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UBC Theses and Dissertations

Bed behaviour in rotary cylinders with applications to rotary kilns Henein, Hani 1981

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BED BEHAVIOUR IN ROTARY CYLINDERS WITH APPLICATIONS TO ROTARY KILNS by HANI HENEIN B. Eng. ( M e t a l l u r g y ) M c G i l l U n i v e r s i t y , 1972 M. Eng. ( M e t a l l u r g y ) M c G i l l U n i v e r s i t y , 1975 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department 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 We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December 1980 0 Hani H e n e i n , 1980 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f fAC^fjrULO^ClcA-^ EzNGl+J e& The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V ancouver, Canada V6T 1W5 DE-6 (2/79) A B S T R A C T Two modes o f t r a n s v e r s e s o l i d s m o t i o n , s l u m p i n g a n d r o l l i n g , in r o t a r y k i l n o p e r a t i o n h a v e b e e n e x p e r i m e n t a l l y c h a r a c t e r i z e d a n d m a t h e m a t i c a l l y m o d e l l e d in t h i s s t u d y . O t h e r modes o f b e d b e h a v i o u r e n c o u n t e r e d in r o t a r y c y l i n d e r s ; s l i p p i n g , c a s c a d i n g , c a t a r a c t i n g a n d c e n t r i f u g i n g h a v e b e e n f o r m u l a t e d m a t h e m a t i c a l l y . T h e m o d e l s h a v e b e e n v e r i f i e d u s i n g e x p e r i m e n t a l o b s e r v a t i o n s r e p o r t e d i n t h e l i t e r a t u r e . An e x p e r i m e n t a l s t u d y o f t h o s e c o n d i t i o n s u n d e r w h i c h t h e b e d c h a n g e d f r o m s l u m p i n g t o r o l l i n g was u n d e r t a k e n a n d t h e c h a r a c t e r i s t i c s o f t h e s e modes o f m o t i o n q u a n t i f i e d u s i n g d i f f e r e n t t y p e s o f s o l i d s i n t h r e e h o r i z o n t a l r o t a r y c y l i n d e r s a n d a s m a l l p i l o t k i l n . A B e d - B e h a v i o u r D i a g r a m w h i c h i s a p l o t o f b e d d e p t h v e r s u s r o t a t i o n a l s p e e d was d e v e l o p e d t o d e l i n e a t e t h e v a r i o u s a r e a s o f d o m i n a n c e o f s l u m p i n g a n d r o l l i n g a n d i t was shown u s i n g t h i s D i a g r a m t h a t b e d b e h a v i o u r o b s e r v a t i o n s made on b a t c h c y l i n d e r s w e r e r e p r e s e n t a t i v e o f s o l i d s m o t i o n i n a c o n t i n u o u s k i l n o p e r a -t i o n , t h e e f f e c t s o f b e d d e p t h , p a r t i c l e s i z e , p a r t i c l e s h a p e a n d c y l i n d e r d i a m e t e r on t h e p o s i t i o n o f t h e s l u m p i n g -r o l l i n g b o u n d a r y w e r e a l s o e x p e r i m e n t a l l y i n v e s t i g a t e d . T h e q u a n t i t a t i v e c h a r a c t e r i z a t i o n o f s l u m p i n g a n d r o l l i n g i n -d i c a t e d t h a t a new i n t e r p r e t a t i o n o f t h e c h a n g e i n b e d m o t i o n f r o m s l u m p i n g t o r o l l i n g was r e q u i r e d . i i A study of s e g r e g a t i o n in the bed r e v e a l e d t h a t w h i l e the p resence of f i n e s a f f e c t e d the s i u m p i n g - r o l 1 i n g boundary , they had l i t t l e e f f e c t on the s lumping f r e q u e n c y , the shear a n g l e , the s t a t i c and dynamic ang les of repose and the a c t i v e l a y e r t h i c k n e s s of the mix tures when compared to those f o r the parent m a t e r i a l s . Th is i n d i c a t e d t h a t f o r the bu lk s o l i d s t e s t e d , s e g r e g a t i o n o c c u r r e d by the p e r c o l a -t i o n and not by the f low mechanism. Sampl ing of the bed r e v e a l e d two s e g r e g a t i o n cores whose f o r m a t i o n and e f f e c t on k i l n o p e r a t i o n s i s d i s c u s s e d . A mathemat ica l model i s a l s o p r e s e n t e d to p r e d i c t the c o m p o s i t i o n and s i z e of the c e n t r a l s e g r e g a t i o n zone . A s e m i - e m p i r i c a1 mathemat ica l model o f the s l u m p i n g -r o l l i n g boundary was deve loped and the e f f e c t s of o p e r a t i n g , m a t e r i a l and c y l i n d e r v a r i a b l e s were i l l u s t r a t e d . S c a l e - u p c r i t e r i a were found to be the f i l l r a t i o , the Froude number, and the minimum shear wedge. For m a t e r i a l s hav ing the same shape but d i f f e r e n t s i z e , t h i s l a t t e r c r i t e r i o n may be r e -p l a c e d by the c y l i n d e r d iameter to p a r t i c l e s i z e r a t i o . S l i p p i n g , c a s c a d i n g and c a t a r a c t i n g were a l s o mode l l ed and t h e i r b o u n d a r i e s i l l u s t r a t e d o n the B e d - B e h a v i o u r D iagram. O b s e r v a t i o n s by o t h e r workers of these modes of bed b e h a v i o u r are compared to the model p r e d i c t i o n s and the a p p r o p r i a t e s c a l e - u p c r i t e r i a are p r e s e n t e d . i i i T A B L E OF CONTENTS Page A b s t r a c t T a b l e o f Contents 11 i v L i s t o f T a b l e s L i s t o f F i g u r e s L i s t o f Symbols Acknowledgments x x i 11 xx i 11 x x i x C h a p t e r INTRODUCTION 1.1 1.2 1.3 1.4 1.5 1.6 I n t r o d u c t i o n D e s c r i p t i o n o f a R o t a r y K i l n R o t a r y K i l n A p p l i c a t i o n s Some I n t r i c a c i e s o f t h e R e a c t o r The Approach o f Research t o Date 1.5.1 Heat Flow A n a l y s e s 1.5.2 Res i d e n c e Time S t u d i e s The O b j e c t i v e s o f t h i s Work 1ATURE REVIEW OF BED BEHAVIOUR I n t r o d u c t i o n The C h a r a c t e r i z a t i o n o f the Modes o f Bed B e h a v i o u r . 2.2.1 S l i p p i n g 2.2.1.1 D e s c r i p t i o n and T r a n s v e r s e M i x i n g . 2.2.1.2 E f f e c t o f V a r i a b l e s 2.2.2 Slumping 2.2.2.1 D e s c r i p t i o n and T r a n s v e r s e M i x i n g . 2.2.2.2 E f f e c t o f V a r i a b l e s i v C h a p t e r Page 2.2.3 R o l l i n g and C a s c a d i n g 2 1 2.2.3.1 D e s c r i p t i o n and T r a n s v e r s e M i x i n g . 2 1 2.2.3.2 E f f e c t o f V a r i a b l e s 2 6 2.3 Bed-Behaviour M o d e l l i n g 3 0 2.3.1 A c t i v e L a y e r T h i c k n e s s 3^ 2.3.2 T r a n s v e r s e M i x i n g 31 2.3.3 The Mechanics o f R i g i d Bodies 3 2 2.4 S c a l e - u p o f S o l i d s Flow i n R o t a t i n g C y l i n d e r s 3 4 2.5 Summary 3 6 3 PARTICLE CHARACTERIZATION AND DESCRIPTION OF APPARATUS.. 3 9 on 3.1 I n t r o d u c t i o n J y oq 3.2 P a r t i c l e C h a r a c t e r i z a t i o n 3.2.1 P a r t i c l e S i z e 3 9 3.2.2 P a r t i c l e Shape 4 4 3.2.3 V o i d F r a c t i o n 4 6 3.2.4 A n g l e o f Repose 3.3 D e s c r i p t i o n s o f the R o t a r y " K i l n and the Ro t a r y , 59 C y l i n d e r s 3.3.1 The P i l o t R o t a r y K i l n 59 3.3.2 Ro t a r y C y l i n d e r s 6 1 3.4 S o l i d s - W a l l F r i c t i o n A n g l e 6 5 v Chapter Page 4 EXPERIMENTAL RESULTS AND ANALYSIS 6 9 4.1 Introduction 4.2 Bed Behaviour: Continuous versus Batch Operation 4.2.1 Reduction of K i ln Variables 4.2.2 Bed Behaviour in a Continuous Operation . 4.2.3 Continuous versus Batch Bed Behaviour . . . 4.3 Instrumentation of Bed Behaviour Observations . . . 4.3.1 Construction and Appl icat ion 4.3.2 Correspondence of Instrumented and Visual Observations 69 70 71 76 79 82 82 84 89 89 93 94 97 100 4.4 Effect of Variables on Bed Behaviour 4.4.1 Wall Ef fects 4.4.2 Pa r t i c l e Shape 4.4.3 Pa r t i c l e Size 4.4.4 Combined Pa r t i c l e Size and Shape Effects . . 4 .4.5 S ta t i c Angle of Repose 4.4.6 Rotary Cyl inder Diameter 1 0 1 4.5 Character izat ion of Slumping and Rol l ing Beds 1 0 2 4.5.1 Slumping 1 0 2 4.5.2 Ro l l ing 1 1 5 4.6 Summary 121 vi Chapter Page 5 SEGREGATION 1 2 5 125 5.1 Introduction 5.2 Previous Work 5.2.1 Types of Segregation 5.2.2 Segregation Kinet ics 5.2.3 Mechanisms of Segregation 5.2.4 Effect of Variables on Segregation 5.3 Material Preparation 5.4 Segregation and Modes of Bed Behaviour 5.5 Segregation and Bed-Behaviour Character izat ion 5.5.1 Slumping Character izat ion 5.5.2 Ro l l ing Character izat ion 5.6 The Segregation Cores 5.6.1 The Segregated Core Model 5.7 Summary 127 127 129 130 132 133 135 139 139 144 144 149 5.6.2 Experimental Procedure, Sampling and Analysis ^ 2 5.6.3 Discussion 1 5 6 5.6.3.1 Radial Segregation 156 5.6.3.1-A Val idat ion of Assumptions and . . . the Second Core 1 ° ' 5.6.3.1-B The Percolat ion Segregation Process 1 5 4 5.6.3.2 Longitudinal Segregation 1 6 8 171 vi i Chapter Page 6 MATHEMATICAL MODELS OF BED BEHAVIOUR 1 7 4 6.1 Introduction 1 7 4 6.2 The Slumping-Roll ing Boundary 1 7 4 6.2.1 Mobi l iza t ion of F r i c t i on and Dilatancy 1 7 5 6.2.1.1 The Slumping Process 1 7 5 6.2.1.2 The Appl icat ion of the Sensor 6.2.2 The Chordal Trajectory of a Slump 6.2.3 The Cr i te r ion of the Slumping-Roll ing Boundary 6.3 The Sl ipp ing Model 6.4 The Cascading and Cataracting Models MODEL PREDICTIONS AND DISCUSSION 7.1 Introduction 7.2 The Slumping-Roll ing Boundary 7.2.1 Predicted versus Measured Results 7.2.2 The Bed Depth 7.2.3 Pa r t i c l e Shape 7.2.4 Pa r t i c l e Size 7.2.5 Combined Effect of Pa r t i c l e Size and Shape. 7.2.6 Cyl inder Diameter 7.2.7 Scale-up 181 6.2.1.3 Times to Maximum and Minimum Bed Inc l ina t ions 1 8 3 190 198 201 204 212 212 212 213 215 217 220 222 223 224 7.2.8 The Temperature of the Sol ids and 233 Gas Evolution v i i i Chapter 7.3 Sl ipp ing 7.4 The Complete Bed-Behaviour Diagram 7.5 Other Aspects 7.5.1 Internal F r i c t i on 7.5.2 The Phase Rule 8 SUMMARY AND CONCLUSIONS 8.1 Summary and Conclusions 8.2 Recommendation for Future Work BIBLIOGRAPHY APPENDICES. A Results of Screen Analyses B Iden t i f i ca t ion of the Run Numbers Corresponding to the Bed-Behaviour Observations C Experimental and Regression Results of the Times to Maximum and Minimum Bed Inc l inat ions D The Co-ordinates of the Wedge Centroids E Program L i s t i ng for Slumping-Roll ing Boundary . . . F Program L i s t i ng for S l ipp ing G Program L i s t i ng for Predict ing Fu l l Bed-Behaviour Diagram ix LIST OF TABLES Chapter 1 5 I Sample d e t a i l s of p rocess k i l n s Chapter 3 II Summary of r e s u l t s f o r the p a r t i c l e c h a r a c t e r i z a t i o n of m a t e r i a l s t e s t e d . III S t a t i c angle of repose measurements . . IV Dimensions of r o t a r y c y l i n d e r s V Angles of s l i p measurements Chapter 4 VI B e d - b e h a v i o u r of sand A in p i l o t k i l n ( k i l n i n c l i n a t i o n = 1 . 2 ° ) VII Shear angles of m a t e r i a l s t e s t e d in c y l i n d e r A , B and C VIII Dynamic angle of repose of m a t e r i a l s t e s t e d Chapter 5 IX Composi t ion of mix tures used f o r the s e g r e g a t i o n study X E f f e c t of f i n e s on the s t a t i c angle of r e p o s e * and the shear a n g l e * f o r the mixtures t e s t e d XI E f f e c t o f f i n e s on the dynamic angle of repose f o r the mix tures t e s t e d C h a p t e r 6 XII X I I I XIV XV XVI Time to maximum bed i n c l i n a t i o n R e g r e s s i o n r e s u l t s f o r the t^ measurements ( t , = * o l + C , ) 6. n R e g r e s s i o n r e s u l t s f o r the t^ measurements ( t . o2 6.n + C 2 + Cg n) R e g r e s s i o n r e s u l t s f o r the s l ump ing f r e q u e n c y measurements ( t T = V 6. n + C 4 + C 5 n) Compar ison o f r e g r e s s i o n c o e f f i c i e n t s o b t a i n e d f rom t , , t~ da ta and f rom the s l ump ing f r e q u e n c y da ta Page 184 191 192 193 194 Append ix A A, 1 Sc reen a n a l y s i s res u l t s f o r g r a v e l A. 2 Sc reen a n a l y s i s r e s u l t s f o r i r o n o x i d e A. 3 Sc reen a n a 1 y s i s r e s u l t s f o r 1 imes tone B A. 4 Sc reen a n a l y s i s r e s u l t s f o r 1 imes tone C A. 5 Sc reen a n a l y s i s r e s u l t s f o r 1 imes tone D A. 6 Sc reen a n a l y s i S ' r e s u l t s f o"r'". l i m e s t o n e E A. 7 Sc reen a n a l y s i s r e s u l t s f o r 1 imes tone F • . . A. 8 Sc reen a n a l y s i s r e s u l t s f o r n i c k e l o x i d e . . A. 9 Sc reen a n a l y s i s r e s u l t s f o r A. 10 Sc reen a n a l y s i s res u l t s f o r sand C 275 276 277 278 279 280 281' 282 283 284 x i Page Append ix B B . l I d e n t i f i c a t i o n o f the run numbers c o r r e s p o n d i n g to the b e d - b e h a v i o u r o b s e r v a t i o n s B . 2 I d e n t i f i c a t i o n o f the run numbers c o r r e s p o n d i n g to the b e d - b e h a v i o u r o b s e r v a t i o n s ( s e g r e g a t i o n t e s t s ) Append ix D D. l V e r t e x c o - o r d i n a t e s D . 2 C e n t r o i d c o - o r d i n a t e s o f component 309 t r i a n g u l a r a reas D.3 A reas o f t r i a n g u l a r components xi i LIST OF FIGURES C h a p t e r 2 F i gure Page 2.1 Schema t i c d iag ram o f s l i p p i n g 14 2 .2 Schema t i c d iag ram of s l ump ing 14 2 .3 R o l l i n g or c a s c a d i n g a c c o r d i n g to Ru tge rs . . . 2 3 2.4 Second f l o w v i s u a l i z a t i o n model o f r o l l i n g • ; or cascading 4^» 51 , 61 ,62 23 2 .5 T h i r d f l o w v i s u a l i z a t i o n model o f r o l l i n g or c a s c a d i n g 3 6 , 5 9 , 6 0 , 6 5 2.6 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 the v e l o c i t y p r o f i l e o f the s o l i d s i n the a c t i v e and p a s s i v e r e g i o n s ^ ! s.52,,67 25 2.7 E f f e c t o f drum speed on the r e l a t i v e a c t i v e l a y e r t h i c k n e s s o f . a m i x t u r e o f p e l l e t s i n a m i x i n g drum 0.8 D x 0 . 5 m 2 ? 28 2 .8 R e l a t i o n between f l o w l i m i t and p a r t i c l e s i z e 5 7 28 Chap te r 3  F i gure 3.1 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 o f m a t e r i a l s t e s t e d 41 3.2 Log p r o b a b i l i t y p l o t o f s c r e e n a n a l y s i s r e s u l t s o f m a t e r i a l s t e s t e d 43 3 .3 A n g u l a r g r a v e l w i t h 3mm ave rage p a r t i c l e s i z e (1 d i v i s i o n = 1 mm) 47 3.4 S p h e r i c a l i r o n o x i d e w i t h 11 .6 mm ave rage p a r t i c l e s i z e (1 d i v i s i o n = 1 mm) 47 3.5 I r r e g u l a r l i m e s t o n e B w i t h 4 . 3 mm average p a r t i c l e s i z e (1 d i v i s i o n = 1 mm) 48 x i i i F i g u r e : 3 .6 I r r e g u l a r l i m e s t o n e C w i t h 1.5 mm ave rage p a r t i c l e s i z e (1 d i v i s i o n = 1 mm) 3.7 I r r e g u l a r l i m e s t o n e D w i t h 0 .58 mm ave rage p a r t i c l e s i z e (1 d i v i s i o n = 1 mm) 3.8 E q u i - d i m e n s i o n a l l i m e s t o n e E w i t h 0 .54 mm r -ave rage p a r t i c l e s i z e (1 d i v i s i o n = 1 mm) . . . 3.9 A n g u l a r l i m e s t o n e F w i t h 8.1 mm ave rage p a r t i c l e s i z e (1 d i v i s i o n = 1 mm) 3.10 S p h e r i c a l n i c k e l o x i d e w i t h 4 .9 mm ave rage p a r t i c l e s i z e (1 d i v i s i o n = 1 mm) 3.11 N o d u l a r sand B w i t h 0 .50 mm ave rage p a r t i c l e s i z e (1 d i v i s i o n = 1 mm) 3.12 N o d u l a r sand C w i t h 0 .23 mm ave rage p a r t i c l e s i z e (1 d i v i s i o n = 1 0 ~ 2 mm) 3 .13 V o i d f r a c t i o n c o m p o s i t i o n d iag ram f o r l i m e s t o n e B and E and sands B and C 3.14 Three methods f o r measu r i ng the s t a t i c a n g l e o f r epose 3.15 The UBC p i l o t k i l n 3 .16 R o t a r y c y l i n d e r A ( 0 .4 m ID x 0 .46 m L) s e t on r o l l s 3.17 Schema t i c d iag ram o f the a p p a r a t u s used f o r measu r i ng the s o l i d s - w a l l f r i c t i o n c o e f -f i c i e n t . . . . Chap te r 4  F i gure 4.1 Bed depth p r o f i l e f o r l i m e s t o n e A i n the ' UBC p i l o t k i l n , i l l u s t r a t i n g the e f f e c t o f k i 1 n i n c l i n a t i o n 4 . 2 Bed depth p r o f i l e f o r l i m e s t o n e B i n the UBC p i l o t k i l n , i l l u s t r a t i n g the e f f e c t o f k i l n r o t a t i o n a l speed x iv F i g u r e 4 . 3 Bed depth p r o f i l e f o r sand A i n the UBC p i l o t k i l n , i l l u s t r a t i n g the e f f e c t o f k i l n dams and s o l i d s feed r a t e 4 .4 B e d - B e h a v i o u r Diagram of sand A v i s u a l l y d e t e r m i n e d i n the UBC p i l o t k i l n 4 . 5 B e d - B e h a v i o u r Diagram v i s u a l l y de te rm ined f o r sand A i n c y l i n d e r A ( 0 .4 m ID x 0 . 4 6 m L ) and compared to the b o u n d a r i e s o b t a i n e d i n the p i l o t k i l n 4 .6 Pho tog raph i l l u s t r a t i n g the p o s i t i o n o f the s e n s o r i n the c y l i n d e r 4 . 7 Sample o u t p u t o f s e n s o r f o r s l u m p i n g and r o l 1 i ng bed 4 . 8 B e d - B e h a v i o u r Diagram of l i m e s t o n e B in c y l i n d e r A ( 0 . 4 m ID x 0 .46 m L ) , v i s u a l l y de te rm i ned 4 .9 B e d - B e h a v i o u r Diagram of n i c k e l o x i d e i n c y l i n d e r A ( 0 . 4 m ID x 0 .46 m L) de te rm ined v i s u a l l y 4 .10 B e d - B e h a v i o u r Diagram of l i m e s t o n e B i n c y l i n d e r A ( 0 . 4 m ID x 0 .46 m L) d e t e r m i n e d by i n s t r u m e n t a t i o n and compared to the v i s u a l d e t e r m i n a t i o n 4.11 B e d - B e h a v i o u r Diagram of n i c k e l o x i d e i n c y l i n d e r A ( 0 .4 m ID x 0 .46 m L) d e t e r m i n e d by i n s t r u m e n t a t i o n and compared w i t h the v i s u a l d e t e r m i n a t i o n 4 .12 B e d - B e h a v i o u r Diagram o f l i m e s t o n e A compar ing the r e s u l t s f rom c y l i n d e r A ( 0 . 4 m ID x 0 .46 m L) and c y l i n d e r B ( 0 .4 m ID x 0 .86 m L) 4 . 1 3 B e d - B e h a v i o u r Diagram o f l i m e s t o n e D i n c y l i n d e r A (0 .4 m ID x 0 .46 m L) 4 .14 B e d - B e h a v i o u r Diagram o f sand B i n c y l i n d e r A ( 0 .4 m ID x 0 .46 m L) xv F i g u r e 4 . 1 5 B e d - B e h a v i o u r Diagram of l i m e s t o n e C i n c y l i n d e r A (0 .4 m ID x 0 .46 m L) 4 .16 B e d - B e h a v i o u r Diagram o f g r a v e l i n c y l i n d e r A ( 0 . 4 m ID x 0 .46 m L) 4 .17 B e d - B e h a v i o u r Diagram of l i m e s t o n e B i n c y l i n d e r C ( 1 . 0 6 m ID x 0 .4 m L) 4 . 1 8 Upper a n g l e o f r epose as a f u n c t i o n o f r o t a t i o n a l speed f o r g r a v e l i n c y l i n d e r A ( 0 .4 m ID x 0 .46 m L) 4 .19 Upper a n g l e of r epose as a f u n c t i o n o f r o t a t i o n a l speed measured a t s e v e r a l bed depths in c y l i n d e r A ( 0 .4 m ID x 0 .46 m L) u s i n g 1 imes tone C 4 .20 S lump ing f r e q u e n c y as a f u n c t i o n o f r o t a t i o n a l speed f o r l i m e s t o n e B t e s t e d i n c y l i n d e r s A ( 0 .4 m ID x 0 .46 m L) and B ( 0 . 4 m ID x 0 .86 m L) 4.21 S lump ing f r e q u e n c y as a f u n c t i o n o f r o t a t i o n a l speed f o r l i m e s t o n e B showing t he ' i n d e p e n d e n c e o f bed depth and the dependence on c y l i nder d i a'meter- • • • -4 . 2 3 S lump ing f r e q u e n c y as a f u n c t i o n o f r o t a t i o n a l speed f o r s e v e r a l m a t e r i a l s i n c y l i n d e r A ( 0 .4 m ID x 0 .46 m L) showing the e f f e c t o f p a r t i c l e shape and s i z e 4 .24 A c t i v e l a y e r t h i c k n e s s as a f u n c t i o n o f bed d e p t h , measured f o r s e v e r a l m a t e r i a l s 4 .25 D i m e n s i o n l e s s p l o t o f a c t i v e l a y e r t h i c k n e s s as a. function of bed d e p t h . 4 . 26 D i m e n s i o n l e s s p l o t of the r e l a t i v e s i z e of t he a c t i v e l a y e r to the bed depth Page 96 99 99 103 103 110 111 4 .22 S lump ing f r e q u e n c y as a f u n c t i o n o f r o t a t i o n a l s p e e d , showing the e f f e c t o f :; ' c y l i n d e r d i a m e t e r 114 119 120 120 xvi C h a p t e r 5 F i g u r e Page 128 5 . 1 Types o f s e g r e g a t i o n e n c o u n t e r e d i n h o r i z o n t a l r o t a r y c y 1 i n d e r s 5 9 ,67 . . 5 . 2 B e d - B e h a v i o u r Diagram f o r sand mix A (3 .6%-70 mesh) i n c y l i n d e r A (0 .4 m ID x 0 .46 m L ) e x p e r i m e n t a l l y de te rm ined and compared to the B e d - B e h a v i o u r Diagram of sand B (1 .1 %-70 mesh) 137 5 . 3 The t r a n s i t i o n - r o l l i n g boundary f o r sand mix B (16.7%-20 mesh) i n c y l i n d e r A ( 0 .4 m ID x 0 .46 m L ) e x p e r i m e n t a l l y de te rm ined and compared to t h o s e f o r sand B (1.135-70 mesh) and sand mix A (3 .6%-E0 mesh) . 1 3 7 5 . 4 B e d - B e h a v i o u r Diagram of l i m e s t o n e mix A (8 .7%-20 mesh) i n c y l i n d e r A ( 0 . 4 m ID x 0 .46 m L ) e x p e r i m e n t a l l y d e t e r m i n e d and compared to the B e d - B e h a v i O u r Diagram of l i m e s t o n e B ( 0 . 735- 20 mesh) 1 3 8 5 . 5 S lump ing f r e q u e n c y v e r s u s r o t a t i o n a l speed f o r sand B ( V . l % - 7 0 mesh) and sand mix A (3.635-70 mesh) and B (1 6. 7%-70 mesh) i n c y l i n d e r A (0 .4 m ID x 0 .46 m L ) 5 . 6 S lump ing f r e q u e n c y v e r s u s r o t a t i o n a l speed f o r l i m e s t o n e B (0 .7%-20 mesh) and l i m e s t o n e mix A ( 8 . 735- 20 mesh) i n c y l i n d e r A ( 0 .4 m ID x 0 .46 m L ) 5 . 7 A c t i v e l a y e r t h i c k n e s s v e r s u s bed depth f o r sand B (1.1 %-70 mesh) and sand mix A ( 3 . 635-70 mesh) and B •'••(16.735-70 mesh) i n c y l i n d e r A ( 0 .4 m ID x 0 .46 m L ) 5 . 8 A c t i v e l a y e r t h i c k n e s s v e r s u s bed depth f o r l i m e s t o n e B (0 .7%-20 mesh) and l i m e s t o n e mix A (8.735-20 mesh) in c y l i n d e r A ( 0 .4 m ID x 0 .46 m L ) 142 143 146 146 5.9 D i m e n s i o n l e s s P l o t of a c t i v e l a y e r t h i c k n e s s v e r s u s bed d e p t h . Compar ison o f s e g r e g a t i o n 1 4 7 r e s u l t s w i t h t hose f rom s e c t i o n 4 . 5 . 2 xv i i F i gure 5.10 R e l a t i v e t h i c k n e s s o f a c t i v e l a y e r w i t h r e s p e c t to the d i m e n s i o n l e s s bed d e p t h . Compar ison o f s e g r e g a t i o n r e s u l t s w i t h t hose f rom s e c t i o n 4 . 5 . 2 5.11 Schema t i c r e p r e s e n t a t i o n o f the c e n t r a l s e g r e g a t i o n c o r e (a) s e g r e g a t i o n c o r e -r e c e s s e d f rom the s u r f a c e o f the bed (b) s u r f a c e o f c o r e a t the bed s u r f a c e . . . . 5.12 S l i c e r made o f 1.6 mm t h i c k s t e e l shee t c o v e r e d w i t h p o s t e r board 5 .13 S l i c e r s i n t r o d u c e d i n t o the bed o f s o l i d s , i n p r e p a r a t i o n f o r s o l i d s removal 5.14 E x p e r i m e n t a l r e s u l t s and c o r e w i d t h model p r e d i c t i o n s f o r s l u m p i n g i n c y l i n d e r A ( 0 . 4 m ID x 0 .46 m L) 5 .15 E x p e r i m e n t a l r e s u l t s and co re w i d t h model p r e d i c t i o n s f o r r o l l i n g i n c y l i n d e r A (0 .4 m ID x 0 .46 m L) 5.16 E x p e r i m e n t a l r e s u l t s and c o r e w i d t h model p r e d i c t i o n s f o r s l u m p i n g in c y l i n d e r B ( 0 . 4 m ID x 0 .86 m L) 5.17 E x p e r i m e n t a l r e s u l t s and c o r e w i d t h model p r e d i c t i o n s f o r r o l l i n g i n c y l i n d e r B ( 0 .4 m ID x 0 .86 m L) 5.18 Pho tog raph o f second s e g r e g a t i o n c o r e at the top o f the bed a d j a c e n t to the c y l i n d e r w a l H 3 2 5.19 S c h e m a t i c o f the p e r c o l a t i o n s e g r e g a t i o n p r o c e s s f o r both c o r e s C h a p t e r . 6  F i g u r e 6.1 T r a c e d bed i n c l i n a t i o n s f o r the f i r s t p a r t o f a slump of l i m e s t o n e B i n c y l i n d e r B (0 .4 m ID x 0 .86 m L) r o t a t e d at 0 .85 r / m i n . . xvi i i F i g u r e Page 6 .2 T r a c e d bed i n c l i n a t i o n s f o r the l a t t e r p a r t o f a slump of l i m e s t o n e B i n c y l i n d e r B ( 0 .4 m ID x 0 .86 m L) r o t a t e d at 0 .85 r / m i n . . 177 6 .3 D i g i t i z e d bed i n c l i n a t i o n s f o r the same slump shown i n F i g u r e s 6.1 and 6 . 2 . The t i m e s of bed mot ion and no mot ion as r e c o r d e d f rom the bed s u r f a c e a re a l s o shown '79 6.4 The t-, and t ? measured r e s u l t s as w e l l as t he r e g r e s s i o n c u r v e s f o r g r a v e l i n c y l i n d e r A ( 0 . 4 m ID x 0 .46 m L) 1 8 6 6 .5 F o r c e b a l a n c e on a s h e a r wedge ^ 6 6 .6 F o r c e b a l a n c e on the bed o f b u l k s o l i d s f o r the s l i p p i n g model 2 ^ 2 6.7 a) F o r c e b a l a n c e on t he bed o f b u l k s o l i d s f o r the c a s c a d i n g model w i t h the apex i n the f i r s t quad ran t 2 0 7 b) Bed c o n f i g u r a t i o n w i t h the apex in the f o u r t h quadran t c) Bed c o n f i g u r a t i o n d e f i n i n g the r o l l i n g -c a s c a d i n g boundary Chap te r 7  F i gure ; 7.1 P r e d i c t e d and e x p e r i m e n t a l b o u n d a r i e s f o r l i m e s t o n e B t e s t e d i n c y l i n d e r s A and B (0 .4 m ID x 0 .46 m L and 0 .86 m L 2 1 4 r e s p e c t i v e l y ) 7 .2 P r e d i c t e d and e x p e r i m e n t a l b o u n d a r i e s f o r n i c k e l o x i d e t e s t e d i n c y l i n d e r A ? 1 4 ( 0 . 4 m ID x 0.46 m L) 7 .3 P r e d i c t e d and e x p e r i m e n t a l b o u n d a r i e s f o r l i m e s t o n e D t e s t e d i n c y l i n d e r A 2ig (0 .4 m ID x 0 .46 m L) x ix F i g u r e 7.4 P r e d i c t e d and e x p e r i m e n t a l b o u n d a r i e s f o r sand B t e s t e d i n c y l i n d e r A ( 0 . 4 m ID x 0 .46 m L) 7 .5 P r e d i c t e d and e x p e r i m e n t a l b o u n d a r i e s f o r l i m e s t o n e C t e s t e d i n c y l i n d e r A ( 0 .4 m ID x 0 .46 m L) 7.6 P r e d i c t e d and e x p e r i m e n t a l b o u n d a r i e s f o r l i m e s t o n e B t e s t e d in c y l i n d e r C (1.06rm ID x 0 .4 m L) 7.7 Minimum shea r wedge v e r s u s ave rage p a r t i c l e s i z e f o r a l l m a t e r i a l s t e s t e d 7 .8 D i l a t a n c y c o e f f i c i e n t v e r s u s ave rage p a r t i c l e s i z e f o r a l l m a t e r i a l s t e s t e d 7.9 D i m e n s i o n l e s s P l o t o f the e x p e r i m e n t a l b o u n d a r i e s o f l i m e s t o n e s B and C t e s t e d i n c y l i n d e r s . A and B (0 .4 m ID x 0 .46 m L and 0 .86 m L r e s p e c t i v e l y ) a) u s i n g Froude number and % F i l l b) u s i n g Froude number, % F i l l and s i z e r a t i o . . 7 .10 D i m e n s i o n l e s s P l o t o f e x p e r i m e n t a l b o u n d a r i e s o f l i m e s t o n e s B, C and D t e s t e d i n c y l i n d e r s A and B, s c a l e - u p , u s i n g the Froude number, the % F i l l and the p a r t i c l e s i z e to c y l i n d e r d i a m e t e r r a t i o , and compared to the b o u n d a r i e s o f l i m e s t o n e B t e s t e d in c y l i n d e r C 7.11 S lump ing f r e q u e n c i e s o f l i m e s t o n e C i n c y l i n d e r A ( 0 . 4 m ID x 0 .46 m L) compared w i t h t h o s e measured i n the p i l o t k i l n d u r i n g hot and c o l d runs 7.12 P r e d i c t i o n s o f the s l i p p i n g model f o r k i l n d i a m e t e r s o f 0 .4 m, 1.0 m and 2 .7 m 7 .13 Comple te B e d - B e h a v i o u r Diagram of g r a v e l i n c y l i n d e r A (0 .4 m ID x 0 .46 m L) xx F i gure 7.14 Complete B e d - B e h a v i o u r Diagram of n i c k e l o x i d e i n c y l i n d e r A ( 0 .4 m ID x 0 .46 m L) . . . 7 .15 Comple te B e d - B e h a v i o u r Diagram of l i m e s t o n e C i n c y l i n d e r A (0 .4 m ID x 0 .46 m L) 7 .16 Comple te B e d - B e h a v i o u r Diagram o f l i m e s t o n e B i n c y l i n d e r C (1 .06 m ID x 0 .4 m L) Append i x C  F i g u r e C l The t , and t 2 measured r e s u l t s as w e l l as t h e r e g r e s s i o n c u r v e s f o r g r a v e l i n c y l i n d e r C (1 .06 m ID x 0 .4 m L) C.2 The t , and tp measured r e s u l t s as w e l l as < the r e g r e s s i o n c u r v e s f o r i r o n o x i d e i n c y l i n d e r A ( 0 .4 m ID x 0 .46 m L) C .3 The t-, and t 2 measured r e s u l t s as w e l l as the r e g r e s s i o n c u r v e s . f o r l i m e s t o n e B i n c y l i n d e r B ( 0 .4 m ID x 0 .86 m L) C.4 The t-, and t 2 measured r e s u l t s as w e l l as the r e g r e s s i o n c u r v e s f o r 1 imes tone B i n c y l i n d e r C ( 1 . 0 6 m ID x 0.4 m L) C .5 The t-. and t~ measured r e s u l t s as w e l l as the r e g r e s s i o n c u r v e s f o r l i m e s t o n e C i n c y l i n d e r A (0 .4 m ID x 0 .46 m L) C.6 The t-, and t 2 measured r e s u l t s as w e l l as the r e g r e s s i o n c u r v e s f o r l i m e s t o n e C i n c y l i n d e r C (1 .06 m ID x 0.4 m L) C.7 The t , and t 2 measured r e s u l t s as w e l l as the r e g r e s s i o n c u r v e s . f o r l i m e s t o n e D i n c y l i n d e r A ( 0 .4 m ID x 0 .46 m L) C.8 The t-, and t 2 measured r e s u l t s as w e l l as the r e g r e s s i o n c u r v e s f o r l i m e s t o n e F in c y l i n d e r C (1 .06 m ID x 0.4 m L) xxi F i gure Page C.9 The t , and t~ measured r e s u l t s as w e l l as the r e g r e s s i o n c u r v e s f o r n i c k e l o x i d e i n c y l i n d e r A ( 0 . 4 m ID x 0 .46 m L) C I O The t , and t ? measured r e s u l t s as w e l l as the r e g r e s s i o n c u r v e s f o r sand B i n c y l i n d e r A ( 0 . 