w e i g h e d . a n d
added t o t h e f e e d h o p p e r by u s e o f a b u c k e t and. c h a i n
h o i s t . , ■ Each t e s t r e q u i r e d a p p r o x i m a t e l y - 2 2 5 . p ounds o f raw
m a t e r i a l , . The o re and w a t e r f e e d r a t e s were a d j u s t e d , and
t h e b a l l - m i l l u n i t s t a r t e d • T h e ■r a t e ' o f . s o l i d s d i s c h a r g e s ,
w a t e r f l o w r a t e , and p e r c e n t s o l i d s i n t h e s l u r r y ;were
r e c o r d e d a t p e r i o d i c i n t e r v a l s t h r o u g h o u t t h e t e s t t im e
p e r i o d . The s o l i d s d i s c h a r g e r a t e was c h e c k e d a g a i n s t t h e
s o l i d s f e e d . r a t e . A f t e r t h e - r u n n i n g tS'me. exceeded , a
minimum o f . e i g h t t i m e s t h e mean m i l l . r e s i d e n c e t i m e ,
r e p l i c a t e - s a m p le s o f t h e d i s c h a r g e s t r e a m were c o l l e c t e d .
- T h e s e : sa m p le s were 1 u s e d t o c h e c k t h e p e r c e n t s l u r r y
• s o l i d s and t h e d i s c h a r g e . - p a r t i c l e - s i z e . d i s t r i b u t i o n . - The
f e e d , w a t e r , and; p o w e r -to t h e m i l l were t h e n s i m u l t a n e o u s l y
■ s to p p e d , . H e a t : lamps • were p laced- , on t h e o u t s i d e o f . t h e m i l l
t o d r y ■ t h e c o n t e n t s ». A f t e r a b o u t . 18 h o u r s , t h e d i s c h a r g e
end ©f. t h e m i l l was removed-, a n d t h e c o n t e n t s s e c t i o n e d : i n t o
• s i x s e g m e n t s a l o n g t h e l o n g i t u d i n a l a x i s o f t h e b a l l m i l l .
S t a r t i n g f rom th e f e e d e n d , t h e . f i r s t f i v e s e c t i o n s were
- 2 .7 5 i n c h e s i n w i d t h , and t h e f i n a l s e c t i o n , i n c l u d i n g t h e
g r a t e , was 2 .2 5 . i n c h e s w i d e .
- A f t e r r e m o v a l o f t h e s o l i d s m i l l i n v e n t o r y , t h e
s o l i d s were d r i e d and. weighed, . ■ The w e i g h t i n e a c h : s e c t i o n
was u s e d t o • d e v e l o p t h e in ven tory - interns i ty - p r o f i l e a l o n g
t h e . l e n g t h o f t h e b a l l m i l l . The p a r t i c l e - s i ze-..- d i s t r i b u
t i o n o f . .each p r o d u c t was t h e n d e te r m in e d , . The . s c r e e n i n g .
t e c h n iq u e ; : f o r d e t e r m i n i n g t h e p a r t i c l e - s i z e d i s t r i b u t i o n
was- s i m i l a r t o t h a t d e v e l o p e d b y K e l l n e r ( 2 0 ) . The . o n l y
m o d i f i c a t i o n i n . t h e s c r e e n i n g t e c h n i q u e was t h a t : a - T y l e r , 2
C s /2 ) s e r i e s was u se d , f o r th e - d r y / s c r e e n a n a l y s i s and d a t a
p r e s e n t a t i o n , . . The T y l e r ( .V 2 )^ . s e r i e s i n c o r p o r a t e s . - e v e r y\
- o t h e r , s c r e e n : i n t h e s t a n d a r d - T y l e r s e r i e s » \\
When t h e b a l l m i l l was s t o p p e d ' a f t e r ^ c o m p l e t i o n o f .
■ a t e s t , m a t e r i a l f l o w e d f ro m t h e l a s t s e c t i o n ■i n t o •th e
g ra te . . . ■ The g r a t e m a t e r i a l was a d d ed t o t h e . m a t e r i a l
o b t a i n e d f ro m th e l a s t s e c t i o n p r i o r t o . s c r e e n a n a l y s i s .
P izzUtO.-Zamani . l lo . (2 7 ) d e m o n s t r a t e d t h a t t h e g r a t e and
f i n a l - s e c t i o n had t h e . same p a r t i c l e - s i z e d i s t r i b u t i o n .
. A t y p i c a l exam ple o f t h e d a t a c o l l e c t e d d u r i n g a
g r i n d i n g t e s t i s s h o w n . i n A p p e n d ix B .
EXPERIMENTAL;. RESULTS
= The e x p e r i m e t i t a l d a t a o b ta in e d : f ro m th e t e n t e s t s
p e r f o r m e d , d u r i n g t h i s i n v e s t i g a t i o n , a r e t a b u l a t e d , i n T a b l e s
1 and 2 . I n . a d d i t i o n , T a b le 1 i n c l u d e s t h e r e s u l t s o f two
o f t h e t e s t s p e r f o r m e d b y / P i z z u t e - Z a m a n i l l o • ( 2 7 ) . These
were u s e d d u r i n g t h e d a t a a n a l y s i s t o i n c r e a s e t h e num ber
' o f . a v a i l a b l e d a t a . The t e s t s made by P i . z z u t o - Zamani. 11.o
■ w i l l be r e f e r r e d t o a s WP-5 a n d WP-7, r e s p e c t i v e l y .
The e x p e r i m e n t a l r e s u l t s were u s e d t o i s o l a t e t h e
i n f l u e n c e of t h e s l u r r y s o l i d s f rom t h e c o m m in u t io n c o e f
f i c i e n t s , f o r t h e w e l l - s t i r r e d - t a n k . and a x ia l - d i f f u s io n
m o d e l s d e v e l o p e d e a r l i e r . A t o t a l o f t w e lv e s e t s o f
e x p e r i m e n t a l d a t a were a v a i l a b l e w i t h many o f t h e ru n s
h a v i n g b e e n made i n r e p l i c a t e . . C a l c u l a t e d r e s u l t s , f o r
t h e s e r u n s a r e shovm i n T a b l e s 3 and 4 .
F o r a l l . o f t h e t e s t s u s e d i n t h i s i n v e s t i g a t i o n
• s o l i d s , i n v e n t o r y d a t a were a v a i l a b l e , . • From: t h e s e d a t a
in v e n to r y - in te n s ity • p r o f i le s - were-.- obtained a l o n g t h e
l e n g t h o f t h e m i l l . The s o l i d s •in v e n to r y - in te n s ity •data
were o b t a i n e d by d i v i d i n g t h e w e i g h t . o f s o l i d s by t h e
l e n g t h ;o f t h e s e c t i o n w i t h i n t h e b a l l m i l l w here t h e
s o l i d s were ob ta ined . . . • T hese d a t a f o r e a c h t e s t were t h e n
f i t t e d w i t h a. f i r s t - o r d e r l e a s t - s q u a t e s c u r v e u s i n g a
11
. 12
T a b le 1» E x p e r i m e n t a l R e s u l t s , Type K . M a t e r i a l
T e s t No. 6 7 ' 8 9 -WP-5 WP-7
P e r :C ent S o l i d s 7 1 .3 0 6 0 .1 8 7 2 ,5 1 5 8 .6 2 6 5 ,1 0 6 7 .0 0 •
Feed Ra te ( Ib /m im ) , 2 . 9 7 . 2 . 8 9 3 , 1 1 2 . 9 4 2 . 8 9 3 , 2 1
Mesh S i z e ..Discharge- S i z e D i s t r i b u t i o n
3 X 6 0 ,1 0 1 6 0 ,1 2 1 9 0 .0 6 3 3 0 .0 8 9 1 0 .1 2 3 5 ■ 0 ,1 3 4 9
6 X 10 0 .0 8 5 2 0 ,1 1 1 3 0 .0 7 0 6 0 .0 9 8 8 .0 ,1 0 9 6 0 .1 1 0 3
10 X 20 0 .0 6 5 2 0 .0 8 1 4 0 , 0 7 0 4 0 .0 7 2 7 0 ,0 7 9 6 • 0 .0 7 7 7
20 X 35 0 ,0 8 5 7 0 .0 9 5 8 0 .1 0 2 1 0 .0 8 9 5 0 .0 9 9 8 0 .0 9 1 2
35 X 65 0 .1 2 8 3 0 ,1 2 4 7 0 .1 5 1 2 0 .1 2 8 8 0 ,1 2 2 9 0 ,1 1 8 6
65 X 150 0 .1 4 2 3 0 .1 3 1 4 0 .1 5 4 1 .0 .1 4 5 2 0 , 1 2 8 4 0 .1 2 8 8
150.X 270 0 ,0 9 8 3 0 .0 9 1 1 0 ,1 0 0 8 0 .1 0 2 9 0 .0 9 0 0 0 ,0 8 9 4
-270 0 .2 9 3 6 0 .2 4 2 4 0 .2 8 7 5 0 .2 7 3 0 0 .2 4 6 2 . 0 , 2 4 9 1
T o t a l 1 .0 0 0 0 1 .0 0 0 0 1 .0 0 0 0 . 1 .0 0 0 0 . 1 .0 0 0 0 1 .0 0 0 0
Feed L o t No. B B ■ r A • A C D
13
. Table..- 2 . E x p e r im e n ta l - R e s u l t s , Type,.,Y M a t e r i a l
T e s t No. 1 2 .. 3 4 5 . 10
Pe r - G e n t S o l i d s 5 8 .9 3 7 0 .6 0 6 6 ,6 3 5 8 .6 6 7 1 .2 3 5 9 .8 3
Feed Ra te ( I b / m i a ) , 3 . 0 6 . 2 . 90 2 .9 5 2 .9 1 2 .8 9 2 .8 2
M e sh ' S i ze ,D i s c h a r g e . 5 i z e - D i s t r i b u t i o n
.6 X 10 0 .0 6 3 8 0 ,0 2 9 3 0 .0 6 7 6 0 .0 7 0 4 0 .0 4 2 3 0 .0 7 2 4
10 X 20 0 .1 0 7 7 0 .0 6 3 6 0 ,1 0 2 7 0 ,1 0 6 2 0 ,0 7 5 7 0 .1 1 1 1
20 X 35 0 .1 2 9 7 0 .1 1 4 7 0 ,1 2 5 6 0 ,1 2 7 4 0 .1 2 2 2 0 .1 2 5 5
.3.5 X 65 0 .1 6 5 2 0 ,1 8 5 8 0 ,1 6 1 3 0 .1 6 2 7 0 ,1 8 0 9 0 .1 5 6 1
65 X .150 0 .1 5 8 8 0 ,1 8 3 2 0 ,1 5 9 9 0 .1 6 0 0 0 .1 7 2 3 0 .1 5 5 1
1 5 0 -X 270 0 .1 0 3 0 0,11.73 0 .1 0 3 6 0 .1 0 5 4 0 ,1 0 8 0 0 . 1 0 2 9
-270 0 .2 7 1 8 0 .3 0 6 1 0 .2 7 9 3 0 .2 6 7 9 0 .2 9 8 6 0 .2 7 3 9
T o t a l 1 .0 0 0 0 1 .0 0 0 0 1 .0 0 0 0 1 .0 0 0 0 1 .0 0 0 0 1 .0 0 0 0
Feed; L o t .N o . . - E F F E G G
T ab le 3 . C a l c u l a t e d R e s u l t s f o r Type X M a t e r i a l
Run
M a t e r i a l T e s t s
Fg ( I b / m i n ) w' ( l b / f t ) w ^ ( l b / f t ) p g C l b / f t 3 ) W( l b )
6 4 . 1 6 11 .59 1 6 .22 1 17 .2 1 4 .7 1
7 4 . 8 0 8 .4 8 1 4 .07 9 8 .9 1 0 .7 3
8 4 . 2 8 1 5 .2 3 2 1 .02 1 1 9 .1 1 9 .3 6
9 5 .0 1 7 .1 8 12 .2 3 9 6 . 5 9 . 1 3
WP-S 4 .4 3 00 00 Ul 1 3 .6 0 106 .8 1 1 .22
WP-7 4 .7 9 1 0 .5 8 15 .79 1 0 9 .0 1 3 .4 1
T ab le 4* C a l c u l a t e d R e s u l t s f o r Type Y M a t e r i a l
. Run
M a t e r i a l T e s t s
, Fg ( I b / m i n ) w ’ ( l b / f t ) w ^ ( l b / f t ) P s C l b / f t 3 ) W(Tb)
1 5 ,19 5 ,98 1 0 .1 6 9 6 . 6 7 .61
. 2 4 /12 1 1 ,8 4 1 6 ,8 0 1 16 .2 1 5 .0 3
3 4 ,4 3 7 ,0 1 1 0 .5 3 1 0 9 .6 8 .9 0
4 4 , 9 6 5 ,62 9 .5 7 9 6 .2 7 .1 4
5 4 , 0 6 1 1 .28 ' 2 0 .8 1 116 .9 14 .32
10 4 .7 2 6 ,2 3 1 3 .2 3 9 8 . 3 7 .9 1
16
FORTRAN p r o g r a m . The e q u a t i o n s f o r t h e s e c u r v e s a r e
p r e s e n t e d i n T a b le 5.