4 m ID x 0 .46 m l ) Append ix D  F i g u r e D. l G e o m e t r i c a l c o n s t r u c t i o n o f the bed f o r the apex i n the f i r s t quad ran t f o r both the <j>y and <j) '^bed i n c l i n a t i o n s D.2 C o n s t r u c t i o n i l l u s t r a t i n g the d e r i v a t i o n o f the c e n t r o i d o f s e c t o r OAB D.3 G e o m e t r i c a l c o n s t r u c t i o n o f the bed f o r the apex i n the f i r s t and f o u r t h q u a d r a n t s a t bo th the <f>u and <t>^  bed i n c l i n a t i o n s ' ? r e s p e c t i veVy , D.5 G e o m e t r i c a l c o n s t r u c t i o n of the bed i l l u s t r a -t i n g the c a l c u l a t i o n o f the c e n t r o i d o f PSC 295 296 299 303 305 D.4 G e o m e t r i c a l c o n s t r u c t i o n of the bed f o r the apex i n the f o u r t h quad ran t a t both the <j>j 30 6 and cf>^ bed i n c l i n a t i o n s 307 xxi i LIST OF SYMBOLS a i n i t i a l accelerat ion of shear wedge m/sec d pa r t i c l e s ize m P 2 g grav i ta t iona l accelerat ion constant m/sec n rotat ional speed of the cy l inder r/min n c c r i t i c a l rotat ional speed of the cy l inder r/min s average distance t rave l led by bulk so l ids in the shear wedge in i t s chordal t ra jectory m s i n i t i a l distance t rave l led by the shear wedge in 0 the d i rec t ion of the average chordal t ra jectory m t time seconds t time required to traverse the l im i t i ng shear 0 wedge seconds t-| time from minimum to maximum bed inc l i na t i on seconds t^ time from maximum to minimum bed inc l i na t ion seconds t<* slumping time sees t T to ta l time per slump seconds slump v l i near ve loc i ty m/sec v Q i n i t i a l granule ve loc i ty in i t s parabol ic t ra jec tory m/sec v j i n i t i a l ve loc i ty of the shear wedge in the 0 d i rec t ion of the average chordal t ra jectory m/sec w^ width of central segregated core m x weight f rac t ion of the f ine component in the s l i c e dimensionless x^g£ abscissa of the centroid of the upper wedge m x n££ abscissa of the centroid of t r iang le DEC m x0AB abscissa of the centroid of sector OAB m xxi i i X g n B abscissa of the centroid of t r iang le ODB m x0DE abscissa of the centroid of t r iang le ODE m x0EA abscissa of the centroid of t r iang le OEA m X p t - £ abscissa of the centroid of the lower wedge m. y weight f rac t ion of the coarse component in the central segregated core dimensionless ^ABC ordinate of the centroid of the upper wedge m yDEC ordinate of centroid of t r iang le DEC m •^ OAB ordinate of centroid of sector OAB m y0DB ordinate of centroid of t r iang le ODB m y Q D E ordinate of centroid of t r iang le ODE m y0EA ordinate of centroid of t r iang le OEA m y~p<.Q ordinate of the centroid of the upper wedge m 2 A A B C area of shear wedge ABC m 2 AD££ area of t r iang le DEC m 2 A Q A B area of sector OAB m A 2 ODB area of t r iang le ODB m 2 A0DE a r e a o f t r i ' a n 9 ' l e 0 D E m 2 ^OEA area of t r iang le OEA m C number of components •' in the Phase Rule dimensionless C, regression coe f f i c ien t from curve f i t on t^ data seconds C 2 regression coe f f i c ien t from curve f i t on t 2 data seconds 2 C- regression coe f f i c ien t from curve f i t on 60 s t 2 data rev C« regression coe f f i c ien t from curve f i t on slumping frequency data seconds xxiv D F FR Fr H H K J K L M P R R A B C S V,. regression coe f f i c ien t from curve f i t on slumping frequency data ins ide diameter of the cy l inder number of degrees of freedom in the Phase Rule centr i fugal force f r i c t i o n a l force gravi ty force normal force resul tant force react ion force o)2R Froude number (•*—) g depth of the bulk so l ids in the rotary cy l inder or rotary k i l n depth of the central core of segregation degree of f i l l constant in equation (7.6) by Oyama length of rotary cyl inder or rotary k i l n to ta l mass of bulk so l ids in the rotary cy l inder number of phases in the Phase Rule ins ide radius of the cy l inder distance between the centroid of the shear wedge from the centre of rotat ion of the cy l inder radius of rotat ion of the centre of gravi ty of the bulk sol ids outer radius of rotat ion of the central core of segregation slumping frequency volume occupied by the bulk so l ids that is associated with the central core of segregation 60 s rev m dimensionless N N N N N dimensionless m m per cent m kg dimensionless m m m siumps/min 3 m XXV to ta l weight of bulk so l ids associated with the central core of segregation kg to ta l weight of bulk so l ids in the bed kg abscissa of parabol ic t ra jectory of granules m abscissa of the mid-point of the bed surface m ordinate of parabol ic t ra jec tory of granules m ordinate of the mid-point of the bed surface m angular measure of the centroid of lower wedge from absc issa , measured in clockwise d i rec t ion degrees di f ference between the maximum and minimum bed inc l i na t ions for a s l ipp ing bed radians angle re la t ing the f r i c t i o n force and the normal force in a force balance on a s l ipp ing bed degrees shear wedge degrees l im i t i ng shear wedge degrees shear wedge degrees regression constant from curve f i t on t-j data degrees regression constant from curve f i t on tr, data degrees angle traversed by rotary cy l inder in time t degrees angular measure of degree of f i l l degrees void f rac t ion of the bulk so l ids dimensionless maximum bed contract ion dimensionless slope of average chordal t ra jectory of the shear wedge degrees angle between the apex of the bed and the abscissa degrees angular measure of the degree of f i l l degrees dynamic coe f f i c ien t of f r i c t i o n of bulk so l ids dimensionless xx vi shearing coe f f i c ien t of f r i c t i o n of bulk so l ids dimensionless W/S ' D aR/C coe f f i c ien t of f r i c t i o n between the cy l inder wall and the bulk so l ids bulk density of bulk so l ids pa r t i c l e density bulk density of central core of segregation resul tant angle of bed inc l i na t i on for the force balance of a s l ipp ing bed dynamic angle of repose of bulk so l ids shear angle s t a t i c angle of repose of the bulk so l ids angle of s l ipp ing f r i c t i o n upper angle of repose resul tant angle of bed i nc l i na t i on for the force balance of a cascading bed angular measure of the sector OAB from the abscissa angular ve loc i ty of the rotary cy l inder rotat ional speed at the ro l l ing-cascading boundary dimensionless kg/m 3 kg/m 3 kg/m 3 degrees degrees degrees degrees degrees degrees degrees degrees radians/sec radians/sec Subscripts P M indicates a prototype indicates a model xxvi i "Man, God created to be a witness and grateful in terpreter of His works." St. Antony the Great xxvi i i ACKNOWLEDGMENTS The a u t h o r w i shes to e x p r e s s h i s s i n c e r e g r a t i t u d e to Dr . J . K. Brimacombe and Dr . A. P. Watk inson f o r t h e i r gu i dance t h r o u g h o u t the c o u r s e o f t h i s r e s e a r c h . Thanks are e x p r e s s e d to D r s . D. A n d e r s s o n , P. Byrne and N. E p s t e i n o f the U n i v e r s i t y o f B r i t i s h Co lumb ia f o r engag ing i n many h e l p f u l and f r u i t f u l d i s c u s s i o n s . The a u t h o r would a l s o l i k e to acknowledge Mr. E. Sunnerg ren o f Be th lehem S t e e l C o r p . , Mr . E. W h i t l o c k o f Domtar C h e m i c a l s L i m i t e d and Mr. W. Zimmer o f Kennedy Van Saun C o r p . f o r t h e i r he lp and c o -o p e r a t i o n d u r i n g the v a r i o u s s t a g e s o f t h i s p r o j e c t . The au tho r i s a l s o i n d e b t e d to Dr . L. M i l l e r and h i s s t u d e n t s f o r t h e i r i n v a l u a b l e a i d i n t r a n s l a t i n g much o f the r e q u i r e d German t e c h n i c a l l i t e r a t u r e . The d i s c u s s i o n s and a s s i s t a n c e o f f e l l o w g radua te s t u d e n t s , t e c h n i c i a n s and f a c u l t y members i s a l s o g r a t e f u l l y a c k n o w l e d g e d . To h i s f r i e n d s and p a r t i c u l a r l y to h i s p a r e n t s and L o u i s e , the a u t h o r w i s h e s to ex tend to them warmest a p p r e c i a t i o n f o r t h e i r unend ing p a t i e n c e and unswerv ing s u p p o r t t h r o u g h o u t the d u r a t i o n o f t h i s e n d e a v o u r . The a u t h o r i s a l s o g r a t e f u l to the S t e e l Company o f Canada and to NSERC f o r p r o v i d i n g f i n a n c i a l s u p p o r t i n the form of a S t e l c o F e l l o w s h i p and a R e s e a r c h A s s i s t a n t s h i p r e s p e c t i v e l y . X X 1 - X Chap te r 1 1 INTRODUCTION 1.1 I n t r o d u c t i o n T h i s work i s conce rned w i t h an a n a l y s i s . o f the d i f -f e r e n t modes o f t r a n s v e r s e s o l i d s mot ion i n r o t a r y k i l n s . The p r i n c i p l e s d e r i v e d and models p r e s e n t e d w i l l a l s o f i n d a p p l i c a t i o n i n o t h e r i n d u s t r i a l p r o c e s s e s where r o t a r y c y l i n d e r s a re u s e d . O ther a p p l i c a t i o n s w i l l i n c l u d e the e f f e c t o f the p h y s i c a l p r o p e r t i e s o f bu l k s o l i d s on t h e i r f l o w c h a r a c t e r i s t i c s . However , s i n c e the r o t a r y k i l n was the main impetus o f t h i s p r o j e c t , r e f e r e n c e w i l l be made p r i m a r i l y to i t s a p p l i c a t i o n . 1 . 2 D e s c r i p t i o n o f a R o t a r y K i l n A r o t a r y k i l n i s a h o r i z o n t a l l y i n c l i n e d s t e e l c y l i n d e r l i n e d w i t h r e f r a c t o r y . The burden i s c o n t i n u o u s l y cha rged at the upper end and moves s l o w l y a l o n g the f u r n a c e by v i r t u e o f the i n c l i n a t i o n and r o t a t i o n o f the k i l n . In most c a s e s , the depth o f the moving bed i s a s m a l l f r a c t i o n o f the k i l n d i a m e t e r . Dams a re o f t e n p l a c e d at the s o l i d s d i s c h a r g e end and a l s o at s e l e c t e d p o i n t s a l o n g the l e n g t h o f the k i l n to 2 i n c r e a s e i t s h o l d - u p . * The l e n g t h to d i a m e t e r r a t i o o f 1 2 t y p i c a l r o t a r y k i l n s v a r i e s f rom 15 to 4 0 . ' As the o b j e c t o f t h e s e r e a c t o r s i s to e f f e c t a c h e m i -c a l o r p h y s i c a l change to the burden by r a i s i n g i t s t e m p e r a -t u r e , a bu rne r f o r f u e l combus t i on i s g e n e r a l l y l o c a t e d a t the l ower end o f the k i l n a l l o w i n g the gases to f l o w c o u n t e r -c u r r e n t to the c h a r g e . To reduce heat l o s s e s th rough the k i l n w a l l s the r e f r a c t o r y l i n i n g may be f rom 0 .15 m to 0 .30 m t h i c k . The c y l i n d e r r e s t s on a s e t o f t r u n i o n s w i t h r o t a t i o n 3 4 b e i n g e f f e c t e d by a v a r i a b l e speed D .C . mo to r . ' A u x i l i a r y equipment commonly found i n r o t a r y k i l n p l a n t s i n c l u d e gas h a n d l i n g and dus t c o l l e c t i n g i n s t a l l a t i o n s as w e l l as heat r e c u p e r a t o r s f o r the hot e x i t gases and d i s c h a r g e s o l i d s . Some common r o t a r y k i l n o p e r a t i n g c o n d i t i o n s a re l i s t e d i n T a b l e I. 1.3 R o t a r y K i l n A p p l i c a t i o n s 3 Due to the m e c h a n i c a l s i m p l i c i t y o f the r o t a r y k i l n i t i s not s u r p r i s i n g t h e r e f o r e , t h a t i t i s w i d e l y used f o r c o n t i n u o u s p r o c e s s i n g i n the c h e m i c a l and m e t a l l u r g i c a l *The r a t i o between the volume o c c u p i e d by the s o l i d s and the t o t a l volume of the k i l n ( a l s o c a l l e d the per cen t f i l l o r the f i l l r a t i o ) . TABLE I Sample D e t a i l s of P r o c e s s K i l n s Material K i ln Par t i c le Size Inner Shell Dia. (•!"-.) Rotational Speed ( r / rn i n ) % F i l l Slope (%) Pebble lime -63 mm 3.5 1 .5 10-12 4.2 Wet Sludge lime - 3.8 1.0 4 3 Magnesite -13 mm 3.0 1 .0 6-10 4.2 SL/RN 13 mm 6.0 0.4 13 3 Petroleum Coke 40% + 10 mesh 4.0 1.0 5-8 . : 5.2 Wet Process Cement - 4.5-5.6 1.0 7-10 4 OJ 4 i n d u s t r i e s . In the p r o d u c t i o n o f cement , r o t a r y k i l n s a re used f o r c a l c i n a t i o n . C a l c i n e d d o l o m i t e , l i m e mud ( i n the K r a f t P u l p i n g p r o c e s s ) and l i m e s t o n e ( f o r d e s u l p h u r i z i n g i n the s t e e l i n d u s t r y ) a r e o t h e r examples o f r o t a r y k i l n a p p l i c a t i o n s i n the c h e m i c a l i n d u s t r y . S e v e r a l p r o c e s s e s have been p roposed f o r d i r e c t r e d u c t i o n o f i r o n i n a r o t a r y 6-11 k i l n , " w h i l e i n the n o n - f e r r o u s i n d u s t r y the k i l n i s used i n t r e a t i n g z i n c o x i d e and p y r r h o t i t e o r e s . The c a l c i n a t i o n o f h y d r a t e d a l u m i n a and o f p e t r o l e u m coke i n the aluminum i n d u s t r y s h o u l d a l s o be c i t e d , as w e l l as the a p p l i c a t i o n o f the r o t a r y k i l n as a p r e h e a t e r and a d r y e r i n the p r o d u c t i o n o f h igh grade f e r r o - c h r o m e and t i t a n i u m r e s p e c t i v e l y . In s h o r t , the r o t a r y k i l n i s among the most common g a s - s o l i d r e a c t o r s used i n the p r i m a r y i n d u s t r i e s . 1.4 Some I n t r i c a c i e s o f the R e a c t o r A l t h o u g h the r o t a r y k i l n i s a v e r y s i m p l e r e a c t o r to o p e r a t e , the p r o c e s s e s o c c u r r i n g i n i t a re q u i t e comp lex . For e x a m p l e , d i f f e r e n t t e m p e r a t u r e s , h o l d - u p and w a l l roughness may p r e v a i l a t d i f f e r e n t p o s i t i o n s of the k i l n , w h i l e the cha rge may undergo p h y s i o c h e m i c a l changes a f f e c t i n g i t s m o t i o n . O f ten the k i l n f e e d c o n s i s t s o f a m i x t u r e o f m a t e r i a l s , each h a v i n g i t s own r e s p e c t i v e s i z e and shape d i s t r i b u t i o n . T h e s e , as w e l l as o t h e r c o m p l i c a t i n g f a c t o r s , have r e n d e r e d many o p e r a t i n g improvements d i r e c t l y a p p l i c a b l e 5 o n l y to the k i l n t e s t e d , even when c o n s i d e r i n g the same p r o c e s s . In a d d i t i o n , r o t a r y k i l n s o f the same d e s i g n f o r the same a p p l i c a t i o n w i l l r e q u i r e d i f f e r e n t o p e r a t i n g s t r a t e g i e s . T h i s i s due to d i f f e r e n c e s i n the p h y s i c a l p r o -p e r t i e s of the m a t e r i a l f e e d ( e . g . d e n s i t y , g r a i n s i z e and shape), ' p a r t i c u l a r l y when the charge i s composed o f s e v e r a l m a t e r i a l s d i f f e r i n g i n p h y s i c a l as w e l l as c h e m i c a l 6 1 4 p r o p e r t i e s . ' Improper b a l a n c e o f f u e l and f eed can l e a d to a c c r e -1 5 t i o n b u i l d - u p on the k i l n w a l l s , wh ich may r e q u i r e c o n -t i n u a l a d j u s t m e n t o f the o p e r a t i n g c o n d i t i o n s . L e f t to a c -c u m u l a t e , t h e s e a c c r e t i o n s w i l l d e c r e a s e the open c r o s s -s e c t i o n a l a rea o f the k i l n and r e s u l t i n s i g n i f i c a n t down t ime f o r c l e a n o u t . In v iew o f c u r r e n t t r e n d s towards 4 16 l a r g e r d i a m e t e r r e a c t o r s , ' t h e s e p o t e n t i a l d i f f i c u l t i e s must be d e a l t w i t h i n the s c a l e - u p and d e s i g n s t a g e s o f an i n s t a l l a t i o n . The r e s u l t i s v e r y c o s t l y i f d e s i g n e x p e c t a -t i o n s a re not met. T h i s was r e c e n t l y seen i n Canada when a r o t a r y k i l n f a c i l i t y f a i l e d a f t e r an i n v e s t m e n t o f $65 m i l l i o n . ^ Fundamenta l r e s e a r c h i n t h i s a rea i s t h e r e f o r e e s s e n t i a l f o r the e f f i c i e n t o p e r a t i o n and sound d e s i g n o f t h e s e k i l n s . 1.5 The Approach of R e s e a r c h to Date The aim of a p r o c e s s r e s e a r c h program i s u s u a l l y to 6 enhance our u n d e r s t a n d i n g o f a p r o c e s s and to p r o v i d e c r i t e r i a f o r e f f i c i e n t o p e r a t i o n , and sound d e s i g n , g i v e n ; the range o f o p e r a t i n g c o n d i t i o n s e x p e c t e d . In a r o t a r y k i l n , e f f i c i e n t o p e r a t i o n means d e v e l o p i n g a p r o c e s s i n g s t r a t e g y t h a t w i l l m a i n t a i n the l o w e s t energy c o s t s f o r a g i v e n p r o d u c t i o n demand. In d e s i g n i n g r o t a r y k i l n p l a n t s , i t i s advan tageous to u t i l i z e the minimum r e q u i r e d p l a n t s i t e a r e a w h i l e p r o p e r l y b a l a n c i n g the c a p i t a l c o s t and the c a p a c i t y o f the k i l n w i t h t h a t o f the a u x i l i a r y e q u i p -ment r e q u i r e d . T h i s l a t t e r may a c c o u n t f o r 60% o f the 1 8 c a p i t a l i n v e s t m e n t o f the p l a n t . Wi th t h e s e f a c t o r s i n m i n d , r e s e a r c h on r o t a r y k i l n s has been c a r r i e d out i n two main a r e a s : heat f l o w a n a l y s e s and r e s i d e n c e t ime s t u d i e s . The fo rmer i s conce rned w i t h the a n a l y s i s o f the heat f l o w s t e p s i n a k i l n to o b t a i n the minimum t ime r e q u i r e d f o r a c h e m i c a l o r p h y s i c a l change to o c c u r . The l a t t e r i n c l u d e s t h e o r e t i c a l and e x p e r i m e n t a l s t u d i e s e s t i m a t i n g the t ime r e q u i r e d f o r the f eed to p h y s i c a l l y move f rom the cha rge end to the d i s c h a r g e end . C l e a r l y , f o r a g i v e n a p p l i c a t i o n t h e s e two t imes must c o r -1 9 r e s p o n d . B e t t e r s t i l l , the two t imes must c o r r e s p o n d f o r each zone i n the k i l n . For examp l e ; i n the S t r a t e g i c -Udy p r o c e s s f o r the r e d u c t i o n o f i r o n o r e , ove r o n e - h a l f the k i l n l e n g t h i s used f o r d r y i n g . I f t h i s can be 7 reduced to o n e - t h i r d , a g r e a t e r p o r t i o n o f the k i l n can then be u t i l i z e d f o r the more heat i n t e n s i v e r e d u c t i o n z o n e . An added b e n e f i t to t h i s app roach would be the p o t e n t i a l i n c r e a s e i n p r o d u c t i o n . 1.5 . 1 Heat Flow A n a l y s e s S e v e r a l m a t h e m a t i c a l models o f the heat f l o w 7 2 0 - 2 2 s t e p s i n a r o t a r y k i l n have been f o r m u l a t e d . ' Heat f l o w between the gas and the s o l i d s i s the o n l y mode o f heat t r a n s f e r o f c o n c e r n i n t h i s d i s c u s s i o n . I t has been 2 3 - 2 8 r e c o g n i z e d to take p l a c e i n two s t a g e s . ~ F i r s t l y , heat i s t r a n s f e r r e d f rom the gases to the s o l i d s on the bed s u r f a c e by r a d i a t i o n and c o n v e c t i o n . S e c o n d l y , heat i s exchanged f rom the h o t t e r to the c o l d e r s o l i d s when the fo rmer a re mixed i n t o the b e d . T h i s second s t e p i s enhanced by the m i x i n g o f the s o l i d s when they r e - s u r f a c e f rom the bu l k o f the bed . In a l l m a t h e m a t i c a l models c i t e d , the f i r s t o f t h e s e two s t e p s was assumed to be r a t e l i m i t i n g ( i . e . ' w e l l - m i x e d ' bed a s s u m p t i o n ) and the mechanisms o f heat t r a n s f e r f o r m u l a t e d were those o f r a d i a t i o n and c o n v e c t i o n to a f l a t p l a n a r s u r f a c e . On the b a s i s o f v i s u a l o b s e r v a t i o n s i n i n d u s t r i a l k i l n s and e x p e r i m e n t a l e v i d e n c e on p i l o t f a c i l i t i e s , some 8 i n v e s t i g a t o r s 5 ' 2 4 ' 2 6 ' 2 9 ' 3 0 have c o n c l u d e d t h a t the ' w e l l -m i x e d ' assump t i on i s not a lways v a l i d . I t has been . , 5 ,1 4 ,24 -26 , 3 1 : . , , . . . , . . , r e c o g n i z e d , as w i l l be d i s c u s s e d e x t e n s i v e l y i n c h a p t e r 2 , t h a t the s o l i d s i n a k i l n can have any one o f t h r e e p o s s i b l e t y p e s o f t r a n s v e r s e m o t i o n : s l i p p i n g , s l ump ing and r o l l i n g . Each o f t h e s e modes wou ld p r e s e n t to the hot gases d i f f e r e n t s u r f a c e s f o r v a r y i n g p e r i o d s o f t ime and t h e i r t r a n s v e r s e m i x i n g b e h a v i o u r s would d i f f e r g r e a t l y . F u r t h e r m o r e , based on heat f l o w measurements i n 5 a p i l o t k i l n , i t has been c o n c l u d e d t h a t the t r e a t m e n t o f the s o l i d s s u r f a c e as a f l a t p l a n e i s i n a d e q u a t e to c h a r a c t e r i z e the heat f l o w s t e p s f rom the hot gases to the s o l i d s . I t i s t h e r e f o r e c l e a r t h a t the t r a n s v e r s e bed mot ion e n c o u n t e r e d i n r o t a r y k i l n o p e r a t i o n must be f u l l y d e f i n e d i n o r d e r to u n d e r s t a n d the degree o f i t s e f f e c t on the heat t r a n s f e r p r o c e s s e s to the s o l i d s i n r o t a r y k i l n s . 1 . 5 . 2 R e s i d e n c e Time S t u d i e s T h e ' s o l i d s r e s i d e n c e t ime i n a r o t a r y k i l n i s a f f e c t e d by k i l n i n c l i n a t i o n and r o t a t i o n . S i n c e the . t e m p e r a t u r e and c o m p o s i t i o n o f the gas w i l l a l s o va ry a l o n g the k i l n l e n g t h , the p h y s i c a l p r o p e r t i e s o f the cha rge may change and hence a l s o a f f e c t the r e s i d e n c e t ime of the s o l i d s . Gases g e n e r a t e d i n the bed w i l l f u r t h e r c o m p l i c a t e the 9 mot ion o f the charge not to ment ion the e f f e c t o f dams, a c c r e t i o n b u i l d - u p , v a r y i n g k i l n d i a m e t e r and feed s i z e and shape d i s t r i b u t i o n s . Many i n v e s t i g a t o r s have t h e r e f o r e a p p l i e d r a d i o a c t i v e t r a c e r t e c h n i q u e s to r e s i d e n c e t ime s t u d i e s i n i n d u s t r i a l k i l n s , 1 , 1 2 ' 1 3 ' 3 2 ' 3 3 T h e i r r e s u l t s a re l i m i t e d to the r e a c t o r s t e s t e d ove r the range o f v a r i a b l e s i n v e s t i g a t e d . O the r fundamenta l a p p r o a c h e s 1 ' 3 ' 1 4 ' 1 9 ' 3 4 " 4 4 have i d e n t i f i e d f o u r mechanisms by wh ich the cha rge moves a l o n g the k i l n l e n g t h . These a r e : movement r a d i a l l y and a x i a l l y by c o n v e c t i o n and d i f f u s i o n . A l t h o u g h t h e s e s t u d i e s p r o v i d e q u a n t i t a t i v e l y the magn i tude o f each o f t hese mechan isms , they have been a p p l i e d o n l y to a r o l l i n g or c a s c a d i n g bed . A s y s t e m a t i c i n v e s t i g a t i o n o f a l l r e a c t o r and m a t e r i a l v a r i -a b l e s , i n the v a r i o u s t ypes o f t r a n s v e r s e bed m o t i o n , ove r the f u l l o p e r a t i n g range o f the k i l n has not y e t been c a r r i e d o u t . I t has been r e c o g n i z e d t h a t each t ype o f t r a n s v e r s e bed mot ion ( s l i p p i n g , s l ump ing and r o l l i n g ) w i l l a f f e c t 44-48 the r e s i d e n c e t ime o f the b u r d e n . Hence the comp le te c h a r a c t e r i z a t i o n o f t r a n s v e r s e bed mot ion i s l a c k i n g even though i t forms one o f the b a s i c b u i l d i n g b l o c k s o f r o t a r y k i l n o p e r a t i o n . 10 1.6 The O b j e c t i v e s o f t h i s Work The f i r s t o b j e c t i v e o f t h i s work i s to i d e n t i f y t hose r o t a r y k i l n o p e r a t i n g and m a t e r i a l v a r i a b l e s t h a t a f f e c t the d i f f e r e n t modes o f t r a n s v e r s e bed b e h a v i o u r : s l i p p i n g , s l u m p i n g and r o l l i n g . These o p e r a t i n g v a r i a b l e s i n c l u d e k i l n bed d e p t h , d i a m e t e r , r o t a t i o n a l speed and w a l l r o u g h n e s s ; the m a t e r i a l v a r i a b l e s a re p a r t i c l e s h a p e , p a r t i c l e s i z e , s i z e d i s t r i b u t i o n , and d e n s i t y . The second o b j e c t i v e i s to i n t e r p r e t and a p p l y the r e s u l t s thus o b t a i n e d to r o t a r y k i l n o p e r a t i n g c o n d i t i o n s and to d e v e l o p s c a l e - u p c r i t e r i a f o r k i l n d e s i g n based on bed b e h a v i o u r s i m i l a r i t y . C h a p t e r 2 n LITERATURE REVIEW OF BED BEHAVIOUR 2.1 I n t r o d u c t i on The a im of t h i s c h a p t e r i s f i r s t l y to p ropose a c o n s i s t e n t t e r m i n o l o g y by wh ich the v a r y i n g t ypes o f bed mot ion w i l l be a d d r e s s e d t h r o u g h o u t the t e x t o f t h i s t h e s i s . In the l i t e r a t u r e the same t ype o f s o l i d s mot ion has been i d e n t i f i e d by d i f f e r e n t t e r m i n o l o g y ; w h i l e d i f f e r e n t t ypes o f s o l i d s mot ion have been a d d r e s s e d by the same t e r m . The second a im w i l l be to r e v i e w the d e s c r i p t i o n s o f the v a r i e d bed.. behav i ours e n c o u n t e r e d i n r o t a r y k i l n o p e r a -t i o n s . T h i r d l y , the d i f f e r e n t methods o f e x p e r i m e n t a l l y c h a r a c t e r i z i n g t h e s e various bed b e h a v i o u r s wi11 be p r e -s e n t e d . Three methods w i l l then be d e v e l o p e d f o r mode l -l i n g s o l i d s f l o w i n r o t a t i n g c y l i n d e r s and f i n a l l y the s c a l e - u p c r i t e r i a t h a t have been p roposed i n the l i t e r a -t u r e f o r bed b e h a v i o u r s i m i l i t u d e w i l l be d e s c r i b e d . 2 . 2 The C h a r a c t e r i z a t i o n o f the Modes o f Bed B e h a v i o u r In a c y l i n d e r o f a g i v e n d i a m e t e r , f i l l e d w i t h v a r y i n g amounts of g r a n u l a r s o l i d s and r o t a t e d ove r a w ide range o f r o t a t i o n a l s p e e d s , the f o l l o w i n g s i x modes o f bed b e h a v i o u r have been o b s e r v e d : s l i p p i n g , s l u m p i n g , r o l l i n g , c a s c a d i n g , c a t a r a c t i n g and c e n t r i f u g i n g . Only the f i r s t f o u r o f t hese w i l l be d i s c u s s e d i n t h i s s e c t i o n , s i n c e c a t a r a c t i n g and c e n t r i f u g i n g a re not n o r m a l l y found i n r o t a r y k i l n o p e r a t i o n . 2 . 2 . 1 S l i p p i n g 2 . 2 . 1 . 1 D e s c r i p t i o n and T r a n s v e r s e M i x i n g Three t ypes o f ;s 1 i p'p.i ng have been 2 46 48 49 o b s e r v e d ! ' ' ' The f i r s t i s an o s c i l l a t o r y t ype o r pendu l um-1 i k e mot ion and i s d e s c r i b e d as f o l l o w s . I f a smooth-wal l e d c i r -c u l a r c y l i nder i s f i 11 ed wi t h a g i ven q u a n t i t y o f bu l k s o l i d s and s l o w l y r o t a t e d , the s o l i d s bed w i l l move w i t h the w a l l o f the c y l i n d e r u n t i l i t r e a c h e s a bed i n c l i n a t i o n l ower 2 48 than the a n g l e o f repose o f the bu l k s o l i d s . ' At t h i s p o i n t , the whole bed w i l l s l i p a g a i n s t the c y l i n d e r w a l l as a s o l i d mass and come to r e s t a t an even l ower a n g l e o f bed i n c l i n a t i o n . The s o l i d s bed w i l l then s t a r t r o t a t i n g w i t h the c y l i n d e r w a l l once more and w i l l a g a i n s l i p when i t r e a c h e s a p p r o x i m a t e l y the same maximum i n c l i n a t i o n as b e f o r e . T h u s , w i t h the c o n t i n u o u s r o t a t i o n o f the c y l i n d e r , 49 a p e n d u m - l i k e or o s c i l l a t o r y t ype o f bed mot ion r e s u l t s . 2 Ru tge rs s t a t e s t h a t t h i s t ype o f s l i p p i n g a c t i o n has a d e f i n i t e f r e q u e n c y and a m p l i t u d e ; however , no s u p p o r t i n g 13 e x p e r i m e n t a l r e s u l t s have been p r e s e n t e d . A 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 t h i s t ype o f a s l i p p i n g bed i s shown i n F i g u r e 2 . 1 . A second t ype o f s l i p p i n g a c t i o n has been r e p o r t e d by Ronco. In t h i s c a s e , the s o l i d s r i s e w i t h the c y l i n d e r w a l l but w i t h a s m a l l e r r o t a t i o n a l s p e e d . T h i s b e h a v i o u r i s due to the l a y e r o f s o l i d s i n c o n t a c t w i t h the w a l l a c t i n g as b e a r i n g s between the burden and the c y l i n d e r w a l l . The s o l i d s r e a c h i n g the apex ( t he upper p o r t i o n o f the bed s u r f a c e ) move s l o w l y down the i n c l i n a t i o n o f the bed towards i t s l owe r e x t r e m i t y , the c h o r d a l b a s e . F i n a l l y , w i t h i n a c e r t a i n range o f r o t a t i o n a l speeds and deg rees o f f i l l , the s o l i d s bed may t a k e up a 2 s t a t i o n a r y c o n d i t i o n . T h i s phenomenon c o n s t i t u t e s a t h i r d t ype o f s l i p p i n g a c t i o n where the bed i s seen to remain m o t i o n l e s s w i t h i n the r o t a t i n g w a l l s . A l l o f the i n v e s t i g a t o r s who have o b s e r v e d the s l i p -p i ng mode r e p o r t t h a t l i t t l e o r no t r a n s v e r s e s o l i d s m i x i n g r e s u l t s . T h i s i s i n agreement w i t h the e x p e r i m e n t a l f i n d i n g s o f R o s e . 5 0 2 . 2 . 1 . 2 E f f e c t o f V a r i a b l e s In the range o f o p e r a t i n g c o n -d i t i o n s f o r r o t a r y k i l n s , no r i g o r o u s e x p e r i m e n t a l Fi gure 2.2 Schematic diagram of slumping. 15 i n v e s t i g a t i o n o f the v a r i a b l e s a f f e c t i n g s l i p p i n g has been c a r r i e d o u t . Some q u a l i t a t i v e o b s e r v a t i o n s , however , have been r e c o r d e d w h i l e o t h e r modes o f bed b e h a v i o u r were be ing i n v e s t i g a t e d . These a re summar ized be low . W i t h i n the wide range o f p r o c e s s a p p l i c a t i o n s and o p e r a t i n g c o n d i t i o n s o f r o t a t i n g c y l i n d e r s , low degrees o f f i l l have been commonly o b s e r v e d to enhance c n ™ - „ „ 2 > 3 , 2 5 , 2 7 , 2 8 , 4 4 ,46 ,51 -55 c n + + u u- u s l i p p i n g . For e x a m p l e , a t the h i g h r o t a t i o n a l speeds o f b a l l m i l l i n g , s l i p p i n g can o c c u r at degrees o f f i l l as h igh as 30 and 40 per c e n t ; w h i l e at 53 55 per c e n t , s l i p p i n g may be e l i m i n a t e d . Smooth w a l l s 2 ' 2 5 ' 4 4 ' 4 7 ' 5 0 ' 5 3 and h igh i n t e r n a l burden f r i c t i o n 2 ' 2 5 are a l s o b e l i e v e d to promote s l i p p i n g . 54 Rose and B l u n t a re the o n l y i n v e s t i g a t o r s who have a t t emp ted to i n t e r p r e t t h e i r e x p e r i m e n t a l o b s e r v a -t i o n s o f s l i p p i n g i n b a l l m i l l s . T h e i r d i m e n s i o n a l a n a l y s i s y i e l d e d the f o l l o w i n g n o n - d i m e n s i o n a l g r o u p s : Act = f (D , y w / s , J) ( 2 . 1 ) d T The d i m e n s i o n l e s s group (D/d ) was g i v e n to m a i n t a i n the g e o m e t r i c a l s i m i l a r i t y o f the number o f c o n t a c t p o i n t s a t the w a l l - b e d i n t e r f a c e . The f u n c t i o n a l r e l a t i o n s h i p o f 16 e q u a t i o n ( 2 . 1 ) was g r a p h i c a l l y p r e s e n t e d by Rose and 54 B l u n t , but has not r e c e i v e d wide a p p l i c a t i o n . T h u s , f o u r v a r i a b l e s have been e x p e r i m e n t a l l y i d e n t i f i e d to a f f e c t the s l i p p i n g mode o f bed b e h a v i o u r : an o p e r a t i n g v a r i a b l e , the degree o f f i l l ; a m a t e r i a l v a r i a b l e , the i n t e r n a l a n g l e o f f r i c t i o n ; and two m a t e r i a l / c y l i n d e r v a r i a b l e s , the b e d - w a l l f r i c t i o n c o e f f i c i e n t and the r a t i o o f c y l i n d e r d i a m e t e r to the ave rage p a r t i c l e s i z e . 2 . 2 . 2 S lump ing S lump ing was b r i e f l y d e s c r i b e d by Dav i s as 56 e a r l y as 1920. Most i n v e s t i g a t o r s s i n c e have c a r r i e d out t h e i r e x p e r i m e n t a l work e i t h e r on smooth w a l l e d c y l i n d e r s or a t r o t a t i o n a l speeds h i g h e r than i s r e q u i r e d f o r a bed o f bu l k s o l i d s to s lump . Hence , the s l u m p i n g bed has r e c e i v e d ve ry l i t t l e a t t e n t i o n d e s p i t e i t s i n -d u s t r i a l i m p o r t a n c e i n r o t a r y k i l n s . 5 , 1 4 ' 2 4 ' 2 6 ' 2 9 ' 3 1 A d e s c r i p t i o n o f the s l u m p i n g bed as r e c o r d e d by Z a b l o t n y , 1 2 7 28 57 R e u t e r , Schnabe l and W a h l s t e r e t a l . . , i s g i v e n be low . 2 . 2 . 2 . 1 D e s c r i p t i o n and T r a n s v e r s e M i x i n g I f a r o t a r y c l y l i n d e r , f i l l e d w i t h bu l k s o l i d s ( F i g u r e 2 . 2 ) , i s r o t a t e d at a low speed and i f t h e r e i s no r e l a t i v e mot ion between the bu l k s o l i d s and the c y l i n d e r w a l l , the s u r f a c e i n c l i n a t i o n o f the bed w i l l i n c r e a s e . When i t r eaches the s t a t i c a n g l e o f r e p o s e , <j>D, o f the s o l i d s , a segment o f the bu l k s o l i d s w i l l de tach f rom the r e m a i n i n g s o l i d s , away f rom the c y l i n d e r w a l l , and w i l l s lump towards the l o w e r e x t r e m i t y o f the bed s u r f a c e . The f i n a l bed i n c l i n a t i o n w i l l be l owe r than the s t a t i c a n g l e o f repose and w i l l be g i v e n by p o s i t i o n BP ( F i g u r e 2 . 2 ) . S i n c e the c y l i n d e r i s c o n t i n u o u s l y r o t a t e d , the i n c l i n a t i o n o f the bed w i l l s t a r t to i n c r e a s e a g a i n u n t i l i t r e a c h e s the s t a t i c a n g l e o f r epose a t wh ich p o i n t a n o t h e r slump w i l l o c c u r . The segment o f s o l i d s s l u m p i n g , ABC, forms an a n g l e c a l l e d the s h e a r i n g a n g l e , y, wh ich has been r e p o r t e d to be 12° - 15° a t 1-2 r / m i n . 1 57 W a h l s t e r e t a l . have c a r r i e d out a s m a l l number o f t r a c e r e x p e r i m e n t s i n the s l u m p i n g mode to d e t e r m i n e the f l o w p a t t e r n o f the p a r t i c l e s i n the s o l i d s b e d . A l t h o u g h t h e i r p r i m a r y o b j e c t i v e was to s t udy s e g r e g a t i o n p a t t e r n s i n the b u r d e n , some of t h e i r i n t e r p r e t a t i o n s a re r e l e v a n t to a d i s c u s s i o n on t r a n s v e r s e m i x i n g i n s l u m p i n g beds and w i l l be p r e s e n t e d be low . S i n c e t h e r e i s no r e l a t i v e mot ion between the s o l i d s bed and the c y l i n d e r w a l l , the s o l i d s i n the bed have a f i x e d c i r c u l a r t r a j e c t o r y whose c e n t r e c o i n c i d e s w i t h the c e n t r e o f r o t a t i o n o f the c y l i n d e r . T h e r e f o r e , no m i x i n g t a k e s p l a c e i n the bed as i t s i n c l i n a t i o n 18 i n c r e a s e s . When a slump o c c u r s , however , the s o l i d s moving down the bed s u r f a c e a re seen to change c i r c u l a r t r a j e c t o r y p o s i t i o n s . The f i n a l r a d i u s o f r o t a t i o n of the s o l i d s in the bed c o u l d e i t h e r be g r e a t e r o r l e s s than the one f rom wh ich they had emerged. T h i s m i x i n g p r o c e s s i s b e l i e v e d to be s low and f o r s o l i d s w i t h c o n s t a n t p h y s i -c a l p r o p e r t i e s , i t i s a l s o t hough t to be random; however , no q u a n t i t a t i v e a n a l y s i s has been r e p o r t e d . 2 . 2 . 2 . 2 E f f e c t of V a r i a b l e s The f i r s t method used by i n v e s t i -g a t o r s to c h a r a c t e r i z e s l u m p i n g i s the measurement o f . the s l u m p i n g f r e q u e n c y o f the bu l k s o l i d s . Z a b l o t n y 1 b e l i e v e d t h a t i t i n c r e a s e d w i t h r o t a t i o n a l speed and was dependent on the p h y s i c a l p r o p e r t i e s o f the bu l k s o l i d s . The o b s e r v a t i o n s o f s l u m p i n g f r e q u e n c y made by Pea rce on 24 s e v e r a l i n d u s t r i a l k i l n s can be p r e s e n t e d as f o l l o w s : S = 60 + 2_4 ( 2 . 2 ) t s n 57 W a h l s t e r e t a l . , on the o t h e r hand , measured the s lump-i n g f r e q u e n c y o f coke b r e e z e , s i 1 i c a , and i r o n o re and found no dependence on p a r t i c l e s i z e , shape or d e n s i t y . 27 More r e c e n t l y , Reu te r a n a l y z e d the s l u m p i n g f r e q u e n c y 19 o f beds w i t h v a r y i n g p a r t i c l e s i z e and m i x t u r e s o f c o a l and i r o n o r e . Three f r e q u e n c y groups were i d e n t i f i e d : ( i ) 0 < n < 0 .04 ( 2 . 3 ) • ( I D 0. 04 < n < 0 .1 0 ( 2 . 4 ) ( i i i ) 0 .10 < n_<. 0 .20 ( 2 . 5 ) The f i r s t group was no ted f o r a s t eep i n c r e a s e i n s l ump ing f r e q u e n c y f o r r e l a t i v e l y s m a l l e r i n c r e a s e s i n r o t a t i o n a l s p e e d . I t was b e l i e v e d to be c h a r a c t e r i s t i c o f the boun-dary between s l i p p i n g and s l u m p i n g . In the second group o f r o t a t i o n a l s p e e d s , s m a l l e r i n c r e a s e s i n s l u m p i n g f r e -quency were o b s e r v e d . R e u t e r a l s o b e l i e v e d t h a t the p a r t i c l e d e n s i t y d e t e r m i n e d the t ype o f p a r t i c l e mot ion i n t h i s second z o n e . For O'.J <n_ <0 .20 , a change i n bed n C mode was seen and a t t r i b u t e d to the i n c r e a s e d e f f e c t o f the c e n t r i f u g a l f o r c e . Renewed, l a r g e i n c r e a s e s i n *The c r i t i c a l r o t a t i o n a l s p e e d , n c , i s the c y l i n d e r r o t a t i o n a l speed at wh ich a p a r t i c l e on the c y l i n d e r w a l l c e n t r i f u g e s . For a s m a l l p a r t i c l e s i z e r e l a t i v e to the c y l i n d e r d i a m e t e r , n g , i s g i v e n by : (2.6) 20 s l u m p i n g f r e q u e n c y r e s u l t wh ich equal the r a t e o f a r r i v a l o f p a r t i c l e s to the bed s u r f a c e when the change i n bed b e h a v i o u r o c c u r s . In the s t u d i e s o f R e u t e r , P e a r c e and W a h l s t e r e t a l . t h e r e appears to be no c o n s i s t e n t bed b e h a v i o u r p a t t e r n o f what v a r i a b l e s a f f e c t the s l ump ing f r e q u e n c y . T h e i r o b s e r v a t i o n s w i l l a g a i n be d i s c u s s e d in the l i g h t o f r e s u l t s f rom the c u r r e n t i n v e s t i g a t i o n . O the r dependent v a r i a b l e s t h a t would e n a b l e a c h a r a c t e r i z a t i o n o f a s l u m p i n g bed to be c a r r i e d out a re the a n g l e o f r epose and the s u r f a c e p a r t i c l e r e s i d e n c e 57 t i m e . W a h l s t e r e t a l . r e p o r t e d no v a r i a t i o n i n the a n g l e o f r epose w i t h i n c r e a s i n g r o t a t i o n a l speed f o r the m a t e r i a l s t e s t e d . They a l s o no ted t h a t the r a t i o o f s u r f a c e r e s i d e n c e t ime to bu l k c i r c u l a t i o n t ime was i n the range o f one-t w e l f t h ( 1 / 1 2 ) to o n e - f i f t h ( 1 / 5 ) . These were not com-pared w i t h t h o s e o f o t h e r bed modes, and no d e s c r i p t i o n was g i v e n o f the e x p e r i m e n t a l c o n d i t i o n s and t e c h n i q u e s u s e d . In c o n c l u s i o n , i t can be s t a t e d t h a t w h i l e some of the pa rame te rs r e q u i r e d to c h a r a c t e r i z e a s l u m p i n g bed are known, t h e i r dependence on m a t e r i a l , c y l i n d e r and o p e r a t i n g v a r i a b l e s i s l a r g e l y unknown. 21 2 . 2 . 3 R o l l i n g and C a s c a d i n g There i s l i t t l e d i f f e r e n t i a t i o n made i n the l i t e r a t u r e between r o l l i n g and c a s c a d i n g . In f a c t , the terms have o f t e n been used i n t e r c h a n g e a b l y . A l t h o u g h , i n t h i s t h e s i s , new c r i t e r i a w i l l be d e v e l o p e d to d i s t i n g u i s h them, they w i l l be d i s c u s s e d t o g e t h e r i n t h i s s e c t i o n . , 2 . 2 . 3 . 1 D e s c r i p t i o n and T r a n s v e r s e M i x i n g At h i g h e r r o t a t i o n a l s p e e d s , the 27 57 s l ump ing mot ion o f the bed i s r e p l a c e d by r o l l i n g . ' T h i s r o l l i n g bed i s c h a r a c t e r i z e d by the c o n t i n u o u s mot ion of a l a y e r o f s o l i d s ove r the bed s u r f a c e . T h i s zone i s c o n t i n u a l l y f ed w i t h s o l i d s f rom the b u l k o f the b u r d e n , wh ich reach the upper p a r t o f the bed by means of the r o t a t i o n o f the c y l i n d e r . The r e m a i n d e r o f the c h a r g e remains below the moving l a y e r . As t h e r e i s no r e l a t i v e mot ion between the c y l i n d e r w a l l and the s o l i d s i n the b u l k of the b e d , t h e s e bu l k p a r t i c l e s t h e r e f o r e t r a v e l i n f i x e d t r a j e c t o r i e s . Wi th the c y l i n d e r r o t a t i n g a t low r o t a t i o n a l s p e e d s , the r o l l i n g bed assumes a c o n s t a n t a n g l e o f i n c l i n a t i o n and i s seen to have a f l a t p l a n a r s u r f a c e . With i n c r e a s i n g speeds the s o l i d s a t the upper p a r t of the c r o s s s e c t i o n 22 o f the b e d , the a p e x , r i d e h i g h e r up the w a l l b e f o r e d e t a c h i n g t h e m s e l v e s f rom i t . The c r o s s s e c t i o n o f the bed then assumes amore 1 c r e s c e n t ' o r ' k i d n e y ' shape . T h i s has been termed c a s c a d i n g . Three models d e s c r i b i n g the f l o w p a t t e r n s o f the 2 s o l i d s have been p r o p o s e d . The f i r s t by Ru tge rs i s p r e s e n t e d i n F i g u r e 2 . 3 , and a p p l i e s at the h i g h e r r o t a -t i o n a l speeds where c a s c a d i n g d o m i n a t e s . The moving l a y e r o f s o l i d s on the bed s u r f a c e i s r e l a t i v e l y t h i n and i s r e s p o n s i b l e f o r a l l the t r a n s v e r s e m i x i n g wh ich o c c u r s by c o n v e c t i o n and d i f f u s i o n . Below t h i s l a y e r o f p a r t i c l e s i s a s t a g n a n t c o r e . When the degree o f f i l l i s g r e a t e r than 50 per c e n t , t h i s co re becomes a dead s p o t . A l t h o u g h 58 Hogg has shown p h o t o g r a p h i c a l l y the p r e s e n c e o f t h i s dead s p o t , f u r t h e r e x p e r i m e n t a l work by Hogg as w e l l as the r e s u l t s of o t h e r w o r k e r s 2 7 ' 3 6 ' 5 9 ' 6 0 have shown t h a t the t r e a t m e n t by Ru tge rs o f the moving l a y e r o f s o l i d s to be l i m i t e d i n a p p l i c a t i o n . In the second f l o w v i s u a l i z a t i o n model ( F i g u r e 2 . 4 ) , s o l i d s a re b e l i e v e d to t r a v e l on f i x e d pa ths both i n the bu l k of" t h e ' b e d as w e l l as. oh t h e ' s u r f a c e Amoving . l a y e r . 4 6 ' 5 1 ' 6 1 ' 6 2 In t h i s r e p r e s e n t a t i o n , p a r t i c l e s do not r 23 24 i n t e r c h a n g e t r a j e c t o r y p a t h s . T h i s i s an i n c o r r e c t r e p r e s e n t a -t i o n because r e c e n t e x p e r i m e n t a l e v i d e n c e has i n d i c a t e d t h a t . . . , 2 7 , 3 6 , 3 7 , 5 2 , 5 9 , 6 3 - 6 6 t r a n s v e r s e m i x i n g does i ndeed o c c u r . In f a c t , when c a s c a d i n g , the bed can reach a s t e a d y s t a t e o f m i x e d n e s s * w i t h i n f i v e (5) c y l i n d e r r e v o l u t i o n s . The t h i r d model o f s o l i d s m o t i o n , wh ich w i l l now be p r e s e n t e d , has been l a r g e l y d e v e l o p e d based on o b s e r v a -t i o n s made on c a s c a d i n g b e d s . P h o t o g r a p h i c and t r a c e r t e c h n i q u e s have shown t h a t the bu l k s o l i d s bed i s e s s e n t i a l l y composed o f two r e g i o n s : a c t i v e and p a s s i v e ( F i g u r e 2 5 ) 2 7 , 3 6 , 5 2 , 5 8 , 5 9 , 6 3 - 6 5 , 67 J h e m a j o r 1 t y o f t h e s o l i d s are i n the p a s s i v e r e g i o n . H e r e , they have no r e l a t i v e mot ion w i t h one a n o t h e r nor w i t h the c y l i n d e r w a l l . They a re t h e r e f o r e on f i x e d t r a j e c t o r i e s r o t a t i n g w i t h the same a n g u l a r v e l o c i t y as the c y l i n d e r . S o l i d s e n t e r the a c t i v e r e g i o n a t AB ( F i g u r e 2 . 5 ) i n the upper h a l f o f t he b e d . Once they c r o s s the m i d - p o i n t o f t he s u r f a c e , OB, they e n t e r the p a s s i v e zone a l o n g CB. T h e i r p o i n t o f e n t r y i n t o the p a s s i v e r e g i o n i s not e q u i d i s t a n t f rom t h e i r p o i n t o f e x i t . Thus s o l i d s mot ion i n the a c t i v e zone i s f a i r l y random, and t h e r e f o r e a l l m i x i n g i s a c h i e v e d i n the a c t i v e r e g i o n .  * M i x e d n e s s i s d e f i n e d as " t h e s t a t e i n wh ich t h e r e i s the same p r o b a b i l i t y o f a p a r t i c l e s e l e c t e d at a g i v e n p o i n t y b e i n g o f a c e r t a i n t y p e , f o r a l l p o i n t s i n the m i x t u r e . " 25 Figure 2.6 Schematic representat ion of the ve loc i t y p r o f i l e of the so l ids in the act ive and passive regions. 26 51 52 67 Some wo rke rs ' ' a l s o s u g g e s t t h a t t h e r e e x i s t s a v e l o c i t y p r o f i l e o f the s o l i d s i n the a c t i v e r e g i o n ( F i g u r e 2 . 6 ) . Those s o l i d s t r a v e l l i n g on the bed s u r f a c e have the h i g h e s t v e l o c i t y down the i n c l i n a t i o n o f the bed . In moving r a d i a l l y i n t o the bed the v e l o c i t y o f the s o l i d s d e c r e a s e s u n t i l i t r eaches z e r o at the a c t i v e - p a s s i v e boundary wh ich i s f i x e d i n s p a c e . The s o l i d s below i t have no r e l a t i v e v e l o c i t y w i t h each o t h e r nor w i t h the c y l i n d e r w a l l ; w h i l e t h o s e above i t a re i n the a c t i v e zone and t h e i r v e l o c i t y down the bed s u r f a c e ( f rom the apex to the c h o r d a l b a s e ) i n c r e a s e s as the s u r f a c e o f the bed i s a p p r o a c h e d . 2 . 2 . 3 . 2 E f f e c t o f V a r i a b l e s A r o l l i n g or a c a s c a d i n g bed i s more d i f f i c u l t to c h a r a c t e r i z e than a s l ump ing b e d . To da te t h i s has been a t t e m p t e d by measu r i ng the bed i n c l i n a t i o n , the r e s i d e n c e t ime o f s o l i d s on the s u r f a c e , the a c t i v e l a y e r t h i c k n e s s and the c o n d i t i o n s at wh ich a bed w i l l change i n t o the r o l l i n g mode. When r o l l i n g , the a n g l e o f i n c l i n a t i o n o f the bed i s b e l i e v e d to be the dynamic a n g l e o f repose o f the bu l k 57 s o l i d s . W a h l s t e r e t a l . r e p o r t no change i n t h i s a n g l e 6 8 w i t h r o t a t i o n a l s p e e d . Wes et a l . , on the o t h e r h a n d , 27 have found i t to d e c r e a s e w i t h i n c r e a s i n g c y l i n d e r r o t a -t i o n a l s p e e d . T h i s i s not c o n s i s t e n t w i t h the f a c t t h a t the s o l i d s r i d e h i g h e r on the c y l i n d e r w a l l as the r o t a t i o n a l speed i n c r e a s e s . The r e s u l t s o f Wes et a l . were p r o b a b l y 2 due to the p r e s e n c e o f l i f t e r s i n t h e i r c y l i n d e r s . R u t g e r s , 27 4 6 Reu te r and Ronco have compared the dynamic ang le o f r epose to the s t a t i c and r e p o r t the fo rmer to be h i g h e r . On the b a s i s o f t h e i r own e x p e r i m e n t a l r e s u l t s , F r a n k l i n and 69 Johanson have r e p o r t e d the o p p o s i t e to be the c a s e , wh ich i s in l i n e w i t h the b a s i c p r i n c i p l e s o f m e c h a n i c s . 7 0 The measurement o f s u r f a c e r e s i d e n c e t imes has been c a r r i e d ou t by R e u t e r 2 7 and O y a m a . 6 0 U n f o r t u n a t e l y , t h e i r da ta i s not i n a u s e a b l e form f o r bed c h a r a c t e r i z a t i o n and t h e i r e x p e r i m e n t a l t e c h n i q u e s a re not d e s c r i b e d . The a c t i v e l a y e r measurements o f R e u t e r , F i g u r e 2 . 7 , i n d i c a t e an i n c r e a s e i n the a c t i v e l a y e r depth w i t h i n -c r e a s i n g r o t a t i o n a l s p e e d . A s m a l l e r i n c r e a s e i n the r a t i o o f the a c t i v e l a y e r t h i c k n e s s ' t o the bed depth w i t h i n -27 c r e a s i n g degree o f f i l l i s a l s o i l l u s t r a t e d . A l t h o u g h no f u r t h e r a n a l y s i s o f t h i s da ta i s p r e s e n t e d by R e u t e r , a more tho rough t r e a t m e n t o f i t w i l l be made i n t h i s t h e s i s . S e v e r a l o b s e r v e r s have r e c o r d e d the o p e r a t i n g 28 > ca: Drum Speed F i g u r e 2.7 E f f e c t o f drum speed on the r e l a t i v e a c t i v e l a y e r t h i c k n e s s 27 o f a m i x t u r e o f p e l l e t s i n a m i x i n g drum 0.8 D x 0.5 m. F i g u r e 2.8 R e l a t i o n between f l o w l i m i t and p a r t i c l e s i z e . 1. P a r t i c l e s i z e 2. K i l n r o t a t i o n speed f o r the f l o w l i m i t 3. Degree o f f i 1 1 i n g 4. Kiruna-D-Ore 5. B i c o r i t e 6. Coke 29 c o n d i t i o n s under wh ich a bed w i l l change f rom s l u m p i n g to 56 r o l l i n g . Dav i s s t a t e d t h a t t h i s change was due to the i n e r t i a o f the p a r t i c l e s as w e l l as to the c e n t r i f u g a l 2 f o r c e s a c t i n g on them. R u t g e r s , however , r e p o r t s t h a t the change i n bed b e h a v i o u r o c c u r s a t a r o t a t i o n a l speed o n e - t e n t h t h a t o f the c r i t i c a l (0.1 n c ) . C a r l e y - M a c a u l y 6 3 and Dona ld c l a i m t h a t t h i s change i n bed b e h a v i o u r o c c u r s a t 0 .056 n^ . T h i s would c o r r e s p o n d to a r a t i o o f c e n t r i -f u g a l to g r a v i t a t i o n a l f o r c e s o f 0.01 and 0 .003 r e s p e c t i v e l y C l e a r l y the c e n t r i f u g a l f o r c e i s n e g l i g i b l e . The most tho rough s tudy o f the change i n the bed b e h a v i o u r f rom s l u m p i n g to r o l l i n g i s t h a t o f W a h l s t e r e t a l however, t h e i r r e s u l t s , shown i n F i g u r e 2 . 8 , a re somewhat c o n f u s i n g . The o r d i n a t e o f the graph as w e l l as the t e x t o f t h e i r paper s t a t e t h a t the r o t a t i o n a l speed i s p l o t t e d 7 1 but the a x i s i n d i c a t e s a l i n e a r s p e e d . Owing to t h i s c o n -f u s i o n o n l y a q u a l i t a t i v e a n a l y s i s o f t h e i r r e s u l t s i s p o s s i b l e . The speed at wh ich the bed changes f rom s l u m p i n g to r o l l i n g , ' t h e f l o w l i m i t * i s p l o t t e d a g a i n s t the p a r t i c l e s i z e f o r v a r y i n g f i l l r a t i o s f o r t h r e e m a t e r i a l s t e s t e d . I t i s e v i d e n t t h a t as the f i l l r a t i o i n c r e a s e s , the ' f l o w l i m i t ' d e c r e a s e s . A l s o f o r s m a l l p a r t i c l e s i z e s , the ' f l o w l i m i t ' i n c r e a s e s u n t i l a maximum i s r eached at a p a r t i c l e s i z e o f 2-4 mm. Wi th f u r t h e r i n c r e a s e s i n p a r t i c l e s i z e , 30 the ' f l o w l i m i t 1 d e c r e a s e s . Fur thermore , the type of m a t e r i a l was not observed to a f f e c t t h i s ' f l o w l i m i t ' . While the concept of a ' f l o w l i m i t ' appears to be u s e f u l in c h a r a c t e r i z i n g r o l l i n g b e d s , the c o n c l u s i o n s and the r e s u l t s of Wahlster et a l . have l i m i t e d usage. In c o n c l u s i o n , of the techn iques d e s c r i b e d in t h i s s e c t i o n , the a c t i v e l a y e r t h i c k n e s s and the ' f l o w l i m i t ' measurements appear to be the two most f r u i t f u l methods of c h a r a c t e r i z i n g the r o l l i n g bed and the v a r i a b l e s a f f e c t -in g i t . 2.3 Bed Behaviour M o d e l l i n g There have been th ree main approaches to the model -l i n g of s o l i d s f low in r o t a t i n g c y l i n d e r s based r e s p e c t i v e l y on the a c t i v e l a y e r t h i c k n e s s , t r a n s v e r s e mix ing and the mechanics of a s t a t i c r i g i d body. Each of these w i l l be d i s c u s s e d in the f o l l o w i n g s e c t i o n s . 2.3.1 A c t i v e Layer T h i c k n e s s The o b j e c t of t h i s f i r s t approach was to p r e -d i c t the a c t i v e l a y e r t h i c k n e s s of a r o l l i n g or a c a s c a d i n g p o C O bed based on f i r s t p r i n c i p l e s . ' Th is approach has not been very s u c c e s s f u l in p r e d i c t i n g rea l systems because of 31 the ve ry complex n a t u r e o f p a r t i c l e k i n e m a t i c s . 2 . 3 . 2 T r a n s v e r s e M i x i n g The p r i m a r y aim o f the m i x i n g s t u d i e s i n 37 72 -77 r o t a r y c y l i n d e r s r e p o r t e d i n the l i t e r a t u r e ' was to d e v e l o p , u s i n g s t a t i s t i c a l t e c h n i q u e s or d i f f u s i o n a n a l o g i e s , a c o e f f i c i e n t o f m ixedness by wh ich the mixed o r un-mixed s t a t e o f the g r a n u l a r s o l i d s c o u l d be e v a l u a t e d . A hos t o f t h e s e c o e f f i c i e n t s have been p r o p o s e d , but no s i n g l e one has found w ide a p p l i c a t i o n . 0 7 C O The work of C a r l e y - M a c a u l y and Donald ' and o f 58 Hogg bears d i r e c t l y on the t r a n s v e r s e m i x i n g o f g r a n u l a r c a s c a d i n g s o l i d s i n c i r c u l a r c y l i n d e r s . The fo rmer a p p l i e d a s t a t i s t i c a l m o d e l l i n g approach w h i l e the l a t t e r used a d i f f u s i o n a n a l o g y . Both show t h a t the t r a n s v e r s e m i x i n g p r o c e s s i s a t l e a s t two o r d e r s o f magn i tude f a s t e r than a x i a l m i x i n g . T h e i r models were used to s tudy the k i n e t i c s o f the m i x i n g p r o c e s s as w e l l as the e f f e c t s o f the o p e r a -t i o n a l pa rame te rs on the s t a t e o f m i x e d n e s s . T h e i r r e s u l t s i n d i c a t e t h a t the m i x i n g k i n e t i c s s a t i s f y a f i r s t ^ o r d e r r a t e e q u a t i on . T h i s f i n d i n g was a l s o c o r r o b o r a t e d by the work o f 32 3 6 Lehmberg et a l . u s i n g a un ique a p p r o a c h . Heated s o l i d p a r t i c l e s were added a x i a l l y to c o l d e r bu lk s o l i d s r o t a t i n g i n a c y l i n d e r . A t h e r m o c o u p l e p l a c e d in the a c t i v e l a y e r measured the t e m p e r a t u r e o f the s o l i d s and i t s t r a n s i e n t r e s p o n s e was s u c c e s s f u l l y f i t t e d to a f i r s t - o r d e r r a t e e q u a t i on . The e f f e c t s o f m a t e r i a l and c y l i n d e r v a r i a b l e s were not i n v e s t i g a t e d by C a r l e y - M a c a u l y and D o n a l d , ' Hogg ' 3 6 or by Lehmberg et a l . T h e i r e x p e r i m e n t a l i n v e s t i g a t i o n s were a l s o r e s t r i c t e d to the c a s c a d i n g mode o f bed b e h a v i o u r . A more i m p o r t a n t drawback to t h e s e app roaches though i s t h a t l i t t l e i n f o r m a t i o n i s p r o v i d e d about the u n d e r l y i n g mechanisms o f the m i x i n g p r o c e s s . H e n c e , w h i l e the m i x i n g speed and the m i x i n g k i n e t i c s may be a d e q u a t e l y e x p r e s s e d ma thema t i -c a l l y , no i n s i g h t i s g a i n e d on the mechanisms i n v o l v e d or on the i n t e r a c t i o n o f the many v a r i a b l e s o f the s y s t e m . 2 . 3 . 3 The Mechan i cs o f R i g i d B o d i e s In a p p l y i n g the p r i n c i p l e s o f s o l i d m e c h a n i c s , the g r a n u l a r bed i s t r e a t e d as a r i g i d body and a l l the f o r c e s a c t i n g on i t a re a c c o u n t e d f o r . The r e s u l t -ant f o r c e o r the c o n f i g u r a t i o n o f the bed f o r wh ich the f o r c e po l ygon i s c l o s e d ( i . e . e q u i l i b r i u m ) i s hence 33 c a l c u l a t e d . T h i s wou ld y i e l d , f o r e x a m p l e , the l i k e l i h o o d o f a bed o f g r a n u l a r s o l i d s s l i p p i n g a g a i n s t the w a l l . However , the assump t i on o f t r e a t i n g the g r a n u l a r s o l i d s as a r i g i d - b o d y i s not t o t a l l y v a l i d and w i l l be f u r t h e r d i s c u s -sed i n C h a p t e r s 6 and 7. In t h i s s e c t i o n , o n l y a r e v i e w of t h o s e models based on t h i s method w i l l be p r e s e n t e d . The s o l i d mechan ics approach has o n l y been a t t emp ted f o r the case o f a s l i p p i n g bed . Wh i l e the f o r c e s a c t i n g on the s o l i d s bed can be e a s i l y i d e n t i f i e d as g r a v i t a t i o n a l 78 and f r i c t i o n a l , many d i f f e r e n t p o i n t s o f a p p l i c a t i o n and magn i tudes have been p r o p o s e d . In f a c t , t h e r e i s s t i l l no t a s a t i s f a c t o r y s o l u t i o n f o r the s l i p p i n g b e d . 79 In 1930 , Ugg la s e t up a f r e e body d iag ram o f . t h e s o l i d s bed w i t h the g r a v i t y f o r c e a c t i n g th rough the c e n t r e o f g r a v i t y o f the b e d , and the c e n t r i f u g a l f o r c e t h rough the c e n t r e o f r o t a t i o n o f the c y l i n d e r ( i n b a l l m i l l a p p l i c a -t i o n the c e n t r i f u g a l f o r c e i s s i g n i f i c a n t ) . He then c a l c u -l a t e d the f r i c t i o n a l f o r c e r e q u i r e d to b a l a n c e the r e s u l t a n t f o r c e due to g r a v i t y and c e n t r i f u g i n g . T h i s a n a l y s i s i s i n -c o m p l e t e , as i t does not t ake i n t o accoun t the f a c t t h a t not a l l f o r c e s a re c o i n c i d e n t , and hence a moment b a l a n c e i s a l s o nc no or\ q i n e c e s s a r y . Subsequent a t t e m p t s ' ' ' a t a n a l y z i n g the f o r c e s a c t i n g on the bed have not advanced U g g l a ' s wo rk , and 34 have e i t h e r f a l l e n s h o r t o f h i s t r e a t m e n t o r have r e p r o d u c e d 27 14 31 i t . More r e c e n t l y , R e u t e r and Cross ' have a t t emp ted to a c c o u n t f o r the moment o f the g r a v i t a t i o n a l f o r c e about the c e n t r e o f r o t a t i o n o f the c y l i n d e r . T h e i r r e s p e c t i v e m a t h e m a t i c a l e x p r e s s i o n s , wh ich d e s c r i b e t he c o n d i t i o n s under wh ich s l i p p i n g would r e s u l t , do not agree w i t h one ano the r and on c l o s e r e x a m i n a t i o n were found to c o n t a i n d e r i v a t i o n a l e r r o r s . In summary t h e r e f o r e , w h i l e the s o l i d s mechan ics ap -p roach has the p o t e n t i a l o f e l u c i d a t i n g the i n t e r d e p e n d e n c e o f the modes o f bed b e h a v i o u r , no s a t i s f a c t o r y a n a l y s i s has been a d e q u a t e l y c a r r i e d out f o r any o f the modes of bed b e h a v i o u r . 2.4 S c a l e - u p o f S o l i d s Flow i n R o t a t i n g C y l i n d e r s In the absence o f m e c h a n i s t i c m o d e l s , the d i m e n s i o n a l a n a l y s i s t e c h n i q u e has been a p p l i e d to d e s c r i b e and p r e d i c t the o c c u r r e n c e o f the v a r i o u s modes o f bed b e h a v i o u r . The d i m e n s i o n a l a n a l y s i s o f Rose and B u l l ( S e c t i o n 2 . 2 . 1 . 2 ) s u g g e s t e d the f o l l o w i n g s c a l e - u p f a c t o r s : the degree of f i l l , the c o e f f i c i e n t o f f r i c t i o n f o r the s o l i d s - w a l l i n t e r -f a c e , and the r a t i o o f c y l i n d e r d i a m e t e r to p a r t i c l e s i z e . Subsequent a p p l i c a t i o n to i n d u s t r i a l b a l l m i l l s showed good agreement w i t h t h e i r l a b o r a t o r y t r i a l s . 35 For s l u m p i n g and r o l l i n g , no s c a l e - u p f a c t o r s a re s p e c i f i c a l l y s u g g e s t e d . I n s t e a d , bed b e h a v i o u r s i m i l i t u d e ( has been g e n e r a l l y t i e d to the s c a l e - u p of the c y l i n d e r d i a m e t e r , o f wh ich t h e r e a re two s c h o o l s of t h o u g h t . The 46 7 3 f i r s t i s to a p p l y equa l p e r i p h e r a l s p e e d s . ' T h i s s c a l i n g c r i t e r i o n w i l l o b v i o u s l y not h o l d a t c e n t r i f u g i n g speeds and i s thus not r e l i a b l e . A more w i d e l y a c c e p t e d c r i t e r i o n i s the Froude number, a r a t i o o f i n e r t i a l to ? 44 fil 7fi 77 ft? g r a v i t a t i o n a l f o r c e s ' ' ' ' ' wh ich i s g i v e n by : Fr = A ( 2 - 7 ) g o r c 2 F r = Tr 900 n 2 R ( 2 . 8 ) The j u s t i f i c a t i o n o f the Froude number c r i t e r i o n i s based on the c e n t r i f u g i n g mode o f bed b e h a v i o u r where the g r a v i t a -t i o n a l f o r c e i s equa l to the c e n t r i f u g a l f o r c e . T h u s : F r c = 1 ( 2 . 9 ) For Froude number s i m i l a r i t y : ( F r ) M = ( F r ) p . ( 2 . 1 0 ) D i v i d i n g e q u a t i o n ( 2 . 1 0 ) by ( 2 . 9 ) , and s u b s t i t u t i n g e q u a t i o n s ( 2 . 6 ) and ( 2 . 8 ) then s i m p l i f y i n g : 36 ( 2 . 1 1 ) M u l t i p l y i n g both s i d e s by one : ( 2 . 1 2 ) Hence , a t equal f r a c t i o n o f c r i t i c a l speed or a t equa l f r a c t i o n o f p e r i p h e r a l c r i t i c a l s p e e d , the bed b e h a v i o u r would be e x p e c t e d to be e q u a l . E q u a t i o n s ( 2 . 1 1 ) and ( 2 . 1 2 ) a re o t h e r forms o f the Froude number c r i t e r i o n wh ich a l s o c o n t r a d i c t s the p e r i p h e r a l speed c r i t e r i o n . I t i s i m p o r t a n t to r e - e m p h a s i z e t h a t t h i s c r i t e r i o n has not been t e s t e d f o r d i a m e t e r e f f e c t s f o r any o f the modes o f bed b e h a v i o u r , and i n the l i g h t o f t h i s work has been found not t o be s u f -f i c i e n t f o r bed b e h a v i o u r s i m i l i t u d e . 2 .5 Summary A d e s c r i p t i o n o f s l i p p i n g , s l ump ing and r o l l i n g has been p r e s e n t e d and a r e v i e w o f the v a r i e d a t t emp ts a t c h a r a c t e r i z i n g the bed mot ion has been made. The degree o f f i l l and some of the p h y s i c a l p r o p e r t i e s o f the m a t e r i a l : s t a t i c and dynamic a n g l e s o f r epose c o e f f i c i e n t of f r i c t i o n o f the s o l i d s - w a l l i n t e r f a c e and p a r t i c l e s i z e , have been found to a f f e c t t r a n s v e r s e bed m o t i o n . Wh i l e a comprehens i ve 37 d e s c r i p t i o n o f s l i p p i n g , s l ump ing and r o l l i n g can be deduced f rom the l i t e r a t u r e , l i t t l e i s u n d e r s t o o d about t h e i r o c c u r -r ence and i n t e r r e l a t i o n s h i p . There i s l i t t l e agreement i n the l i t e r a t u r e on the c o n d i t i o n s under wh ich a bed changes f rom one t ype o f bed mot ion to a n o t h e r . Three m o d e l l i n g approaches of bed be-h a v i o u r t h a t have been used a r e : the a c t i v e l a y e r t h i c k n e s s , the t r a n s v e r s e m i x i n g and the mechan ics o f r i g i d b o d i e s . These app roaches have so f a r y i e l d e d l i t t l e fundamenta l u n d e r s t a n d i n g o f the e f f e c t o f m a t e r i a l , c y l i n d e r o r o p e r a t i n g v a r i a b l e s on the t r a n s v e r s e f l o w c h a r a c t e r i s t i c s o f bu l k s o l i d s i n r o t a r y k i l n s . The Froude number has been s u g g e s t e d as a s c a l e - u p c r i t e r i o n f o r s l u m p i n g and r o l l i n g b e d s . I t has been j u s t i f i e d on the b a s i s o f c e n t r i f u g i n g c a l c u l a t i o n s but has not been v e r i f i e d . For s l i p p i n g , s c a l e - u p c r i t e r i a have been p roposed to be the c o e f f i c i e n t o f f r i c t i o n between the 54 bu lk s o l i d s and the c y l i n d e r w a l l and the degree o f f i l l . An i n depth e x p e r i m e n t a l and m a t h e m a t i c a l a n a l y s i s o f s l i p p i n g , s l u m p i n g and r o l l i n g w i l l t h e r e f o r e be c a r r i e d out i n t h i s s t u d y . The r e l a t i v e c o n t r i b u t i o n o f the m a t e r i a l , c y l i n d e r and o p e r a t i n g v a r i a b l e s w i l l be i n v e s t i g a t e d i n an a t tempt to enhance the c u r r e n t l e v e l o f u n d e r s t a n d i n g of the mechanisms i n e f f e c t . T h i s would a l s o p r o v i d e more f u l t e c h n i q u e s o f c h a r a c t e r i z i n g the f l o w o f b u l k s o l i d s r o t a r y k i l n s and o t h e r p r o c e s s a p p l i c a t i o n s . 39 C h a p t e r 3 PARTICLE CHARACTERIZATION AND DESCRIPTION OF APPARATUS 3.1 I n t r o d u c t i on A d e s c r i p t i o n o f the m a t e r i a l s and a p p a r a t u s used f o r the s t udy o f bed b e h a v i o u r i n r o t a r y k i l n s w i l l be p r e s e n t e d i n t h i s c h a p t e r . A t o t a l o f ten m a t e r i a l s were c h a r a c t e r i z e d . The bed b e h a v i o u r s t u d i e s o f t h e s e m a t e r i a l s were c a r r i e d out i n the UBC p i l o t k i l n as w e l l as i n t h r e e ba t ch r o t a t i n g c y l i n d e r s . The c h a r a c t e r i z a -t i o n o f the b e d - w a l l s t a t i c f r i c t i o n c o n d i t i o n s i s a l s o p r e s e n t e d . 3 . 2 P a r t i c l e C h a r a c t e r i z a t i o n The p a r t i c l e c h a r a c t e r i z a t i o n was c a r r i e d out by measur ing o r i d e n t i f y i n g the f o l l o w i n g p r o p e r t i e s : p a r t i c l e s i z e , p a r t i c l e s h a p e , v o i d f r a c t i o n and s t a t i c a n g l e o f r e p o s e . The t e c h n i q u e s a p p l i e d w i l l be p r e s e n t e d f o r each r e s p e c t i v e p r o p e r t y . 3.2 .1 P a r t i c l e S i z e The p a r t i c l e s i z e a n a l y s i s was unde r taken i n 40 s t a n d a r d w i r e c l o t h s i e v e s u s i n g the US S t a n d a r d S i e v e p o on S e r i e s . ' The s a m p l i n g and s i e v i n g methods used were i n a c c o r d w i t h ASTM p roposed p r o c e d u r e s . ^ 5 At l e a s t t h r e e samples were r i f f l e d f rom i n d u s t r i -a l l y bagged m a t e r i a l s . The samples were s i e v e d i n s t a n d a r d 203 mm t e s t s i e v e s and p l a c e d i n a Ro-Tap t e s t s i e v e s h a k e r f o r 20 - 25 m i n u t e s . The w e i g h t r e t a i n e d on each s c r e e n was r e c o r d e d and an ave rage s i z e d i s t r i b u t i o n was c a l c u l a t e d on the b a s i s o f a l l the samples s i e v e d . G r a p h i c a l r e p r e -s e n t a t i o n s o f the d i s t r i b u t i o n s o f a l l m a t e r i a l s t e s t e d a re shown i n F i g u r e 3.1 and the d e t a i l e d s i e v e a n a l y s i s r e s u l t s f o r each m a t e r i a l a re l i s t e d i n T a b l e s A . l to A .10 (Append ix A ) . The ave rage p a r t i c l e s i z e f o r each m a t e r i a l wh ich was c a l c u l a t e d u s i n g the w e i g h t e d a r i t h m e t i c mean method i s a l s o l i s t e d i n T a b l e I I . F i n a l l y , the s c r e e n a n a l y s i s da ta was p l o t t e d on a l o g p r o b a b i l i t y p l o t , as shown i n F i g u r e 3 . 2 . Wi th the e x c e p t i o n o f sand C, a l l the m a t e r i a l s used had ve ry narrow 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 i g u r e 3 . 1 ) . Sand C was a l s o o b s e r v e d to have a m u l t i m o d a l d i s t r i - • 86-89 b u t i o n " as may be seen i n F i g u r e 3 . 2 . L imes tone D a l s o seems to have a s l i g h t m u l t i m o d a l d i s t r i b u t i o n . Both n i c k e l o x i d e and l i m e s t o n e F appear to have l o g - n o r m a l 170 Equivalent sieve mesh numbers (U.S. series) 70 3 5 - 1 6 8 5/16 5 / 8 5 0 I I I I 3 0 0 Iron oxide A Nickel ox ide • S a n d B • S a n d C • V I I I I I ~i r sz. CP 4 0 2 0 Gravel O L imes tone B O II C A D • E • n F O \ o1 A A" — A . \ o I I n l I 1 I 100 1000 Particle size (ym) 10,000 Figure 3.1 Part icle size distributions of materials tested. TABLE II Summary of Resu l ts f o r the P a r t i c l e C h a r a c t e r i z a t i o n of the M a t e r i a l s Average Size Particle Particle Loose Bulk Dense Bulk Loose Void Dense Void Static Material Shape Density Oensity Density Fraction Fraction Angle of Repose (mm) (kg/m3) (kg/mJ) (kg/mJ) (degrees) Gravel . 3.0 Angular 2,870 1,560 1,690 0.46 0.41 , 40.7 Iron Oxide 11.6 Spherical - - - - 31.5 Limestone B 4.3 Irregular 2,700 1,450 1,610 0.46 0.40 40.3 Limestone C 1.5 Irregular 2,690 1,520 1,600 0.43 0.40 37.8 Limestone D 0.58 Irregular 2,680 1,490 1,570 0.44 0.41 35.6 Limestone E 0.54 Equi-dimensional 2,670 1,680 1,860 0.37 0.30 38.6 Limestone F 8.1 Angular 2,690 - - - - 42.8 Nickel Oxide 4.9 Spherical - 870 900 - - 32.5 Sand B 0.50 Nodular 2,660 1,640 1.740 0.38 0.35 33.4 Sand C 0.23 Nodular 2.730 1.710 1.810 0.37 0.34 32.2 - p i IX) 43 10 E o 99.99 O Limestone B A " C • •' D » E O F O Gravel O S a n d B • C 4^ Nickel oxide X Iron oxide ^5/8 J3/8" O =1 9 8 9 0 .50 10 Cumulative % retained .2 a) tn 7 aj E C .e tn <u 18 E CD > CU 'tn , 40 Ion O > '5 cr Ul 70 H l 2 0 2 0 0 0.01 Figure 3.2 Log p robab i l i t y p lo t of .screen ana lys is resu l t s of mater ia ls tes ted. 44 8 6 - 91 d i s t r i b u t i o n s as i n d i c a t e d by t h e i r s t r a i g h t l i n e p l o t s i n F i g u r e 3 . 2 . The l o g - n o r m a l d i s t r i b u t i o n p l o t s o f the o t h e r m a t e r i a l s t e s t e d i n d i c a t e t h a t they r e s u l t f rom a c o a r s e s i z e s e p a r a t i o n p r o c e s s . T h i s c o n c l u s i o n was c o n -f i r m e d f o r the l i m e s t o n e s and the g r a v e l by the r e s p e c t i v e s u p p l i e r s . 3 . 2 . 