An e q u a t i o n d e v e l o p e d by Wada, H i r a n o , and Homma
( 3 1 ) was u s e d t o c a l c u l a t e t h e a p p a r e n t s l u r r y v i s c o s i t y
f o r e a c h o f t h e tw e lv e t e s t s . The e q u a t i o n h a s t h e forms
e = Sw[ l + K{6 - 1 + P ( CT/ D ) k } C +
q e x p ( n , 6) ( ct/D) 2 {6 - 1 + ( o / D ) 11] Cn ] ( 1 )
where C^ = v i s c o s i t y o f w a t e r ( o p )
C = volume f r a c t i o n o£ s o l i d
6 = s o l i d s d e n s i t y (g /cm )
a = mean p a r t i c l e d i a m e t e r ( p )
D = c a p i l l a r y d i a m e t e r ( p )
K , P , k , g , n , n ^ , n 2 = c o n s t a n t s
The a p p a r e n t s l u r r y v i s c o s i t i e s t h a t r e s u l t e d f rom th e u se
o£ E q . 1 a r e p r e s e n t e d i n T ab le 6.
17
T a b le 5 . F i r s t - O r d e r L e a s t - S q u a r e s E q u a t i o n s f o r I n v e n t o r y - I n t e n s i t y P r o f i l e s
w? = a Q + a-^Z
S td DivT e s t No. a Q a a^ l b / f t
1 5 .5 0 4 - 0 . 2 4 2 0 .2 9 5
2 11 .325 - 0 .9 5 0 0 .4 5 2
3 6 .7 0 0 - 0 . 5 3 6 0 .3 1 6
4 5 .6 2 0 - 0 . 7 7 4 0 .3 3 5
5 9 .7 6 9 0 .8 7 2 0 .5 4 0
6 1 0 .4 8 6 0 .0 1 1 0 .6 5 7
7 8 .1 8 6 - 0 . 9 8 7 0 .5 9 1
8 1 5 .995 - 2 . 9 9 5 0 .4 6 1
9 7 .2 3 8 - 1 . 2 1 4 0 .3 6 4
10 6 .0 1 0 0 .0 5 1 0 .1 1 8
WP-5 9 .4 3 0 - 1 . 4 2 0 0 .9 1 0
WP-7 1 0 .6 7 0 - 1 . 9 6 0 0 .9 5 0
18
T a b le 6 . C a l c u l a t e d A p p a r e n t S l u r r y V i s c o s i t yf rom E q . 1
T e s t No. ( c p ) T e s t No. C (c p )
1 7 .5 9 6 1 2 .2 5
2 11 .82 7 7 .8 3
3 9 .6 9 8 12 .3 7
4 7 .4 3 9 7 .6 5
5 1 1 .9 6 WP-5 8 .5 3
10 7 .8 6 WP-7 9 .1 3
K = 3 .3 2 n | = - 1 . 6 1
P = 0 .0 6 r\2 ~ - 1 . 0 8
k = - 1 . 2 7 n = 6 .0
q = 1 4 6 .0
COMPUTATIONAL TECHNIQUES
I n th e d e v e lo p m e n t o f m a t h e m a t i c a l m odels i t i s
n e c e s s a r y to a d j u s t c e r t a i n o f t h e p a r a m e t e r s i n th e model
to f i t t h e e x p e r i m e n t a l d a t a . In t h e d e v e lo p m e n t o f a
model f o r m ost s y s te m s a n o n - l i n e a r a n d / o r i m p l i c i t model
i s p r e d i c t e d . There a r e s e v e r a l ways to accommodate a
model o f t h i s t y p e . One m ethod i s to r e a r r a n g e t h e model
so t h a t i t i s i n a l i n e a r fo rm . However, t h i s method may
l e a d to u n s a t i s f a c t o r y r e s u l t s ( 1 1 , 2 6 ) , N o n - l i n e a r
d i g i t a l program m ing h a s im proved th e number and q u a l i t y o f
t h e a v a i l a b l e t e c h n i q u e s to d e a l w i t h n o n - l i n e a r m o d e l s .
Two o f t h e m ost p o p u l a r n o n - l i n e a r m ethods a r e s t e e p - a s c e n t
and th e G auss -N ew ton n o n - l i n e a r e s t i m a t i o n p r o c e d u r e . The
u s u a l c r i t e r i o n f o r th e g o o d n e ss - o f - f i t o f t h e model to
t h e e x p e r i m e n t a l d a t a i s t h e l e a s t - s q u a r e s c r i t e r i o n , o r
m i n i m i z a t i o n o f the e r r o r s q u a r e d f u n c t i o n
Q = z (Y. - y , ) 2 ( 2 )i = l 1 1
where i s th e e x p e r i m e n t a l v a lu e and y^ i s t h e c o r
r e s p o n d i n g v a lu e d e t e r m i n e d by t h e m o de l . F o r l i n e a r
models th e minimum i s r e a c h e d when t h e p a r t i a l d e r i v a t i v e s
o f Q w i t h r e s p e c t to e ac h o f th e p a r a m e t e r s i s z e r o . For
n o n - l i n e a r m odels t h e e q u a t i o n s f o r t h e s y s t e m a r e o f t e n
19
20
com plex and. .may be d i f f i c u l t t o e v a l u a t e . C o n s e q u e n t l y ,
n o n - l i n e a r m ethods a r e r e q u i r e d t o s o l v e t h e s e e q u a t i o n s .
The p ro g ra m u s e d i n t h i s i n v e s t i g a t i o n i s . . a
m o d i f i c a t i o n o f th e p ro g ra m d e v e l o p e d by F l a n a g a n a n d . a
more c o m p le t e d e s c r i p t i o n i s a v a i l a b l e i n h i s d i s s e r t a t i o n
( 1 4 ) . The a p p r o a c h u s e d com bines two o f t h e p r i m a r y n o n
l i n e a r o p t i m i z a t i o n methods.. . The p ro g ram , f i r s t u s e s th e
t e c h n i q u e o f s t e e p - a s c e n t t o s t a r t t h e s e a r c h , and. a p p r o a c h
t h e opt imum p a r a m e t e r v a l u e s . The method o f s t a . e p - a s . c e n t
c o n s i d e r s t h e e r r o r s q u a r e d f u n c t i o n , Q, as. a s u r f a c e i n
h y p e r s p a c e . The p a r a m e t e r s t o be a d j u s t e d ( c o e f f i c i e n t s ,
e x p o n e n t s , e t c . ) a r e t h e i n d e p e n d e n t v a r i a b l e s . I n t h e
s e a r c h p r o c e d u r e , t h e m ethod o f s t e e p - a s c e n t i s a p p l i e d to
t h e s u r f a c e t o f i n d t h e v a l u e s o f t h e p a r a m e t e r s t h a t
m in im iz e t h e e r r o r , Q. The m ethod i s a v e c t o r s e a r c h
m ethod w h ich r e q u i r e s t h e d e f i n i t i o n o f a s e a r c h v e c t o r i n
t e rm s o f t h e r a t e o f change o f t h e o b j e c t i v e f u n c t i o n w i t h
r e s p e c t to e a c h p a r a m e t e r . C o n vergence i s u s u a l l y a s s u r e d
f o r any r e a s o n a b l e i n i t i a l e s t i m a t e o f t h e p a r a m e t e r s ; how
e v e r , a l a r g e number o f i t e r a t i o n s may be r e q u i r e d .
As t h e p r o g r e s s to w a rd t h e minimum v a l u e f o r t h e
e r r o r f u n c t i o n s l o w s , t h e p r o g r a m s w i t c h e s to a s e c o n d
s e a r c h t e c h n i q u e . A s e c o n d - o r d e r model i s u s e d t o a p p r o x i
mate t h e . r e s p o n s e s u r f a c e o f t h e e r r o r f u n c t i o n , Q. T h is
i s i t e r a t e d i n t h e Gauss m ethod u n t i l c o n v e r g e n c e i s
r e a c h e d . The Gauss i t e r a t i v e s o l u t i o n c o n s i s t s b a s i c a l l y
21
o f a f i r s t - o r d e r T a y l o r s e r i e s e x p a n s i o n o f t h e g i v e n
f u n c t i o n a b o u t some b a se p a r a m e t e r v a l u e ( 2 5 ) . The
r e s u l t i n g e q u a t i o n i s l i n e a r and l e a s t - s q u a r e s t e c h n i q u e s
a r e u s e d t o d e t e r m i n e t h e c o r r e c t i o n v e c t o r f o r t h e
p a r a m e t e r e s t i m a t e s . T h is e s t a b l i s h e s a new b a s e p o i n t
•which i s a g a i n i t e r a t e d u n t i l c o n v e r g e n c e i s e s t a b l i s h e d . .
T h is m ethod c o n v e r g e s o n l y f o r good i n i t i a l p a r a m e t e r
e s t i m a t e s , t h u s t h e i n i t i a l u s e o f s t e e p a s c e n t t o s t a r t
t h e m eth o d .
The i m p o r t a n t f e a t u r e s o f t h i s s e a r c h p ro g ra m ,
r e f e r r e d t o as S t e e p - A s c e n t 67 ( 1 4 ) , a r e g i v e n i n
A p p e n d ix E.
. • SLURRY ..' INFLUENCE - IN.: WELL- STIRRED-TANK GOMMINUTIQN:MODEL '
. .D ev e lo pm en t ' .o f t h e Model t.—— - v- - r- , —---- —
The we 1 1 - s t i r r e d - t a n k co m m in u t io n m ode l was u se d
: i n i t i a l l y t o d e v e l o p t h e d e p e n d e n c y - o f t h e r a t e , o f . cem-
m in u t 1 on on t h e s l u r ry s © l i d s . T h e r e . a r e ce r t a i n
• a d v a n t a g e s t o u s i n g t h e '• we 11 - s t i r r e d - t a n k .model:. S in c e
t h e w e l l - s t i r r e d - t a n k m odel i s a n . a l g e b r a i c m ode l i t . c a n
be im p le m e n te d on t h e d i g i t a l c o m p u te r more s i m p ly t h a n
t h e a x i a l - d i f f u s i o n m o d e l . - I n :a d d i t i o n , t h e w e l l - s t i r r e d -
t a n k . m o d e l u s e s , much l e s s , c o m p u t e r t im e d u r i n g t h e compu
t a t i o n a l p r o c e s s , s i n c e n o ■n u m e r i c a l i n t e g r a t i o n .p r o c e s s e s
a r e r e q u i r e d . a s t h e y • a r e w i t h t h e . a x i a l - d l f f u s i o n a n o d e l .
The b a s i c m odel f o r t h e we1 1 - s t i r r e d - t a n k d e s c r i p
t i o n : o f com m in u t io n was d e v e l o p e d by - H o r s t (1.9) . and K e l ln e r
( 2 0 ) S i n c e t h i s m odel was d e v e l o p e d e a r l i e r i t w i l l o n ly '
be p r e s e n t e d i n summary fo rm he r e .
- A w e l l - s t i r r e d t a n k c a n be d e f i n e d a s a t a n k ;w i t h a
- homogeneous volume, and w i t h t h e o u t p u t .c o n c e n t r a t i o n . e q u a l
t o t h e c o n c e n t r a t i o n o f t h e i n t e r i o r v o lum e . I n t h e c a s e
• o f t h e b a l l m i l l . t h e p a r a m e t e r b e in g d e s c r i b e d i s t h e
p a r t i c l e - s i z e d i s t r i b u t i o n .
The c o n c e p t o f f e e d - f o r w a r d one i s u t i l i z e d i n t h e
model,. • When a p a r t i c l e i s b r o k e n i t c o u l d be e x p e c t e d t o
22
23
be b r o k e n i n t o ' ©me : ©r a l l .o f t h e s m a l l e r s i z e s , . However,
i t i s ■ assumed, i n t h i s m odel t h a t , t h e f l o w o f m a t e r i a l o f a
g i v e n :s i z e a f t e r i t i s b r o k e n i s t o ■t h e n e x t s m a l l e r s i z e
o n l y . T h a t i s n o t t o im p ly t h a t . a l l m a t e r i a l i s b r o k e n
o n ly i n t o ■ t h e t e x t . . ' sm a l le r , s i ze y b u t von ly t o s a y t h a t t h i s
c o n c e p t c an be u s e d t o d e s c r i b e t h e c o m m i n u t i o n ■t a k i n g
p lace, . . T h i s i s r e f e r r e d t o a s f e e d - f o r w a r d - o n e , and was
d e m o n s t r a t e d t o - be v a l i d by H o r s t ( 1 9 ) .
- The f o l l o w i n g , a s s u m p t io n s ' have b e en made c o n c e r n i n g
t h i s a p p r o a c h . -The m odel i s . a f e e d - f o r w a r d t o n e s y s t e m .a s
d e s c r i b e d a b o v e . • A s i n g l e w e l l - s t i r r e d tank.: I s . u s e d t o
d e s c r i b e t h e s y s t e m and t h e r e f o r e t h e i n t e r i o r - s i z e d i s
t r i b u t i o n i s r e l a t e d t o d i s c h a r g e p a r t i c l e - s l z e d i s t r i b u
t i o n t h r p u g h t h e g r a t e . d i s c h a r g e c o e f f i c i e n t .
R e f e r r i n g to- F i g . -3 - t h e . m a t e r i a l b a l a n c e . f o r • a
g i v e n , s i z e i a t s t e a d y •s t a t e y i e l d s 8
(FXo i + - (FXd i + Rt ) . = 0 , ( 3 )
■Xd iS in c e ' a = .^— and X . i s known f o r a l l t e s t s i t i s n o t - i . a • mimin e c e s s a r y t o compute a n a v e c t o r t o - d e s c r i b e t h e . e f f e c t s , o f
t h e g r a t e - d i s c h a r g e . , s e c t i o n .
■ R e a r r a n g e m e n t - o f E q . •3 y i e l d s t h e e x p r e s s i o n f o r
t h e mass f r a c t i o n o f s i z e i i n t h e d i s c h a r g e a s - a f u n c t i o n
- o f . f e e d r a t e : and co m m in u t io n r a t e
24
FX
i - 1
01
mi
d i+ 1
i+1 mii+1
G r a teD i s c h a r g e
S e c t i o nG r i n d i n gS e c t i o n
mi
wne re F
W
Xo i
Xmi
Xd i
Ria .
s o l i d s f e e d r a t e
b a l l m i l l i n v e n t o r y of s o l i d s
mass f r a c t i o n o f s i z e i i n f e e d
mass f r a c t i o n of s i z e i i n i n v e n t o r y
mass f r a c t i o n o f s i z e i i n d i s c h a r g e
c o m m inu t io n r a t e
d i s c h a r g e c o e f f i c i e n t f o r s i z e i
F i g . 3 . S c h e m a t i c r e p r e s e n t a t i o n of mass f l o w t h r o u g h t h e b a l l m i l l .
25
Xd i = F ^FXoi * Ri - 1 Ri )
and
Ri = ki wi ( 5 )
where = co m m in u t io n r a t e f o r s i z e i
k 1 = co m m in u t io n c o e f f i c i e n t f o r s i z e i
= mass of s i z e i i n t h e b a l l m i l l
E q . 4 i s t h e bas i . s f o r th e f o r m u l a t i o n o f th e
we 1 1 - s t i r r e d - t a n k m o d e l . However , t h e fo rm shown i n Eq.
4 d o e s n o t a c c o u n t f o r v a r i a t i o n s i n p e r c e n t s l u r r y
s o l i d s , f e e d r a t e , o r g r i n d a b i l i t y . F o r t h i s i n v e s t i g a t i o n
th e f e e d r a t e r em a in e d e s s e n t i a l l y c o n s t a n t a t t h r e e pounds
p e r m in u te and t h e same ore was u se d f o r a l l t e s t s .
I n p r e v i o u s work H o r s t ( 1 9 ) i n c o r p o r a t e d t h e s l u r r y
s o l i d s i n t o t h e we 11 - s t i r r e d - t a n k model by u s i n g an e x p r e s
s i o n f o r t h e a p p a r e n t s l u r r y v i s c o s i t y . The a p p a r e n t
s l u r r y v i s c o s i t y was l i n e a r i z e d and a p p l i e d o v e r an
a p p r e c i a b l e r a n g e o f s l u r r y s o l i d s .
Based on t h e p r e v i o u s work th e i n f l u e n c e o f th e p e r
c e n t s o l i d s was i s o l a t e d f rom t h e c o m m in u t io n c o e f f i c i e n t k*
by an e q u a t i o n of t h e fo rm ;
Rj = k^w.fB (6)
where (3 = s l u r r y s o l i d s i n f l u e n c e
k^ = co m m in u t io n c o e f f i c i e n t i n d e p e n d e n t o f p e r c e n t s o l i d s i n th e s l u r r y
E v a l u a t i o n o f P e r Gent S o l i d s
In a c o m m e r c i a l o p e r a t i o n i t would be a d v a n t a g e o u s
t o u se p e r c e n t s o l i d s by w e i g h t ( s ) t o d e f i n e t h e s l u r r y
i n f l u e n c e on t h e com m inu t io n r a t e , s i n c e t h i s p r o p e r t y i s
t h e one w h ic h i s n o r m a l l y m e a s u r e d i n a c o n t i n u o u s m i l l i n g
o p e r a t i o n . T h e r e f o r e , S was u s e d i n i t i a l l y t o d e t e r m i n e
i t s i n f l u e n c e on th e g r i n d i n g t a k i n g p l a c e w i t h i n th e b a l l
m i l l .
The s t e e p - a s c e n t o p t i m i z a t i o n t e c h n i q u e ( 1 4 )
d e s c r i b e d p r e v i o u s l y was u s e d t o d e t e r m i n e one c o m m in u t ion
c o e f f i c i e n t ( £ ) f o r e a c h e x p e r i m e n t a l t e s t a T h a t i s , one
co m m inu t io n c o e f f i c i e n t was d e t e r m i n e d r a t h e r t h a n an
i n d i v i d u a l c o e f f i c i e n t k j f o r e ac h s i z e f r a c t i o n . These
v a l u e s a r e shown i n T a b le 7 . The v a l u e s f o r k were t h e n
p l o t t e d a g a i n s t t h e p e r c e n t s o l i d s t o show t h e Tc vs.. S
r e l a t i o n s h i p . These r e s u l t s a r e shown i n F i g . 4 . I t can
be s e e n t h a t t h i s c u r v e i s a p p r o x i m a t e l y l i n e a r o v e r t h e
m a j o r p o r t i o n of t h e v a r i a t i o n s i n p e r c e n t s o l i d s . I t was
t h e r e f o r e d e c i d e d t o u se a p o l y n o m i a l a p p r o x i m a t i o n f o r t h e
s l u r r y - s o l i d s i n f l u e n c e . From a p r a c t i c a l p o i n t o f v iew i t
would be d e s i r a b l e t o r e d u c e t h e o r d e r o f th e p o l y n o m i a l
a p p r o x i m a t i o n t o a minimum.
The shape o f t h e c u rv e i n F i g . 4 i s c o n fo u n d e d
somewhat by u s i n g one c o m m in u t ion c o e f f i c i e n t s i n c e t h e
c o m m in u t io n c o e f f i c i e n t i s a f u n c t i o n of s i z e . H o w ev e r ,
t h e t r e n d i s t h e r e s u l t o f two o p p o s in g f a c t o r s . H o r s t
27
T a b le 7. Comminution C o e f f i c i e n t s (E)
P e r Cent S o l i d s E T e s t No.
Iyp.e . X Mate r i a l
5 8 .6 0 . 4 4 9
6 0 .2 0 . 3 0 7
6 5 .1 0 .3 1 WP-5
6 7 . 0 0 .2 7 WP-7
7 1 .3 0 . 2 5 6
7 2 .5 0 . 2 3 8
Type Y Mate r i a l
5 8 .7 0 . 4 8 4
5 8 .9 0 . 5 1 1
5 9 .8 0 . 5 2 10
6 6 .6 0 .3 7 3
7 0 .6 0 .2 1 2
7 1 .2 0 . 2 5 5
Com
min
utio
n C
oeff
icie
nt
(It)
28
A Type X m a t e r i a l ° Type Y m a t e r i a l
0 . 4 -
56 60 64 68 72P e r C en t S o l i d s
F i g . 4 . Comminution c o e f f i c i e n t (IE) vs_. p e r c e n t s o l i d s i n s l u r r y .
29
(1 9 ) showed t h a t t h e m i l l i n v e n t o r y f o r a g r a t e - d i s c h a r g e
b a l l - m i 11 i n c r e a s e s a s th e p e r c e n t s o l i d s i n t h e s l u r r y
i n c r e a s e s . T h u s , a t c o n s t a n t s o l i d s f e e d r a t e , t h e mean
mi 1 1 - r e s i d e n c e t im e i n c r e a s e s w i t h t h e i n c r e a s e i n p e r c e n t
s o l i d s and t h i s i n f l u e n c e s h o u l d i n c r e a s e t h e com m inu t ion
r a t e . However , t h e a p p a r e n t s l u r r y v i s c o s i t y a l s o i n c r e a s e s
w i t h th e i n c r e a s i n g p e r c e n t s o l i d s and t h e c u s h i o n i n g
e f f e c t f rom t h i s v i s c o s i t y i n c r e a s e would t e n d t o d e c r e a s e
t h e c o m m in u t io n r a t e . The n e t r e s u l t w i l l t h e n be what i s
shown i n F i g . 4 . The r e s u l t s shown i n F i g . 4 and t h e
f o l l o w i n g e v a l u a t i o n s show t h a t t h e o v e r a l l i n f l u e n c e of
t h e i n c r e a s i n g p e r c e n t s o l i d s i n th e s l u r r y i s t o d e c r e a s e
t h e c o m m in u t io n r a t e .
I n c o m m e rc ia l c o n t i n u o u s - g r i n d i n g o p e r a t i o n s t h e
r an g e o f p e r c e n t s o l i d s i n th e s l u r r y i s g e n e r a l l y f rom
60 t o 75 p e r c e n t . At s t e a d y s t a t e t h e r an g e of p e r c e n t
s o l i d s i n any one g r i n d i n g u n i t would be c o n s i d e r a b l y l e s s .
T h i s s t u d y v a r i e d th e p e r c e n t s o l i d s f ro m 55 t o 73 p e r
c e n t , w h ich s h o u l d be a d e q u a t e f ro m an a p p l i c a t i o n v ie w
p o i n t .
As s t a t e d p r e v i o u s l y , t h e t e r m (3 was u s e d t o
i s o l a t e t h e s l u r r y s o l i d s i n f l u e n c e f rom t h e com m inu t io n
c o e f f i c i e n t . I n t h e g e n e r a l fo rm
P = a Q - a^A . . . - a nAn , ( 7 )
30
where a 0 , . . . a n a r e e m p i r i c a l c o n s t a n t s d e t e r m i n e d by
s t e e p - a s c e n t m e t h o d s , and A i s e i t h e r p e r c e n t s o l i d s o r
volume f r a c t i o n of s o l i d s .
Based on t h e c u rv e shown i n F i g . 4 f o r t h e com
m i n u t i o n c o e f f i c i e n t , k , vs.. p e r c e n t s o l i d s t h e f i r s t
i m p l e m e n t a t i o n f o r t h e p e r c e n t s o l i d s f u n c t i o n a l i t y w as :
(3 = a 0 - a^S ( 8 )
The r e s u l t s of t h i s e v a l u a t i o n a r e g i v e n i n T a b le 8. The
r e s u l t s w i t h E q . 8 h av e an a v e r a g e d e v i a t i o n of 0 .0 2 0 5 m ass
f r a c t i o n u n i t s f o r Type X m a t e r i a l , w hich i s w i t h i n e x p e r i
m e n t a l e r r o r . The Type Y m a t e r i a l h a s an a v e r a g e d e v i a t i o n
of 0 .0 3 3 6 mass f r a c t i o n u n i t s . E i t h e r of t h e r e s u l t s c o u ld
be im proved by r e d u c i n g t h e r a n g e of th e p e r c e n t s o l i d s i n
t h e s l u r r y w h ich i s s c a n n e d . As s t a t e d e a r l i e r , i t would
be e x p e c t e d t h a t i n a c o m m e rc ia l o p e r a t i o n t h e range of t h e
p e r c e n t s o l i d s would be l e s s .
In an a t t e m p t t o f u r t h e r r e d u c e t h e number of
p a r a m e t e r s Eq. 8 was e v a l u a t e d f i r s t w i t h a^ = 1 . 0 and t h e n
w i t h a-^ and a 2 = 1 . 0 . These r e s u l t s a r e r e p o r t e d i n T a b l e s
9 and 10. The rang e o f th e p e r c e n t s o l i d s u s e d i n th e
e v a l u a t i o n was 18 p e r c e n t ; i f t h i s r ang e were l e s s i t may
be p o s s i b l e t o u se Eq. 8 w i t h a^ a n d / o r a ^ e q u a l to 1 . 0 .