2 P a r t i c l e Shape The most t ime consuming and d i f f i c u l t p a r a -meter o f a p a r t i c l e to be q u a n t i t a t i v e l y measured i s the 92 p a r t i c l e s h a p e . S e v e r a l i n v e s t i g a t o r s have a t t emp ted to 9 3-96 d e v e l o p t h e o r e t i c a l models f o r d e s c r i b i n g i t . The c o m p l e x i t y o f the e x p r e s s i o n s and t h e i r l a c k o f immedia te p r a c t i c a l i t y i n q u a n t i f y i n g the p a r t i c l e shape has p r e -ven ted o t h e r i n v e s t i g a t o r s f rom a p p l y i n g them. E m p i r i c a l t e c h n i q u e s have a l s o been used to c h a r a c t e r i z e p a r t i c l e s h a p e s . These have i n c l u d e d f l o w p r o p e r t y measurements 97 98 99 f rom b i n s , s i e v e a n a l y s i s ' and the r a t i o o f p a r t i c l e o 7 s i z e s d e t e r m i n e d by two d i f f e r e n t s i z i n g methods . F i n a l l y , shape measurements made d i r e c t l y on the p a r t i c l e s i n q u e s t i o n have g e n e r a l l y been p r e f e r r e d , p a r t i c u l a r l y the shape f a c t o r s termed " s p h e r i c i t y " 8 9 ' 9 0 , 1 0 0 ' 1 0 1 and the "Heywood : r a t i o s . " 1 0 2 ' 1 0 3 The " s p h e r i c i t y " , wh ich has been a p p l i e d where the 45 s u r f a c e a rea o f the p a r t i c l e i s o f i m p o r t a n c e , may be measured s e v e r a l ways . The e a s i e s t i n v o l v e s measurement o f the p o r o s i t y o f a randomly packed bed o f u n i f o r m l y s i z e d 1 04 p a r t i c l e s . However , s i n c e the m a t e r i a l s used i n t h i s s tudy have s i z e d i s t r i b u t i o n s , t h e i r s p h e r i c i t y canno t be measured u s i n g t h i s t e c h n i q u e . Ano the r t e c h n i q u e i n v o l v e s p a r t i c l e s u r f a c e a rea and volume measurements wh ich a re ve ry t e d i o u s and t ime c o n s u m i n g . The measurement o f the "Heywood r a t i o s " would a l s o be as i n v o l v e d as the l a t t e r measurement o f " s p h e r i c i t y " , p a r t i c u l a r l y when d e a l i n g w i t h i n d u s t r i a l p r o d u c t s . N o t w i t h s t a n d i n g , i t has y e t to be f i r m l y e s t a b l i s h e d i n the l i t e r a t u r e whether e i t h e r the " s p h e r i c i t y " o r the "Heywood r a t i o s " r e f l e c t the e f f e c t o f p a r t i c l e shape on t he f l o w a b i l i t y o f p a r t i c l e s , a l t h o u g h 9 7 the r e s u l t s o f S u t h e r l a n d and Nea le seem to s u p p o r t the use o f " s p h e r i c i t y " . The q u a n t i t a t i v e c h a r a c t e r i z a t i o n o f p a r t i c l e shape was not pu rsued i n t h i s s t u d y , however the t r e a t m e n t o f p a r t i c l e shape w i l l be f u r t h e r d i s c u s s e d i n C h a p t e r s 4 and 7. The q u a l i t a t i v e d e s c r i p t i o n o f p a r t i c l e shape o f the m a t e r i a l s used was c a r r i e d ou t i n a c c o r d a n c e w i t h the d e f i n i t i o n s o f p a r t i c l e shape p r e s e n t e d i n the B r i t i s h 1 05 S t a n d a r d 2955. They were more a p p l i c a b l e to the < m a t e r i a l s used i n . t h i s s t u d y than the c l a s s i f i c a t i o n 46 106 p roposed by Hausner f o r powdered m e t a l s . The c l a s s i f i c a -t i o n o f the m a t e r i a l s used i s l i s t e d i n T a b l e I I , w h i l e pho tog raphs o f r e p r e s e n t a t i v e samples a re shown i n F i g u r e s 3 .3 to 3 . 1 2 . O b s e r v a t i o n s o f the p a r t i c l e s b e f o r e and a f t e r t u m b l i n g i n the r o t a t i n g c y l i n d e r s showed no d e t e c t a b l e d e g r a d a t i o n or change i n p a r t i c l e shape . 3 . 2 . 3 V o i d F r a c t i o n Whenever a q u a n t i t y o f g r a n u l a r m a t e r i a l i s p l a c e d i n a g i v e n vo lume , the r e s u l t i n g d e n s i t y o f the m a t e r i a l i s l e s s than the a p p a r e n t p a r t i c l e d e n s i t y , and i s termed the b u l k d e n s i t y . The c l o s e r the v a l u e o f the bu l k d e n s i t y i s to t h a t o f the appa ren t p a r t i c l e d e n s i t y , the dense r the p a c k i n g c h a r a c t e r i s t i c s o f the p a r t i c l e s and the s m a l l e r the i n t e r p a r t i c l e v o i d a g e . .Many a t t emp ts have been;, made t o . • t h e o r e t i c a l l y c a l c u l a t e the v o i d f r a c t i o n f o r a g i v e n m a t e r i a l o r f o r a m i x t u r e o f m a t e r i a l s . However , f o r i n d u s t r i a l p r o d u c t s w i t h 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 d i r e c t e x p e r i m e n t a l app roach i s s t i l l the most r e l i a b l e . The v o i d f r a c t i o n can be c a l c u l a t e d f rom e x p e r i m e n t a l r e s u l t s as f o l 1 o w s : 1 P B ( 3 . 1 ) Pp The bu l k d e n s i t y o f the m a t e r i a l can be measured u s i n g F i g u r e 3.3 A n g u l a r g r a v e l w i t h 3mm average p a r t i c l e s i z e ( 1 d i v i s i o n = 1 mm) F i g u r e 3.4 S p h e r i c a l i r o n o x i d e w i t h 11.6 mm average p a r t i c l e s i z e (1 d i v i s i o n = 1 mm) F i g u r e 3.5 I r r e g u l a r l i m e s t o n e B w i t h 4.3 mm a v e r a g e p a r t i c l e s i z e (1 d i v i s i o n = 1 mm) F i g u r e 3.6 I r r e g u l a r l i m e s t o n e C w i t h 1.5 mm a v e r a g e p a r t i c l e s i z e (1 d i v i s i o n = 1 mm) Figure 3.7 I r regular limestone D with 0.58 mm average pa r t i c l e s i ze (1 d i v i s i on = 1 mm) Figure 3.8 Equi-dimensional limestone E with 0.54 mm average p a r t i c l e s i ze (1 d i v i s ion = 1 mm) 50 Figure 3.9 Angular limestone F with 8.1 mm average p a r t i c l e s i ze (1 d i v i s ion = 1 mm) Figure 3.10 Spherical n ickel oxide with 4.9 mm average pa r t i c l e s i ze (1 d i v i s ion = 1 mm) Figure 3.12 Nodular sand C with 0.23 mm average _2 pa r t i c l e s i ze (1 d i v i s ion = 10 mm) 52 the method p roposed by Eastwood e t a l , ^ ° 7 who sugges t the use o f measu r i ng c o n t a i n e r s w i t h a d i a m e t e r 50 to 100 t imes t h a t o f the ave rage p a r t i c l e s i z e o f the s o l i d s to be measured . Hence to m i n i m i z e w a l l e f f e c t s , 1 0 7 ' 1 0 ^ two 0 .285m d i a m e t e r c o n t a i n e r s were u s e d . The i r o n o x i d e and l i m e s t o n e F were not t e s t e d as t h e i r w a l l e f f e c t s would have been s i g n i f i c a n t . I n i t i a l l y , the volume of one of the c o n t a i n e r s was measured w i t h wa te r and was used as a r e f e r e n c e . To measure the bu l k d e n s i t y o f a m a t e r i a l , excess s o l i d s were p l a c e d i n one o f the c o n t a i n e r s . I t was c o v e r e d by the second c o n t a i n e r and both were i n v e r t e d s e v e r a l t imes i n t o each o t h e r . The c o n t e n t s o f the r e f e r e n c e c o n t a i n e r were l e v e l l e d , we ighed and the random l o o s e b u l k d e n s i t y c a l c u l a -t e d . The random dense bu l k d e n s i t y was o b t a i n e d i n a s i m i -l a r manner excep t r a t h e r than i n v e r t i n g the c o n t a i n e r , i t was tapped up to ten t imes on the f l o o r . S e v e r a l measure -ments o f each m a t e r i a l were taken f o r the random l o o s e and random dense b u l k d e n s i t i e s . The ave raged r e s u l t s a re l i s t e d i n T a b l e 11 . The a p p a r e n t p a r t i c l e d e n s i t i e s were measured u s i n g a p y c n o m e t r i c t e c h n i q u e i n a 1 a v o l u m e t r i c f l a s k . The r e s u l t s , a l o n g w i t h the c a l c u l a t e d v o i d f r a c t i o n s , a re a l s o l i s t e d i n T a b l e I I . In o r d e r to f u l l y c h a r a c t e r i z e the v o i d f r a c t i o n o f the m a t e r i a l s to be used i n the s e g r e g a t i o n s t u d i e s , the v o i d f r a c t i o n as a f u n c t i o n o f c o m p o s i t i o n was measured f o r sands B and C and l i m e s t o n e s B and E. The appa ren t p a r t i -c l e d e n s i t y o f each m i x t u r e was c a l c u l a t e d u s i n g the we igh ted a p p a r e n t p a r t i c l e d e n s i t i e s o f the i n d i v i d u a l components f o r a g i v e n m i x t u r e c o m p o s i t i o n . Only the random l o o s e v o i d f r a c t i o n was measured f o r the m i x t u r e s . The r e s u l t s a re shown i n F i g u r e 3 . 1 3 . The p a r t i c l e shapes o f the m a t e r i a l s used i n t h i s s tudy d i f f e r e d f rom the g l a s s s p h e r e s o f Eastwood e t a l . A l s o the i n d u s t r i a l m a t e r i a l s had 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 , w h i l e i n the s t udy o f Eastwood e t a l . . , m i x t u r e s o f two e q u a l l y s i z e d p a r t i c l e s had been t e s t e d . D e s p i t e t h e s e d i f f e r e n c e s the o b s e r v a t i o n s o f the v o i d f r a c t i o n -c o m p o s i t i o n d iag ram o b t a i n e d i n t h i s s tudy agree w e l l w i t h t h o s e o f Eastwood e t a l . Name ly , as the ave rage p a r t i c l e s i z e o f the two components i n c r e a s e , the maximum bed c o n t r a c t i o n , Ae , o f the r e s u l t i n g m i x t u r e i n c r e a s e s . max 3 A c o a r s e to f i n e ave rage s i z e r a t i o o f 8 .0 f o r the l i m e -s t o n e s y i e l d e d a Ae o f 0 . 1 1 4 , w h i l e f o r the sands Ae „ w max max was 0 .054 f o r a c o a r s e to f i n e ave rage p a r t i c l e s i z e r a t i o o f 2 . 2 . The second c o n c l u s i o n o f Eastwood e t a l . was t h a t the maximum bed c o n t r a c t i o n o c c u r s a t a m i x t u r e c o m p o s i t i o n o f 50-70% o f the l a r g e r component . From F i g u r e 3 .13 i t appears t h a t t h i s c o n c l u s i o n i s i n agreement w i t h the C o a r s e F i n e s o L i m e s t o n e B L i m e s t o n e E A S a n d B S a n d C 0 2 0 4 0 6 0 Weight % f iner component 100 Figure 3.13 Void fraction composition diagram for limestone B and E and sands B and C. 55 1 i m e s t o n e r e s u l t s but not w i t h t h o s e f o r the s a n d s . On c l o s e r e x a m i n a t i o n though i t i s found t h a t a t a m i x t u r e c o m p o s i t i o n o f 40% f i n e s , the bed c o n t r a c t i o n i s 0 . 0 4 6 ; w h i l e a t 60% f i n e s c o m p o s i t i o n , i t i s 0 . 0 5 4 . T h i s d i f f e r e n c e i n Ae i s w i t h i n the measured w e i g h i n g a c c u r a c y o f +1%. The max 3 3 v o i d f r a c t i o n o f m i x t u r e s o f g r a n u l a r s o l i d s w i l l a g a i n be d i s c u s s e d i n C h a p t e r 5 w i t h r e f e r e n c e to the s e g r e g a t i o n s t u d i e s . 3 . 2 . 4 Ang le o f Repose S e v e r a l t ypes o f a n g l e s o f repose have been o b s e r v e d : the d r a i n e d , the p i l e d and the dynamic a n g l e s o f r e p o s e . In t h i s s e c t i o n , the measurements r e p o r t e d c o r r respond to the p i l e d or s t a t i c a n g l e o f r e p o s e . There a re t h r e e methods o f measu r ing the s t a t i c a n g l e o f r e p o s e : the f i x e d c o n e , the t i l t i n g box and the r o t a t i n g c y l i n d e r methods ( F i g u r e 3 . 1 4 ) . 7 0 > 9 1 > 9 2 > 1 0 6 • 1 0 8 - 1 1 0 In the f i x e d - c o n e method , m a t e r i a l i s passed th rough a f u n n e l o r p i p e and p i l e d on to a f l a t s u r f a c e . The d i s - ; cha rge p o i n t o f s o l i d s must be r e l a t i v e l y c l o s e to the h e i g h t o f the p i l e i n o r d e r t h a t the p a r t i c l e s do not have too g r e a t a momentum when p i l e d . The a n g l e w i t h the h o r i z o n t a l i s measured when a s i g n i f i c a n t q u a n t i t y o f m a t e r i a l has been p o u r e d . ( c ) Rotary cyl inder Figure 3.14 Three methods for measuring the s ta t i c angle of repose. 57 In the t i l t e d - b o x method , m a t e r i a l i s p l a c e d i n a h o r i z o n t a l l y - p o s i t i o n e d box . The s u r f a c e o f the m a t e r i a l i s a l s o h o r i z o n t a l , hence p a r a l l e l to the base o f the box . One end o f the box i s t i l t e d upwards ve ry s l o w l y by means o f a w inch u n t i l a q u a n t i t y of m a t e r i a l s lumps to the bot tom o f the b^X. At t h i s p o i n t , t he w inch i s t u r n e d o f f and the a n g l e o f i n c l i n a t i o n of the box i s measured . S i n c e the s u r -f a c e o f the bot tom o f the box i s p a r a l l e l to the m a t e r i a l s u r f a c e a t the s t a r t o f the t e s t , the measured a n g l e o f the box i s the maximum a n g l e o f repose the m a t e r i a l had p r i o r to i t s s l u m p i n g ( i . e . the s t a t i c a n g l e o f r e p o s e ) . The t h i r d method i s s i m i l a r to the i n c l i n e d box t e c h n i q u e . S o l i d s a re p l a c e d i n s i d e a c y l i n d e r w i t h a p l e x i g l a s end p l a t e which i s s l o w l y r o t a t e d by an e l e c t r i c moto r . The i n c l i n a t i o n of the s u r f a c e o f the bed i s f o l l o w e d m a n u a l l y w i t h a l ong arm p r o t r a c t o r measu r i ng the i n c l i n a t i o n o f the bed th rough the p l e x i g l a s . " end p l a t e . The maximum i n c l i n a t i o n o f the bed i s thus r e c o r d e d as the s t a t i c a n g l e o f r epose o f the m a t e r i a l . The measured r e s u l t s f o r the t e s t i n g methods d e s c r i b e d above a re l i s t e d i n T a b l e I I I f o r some of the m a t e r i a l s u s e d . A l l da ta was measured to an a c c u r a c y o f 0 . 5 ° . I t i s seen t h a t a l l t h r e e methods y i e l d r e s u l t s t h a t a re c o m p a r a b l e . 58 T A B L E ' I I I S t a t i c Ang le o f Repose Measurements Material Method Angle of Repose (degrees) Standard Deviation (degrees) Number of Observa-t ions Gravel Cone 40.7 0.8 6 Incl ined Box 40.7 0.7 6 Rotary Cyl inder (D = Im) 40.7 0.3 5 Limestone B Cone 40.3 1.5 9 Incl ined Box 41.7 0.9 6 Rotary Cyl inder (D = 0.4m) 41.6 1.2 11 Limestone C Cone 37.8 1.3 17 Rotary Cyl inder (D = 0.4m) 38.1 0.6 11 Rotary Cyl inder (D = Im) 37.6 0.5 8 Nickel Oxide Cone 32.5 1.6 7 Incl ined Box 33.3 0.6 6 Rotary Cyl inder (D = 0.4m) 31.9 1.2 6 59 However , i n a tho rough d i s c u s s i o n of t h e s e t h r e e me thods , 1 08 R i c h a r d s and Brown f a v o u r e d the cone method because o f d i m e n s i o n a l c o n s t r a i n t s o f f e r e d by the v e s s e l s o f the o t h e r two t e c h n i q u e s . The t h r e e d i m e n s i o n a l n a t u r e o f the f o r c e d i s t r i b u t i o n i n the cone as opposed to the two d i m e n s i o n a l f o r c e d i s t r i b u t i o n i n the o t h e r two methods was a l s o c i t e d as an a d v a n t a g e . D e s p i t e R i c h a r d s and B rown ' s d i s c u s s i o n , l i t t l e i s u n d e r s t o o d o f the d i f f e r e n c e s between the t h r e e measu r i ng t e c h n i q u e s . Due to i t s w ide a p p l i c a t i o n , the cone method was used i n t h i s s tudy f o r a l l m a t e r i a l s . The r e s u l t s a re l i s t e d i n T a b l e I I . 3 .3 D e s c r i p t i o n s of the R o t a r y K i l n and the R o t a r y C y l i nders 3 .3 .1 The P i l o t R o t a r y K i l n The r o t a r y k i l n used f o r the bed b e h a v i o u r o b s e r v a t i o n s has been d e s c r i b e d i n d e t a i l by Brimacombe and W a t k i n s o n 1 1 1 and i s shown i n F i g u r e 3 . 1 5 . I t i s s u f f i c i e n t here to s t a t e t h a t the k i l n has an i n t e r n a l d i a m e t e r o f 0 .4 m and i s 5 .5 m i n l e n g t h w i t h v a r i a b l e i n c l i n a t i o n and a c a s t a b l e r e f r a c t o r y l i n i n g . C a s t a b l e dams of v a r i a b l e geometry can be p l a c e d a t the s o l i d s d i s c h a r g e e n d . The r o t a t i o n a l speed as w e l l as the s o l i d s f eed r a t e a re a l s o v a r i a b l e . A r u l e r we lded p e r p e n d i c u l a r l y to a l o n g 61 h o r i z o n t a l p i p e was i n t r o d u c e d a x i a l l y i n t o the k i l n t o manua l l y measure the bed depth a l o n g the k i l n l e n g t h . 3 . 3 . 2 R o t a r y C y l i n d e r s Three b a t c h r o t a t i n g c y l i n d e r s were c o n -s t r u c t e d f o r the d e t a i l e d bed b e h a v i o u r o b s e r v a t i o n s . T h e i r d i m e n s i o n s a re l i s t e d i n T a b l e IV . Both c y l i n d e r s A and B were c o n s t r u c t e d w i t h the same i n t e r n a l d i a m e t e r as the p i l o t k i l n . T h i s a l l o w e d da ta o b t a i n e d on the r o t a r y c y l i n d e r s to be r e l a t e d to a c o n t i n u o u s o p e r a t i o n a n d w i l l be f u r t h e r d i s c u s s e d i n C h a p t e r 4 . C y l i n d e r A was c o n s t r u c t e d f rom 9 .5 mm t h i c k m i l d s t e e l p l a t e , whereas c y l i n d e r s B and C were c o n s t r u c t e d f rom Perma T u b e , * a d i m e n s i o n a l l y s t a b l e c a r d b o a r d . To ensu re a h igh f r i c t i o n a l c o e f f i c i e n t between the c y l i n d e r w a l l and the s o l i d s , as w e l l as to ensure s i m i l a r w a l l c o n -d i t i o n s i n a l l t h r e e c y l i n d e r s , a 24 -3 g r i t t ype E s i l i c o n c a r b i d e a b r a s i v e paper was g l u e d to the i n s i d e w a l l s o f the t h r e e c y l i n d e r s . One o f the two c y l i n d e r end p l a t e s was made o f p l e x i g l a s : and was p l a c e d on the f r o n t end o f the c y l i n d e r thus e n a b l i n g the c r o s s s e c t i o n o f the bed to be *Perma Tubes L t d . , 4751 V a n g u a r d , R ichmond, B . C . , Canada . TABLE IV Dimensions of Rotary Cylinders Cylinder Inside Diameter Cm;) Length (m) A 0.4 0.46 B 0.4 0.86 C 1.0 0.41 63 v iewed and p h o t o g r a p h e d . A h o l e was cut out i n the c e n t r e o f the end p l a t e s to a l l o w easy a c c e s s to the s o l i d s bed f o r c h a r g i n g and d i s c h a r g i n g . The c y l i n d e r s were s u p p o r t e d on two f r i c t i o n a l r o l l s , o f wh ich one was the d r i v e r o l l . These r o l l s were b o l t e d to a s u p p o r t s t a n d on wh ich two t h r u s t b e a r i n g s were a l s o a t t a c h e d . At the v i e w i n g end o f the c y l i n d e r , a s m a l l v a r i a b l e h e i g h t p l a t f o r m was a t t a c h e d to a l l o w the measure -ments o f the i n c l i n a t i o n o f the b e d . The h o r i z o n t a l p o s i t i o n o f t h i s p l a t f o r m was a d j u s t a b l e to ensu re t h a t i t was h o r i -z o n t a l l y l e v e l whenever a n g u l a r measurements were made. The c y l i n d e r s were r o t a t e d by a 1/6 HP GE e l e c t r i c motor a t t a c h e d to a Ze ro -Max D r i v e Power B l o c k , Model 2 2 , w i t h a speed range f rom 0 to 400 r e v o l u t i o n s per m i n u t e . The speed was s tepped down by means o f a p u l l e y ar rangement a t t a c h e d to the d r i v e r o l l e r . The maximum r o t a t i o n a l speed i m p a r t e d to the r o t a t i n g c y l i n d e r was about seven r e v o l u t i o n s per m inu te f o r c y l i n d e r s A and B. The r o t a t i o n a l speed of the c y l i n d e r was measured d i r e c t l y o f f the c y l i n d e r o u t e r c i r -cumfe rence u s i n g a t achome te r w i t h an a c c u r a c y o f 0 .05 r e v o l u t i o n s per minu te and a maximum range o f 100 r e v o l u t i o n s per m i n u t e . A pho tog raph of the equipment i s shown i n F i g u r e 3 . 1 6 . 54 Figure 3.16 Rotary cylinder A (0.4 m ID x 0.46 m L) set on rol ls . 65 3.4 S o l i d s - W a l l F r i c t i o n A n g l e S e v e r a l methods have been s u g g e s t e d f o r measu r i ng the 81 1 1 2 - 1 1 6 c o e f f i c i e n t o f f r i c t i o n o f an a b r a s i v e s u r f a c e . ' 81 For e x a m p l e , H a l b a r t and Freymann s u g g e s t a f r i c t i o n c o -e f f i c i e n t based on the p a c k i n g d e n s i t y o f the m a t e r i a l i n c o n t a c t w i t h the w a l l . However each o f t h e s e methods c h a r a c t e r i z e s o n l y the s u r f a c e o f the w a l l o r s o l i d s , w h i l e the c o m b i n a t i o n o f the m a t e r i a l and w a l l s u r f a c e s would not be a c c o u n t e d f o r . O the r s u r f a c e roughness measu r i ng t e c h n i -ques must t h e r e f o r e be p u r s u e d . To measure the s o l i d s - w a l l f r i c t i o n c o e f f i c i e n t , Rose 54 and B l u n t d e v e l o p e d t h e i r own e x p e r i m e n t a l method. A f l a t s u r f a c e was p r e p a r e d r e p l i c a t i n g the c y l i n d e r w a l l s u r f a c e . A b o t t o m l e s s box was p l a c e d on top o f t h i s s u r f a c e i n a h o r i z o n t a l p o s i t i o n and was f i l l e d w i t h the m a t e r i a l o f i n t e r e s t . A s p r i n g l o a d was used to push the box u n t i l the l a t t e r s l i p p e d a g a i n s t the f l a t s u r f a c e . The w a l l - s o l i d s c o e f f i c i e n t o f f r i c t i o n was then c a l c u l a t e d u s i n g a f o r c e b a l a n c e . Conrad e t a l . ^ ^ used a d i f f e r e n t t e c h n i q u e . A g i v e n q u a n t i t y o f the m a t e r i a l to be s t u d i e d was s p r i n k l e d onto a g l a s s p l a t e , wh ich was s l o w l y r o t a t e d f rom a h o r i z o n t a l p o s i t i o n . The a n g l e of i n c l i n a t i o n o f the p l a n e was r e c o r d e d 66 when a l l o f the m a t e r i a l s l i d o f f i t s i m u l t a n e o u s l y . The p a r t i c l e s i z e and the q u a n t i t y o f s o l i d s p l a c e d on the g l a s s p l a t e were v a r i e d . T h e i r r e s u l t s r e v e a l e d t h a t f o r a m a t e r i a l h a v i n g a p a r t i c l e s i z e g r e a t e r than 0 .25 mm, the w a l l - s o l i d s f r i c t i o n c o e f f i c i e n t was i ndependen t o f the mass p l a c e d on the f l a t s u r f a c e , and t h a t the i n t e r - p a r t i c l e c o h e s i v e f o r c e was n e g l i g i b l e . T h e r e f o r e , the c o e f f i c i e n t o f f r i c t i o n f o r the w a l l - s o l i d s . ' s u r f ace was s i m p l y g i v e n by : u w / s = tan <j>s ( 3 . 2 ) The method o f Conrad et a l . i s much s i m p l e r than t h a t o f Rose and B l u n t and was adap ted f o r c h a r a c t e r i z i n g the w a l l - s o l i d s f r i c t i o n c o e f f i c i e n t i n t h i s s t u d y . The e x p e r i -menta l s e t up u s e d , shown s c h e m a t i c a l l y i n F i g u r e 3 . 1 7 , c o n s i s t e d of a f l a t wooden board to wh ich the a b r a s i v e paper used f o r the i n s i d e c y l i n d e r s u r f a c e was s e c u r e d . The g r a n u l a r m a t e r i a l s o f i n t e r e s t were e p o x i e d onto c a r d b o a r d b a c k i n g s wh ich were g l u e d to a s m a l l w e i g h t to ensu re good c o n t a c t between the s o l i d s and the a b r a s i v e p a p e r . The board was r a i s e d by a v a r i a b l e speed e l e c t r i c motor (see S e c t i o n 4 . 2 . 2 ) a t a maximum l i f t i n g r a t e o f 0 .17 m/min , u n t i l the g r a n u l a r s o l i d s s l i p p e d . The a n g l e o f i n c l i n a -t i o n o f the board was measured to w i t h i n 0 . 5 ° . The ave rage s l i p p i n g a n g l e s f o r the m a t e r i a l s t e s t e d a re l i s t e d i n T a b l e V. These v a l u e s a re g r e a t e r than the s t a t i c a n g l e s of r e p o s e , g i v e n i n T a b l e I I , f o r each r e s p e c t i v e m a t e r i a l . ( 67 Winch Figure 3.17 Schematic diagram of the apparatus used for measuring the so l ids -wa l l f r i c t i o n coe f f i c i en t . 68 TABLE V A n g l e s o f S l i p Measurements M a t e r i a l Ang le o f S l i p ( d e g r e e s ) S t a n d a r d D e v i a t i o n ( d e g r e e s ) Number o f O b s e r v a -t i ons G r a v e l 48 .8 1 .2 5 Iron Ox ide 38 .9 2 .9 8 L imes tone B 4 3 . 9 1 .6 8 L imes tone C 43 .2 2.4 8 L imes tone D 40 .9 1 .3 8 L imes tone F 4 3 . 3 1 .9 8 N i c k e l Ox ide 41 .2 2 . 3 8 Sand B 38 .8 1.1 8 69 C h a p t e r 4 EXPERIMENTAL RESULTS AND ANALYSIS 4.1 I n t r o d u c t i o n The o b s e r v a t i o n s and the e x p e r i m e n t a l c h a r a c t e r i z a t i o n o f s l u m p i n g and r o l l i n g w i l l be p r e s e n t e d i n t h i s c h a p t e r . The e x p e r i m e n t s were m o s t l y c a r r i e d out on ba tch r o t a t i n g c y l i n d e r s . S i n c e r o t a r y - k i l n a p p l i c a t i o n i s the p r i m a r y t h r u s t o f t h i s r e s e a r c h , a method w i l l be p r e s e n t e d i n S e c t i o n 4 .2 to r e l a t e the r e s u l t s o b t a i n e d on the ba tch r o t a r y c y l i n d e r s to a c o n t i n u o u s r o t a r y k i l n . T h i s w i l l be f o l l o w e d by a p r e s e n t a t i o n o f the t r a n s v e r s e bed mot ion o b s e r v a t i o n s on a B e d - B e h a v i o u r Diagram wh ich was d e v e l o p e d to d e l i n e a t e the v a r i o u s modes o f bed b e h a v i o u r e n c o u n t e r e d : s l i p p i n g , s l ump ing and r o l l i n g . An i n s t r u m e n t e d t e c h n i q u e f o r the d e t e r m i n a t i o n o f bed b e h a v i o u r w i l l be a p p l i e d to an i n v e s t i g a t i o n o f the e f f e c t o f c y l i n d e r and m a t e r i a l v a r i a b l e s on the s l u m p i n g -r o l l i n g boundary . The v a r i a b l e s s t u d i e d a re c y l i n d e r w a l l e f f e c t s , p a r t i c l e s i z e , p a r t i c l e s h a p e , combined e f f e c t s o f s i z e and s h a p e , the s t a t i c a n g l e o f r e p o s e , the degree o f f i l l , the r o t a t i o n a l speed and the c y l i n d e r d i a m e t e r . 70 F i n a l l y , an a t tempt a t q u a n t i t a t i v e l y c h a r a c t e r i z i n g s l u m p i n g and r o l l i n g w i l l be p r e s e n t e d i n S e c t i o n 4 . 5 . Fo r the s l u m p i n g b e d , the upper a n g l e o f r e p o s e , the s h e a r ang le and the s l ump ing f r e q u e n c y were m e a s u r e d ; w h i l e the dynamic a n g l e o f r epose and the a c t i v e l a y e r t h i c k n e s s were r e p o r t e d f o r the r o l l i n g bed . Whenever a p p l i c a b l e , t h e s e v a r i a b l e s w i l l be r e l a t e d to the m a t e r i a l v a r i a b l e s and hence to the B e d - B e h a v i o u r D iag rams . T h i s a n a l y s i s w i l l show t h a t the t h e o r i e s a l r e a d y put f o r w a r d by p r e v i o u s worke rs ( see C h a p t e r 2) do not p r o v i d e a s a t i s f a c t o r y i n t e r -p r e t a t i o n o f the change i n bed b e h a v i o u r f rom s l u m p i n g to r o l l i n g and t h a t the q u a n t i t a t i v e c h a r a c t e r i z a t i o n methods o f the bed mot ion do not i d e n t i f y the c o n d i t i o n s f o r t h i s change . 4 .2 Bed B e h a v i o u r : C o n t i n u o u s v e r s u s Ba tch O p e r a t i o n s A s tudy o f bed b e h a v i o u r i n r o t a r y k i l n s n a t u r a l l y i m p l i e s making d i r e c t o b s e r v a t i o n s o f v a r i o u s t y p e s o f bed mot ion i n a c o n t i n u o u s p r o c e s s . Due to the l a r g e number o f v a r i a b l e s i n c o n t i n u o u s k i l n o p e r a t i o n , and due to the t ime r e q u i r e d f o r the s o l i d s d i s c h a r g e to reach s t e a d y s t a t e (3 to 7 h o u r s , depend ing on the o p e r a t i n g c o n d i t i o n s ) , i t wou ld be advan tageous to be a b l e to c a r r y out e x p e r i m e n t s on a ba t ch c y l i n d e r and r e l a t e the r e s u l t s to c o n t i n u o u s o p e r a -t i o n . The k i l n v a r i a b l e s s t u d i e d were the f e e d r a t e , 71 the r o t a t i o n a l s p e e d , t he i n c l i n a t i o n o f the k i l n , the h e i g h t of dam and the a x i a l bed depth p r o f i l e . The e f f e c t s of t e m p e r a t u r e and o f k i l n i n t e r n a l s w i l l be b r i e f l y d i s c u s s e d i n C h a p t e r s 5 and 7. 4 . 2 . 1 R e d u c t i o n o f K i l n V a r i a b l e s I t has been o b s e r v e d i n i n d u s t r i a l k i l n s t h a t changes in o p e r a t i n g v a r i a b l e s r e s u l t i n an i n c r e a s e o r a d e c r e a s e i n the r e t a i n e d volume of the k i l n ( i . e . v a r y i n g bed dep ths a l o n g the k i l n l e n g t h ) . 4 , 1 0 , 2 5 ' 5 7 ' 1 1 1 ' 1 1 8 S i n c e t h e r e i s no e x i s t i n g comprehens i ve fundamenta l e q u a t i o n to r e l a t e the bed depth p r o f i l e to the geometry and o p e r a t i n g c o n d i t i o n s o f a k i l n , i t was d e c i d e d to e x p e r i m e n t a l l y i n -v e s t i g a t e the i n t e r r e l a t i o n s h i p o f t h e s e v a r i a b l e s , u s i n g the UBC p i l o t k i l n . T h u s , f o r the f i r s t s t a g e o f the i n -v e s t i g a t i o n , subsequen t o b s e r v a t i o n s c o u l d be d i r e c t l y r e -l a t e d to the o p e r a t i n g v a r i a b l e s o f the p i l o t k i l n . The dependence o f the bed p r o f i l e on the p i l o t k i l n geometry and i t s o p e r a t i n g v a r i a b l e s was t h e r e f o r e i n v e s t i -g a t e d . Only a s y n o p s i s o f the r e s u l t s , wh ich i l l u s t r a t e the o b s e r v e d t r e n d s , w i l l be p r e s e n t e d . The e x p e r i m e n t s c o n s i s t e d o f f e e d i n g the cha rge a t a g i v e n r a t e and r o t a t i o n a l speed under v a r y i n g k i l n c o n f i g u r a t i o n s ( i . e . i n c l i n a t i o n and 72 d i s c h a r g e dam). Once the s o l i d s d i s c h a r g e r a t e had reached s t e a d y s t a t e , the bed p r o f i l e i n the k i l n was m e a s u r e d , u s i n g the r u l e r d e s c r i b e d i n S e c t i o n 3 .3 .1 wh ich was m a n u a l l y i n -s e r t e d i n t o the bed o f s o l i d s a t a g i v e n a x i a l p o s i t i o n and the depth r e c o r d e d . Thus f o r any s e t o f o p e r a t i n g c o n d i t i o n s , the bed depth p r o f i l e a l o n g the k i l n l e n g t h c o u l d be d i r e c t l y measured . The l i m e s t o n e bed p r o f i l e s ( F i g u r e s 4.1 and 4 . 2 ) were measured subsequen t to c a l c i n a t i o n e x p e r i m e n t s , w h i l e the sand e x p e r i m e n t s ( F i g u r e 4 . 3 ) were c a r r i e d out i n a c o l d t e s t . E a r l y i n t h i s e x p e r i m e n t a l c a m p a i g n , s e v e r a l bed p r o f i l e measurements were pe r fo rmed b e f o r e and a f t e r the c a l c i n a -t i o n o f l i m e s t o n e s A and B. No d i f f e r e n c e i n bed p r o f i l e s was o b s e r v e d . The e f f e c t o f k i l n i n c l i n a t i o n i s seen i n F i g u r e 4 . 1 . As the k i l n i n c l i n a t i o n i n c r e a s e s , the bed depth a t the cha rge end d e c r e a s e s , w h i l e a t the d i s c h a r g e end the d e c r e a s e i n bed depth i s not as p r o n o u n c e d . A l s o , the r e s u l t a n t bed p r o f i l e c o n t o u r s a re marked l y d i f f e r e n t . I l l u s t r a t e d i n F i g u r e 4 . 2 i s the e f f e c t o f k i l n r o t a t i o n a l s p e e d . In the o p e r a t i n g range chosen f o r l i m e s t o n e B, a l i n e a r bed p r o f i l e r e s u l t e d f o r the t h r e e r o t a t i o n a l speeds t e s t e d . The o n l y obse rved e f f e c t s a re t h a t a t a g i v e n a x i a l l o c a t i o n the Kiln length ( f t ) 0 8 0.12h 0.09 CL CD "O 0 0.06 X3 CD CD 0.03 Feed rate R P M Incl inat ion Dam Run ( k g / h r ) ( r /min ) (degree) t y p e •oo 7 5 7 5 7 5 3 3 3 0.7 1.2 2.4 C C C 163 161 162 Figure 4 . 1 2 4 16 9 .o length (m) Bed depth p ro f i l e for limestone A in the UBC p i l o t k i l n , i l l u s t r a t i n g the e f fec t of k i l n i n c l i n a t i o n . 0 0.12 r 0.09 2" 0.061 XJ cu CD 0.03 0 0 F i g u r e 4 . 2 Ki ln length (ft ) 8 _ • — • • L i m e s t o n e F e e d rate R P M Incl inat ion D a m R u n ( k g / h r ) ( r / m i n ) ( d e g r e e ) t y p e • 4 0 0.7 5 1.2 C 8 9 , 9 4 0 4 0 1.5 1.2 C 7 6 , 7 7 , 9 7 A 4 0 Z 2 5 1.2 C 8 8 2 4 16 I 2 3 . 4 Ki ln length ( m) Bed depth p ro f i l e for limestone B in the UBC p i l o t k i l n , i l l u s t r a t i n g the e f fec t of k i l n rotat ional speed. Figure 4.3 Bed depth p ro f i l e for sand A in the UBC p i l o t k i l n , i l l u s t r a t i n g the e f fec t of k i l n dams and so l ids feed rate. 76 bed depth d e c r e a s e s as the r o t a t i o n a l speed i n c r e a s e s and t h a t the s l o p e o f the bed p r o f i l e s i n c r e a s e s w i t h i n c r e a s i n g r o t a t i o n a l s p e e d . F i n a l l y , the e f f e c t s o f f eed r a t e and d i s c h a r g e dam h e i g h t a re i l l u s t r a t e d i n F i g u r e 4 . 3 u s i n g sand A. An i n c r e a s e i n bed depth i s o b s e r v e d as a r e s u l t o f i n c r e a s e s i n both f eed r a t e and dam h e i g h t . T h e r e f o r e , changes i n the k i l n geometry o r i n i t s o p e r a t i n g v a r i a b l e s r e s u l t i n a change in the bed depth p r o f i l e o f the k i l n . Hence , i t would be a c c e p t a b l e to r e p o r t subsequen t bed b e h a v i o u r o b s e r v a t i o n s ; a s a f u n c t i o n o f the bed depth p r o f i l e . The number o f k i l n v a r i a b l e s to be i n -v e s t i g a t e d i n a s t udy o f bed b e h a v i o u r may thus be g r e a t l y r e d u c e d . 4 . 2 . 2 Bed B e h a v i o u r i n a C o n t i n u o u s O p e r a t i o n W h i l e measu r i ng the bed depth p r o f i l e s d e s c r i -bed in S e c t i o n 4 . 2 . 