31
T ab le 8. Summary E q . 8
o f R e s u l t s f o r (3 = a Q - (9 P a r a m e t e r s )
a^S
Type X M a t e r i a l Type Y M a t e r i a l A l l T e s t s
k i 1 .6 9 3 — — 2 .1 6 3
k 2 3 .2 0 9 3 .7 2 6 4 .0 0 2
k 3 5 .3 2 4 5 .0 8 8 6 .0 0 2
k4 4 .5 2 7 4 .7 1 9 5 .3 0 9
k 5 2 .7 7 6 2 .9 6 3 3 .2 8 8
k 6 1 .9 1 9 2 .2 4 9 2 .4 0 1
k 7 2 .0 4 8 2 .6 5 4 2 .7 5 8
a o 0 .7 5 2 0 .7 3 6 0 .5 7 1
a l 0 .6 7 7 0 .7 6 5 0 .5 2 7
Var 0 .0 0 0 6 5 0 .0 0 1 7 7 0 .0 0 2 1 8
S .D . 0 .0 2 5 6 0 .0 4 2 1 0 .0 4 6 6
A.D. 0 .0 2 0 5 0 .0 3 3 6 0 .0 3 7 4
No T e s t s 5 6 11
32
T a b le 9. Summary of R e s u l t s f o r {3 = ( a - a-jS)E q . 8 (8 P a r a m e t e r s ) ( a Q = 1 . 0 0 )
Type X M a t e r i a l Type Y M a t e r i a l A l l T e s t
k l 2 .1 0 3 — — 2 .1 6 4
k 2 3 .8 0 5 4 .3 4 9 3 .8 4 8
k 3 6 .3 4 9 5 .3 5 8 5 .59 7
k 4 5 .3 1 3 5 .0 1 8 . 5 .0 6 2
k 5 3 .2 3 7 3 .2 0 1 3 .1 9 4
k 6 2 .2 4 1 2 .4 0 8 2 .37 3
k 7 2 .4 3 7 2 .7 8 0 2 .7 1 4
a o 0 .9 4 1 0 .9 0 0 0 .9 1 3
Var 0 .0 0 0 6 3 0 0 .0 0 1 7 1 0 .0 0 2 3 4
S .D . 0 .0 2 5 1 0 .0 4 1 4 0 .0 4 8 4
A.D. 0 .0 2 0 1 0 .0 3 3 1 0 .0 3 8 7
No T e s t s 5 6 11
3 3
T a b le 10. Summary o f R e s u l t s f o r (3 = 1 - SE q . 8 (7 P a r a m e t e r s )
Type X M a t e r i a l
k l 1 .6 3 2
k 2 2 .9 1 9
k 3 4 .8 3 1
k 4 3 .9 5 3
k 5 2 .3 3 1
k 6 1 .5 7 1
k 7 1 .7 3 0
Var 0 .0 0 0 8 6 1
S.D . 0 .0 2 9 4
A. D. 0 .0 2 3 5
No T e s t s 5
Type Y M a t e r i a l A l l T e s t s
--- 1 .568
3 .0 9 4 2 .7 8 2
3 .7 6 0 3 .9 8 1
3 .4 3 7 3 .5 2 3
2 .1 6 2 2 .1 7 4
1 .6 6 1 1 .5 9 1
1 .1 9 9 1 .8 3 4
0 .0 0 1 9 5 0 .0 0 5 4 6
0 .0 4 4 2 0 .0 7 3 9
0 .0 3 5 3 0 .0 5 9 0
6 11
34
E v a l u a t i o n of Volume F r a c t i o n of S o l i d s
Brown and a s s o c i a t e s ( 1 0 ) i n d i c a t e d t h a t th e
a p p a r e n t s l u r r y v i s c o s i t y i s b e t t e r d e f i n e d by t h e volume
f r a c t i o n r a t h e r t h a n t h e w e i g h t f r a c t i o n of s o l i d s . S i n c e
a p p a r e n t s l u r r y v i s c o s i t y i s wha t i t i s d e s i r e d t o a p p r o x i
m a t e , a c o r r e l a t i o n of t h e co m m in u t io n r a t e w i t h volume
f r a c t i o n of s o l i d s was p e r f o r m e d u s i n g E q . 9
where X i s d e f i n e d a s t h e volume f r a c t i o n o f s o l i d s i n t h e
s l u r r y . An a d d i t i o n a l a d v a n t a g e t o u s i n g volume f r a c t i o n
i s t h a t i t i m p l i c i t l y i n c l u d e s t h e s o l i d s d e n s i t y . The
i n c l u s i o n of s o l i d s d e n s i t y in t h e c o m m i n u t i o n - r a t e s l u r r y -
s o l i d s r e l a t i o n s h i p s h o u l d make t h e r e s u l t s more g e n e r a l .
The volume f r a c t i o n i s r e l a t e d t o t h e w e i g h t f r a c t i o n , S ,
by Eq. 10
- l i q u i d d e n s i t y
The r e s u l t s of e v a l u a t i o n s u s i n g Eq. 9 a r e g i v e n i n
T a b le 11. As i n t h e p r e v i o u s c o r r e l a t i o n s f o r t h e com
m i n u t i o n r a t e , t h e s t e e p - a s c e n t m e th o ds were u s e d t o
( 9 )
X S / p s + (1 - s ) / p L ( 10 )
where X = volume f r a c t i o n of s o l i d i n th e s l u r r y
S = w e i g h t f r a c t i o n o f s o l i d i n t h e s l u r r y
Ps = s o l i d s d e n s i t y
35
T ab le 11. Summary of R e s u l t s f o r (3 = a - a-,X - a^X^E q . 9 (12 P a r a m e t e r s )
Type X M a t e r i a l Type Y M a t e r i a l A l l T e s t s
k l 1 .9 8 0 — — 2 .1 1 9
k 2 3 .6 3 0 4 . 2 3 4 3 .8 3 9
k 3 6 .0 4 4 5 .243 5 .5 1 7
k 4 5 .0 2 8 4 .6 8 5 4 .7 6 3
k 5 3 .0 2 3 2 .9 4 7 2 .9 3 2
k 6 2 .0 7 7 2 .2 5 2 2 .1 0 9
k 7 2 .2 6 0 2 .6 3 3 2 .3 5 8
a o 0 .4 3 8 0 .4 4 6 0 .4 1 0
a l 0 .0 0 8 5 2 0 .0 0 8 1 1 0 .0 0 8 9 5
a 2 0 .8 6 7 1 .0 2 2 0 .8 0 6
Var 0 .0 0 0 5 8 5 0 .0 0 1 7 0 0 .0 0 2 0 5
S.D . 0 .0 2 4 2 0 .0 4 1 3 0 .0 4 3 5
A.D. 0 .0 1 9 3 0 .0 3 2 9 0 .0 3 4 7
No T e s t s 5 6 11
3 6
m in im iz e t h e e r r o r : b e tw e e n , the. ' computed and .e ,xper imenta 1
d i s c h a r g e - p a r t i c l e - s i ze d i s t r i b u t i o n s ;
The u se ..of- Eq. 9 p r o d u c e d . a v e r a g e d e v i a t i o n s o f
0 .0 1 9 3 and 0 .0329 m ass f r a c t i o n u n i t s f o r - Type-. 5C. and:. Type Y
. m ate r i a l , ■ r e s p e c t i v e l y - The • im pr ovement i n t h e ,ave rag e
d e v i a t i o n s o b t a i n e d u s i n g . E q . 9 . o v e r t h e : d e v i a t i o n s .
o b t a i n e d u s in g . Eq. 8 , c a n ' b e . a t t r i b u t e d p r i m a r i l y t o t h e
. a d d i t i on o f . one more p a r a m e t e r , name l y ■ a^ , i n - Eq,.., 9.,
• Eq. 9 is. more g e n e r a l b e c a u s e . o f t h e i m p l i c i t i n c l u s i o n
of t h e - s o l i d s , d e n s i t y ; h o w e v e r , i n m os t - c o m m e r c i a l o p e r ac
t i o n s . S i s w h a t i s m e a s u r e d c o n v e n ie n t ly . , and. t h e , d e n s i t y
m u s t be known t o . c a l c u l a t e X. ■ T h e r e f o r e m o s t o f _.t h e
a d v a n t a g e s - o f . t h e i m p l i c i t , i n c l u s i o n of t h e o re d e n s i t y
w ould be l o s t .
E v a l u a t i o n . ' . o f A p p a r e n t ' S l u r r y . V i s c o s i t y
P r o m . t h e c o r r e l a t i o n s d e r i v e d : f o r t h e i n f l u e n c e o f
t h e p e r c e n t . s o l i d s . and volume f r a c t i o n o f s o l i d s i n t h e
. s l u r r y - on the., co m m in u t io n , r a t e , , one p o i n t ; s h o u l d be
- n o t i c e d . ■©ver t h e r a n g e - o f s l u r r y s o l i d s - i n v e s t i g a t e d
- (55 t o 73 p e r - c e n t ) . t h e r e , were two . d i s t i n c t p a r t i c l e - s i z e
. d i s t r i b u t i ©ms. - O n e - h a l f . of . t h e t e s t s were made w i t h ' Type .... -
- X m ate r i a l ,; and t h e - o t h e r h a l f .. o f t h e t e s t s we re made • w i t h
-; Type . Y.: m ate r i a l . - As shown ■ by t h e r e s u l t s . • summari zed i n
. T a b l e s 8 , 9 , 10 , and 1 1 . t h e e r r o r b e tw e e n t h e . c o m p u te d . and!
.. e x p e r i m e n t a l d i s c h a r g e - p a r t i c l e s i z e d i s t r i b u t i o n s , i s
!- ir
. 37
w i t h i n : t h e : e x p e r i m e n t a l e r r o r ' when t h e 'Type- X a n d , T y p e ;Y
m ate r i a l i s t r e a t e d . s e p a r a t e T y . Howe ve ry , w h e n r a l l , t h e
t e s t s a r e c o m b i n e d . t h e e r r o r i n c r e a s e s . ...................
The i n c r e a s e i n t h e . e r r o r f o r th e - combined: t e s t s i s
r e l a t e d . t o • t h e f a c t : t h a t .the", s l u r r y ’ v i s c o s i t y i s , t h e p r o c e s s
v a r i a b l e 1 t h a t : i s r e a l l y i n f l u e n c i n g - t h e : com m inu t i on ■ r a t e ,
- w h i l e r the , mass o r v o lu m e . f r a c t i o n o f s o l i d s ' i s b e in g
: c o r r e l a t e d w i t h the- co m m in u t io n i rate"." I t has" b e e n «shown
by -Brown - and. a s s o c i a t e ' s ( 1 0 ) t h a t a n ' ' e x p r e s s i on : can ■ be
d e r i v e d . f o r a p p a r e n t . s l u r r y v i s c o s i t y , f o r p a r t i c u l a t e
s y s t e m s . However , t h i s - r e l a t i o n s h i p d o e s ' n o t t a k e i n t o
-• a c c o u n t . v a r i a t i o n s i n t h e p a r t i c l e . s i z e . d i s t r i b u t i o n .
• O t h e r i n v e s t i g a t o r s ( 2 9 , 30 ) h a v e . s t u d i e d t h e e f f e c t . o f t h e
s i z e ■ o f : g l a s s , s p h e r e s on i s l u r r y v i s c o s i t y and h ave : f ound
t h a t t h e / v i s c o s i t y / o f th e , m i x t u r e i s r a d i c a l l y a f f e c t e d by
p a r t i c l e s i z e . •I t w ould b e ; e x p e c t e d t h a t t h e s e . same
. e f f e c t s would i n f l u e n c e m i n e r a l - s l u r r y s y s te m s e v e n ; th o u g h
m o s t . work : r e p o r t e d on v i s c o s i t y c h a n g e s . w i t h p a r t i c l e . , s i z e
h a s b e en ,done, u s i n g c l o s e l y s i z e d , m i x t u r e s , o f g l a s s
s p h e r e s . . C o n v e r s e l y , i n m os t mine r a l - s l u r r y s y s t e m s t h e
:• p a r t i c l e - s i z e •• d i s t r i b u t i o n i s v e r y wide . I t . would t h e r e
f o r e be v e r y d e s i r a b l e t o - u s e ; a. r e l a t i o n s h i p ■f o r a p p a r e n t
- s l u r r y v i s c o s i t y t h a t t a k e s i n t o ,a c c o u n t t h e ; p a r t i c l e - s i z e
d i s t r i b u t i o n . a n d . t h e mass f r a c t i o n o f . s o l i d s , i n . t h e . s l u r r y .
• A r e v i e w of t h e l i t e r a t u r e . showed v e r y fe w r e l a
t i o n s h i p s w h i c h •i n c o r p o r a t e t h e p a r t i c l e - s i z e f a c t o r i n t h e
38
v i s c o s i t y r e l a t i o n s h i p . The o n ly e x p r e s s i o n w h ich a p p e a r e d
t o have p r o m is e was t h e e x p r e s s i o n by Wada e t a l . ( 3 1 )
g i v e n i n E q . 1. T h i s e x p r e s s i o n was u s e d i n two w a y s .
F i r s t , t h e v i s c o s i t y was c a l c u l a t e d u s i n g t h e o r i g i n a l
e m p i r i c a l c o n s t a n t s d e v e l o p e d by Wada e t a l . ( 3 1 ) . T h i s
was t h e n i n c o r p o r a t e d i n t o t h e co m m in u t io n r a t e e q u a t i o n
a s :
P = i / ; ( i i )
where Q was t h e v i s c o s i t y i n c e n t i p o i s e f ro m Eq. 1.
The r e s u l t s o f t h i s i n v e s t i g a t i o n a r e p r e s e n t e d i n
T a b le 12.
S in c e c o r r e l a t i o n s u s i n g Eq. 11 d i d n o t show any
a p p r e c i a b l e im provem en t o v e r t h o s e p r e v i o u s l y d e v e l o p e d f o r
s l u r r y s o l i d s , t h e v a l u e s o f t h e e m p i r i c a l c o n s t a n t s
d e t e r m i n e d by Wada e t a l . ( 3 1 ) were a l t e r e d . The a l t e r a
t i o n was done by u t i l i z i n g t h e s t e e p - a s c e n t o p t i m i z a t i o n
t e c h n i q u e s t o s e a r c h f o r t h e s e t o f p a r a m e t e r s f o r Eq. 1
t h a t would m in im iz e t h e e r r o r b e tw ee n t h e com puted and
e x p e r i m e n t a l p a r t i c l e - s i z e d i s t r i b u t i o n s . The r e s u l t s of
t h i s e v a l u a t i o n f a i l e d t o improve t h e c o r r e l a t i o n ; h o w e v e r ,
i t a p p e a r e d t h a t Eq. 1 had been d e v e l o p e d f o r a s l u r r y
s y s te m w i t h a r a t h e r n a r r o w ra n g e of p a r t i c l e d i a m e t e r s .
The r e s u l t s o b t a i n e d by a l t e r i n g E q . 1 a r e sum m arized i n
T a b le 13.
39
T ab le 12. Summary • o f R e s u l t s f o r (3 = 1 . 0 / £Based on E q . 1 (7 P a r a m e t e r s )
Type X M a t e r i a l Type Y M a t e r i a l A l l T e s t s
k l 2 .2 3 5 — — 2 .2 0 8
k 2 4 .0 9 7 4 .5 4 1 4 .0 5 8
k 3 6 .7 7 9 5 .6 4 7 5 .8 3 1
k 4 5 .6 1 4 4 .9 1 5 4 .9 9 8
k 5 3 .3 6 7 3 .0 0 6 3 .0 5 2
k 6 2 .3 2 0 2 .2 6 0 2 .2 1 0
k 7 2 .5 3 1 2 .6 2 0 2 .5 1 4
Var 0 .0 0 0 6 0 2 0 .0 0 1 7 8 0 .0 0 1 9 0
S .D . 0 .0 2 4 6 0 .0 4 2 3 0 .0 4 3 6
A.D. 0 . 0 1 9 6 0 .0 3 3 7 0 .0 3 4 7
No T e s t s 5 6 11
40
T a b le 13. Summary of R e s u l t s f o r (3 = 1 . 0 / £ w i t hA d j u s t e d P a r a m e t e r s Eq. 1 (15 P a r a m e t e r s )
Type X M a t e r i a l Type Y M a t e r i a l- — — — - =3
A l l T e s t
k l 2 .0 6 9 — — 2 .2 6 0
k 2 3 .7 8 8 4 .4 0 6 4 .0 6 6
k 3 6 .2 4 9 5 .3 7 8 5 .8 2 1
k 4 5 .1 9 7 4 .7 1 1 4 .9 0 2
k 5 3 .126 2 .8 8 5 3 .0 3 7
k 6 2 .1 5 6 2 .1 7 2 2 .4 1 4
k 7 2 .3 5 3 2 .5 3 3 2 .5 8 7
Var 0 .0 0 0 6 1 5 0 .0 0 1 6 1 0 .0 0 2 0 1
S .D . 0 .0 2 4 8 0 .0 4 0 2 0 .0 4 4 9
A.D. 0 .0 1 9 7 0 .0 3 2 1 0 .0 3 5 8
No T e s t s 5 6 11
41
What s h o u l d be n o t e d i s t h a t E q . 1 i n c l u d e s t h e
i n f l u e n c e o f t h e p a r t i c l e s i z e d i s t r i b u t i o n i n t h e t e r m
( ct/ D ) , i . e . , mean p a r t i c l e d i a m e t e r d i v i d e d by c a p i l l a r y
d i a m e t e r . T h i s s h o u l d a c c o u n t f o r th e v a r i a t i o n i n th e
a p p a r e n t s l u r r y v i s c o s i t y w i t h p a r t i c l e s i z e . However , f o r
a m i n e r a l - s l u r r y s y s te m where th e r a n g e of p a r t i c l e s i z e i s
s e v e r a l o r d e r s of m a g n i tu d e and t h e mean d i a m e t e r i s s m a l l ,
t h e t e r m ( a / D ) becomes r e l a t i v e l y u n i m p o r t a n t . (D m ust be
g r e a t e r t h a n th e l a r g e s t p a r t i c l e p r e s e n t . ) T h i s h e l p s
e x p l a i n th e l a c k o f im provem en t i n c o r r e l a t i o n by u s i n g
Eq. 1 . As d e m o n s t r a t e d by Wada e t a l . ( 3 1 ) t h e e q u a t i o n i s
v a l i d f o r a n a r r o w p a r t i c l e - s i z e rang e i n the s l u r r y .
The t im e a l l o t t e d t o t h e i n v e s t i g a t i o n o f th e
a p p a r e n t s l u r r y v i s c o s i t y d i d n o t a l l o w f o r a c o m p le te
i n v e s t i g a t i o n o f E q . 1 a f t e r i t s u se i n t h e we 11 - s t i r r e d -
t a n k model f a i l e d t o r e d u c e t h e e r r o r b e tw een th e computed
and e x p e r i m e n t a l d i s c h a r g e - p a r t i c l e - s i z e d i s t r i b u t i o n s .
However , two m o d i f i c a t i o n s of E q . 1 which m ig h t be u s e f u l
were n o t e d . The a p p a r e n t s l u r r y v i s c o s i t y c o u l d be c a l c u
l a t e d u s i n g E q . 1 and n a r r o w s i z e r a n g e s . The e q u a t i o n
c o u l d t h e n be i n t e g r a t e d t o c o v e r t h e e n t i r e s i z e d i s t r i b u
t i o n . A seco n d a r e a o f p o s s i b l e i n v e s t i g a t i o n would be t o
c o n s i d e r t h e c o n t r i b u t i o n t o t h e a p p a r e n t s l u r r y v i s c o s i t y
o f o n ly t h e m a t e r i a l f i n e r t h a n a g i v e n s i z e . F o r e x a m p le ,
t h e m a t e r i a l f i n e r t h a n 150 mesh (105 m i c r o n s ) c o u ld be
u s e d t o a p p r o x i m a t e t h e a p p a r e n t s l u r r y v i s c o s i t y . I t c an
b e : .dem©ns t r a t e d - t h a t t h e f i n e m a t e r i a l is:- t h e maj o r
e o n t r i b u t ©r t o ■ t h e t ©t a l a p p a T e n t . s l u r r y •v i s e © s i t y ♦
As - s t a t e d e a r l i e r , • the-, lumped-‘p a x a m e t e r * o r w e l l -
s t i r r e d - t a n k . r r a o d e 1 was u s e d f ©r c o n v e n i e n c e t o . d e ve lop . , t h e
c o r r e l a t i o n s : b e tw e e n t h e s l u r r y s o l i d s and:, commrnutl©n
r a t e s . The . r e s u l t s showed t h a t : a c o r r e l a t i o n ; , v which, was.
■ w i t h i n . e x p e r i m e n t a l . e r r o r - c o u ld • be 1 o b t a i n e d u s i n g . e i t h e r
• t h e w e i g h t - p e r c e n t , o r - volume f r a c t i o n o f . a o l i d s i n . th e .
s l u r r y . - O n e . o f . t h e m a in . o b j e c t i v e s o f . t h i s i n v e s t i g a t i o nA ar
was. t o d e v e l o p , a. workable. , e x p r e s s i o n f o r t h e s l u r r y - s o l i d s
i n f l u e n c e • on ■ t h e communi t i o n : r a t e w h ic h . c o u l d be u s e d f o r
a u t o m a t i c p r o c e s s c o n t r o l . From . t h i s , s t a n d p o i n t , i t i s
b e l i e v e d t h a t t h e : c o r r e l a t i o n o b t a i n e d u s i n g - t h e p e r c e n t
s o l i d s s h o u l d . p r o v e t o be a d e q u a t e . - I n m o s t . c o m m e r c i a l
o p e r a t i o n s t h e . r a n g e o f . t h e p e r . - c e n t . s o l i d s , i n :any. g i v e n
mi 11 a t s t e a d y s t a t e j would n o r m a l l y be l e s s t h a n ; t h a t
c o v e r e d i n t h e i n v e s t i g a t i o n .
AXIAL-DIFFUS10#'MODEL
App1x c a t 1 e n .o £ t h e ■M odel .
I t was d e m o n s t r a t e d b y H o r s t ( 1 9 ) ; and. P i z z u t .o -
Z a m a n i l l o • (2 7 ) t h a t t h e b a l l m i l l c a n be d e s c r i b e d . more
p r e c i s e l y by a complex, mode 1 t h a t t a k e s i n t o . a c c o u n t t h e
! v a r i a t i o n . . i n p a r t i c l e “- s i z e w i t h a x i a l p o s i t i o n . T h i s
m o d e l , t h e a x i a l - d i f f u s i o n m o d e l , was im p lem en ted , t o
i n c o r p o r a t e t h e . s l u r r y , s o l i d s r e l a t i o n s h i p s d e v e l o p e d i n
: t h e p r e v i o u s s e c t i o n s .
T h e . a s s u m p t io n s , and t h e b a s i c e q u a t i o n s p e r t a i n i n g
t o t h e a x i a l - d i f f u s i o n model a r e p r e s e n t e d - b e lo w . The
a s s u m p t i o n s t h a t app 1 y , . to , -L h is model a r e »
1. N e g l i g i b l e .. r a d i a l v a r i a t i o n i n . c o m p o s i t i o n .
2 . ■N e t . a x i a l - d i f f u s i o n . m u c h g r e a t e r t h a n r a d i a l
.d i f f u s i o n and i n d e p e n d e n t o f . a x i a l p o s i t i o n .
3 . U n i fo rm b u l k - f l o w v e l o c i t y o v e r t h e c r o s s s e c t i o n
o f the : v o l u m e o c c u p i e d by t h e . s l u r r y .
The: f o l l o w i n g . n o t a t i o n p e r t a i n s , t o t h e a x i a l -
d i f f u s i o n m odels
; F = f e e d , r a t e , : I b / m i n .
A' - c r o s s : s e c t i o n a l , a r e a of. t h e . s l u r r y i n t h e b a l l m i l l
V = a v e r a g e •s l u r r y v e l o c i t y i n 2 d i r e c t i o n f t / m i n .
43
44
pg = s l u r r y d e n s i t y , l b / f
w 1 = i n v e n t o r y i n t e n s i t y , l b / f t
X = mass f r a c t i o n of s i z e i , l b ^ / l b
X = dX/dZ
S = s l u r r y s o l i d s , w e i g h t f r a c t i o n
= a x i a l - d i f f u s i o n c o e f f i c i e n t , f t ^ / m i n
W = s o l i d s i n v e n t o r y , l b
Z = a x i a l d i r e c t i o n , f t .
r . = s p e c i f i c c o m m in u t io n r a t e o f s i z e i ,1 1 L_ • — \I b u / ( f t ^ - m i n . )
N . - Net com m inu t ion r a t e of s i z e i ,C l - i i / 1 / _ « __ \l b ^ / ( f t'Vmin. )
k! = com m inu t io n c o e f f i c i e n t c o n fo u n d e d by s l u r r y s o l i d s
= co m m in u t io n c o e f f i c i e n t f o r s i z e i , min ( f e e d - f o r w a r d - o n e s y s te m )
The f o r m u l a t i o n of t h e a x i a l - d i f f u s i o n model
i n v o l v e s t a k i n g a m a t e r i a l b a l a n c e a c r o s s a d i f f e r e n t i a l
a x i a l e l e m e n t of t h e b a l l m i l l a s shown i n F i g . 5.