1 , the v a r i o u s modes o f bed b e h a v i o u r e n c o u n t e r e d were a l s o r e c o r d e d . Four t y p e s were o b s e r v e d : s l i p p i n g , s l u m p i n g , r o l l i n g and t r a n s i t i o n a l , wh ich e x h i b i t e d mixed c h a r a c t e r i s t i c s o f s l u m p i n g and r o l l i n g . The c o r r e s p o n d -i n g k i l n o p e r a t i n g v a r i a b l e s were a l s o r e c o r d e d ; some of the r e s u l t s f o r sand A a re p r e s e n t e d i n T a b l e VI f rom wh i ch i t i s c l e a r t h a t the bed depth and r o t a t i o n a l speed s u f f i c e to i d e n t i f y the mode o f bed b e h a v i o u r . That i s , f o r q u i t e TABLE VI Bed B e h a v i o u r o f Sand A i n P i l o t K i l n ( K i l n I n c l i n a t i o n = 1 .2 ° ) Run # Rotational Bed Depth Feed Rate Dam Bed Behaviour Speed (r/min) (m) (kg/h) 19 0.5 0.085 29 A Transi t ion 21 0.5 0.085 61 A Transi t ion 21 0.5 0.102 61 A Transi t ion 34 0.5 0.103 38 B Transi t ion 14 1.0 0.030 38 D Slumping 18 1.0 0.032 32 A Slumping 16 1.0 0.045 67 D Slumping 18 1.0 0.045 32 A Slumping 18 1.0 0.054 32 A Transi t ion 20 1.0 0.054 69 A Transi t ion 35 1.0 0.055 79 B Transi t ion 18 1.0 0.064 32 A Transi t ion 20 1.0 0.064 69 A Transi t ion 27 2.0 0.025 29 A Ro l l ing 45 2.0 0.025 94 B Rol l ing 46 2.0 0.025 100 B Rol l ing 45 2.0 0.029 94 B Rol l ing 46 2.0 0.029 100 B Rol l ing 28 2.0 0.032 68 A Rol l ing 45 2.0 0.032 94 B Rol l ing 27 2.0 0.038 29 A Ro l l ing 28 2.0 0.038 68 A Rol l ing 45 2.0 0.038 94 B Rol1ing 27 2.0 0.051 29 A Rol l ing 28 2.0 0.051 68 A Rol l ing 78 d i f f e r e n t f eed r a t e s and dam h e i g h t s , the bed b e h a v i o u r o f sand A was the same when the r e s u l t i n g bed depths and r o t a t i o n a l speeds were e q u a l . The same c o n c l u s i o n was a l s o reached u s i n g l i m e s t o n e A when the a n g l e o f i n c l i n a t i o n o f the k i l n was v a r i e d . T h e r e f o r e , i t appears t h a t t h e r e a r e no downstream e f f e c t s o f bed b e h a v i o u r in the k i l n and t h a t o n l y the bed depth and r o t a t i o n a l speed need be s p e c i f i e d . To f u r t h e r t e s t t h i s c o n c l u s i o n , a l l o f the bed b e h a v i o u r o b s e r v a t i o n s f o r sand A have been p r e s e n t e d i n F i g u r e 4.4 wh ich i s a p l o t o f bed depth v e r s u s r o t a t i o n a l speed and i s c a l l e d a B e d - B e h a v i o u r D iag ram. S i n c e the Froude number has been p roposed by o t h e r wo rke rs (see C h a p t e r 2) as the s c a l e - u p c r i t e r i o n o f bed b e h a v i o u r , i t i s a l s o shown on the d i a g r a m . The bed depth measurements r e c o r d e d i n the bed p r o f i l e r e s u l t s ( e . g . F i g u r e 4.3) were r e p l o t t e d i n F i g u r e 4.4,at t h e i r r e s p e c t i v e r o t a t i o n a l s p e e d s . S l i p p i n g , s l u m p i n g , t r a n s i t i o n and r o l l i n g modes o f bed b e h a v i o u r were l a b e l e d a c c o r d i n g l y . Wh i l e making t h e s e o b s e r v a t i o n s , i t was not unusua l to o b s e r v e two or t h r e e t ypes o f bed modes o c c u r r i n g a t d i f -f e r e n t a x i a l p o s i t i o n s . In such c a s e s , the bed dep ths were p l o t t e d , p r o p e r l y coded f o r the t ype o f bed b e h a v i o u r t h a t had been o b s e r v e d at the r e s p e c t i v e a x i a l p o s i t i o n s . I t i s 79 e v i d e n t f rom F i g u r e 4 .4 t h a t t h e r e a re r e g i o n s on the d iag ram where a p a r t i c u l a r t ype o f bed mode d o m i n a t e s . Fo r e x a m p l e , ove r the range o f r o t a t i o n a l speeds t e s t e d , the s l i p p i n g mode o c c u r s whenever the bed depth was l e s s than 0 .03 m. At r o t a t i o n a l speeds l e s s t han 0 . 3 r / m i n and a t bed dep ths g r e a t e r than 0 .03 m, the s o l i d s bed was a lways o b s e r v e d to s lump . As the r o t a t i o n a l speed i n c r e a s e s , s l u m p i n g i s f a v o u r e d a t the l o w e r bed d e p t h s ; w h i l e a t the h i g h e r bed dep ths the o b s e r v e d b e h a v i o u r i s the t r a n s i t i o n mode. No s l u m p i n g i s seen to o c c u r a t r o t a t i o n a l speeds g r e a t e r than 1 .0 r / m i n . L i n e s d e m a r c a t i n g t he b o u n d a r i e s between the v a r i o u s bed modes were v i s u a l l y drawn and d e l i n e a t e a bed depth dependency f o r the s i u m p i n g - t r a n s i t i o n - r o l 1 i n g b o u n d a r i e s ( F i g u r e 4 . 4 ) . T h e r e f o r e f o r sand A t e s t e d i n the p i l o t k i l n , the s o l i d s bed b e h a v i o u r i s s o l e l y a f u n c t i o n o f the bed depth and the r o t a t i o n a l s p e e d . T h i s c o n c l u s i o n though needs to be f u r t h e r t e s t e d by compar ing the bed b e h a v i o u r i n a c o n t i n u o u s i n c l i n e d o p e r a t i o n w i t h t h a t i n a ba tch h o r i z o n t a l c y l i n d e r . 4 . 2 . 3 C o n t i n u o u s v e r s u s Ba tch Bed B e h a v i o u r In t h i s s e c t i o n , the p o s s i b i l i t y o f s i m p l i f y -i n g the c o n t i n u o u s a p p a r a t u s to a ba tch o p e r a t i o n w i l l be i n v e s t i g a t e d . The l a t t e r would o f f e r the added advan tage o f e x p e r i m e n t a l l y t r e a t i n g the bed depth and the r o t a t i o n a l Froude number ( F r » O J R / g ) -6 -5 IxlO Ix10 -4 I x l O 0.16h-0.12 CL V •D •o CD 0.08 0.04 I X l O ' Sand A 0.5 Slump ing O Transition 9 Rolling • Slipping • Runs 13 - 21 ,27 -30 ,32 , 34 -40 ,45 ,46 • I 1.5 Rotationol speed ( r / m i n ) 80, 35 25 15 ?! Figure 4.4 Bed-Behaviour Diagram of sand A v i s u a l l y determined in the the UBC p i l o t k i l n . Froude number (Fr=OJ R/g) 0.15 0.12 0.09 E n f 0.06 13 CD - 6 - 5 IxlO IxlO IXIO IxlO 0.03 Slumping i l( Transition \ \ \ Rolling Sand A 0.4 m X 0.46m L Pilot kiln r -o CSDO aooo 9 O O O 9 W 9 9 9 9 \ \ N I O a » > ^ 3 9 » O v » » » f « « \ \ X I O O OCED 00s O O 9 9 0 9 9 » O M • • O O O O O ^ O & W 9 9 9 Otito* • • • • 25 2 0 5 = LL. O O O O^Nrjaact - 3 9 9 9 999 OS* • • • O C O O ' O ^ ' 10 0.5 1.5 2 0 Rotational speed (r/min) Figure 4.5 Bed-Behaviour Diagram v i sua l l y determined for sand A in cy l inder A (0.4 m ID x 0.46 m L) and compared to the boundaries obtained in the p i l o t k i l n . 81 speed as two i ndependen t v a r i a b l e s . T h i s was not a lways p o s s i b l e i n the c o n t i n u o u s o p e r a t i o n , as a change i n the r o t a t i o n a l speed had a lways r e s u l t e d i n an accompany ing change i n the bed d e p t h . A B e d - B e h a v i o u r Diagram was e x p e r i m e n t a l l y d e t e r m i n e d i n c y l i n d e r A u s i n g sand A. V a r i o u s q u a n t i t i e s o f the g r a n u l a r m a t e r i a l were p l a c e d i n the c y l i n d e r wh ich was t e s t e d i n the h o r i z o n t a l p o s i t i o n . For each s o l i d s bed depth the r o t a t i o n a l speed o f the c y l i n d e r was v a r i e d ove r the s i u m p i n g - r o l 1 i n g range o f the m a t e r i a l . The run number o f each o f t h e s e t r a v e r s e s i s - l i s t e d i n Append ix B. In i t s t u r n , the bed depth was v a r i e d ove r the range o f 3 to 20% f i l l wh ich would be found i n r o t a r y k i l n o p e r a t i o n . The bed b e h a v i o u r o b s e r v a t i o n s were made v i s u a l l y and were r e c o r d e d i n F i g u r e 4 . 5 . A l s o shown i n F i g u r e 4 . 5 a re the s i u m p i n g - t r a n s i t i o n - r o l 1 i n g b o u n d a r i e s d e t e r m i n e d on the p i l o t k i l n . No s l i p p i n g was obse rved i n c y l i n d e r A w i t h sand A , as i t s w a l l was much rougher than t h a t o f the p i l o t k i l n . Hence , the s l i p p i n g boundary f o r the p i l o t k i l n was not i n c l u d e d i n F i g u r e 4 . 5 . At f i r s t i n s p e c t i o n , the agreement between the s i u m p i n g - t r a n s i t i o n - r o l 1 i n g b o u n d a r i e s does not appear to be ve ry good . However , i n the p i l o t k i l n the p r o t r u d i n g 82 t he rmocoup le s h e a t h s were c o n t i n u o u s l y d i s t u r b i n g the s o l i d s . T h i s r e s u l t e d , as o b s e r v e d , i n the bed r o l l i n g a t l owe r r o t a t i o n a l speeds than would o t h e r w i s e have been the case i n the absence o f t h e s e i n t e r n a l d e v i c e s . The s l u m p i n g -t r a n s i t i o n boundary shows ve ry good agreement between the c o n t i n u o u s and the b a t c h o p e r a t i o n s . I t i s t h e r e f o r e f e l t t h a t bed b e h a v i o u r o b s e r v a t i o n s of s l ump ing and r o l l i n g , made on ba tch c y l i n d e r s a re r e p r e s e n t a t i v e o f t h o s e made i n c o n t i n u o u s o p e r a t i o n s . E x p r e s s e d d i f f e r e n t l y , g i v e n a s e t o f k i l n o p e r a t i n g v a r i a b l e s and the r e s u l t i n g bed p r o f i l e , the o b s e r v e d bed modes, s l u m p i n g and r o l l i n g , a t each a x i a l p o s i t i o n would be the same as t h o s e o c c u r r i n g i n a ba t ch c y l i n d e r o f the same i n t e r n a l d i a m e t e r o p e r a t e d under equa l c o n d i t i o n s of bed depth and r o t a t i o n a l s p e e d . Subsequent B e d - B e h a v i o u r Diagrams were t h e r e f o r e d e t e r m i n e d on ba tch r o t a r y c y 1 i n d e r s . 4 . 3 I n s t r u m e n t a t i o n o f Bed B e h a v i o u r O b s e r v a t i o n s 4 . 3 . 1 C o n s t r u c t i o n and A p p l i c a t i o n The v i s u a l d e t e r m i n a t i o n o f bed b e h a v i o u r p r e s e n t s two d i s a d v a n t a g e s : f i r s t l y , near the bed b e h a v i o u r b o u n d a r i e s , the judgement o f the o b s e r v e r l a c k s a b s o l u t e o b j e c t i v i t y and r e p r o d u c i b i l i t y and s e c o n d l y , a permanent r e c o r d o f the bed mot ion i s not o b t a i n e d . An i n s t r u m e n t e d 83 t e c h n i q u e f o r o b s e r v i n g and r e c o r d i n g bed b e h a v i o u r was t h e r e f o r e s o u g h t . A p h o t o g r a p h i c method was f i r s t a t t e m p t e d . W h i l e i t p r o v i d e d a permanent r e c o r d o f the bed m o t i o n , the o b j e c t i v -i t y i n the judgement o f the bed mode i d e n t i f i c a t i o n was . * s t i l l l a c k i n g . A L i g h t R e f l e c t i o n D i s t a n c e Gauge ( h e n c e -f o r t h r e f e r r e d to as an IR s e n s o r o r j u s t s i m p l y as the s e n s o r ) was t h e r e f o r e d e v e l o p e d . I t c o n s i s t e d o f an I n f r a Red E m i t t e r / S e n s o r A r r a y ( F a i r c h i l d Model FPA 104) wh ich e m i t t e d I n f r a Red S i g n a l s and d e t e c t e d t hose r e f l e c t e d f rom the r e b o u n d i n g s u r f a c e . The c u r r e n t ou tpu t f rom the s e n s o r was p r o p o r t i o n a l to the d i s t a n c e between the s e n s o r and the r e b o u n d i n g s u r f a c e . T h u s , a s l ump ing bed c h a r a c t e r i z e d by a c o n s t a n t l y chang ing bed i n c l i n a t i o n would r e s u l t i n a v a r i a b l e s i g n a l w h i l e a r o l l i n g bed would y i e l d a more c o n -s t a n t s i g n a l . The s e n s o r was mounted i n an a c r y l i c tube i n o r d e r to suspend i t above the s o l i d s bed s u r f a c e i n s i d e the r o t a t i n g c y l i n d e r ; t h e r e b y not i n t e r f e r i n g w i t h the s o l i d s mo t ion . , . The a x i a l and r a d i a l p o s i t i o n o f the s e n s o r was v a r i a b l e . The ou tpu t s i g n a l o b t a i n e d was a m p l i f i e d and f e d to a H e a t h k i t r e c o r d e r , model E U - 2 0 B , w i t h a 250 mv f u l 1 r a n g e . 84 The roughness o f the bed s u r f a c e (due to the i n -d i v i d u a l p a r t i c l e s ) was o n l y seen to a d v e r s e l y a f f e c t the o u t p u t s e n s o r s i g n a l when l a r g e r s i z e d p a r t i c l e s were t e s t e d , namely l i m e s t o n e F and the i r o n o x i d e . For a l l o t h e r m a t e r i a l s , the e f f e c t s o f s u r f a c e roughness were l e s s than the changes i n bed i n c l i n a t i o n r e s u l t i n g f rom a s lump . The p o s i t i o n i n g o f the s e n s o r i s shown in F i g u r e 4 . 6 , w h i l e a sample c h a r t r e c o r d e r o u t p u t f rom the s e n s o r f o r both s l ump ing and r o l l i n g i s shown i n F i g u r e 4 . 7 . The t r a n s i t i o n mode was i d e n t i f i e d by a s e n s o r s i g n a l showing c h a r a c t e r -i s t i c s o f both s l ump ing and r o l l i n g s i g n a l s . Hence , the s e n s o r overcame the d i s a d v a n t a g e s o f the v i s u a l t e c h n i q u e . As w i l l be shown i n S e c t i o n 4 . 4 . 3 and i n C h a p t e r 6 , f u r t h e r a p p l i c a t i o n o f the s e n s o r s ou tpu t da ta a l l o w e d the s u c c e s s -f u l q u a n t i t a t i v e c h a r a c t e r i z a t i o n o f a s l u m p i n g bed as w e l l as the deve lopment o f a s e m i - e m p i r i c a l m a t h e m a t i c a l model p r e d i c t i n g the s i u m p i n g - r o l 1 i n g boundary . 4 . 3 . 2 C o r r e s p o n d e n c e o f I n s t r u m e n t e d and  V i s u a l O b s e r v a t i o n s B e d - B e h a v i o u r Diagrams were d e t e r m i n e d f o r l i m e s t o n e B and n i c k e l o x i d e i n c y l i n d e r A u s i n g both the v i s u a l t e c h n i q u e ( F i g u r e s 4 . 8 and 4 . 9 ) and t he i n s t r u m e n t e d t e c h n i q u e ( F i g u r e s 4 .10 and 4 . 1 1 ) . The b o u n d a r i e s o b t a i n e d 85 F i g u r e 4.6 P h o t o g r a p h i l l u s t r a t i n g t h e p o s i t i o n o f t h e s e n s o r i n t h e c y l i n d e r . Figure 4.7 Sample output of sensor fo r slumping and r o l l i n g . oo cn 87 2 F r o u d e n u m b e r ( F r = C U R / g ) -6 - 5 - 4 -IxlO IxlO ixiO IX10 Limestone B 0.12 Size 4.3 mm Shape irreguior Q5 1.0 1.5 Rotational 'speed ( r / m i n ) 2.0 F i g u r e 4.8 Bed-Behaviour Diagram o f l i m e s t o n e B i n c y l i n d e r A (0.4 m ID x 0.46 m L ) , v i s u a l l y d e t e r m i n e d . -6 -5 1X10 IxlO F r o u d e n u m b e r ( F r = 0 J R / g ) IXIO 0 . 1 2 -S l u m p i n g 0.09 f 0.06 •o •o m Trans i t ion c^a> a o ao azoa o* o GO 9 3 ooo ooa>3 o oo.» R o l l i n g • • • • 0.03 Nickel oxide Size 4.9 mm Shape spherical 0.5 1.0 1.5 Rota t iona l speed ( r / m i n ) 1X10' © 3 2.0 F i g u r e 4.9 Bed-B e h a v i o u r Diagram o f n i c k e l o x i d e i n c y l i n d e r A (0.4 m ID x 0.46 m L) d e t e r m i n e d v i s u a l l y . • 2 5 • 2 0 •15 •10 Froude number (Fr =CU R/g) - 6 ~ 5 1 X 1 0 ix 10 IxlO ix io3 0.12h - 0.09 e Q. 0.06 i — r Limestone B Instrumented Visual Slumping O O O O Transition 0\» \ ' 0 > Rolling O •• O O A O03 O C O O O 3 3 0 X X D 3 - ® I o.03r-88 25 20 15 ii 10 * o1 1 1 • •— 0 0.5 I 1.5 2 Rotational speed (r /min) Figure 4.10 Bed-Behaviour Diagram of limestone B in cy l inder A (0.4 m ID x 0.46 m L) determined by instrumentation and compared to the v isual determination. Froude number ( F r = G J R / g ) IxlO6 IXIO5 IXIO4 0.12 0.09b OCCQD0»C f Q ° 6 •o 0.03 Nickel oxide \ Slumping Transition ^ Rolling , 3 K M * • \ \ \ o o o a ^ r a w O O C O O O < M » 3 f O O C X O O 3 0 OO O O O OID^J 3 3 3 3 l T 3 - 3 - » • 3 -Instrumental Visual 0.5 I 1.5 Rotational speed (r/min) IxlO - 2 5 - 2 0 - 15 \\0 Figure 4.11 Bed-Behaviour Diagram of nickel oxide i n . cy l i nde r A (0.4 m ID x 0.46 m L) determined by instrumentation and compared with the v isual determination. 89 u s i n g both t e c h n i q u e s a re i n e x c e l l e n t agreement . The l a r g e s t d i f f e r e n c e i n the b o u n d a r i e s was o b t a i n e d w i t h the t r a n s i t i o n T r o l 1 i n g boundary o f the n i c k e l o x i d e p a r t i c l e s , which was o n l y o f the o r d e r o f 0 .2 r / m i n . S i n c e good c o r r e s p o n d e n c e i s o b t a i n e d ; between the i n s t r u m e n t and v i s u a l o b s e r v a t i o n s and s i n c e the s e n s o r s p roved u s e f u l i n q u a n t i -f y i n g o t h e r pa ramete rs ( S e c t i o n s 4 . 4 . 3 and 6 .2 ) a l l s u b - • sequent B e d - B e h a v i o u r Diagrams were d e r i v e d u s i n g the s e n -s o r s and a re i l l u s t r a t e d i n F i g u r e s 4 . 1 0 to 4 . 1 7 . 4 .4 E f f e c t o f - V a r i a b l e s on Bed B e h a v i o u r 4 . 4 . 1 Wal l E f f e c t s B e f o r e i n v e s t i g a t i n g the e f f e c t o f m a t e r i a l and c y l i n d e r v a r i a b l e s on the s 1 u m p i n g - r o l l i n g b o u n d a r y , the i n t e r a c t i o n o f the c y l i n d e r w a l l w i t h the s o l i d s mot ion when s l u m p i n g and r o l l i n g must f i r s t be e l u c i d a t e d . To a c c o m p l i s h t h i s , the s e n s o r p o s i t i o n i n s i d e the r o t a r y c y l i n d e r was v a r i e d r a d i a l l y and a x i a l l y and t h e s e t e s t s were c a r r i e d out u s i n g l i m e s t o n e B i n c y l i n d e r s A and B ( F i g u r e s 4 .10 and 4 . 1 2 , r e s p e c t i v e l y ) . For the f i r s t p a r t o f t h i s i n v e s t i g a t i o n , two s e n s o r s were p o s i t i o n e d a x i a l l y i n the c e n t r e o f the c y l i n d e r w i t h one o f them p o i n t e d a t the apex o f the bed and the o t h e r a t the c h o r d a ! b a s e . At equal b e d i d e p t h s , both s e n s o r s i n d i c a t e d a change o f bed b e h a v i o u r , 90 Froude number ( F r = C U R / g ) - 6 - 5 1X10 IxlO 1X10 IxlO" 0.12 - 0.09 E a QJ •c 0.06 •o a> 0.03 Limestone B 0.4mlD.X0.8 6mL 0.4m ID. X 0 . 4 S m L O O O O o o o Slumping Transition o o o 6 o o o o Rolling JOOK».« Q 0 3 > 3 Q » « ^ » y v oj9o o ao ao 9 9 9 3 0.5 I 1.5 Rotationol speed ( r /m in ) 2.5 Figure 4.12 Bed-Behaviour Diagram of limestone A comparing the resu l ts from cy l inder A (0.4 m ID x 0.46 m L) and cy l inder B (0.4 m ID x 0.86 m L) . ixib 6 ixib5 Froude number (Fr = 0 J R / g ) IXIO 4 IXI0 J 0.1 2 h -— 0.09 E JC ' Q. •° 0.06 TD 0) CD 0.03' Slumping Transition Roll ing " C O Q D O C D O C X J Q» 3 0 9 3 W • \ \ o o c r j o o o c o o o 9 3 0 3 * 0 • O O CD C©CXDCt» ^99 9 O O O ^ e • / Visual \ . OOOO O OO^OOKIPHD a O O O d a 99* 0 0 0 0 0 ^ 3 3 00 3 9 9 9 _L Limesione D Size 0.58mm Shope irregular 3 9~9jm> 1 0.5 I 1.5 Rotational speed ( r / m i n ) •25 •20 15 HO Figure 4.13 Bed-Behaviour Diagram of limestone D in cy l inder A (0.4 m ID x 0.46 m L) . 91 whether s l u m p i n g , t r a n s i t i o n or r o l l i n g , a t the same c y l i n d e r r o t a t i o n a l speed and at the same moment. Hence , the s 1 u m p i n g - r o l 1 i n g boundary i s i ndependen t o f the r a d i a l p o s i t i o n o f the s e n s o r s . As a slump i s i n i t i a t e d at the top h a l f of the bed s u r f a c e , i n subsequen t runs the s e n s o r was a lways p o i n t e d a t the apex of the bed . Wi th both s e n s o r s p o s i t i o n e d at the apex but a t d i f -f e r e n t c y l i n d e r a x i a l p o s i t i o n s , a d i f f e r e n c e i n the l o c i o f the bed b e h a v i o u r b o u n d a r i e s was o b s e r v e d . The s o l i d s a d j a c e n t to the end p l a t e s tended to r o l l a t l ower r o t a - ! t i o n a l speeds than those l o c a t e d at the a x i a l m i d - p o i n t . T h i s i s due to the l i f t i n g a c t i o n o f the end w a l l s on the p a r t i c l e s . Hence , the a x i a l c e n t r e of the c y l i n d e r was p r e f e r r e d f o r making bed b e h a v i o u r o b s e r v a t i o n s . In o r d e r to f u r t h e r c o n f i r m the i n f l u e n c e o f the end w a l l s on the bed b e h a v i o u r r e s u l t s , c y l i n d e r B was c o n -s t r u c t e d w i t h a l e n g t h to d i a m e t e r r a t i o , o f 2:1 as compared to 1:1 f o r c y l i n d e r A. Wi th a s e n s o r p o s i t i o n e d at the a x i a l c e n t r e o f each o f the c y l i n d e r s , a B e d - B e h a v i o u r Diagram was e x p e r i m e n t a l l y d e t e r m i n e d w i t h l i m e s t o n e B ( F i g u r e 4 . 1 2 ) . At degrees o f f i l l g r e a t e r than 6%, the t r a n s i t i o n zone o f the l o n g e r c y l i n d e r B was s m a l l e r than t h a t o f c y l i n d e r A , wh ich c o n f i r m s the p r e s e n c e o f the w a l l e f f e c t s d i s c u s s e d 92 e a r l i e r . However , the d i f f e r e n c e i n the p o s i t i o n o f the bed b e h a v i o u r b o u n d a r i e s i s not s i g n i f i c a n t . T h e r e f o r e , c y l i n d e r A was s u b s e q u e n t l y used f o r a l l B e d - B e h a v i o u r Diagram d e t e r m i n a t i o n s w i t h a s e n s o r p o s i t i o n e d a t the apex o f the bed i n t he a x i a l c e n t r e o f the c y l i n d e r . In F i g u r e s 4 . 1 0 , 4.11 and 4 .12 i t i s o b s e r v e d t h a t as the bed depth d e c r e a s e s , a h i g h e r r o t a t i o n a l speed i s r e q u i r e d f o r the bed to d i s p l a y t r a n s i t i o n a l b e h a v i o u r * ' "At low bed d e p t h s , some m a t e r i a l s d i s p l a y a r e v e r s a l o f t h i s t r e n d wh ich i s accompan ied by a w i d e n i n g of the t r a n s i t i o n zone ( F i g u r e 4 . 1 2 ) . The f o l l o w i n g e x p l a n a t i o n i s s u g g e s t e d as the cause o f t h i s phenomenon. When a m a t e r i a l s lumps i n a c y l i n d e r , t he e n t i r e bed a l o n g the c y l i n d e r l e n g t h does not n e c e s s a r i l y s lump s i m u l t a n e o u s l y . At s u f f i c i e n t l y low bed depths («5% f i l l ) the s l u m p i n g sequence o f two a d j a c e n t r e g i o n s o f the bed o f some m a t e r i a l s c o u l d r e s u l t i n t h e i r i n t e r f a c e h a v i n g a c o n s t a n t i n c l i n a t i o n . I f t h i s i n t e r f a c e o c c u r s below the s e n s o r , t he ou tpu t would i n d i c a t e a r o l l i n g b e d , s i n c e the bed s u r f a c e i s a r e l a t i v e l y c o n s t a n t d i s t a n c e away f rom the s e n s o r . F u r t h e r m o r e , t h i s i n t e r f a c e does not a lways o c c u r a t the same c y l i n d e r a x i a l p o s i t i o n and at t imes does not e x i s t , as the whole bed c o u l d be s l ump ing s i m u l t a n e o u s l y . 93 T h e r e f o r e , the o u t p u t o f a s e n s o r at a f i x e d a x i a l p o s i t i o n would i n d i c a t e a bed i n the t r a n s i t i o n mode, w h i l e v i s u a l l y i t was o b s e r v e d to be s l u m p i n g w i t h no e v i d e n c e of any r o l -l i n g a c t i o n . For t h e s e minor number o f c a s e s , both v i s u a l and i n s t r u m e n t e d o b s e r v a t i o n s were no ted and w i l l be i d e n t i -f i e d a c c o r d i n g l y , f o r example F i g u r e 4 . 1 3 . T h i s v i s u a l l y r e c o r d e d bed depth dependency f o r the s 1 u m p i n g - t r a n s i t i o n b o u n d a r y , a lways i n d i c a t e d i n c r e a s i n g r o t a t i o n a l speeds f o r d e c r e a s i n g bed depths ( i . e . the same t r e n d o b s e r v e d a t h i g h e r f i 1 1 r a t i os ). The t r a n s i t i o n - r o l l i n g boundary was a l s o bed depth dependen t , but c o u l d not be a t t r i b u t e d to e i t h e r c i r c u m -f e r e n t i a l o r e n d - p l a t e w a l l e f f e c t s . 4 . 4 . 2 P a r t i c l e Shape Wh i l e the ave rage p a r t i c l e s i z e s o f n i c k e l o x i d e and o f l i m e s t o n e B a r e about e q u a l , 4 . 9 mm and 4 . 3 mm r e s p e c t i v e l y , t h e i r B e d - B e h a v i o u r Diagrams d i f f e r ma rked l y ( F i g u r e s 4 . 1 0 and 4 . 1 1 ) . The s 1 u m p i n g - t r a n s i t i o n - r o l 1 i n g b o u n d a r i e s f o r the n i c k e l o x i d e o c c u r at much lower r o t a t i o n a l speeds than f o r l i m e s t o n e B. T h i s d i f f e r e n c e i s a t t r i -b u t a b l e t o d i f f e r e n c e s of p a r t i c l e shapes between the m a t e r i a l s . The one a s p e c t wh ich i s common to both d iagrams 94 i s the bed depth dependency o f the b o u n d a r i e s . The B e d - B e h a v i o u r Diagrams f o r l i m e s t o n e D and sand B a re shown i n F i g u r e s 4 . 1 3 and 4 .14 r e s p e c t i v e l y . These two m a t e r i a l s a l s o have a p p r o x i m a t e l y equa l ave rage p a r t i c l e s i z e - 0 .58 mm f o r l i m e s t o n e D and 0 .50 mm f o r sand B - but d i f f e r e n t p a r t i c l e s h a p e s . The n o d u l a r sand i s seen to have s 1 u m p i n g - t r a n s i t i o n - r o l 1 i n g b o u n d a r i e s a t l ower r o t a -t i o n a l speeds than the i r r e g u l a r l i m e s t o n e ; a g a i n , bo th t h e i r b o u n d a r i e s a re o b s e r v e d to be bed depth dependen t . I t may t h e r e f o r e be c o n c l u d e d f rom t h i s a n a l y s i s t h a t the B e d - B e h a v i o u r Diagram does show the e f f e c t o f p a r t i c l e s h a p e , w i t h s p h e r i c a l and n o d u l a r shaped p a r t i c l e s r o l l i n g more e a s i l y than i r r e g u l a r l y shaped p a r t i c l e s . T h i s i s .in agreement w i t h g e n e r a l o b s e r v a t i o n s made on the f l o w -a b i l i t y o f g r a n u l a r m a t e r i a l s i n b i ns and h o p p e r s . 9 0 ' 9 7 ' 1 0 8 4 . 4 . 3 P a r t i c l e S i z e F i g u r e s 4 . 1 0 , 4 . 1 3 and 4 . 1 5 i l l u s t r a t e the e f f e c t of p a r t i c l e s i z e on the bed b e h a v i o u r b o u n d a r i e s . The p a r t i c l e shape o f both l i m e s t o n e s B and D i s i r r e g u l a r a l t h o u g h t h e r e i s a lmos t an o r d e r o f magn i tude d i f f e r e n c e between t h e i r ave rage p a r t i c l e s i z e s , 4 . 3 mm and 0.58,mm Froude n u m b e r ( F r = G J R / g ) IXIO6 IXIO 5 IXIO 4 0.12 0.09 -g 0.06 co DO 0.03 0 Slumping Rolling iTransilion . (JDCBBB93 ( )33@© ® 1 OO0D3 0XJ0)33> 0)®® \ - 0DO0XX> 0S3 O C 3 O 0Q9 30)3 3 3 ® ® . i \ O C Q B O ) 3 030)330®® Sand B Size Q 5 m m S h a p e nodular 0.5 IX IO 3 — 2 5 2 0 15 10 H 5 I 1.5 2 Rotational speed ( r / m i n ) Figure 4.14 Bed-Behaviour Diagram of sand B in cylinder A (0.4 m ID x 0.46 m L). Froude number (Fr=CUR/g) -6 - 5 1X10 IXIO -4 IXIO IXIO' 0.12 0 .09 £ 0.06 OJ m 0.03 Slumping O O O O Transition Rol l ing O O O O O O O O O O O O O O . O O (39 3 9 \ 003 0> 08\ OQS> Limestone C Size 1.5mm Shape irregular © • O O 0) 0)0>®KK® ) O O S S ^ Q ) O O O O-CX® O (BO (5 0 0.5 I 1.5 Rotational speed ( r / m i n ) 2.5 — 2 5 2 0 15 10 ^ F igu re 4 .15 Bed-Behaviour Diagram of l imestone C i n c y l i n d e r A (0.4 m ID x 0.46 m L ) , CTl 97 r e s p e c t i v e l y . D e s p i t e the l a r g e e x p e r i m e n t a l t r a n s i t i o n zone o b t a i n e d f o r l i m e s t o n e D, i t can be seen t h a t s m a l l e r p a r t i c l e s r o l l more e a s i l y than c o a r s e r o n e s . To f u r t h e r i l l u s t r a t e t h i s p o i n t , a B e d - B e h a v i o u r Diagram was e x p e r i -m e n t a l l y d e t e r m i n e d f o r l i m e s t o n e C wh ich has the same p a r t i c l e shape as l i m e s t o n e s B and D but an i n t e r m e d i a t e ave rage p a r t i c l e s i z e o f 1.5 mm. The r e s u l t a n t b o u n d a r i e s shown i n F i g u r e 4 . 1 5 a re a t r o t a t i o n a l speeds wh ich f a l l between t hose f o r l i m e s t o n e B ( F i g u r e 4 . 1 0 ) and l i m e s t o n e D ( F i g u r e 4 . 1 3 ) . The b o u n d a r i e s f o r l i m e s t o n e C a l s o show a bed depth dependency wh ich i s i n agreement w i t h the o b s e r v a -57 t i o n s o f W a h l s t e r et a l . 4 . 4 . 4 Combined P a r t i c l e S i z e and Shape E f f e c t s The aim o f t h i s s e c t i o n i s t o s t u d y the combined e f f e c t s o f p a r t i c l e s i z e and shape and to o b s e r v e wh ich o f the two v a r i a b l e s i s dominant i n d e t e r m i n i n g the f l o w c h a r a c t e r i s t i c s o f a m a t e r i a l . F i g u r e s 4.11 and 4 .14 w i l l f i r s t be used to i l l u s t r a t e t h e s e e f f e c t s . H e r e , a c o a r s e s p h e r i c a l n i c k e l o x i d e i s compared to a f i n e r n o d u l a r s a n d . Had both m a t e r i a l s the same p a r t i c l e s i z e , "the d i f f e r e n c e i n t h e i r shape would sugges t t h a t the b o u n d a r i e s o f sand B l i e to the r i g h t o f t h o s e f o r the n i c k e l o x i d e . On the o t h e r hand , i f the shapes were 98 the same, the d i f f e r e n c e i n s i z e would i n d i c a t e t h a t the r e l a t i v e p o s i t i o n s of the boundaries s h o u l d be r e v e r s e d , i . e . the boundaries f o r sand B s h o u l d be to the l e f t of those f o r n i c k e l o x i d e . When comparing the e x p e r i m e n t a l l y determined boundaries shown i n F i g u r e s 4.11 and 4.14, i t i s seen t h a t the t r a n s i t i o n - r o l l i n g b oundaries of both m a t e r i a l s n e a r l y c o i n c i d e , but the s 1 u m p i n g - t r a n s i t i o n boundary of sand B i s to the l e f t of t h a t f o r n i c k e l o x i d e . T h e r e f o r e , i t appears t h a t i n t h i s case the p a r t i c l e s i z e has p a r t i a l l y dominated. The r e s u l t s of the a n g u l a r g r a v e l ( F i g u r e 4.16) are now a n a l y s e d w i t h r e s p e c t to those f o r l i m e s t o n e s B ( F i g u r e 4.10) and C ( F i g u r e 4.15). With a c o a r s e r p a r t i c l e s i z e of 3.0 mm, as compared to the 1.5 mm of l i m e s t o n e C, and w i t h -an a n g u l a r p a r t i c l e shape r a t h e r than an i r r e g u l a r one, i t would be expected t h a t the bed beh a v i o u r boundaries of the g r a v e l would l i e to the r i g h t of those f o r l i m e s t o n e C. Th i s r e s u l t was indeed o b t a i n e d when comparing F i g u r e s 4.15 and 4.16. In comparing g r a v e l and l i m e s t o n e B based on the p a r t i c l e shape d i f f e r e n c e , the bed b e h a v i o u r boundaries f o r g r a v e l s h o u l d l i e to the r i g h t of those f o r l i m e s t o n e B ( F i g u r e 4.10). On the o t h e r hand, the p a r t i c l e s i z e d i f -f e r e n c e f a v o u r s the o p p o s i t e e f f e c t . The e x p e r i m e n t a l r e s u l t s suggest t h a t these e f f e c t s have c a n c e l l e d each o t h e r out s i n c e the bo u n d a r i e s are c o i n c i d e n t . 9 9 Froude number (Fr»GJ R/g) -6 "5 IxlO IxlO -4 IxlO IXIO 0.12 J Z o. CO 0.03 Grovel Si2e 3mm Shape angular Slumping Transition O 03 0» R o l l i n 9 0 . 0 9 h - O O 0 D O O C O O O O O O O O O 0.06 2 0 15 10 0 5 I 1.5 Rotational speed (r/min) 2 5 Figure 4.16 Bed-Behaviour Diagram of gravel in cy l inder A (0.4 m ID x 0.4 m L ) . Froude number (Fr=CJR/g) -6 -5 IXIO IXIO IXIO IXIO3 0.17 I | Transition I Slumping I \ Rolling O O O© 0 • O 0.13 L P O O O OOO 3 © • © 0.09 CD 0.05 o o o o o o ? ooa>©o O O O O O O O O O C t > 9 W » Limestone B Size-4.3mm Shope - irregulor 0.5 I Rotational speed (r/min) 1.5 10 Figure 4.17 Bed-Behaviour Diagram of limestone B in cy l i nce r C " (1.06 m ID x 0.4 m L ) . 100 These a n a l y s e s show t h a t w h i l e the r e s p e c t i v e q u a l i -t a t i v e e f f e c t s o f p a r t i c l e s i z e and shape on bed b e h a v i o u r a re c l e a r l y i l l u s t r a t e d on a B e d - B e h a v i o u r D i a g r a m , t h e i r combined e f f e c t i s not e a s i l y p r e d i c t e d . These e x p e r i m e n t a l r e s u l t s w o u l d , t h u s , not p r o v i d e a p r e d i c t i v e method o f d e t e r m i n i n g whether the e f f e c t o f s i z e or shape would d o m i n a t e . A m a t h e m a t i c a l model, wh ich p r e d i c t s the s l u m p i n g and r o l l i n g boundary would be r e q u i r e d . The model d e v e l o p e d i n t h i s t h e s i s i s p r e s e n t e d and v e r i f i e d i n C h a p t e r s 6 and 7. 4 . 4 . 5 S t a t i c Ang le o f Repose The s t a t i c a n g l e o f repose has been c o n s i d e r e d to be an e m p i r i c a l measure o f the f l o w p r o p e r t i e s o f g r a n u l a r 90 91 m a t e r i a l s . ' I f the s t a t i c a n g l e o f r epose da ta l i s t e d i n T a b l e II a re compared w i t h the bed b e h a v i o u r b o u n d a r i e s e x p e r i m e n t a l l y d e r i v e d a t r e n d can be s e e n . The s m a l l e r the s t a t i c a n g l e o f r epose the more the bed b e h a v i o u r b o u n d a r i e s a re s h i f t e d towards lower r o t a t i o n a l s p e e d s , i . e . the e a s i e r i t i s to change f rom s l ump ing to r o l l i n g . T h i s t r e n d does not a t f i r s t appear to h o l d f o r sand B and the n i c k e l o x i d e s p h e r e s . A l t h o u g h the o r d e r o f t h e i r bed b e h a v i o u r b o u n d a r i e s was not i n a c c o r d w i t h the r e s p e c t i v e magn i tudes o f t h e i r s t a t i c a n g l e s o f r e p o s e , the d i f f e r e n c e s a re w i t h i n the e x p e r i m e n t a l e r r o r s . T h e r e f o r e , w h i l e a p p e a r i n g to be a s i m p l i s t i c measurement , the a n g l e o f r e p o s e 101 p r o v i d e s an easy and q u i c k method o f c l a s s i f y i n g the r e l a -t i v e p o s i t i o n o f the s i u m p i n g - r o l 1 i n g b o u n d a r i e s f o r m a t e r i a l s w i t h combined p a r t i c l e s i z e and shape e f f e c t s . 4 . 4 . 6 R o t a r y C y l i n d e r D iamete r A B e d - B e h a v i o u r Diagram was de te rm ined f o r l i m e s t o n e B i n c y l i n d e r C. I t has a l r e a d y been s u g g e s t e d i n the l i t e r a t u r e ( S e c t i o n 2 .4 ) t h a t bed b e h a v i o u r s i m i l i -tude i s o b t a i n e d at equa l Froude n u m b e r s . 2 , 4 4 , 6 1 ' 7 7 , 8 2 ' 1 1 9 based on c a l c u l a t i o n s f o r c e n t r i f u g i n g . However , compar ing F i g u r e s 4 . 1 0 and 4 .17 i.t i s e v i d e n t t h a t the Froude c r i t e r i a i s i n s u f f i c i e n t . F i r s t l y , i n both d iagrams the bed b e h a v i o u r b o u n d a r i e s show a bed depth dependency ; hence the degree o f f i l l must i n some way appear i n the s c a l e - u p c r i t e r i a . . S e c o n d l y , a t equa l degrees o f f i l l , t h e b o u n d a r i e s i n F i g u r e 4 . 1 0 o c c u r a t h i g h e r Froude numbers than t h o s e i n F i g u r e 4 . 1 7 . T h e r e f o r e , the deve lopment o f s c a l e - u p c r i t e r i a d i r e c t l y based on an a n a l y s i s o f the s i u m p i n g - r o l 1 i n g bed modes i s i m p e r a t i v e , s i n c e the c r i t e r i o n based on c e n t r i f u g i n g i s not r e l i a b l e . T h i s w i l l be a c h i e v e d by the q u a n t i t a t i v e c h a r a c t e r i z a t i o n o f the s l u m p i n g and r o l l i n g beds and by the m a t h e m a t i c a l d e s c r i p t i o n of the s l u m p i n g - r o l 1 i n g " . boundary . 102 4.5 C h a r a c t e r i z a t i o n of Slumping and R o l l i n g Beds  4.5.1 S I u m p i ng For a s lumping bed , the parameters i n v e s t i g a t e d were: the upper angle of repose of the b e d , the shear angle and the s lumping f r e q u e n c y . The maximum angle of bed i n c l i n a t i o n (the upper angle of repose) was measured on a s lumping bed f o r v a r i o u s m a t e r i a l s in the manner d e s c r i b e d in S e c t i o n 3 . 3 . 2 . The r e s u l t s f o r g rave l are shown in F i g u r e 4.18 and i n d i c a t e a s l i g h t de-pendence of the upper angle of repose on the c y l i n d e r r o t a -t i o n a l speed . A l i n e a r l e a s t - s q u a r e f i t was c a r r i e d out on t h i s data and the f o l l o w i n g equat ion was o b t a i n e d : (fry = 0.596 n + 40.6 (4.1) A l though the m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t was equal to 0.342 , i n d i c a t i n g a poor f i t , the asympto t ic va lue of <(>y, 4 0 . 6 ° , i s in good agreement wi th the measured s t a t i c angle of repose (<j>^  = 4 0 . 7 ° ) . It i s observed in F i g u r e 4.18 tha t there i s a g r e a t e r s c a t t e r in the data at h igher than at lower r o t a t i o n a l speeds^ which would c o n t r i b u t e to the poor f i t . The e f f e c t o f bed depth on the upper angle of repose 103 QJ OJ cn -S 4 5 3 OJ to O Q. QJ 4 3 4 I — _0> ° 39 OJ CL CL 37 1 1 1 Gravel — 0 4 m 0 X 0 4 6 m L — Bed depth 45mm o — o  o o o O O O O 0 o o o o o o o o o o o o — o o 0 o 1 1 1 0.5 1.5 Rotational speed (r/min) Figure 4.18 Upper angle of repose as a funct ion of ro ta t ional speed for gravel in cy l inder A (0.4 m ID x 0.46 m L ) . QJ QJ l_ cn QJ 3 -e-OJ If. o Q . CJ _QJ c o OJ Q . CL ID 4 2 4 0 38 36 L i m e s t o n e C 0 . 4 m <f)X 0 . 4 6 m L O • o A O V 9 4 81 6 8 5 2 39 2 9 Bed depth (mm) o o A • - n> o o o • o o A A O • O o C O O A AD O O A o o n n o o 0.5 I Rotational speed ( r / m i n ) Figure 4.19 Upper angle of repose as a funct ion of ro ta t i ona l speed measured at several bed depths in cy l inder A (0.4 m ID x 0.46 ni L) using limestone C. 104 was i n v e s t i g a t e d u s i n g l i m e s t o n e C ( F i g u r e 4 . 1 9 ) . No dependence was o b s e r v e d . The l i n e a r r e g r e s s i o n , a p p l i e d to a l l the d a t a , y i e l d e d the f o l l o w i n g e q u a t i o n : $ u = 1.01 n + 37 .7 ( 4 . 2 ) A g a i n , the m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t w i t h a v a l u e o f 0 .599 was poor but the o r d i n a t e i n t e r c e p t o f ^ i s in good agreement w i t h the measured s t a t i c a n g l e o f r epose (4>p = 3 7 . 8 ° ) , . For t h i s m a t e r i a l , the s c a t t e r i n the e x p e r i -menta l r e s u l t s was about 1 .5° and 2 . 5 ° f o r the low and h igh r o t a t i o n a l speeds r e s p e c t i v e l y . The l a r g e s c a t t e r i n t he r e s u l t s has been a t t r i b u t e d to the f o l l o w i n g f a c t o r s . F i r s t l y , the p r o t r a c t o r used f o r t hese measurements c o u l d o n l y be read to the n e a r e s t 0 . 5 ° . S e c o n d l y , the s u r f a c e o f the bed was not a lways p e r f e c t l y p l a n a r due to the b r i d g i n g o f p a r t i c l e s a t the e n d - w a l l . T h i r d l y , ove r the range o f r o t a t i o n a l s p e e d s , the maximum r e c o r d e d i n c r e a s e i n the upper 1 a n g l e o f repose was o n l y 3 ° . In v iew of the a c c u r a c y o f measurement d i s c u s s e d above i t i s not s u r p r i s i n g t h a t poor f i t s were o b t a i n e d . The l a s t c o n t r i b u t i n g f a c t o r l i e s i n the d i f f i c u l t y o f l o c a t i n g the p r e c i s e p o s i t i o n o f the upper a n g l e o f r e p o s e , due to the c o n t i n u o u s c y l i n d e r r o t a t i o n . T h i s d i f f i c u l t y , o f c o u r s e , i n c r e a s e s w i t h i n c r e a s i n g r o t a t i o n a l s p e e d . 105 In c o n c l u s i o n , ove r the s l ump ing range o f a m a t e r i a l , the upper a n g l e o f r epose i n c r e a s e s w i t h i n c r e a s i n g rbta.T t i o n a l s p e e d . U n f o r t u n a t e l y , the measur ing t e c h n i q u e used was hot s e n s i t i v e enough to r e l i a b l y q u a n t i f y t h i s dependence . The upper a n g l e o f r e p o s e was a l s o not o b s e r v e d to be a f u n c t i o n o f the q u a n t i t y o f m a t e r i a l i n the c y l i n d e r . In g e n e r a l , the measurement o f upper a n g l e o f r epose i s not s u i t a b l e f o r s t u d y i n g the f l o w p r o p e r t i e s o f m a t e r i a l s . The c h a r a c t e r i z a t i o n o f a s l u m p i n g bed u s i n g the s h e a r a n g l e w i l l now be d e s c r i b e d . Subsequent to a m a t e r i a l r e a c h -i n g i t s upper a n g l e o f r e p o s e , a slump o c c u r s . The shea r wedge o f s o l i d s i n the slump moves f rom the apex to the c h o r d a l base of the bed w h i l e the c y l i n d e r i s c o n t i n u o u s l y r o t a t i n g . The p l a n e i n the bed d i v i d i n g the p a r t i c l e s moving w i t h the slump and t hose moving w i t h the c y l i n d e r w a l l i s 1 g g c a l l e d the s h e a r p l a n e . I t s p o s i t i o n has been v i s u a l l y obse rved to be f i x e d i n space d u r i n g a slump and to form the f i n a l bed i n c l i n a t i o n once the s o l i d s in the slump have come to r e s t . The measurement o f the s h e a r a n g l e i s made i n t h i s l a t t e r p o s i t i o n , u s i n g the t e c h n i q u e d e s c r i b e d i n S e c t i o n 3 . 2 . 2 . The s h e a r a n g l e was measured f o r a l l m a t e r i a l s and i n the t h r e e c y l i n d e r s . The r e s u l t s a re p r e s e n t e d i n T a b l e VI I 106 where i t can be seen t h a t i n a g i v e n c y l i n d e r the s h e a r ang le was found to be i ndependen t of the r o t a t i o n a l speed and o f the bed d e p t h . On c l o s e r e x a m i n a t i o n o f the r e s u l t s , the dependence o f the s h e a r a n g l e on the c y l i n d e r d i a m e t e r i s a p p a r e n t , p a r t i c u l a r l y f o r l i m e s t o n e B. In g e n e r a l , i t was o b s e r v e d t h a t the s h e a r a n g l e was l ower i n the l a r g e r d i a m e t e r c y l i n d e r . T h i s c o n c u r s w i t h the r e s u l t s f o r the s t a t i c a n g l e o f r epose o b t a i n e d by o t h e r wo rke rs wh ich 108 R i c h a r d s and Brown have a t t r i b u t e d to w a l l e f f e c t s . When compar ing the shea r a n g l e s o f a l l m a t e r i a l s t e s t e d i n the same c y l i n d e r , the e f f e c t s o f p a r t i c l e s i z e and shape a re e v i d e n t . For examp le , the i r o n o x i d e and the n i c k e l o x i d e have l ower s h e a r a n g l e s than l i m e s t o n e s B, C and D wh ich a l l have i r r e g u l a r s h a p e s . W i t h i n each of t hese two groups o f p a r t i c l e s h a p e s , g r e a t e r p a r t i c l e s i z e s r e s u l t i n g r e a t e r r e s u l t a n t s h e a r a n g l e s . A l t h o u g h the s h e a r a n g l e may be used to c h a r a c t e r i z e a m a t e r i a l , i t r e v e a l s no i n s i g h t i n t o the f l o w p r o p e r t i e s o f m a t e r i a l s . However , i t s l a c k o f d e p e n d e n c e on bed depth and r o t a t i o n a l speed in a g i ven c y l i n d e r p roved i n v a l u a b l e i n the deve lopment o f the bed b e h a v i o u r m a t h e m a t i c a l m o d e l . I t has been shown t h a t the upper a n g l e o f r epose has not been u s e f u l to a d e q u a t e l y q u a n t i f y the s iumprng ' bed 107 TABLE VII Shear Angles of Mater ials Tested in Cyl inders A, B and C Material Cyl inder Diameter (m) Shear ; Angle (degrees) Standard Deviation (degrees) Number of Observa-t ions Gravel 0.40 1.06 34.7 34.4 0.8 0.6 53 26 Iron Oxide 0.40 33.3 0.4 7 Limestone B 0.40 1.06 37.7 34.5 1.1 0.7 51 39 Limestone C 0.40 1.06 33.6 32.5 0.5 0.8 37 16 Limestone D 0.40 33.5 0.3 51 Limestone F 1.06 38.5 1.2 6 Nickel Oxide 0.40 29.9 0.7 56 Sand A 0.40 32.4 0.7 5 Sand B 0.40 32.2 0.5 44 Sand C 0.40 33.0 - 1 108 m o t i o n , to c h a r a c t e r i z e the m a t e r i a l o r to e l u c i d a t e the causes f o r the change i n bed mot ion f rom s lump ing to r o l l i n g , w h i l e the s h e a r a n g l e has o n l y a c h i e v e d the second o f t h e s e o b j e c t i v e s ; The s l u m p i n g f r e q u e n c y measurements to be d e s c r i b e d w i l l f u l f i l the f i r s t two o b j e c t i v e s and w i l l i n d i c a t e t h a t a new i n t e r p r e t a t i o n o f the reason m a t e r i a l s change f rom a s l u m p i n g to a r o l l i n g bed i s r e q u i r e d . As was men t ioned i n S e c t i o n 2 . 2 . 2 . 2 , the e a s i e s t and most o b v i o u s method o f c h a r a c t e r i z i n g a s l u m p i n g bed i s by measu r i ng i t s s l u m p i n g f r e q u e n c y . A sample o f a s e n s o r o u t -put was d i s p l a y e d i n F i g u r e 4 . 7 where each peak seen on the s l u m p i n g c h a r t r e p r e s e n t s one slump of the m a t e r i a l . Thus the s l u m p i n g f r e q u e n c y c o u l d be o b t a i n e d f rom the r e c o r d o f the s e n s o r o u t p u t . The s l u m p i n g f r e q u e n c y was measured f rom a c o n t i n u o u s s e n s o r ou tpu t r a n g i n g u s u a l l y f rom 5 to 30 m i n u t e s . On ly d u r i n g the t r a n s i t i o n mode was the c o n t i n u o u s measu r i ng p e r i o d below 5 m i n u t e s . In t h e s e c i r c u m s t a n c e s , the t o t a l measu r i ng t ime o f a t l e a s t 5 m inu tes i n c o r p o r a t e d s e v e r a l i n d i v i d u a l s l u m p i n g p e r i o d s ; each h a v i n g a minimum of one m inu te d u r a t i o n . The e f f e c t o f s e n s o r p o s i t i o n i n g on the measurement o f s l ump ing f r e q u e n c y was f i r s t i n v e s t i g a t e d . The p o s i t i o n s t e s t e d i n c y l i n d e r A v a r i e d a x i a l l y and r a d i a l l y . For the 109 same t e s t i n g c o n d i t i o n s , a l l s e n s o r p o s i t i o n s r e c o r d e d the same s l u m p i n g f r e q u e n c y . The s e n s o r was t h e r e f o r e p o s i t i o n e d a x i a l l y i n the c e n t r e o f the c y l i n d e r a t the apex o f the bed f o r a l l measurements . In f a c t , the s l ump ing f r e q u e n c y measurements were made f rom the same s e n s o r p o s i t i o n as the bed b e h a v i o u r o b s e r v a t i o n s . For the sake o f c l a r i t y , o n l y a f r a c t i o n o f the r e s u l t s have been p r e s e n t e d i n F i g u r e s 4 . 2 0 to 4 . 2 3 . The e f f e c t o f c y l i n d e r l e n g t h and bed depth on the s l ump ing f r e q u e n c y i s shown i n F i g u r e s 4 .20 and 4 . 2 1 . N e i t h e r v a r i a b l e i s seen to a f f e c t the s l u m p i n g f r e q u e n c y . In c o n -t r a s t , t he s l u m p i n g f r e q u e n c y i s dependent on the c y l i n d e r d i a m e t e r . T h i s i s i l l u s t r a t e d i n F i g u r e 4.21 u s i n g l i m e s t o n e B t e s t e d i n c y l i n d e r s A and C. These f i n d i n g s were f u r t h e r c o n f i r m e d by the r e s u l t s o f the r e m a i n i n g m a t e r i a l s t e s t e d , as i s d i s c u s s e d b e l o w . The g e n e r a l c h a r a c t e r i s t i c s o f t h e s e s l u m p i n g f r e -quency c u r v e s a re as f o l l o w s . At low r o t a t i o n a l s p e e d s , s m a l l i n c r e a s e s i n r o t a t i o n a l speed r e s u l t i n l a r g e i n c r e a s e s i n s l u m p i n g f r e q u e n c y . As the r o t a t i o n a l speed i n c r e a s e s , the s l ump ing f r e q u e n c y i n c r e a s e s by s m a l l e r i n c r e m e n t s . The l a r g e s t s l u m p i n g f r e q u e n c i e s a re o b t a i n e d f o r the l o w e s t bed d e p t h s . T h i s i s c o n s i s t e n t w i t h the f i n d i n g t h a t a t no 4 0 I 301 £ c CO cr 2 0 c 'Q. E 00 A O 10 L imes tone B 0 4 m 1 D. X 0 4 6 m L Bed depth (mm) O 8 8 A 6 6 • 4 6 O 3 5 0 4 m 1 .D. X 0 . 8 6 m L f l 8 5 A 7 0 B 5 5 4 3 3 0 o1 0 0.5 I 1.5 2 2.5 Rotat ional speed ( r / m i n ) Figure 4.20 Slumping frequency as a funct ion of ro ta t iona l speed for limestone B tested in cy l inders A (0.4 m ID x 0.46 m L) and B (0.4 m ID x 0.86 m L ) . 4 0 3 0 2 0 cn c Q . E £ I Of A o ft • • A B • Limestone B Cylinder <|> 0.4 m Bed depth % Fil l Run (m) O 0.030 3 182 o 0.043 6 175 0 0.0 55 8 181 A 0.0 7 0 1 2 1 7 9 • 0.085 .1 5 1 8 0 Cylinder <p 1.06 m • 0.078 3 216 • 0.106 5 217 E 0.1 3 2 7 2 1 8 O.I 6 5 10 2 1 9 o 1 It 0 0.5 1.5 Rotational speed (r /min) ure 4.21 Slumping frequency as a function of ro tat ional speed for limestone B showing the independence of bed depth and the dependence on cy l inder diameter. 112 s m a l l e r bed depths the s l u m p i n g zone o c c u r s a t h i g h e r r o t a -t i o n a l speeds than i t does a t h i g h e r bed d e p t h s . T h e r e f o r e , the bed changes f rom s l u m p i n g to r o l l i n g w i t h o u t a r a p i d i n -c r e a s e i n s l u m p i n g f r e q u e n c y . " T h i s o b s e r v a t i o n i s c o n t r a r y to t h a t o f Reu te r in h i s c a t e g o r i z a t i o n o f s l ump ing f r e q u e n c y 2 7 i n t o t h r e e p a r t s . At low r o t a t i o n a l s p e e d s , a m a t e r i a l d i s p l a y s the same s l u m p i n g f r e q u e n c y i n c y l i n d e r s o f v a r y i n g d i a m e t e r s ( F i g u r e s 4.21 and 4 . 2 2 ) . With i n c r e a s i n g r o t a t i o n a l s p e e d s , the f r e q u e n c i e s in the l a r g e r d i a m e t e r c y l i n d e r are l owe r than those in the s m a l l e r d i a m e t e r c y l i n d e r . T h i s e f f e c t o f c y l i n d e r d i a m e t e r i s e v i d e n t f o r c y l i n d e r s A and C a t r o t a -t i o n a l speeds above 0 . 3 r /m in f o r l i m e s t o n e B , above 0 .75 r /min f o r l i m e s t o n e C and above 0 .65 r /m in f o r g r a v e l . T h i s e f f e c t o f d i a m e t e r i s no t e l i m i n a t e d by p l o t t i n g the s lump-i n g f r e q u e n c y as a f u n c t i o n o f the Froude number. The f o l l o w i n g c o n c l u s i o n s can be drawn f rom the e f f e c t s of p a r t i c l e s i z e and shape on the s l ump ing f r e q u e n c y as i l l u s t r a t e d i n F i g u r e 4 . 2 3 . F i r s t l y , m a t e r i a l s h a v i n g the same p a r t i c l e shape w i l l e x h i b i t i n c r e a s e d s l ump ing f r e q u e n -c i e s w i t h d e c r e a s i n g average p a r t i c l e s i z e . S e c o n d l y , m a t e r i a l s h a v i n g the same p a r t i c l e s i z e w i l l e x h i b i t i n c r e a s e d 113 4 0 r — E \ V) Q . E 3 >> O c 0> Z3 CT 4> 3 0 0= 20 cn C CL E CO 0 A A . A • • SEP 1^ • Mater ia l dp D S h a p e (mm) (m) A Limestone 1.5 0 .40 irregular A C 1.06 • Grave l 3 D 0.4 0 angular • 1.06 • L imestone 8.1 1.06 irregular o i 0 0.5 1.5 2.5 Rotat iona l speed ( r / m i n ) F i g u r e 4 . 2 2 S l u m p i n g f r e q u e n c y as a f u n c t i o n o f r o t a t i o n a l s p e e d , s h o w i n g t h e e f f e c t o f c y l i n d e r d i a m e t e r . 114 4 5 4 0 3 5 • • D D § 2 ^ Q5 I 1.5 Rotational speed ( r /min) Cylinder A 0.4 m 1 D x 0.46 m L Material dp (mm) Shape • Sand 8 0.5 Nodular • Nickel oxide 4.9 Spherical ^ Iron oxide 1 1.6 n — O Limestone D Q58 Irregular a » C 1.5 it 'o Grovel 3.0 Angular O Limestone B 4.3 Irregular -Figure 4.23 Slumping frequency as a funct ion of ro ta t iona l speed fo r several mater ia ls in cy l inder A (0.4 m ID x 0.46 m L) showing the e f fec t of p a r t i c l e shape and s i z e . 115 s l u m p i n g f r e q u e n c i e s as the p a r t i c l e shape becomes more n e a r l y ' s p h e r i c a l . T h e r e f o r e , the s l u m p i n g f r e q u e n c y o f a m a t e r i a l i s a f u n c t i o n of i t s p h y s i c a l p r o p e r t i e s as s u g g e s t e d by Z a b l o t n y . ^ F u r t h e r m o r e , the combined e f f e c t o f p a r t i c l e s i z e and shape a re a c c o u n t e d f o r by the r e l a t i v e p o s i t i o n o f the s l u m p i n g f r e q u e n c y c u r v e s . Fo r e x a m p l e , no te the r e s u l t s o f g r a v e l and l i m e s t o n e B and C. A l s o , t h e r e does appear to be a r e l a t i o n s h i p between the o r d e r o f s l u m p i n g f r e q u e n c y cu rves in F i g u r e 4 . 2 3 and the r e l a t i v e p o s i t i o n o f the s i u m p i n g - r o l 1 i n g b o u n d a r i e s f o r the m a t e r i a l s t e s t e d . The h i g h e r the s l ump ing f r e q u e n c y c u r v e , the f u r t h e r the bed b e h a v i o u r boundary w i l l be s h i f t e d to the l e f t on the B e d -B e h a v i o u r D iag ram. T h i s c r i t e r i o n seems to be more c o n -s i s t e n t t han t h a t o b s e r v e d f o r the s t a t i c a n g l e o f repose d e s c r i b e d i n S e c t i o n 4 . 3 . 5 . The s l ump ing f r e q u e n c y w i l l be f u r t h e r d i s c u s s e d in Chap te r 6 in c o n n e c t i o n w i t h the mathe-m a t i c a l m o d e l l i n g o f bed b e h a v i o u r . 4 . 5 . 2 R o l l i n g The r o l l i n g bed was c h a r a c t e r i z e d by i t s dynamic ang le o f repose and the t h i c k n e s s o f the a c t i v e bed l a y e r . The e f f e c t o f the sys tem v a r i a b l e s were i n v e s t i g a t e d 2 7 and the a c t i v e l a y e r t h i c k n e s s measurements o f R e u t e r were compared to those o b t a i n e d in t h i s s t u d y . The dynamic ang le o f repose was measured u s i n g t he same t e c h n i q u e d e s c r i b e d f o r the s h e a r ang le and the upper ang le o f r e p o s e . The c y l i n d e r r o t a t i o n a l speed was v a r i e d between the s 1 u m p i n g - r o l 1 i n g boundary o f a m a t e r i a l and a p p r o x i m a t e l y 3 r / m i n . The r e s u l t s a re summar ized i n Tab le V I I I and do no t r e f l e c t a r o t a t i o n a l speed o r a bed depth dependency ; w h i l e the e f f e c t s o f p a r t i c l e s i z e and shape are the same as t h o s e o b s e r v e d f o r the s h e a r a n g l e and the s t a t i c ang le o f r e p o s e . Combined p a r t i c l e shape and s i z e e f f e c t s as w e l l as the e f f e c t of c y l i n d e r d i a m e t e r on the dynamic a n g l e o f repose are a l s o s i m i l a r to those f o r the s h e a r a n g l e . The magn i tude of the dynamic ang le o f repose was a lways h i g h e r than the s h e a r a n g l e , but l e s s than the s t a t i c ang le o f r e p o s e . The r e l a t i v e magn i tude o f the dynamic a n g l e s o f repose o f v a r i o u s m a t e r i a l s does no t r e f l e c t t h e i r ' f l o w p r o p e r t i e s , as shown on t h e i r r e s p e c t i v e B e d - B e h a v i o u r D iag rams . As v iewed from the p l e x i g l a s s end p l a t e , the c i r c u -l a t i o n p a t t e r n o f the g r a n u l a r s o l i d s was in agreement w i t h 36 t h a t p roposed by Lehmberg e t a l . (F igure 2 . 5 ) as w e l l as w i t h the p a r t i c l e v e l o c i t y p r o f i l e s shown in F i g u r e 2 . 6 . The a c t i v e l a y e r depth was m a n u a l l y measured u s i n g a m i l l i -meter s c a l e a t the maximum t h i c k n e s s o f the a c t i v e l a y e r , namely a t the c l o s e s t d i s t a n c e o f the bed to the c e n t r e of 117 TABLE V I I I Dynamic Ang le o f Repose o f M a t e r i a l s T e s t e d Material Cyl inder Diameter (m) Dynamic Angle (degrees) Standard Deviation (degrees) Number of Observa-t ions Gravel 0.40 1.06 37.5 37.0 0.4 10 1 Iron Oxide 0.40 35.2 0.3 3 Limestone B 0.40 1.06 39.6 36.5 1.3 1.2 19 12 Limestone C 0.40 36.0 0.6 21 Limestone D 0.40 34.9 0.3 31 Limestone F 1.06 41.5 0.5 3 Nickel Oxide 0.40 30.2 0.5 6 Sand A 0.40 33.8 0.9 5 Sand B 0.40 33.6 0.8 41 Sand C 0.40 34.0 - 1 118 r o t a t i o n o f the c y l i n d e r . The r e s u l t s p l o t t e d as a f u n c t i o n o f the bed depth f o r v a r i o u s m a t e r i a l s t e s t e d a re shown in F i g u r e 4 . 2 4 . The f o l l o w i n g c o n c l u s i o n s can be d rawn: f i r s t l y , a t e q u a l bed d e p t h s , the s m a l l e r the p a r t i c l e s i z e o f the m a t e r i a l , the t h i n n e r i s the a c t i v e l a y e r . S e c o n d l y , f o r a g i ven m a t e r i a l , the h i g h e r the r o t a t i o n a l speed the t h i c k e r i s t h e a c t i v e l a y e r , and t h i r d l y , as the bed depth i n c r e a s e s the a c t i v e l a y e r t h i c k n e s s o f a l l m a t e r i a l s a l s o i n c r e a s e s . An a t t emp t was made to c l a r i f y the e f f e c t o f m a t e r i a l v a r i a b l e s on the a c t i v e l a y e r d e p t h ; a d i m e n s i o n l e s s p l o t o f the maximum number o f p a r t i c l e s in the a c t i v e l a y e r v e r s u s the maximum number i n the bed depth i s shown in 2 7 F i g u r e 4 . 2 5 . The da ta o f R e u t e r ( F i g u r e 2 . 7 ) h a v e a l s o been p l o t t e d on t h i s d i a g r a m . The e f f e c t o f p a r t i c l e s i z e and c y l i n d e r r o t a t i o n a l speeds are c l e a r l y e v i d e n t ; w i t h i n c r e a s i n g r o t a t i o n a l speeds and d e c r e a s i n g p a r t i c l e s i z e , the number o f p a r t i c l e s in the a c t i v e l a y e r t h i c k n e s s i n -c r e a s e s . On the o t h e r h a n d , the p a r t i c l e shape does no t seem to have an e f f e c t on the a c t i v e l a y e r t h i c k n e s s . F i g u r e 4 .26 i s a p l o t o f the r e l a t i v e d i s t r i b u t i o n o f the p a r t i c l e s between the a c t i v e and p a s s i v e z o n e s . I t I 5 1 . 2 </> OJ C o OJ a OJ > o < 0 RPM >2 2>RPM^I l>RPM Material • © Limestone B • C Hi E • Sand B • B B • • B • • • © A • ©" . • • • 2 0 4 0 6 0 8 0 B e d d e p t h H ( m m ) 1 0 0 Figure 4.24 Active layer thickness as a function of bed depth, measured for several materials 120 16 14 12 I oh-a. - 8 Moteriol 0(m )| Limestone F 8.1 1.06 o B 4.3 0.4 e © " ' C 1.50 0.4 A Sond B 0.5 0.4 B E • Reuter2 7 8.0 0.8 ® e o O J A l E o. r pm >l >r pm A CM o E * tb tb A o • • • H 4 0 8 0 120 160 2 0 0 H / d r Figure 4.25 Dimensionless p lo t of act ive layer thickness as a funct ion of bed depth. 0.3 2 0.24 0.1 6 0.0 8 tb Materiol i p l m r n ) D (m ) o Limestone F S.I 1.06 • A B e B • B . " C Sand R 27 Reuter 4.3 1.5 0.5 0.4 0 0.4 0 0.40. — e © o 8.0 0.80 CvJ A l A l E E c E o. A A B — o tt d- tt & t, a tb • D E • a o 5 0 100 H / d . E • D B 150 2 0 0 Figure 4.26 Dimensionless plot of the re la t i ve s ize of the act ive layer to the bed depth. 121 i s seen t h a t as t he number o f p a r t i c l e s in the bed depth i n c r e a s e s , the number of p a r t i c l e s in the a c t i v e l a y e r d e c r e a s e s u n t i l the H / d p r a t i o i s e q u a l to 50 . G r e a t e r i n c r e a s e s i n t h i s r a t i o r e s u l t in the r e l a t i v e volume o c -c u p i e d by the a c t i v e and p a s s i v e l a y e r s becoming i n s e n s i t i v e to changes in the p a r t i c l e s i z e . The e f f e c t o f r o t a t i o n a l speed i s a l s o a p p a r e n t . The h i g h e r the r o t a t i o n a l s p e e d , the g r e a t e r the amount o f p a r t i c l e s in the a c t i v e l a y e r r e l a t i v e to t hose i n t h e p a s s i v e l a y e r . I f e x t e n d e d to the h i g h e r r o t a t i o n a l speeds where c a s c a d i n g and c a t a r a c t i n g o c c u r , t h i s t r e n d wou ld be found to be in agreement w i t h the e x p e r i m e n t a l o b s e r v a t i o n s made i n t hese modes by o t h e r , 4 , 5 3 , 5 5 ,56 .61 , 120„ r . , ', , ^ . . . w o r k e r s . ' ' '<- u° F i n a l l y , no r e l a t i o n s h i p i s o b s e r v e d between the a c t i v e l a y e r t h i c k n e s s and the s l ump ing -r o l l i n g boundary on a B e d - B e h a v i o u r D iag ram. 4 .6 S umma ry Three modes o f bed b e h a v i o u r : s l i p p i n g , s l u m p i n g and r o l l i n g , were r e p r e s e n t e d on a B e d - B e h a v i o u r Diagram which c o n s i s t s o f a p l o t o f bed depth v e r s u s r o t a t i o n a l s p e e d . I t was shown t h a t f o r a r o t a r y k i l n where the r e l a t i o n s h i p between the bed p r o f i l e , the f e e d r a t e and the r o t a t i o n a l speed i s known, then the bed b e h a v i o u r c o u l d be d e t e r m i n e d f rom the B e d - B e h a v i o u r Diagram o f the same m a t e r i a l d e t e r -mined i n a b a t c h h o r i z o n t a l r o t a r y c y l i n d e r o f the same i n t e r n a l d i a m e t e r . An i n s t r u m e n t a l t e c h n i q u e f o r i d e n t i f y i n g and r e c o r d -i n g bed b e h a v i o u r has been d e v e l o p e d . The range o f o p e r a -t i n g c o n d i t i o n s i n wh ich t r a n s i t i o n i s o b s e r v e d was i n -c r e a s e d by c i r c u m f e r e n c i a l and e n d - w a l l e f f e c t s . Us ing these s e n s o r s B e d - B e h a v i o u r Diagrams were de te rm ined f o r s e v e r a l m a t e r i a l s . P a r t i c l e s i z e and shape were both found to a f f e c t the l o c u s o f the s 1 u m p i n g - r o l 1 i n g b o u n d a r y . S p h e r i c a l shape and s m a l l e r p a r t i c l e s i z e s f a v o u r e d a boundary a t l o w e r r o t a t i o n a l s p e e d s ; w h i l e a n g u l a r and c o a r s e r p a r t i c l e s r o l l e d at h i g h e r r o t a t i o n a l s p e e d s . The s t a t i c a n g l e o f r e s p o s e o f the m a t e r i a l s p roved e f f e c t t i v e i n c a t e g o r i z i n g the r e l a t i v e p o s i t i o n o f the bed b e -h a v i o u r b o u n d a r i e s . I t was most u s e f u l when c o u p l e d e f -f e c t s o f p a r t i c l e s i z e and shape were e n c o u n t e r e d . The s 1 u m p i n g - r o l l i n g boundary c o n s i s t e n t l y d i s p l a y e d a bed depth dependency wh ich was not e x p l a i n e d by any o f the e x p e r i m e n t a l r e s u l t s and w i l l be e x p l o r e d l a t e r i n the t h e s i s . F u r t h e r m o r e , the Froude number has p roved to be an i n s u f f i c i e n t s c a l e - u p c r i t e r i a f o r . b e d b e h a v i o u r s i m i l i -t u d e . S e v e r a l f l o w pa rame te rs o f g r a n u l a r s o l i d s were measured f o r the s l ump ing and r o l l i n g modes. S lump ing was c h a r a c t e r i z e d u s i n g the upper a n g l e o f r e p o s e , the s h e a r a n g l e and the s l ump ing f r e q u e n c y . The dynamic ang le o f 123 repose and the a c t i v e l a y e r t h i c k n e s s were measured f o r r o l l i n g b e d s . The upper ang le o f r epose i n c r e a s e d w i t h i n c r e a s i n g r o t a t i o n a l s p e e d . However , the measur ing t e c h n i q u e used was not a c c u r a t e enough to r e l i a b l y d e t e c t t h i s i n c r e a s e . The shea r ang le and the dynamic ang le o f repose were shown to be i ndependen t o f the r o t a t i o n a l speed and the bed depths o f the c y l i n d e r ove r the range t e s t e d . They w e r e , however , dependent on the c y l i n d e r d i a m e t e r . L a r g e r a n g l e s were o b t a i n e d f o r s m a l l e r d i a m e t e r c y l i n d e r s . When com-p a r i n g both a n g l e s i n the same c y l i n d e r , t h o u g h , the e f f e c t s o f p a r t i c l e s i z e and shape were o b s e r v e d . The l a r g e r the p a r t i c l e s i z e and the g r e a t e r the d e v i a t i o n f rom s p h e r i c a l s h a p e , the g r e a t e r the s h e a r ang le o f the m a t e r i a l . The dynamic a n g l e o f repose was a lways h i g h e r than the shea r a n g l e . A s l ump ing bed was c h a r a c t e r i z e d by i t s s l u m p i n g f r e q u e n c y wh ich i n c r e a s e d w i t h r o t a t i o n a l speed and was i ndependen t o f the bed d e p t h . A dec rease i n c y l i n d e r <.iv. d i a m e t e r , ave rage p a r t i c l e s i z e or a change i n p a r t i c l e shape f rom a n g u l a r to s p h e r i c a l r e s u l t e d i n an i n c r e a s e i n s l ump ing f r e q u e n c y a t a g i v e n r o t a t i o n a l s p e e d . The r e l a -t i v e o r d e r o f the s l ump ing f r e q u e n c i e s o f a group o f m a t e r i a l s t e s t e d i n a c y l i n d e r i s the same as the r e l a t i v e p o s i t i o n o f the s i u m p i n g - r o l l i n g b o u n d a r i e s o f these m a t e r i a l s on a 124 B e d - B e h a v i o u r D iag ram. The s c a l e - u p c r i t e r i a o f the Froude number d i d not a p p l y f o r the s l ump ing f r e q u e n c i e s o f v a r y i n g c y l i n d e r d i a m e t e r s . F i n a l l y , the a c t i v e l a y e r t h i c k n e s s was ^shown "to •• be i ndependen t o f p a r t i c l e s h a p e , but dependent on p a r t i c l e s i z e a t r a t i o s o f H/d • l e s s than 50 . An i n c r e a s e i n r o t a -P t i o n a l speed r e s u l t e d i n an i n c r e a s e i n the a c t i v e l a y e r t h i c k n e s s . The c h a r a c t e r i s t i c s o f s l ump ing and r o l l i n g have thus been q u a n t i f i e d . The b a s i c cause o f the bed mot ion chang ing from s l u m p i n g to r o l l i n g w i l l be pu rsued i n C h a p t e r s 6 and 7 where a s e m i - e m p i r i c a 1 m a t h e m a t i c a l model o f bed mot ion w i l l be p r e s e n t e d . 125 Chap te r 5 SEGREGATION 5 .1 I n t r o d u c t i o n An a n a l y s i s o f t he e f f e c t o f m a t e r i a l p r o p e r t i e s on r the f l o w o f g r a n u l a r s o l i d s would not be comple te w i t h o u t i n v e s t i g a t i n g the r o l e p l a y e d by 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 . P r e v i o u s work has shown t h a t m a t e r i a l s h a v i n g a s i z e d i s t r i b u t i o n w i l l de -mix (o r s e g r e g a t e ) when s u b j e c t e d to f l o w . That i s , the c o m p o s i t i o n o f the m a t e r i a l w i l l not remain u n i f o r m t h r o u g h o u t i t s t o t a l vo lume. T h i s s t a t e of s e g r e g a t i o n can be a f f e c t e d by d i f f e r e n c e s in m a t e r i a l d e n s i t y and s h a p e , and can s e r i o u s l y hamper i n d u s t r i a l o p e r a t i o n s . In r o t a r y k i l n s , f o r e x a m p l e ; the r e s i d e n c e t ime of c o a r s e r s i z e d o r l e s s dense p a r t i c l e s has been r e p o r t e d to be l e s s than the r e s i d e n c e t ime o f the f i n e r s i z e d or the denser p a r t i c l e s . 1 2 ' 3 2 ' 4 4 ' 4 5 ' 1 1 8 ' 1 2 1 ' 1 2 2 T h i s d i f f e r e n c e in r e s i d e n c e t i m e , can r e s u l t i n the p r o d u c t i o n o f an un-32 12 3 a c c e p t a b l e p r o d u c t . ' For e x a m p l e , i n the d i r e c t r e -d u c t i o n of i r o n ore u s i n g a r o t a r y k i l n , c o a l has been seen to s e g r e g a t e to the s u r f a c e o f the b e d , l e a v i n g the i r o n o x i d e in the c o r e . / » J / » ' ^ t f Jhis r e s u l t s i n e i t h e r 126 unreduced i r o n d i s c h a r g i n g f rom the k i l n o r an e x c e s s i v e l y h igh c o a l c o n s u m p t i o n . A s i m i l a r p rob lem can a r i s e in l i m e -stone c a l c i n a t i o n due to s i z e r a t h e r than d e n s i t y s e g r e g a -t i o n . S m a l l e r p a r t i c l e s i z e s have been o b s e r v e d to r e q u i r e 47 125 a l o n g e r t ime to c a l c i n e than the c o a r s e r f r a c t i o n . ' F i n a l l y , i n d u s t r i a l o b s e r v a t i o n s on l i m e r o t a r y k i l n s have r e v e a l e d t h a t s l i p p i n g has been a s s o c i a t e d w i t h the p r e -47 sence o f s i g n i f i c a n t q u a n t i t i e s o f f i n e s . These f i n d i n g s have p r o v i d e d the impetus f o r r e -s e a r c h in s e g r e g a t i o n . In the f o l l o w i n g s e c t i o n a b r i e f p r e s e n t a t i o n w i l l be made o u t l i n i n g the c u r r e n t s t a t e o f knowledge in t h i s a r e a . Those a r e a s r e q u i r i n g g r e a t e r a t t e n t i o n w i l l then be more c l o s e l y examined to c l e a r l y i d e n t i f y s e g r e g a t i o n p a t t e r n s and mechanisms in r o t a r y c y l i n d e r s . The e x p e r i m e n t a l t e c h n i q u e s , a l r e a d y d e v e l o p e d and p r e s e n t e d in c h a p t e r 4 , w i l l be a p p l i e d in t h i s s t udy and the e f f e c t s of s e g r e g a t i o n on the s i u m p i n g - r o l 1 i n g boundary in a B e d - B e h a v i o u r Diagram w i l l be s t u d i e d . Changes in the s t a t i c ang le of r e p o s e , the shea r a n g l e , dynamic a n g l e o f r e p o s e , s l u m p i n g f r e q u e n c y and the a c t i v e l a y e r t h i c k n e s s w i l l a l s o be r e p o r t e d . A t h e o r e t i c a l and an e x p e r i m e n t a l a n a l y s i s o f t he s e g r e g a t e d co re in the s o l i d s bed w i l l 127 f o l l o w , and w i l l c l e a r l y i d e n t i f y the s e g r e g a t i o n mechan ism. In t h i s s t u d y , more emphas is w i l l be p l a c e d on s i z e s e g r e g a -t i o n as i t i s the most common. However , the p r i n c i p l e s d e r i v e d f rom t h i s c u r r e n t a n a l y s i s may a l s o be a p p l i e d to o t h e r sys tems where d e n s i t y and shape d i f f e r e n c e s o r c o u p l e d e f f e c t s a re e n c o u n t e r e d . The i n d u s t r i a l p r o c e s s i m p l i c a -t i o n s of these f i n d i n g s w i l l a l s o be d i s c u s s e d . 