The s t e a d y - s t a t e e q u a t i o n f o r t h e a x i a l - d i f f u s i o n
model i s
The m odel i s b a sed on a f i r s t - o r d e r c o m m inu t io n
r a t e and f e e d - f o r w a r d - o n e of t h e i n t e r - s i z e m a t e r i a l .
T h e r e f o r e th e n e t co m m in u t ion r a t e becomes:
45
B a l l M i l l
F lowDi r e c t i o n — >
G r a te D i s c h a r g e
Z+AZZ
F i g . 5 . E le m e n t o f s l u r r y and g r i n d i n g m ed ia a l o n g t h e a x i a l d i r e c t i o n of t h e b a l l m i l l .
Nci ki psSXi " ki - l psSXi- l
S u b s t i t u t i o n of Eq. 13 i n t o E q « 12 y i e l d s E q . 14
dX.dZ"
Dl w' d 2X. w ' ( k ! X i - k ! . 1Xi _ 1 )
i d w ' \ dZF ( 1 ' r h z _> f ( i - r 3 ^ )
( 1 3 )
( 1 4 )
w h ich d e s c r i b e s t h e change i n s i z e i w i t h r e s p e c t t o a x i a l
p o s i t i o n w i t h i n th e b a l l m i l l g r i n d i n g s e c t i o n .
The fo rm of t h e a x i a l - d i f f u s i o n model u s e d h e r e h a s
t h e so l i d s - i n v e n t o r y i n t e n s i t y , w' , im p le m e n te d a s a f i r s t -
o r d e r l e a s t - s q u a r e s e q u a t i o n . The u se o f t h e f i r s t - o r d e r
e q u a t i o n i s b a se d on t h e r e s u l t s o f e x p e r i m e n t a l T e s t 10.
T e s t 10 was p e r f o r m e d w i t h a m o d i f i e d g r a t e t o p r e v e n t t h e
f l o w of s l u r r y i n t o t h e g r a t e - d i s c h a r g e s e c t i o n o f th e
46
b a l l m i l l a f t e r t h e t e s t was t e r m i n a t e d . T h i s was
a c c o m p l i s h e d by b l o c k i n g o n e - t h i r d o f t h e g r a t e w i t h p l y
wood . The m i l l was t h e n s t o p p e d so t h a t t h e b l o c k e d p o r
t i o n of t h e g r a t e was i n a p o s i t i o n t o p r e v e n t f l o w i n t o
t h e g r a t e s e c t i o n . I t i s b e l i e v e d t h a t t h i s c o n d i t i o n more
n e a r l y m a i n t a i n s t h e m i l l c o n t e n t s i n t h e o r i g i n a l s t e a d y -
s t a t e c o n f i g u r a t i o n a f t e r t h e m i l l i s s t o p p e d . The
e x p e r i m e n t a l i n v e n t o r y i n t e n s i t y p r o f i l e f o r T e s t 10 i s
shown i n F i g . 6 and c l e a r l y d e m o n s t r a t e s t h e l i n e a r i t y o f
t h e i n v e n t o r y i n t e n s i t y .
a c c o m p l i s h e d by i n c o r p o r a t i n g a t h i r d - o r d e r R u n g e -K u t ta
i n t e g r a t i o n s u b r o u t i n e i n t h e s t e e p - a s c e n t o p t i m i z a t i o n
p r o g r a m . The s e t o f e q u a t i o n s u s e d had t h e g e n e r a l fo rm of
E q s . 15 and 16.
The bou n da ry c o n d i t i o n s f o r E q s . 15 and 16 c o n
s i s t e d of t h e mass f r a c t i o n o f s i z e i a t z = 0**" and i t s
f i r s t d e r i v a t i v e a t t h i s same p o s i t i o n . The v a l u e of
c o r r e l a t i o n o f t h e i n t e r i o r m a s s - f r a c t i o n p r o f i l e s . The
The i m p l e m e n t a t i o n of t h e a x i a l - d i f f u s i o n model was
dX.(1 5 )
dZ Dj^w
( 0 * ) was o b t a i n e d f ro m t h e s e c o n d o r d e r l e a s t - s q u a r e s
Inve
ntor
y In
tens
ity
(w’)
, lb
/ft
47
6 . 6
6 . 4
6 . 2
6 . 0
5 .8
5 .6
5 . 4
5 .20 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0
A x i a l P o s i t i o n ( Z ) , f t
F i g . 6. I n v e n t o r y i n t e n s i t y f o r T e s t 10 ( m o d i f i e d g r a t e d i s c h a r g e ) .
48
v a l u e of t h e f i r s t d e r i v a t i o n was e s t i m a t e d by
d X . ( 0 + ) F .- l b — = LX^O ) - Xo i ] (17)
The c o m m in u t io n c o e f f i c i e n t s ( k | ) were m o d i f i e d t o
i s o l a t e t h e i n f l u e n c e of t h e s l u r r y s o l i d s i n t h e same
m anner a s f o r t h e l u m p e d - p a r a m e t e r i m p l e m e n t a t i o n . The
e q u a t i o n w hich i s o l a t e s t h e s l u r r y s o l i d s e f f e c t i s E q . 7 .
The f o l l o w i n g e q u a t i o n s were u se d t o d e s c r i b e t h e
i n f l u e n c e of t h e s l u r r y s o l i d s on t h e co m m in u t io n r a t e i n
t h e a x i a l - d i f f u s i o n m odels
P = a “ a , S ( 8 )o a L
Lo a l S " =2P - - a-,S - a 0S^ ( 1 8 )
and
P = a 0 - a LX ( 1 9 )
where a l l t e r m s have t h e same m ean ing a s p r e v i o u s l y .
The m ethod of s t e e p - a s c e n t was u s e d t o a d j u s t t h e
p a r a m e t e r s i n t h e a x i a l - d i f f u s i o n m o d e l . The p a r a m e t e r s t o
be a d j u s t e d were t h e co m m in u t io n c o e f f i c i e n t s , t h e a x i a l -
d i f f u s i o n c o e f f i c i e n t s , and t h e e m p i r i c a l c o n s t a n t s i n t h e
s l u r r y s o l i d s r e l a t i o n s h i p s . The e x p e r i m e n t a l and computed
v a l u e s u s e d t o compute t h e e r r o r - s q u a r e d f u n c t i o n i n th e
s t e e p - a s c e n t p ro g ra m were t h e f e e d and i n t e r i o r p a r t i c l e -
s i z e d i s t r i b u t i o n s . To d e m o n s t r a t e t h e v a l i d i t y o f t h e
model t h e e x p e r i m e n t a l and computed d i s c h a r g e - p a r t i c l e - s i z e
49
d i s t r i b u t i o n s , we r e •» net". G0mpiared...'.miti 1 . t h e . s t e e p - a s c e n t
: e r r o r . , c r i t e r i o n h a d . b e e n ; s a t i s f i e d , a n d t h e m odel p a r a m e t e r s
we r e .* f i xed . i n = t h e i r f i n a l . f o r m .
- T h e . a x i a l - d i f f u s i o n c o e f f i c i e n t s i n t h e a x i a l -
d i f f u s i o n m odel a r e lu m p e d . c o e f f i c i e n t s , s i n c e t h e . d i f f u s i o n
. c o e f f i c i e n t is . a. f u n c t i o n , o f . p a r t i c l e s i z e « B e ca u se . i s
a.: f u n c t i o n o f s i z e , t h e a x i a l - d i f f u s i o n m ode l was. i m p l e
m en ted w i t h - a . s e p a r a t e : . f o r . Type X m a t e r i a l and Type Y
m a t e r i a l . - A common k - v e c t o r - w a s u s e d when a l l t e s t s were
s c a n n e d .
The: r e s u l t s ■ o b t a i n e d . when .Eqs . 8 , 18 , and 19 were
: i n c o r p o r a t e d i n t o t h e - . a x i a l - d i f f u s i o n .model a r e - sum m arized
: i n T a b l e s 14, 15 , and 16 . A l l v a r i a n c e s t h a t are .; r e p o r t e d
i n t h e s e t a b l e s a r e t h e v a r i a n c e s , b e t w e e n . t h e e x p e r i m e n t a l
and. computed d i s c h a r g e - p a r t i c l e - s i z e d i s t r i b u t i o n s . T hese
v a r i a n c e s a r e • t h e - m o s t : i m p o r t a n t s i n c e t h e y show ' t h a t . t h e
v a r i a n c e ■o f t h e - d i s c h a r g e p a r t i c l e - s i z e d i s t r i b u t i o n :a s
w e l l w i t h i n , t h e - e x p e r i m e n t a l . e r r o r . ■ T h e . computed d i s c h a r g e
- s i z e d i s t r i b u t i o n i s . f rom .a. m odel b a s e d : on t h e f e e d . and
: i n t e r i o r , d a t a o n l y .
- The . v a r i a n c e s o f th e . c a l c u l a t e d , d i s c h a r g e , p a r t i c l e -
s i z e - d i s t r i b u t i o n s . : f o r ' t h e a x i a . l - d i f f u s i on m o d e l . a r e
. s i g n i f i c a n t l y - l o w e r t h a n t h e v a l u e s o b t a i n e d u s i n g ; t h e
l u m p e d - p a r a m e b e r mode1 . - I n t h e ; c a s e where- E q . 8 was u s e d
t h e . a v e r a g e d e v i a t i o n f o r t h e iType ;X m a t e r i a l . u s i n g t h e
a x i a l - d i f f u s i o n . m ode l was 0 .0105 mass f r a c t i o n u n i t s , a s
50
T a b le 14. Summary of R e s u l t s f o r A x i a l D i f f u s i o n Modelw i t h (3 = ao - a fS - a2$2 E q . 18 and
Two D ^ ' s t 12 P a r a m e t e r s )
Type X M a t e r i a l Type Y M a t e r i a l A l l T e s t s
dt(3 m esh)
DT(6 mesh)
Var
S . D.
A.D.
No T e s t s
0 .4 3 3 5
0 .8 2 0 5
1 .53 91
1 .4 3 3 8
0 .8 6 1 5
0 .5 5 1 5
0 .5 3 1 9
1 .0 9 9 0
2 .3 0 9 6 9
1 .8 3 6 6
2 .2 5 7 5
0 .0 0 0 2 0 3
0 .0 1 4 3
0 .0 1 1 4
6
0 .7 9 6 9
1 .3478
1 .4003
0 .8 5 8 7
0 .5 5 5 2
0 .5 3 2 8
1 .3 1 0 1 9
1 .7003
1 .5 9 7 4
3 .2 7 7 2
0 .0 0 0 1 2 5
0 .0 1 1 2
0 .0 0 0 8 9
' 6
0 .4 3 4 7
0 .8 2 5 5
1 .4 7 3 0
1 .4443
0 .8 6 3 6
0 .5 5 3 8
0 .5 3 2 1
1 .1 3 7 8
2 .1 6 1 3
1 .8 0 0 4
2 .1 7 7 9
3 .9 5 4 9
0 .0 0 0 1 2 0
0 .0 1 0 9
0 .0 0 8 8
12
51
T a b le 15. Summary o f R e s u l t s f o r A x i a l D i f f u s i o n Modelw i t h (3 = a 0 - a iS E q 0 8 and Two
D ^ ' s ( 1 1 P a r a m e t e r s )
Type X M a t e r i a l Type Y M a t e r i a l A l l T e s t s
k l 0 .4 5 3 7 — — 0 .4 5 5 1
k 2 0 . 8 5 7 4 0 .8 6 89 0 .8 6 9 8
k 3 1 o 6565 1 .3 7 7 2 1 .5 8 0 0
k4 1 .5 2 5 5 1 .5 4 6 5 1 .5377
k 5 0 .9 1 0 9 1 0 . 8 9 3 4 0 .9 1 6 8
k6 0 .5 5 9 2 5 0 .5 6 0 5 0 .5 6 1 1
k 7 0 .5 1 0 7 5 0 .5 1 7 2 0 .5 0 9 9
a o 1 .0 2 5 1 1 .0729 1 .0727
a l 3 . 2 8 8 4 2 o6 366 3 .1 8 1 5
D.(3 mesh) 2 .1 4 4 8 — — 2 .1399
dl( 6 mesh) — — 3 .4 4 1 7 3 .9 4 1 3
Var 0 .0 0 0 1 7 4 0 .0 0 0 1 2 5 0=000156
S oD. 0 .0 1 3 2 0 .0 1 1 2 0 .0 1 2 5
A .D e 0 .0 1 0 5 0 .00 8 9 0 .0 0 9 9
No T e s t s 6 6 12
52
T a b le 16 . Summary o f R e s u l t s f o r A x i a l D i f f u s i o n Modelf o r (3 = a 0 - a^X Eq. 19 and Two
D ^ ’ s (1 1 P a r a m e t e r s )
Type X M a t e r i a l Type Y M a t e r i a l A l l T e s t s
(3 mesn)
Dl( 6 m esh)
Var
S ,D 0
A D.
No T e s t s
0 .4 6 7 2
0 .8 9 3 6
1 .6 8 2 2
1 .5 5 7 2
0 .9 1 1 9
0 .5 5 7 3
0 .5 1 1 4
0 .7 8 32
4 .8 3 2 0
2 .2 7 6 6
0 .0 0 0 1 7 7
0 .0 1 3 3
0 .0 1 0 6
6
0 .9 0 6 7
1 .5 6 0 1
1 .6 0 2 0
0 .9 2 6 4
0 .5 6 4 5
0 o 5043
0 .9 3 2 8
3 .6 5 2 3
3 .3 5 3 1
0 .0 0 0 1 1 5
0 .0 1 0 7
0 .0 0 8 6
6
0 .4 5 6 5
0 . 8 9 3 4
1 .6 1 7 4
1 .5599
0 . 9 1 2 4
0 .5 6 0 2
0 .50 9 9
0 .9 1 3 6
3 .9 5 8 7
1 .8 8 5 1
3 .4 9 23
0 .0 0 0 1 6 6
0 .0 1 2 9
0 .0 1 0 3
12
53
com pared t o 0 . 0 2 0 5 f o x • t h e lu m p e d x p a ra m e te r : .m o d e l . This , i s
■ t o b e . / e x p e c te d , s i n c e t h e a x i a l - d i f f u s i o n : m o d e l i s more
- a c c u r a t e b e c a u s e i t accommodate s .: c h a n g e s . • i n . p a r t i c l e - s i ze
d i s t r i b u t i o n a l o n g , t h e . a x i a l l e n g t h o f t h e mi 11.
- S i n c e t h e v a r i a n c e s ; o b t a i n e d u s i n g ; t h e a x i a l -
d i f f u s i o n mode 1 t o d e s c r i b e t h e i n f l u e n c e .o f t h e . s l u r r y
s o l i d s © n:t h e c o m m in u t io n . r a t e a r e w e l l w i t h i n . e x p e r i m e n t a l
. e r r o r , t h i s d e m o n s t r a t e d t h a t t h e u se o f e q u a t i o n s o f . t h e
■ f o rm o f E q s . 8 , - 1 8 , and 1 9 . a r e v a l i d . e x p r e s s i o n s f o r t h e
i n f l u e n c e of t h e s l u r r y s o l i d s : on t h e - c o m m i n u t i o n ■r a t e . '
These r e s u l t s e x t e n d t h e . a x i a l - d i f f u s i o n m odel so t h a t i t
-now i s c a p a b l e o f d e s c r i b i n g t h e . e f f e c t s of . c h a n g e s i n f e e d
. r a t e and s lu r r y . , so l i d s on t h e . c o m m i n u t i o n . r a t e .
. CONCLUSIONS:AND BEG0MMENPAT1ONS
... The:, r e s u l t s o f t h i s i n v e s t i g a t i o n showed, t h a t t h e
- i n f l u e n c e of p e r - c e n t s o l i d s i n t h e . s l u r r y >c o u l d be
i n c o r p o r a t e d i n ' t h e . c o m m i n u t i o n . r a t e . e x p r e s s i o n s f o r b o th
lum ped- and d i s t r l h u t e d -para toe t e r . mode I s ° This , work
e x t e n d s t h e . a x i a l - d i f f u s i o n m o d e l . so t h a t i t i s now c a p a b l e
o f d e s c r i b i n g V a r i a t i o n s i n s l u r r y , s o l i d s e x p l i c i t l y .
- E i t h e r .the•.•mass f r a c t i o n o r t h e volume f r a c t i o n o f
s o l i d s i n t h e . s l u r r y >■ c a n be . c o r r e l a t e d . t o t h e . co m m in u t io n .
r a t e . w i t h i n .e x p e r i m e n t a l , e r r o r . - The d i s t r l b u t e d - p a r a m e t e r
m odel t h a t d e s c r i b e s t h e . c h a n g e ; o f p a r t i c l e s i z e . d i s t r i b u
t i o n a l o n g t h e =a x i a l l e n g t h .o f . t h e m i l l p r o v i d e d a b e t t e r
. c o r r e l a t i o n t h a n t h e stirred -ta n k . :model.
The p a r t i c l e - s i z e d i s t r i b u t i o n .of t h e d i s c h a r g e
f r o m . t h e . b a l l - m i 11 c a n be p r e d i c t e d b a s e d on t h e . f e e d . s i z e
d i s t r i b u t i o n . , s l u r r y - s o l i d s , : and. f e e d rate- , ever.:a.- r an g e i n
s l u r r y s o l i d s of 5 8 . t o 73 p e r c e n t . In . a d d i t i o n , : l a r g e
1 v a r i a t i o n s : i n t h e - f e e d - s i z e d i s t r i b u t i o n . - c a n be i n c o r p o
r a t e d i n : the - a x ia l -diffusion.'.m ode 1. w i t h t h e . a x i a l d i f f u s i o n
c o e f f i c i e n t s »
- R e s u l t s , o b t a i n e d i n t h i s . , s t u d y - i n d i c a t e t h a t . a. n e e d
. e x i s t s t o .d e f i n e t h e s l u r r y v i s c o s i t y i n = t e r m s ■of p a r t i c l e -
s i z e d i s t r i b u t i o n and p e r c e n t s o l i d s i n •t h e , s l u r r y . - I n
54
55
a d d ! fc io n , t h e in c © rp © r a t i® n of _ a f u n c t i o n a l i t y t h a t
d e s c r i b e s t h e e f f e c t s ■ of t h e g r i n d a b i l i t y • o f . t h e o re would
f u r t h e r , expand t h e -use ■ of . t h e s e m od e ls .
APPENDIX A
NOMENCLATURE
2C r o s s - s e c t i o n a l a r e a of s l u r r y i n b a l l m i l l , f t
C o n s t a n t
C o n s t a n t
C o n s t a n t
A x i a l - d i f f u s i o n c o e f f i c i e n t , f t /m in
V o l u m e - s u r f a c e mean d i a m e t e r , f t
S o l i d s f e e d r a t e , l b / m i n
S l u r r y f e e d r a t e , I b / m i n2
Mass v e l o c i t y , I b / m i n / f t
Comminut ion c o e f f i c i e n t f o r s i z e i , min
Net com m in u t io n r a t e of s i z e i , l b ^ / ( f t -m in)
Comminution r a t e , f e e d f o r w a r d o n e , I b ^ / m i n
Net c o m m in u t ion r a t e , I b u / m i n
S l u r r y s o l i d s , w e i g h t f r a c t i o n
dX^/dZ i n t h e R u n g e - K u t ta i n t e g r a t i o n f o r m u l a
V e l o c i t y i n Z d i r e c t i o n , f t / m i n
B a l l - m i 11 i n v e n t o r y ( s o l i d s ) , l b
B a l l - m i 11 i n v e n t o r y ( s l u r r y ) , l b
S o l i d s i n v e n t o r y s i z e i , lb ^
I n v e n t o r y i n t e n s i t y , l b s o l i d s / f t of m i l l l e n g t h
I n v e n t o r y i n t e n s i t y , l b s l u r r y / f t of m i l l l e n g t h
56
57
X’ = Volume f r a c t i o n o f s l u r r y o c c u p i e d by t h e f l u i d
Xi = Mass f r a c t i o n o f s i z e i , l b ^ / l b
X. = d X . /d Z
X-,. = Mass f r a c t i o n of s i z e i i n t h e b a l l - m i 11 p r o d u c t , dl i b j / i b
XQi = Mass f r a c t i o n o f s i z e i i n t h e g r i n d i n g f e e d , l b ^ / l b
Z = A x i a l d i r e c t i o n o f t h e m i l l , f t
/ = Mass f r a c t i o n o f t h e l a r g e s t s i z e m a t e r i a l (3 X 6mesh) i n t h e g r i n d i n g f e e d , l b ^ / l b
= A p p a r e n t s l u r r y v i s c o s i t y , l b / ( f t - s e c )
pg - S l u r r y d e n s i t y , l b s l u r r y / f t
0 = R a t i o of volume f r a c t i o n o f s l u r r y o c c u p i e d bys o l i d s t o volume f r a c t i o n o c c u p i e d by f l u i d
k l = Comminut ion c o e f f i c i e n t f o r s i z e i ( s l u r r y d e p e n d e n t ) , m i n ' l
APPENDIX B
EXAMPLE OF DATA COLLECTED DURING AN EXPERIMENTAL RUN
T e s t HC-8
Feed Drum 2 -5p s = 7 1 6 / 2 5 / 6 8
T e s t S t a r t 4 : 3 0 pm
End 5 :38 pm
Time 68 min
M easu red Feed Rate S t a r t 1380 g /m in
7 0 .57 5 .06 8 .54 3 . 0
h2o Added
Time % S c a l e
0 3310 3230 ' 3240 3250 3260 3265 32
Sample s
No.HC-8-1HC-8-2HC-8-3
Ore R em ain ing a t End 4 5 . 5
A c t u a l Feed Rate 3 . 1 1 I b / m i n
D i s c h a r g e Rate
Time g /m in
Wet. Wt624739659
15253040506065
Dry Wt453535478
1305135313531353135313531353
% S o l i d s7 2 .6 07 2 . 3 97 2 .5 3
Avg
7 2 .5 1
58
APPENDIX C
FEED MATERIAL
T a b le 17* . Feed.. S ize-.-Di s t r i b u t l o n -
-Lot . A Lo t .B . Lo t C Xo t - D
3 .X 6 0 .5 0 7 3 0 .4 4 7 8 0 .4 7 6 4 0 .4 6 1 46 .X 10 0 .2 7 7 3 0 .2 6 4 9 0 ,2 7 9 1 0 ,2 6 3 7
10 X 20 0 .0 9 0 3 0 . 1 0 7 4 0 .0 9 5 1 0 .1 0 1 020 X 35 0 .0 4 1 7 0 .0 6 0 4 0 .0 4 9 3 0 .0 5 6 03$ .X 65 0 ,0 2 5 6 0 .0 3 8 4 0 .0 3 1 7 0 .0 3 7 265. X 150 0 .0 2 1 1 0 ,0 3 0 3 0 .0 2 5 1 0 .0 3 0 3
150.X 270 0 .0 1 4 8 0 .0 2 1 7 0 .0 1 7 3 0 .0 2 0 8-270 0 .0 2 1 9 0 .0 2 1 9 0 .0 2 6 0 0 .0 3 0 0
L e t . E L© t ... F Lo t . G
3 X 6 . 0 , 0 0 0 0 0 ,0 0 0 0 0 ,0 0 0 06 X 10 0 .3 8 4 0 0 .3 2 3 1 0 .4 8 4 5
10 X 20 0 ,2 8 6 5 . 0 .3 1 0 4 0 ,2 7 5 220 X 35 0 ,1 3 2 7 0 .1 4 9 3 0 ,0 9 9 635. X 65 0 .0 7 4 0 0 ,0 8 2 5 0 . 0 5 2 2
S 5 X 150 0 .0 5 1 2 0 , 0 5 6 4 0 . 0 3 6 2150 .X 270 0 .0 3 1 0 0 ,0 3 4 1 0 , 0 2 2 4
-2 7 0 0 .0 4 0 6 0 .0 4 4 2 0 .0 2 9 9
59
APPENDIX D
EXPERIMENTAL AND COMPUTED DISCHARGE PARTICLE-SIZE DISTRIBUTIONS
The c a l c u l a t e d , and e x p e r i m e n t a l p a r t i c l e - s i z e d i s
t r i b u t i o n s • f o r a l l t e s t s u s i n g t h e a x i a l - d i f f u s i o n model
and Eq. 18. a r e - p r e s a n t e d .
The s i z e . f r a c t i o n s a r e . a s f o l l o w s ?