5 .2 P r e v i o u s Work In t h i s s e c t i o n , o n l y t hose i n v e s t i g a t i o n s on the s e g r e g a t i o n o f b u l k s o l i d s c a r r i e d ou t p r i m a r i l y i n r o t a r y c y l i n d e r s w i l l be d i s c u s s e d . 5 . 2 . 1 Types o f S e g r e g a t i o n Three t ypes o f s e g r e g a t i o n p a t t e r n s have been o b s e r v e d i n h o r i z o n t a l r o t a r y d e v i c e s . A 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 t hese i s i l l u s t r a t e d i n F i g u r e 5 . 1 . B r o a d l y they can be c a t e g o r i z e d as r a d i a l and a x i a l o r l o n g i t u d i n a l s e g r e g a t i o n z o n e s . A x i a l s e g r e g a t i o n may in tu rn be s u b d i v i d e d i n t o band ing and end-? l o n g i t u d i n a l . In r a d i a l s e g r e g a t i o n , s m a l l e r o r dense r p a r t i c l e s are o b s e r v e d to . fo rm a h o r i z o n t a l core in the b e d 27,37,47,52,55,57,59,67,126-128 whereas in band ing s e g r e g a t i o n , a l t e r n a t e bands o f c o a r s e and f i n e o r l i g h t and heavy p a r t i c l e s form down the l e n g t h I i i 1111 I > 1 > > > > 11J J , , u ! T T ( i ) Radial ( i i ) Banding long i tud ina l V, >m u 111), /1 > 11 ? V. A ( i i i ) End- longi tudinal coarser or l i gh te r so l i ds mm f i ne r or denser so l ids Figure 5.1 Types of segregation encountered in hor izonta l rotary c y l i n d e r s . 5 9 ' 6 7 129 o f the r o t a r y c y 1 i n d e r . 6 9 ' 6 6 ' 6 7 ' 7 2 F i n a l l y , the e n d -longitudiinal i s e g r e g a t i o n i s such t h a t the f i n e r o r dense r p a r t i c l e s form two end-bands a t the c y l i n d e r w a l l s w i t h the c o a r s e o r l i g h t e r p a r t i c l e s in the c e n t r e o f the c y l i n d e r a x i s . 3 7 ' 5 9 ' 6 6 ' 6 7 ' 1 2 9 The most comprehens i ve a n a l y s i s of t hese t ypes o f s e g r e g a t i o n has been made by Dona ld and R o s e m a n . 6 7 ' 1 3 0 ' 1 3 1 T h e i r f i n d i n g s showed t h a t r a d i a l s e g r e g a t i o n a lways o c c u r s w h i l e the type o f a x i a l s e g r e g a t i o n , i f i t o c c u r s , may be p r e d i c t e d , u s i n g the s t a t i c a n g l e s o f repose o f the com-ponents o f the m i x t u r e . When the s m a l l e r s i z e d p a r t i c l e s have a s t a t i c a n g l e o f repose g r e a t e r than the c o a r s e , the band ing type o f a x i a l s e g r e g a t i o n r e s u l t s . The end -l o n . g i t u d i h a l s e g r e g a t i o n o c c u r s when the s t a t i c ang le o f repose o f the s m a l l e r p a r t i c l e s i s l e s s than t h a t o f the c o a r s e r p a r t i c l e s . These c o n c l u s i o n s were based on an e x p e r i m e n t a l program i n v o l v i n g ove r 30 b i n a r y m i x t u r e s wh ich were t e s t e d f o r a r o l l i n g bed in h o r i z o n t a l l y r o t a -t i n g c y l i n d e r s . 5 . 2 . 2 S e g r e g a t i o n K i n e t i c s S e g r e g a t i o n l i k e m i x i n g i s a f i r s t o r d e r p r o c e s s . R a d i a l s e g r e g a t i o n a lways o c c u r s much f a s t e r than a x i a l s e g r e g a t i o n . 6 6 ' 1 3 0 Rogers and C l e m e n t s 6 6 show r a d i a l 130 s e g r e g a t i o n to be w e l l e s t a b l i s h e d f o l l o w i n g 10 c y l i n d e r 2 7 r e v o l u t i o n s . R e u t e r r e p o r t s i t to be comple te a f t e r o n l y 2 r e v o l u t i o n s . These r e s u l t s have been r e c e n t l y c o n f i r m e d 1 32 in t e s t s c a r r i e d out on c o n t i n u o u s r o t a r y k i l n s . A x i a l s e g r e g a t i o n , on the o t h e r h a n d , o n l y r eaches s t e a d y s t a t e a f t e r 500 to 10 ,000 r e v o l u t i o n s . 3 7 ' 6 6 ' 1 2 9 ' 1 3 0 A l l i n v e s t i -g a t i o n s were c a r r i e d out w i t h the bed r o l l i n g o r c a s c a d i n g . R e u t e r has s u c c e s s f u l l y d e s c r i b e d the r a d i a l s e g r e g a -2 7 t i o n p r o c e s s by a f i r s t o r d e r r a t e e q u a t i o n . However , no a t t emp t has y e t been made to m a t h e m a t i c a l l y d e s c r i b e the k i n e t i c s o f a x i a l s e g r e g a t i o n . 5 . 2 . 3 Mechanisms o f S e g r e g a t i o n There are t h r e e mechanisms by wh ich g r a n u l a r m a t e r i a l s s e g r e g a t e : per c o l a t i 6n>f 1 ow and v i b r a t i on ^7,90,130,133 P e r c o l a t i o n s e g r e g a t i o n i s f a v o u r e d when the s i z e o f the v o i d s between c o a r s e r p a r t i c l e s i s l a r g e enough to a l l o w s m a l l e r p a r t i c l e s to pass th rough them. I t o c c u r s in the r e g i o n o f an i n c l i n e d s h e a r p l a n e . The s m a l l e r p a r t i c l e s in mot ion can have t h e i r t r a j e c t o r y a b r u p t l y changed by f a l l i n g i n t o a v o i d on the s t a t i o n a r y s i d e o f the s h e a r p l a n e . Th i s type o f s e g r e g a t i o n i s t h e r e f o r e a f u n c t i o n o f the p a c k i n g c h a r a c t e r i s t i c s of g r a n u l a r s o l i d s , 131 the p a r t i c l e shape and the p a r t i c l e s i z e . S e g r e g a t i o n by f l ow a l s o o c c u r s when g r a n u l a r s o l i d s are s e t i n mot ion ove r an i n c l i n e d s u r f a c e . I t a r i s e s f rom c o a r s e p a r t i c l e s t r a v e l l i n g f u r t h e r than s m a l l e r p a r t i c l e s down an i n c l i n e d s u r f a c e . U n f o r t u n a t e l y , t h e r e are ve ry few e x p e r i m e n t a l r e s u l t s r e p o r t e d in the l i t e r a t u r e to 90 i l l u s t r a t e t h i s mechan ism. Those a v a i l a b l e have been c a r r i e d ou t by p o u r i n g b i n a r y m i x t u r e s i n t o h e a p s . Th i s does no t p r o v i d e c o n c l u s i v e e v i d e n c e however , s i n c e s e g r e g a -t i o n by p r e c o l a t i o n wou ld a l s o o c c u r . N e v e r t h e l e s s , based on the s l u m p i n g f r e q u e n c y r e s u l t s shown i n F i g u r e s 4 .20 to 4 . 2 3 i t can be c o n f i r m e d t h a t t h e r e are d i f f e r e n c e s between the f l o w c h a r a c t e r i s t i c s o f c o a r s e and f i n e , s p h e r i c a l and a n g u l a r p a r t i c l e s . Fu r the rmore i t was o b s e r v e d t h a t s lumps o f the s m a l l e r s i z e d m a t e r i a l s , i n i t i a t i n g a t the apex o f the b e d , came to r e s t on the bed s u r f a c e a t some d i s t a n c e f rom the c h o r d a l b a s e . On the o t h e r h a n d , s lumps f rom c o a r s e s i z e d m a t e r i a l s r eached the c h o r d a l base o f the b e d , and i n some cases c e r t a i n p a r t i c l e s even impac ted the c y l i n d e r w a l l . Hence , s e g r e g a t i o n by f l o w may o c c u r in r o t a r y c y l i n d e r s . a S e g r e g a t i o n by c o n t a i n e r i s f i l l e d v i b r a t i o n i s the t h i r d mechan ism. I f w i t h a m i x t u r e o f p a r t i c l e s i z e s and 132 v i b r a t e d , i t wou ld be o b s e r v e d t h a t the l a r g e r p a r t i c l e s would s e g r e g a t e to the top o f the c o n t a i n e r and the s m a l l e r ones to the b o t t o m . S i n c e i n r o t a r y k i l n s , no v i b r a t i o n i s impa r t ed to the s y s t e m , s e g r e g a t i o n by v i b r a t i o n i s u n l i k e l y and w i l l no t be d i s c u s s e d f u r t h e r . 5 . 2 . 4 E f f e c t o f V a r i a b l e s on S e g r e g a t i o n S e g r e g a t i o n in r o t a r y c y l i n d e r s i s enhanced by l a r g e d i f f e r e n c e s in . p a r t i c l e ; s i z e , , shape and : . . . 2 7 , 5 2 , 5 7 , 6 6 , 1 2 4 , 1 2 6 - 1 3 0 T U f f + „ - „ d e n s i t y . The e f f e c t s o f t hese m a t e r i a l v a r i a b l e s s h o u l d be r e f l e c t e d i n the f l o w p r o p e r -t i e s and p a c k i n g c h a r a c t e r i s t i c s o f g r a n u l a r s o l i d s i n r o t a r y c y l i n d e r s . For e x a m p l e , f o r any g i v e n m a t e r i a l the s l u m p i n g f r e q u e n c y and ang le o f repose are b e l i e v e d to change w i t h the a d d i t i o n o f f i n e r s i z e d s o l i d s . 2 6 ' 1 1 0 How-e v e r , no c o n c l u s i v e e x p e r i m e n t a l e v i d e n c e o f t h i s has been f o u n d . O b s e r v a t i o n s on the f l o w o f g r a n u l a r s o l i d s f rom b i n s and hoppers i n d i c a t e t h a t the a d d i t i o n o f f i n e s to a c o a r s e r grade o f m a t e r i a l does a f f e c t the f l o w c h a r a c t e r -1 34 i s t i c s . by e n h a n c i n g or a b a t i n g the d i s c h a r g e o f the s o l i d s . S i m i l a r e f f e c t s have n o t y e t been r e p o r t e d f o r s o l i d s f l o w i n r o t a r y c y l i n d e r s . No r i g o r o u s s tudy o f the p a c k i n g c h a r a c t e r i s t i c s of the s e g r e g a t e d co re (commonly c a l l e d the ' k i d n e y ' i n : 133 i n d u s t r y ) i n a r o t a r y c y l i n d e r has been c a r r i e d o u t . Some v i s u a l o b s e r v a t i o n s have been r e p o r t e d , a l t h o u g h many o f them are c o n t r a d i c t o r y . Most i n v e s t i g a t o r s b e l i e v e t h a t the c o m p o s i t i o n o f the s e g r e g a t e d co re i s made-up o f the f i n e r , o r dense r m a t e r i a l , s u r r o u n d e d by the c o a r s e r , o r 2 7 52 55 57 128 l i g h t e r p a r t i c l e s . » » » » However , the e x p e r i -menta l o b s e r v a t i o n s o f s e v e r a l i n v e s t i g a t o r s i n d i c a t e t h a t t h i s co re i s c o m p r i s e d o f bo th f i n e and c o a r s e , dense and l i g h t p a r t i c l e s . 6 6 , 1 2 8 , 1 2 9 , 1 3 2 In c o n c l u s i o n , i t i s c l e a r t h a t more e x p e r i m e n t a l measurements on s e g r e g a t e d m a t e r i a l s i n r o t a r y c y l i n d e r s are r e q u i r e d . Based on c u r r e n t k n o w l e d g e , the mechanisms o f s e g r e g a t i o n in these d e v i c e s canno t be f u l l y e l u c i d a t e d . 5 . 3 M a t e r i a l P r e p a r a t i o n Four g r a n u l a r s o l i d s m i x t u r e s were used in t h i s i n -v e s t i g a t i o n , two were p r e p a r e d f rom sands B and C and two from l i m e s t o n e B and E. T h e i r c o m p o s i t i o n s a re l i s t e d i n Tab le I X . Seven ty mesh and 20 mesh s i e v e s were used to measure the f i n e s c o n t e n t in the sand and l i m e s t o n e m i x t u r e s r e s p e c t i v e l y . Both i d e n t i f y the f i n e s end o f the s i z e d i s t r i -b u t i o n f o r sand B and l i m e s t o n e B. Sand B c o n t a i n e d 1.1% -70 mesh and l i m e s t o n e B, 0.7% - 20 mesh. TABLE IX C o m p o s i t i o n o f M i x t u r e s Used f o r the S e g r e t a t i o n Study Mixture Coarser Component Weight % of Coarser - Component Finer Component Weight % of. Finer Component Overal l Mixture Composition d P coarse °P fine Sand: Mix A Mix B Sand B Sand B 94.5 72.6 Sand C Sand C 5.5 27.4 3.6% - 70 mesh 16.7% - 70 mesh 2.2 2.2 Limestone: Mix A Mix B Limestone B : Limestone 'B 91.0 78.8 Limestone E Limestone E 9.0 21.2 8.7% - 20 mesh 19.6% - 20 mesh 8.0 8.0 135 The f o l l o w i n g method o f m a t e r i a l p r e p a r a t i o n was f o l l o w e d f o r the f i r s t s t age o f the s t u d y . A s u f f i c i e n t q u a n t i t y of m a t e r i a l was we ighed and mixed to f i l l 20% of the volume of c y l i n d e r s A ( 0 .4 m ID x 0 .4 m L) and B (0 .4 m ID x 0 .86 m L ) . S u b s e q u e n t l y , the t o t a l p r e p a r e d m i x t u r e was p l a c e d in the a p p r o p r i a t e ; cy l i n d e r and the r e q u i r e d o b s e r v a t i o n s and da ta were r e c o r d e d f o r t h a t bed d e p t h . A f t e r the 20% f i l l r u n , f u r t h e r removal o f s o l i d s was..made by m a n u a l l y m i x i n g the c y l i n d e r c o n t e n t s and randomly remov ing a c e r t a i n p o r t i o n o f the m i x t u r e , f rom d i f f e r e n t l o c a t i o n s in the c y l i n d e r . For the i n v e s t i g a t i o n o f the c o m p o s i t i o n and s i z e of the s e g r e g a t e d c o r e , the second s tage of the s t u d y , o n l y one bed depth was t e s t e d in both c y l i n d e r s A and B. The m i x t u r e was reused f o r the s l u m p i n g and r o l l i n g e x p e r i m e n t s p e r -formed i n the same c y l i n d e r . 5 .4 S e g r e g a t i o n and Modes o f Bed B e h a v i o u r I t has a l r e a d y been shown in S e c t i o n 5.2 t h a t s e g r e g a t i o n due to s i z e d i f f e r e n c e s o c c u r s in the s o l i d s b e d . I t i s no t known, however , whe ther t h i s s e g r e g a t i o n w i l l a f f e c t the p o s i t i o n o f the s i u m p i n g - r o l 1 i n g boundary on a B e d - B e h a v i o u r D iag ram. Us ing the s e n s o r s , B e d - B e h a v i o u r Diagrams were t h e r e f o r e e x p e r i m e n t a l l y de te rm ined f o r sand 136 mix A and B and f o r l i m e s t o n e mix A . The r e s u l t s a re shown in F i g u r e s 5 .2 to 5 . 4 . a n d the run numbers c o r r e s p o n d i n g t o each o f the bed depths t e s t e d are l i s t e d i n Tab le B.2 (Append ix B ) . In t hese t e s t s the s o l i d s bed was m a n u a l l y mixed a t r e g u l a r i n t e r v a l s and each bed b e h a v i o u r o b s e r v a -t i o n was made ove r a r e l a t i v e l y s h o r t p e r i o d o f t ime ( f rom 5-30 m i n u t e s ) . T h e r e f o r e , on the b a s i s of the k i n e t i c s o f s e g r e g a t i o n , the measured bed b e h a v i o u r s h o u l d o n l y be a f f e c t e d by r a d i a l s e g r e g a t i o n . That r a d i a l s e g r e g a t i o n had o c c u r r e d was c o n f i r m e d by m a n u a l l y d i s p l a c i n g l a y e r s o f p a r t i c l e s f rom t h e bed s u r f a c e . The p a r t i c l e s on the s u r -f ace were a lways t he c o a r s e s t components o f the m i x t u r e and the f i n e r components were seen o n l y a f t e r s e v e r a l p a r t i c l e l a y e r s had been removed. For the s a n d . m i x t u r e s , no a p p r e c i a b l e e f f e c t o f f i n e s was a p p a r e n t w i t h mix A wh ich c o n t a i n e d 3.6% - 70 mesh ( F i g u r e 5 . 2 ) , a l t h o u g h the b o u n d a r i e s d i d s h i f t s l i g h t l y to the l e f t . At a f i n e s c o n t e n t o f 16.7% - 70 mesh, a s i g n i f i c a n t s h i f t in the bed b e h a v i o u r b o u n d a r i e s r e s u l t e d ( F i g u r e 5 . 3 ) . S i n c e the s 1 u m p i n g - t r a n s i t i o n boundary o f sand mix B (16.7% - 70 mesh) o c c u r r e d a t under 0.1 r / m i n , i t was not r e c o r d e d ; bu t c l e a r l y t r a n s i t i o n -r o l l i n g boundary had a l s o s h i f t e d to l ower r o t a t i o n a l s p e e d s . N o t e j t h a t a l l b o u n d a r i e s s t i l l show a bed depth dependency . 137 IxlO 6 1X10* Froude number (Fr=CUR/g) IXIO4 IXIO3 0.0 £ f 0.06 "D -D 0) CD 0.03 Slumping Rolling I |Tronsition 0 9 9 Sond mix A •• B 9 © © © DO 0DOQ9©© • cmo 3 ca • , ' \ \> i i osoaoKiatxK>©9© 0.5 I 1.5 Rotational speed (r /min) F i g u r e 5.2 Bed-B e h a v i o u r Diagram f o r sand mix A (3.6%-70 mesh)in c y l i n d e r A (0.4 m ID x 0.46 m L) e x p e r i m e n t a l l y d e t e r m i n e d and compared to the B e d - B e h a v i o u r Diagram o f sand B •(!.!%-70 mesh). 2 Froude number (Fr=OJR/g) Ixi0 6lxl6 5 IxlO 4 Ix lO 3 O 0.5 I 1.5 2 Rotationol speed (r /min) F i g u r e 5.3 The t r a n s i t i o n - r o l 1 i n g boundary f o r sand mix B (16.7%-70 mesh) i n c y l i n d e r A (0.4 m ID x 0.46 m L) e x p e r i m e n t a l l y d e t e r m i n e d and compared t o th o s e f o r sand B ( l . l % - 7 0 mesh) and sand mix A (3.6%-70 mesh). -6 -5 IXIO IXIO - 4 IXIO Froude number ( Fr =OJ R/g) - 3 IXIO 0 0 9 E JC CL OJ XJ X3 OJ m 0.06 0.03 1 1 1 Slumping Transition O O OO OCXJOOOCOOO 3 1 Rolling • • — 0 o \ v o x o \ \ 0 0 0 0 O (TJCX*» \ \ \ \ \ \ — o o \ xO 00(p3«($» v \ o O O \ 0 \ o o o o o — O Q & Limestone mix A » B 1 1 1 1 1 1 0 0.5 2.5 20 15 10 I 1.5 2 Rotational speed (r/min) Figure 5.4 Bed-Behaviour Diagram of limestone mix A (8.7%-20 mesh) in cy l inder A (0.4 m ID x 0.46 m L) experimentally determined and compared to the Bed-Behaviour Diagram of limestone B (0.7%-20 mesh). oo Q u i t e a d i f f e r e n t r e s u l t was o b t a i n e d f o r the l i m e s t o n e mix A wh ich c o n t a i n e d 8.7% - 20 mesh. H e r e , the bed b e h a v i o u r b o u n d a r i e s have s i g n i f i c a n t l y s h i f t e d to the r i g h t . The bed depth dependency has a l s o been g r e a t l y e n h a n c e d . In c o n c l u s i o n , f i n e s do a f f e c t the bed b e h a v i o u r o f g r a n u l a r m a t e r i a l s and the B e d - B e h a v i o u r Diagram a d e q u a t e l y r e f l e c t s i t . However , the cause o f the s h i f t i n the s i u m p i n g - r o l 1 i n g boundary i s no t c l e a r a t t h i s p o i n t . 5 . 5 S e g r e g a t i o n and B e d - B e h a v i o u r C h a r a c t e r i z a t i o n The e f f e c t o f f i n e s on the c h a r a c t e r i z a t i o n v a r i a b l e s of s l u m p i n g and r o l l i n g w i l l now be i n v e s t i g a t e d and w i l l show whe the r f l ow s e g r e g a t i o n i s one o f the mechanisms r e s p o n s i b l e f o r changes in bed b e h a v i o u r . The v a r i a b l e s measured are the s t a t i c and dynamic a n g l e s o f r e p o s e , the s h e a r a n g l e , the s l u m p i n g f r e q u e n c y and the a c t i v e l a y e r th i c k n e s s . 5 .5 .1 S lump ing C h a r a c t e r i s t i c s The s t a t i c a n g l e s o f r e p o s e , measured i n the r o t a t i n g c y l i n d e r and the s h e a r a n g l e s o f the components and mixes used in t h i s s t u d y are l i s t e d in Tab le X. The s t a t i c ang le of repose was no t a f f e c t e d by the f i n e s ; on the o t h e r hand , a s l i g h t t r e n d i s appa ren t in the s h e a r TABLE X E f f e c t o f F i n e s on the S t a t i c Ang le o f Repose* and the Shear A n g l e * f o r the M i x t u r e s T e s t e d Type Composition S ta t i c Angle of Repose (degrees) Standard Deviation (degrees) Number of Observations Shear Angle (degrees) Standard Deviation (degrees) Number of Observations Sand: %-70 mesh B 1.1 34.2 0.6 7 32.2 0.5 44 Mix A 3.6 34.0 0.7 6 32.8 0.3 6 Mix B 1.6.7 34.2 0.3 5 33.0 0.5 5 C 61.4 34.0 - 1 33.0 - 1 Limestone: %-20 mesh B 0.7 42.0 1.1 29 37.7 1.1 51 Mix A 8,7 41.4 0.8 23 36.9 0.7 23 * - measured from c y l i n d e r A ( 0 . 4 mID x 0.46 m l ) O a n g l e . Wi th i n c r e a s i n g f i n e s c o n t e n t , the shea r a n g l e seems to i n c r e a s e f o r the sand and d e c r e a s e f o r the l i m e -s t o n e . T h i s t r e n d t h o u g h , i s not s t a t i s t i c a l l y s i g n i f i -can t . The s l ump ing f r e q u e n c y r e s u l t s a re p r e s e n t e d i n F i g u r e s 5.5 and 5.6 and show t h a t the s l ump ing f r e q u e n c i e s o f the mixes were the same as those o f t h e i r r e s p e c t i v e c o a r s e components . A t temp ts to measure the s l ump ing f r e -quency o f the f i n e r components o f the sand and l i m e s t o n e mixes p roved u n s u c c e s s f u l . Both sand C and l i m e s t o n e E changed f rom s l u m p i n g to r o l l i n g a t such low r o t a t i o n a l s p e e d s , t h a t s u f f i c i e n t da ta ove r a wide enough range o f speeds c o u l d not be o b t a i n e d . I t can be deduced however , f rom the r e s u l t s i n S e c t i o n 4.5.1, t h a t the s l ump ing f r e -q u e n c i e s would have been h i g h e r than those f o r t h e i r r e -s p e c t i v e components . The measurements o f the s t a t i c a n g l e s , s h e a r a n g l e s and the s l ump ing f r e q u e n c i e s i n d i c a t e t h a t f o r the sys tems t e s t e d , the f l o w c h a r a c t e r i s t i c s o f s l ump ing beds were not s i g n i f i c a n t l y a f f e c t e d by the p r e s e n c e o f f i n e s . The f l o w mechan ism, t h e r e f o r e , d i d not c o n t r i b u t e to the s e g r e g a t i o n o f t h e s e mixes when s l u m p i n g . 142 Figure 5.5 Slumping frequency versus ro ta t iona l speed for sand B ( l . l%-70 mesh) and sand mix A (3.6%-70 mesh) and B (16.7%-70 mesh) in cy l inder A (0.4 m ID x 0.46 m L ) . 0 0.5 I 1.5 2 Rotat iona l s p e e d ( r / m i n ) Figure 5.6 Slumping frequency versus rotational speed for limestone B (0.7%-20 mesh) and limestone mix A (8.7%-20 mesh) in cylinder A (0.4 m ID x 0.46 m L). CO 144 5 . 5 . 2 R o l l i n g C h a r a c t e r i z a t i o n The dynamic a n g l e of repose and the a c t i v e l a y e r t h i c k n e s s were measured f o r the mixes under s t udy f o r a r o l l i n g b e d . I t i s seen f rom the r e s u l t s l i s t e d i n Tab le XI t h a t the dynamic ang le o f repose was no t a f f e c t e d by the f i n e s . The a c t i v e l a y e r t h i c k n e s s measurements shown i n F i g u r e s 5 .7 and 5 .8 f o r the sands and l i m e s t o n e s a l s o show no i n d i c a t i o n o f the p r e s e n c e o f f i n e s . These r e s u l t s were then made n o n - d i m e n s i o n a l w i t h r e s p e c t to the c o a r s e p a r t i -c l e s i z e , and the r e s u l t s compared w i t h those p r e s e n t e d i n F i g u r e s 4 . 2 5 and 4 .26 ( F i g u r e s 5.9 and 5 . 1 0 ) . A g a i n , the p resence o f f i n e s i s no t d e t e c t e d . I t i s t h e r e f o r e c l e a r t h a t f o r the mixes u s e d , the f l o w mechanism was r e s p o n s i b l e f o r n e i t h e r the change in bed b e h a v i o u r r e p o r t e d i n S e c t i o n 5.4 nor f o r the r e s u l t i n g r a d i a l s e g r e g a t i o n . 5.6 The S e g r e g a t i o n Cores There r e m a i n s , t h e r e f o r e , o n l y one mechanism by wh ich s e g r e g a t i o n c o u l d have o c c u r r e d : the p e r c o l a t i o n mechan ism. S i n c e none o f the i n v e s t i g a t o r s c i t i n g the p resence o f the s e g r e g a t i o n co re have e x p l a i n e d the p r o c e s s by which s e g r e g a -t i o n o c c u r r e d , an a t tempt w i l l be made to d e s c r i b e i t i n t h i s s e c t i o n . P r i o r to t h a t d i s c u s s i o n , howeve r , i t i s i m p o r t a n t to u n d e r s t a n d the f i n a l c o n f i g u r a t i o n and c o m p o s i t i o n o f the TABLE XI E f f e c t o f F i n e s on the Dynamic Ang le o f Repose f o r the M i x t u r e T e s t e d Type Compos i t i o n Dynami c S t a n d a r d Number Ang le o f D e v i a t i o n of Repose ( d e g r e e s ) O b s e r v a t i ons (deg rees ) %-70 mesh Sand : B 1.1 33 .6 0 .8 41 Mix A 3.6 33 . 6 0.4 16 Mix B 16.7 33 .8 0 .3 12 C 61.4 34 .0 - 1 %-ZO mesh L i m e s t o n e : B 0.7 39. 6 1 . 3 19 Mix A 8.7 39 .2 1 . 0 10 on 146 10 . 8 > rpm >2 2>rpm>l 1 >rpm Sond © © o B • tt A Mix A B a • Mix B © o o © o E D D o o B • 20 40 60 80 Bed depth H (m m) a a 9 • © • O O o tt A — | A 100 Figure 5 - 7 Act ive layer thickness versus bed depth for sand B p.1%-70 mesh) and sand mix A (3.6%-70 mesh) and B (16.7%-70 mesh) in cy l inder A (0.4 m ID x 0.46 m L) V c 20 16 12 1 1 1 1 rpm 2 2 2>rpm>l Limestone A — • © B • • — • tt Mix A A • — A • • e tt • tt • © — © — © 1 1 1 1 20 40 60 Bed depth H (mm) 80 100 Figure 5.8 Act ive layer thickness versus bed depth for limestone B (0.7%-20 mesh) and limestone mix A (8.7%-20 mesh) in cy l inder A (0.4 m ID x 0.46 m L ) . 18 15 12 a ? 9 147 A © o <> o 8 © © A © o A A <> O O o o e © © • T <> A o ^ O A ©O O O o rpm > 2 2 >rpm > 1 1 >rpm • © 0 • O o Sand mix A • A " " B • B Q Limestone mix A 50 1 0 0 1 5 0 200 H / d , Figure 5.9 Dimensionless P lo t of act ive layer thickness versus bed depth. Comparison of segregation resu l ts with those from Section 4 .5 .2 . 0.32 0.24 I o 0.1 6 0.0 8 rpm£ 2 2>rpm>l 1 > rpm • © o • O o Sand mix A . A Jb. A " B • E D Limestone mix A © © © © © ©© • © * 5 © © O © • o s o \ 8 • 4 A A O O 50 100 H/dp 150 200 Figure 5.10 Relat ive thickness of act ive layer with respect to the dimension-less bed depth. Comparison of segregation resu l ts with those from Section 4 .5 .2 . 148 c o r e , as i t c o u l d w e l l shed l i g h t on t h i s s e g r e g a t i o n p r o c e s s . The f i r s t s t e p i n t h i s a n a l y s i s i s to de te rm ine whe the r the f i r s t c r i t e r i o n o f the p e r c o l a t i o n mechanism i s met by the m a t e r i a l s used i n t h i s s t u d y . T h i s c r i t e r i o n s t a t e s t h a t the v o i d s between the c o a r s e r p a r t i c l e s must be o f a l a r g e r s i z e than t he f i n e r p a r t i c l e s . Dona ld and R o s e m a n 6 7 ' 1 3 0 have c l a i m e d t h a t i f the ave rage c o a r s e - t o - f i n e s i z e r a t i o i s g r e a t e r than 1 . 2 , r a d i a l s e g r e g a t i o n w i l l o c c u r s i n c e the s i z e o f v o i d s w i l l be g r e a t e r than t h a t of the f i n e s . The 2 7 e x p e r i m e n t a l r e s u l t s o f Reu te r a l s o v e r i f y t h i s f i n d i n g . As shown i n Tab le I X , both s e t s o f components used i n t h i s s t udy had s i z e r a t i o s g r e a t e r than 1 . 2 . That the s e g r e g a t i o n c r i t e r i o n o f v o i d s i z e has been met was f u r t h e r c o n f i r m e d by the v o i d f r a c t i o n v e r s u s c o m p o s i t i o n p l o t o f F i g u r e 3 . 1 3 . Note t h a t the l i m e s t o n e m i x t u r e has a l a r g e r c o a r s e to f i n e s i z e r a t i o than the sand m i x t u r e and i t a l s o e x p e r i e n c e s a g r e a t e r maximum bed c o n t r a c t i o n , Ae x < H e n c e , the f i r s t c r i t e r i o n o f the p e r c o l a t i o n mechanism i s s a t i s f i e d . The second s tage o f t h i s a n a l y s i s wou ld be to ' d e t e r -mine the l o c a t i o n and c o m p o s i t i o n o f the s e g r e g a t e d c o r e . For t h i s p a r t o f the s t u d y , l i m e s t o n e s B and E were chosen as they d i s p l a y e d the h i g h e r component s i z e r a t i o , as w e l l as the l a r g e s t A e m a x - H e n c e , they show the g r e a t e r s e g r e g a -t i o n t e n d e n c y . The most d i r e c t method o f l o c a t i n g the co re 149 and i d e n t i f y i n g i t s c o m p o s i t i o n i s by c a r e f u l removal o r s a m p l i n g o f s o l i d g r a n u l e s both i n the bed depth and a l o n g the bed s u r f a c e . For the sys tem u s e d , however , the removal o f 1ayers o f s o l i d s i n the bed depth d i r e c t i o n i m p l i e s t h a t each sample l a y e r c o l l e c t e d would o n l y be a few c o a r s e s i z e d p a r t i c l e s . d e e p . The r e s u l t s would not be m e a n i n g f u l . I n -s t e a d , p o r t i o n s o f the bed were removed f rom i t s s u r f a c e to the i n n e r c y l i n d e r w a l l and a n a l y z e d f o r the f i n e s com-ponen t . A t h e o r e t i c a l model was a l s o i n d e p e n d e n t l y d e -v e l o p e d to c a l c u l a t e the d i m e n s i o n s o f the c o r e . The model p r e d i c t i o n s were then compared w i t h the e x p e r i m e n t a l r e s u l t s . 5 .6 .1 The S e g r e g a t e d Core Model To d e v e l o p the model of the s e g r e g a t e d c o r e , i t s shape must f i r s t be assumed. S e v e r a l c o n f i g u r a t i o n s a re 1 35 s u g g e s t e d by V a i l l a n t , r a n g i n g f rom the e l l i p t i c a l and c i r -c u l a r to the r e c t a n g u l a r . None o f t h e s e , t h o u g h , a re i n a g r e e -ment w i t h the c i r c u l a t i o n p a t t e r n s o f the s o l i d s d i s c u s s e d i n S e c t i o n s 2 . 2 . 3 and 4 . 5 . 2 . The co re i s e x p e c t e d to t h e r e f o r e have a shape s i m i l a r to t h a t o f the s o l i d s bed but r e c e s s e d f rom the bed s u r f a c e as shown i n F i g u r e 5 .11a ( r e c a l l t h a t f i n e s were not v i s i b l e on the bed s u r f a c e i n the B e d - B e h a v i o u r Diagram d e t e r m i n a t i o n ) . Thus the co re w i l l be sym-m e t r i c a l about the p e r p e n d i c u l a r OA ( F i g u r e 5 . 1 1 a ) . The 150 Figure 5.11 Schematic representat ion of the centra l segregation core (a) segregation core recessed from the surface of the bed (b) surface of core at the bed sur face. 151 s i z e o f t h e c o r e w i l l o f c o u r s e be d e p e n d e n t on i t s c o m p o s i -t i o n a n d on t h e t o t a l f i n e s c o n t e n t . A s s u m i n g t h a t a l l t h e f i n e s a r e s e g r e g a t e d i n t h e c o r e a n d t h a t l o n g i t u d i n a l s e g r e g a t i o n i s n e g l i g i b l e , t h e w e i g h t f r a c t i o n o f t h e f i n e r c o m p o n e n t i n t h e c o r e i s g i v e n b y : 1 - y = xW T ( 5 . 1 ) T h e v o l u m e V^, o c c u p i e d b y t h e c o r e i s t h e r e f o r e : V K = xW T ( 5 . 2 ) S i n c e , i n t h e e x p e r i m e n t a l p h a s e o f t h i s s t u d y , t h e s o l i d s b e d was n o t s a m p l e d r a d i a l l y , t h e d i s t a n c e t h e c o r e i s r e c e s s e d f r o m t h e b e d s u r f a c e c a n n o t be v e r i f i e d . O n l y i t s w i d t h c a n t h e r e f o r e be c o m p a r e d t o t h e e x p e r i m e n t a l r e s u l t s . F o r t h e p u r p o s e s o f t h e m o d e l c a l c u l a t i o n s , t h e s u r f a c e o f t h e c o r e was c o n s i d e r e d t o be a t t h e s u r f a c e o f t h e b e d i n o r d e r t o s i m p l i f y t h e d e r i v e d f o r m u l a e ( F i g u r e 5 . 1 1 b ) . T h i s d o e s n o t a p p r e c i a b l y a f f e c t t h e m a g n i t u d e s o f t h e c a l c u l a t e d w i d t h o f t h e c o r e v o l u m e . Th e r a d i u s o f t h e o u t e r c o r e b o u n d a r y R^, i s t h e r e f o r e g i v e n b y : ( 5 . 3 ) a n d t h e c o r e v o l u m e i s t h e n c a l c u l a t e d as f o l l o w s 152 L( R - H ) L (5.4) F i n a l l y , the depth of the c o r e , H may be ob ta ined f r o m : = - (R-H) (5 .5) Equat ions (5.2) to (5 .4) are now s o l v e d by t r i a l and e r r o r f o r w^. Given a core c o m p o s i t i o n , i t s c o r r e s p o n d i n g bulk d e n s i t y may be measured as d e s c r i b e d in S e c t i o n 3 . 2 . 3 . The f i n e s content is g iven by the mixture compos i t ion and the t o t a l w e i g h t ; both of which are measured. Using Equa-t i o n s (5 .3) and ( 5 . 4 ) , an a p p r o p r i a t e value of w^ i s chosen to y i e l d a V K equal to that c a l c u l a t e d from Equat ion ( 5 . 2 ) . Two core c o m p o s i t i o n s were assumed f o r the model p r e d i c t i o n s . For the f i r s t , the core was c o n s i d e r e d to c o n s i s t of on ly the f i n e r component of the l imestone mix B. For the s e c o n d , i t was assumed that the core compos i t ion was equal to that f o r which maximum bed c o n t r a c t i o n , A e m a x , o c c u r r e d in F igure 3 .13 . The core width p r e d i c t i o n s from both these cases were compared to the exper imenta l r e s u l t s o b t a i n e d and d e s c r i b e d in the f o l l o w i n g s e c t i o n . 5 .6 .2 Exper imenta l P r o c e d u r e , Sampling and A n a l y s i s The s e g r e g a t i o n t e s t s were c a r r i e d out at on ly 153 one bed d e p t h , 0 .08 tn o r 14% degree o f f i l l , u s i n g l i m e s t o n e mix B (19.6% - 20 mesh) i n both c y l i n d e r s A and B (0 .4 m ID x 0 .46 m L and 0 .86 m L r e s p e c t i v e l y ) . The r e q u i r e d w e i g h t s o f each o f the m i x t u r e componen ts , l i m e -s tones B and E , were p l a c e d i n the c y l i n d e r s and m a n u a l l y m i x e d . To t e s t t h e s e g r e g a t i o n p a t t e r n f o r s l u m p i n g , and r o l l i n g , two r o t a t i o n a l speeds were chosen f o r the t r i a l s , 0 .6 and 3.1 r / m i n . The d u r a t i o n o f each run was a t l e a s t 4 h o u r s . L o n g i t u d i n a l s e g r e g a t i o n was o b s e r v e d to o c c u r f o r both the s l u m p i n g and r o l l i n g beds and was v i s u a l l y d e t e r -mined to be comple te w i t h i n the 4 hours o f c y l i n d e r r o t a t i o n . F o l l o w i n g the c o m p l e t i o n o f a r u n , m a t e r i a l had to be sampled f o r a n a l y s i s . To a c c o m p l i s h t h i s , s e v e r a l t e c h n i q u e s r e p o r t e d in the l i t e r a t u r e 2 7 ' 3 7 ' 4 1 ' 4 2 ' 5 7 ' 1 2 4 ' 1 3 6 " 1 3 8 were a t t e m p t e d bu t were no t found a d a p t a b l e to the c u r r e n t s y s t e m . A d i f f e r e n t approach was t h e r e f o r e d e v e l o p e d . Four s t e e l p l a t e s each w i t h a measu r i ng g r i d , g l u e d i n p l a c e , were cu t to the shape o f the i n s i d e c i r c u m f e r e n c e of the r o t a t i n g c y l i n d e r ( F i g u r e 5 .12 ) and were suspended above the s o l i d s bed i n s i d e the c y l i n d e r . At the end o f the r u n , these p l a t e s were l o w e r e d i n t o the s o l i d s bed by h o l d i n g them a g a i n s t the c y l i n d e r w a l l o p p o s i t e the s o l i d s bed and r o t a t i n g the c y l i n d e r th rough h a l f a r e v o l u t i o n . Thus , the whole l e n g t h o f the bed was d i v i d e d a x i a l l y i n t o f i v e s l i c e s . The m i d d l e 154 Figure 5.12 S l i c e r made of 1.6 mm th ick steel sheet covered with poster board. Figure 5.13 S l i c e r s introduced into the charge, in preparation for so l ids removal. 