, S i z e Me sh
12345678
3 X 6 6 X 10
10 X 20 20 X 35 35 X 65 65. X 150
150 X 270 -27 0
60
T a b le 18„ E x p e r i m e n t a l and C a l c u l a t e d S i z e D i s t r i b u t i o n
T e s t Number
1 2 3 4
S i z e Exp Glcd Exp . m c d - Exp - C lcd Exp . G lc d
1 0 ,0 0 0 0 0 ,0 0 0 0 0 ,0 0 0 0 0 ,0 0 0 0 0 ,0 0 0 0 >0.0000 0 ,0 0 0 0 0 ,0 0 0 0
2 0 .06 3 8 0 ,0 6 3 5 0 ,0 2 9 3 0 ,0 4 3 1 0 ,06 7 6 0 ,0377 0 ,0 7 0 4 0 ,06 3 2
3 0o1077 0 ,0 9 8 3 0 ,0 6 3 6 0 ,0 6 6 0 0 ,1027 0 ,0767 0 ,1 06 2 0 ,0 7 8 0
4 0 ,12 9 7 0 ,1 3 4 6 0 ,1 14 7 0 ,1 1 4 0 0 ,12 5 6 0 ,1339 0 ,1 2 7 4 0 ,1 1 1 6
5 0 ,1652 0 ,1729 0 ,18 5 8 0 ,1 8 3 0 0 .16 1 3 0 ,1 7 7 3 0 ,1627 0 ,1 6 4 4
6 0 ,1 5 8 8 0 ,1 6 0 5 0 ,18 3 2 0 ,1 8 3 8 0 .1599 0 ,1 7 1 5 0 ,1 6 0 0 0 .1 7 4 5
7 0 ,1 0 3 0 0 ,1 0 3 0 0 ,1 1 7 3 0 ,1 1 7 3 0 ,1 0 3 6 0 ,1 1 5 0 0 ,1 0 5 4 0 ,1 1 9 5
8 0 .2 7 1 8 0 ,2 6 6 8 0 ,3 0 6 1 0 ,2 9 2 5 0 ,2 7 93 0 ,2 8 7 6 0 .2679 0 ,2 8 9 3
T ab le 1 8 . - - C o n t i n u e d E x p e r i m e n t a l and C a l c u l a t e d S i z e D i s t r i b u t i o n
T e s t Number
5 6 7 8
S i z e Exp t i e d Exp . t i e d Exp t i e d .. Exp .. t i e d
1 0 ,0 0 0 0 0 ,0 0 0 0 • 0 .1 0 1 6 0 .1 3 4 0 0 ,1219 0 ,1642 0 .0 6 3 3 0 .0742
2 0 ,0 4 2 3 0 ,0 6 3 6 0 ,08 5 2 0 .1 1 7 6 0 .1 1 1 3 0 ,1 0 53 0 .0 7 0 6 0 .0873
3 0 ,0 7 57 0 ,0 8 2 3 0 ,0652 0 ,0 7 5 5 0 .0 8 1 4 0 .0818 0 .0 7 0 4 0 .0 6 8 4
4 0 ,1222 0 ,1 1 8 1 ; 0 ,0 8 57 0 ,0 7 9 4 0 .09 5 8 0 .0979 0 .1 0 2 1 0 .0 8 6 0
5 0 ,1809 0 ,1 7 2 6 0 .1 2 8 3 0 .1189 0 .1247 0 .1 2 4 6 0 .1512 0 .14 31
6 0 ,1 7 2 3 0 ,1709 0,, 1423 0 .1 3 4 5 . 0 .1 3 1 4 0 .1 22 3 0 ,1 5 4 1 0.1579
7 0 ,1 0 8 0 0 ,1 0 7 4 0 .0 9 8 1 0 .0942 0 .0 9 1 1 0 .0 8 3 4 0 .1 0 0 8 0 .1 0 7 4
. 8 0 ,2 9 8 6 , 0 ,2 8 4 8 0 ,2 9 3 6 0 .2 4 4 6 0 .2 4 2 4 0 .2189 0 .2 8 7 5 0 .27 5 1
O'-ho
T ab le 18«, - - C o n t in u e d E x p e r i m e n t a l and C a l c u l a t e d S i z e . D i s t r i b u t i o n
T e s t Number
9 10 W - 5 : WP-7
S i z e Exp Clcd Exp Clcd Exp Clcd Exp Clcd
1 0 .0 8 9 1 . 0 .0 6 3 6 0 .0 0 0 0 0 .0 0 0 0 0 .1 2 3 5 0 .1 2 4 3 0 .1349 0 .1 0 2 2
2 0 .0 9 8 8 0 .0 9 5 1 0 .0 7 2 4 0 .0 9 4 5 0 .1 0 9 6 0 .1139 0 .1103 0 .1109
3 0 .07 2 7 0 .0 7 8 4 0 .1 1 1 1 0 .1247 0 .0 7 9 6 0 .0 8 0 4 0 .0777 0 .0 8 0 3
4 0 .0 8 9 5 0 .0 9 2 3 0 .1 2 5 5 0 .1378 0 .09 9 8 0 .0 9 0 1 0 .0912 0 .0 9 0 0
5 0 .1 2 8 8 0 .1 2 9 6 0 .1 5 6 1 0 .1 5 2 0 0 .1229 0 .1 2 6 1 0 .1 1 8 6 0 .1259
6 0 .1 4 5 2 0 .1 4 7 6 0 .1 5 5 1 0 .14 3 8 0 .1 2 8 4 0 .1 3 22 0 .1288 0 .1327
7 0 .1029 0 .1097 0 .1059 0 .09 61 0 .0 9 0 0 0 .0 9 32 0 .0 8 9 4 0 .0 9 3 4
8 0 .2 7 3 0 0 .2 8 3 1 0 .2739 0 .2508 0 .2462 0 .2392 0 .2491 0 .26 4 2
. APPENDIX E
DIGITAL . COMPUTER ■ PROGRAMS
The s a l i e n t - - f e a t u r e s o f . t h e . s t e e p - a s c e m t 67 o p t i m i
z a t i o n p ro g ra m a r e •p r e s e n t e d . i n .F i g s . 7, 8 , a n d .9 .
64
65
READDATA,PARAMETERS
CALLRSTAR
SET13=0,CALC.
CALLRSTAR
11=012=012
SHIFT
KMOVBRKS
DXRV
ERRORMESSAGECALL
SLOPECALLRSTAR
CALLQCAL
12 =
12+1
K1
I I sO X=X/211 = 11+1
F i g . 7 . FORTRAN p ro g ra m f o r s t e e p e s t a s c e n t m a i n l i n e .
RETURNTRANSFER Q- DATA TO Y. CALC. LOCAL MIN. Q
CALCULATE 1 s t & 2nd
DIFFo RSTAR
X = 4X
ADJUST X TO SIEW BASE PT. CALC o QB
12
X = X/2 12
0H2 = 1
TRANSFER TO NONLINEAR
TRANSFER TO NONLINEAR 13 s K3
F i g . 8 . FORTRAN p ro g ra m f o r s u b r o u t i n e RSTAR.
DXMX(I) = > 14 = 14 + 1 CALL DRV CALC. FORM0 .1 /B R K S (I ) AUG ( I , J ) =■ 0 1 s t & 2nd DERV AUG(I,K+1)
ENTRYPOSITION
LIMITINGCOMPLETED
USED DURING LIMITING CYCLES
LIMITING PROCESS? UTILIZATION OF M ( J ) ; SHIFTING OF DXA'S BASED ON LIMITING SIGN HISTORY. FORMATION OF NEW X ( I ) ' S
CALL GAUSS TO OBTAIN
DXA(I)
CALLQCAL(QNL>
vQNL
& QBgFERR >
PRINT Q, X ( I ) ’S , plL^GO TO COMPARISON
OF DATA & CALC. _______DATA
(™ r)
GO TO ENTRY STATEMENT 39
GO TO KMOV TO
RESET BRKS(I)
F i g . 9. FORTRAN p rog ram f o r n o n l i n e a r e s t i m a t i o n m a i n l i n e .
68
L e a s t S q u a r e s C o r r e l a t i o n P rogram
The p ro g ram e v a l u a t e s f i r s t - and s e c o n d - o r d e r
c o r r e l a t i o n b e tw e e n two v a r i a b l e s .
SUBROUTINE R10LS DIMENSION DATA(1 0 0 ,1 0 )COMMON DATA,ND,A,A1, A2, VAR,SDEV,ADEVSXO = NDSY = 0 . 0SX = 0 . 0SYX = 0 . 0SX2 = 0 . 0DO 3 J = 1, NDSY = SY + DATA(J,1 )SX = SX + DATA(J,2 )SYX = SYX + DATA(J ,1)*DATA(J,2)
3 SX2 = SX2 + DATA(J, 2 )*DATA(J,2 )A1 = (SX*SY-SYX*SX0)/(SX*SX-SX2*SX0) A = (SX2*SY-SYX*SX)/(SXO*SX2-SX*SX) SUM = 0 . 0 ASUM = 0 . 0 DO 4 J = 1 , NDDATA(J,3) = A + A1*DATA(J, 2 ) DATA(J,4) = DATA(J,1 ) - DATA(J.3 ) DATA(J, 5 ) = DATA(J,4)*DATA(J,4)ASUM = ASUM + (ABSF(DATA(J, 4 ) ) )
4 SUM = SUM + DATA(J,5 )VAR = SUM/SXOSDEV = VAR**0.5 ADEV = ASUM/SXO RETURN ENDSUBROUTINE R20LS DIMENSION DATA(1 0 0 ,1 0 )COMMON DATA, ND, A, A 1,A 2 , VAR, SDEV,ADEVSXO = NDSY = 0 . 0SX = 0 . 0SYX = 0 . 0SX2 = 0 . 0SX3 = 0 . 0SX4 = 0 . 0SYX2 = 0 . 0DO 3 J = 1, NDX3‘ = DATA( J , 2 )*DATA( J , 2 )
69
SY = SY + DATA(J , 1 )SX = SX-+ DATA(J,2)SYX = SYX + DATA(J,1)*DATA(J,2)SX2 = SX2 4- X3SX3 = SX3 + X3*DATA(J,2)SX4 = SX4 + X3*X3
3 SYX2 = SYX2 + DATA(J,l)*X3DEN=S XO *S X2*S X4-S XG *S X3 *S X3-S X*S X*S X4
1+SX*SX3*SX2+SX2*SX*SX3-SX2*SX2*SX2 • A=(- S Y*S X2 *S X4 -S Y*S X3 *S X3-5 YX*S X*S X4+S YX*S X3 *S X2 1+SYX2*S X*SX3 - SYX2*SX2*SX2 ) /DEN
A1=(SXG^SYX*SX4-SXG *S Y X2 *S X3-SX*S Y*S X4+SX*S YX2 *S X2 1+SX2*S Y*SX3-SX2*SYX*S X2) /DEN
A2=(SXO*S X2*S YX2- S XO *SX3 *S YX-SX*S X*S YX2+SX*S X3*S Y 1+S X2*S X*S YX-SX2*SX2 *S Y) /DEN
SUM = 0 . 0 ASUM = 0 . 0 DO 4 J = 1 , NDDATA(J,3)=A+A1*DATA(J,2)+A2*DATA(J,2)*DATA(J, 2 )DATA(J,4 ) = DATA(J,1 ) - DATA(J,3)DATA(J,5 ) = DATA(J, 4 ) *DATA(J. 4 )ASUM = ASUM + (ABSF(DATA(J,4)) )
4 SUM = SUM + DATA(J, 5 )VAR = SUM/S XOSDEV = VAR**0.5 ADEV = ASUM/SXO RETURN ENDDIMENSION DATA(1 0 0 ,1 0 )COMMON DATA,ND,A,A1,A2,VAR, SDEV, ADEV
6 N=0READ 80 , ND IF (N D )7 0 ,7 0 ,8
8 READ 8 1 . ( DATA(K, 2 ) , K=1 , ND)7 IF( 8 - N ) 7 0 , 6 , 99 CONTINUE
N=N+1 PRINT 89 PRINT 71, NREAD 8 1 , ( DATA(K, 1 ) ,K=1, ND)1=1
12 CALL R10LS PRINT 84 PRINT 85, A , A1 GO TO 14
16 1=213 CALL R20LS
PRINT 86
14
15
71808182
8384858687888970
70
PRINT 87, A, A 1 , A2 PRINT 82 , VAR, SDEV, ADEV DO 15 K=1, NDPRINT 83 , (D A TA(K ,L) ,L=1 ,5 ) I F ( 1 - 1 ) 7 0 , 1 6 , 7FORMAT FORMAT FORMAT FORMAT
1 , F12 . 1 FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT STOP END
10X,7H W P - , 1 3 / )15)8 F 1 0 .0 )5X,12HVAR1ANCE = , F 1 2 . 8 , 5X,22HSTANDARD DEVIATION 5X, 2 1HAVERAGE DEVIATION = , F 1 2 . 8 / / / )5 F 2 0 . 8 / )/ / 5X, 3 THEIRST ORDER REGRESSION EQUATION//)10X», 5HY- = y2F2 5 . 8 , 3H X / / )/ / 5 X , 3 2HSEGOND ORDER REGRESSION EQUATION//)10X, 5HY = ,2 F 2 5 .8 ,4 H X ,F2 5 .8 ,6 H X * * 2 / / ) 215)1H1)
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