155 th ree s l i c e s , i n c y l i n d e r A , were s a m p l e d , each h a v i n g 76 mm t h i c k n e s s ( F i g u r e 5 . 1 3 ) . In c y l i n d e r B , t h e i r t h i c k n e s s e s were i n c r e a s e d to 144 mm in o r d e r to equa l the same t h i c k n e s s to c y l i n d e r l e n g t h r a t i o as i n c y l i n d e r A . Ra the r than remov ing s m a l l f r a c t i o n s of s o l i d s f rom s e v e r a l l o c a t i o n s o f the bed and a n a l y z i n g the r e s u l t s s t a t i s t i c a l l y , as has been commonly done by o t h e r i n v e s t i -g a t o r s , 7 2 ' 7 3 ' 9 0 ' 1 2 4 ' 1 2 7 ' 1 2 9 ' 1 3 6 - 1 3 9 the t o t a l c o n t e n t s o f each o f the m idd le t h r e e s l i c e s were removed f o r a n a l y s i s . The fo rmer method wou ld have r e q u i r e d more t ime to implement and to a n a l y z e the d a t a . F u r t h e r m o r e , u s i n g the l a t t e r a p p r o a c h , the a n a l y s i s wou ld be f r e e o f a n y ' o f t h e . ' '. s t a t i s t i c a l u n c e r t a i n t i e s u s u a l l y a s s o c i a t e d w i t h s a m p l i n g „ c 72 , 7 3 , 76 , 90 ,129 , 1 36 , 140-142 _. , . . . . t e c h n i q u e s . The s o l i d s bed was t h e r e f o r e l e f t i n i t s i n c l i n e d p o s i t i o n and the s o l i d s in the bed were removed i n s e v e r a l p o r t i o n s s t a r t i n g a t the apex and w o r k i n g down to the c h o r d a l b a s e . Each p o r t i o n sampled r e p r e s e n t e d the s o l i d s f rom a g i ven i n c r e m e n t o f the bed s u r f a c e . They were c o l l e c t e d u s i n g a vacuum c l e a n e r e q u i p p e d w i t h a c l o t h bag s t a r t i n g a t the bed s u r f a c e and w o r k i n g towards t he c i r c u m f e r e n t i a l w a l l . T h i s a l l o w e d the s e l e c t i v e removal o f the bu l k s o l i d s . The c o l l e c t e d p o r t i o n s o f the bed were a n a l y z e d f o r t h e i r - 20 mesh c o n t e n t . The s c r e e n a n a l y s i s r e s u l t s are r e p o r t e d as a w e i g h t pe r cen t -o f the bed p o r t i o n 156 20 mesh w i t h r e s p e c t to the t o t a l we igh t s c r e e n e d . The l o c a t i o n o f each o f these p o r t i o n s had a l s o to be i d e n t i f i e d . By means o f both a m i r r o r and the g r i d s on t h e s t e e l p l a t e s l i c e r s , the c o - o r d i n a t e s o f the s o l i d s l e f t i n the c y l i n d e r were r e c o r d e d . These were then r e -p l o t t e d on graph paper u s i n g a 1:1 s c a l e . F i r s t l y , the a rea o f t h e bed was measured u s i n g a Compensa t ing P l a n i m e t e r (Ko i zum i Type K P - 2 7 ) , and the bed depth o f the s l i c e c a l c u -l a t e d by t r i a l and e r r o r u s i n g the f o l l o w i n g e q u a t i o n : The d i s t a n c e f rom the apex o f each c o l l e c t e d p o r t i o n was a l s o measured f rom the g raphed g r i d p l o t , and was r e p o r t e d as a p e r c e n t a g e o f the t o t a l cho rd l e n g t h , f o r ease i n com-p a r i n g the r e s u l t s f rom v a r i o u s t r i a l s . The e x p e r i m e n t a l r e s u l t s are p r e s e n t e d f o r each s l i c e on a ba r graph as shown i n F i g u r e s 5 .14 to 5 . 1 7 . 5 . 6 . 3 D i s c u s s i o n 5 . 6 . 3 . 1 R a d i a l S e g r e g a t i o n From the e x p e r i m e n t a l r e s u l t s shown in F i g u r e s 5 .14 to 5 . 1 7 , i t i s seen t h a t r a d i a l s e g r e g a t i o n ( 5 . 6 ) 157 I02mm76mm 127mm 1 76mm 76mm , | 00% 2 0 01 E O CM I 6 0 40F CO c 2 0 S l i c e 3 -> 1 1'— L i m e s t o n e mix B 0 .57 r / m i n _ 1 4 % fill A v g % F ines in slice " " " "cyl inder | P r e d i c t i o n s m Q X composit ion. 1 0 0 % f ines S l i c e 2 4 0 2 0 h S l i c e I 1 0 0 8 0 6 0 4 0 2 0 % C h o r d l eng th F i g u r e 5.14 E x p e r i m e n t a l r e s u l t s and c o r e w i d t h model p r e d i c t i o n s f o r s l u m p i n g i n c y l i n d e r A (0.4 m ID x 0.46 m L ) . 158 2" 20 lOr-_ 2 0 JZ 1/7 CD E o CM ^ 10 c 0 2 0 1 i r Slice 3 Slice 2 Slice H 116 mm 76 mm 76 mm 76mm 113mm Viewing end o% Limestone mix B 3.11 r /min 16 % fi l l —Avg % fines in slice -Avg " " " cylinder Predictions —i A£ m Q x compos i t ion 100% fines 100 80 60 4 0 20 % Chord length Figure 5.15 Experimental results and core width model predictions for r o l l i n g in cylinder A ( 0 . 4 m ID x 0 . 4 6 m L). 159 4 0 2 0 CO 0) E O C\J i >L 4 0 to cu c 2 0 2 0 S l i c e 3 S l i c e 2 S l i c e I S l ice 3 2 211 m m 1 4 4 m m 1 4 4 m m 1 4 4 m m 21'5 m m V iewing e n d I 0 0 % L i m e s t o n e mix B 0 . 5 9 r / m i n 14 % f i l l A v g % f i n e s in sl ice II II n II cylinder L^€fmQxcomposition I 1 1 0 0 % f i n e s 1 0 0 8 0 6 0 4 0 2 0 % C h o r d length Figure 5.16 Experimental resu l ts and core width model predict ions for slumping in cy l inder B (0.4 m ID x 0.86 m L ) . 160 12 8 4 h -JZ co E o OJ I tn a> c 0 I 2 8 5 Sl ice 3 S l i c e 2 S l i c e J 1 ll I L imestone mix B 3.19 r / m i n 1 4 % f i l l P r e d i c t i o n s A 6 m a x c o m p o s i t i o n — 1 0 0 % f i n e s - - - - A v g % f ines in slice 2 0 2 m m 2 2 2 m m 100% 1 0 0 8 0 6 0 4 0 2 0 0 % C h o r d length ure 5.17 Experimental resu l ts and core width model predic t ions for r o l l i n g in cy l inder B (0.4 m ID x 0.86 m L ) . 161 o c c u r r e d f o r bo th s l u m p i n g and r o l l i n g b e d s . S i n c e l o n g i -t u d i n a l s e g r e g a t i o n a l s o o c c u r r e d , the co re w i d t h model was a p p l i e d to each s l i c e r a t h e r than to the t o t a l c y l i n d e r l e n g t h . The p r e d i c t i o n s o f the core w i d t h model f o r each of the s l i c e s are a l s o shown i n F i g u r e s 5.14 to 5 . 1 7 . I t i s q u i t e e v i d e n t t h a t the model based on a core c o m p o s i t i o n of o n l y f i n e s does no t show good agreement w i t h the e x p e r i -menta l r e s u l t s . I r r e s p e c t i v e o f bed b e h a v i o u r , s l ump ing or r o l l i n g , as w e l l as c y l i n d e r l e n g t h to d i a m e t e r r a t i o , c o a r s e p a r t i c l e s must form a p a r t of the core i n i t s s t e a d y s t a t e c o n f i g u r a t i o n . E x c e l l e n t agreement w i t h the e x p e r i m e n t a l r e s u l t s was o b t a i n e d f o r a l l the p r e d i c t e d co re w i d t h s u s i n g the A £ m a x c o m p o s i t i o n . T h i s i m p l i e s t h a t the s o l i d s a lways took on a c o n f i g u r a t i o n t h a t r a d i a l l y m i n i m i z e d the t o t a l volume o f the s y s t e m . 5.6.3.1 -A V a l i d a t i o n o f Assump t i ons and the  Second Core Two o f the assump t i ons o f the co re w i d t h model a re t h a t a l l f i n e s a re a s s o c i a t e d w i t h the co re and t h a t i t l i e s s y m m e t r i c a l l y about the p e r p e n d i c u l a r to the bed s u r f a c e f rom the c e n t r e o f r o t a t i o n o f the c y l i n d e r . The e x p e r i m e n t a l r e s u l t s r e v e a l t h a t t hese two assump t i ons are no t s t r i c t l y adhered to in the s y s t e m . R e g a r d i n g the f i r s t a s s u m p t i o n , a more d e t a i l e d a n a l y s i s o f the r e s u l t s r e v e a l s 162 t h a t the p e r c e n t o f -20 mesh in a s l i c e l y i n g o u t s i d e the p r e d i c t e d co re ranges f rom 3 to 13% in the apex and f rom 0 .9 to 3.0% a t the c h o r d a l b a s e . T h i s i s because the bu l k o f the bed volume l i e s between 30% and 70% of the cho rd l e n g t h . Hence , the f i r s t assumpt ion i s v a l i d . The second a s s u m p t i o n , however , t h a t of symmet ry , p r e s e n t s a more i n t e r e s t i n g p r o b l e m . The l a c k o f symmetry i s c l e a r l y appa ren t i n a l l s l i c e s , r e g a r d l e s s o f bed b e h a v i o u r and c y l i n d e r l e n g t h . The core model had i m p l i e d t h a t the f i n e s are s u r r o u n d e d by the c o a r s e r p a r t i c l e s and hence s h o u l d have no c o n t a c t w i t h the c y l i n d e r w a l l . Du r i ng the e x p e r i m e n t a l t r i a l s , though* i t was o b s e r v e d t h a t the a b r a s i v e l i n i n g in the c y l i n d e r became b l i n d e d w i t h f i n e s s h o r t l y a f t e r t he s t a r t o f runs where ca re had been taken to c l e a n the w a l l . T h e r e f o r e , t he re must be a second c o n c e n t r a t i o n of f i n e s a t the top o f the bed near the w a l l . T h i s was f u r t h e r c o r r o b o r a t e d by Sunne rg ren based on h i s o b s e r v a t i o n s o f a 1 32 s i m i l a r sys tem and u s i n g s i m i l a r s a m p l i n g t e c h n i q u e s ( F i g u r e 5 . 1 8 ) . I t i s b e l i e v e d t h a t t h i s second core c o n s i s t s o f the f i n e s t p a r t i c l e s wh ich do not move down the bed s u r -f ace when t hey reach the apex . They are so s m a l l t h a t when s e t i n m o t i o n , they are a b l e to p e r c o l a t e th rough the c o a r s e r s i z e s even when the l a t t e r are s t a t i o n a r y . These f i n e p a r t i -c l e s f i n a l l y come to r e s t a t the c y l i n d e r w a l l where they a c c u m u l a t e , f o rm ing t h i s second s e g r e g a t i o n c o r e . 164 The f i n e p a r t i c l e s in t h i s co re c o u l d e a s i l y f i l l w a l l i m p e r f e c t i o n s and s u b s e q u e n t l y be compacted i n t o p l a c e by the w e i g h t of the b e d . In r o t a r y k i l n o p e r a t i o n s , t h i s wou ld r e s u l t i n a s i g n i f i c a n t smoothen ing of the w a l l t h a t c o u l d then r e s u l t in a change i n bed b e h a v i o u r o c c u r r i n g from s l u m p i n g or r o l l i n g t o s l i p p i n g . T h i s phenomenon was o b s e r v e d i n c y l i n d e r A when d e t e r m i n i n g a B e d - B e h a v i o u r Diagram f o r i r o n - o x i d e p e l l e t s . S i n c e t h e i r s t a t i c ang le o f repose and ang le o f s l i p p i n g were 31 .5 ° and 3 8 . 9 ° r e -s p e c t i v e l y , no s l i p p i n g had been e x p e c t e d . However s l i p p i n g o c c u r r e d p a r t way th rough the t e s t , when the w a l l s had been smoothened and c o v e r e d by the ve ry f i n e i r o n o x i d e p a r t i c l e s . Hence , the f o r m a t i o n o f t h i s second s e g r e g a t i o n co re seems to be one o f the f a c t o r s r e s p o n s i b l e f o r the o n s e t o f s l i p -p ing wh ich o c c u r s in i n d u s t r i a l k i l n s . 5.6.3.1-B The P e r c o l a t i o n S e g r e g a t i o n P r o c e s s The p e r c o l a t i o n s e g r e g a t i o n p r o c e s s w i l l be p r e s e n t e d in t h i s s e c t i o n w i t h r e f e r e n c e to the r o l l i n g mode r a t h e r than the s l ump ing mode, both f o r the sake o f b r e v i t y and because i t w i l l be e a s i e r to d e s c r i b e . The d i s c u s s i o n , t h o u g h , a p p l i e s e q u a l l y to b o t h . The path o f the f i n e s w i l l be d e s c r i b e d f o r an i n i t i a l l y w e l l mixed s o l i d s bed composed o f a c o a r s e and a 165 f i n e component s i m i l a r to l i m e s t o n e s B and E r e s p e c t i v e l y . Due to the r o t a t i o n o f the c y l i n d e r f i n e p a r t i c l e s i n p o s i t i o n 1 ( F i g u r e 5 .19 ) w i l l f o l l o w a t r a j e c t o r y w i t h the c y l i n d e r w a l l i n the p a s s i v e r e g i o n u n t i l i t c r o s s e s the s h e a r p l a n e AB and e n t e r s the a c t i v e r e g i o n a t 2 . I n i t i -a l l y , they wou ld t r a v e l down the s u r f a c e bed i n c l i n a t i o n bu t wou ld soon f i n d as the a c t i v e l a y e r i n c r e a s e s i n t h i c k -ness t h a t t h e r e were c o a r s e r p a r t i c l e s i n mot ion below i t . When the v o i d s between these c o a r s e r p a r t i c l e s are l a r g e r than the f i n e p a r t i c l e s i z e , the l a t t e r wou ld f a l l i n t o these v o i d s . F u r t h e r m o r e , i f t hese f i n e p a r t i c l e s are s u f -f i c i e n t l y f i n e , they c o u l d e a s i l y f a l l ' p a s t the a c t i v e l a y e r and t r i c k l e th rough the v o i d s u n t i l they reach the c y l i n d e r w a l l , p o s i t i o n 3 . T h i s wou ld r e s u l t i n the f o r m a -t i o n o f the s e g r e g a t i o n zone o f ve ry f i n e p a r t i c l e s as e a r l i e r d e s c r i b e d . On the o t h e r h a n d , i f the f i n e p a r t i -c l e s are no t a b l e to p e r c o l a t e th rough t he s h e a r p l a n e A B , they must remain i n the a c t i v e r e g i o n and w i l l seek the f i r s t o p p o r t u n i t y t o f a l l i n t o the p a s s i v e z o n e . These f i n e p a r t i -c l e s w i l l t h e r e f o r e t r a v e l a d j a c e n t to the s h e a r p l a n e , between c o a r s e p a r t i c l e s , u n t i l 1 ine OB i s c r o s s e d . Here p a r t i c l e s may b e g i n to r e - e n t e r the p a s s i v e z o n e . The f i r s t one to do s o , o f c o u r s e , are those p a r t i c l e s , f i n e s and c o a r s e , t r a v e l l i n g j u s t above the s h e a r p l a n e . Once they have e n t e r e d the p a s s i v e z o n e , they must t r a v e l w i t h the 166 Figure 5.19 Schematic of the percolat ion segregation process for both cores. 167 c y l i n d e r wal l and reappear in the a c t i v e l a y e r e q u i d i s t a n t from t h e i r p r e v i o u s p o i n t of en t ry i n t o the p a s s i v e r e g i o n . Subsequent f i n e s c r o s s i n g OB and t r a v e l l i n g near the s l i p p lane would f i n d the zone around B s a t u r a t e d wi th f i n e s a c c o r d i n g to the Ae c o m p o s i t i o n , and would be f o r c e d to 3 max t r a v e l f u r t h e r down the s l i p p lane BC u n t i l the next f a v o u r -ab le s i t e i s e n c o u n t e r e d . As the f i n e s content i n c r e a s e s t h e r e f o r e , i t i s c l e a r that the c e n t r a l core w i l l i n c r e a s e in s i z e . The f i n e p a r t i c l e s w i l l t h e r e f o r e not be v i s i b l e at any time on the bed s u r f a c e except at very high f i n e s content which agrees with the exper imenta l f i n d i n g s . F ines were on ly v i s i b l e on the bed s u r f a c e when t h e i r content in a g iven s l i c e was between 40 and 50% by we igh t . It t h e r e -f o r e f o l l o w s t h a t , at lower f i n e s c o n t e n t , the f i n e s a c t i v e l a y e r , w i l l be s m a l l e r than the t o t a l a c t i v e l a y e r , and the core w i l l be r e c e s s e d from the bed s u r f a c e . In i n d u s t r i a l k i l n s those f i n e s a s s o c i a t e d wi th the c e n t r a l core w i l l not be exposed to gas in the f r e e b o a r d . They w i l l t h e r e f o r e be c o l d e r than the coarse p a r t i c l e s and may reach the d i s c h a r g e of the k i l n on ly p a r t l y r e a c t e d . The f i n e s a s s o c i a t e d wi th the second s e g r e g a t i o n core may r e c e i v e s u f f i c i e n t heat from t h e i r c o n t a c t wi th the w a l l . However, high l o c a l walV temperatures c o u l d r e s u l t in these f i n e s s t i c k i n g to the wal l and enhancing a c c r e t i o n b u i l d - u p , or g l a z i n g the wal l enhancing s l i p p i n g . These temperature 168 e f f e c t s a r e , o f c o u r s e , most l i k e l y t o o c c u r a t the s o l i d s d i s c h a r g e . O the r o p e r a t i n g e f f e c t s o f s e g r e g a t i o n have a l r e a d y been d i s c u s s e d i n the p r e v i o u s s e c t i o n . To overcome these many p r o b l e m s , r e c e n t i n d u s t r i a l r e s e a r c h has s u g g e s t e d the i n t r o d u c t i o n of s p e c i a l l y de-s i g n e d p r o t r u s i o n s to c o n t i n u a l l y expose the f i n e s o f both 47 s e g r e g a t e d zones to the gas f r e e b o a r d . The r e s u l t s o f 1 32 the i n d u s t r i a l t r i a l s are r e p o r t e d to be ve ry f a v o u r a b l e . 5 . 6 . 3 . 2 L o n g i t u d i n a l S e g r e g a t i o n From the r e s u l t s o b t a i n e d f rom c y l i n d e r A ( F i g u r e 5 . 1 4 ) , i t appears t h a t when the bed i s s l u m p i n g , f i n e s tend to s e g r e g a t e i n the c e n t r e of the c y l i n d e r a x i s . For a r o l l i n g bed ( F i g u r e 5 . 1 5 ) , the f i n e s s e g r e g a t e d to the c y l i n d e r end w a l l s wh ich i s i n agreement w i t h the i n v e s t i -g a t i o n s o f o t h e r w o r k e r s d e s c r i b e d e a r l i e r . The t e s t s o f s l u m p i n g and r o l l i n g beds were r e p e a t e d under i d e n t i c a l c o n d i t i o n s i n c y l i n d e r B h a v i n g a l e n g t h to d i a m e t e r r a t i o o f 2 : 1 . S i m i l a r s e g r e g a t i o n p a t t e r n s r e s u l t e d w i t h a r e a s o f l o n g i t u d i n a l f i n e s s e g r e g a t i o n b e i n g v i s i b l e on the s u r f a c e . To recheck the r e s u l t s o b t a i n e d , the bed was o f t e n m a n u a l l y rem ixed i n t h e c y l i n d e r and the t e s t r e s t a r t e d . The r e s u l t -i ng s e g r e g a t i o n d i f f e r e d o n l y i n the case o f a r o l l i n g b e d . H e r e , f i n e s were seen to s e g r e g a t e e i t h e r to both e n d - w a l l s 169 o f the c y l i n d e r or to e i t h e r e n d - w a l l . No reason was found f o r t h i s b e h a v i o u r . A f i n a l t r i a l was made i n wh ich a w e l l mixed bed was a l l o w e d to slump a t 0 .6 r /m in f o r a minimum o f f o u r h o u r s . When the f i n e s had s e g r e g a t e d to the c e n t r e o f the c y l i n d e r a x i s , the r o t a t i o n a l speed was i n c r e a s e d to 3 r /m in and l e f t f o r a t l e a s t a n o t h e r f o u r h o u r s . The f i n e s then moved to e i t h e r o r both end w a l l s . The r o t a t i o n a l speed was a g a i n reduced to 0 .6 r /m in and the f i n e s were seen to move back to the c e n t r e o f the c y l i n d e r a x i s . The bed was rem ixed and the sequence o f t r i a l s was r e p e a t e d ; t h i s t ime s t a r t i n g w i t h a r o l l i n g b e d . The same r e s u l t s were o b s e r v e d . The s l u m p i n g f r e q u e n c y was measured at v a r i o u s t imes d u r i n g t h e s l ump ing t e s t s and was no t found to va ry ove r the t e s t i n g p e r i o d . 6 7 A c c o r d i n g to the a n a l y s i s o f Donald and Roseman, the s t a t i c a n g l e s o f repose o f l i m e s t o n e s B and E ( T a b l e I I ) i n d i c a t e t h a t , f o r a r o l l i n g b e d , e n d - l o n g i t u d i n a l s e g r e g a -t i o n s h o u l d have taken p l a c e ; as was i ndeed the c a s e . They p r o v i d e d the f o l l o w i n g e x p l a n a t i o n f o r i t s o c c u r r e n c e . The r a d i a l v e l o c i t y o f p a r t i c l e s f a l l i n g down the bed i n -c l i n a t i o n v a r i e s a l o n g t he c y l i n d e r l e n g t h w i t h a maximum o c c u r r i n g some s h o r t d i s t a n c e away f rom the c y l i n d e r e n d -w a l l . A f i n e p a r t i c l e i n a band a d j a c e n t to t h i s zone of maximum v e l o c i t y wou ld see more v o i d s p a s s i n g i t on one s i d e 170 than on t h e o t h e r . Hence , the f i n e p a r t i c l e wou ld be f a v o u r e d to move a x i a l l y to the zone o f maximum v e l o c i t y . S i n c e the s t a t i c ang le o f repose o f the f i n e s i s l e s s t h a n t h a t f o r the c o a r s e , the l a t t e r was b e l i e v e d to be r a d i a l l y more m o b i l e ; t h e r e f o r e the f i n e s would o n l y be a b l e to form end-wa l1 b a n d s . I t was o b s e r v e d t h a t when s i u m p i n g , t h e f i n e s a x i a l l y s e g r e g a t e d to the c e n t r e o f the c y l i n d e r a x i s . T h e r e f o r e i t a p p e a r s , f o l l o w i n g the argument o f Donald and Roseman, t h a t f o r a s l u m p i n g bed the r a d i a l v e l o c i t y a t the c e n t r e o f the a x i s i n a h o r i z o n t a l l y r o t a t i n g c y l i n d e r i s a m a x i -mum and t h a t the r a d i a l v e l o c i t y p r o f i l e a l o n g a c y l i n d e r a x i s i s dependent on bed b e h a v i o u r . T h i s was not c o n f i r m e d , however , u s i n g the s l u m p i n g f r e q u e n c y measurements a l o n g the c y l i n d e r a x i s . In an i n c l i n e d c y l i n d e r , i t wou ld be e x p e c t e d t h a t the f i n e s a c c u m u l a t i o n wou ld o c c u r a t the l owe r end o f the c y l i n d e r i r r e s p e c t i v e o f bed b e h a v i o u r . The p rob lem of a x i a l s e g r e g a t i o n r e p o r t e d i n t h i s s tudy may t h e r e f o r e not d i r e c t l y a p p l y to i n d u s t r i a l r o t a r y k i l n p r a c t i c e . I t wou ld be o f i n t e r e s t , however , i n i n d u s t r i e s c o n c e r n e d w i t h the m i x i n g o f g r a n u l a r m a t e r i a l s , where i t i s i m p o r t a n t to o b t a i n a p r o d u c t o f c o n s i s t e n t c o m p o s i t i o n , one t h a t w i l l 171 show minimum s e g r e g a t i o n t e n d e n c i e s upon subsequen t h a n d l i n g . Many o f t hese m a t e r i a l s are mixed i n ba t ch h o r i z o n t a l r o t a r y c y l i n d e r s . Based on t he r e s u l t s o f t h i s a n a l y s i s , an o p e r a -t i n g s t r a t e g y c o u l d be d e v e l o p e d f o r these a p p l i c a t i o n s . I f a g i v e n m i x t u r e o f s o l i d s i s p l a c e d i n a r o t a r y c y l i n d e r and i s a l l o w e d to s e g r e g a t e r a d i a l l y and a x i a l l y , t he r e s u l t i n g l o n g i t u d i n a l l y s e g r e g a t e d zone would have the c o m p o s i t i o n c o r r e s p o n d i n g to t h a t m i x t u r e ' s A e m a x - I f i t can be s e p a r a t e d f rom the r e m a i n i n g c o n t e n t s o f the c y l i n d e r , i t can be h a n d l e d w i t h o u t f u r t h e r s e g r e g a t i o n . I t wou ld a l s o be the d e n s e s t volume o f s o l i d s a t t a i n a b l e f o r t h a t m i x t u r e ; a p r o p e r t y much sought a f t e r i n p h a r m a c e u t i c a l , powder m e t a l l u r g i c a l and c e r a m i c p r o c e s s i n g , f o r e x a m p l e . The o n l y p r o c e s s i n g d i f f i c u l t y e n v i s a g e d w i t h t h i s s t r a t e g y l i e s i n t h e r e c y c l i n g o f the r e m a i n i n g c o n t e n t s o f the m i x e r . In some i n s t a n c e s , t h i s c o u l d be very c o s t l y . The a d -van tages and d i s a d v a n t a g e s o f t h i s approach must t h e r e f o r e be a n a l y z e d f o r each s p e c i f i c a p p l i c a t i o n i n the c o n t e x t o f l o c a l p l a n t e c o n o m i c s . 5 . 7 Summary The e f f e c t o f 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 on the f l ow p r o p e r t i e s o f g r a n u l a r s o l i d s has been i n v e s t i g a t e d . Two t ypes o f s e g r e g a t i o n t h a t o c c u r i n h o r i z o n t a l r o t a r y 172 c y l i n d e r s were i 1 1 u s t r a t e d : ; r a d i a l , e n d - l o n g i t u d i n a l and band l o n g i t u d i n a l . B e d - B e h a v i o u r D i a g r a m s , s l u m p i n g and r o l l i n g c h a r a c t e r i s t i c s were d e t e r m i n e d f o r two sand and one l i m e -s tone m i x t u r e s . Wh i l e the f i n e s r e s u l t e d i n s i g n i f i c a n t changes i n bed b e h a v i o u r , no e f f e c t s were o b s e r v e d i n e i t h e r s t a t i c o r dynamic a n g l e s o f r e p o s e , shea r a n g l e s , s l u m p i n g f r e q u e n c i e s or a c t i v e l a y e r t h i c k n e s s e s . I t was t h e r e f o r e c o n c l u d e d t h a t the f l ow mechanism o f s e g r e g a t i o n d i d not a f f e c t the sys tems s t u d i e d . Hence , the v i s u a l l y o b s e r v e d r a d i a l s e g r e g a t i o n must have o c c u r r e d by means o f the p e r -c o l a t i o n mechan ism. A t h e o r e t i c a l model was d e v e l o p e d to p r e d i c t t he w i d t h o f the c e n t r a l s e g r e g a t e d c o r e , the ' k i d n e y 1 , wh ich was assumed to have a shape s i m i l a r to t h a t o f the s o l i d s b e d . I t s d imens ions were dependent on i t s c o m p o s i t i o n and on the t o t a l f i n e s c o n t e n t . The p r e d i c t e d co re w i d t h s were shown to agree w i t h the e x p e r i m e n t a l r e s u l t s when the c o r e c o m p o s i t i o n was taken to be equa l to t h a t f o r the £ e m a x r e l a t i o n s h i p . The r o t a r y c y l i n d e r bed was sampled f rom the apex to the c h o r d a l b a s e . The e x p e r i m e n t a l r e s u l t s i n d i c a t e d the e x i s t e n c e of a second s e g r e g a t i o n zone a t the top of the b e d . T h i s was c o n f i r m e d when t h e s e r e s u l t s were compared to t hose o b t a i n e d r e c e n t l y by Sunne rg ren on a s i m i l a r sys tem u s i n g a s i m i l a r s a m p l i n g t e c h n i q u e . Us ing the e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s o f t h i s c h a p t e r as w e l l as the s o l i d s c i r c u l a t i o n model f o r a r o l l i n g bed p r e s e n t e d i n S e c t i o n s 2 . 2 . 3 and 4 . 5 . 2 , the p e r c o l a t i o n s e g r e g a t i o n mechanism p r o c e s s f o r bo th s e g r e g a t i o n zones was a l s o d e s c r i b e d . I n d u s t r i a l i mpl i c a t i ons o;f segregation i n c l u d e poor r e a c t i v i t y o f the p a r t i c l e s i n the c e n t r a l c o r e , due to the c o a r s e p a r t i c l e s s h i e l d i n g them from the ho t gases i n the f r e e b o a r d . The second s e g r e g a t i o n zone i s b e l i e v e d to be r e s p o n s i b l e f o r the o n s e t o f s l i p p i n g f rom s l u m p i n g o r r o l -l i n g b e d s , wh ich i s d e t r i m e n t a l to r o t a r y k i l n o p e r a t i o n . Chap te r 6 174 MATHEMATICAL MODELS OF BED BEHAVIOUR 6 .1 I n t r o d u c t i on A d e t a i l e d a n a l y s i s o f the s l ump ing p r o c e s s w i l l be p r e s e n t e d based on an e x p e r i m e n t a l and a s t a t i s t i c a l a n a l y s i s u s i n g p h o t o g r a p h i c and i n s t r u m e n t a l t e c h n i q u e s . The da ta w i l l be s t a t i s t i c a l l y a n a l y s e d and w i l l be used i n the d e r i -v a t i o n o f the m a t h e m a t i c a l model o f the s i u m p i n g - r o l 1 i n g boundary to be d e s c r i b e d in t h i s c h a p t e r . S l i p p i n g , c a s c a d -i n g and c a t a r a c t i n g w i l l a l s o be m a t h e m a t i c a l l y m o d e l l e d . The d i m e n s i o n l e s s form o f the e q u a t i o n s w i l l be used to r e v e a l those v a r i a b l e s a f f e c t i n g the v a r i o u s b o u n d a r i e s o f bed b e h a v i o u r on a B e d - B e h a v i o u r D iag ram. 6 .2 The S I u m p i n g - R o l 1 i n g Boundary The m a t h e m a t i c a l model o f the s i umpi r i g - r o l l i ng boundary i s based on an a n a l y s i s o f a s l u m p i n g b e d . An a t t emp t w i l l be made to d e s c r i b e the model i n terms o f the t h r e e g e n e r a l p r i n c i p l e s o f bu l k s o l i d s f l o w : m o b i l i z a t i o n o f f r i c t i o n , d i l a t a n c y and minimum e n e r g y . 7 0 The f i r s t two o f t hese w i l l be d i s c u s s e d i n the f o l l o w i n g s e c t i o n ; w h i l e the minimum energy p r i n c i p l e w i l l be c o n s i d e r e d i n S e c t i o n 6 . 2 . 2 . The c r i t e r i o n o f the s 1 u m p i n g - r o l 1 i n g boundary w i l l then be d e v e l o p e d . 6 .2 .1 M o b i l i z a t i o n o f F r i c t i o n and D i l a t a n c y The m o b i l i z a t i o n o f f r i c t i o n i s c o n c e r n e d w i t h the magn i tude o f the f r i c t i o n f o r c e between the g r a n u l e s wh ich c o u l d va ry between z e r o and some l i m i t i n g v a l u e . Due to the n a t u r e o f bu l k s o l i d s , the i n t e r n a l s t r e s s d i s t r i b u t i o n i s no t u n i f o r m i n a bed and i s i n -d e t e r m i n a t e u n t i l the l i m i t i n g c o n d i t i o n o f f r i c t i o n i s a t t a i n e d when a shea r p l ane i s f o r m e d . The bu l k s o l i d s l y i n g above the s h e a r p l a n e w i l l be termed the s h e a r wedge. P r i o r to t h i s c o n d i t i o n , the p r o c e s s o f d i l a t a n c y t a k e s p l a c e w h e r e i n b u l k s o l i d s i n c r e a s e o r d e c r e a s e the volume they o c c u p y . In a n a l y z i n g a s l u m p , in terms o f these two p r i n c i p l e s , bo th the p o s i t i o n o f the s h e a r p l a n e and the s i z e o f the s h e a r wedge must be i d e n t i f i e d . S i n c e the s t a t e o f knowledge o f the f l o w o f bu l k s o l i d s , however , i s no t w e l l enough advanced to p r e d i c t m e c h a n i s t i c a l l y the o n s e t o f s l u m p i n g , an e m p i r i c a l approach was t h e r e f o r e a d o p t e d . 6 . 2 . 1 . 1 The S lump ing P r o c e s s The f i r s t s tage in t he deve lopment o f t h i s e m p i r i c a l model was to examine the v a r i o u s p r o c e s s e s 176 a s s o c i a t e d w i t h the o c c u r r e n c e of a s l u m p . T h i s e x a m i n a -t i o n was unde r t aken i n t h r e e p h a s e s . F i r s t l y , t r a c e s of the bed p r o f i l e were supe r imposed on each o t h e r to obse rve the changes in bed i n c l i n a t i o n d u r i n g s l u m p i n g . S e c o n d l y , the bed i n c l i n a t i o n s o f the same slump were then d i g i t i z e d . The d a t a t hus c o l l e c t e d were i n c o r p o r a t e d w i t h the t ime measurements o f bu l k s o l i d s mot ion and no mot ion as was o b s e r v e d f rom the bed s u r f a c e . S i n c e a p h o t o g r a p h i c r e c o r d o f s l u m p i n g was r e q u i r e d f o r a l l t h r e e a n a l y s e s the bed o f bu l k s o l i d s was f i l m e d th rough the p l e x i g l a s s e n d - p l a t e u s i n g a HYCAM mot ion p i c t u r e camera o p e r a t e d at 40 f r a m e s / s . C y l i n d e r B ( 0 .4 m ID x 0 .86 m L) was f i l l e d w i t h l i m e s t o n e B at 5.7% f i l l f o r these t r i a l s . H o r i z o n t a l and v e r t i c a l measu r i ng g r i d s were a l s o pho tog raphed f o r a d i g i t i z i n g r e f e r e n c e , and d i r e c t o b j e c t l i g h t i n g was used f o r the f i l m -i n g . The bed i n c l i n a t i o n was s u b s e q u e n t l y t r a c e d f rom the s c r e e n o f an e l e c t r o n i c f i l m d i g i t i z i n g u n i t and was measured u s i n g the ang le d i g i t i z i n g f e a t u r e . T y p i c a l r e s u l t s o f the t r a c e d i n c l i n a t i o n s are shown i n F i g u r e s 6.1 and 6 .2 f o r one slump when the r o t a t i o n a l speed o f the c y l i n d e r was 0 .85 r / m i n . The s t a r t i n g p o i n t chosen f o r t h i s s e q u e n c e , p o s i t i o n 1 ( F i g u r e 6 . 1 ) , i s the l o w e s t i n c l i n a t i o n t h a t the bed a t t a i n e d f o l l o w i n g the o c c u r r e n c e of a p r e v i o u s s l ump . P o s i t i o n 1 w i l l be used as Figure 6.1 Traced bed inc l i na t i ons for the f i r s t part of a slump of limestone B in cy l inder B (0.4 m ID x 0.86 m L) rotated at 0.85 r/min. Figure 6.2 Traced bed i nc l i na t i ons for the l a t t e r part of a slump of limestone B in cy l inder B (0.4 m ID x 0.86 m L) rotated at 0.85 r/min. 178 a r e f e r e n c e f o r these bed p r o f i l e t r a c e s . P o s i t i o n 2 , wh ich occurs.-. 0 .7 seconds a f t e r p o s i t i o n 1 , has a h i g h e r bed i n -c k i n a t i o n than the r e f e r e n c e p o s i t i o n . F u r t h e r i n c r e a s e s i n t ime ( p o s i t i o n s 3 and 4 ) ' r e v e a l t h a t the apex o f the bed does not r i s e s i g n i f i c a n t l y . A l t h o u g h , the c h o r d a l base o f the bed seems to have g r e a t l y s h i f t e d between p o s i t i o n s 2 and 3 , the r e c o r d e d b a s a l p o s i t i o n o f f rame number 2 8 , p o s i t i o n 2 , i s not r e l i a b l e due to the p r e s e n c e o f l o o s e g r a n u l e s . T h e r e f o r e , i n e q u a l t ime i n c r e m e n t s , 0 .7 s e c o n d s , l a r g e r i n c r e a s e s i n bed i n c l i n a t i o n o c c u r f rom p o s i t i o n 1 to 2 as compared to p o s i t i o n 2 to 4 . The f u r t h e r l a p s e o f t ime i s i l l u s t r a t e d i n F i g u r e 6 .2 when the apex and the c h o r d a l base approach the s t a r t i n g i n c l i n a t i o n o f the b e d . The t o t a l d u r a t i o n o f t h i s slump i s 2 .3 seconds and the maximum bed i n c l i n a t i o n t a k e s p l a c e between 1.4 and 1.7 s e c o n d s , wh ich i s ove r t w i c e as l o n g as the t ime o f de-c r e a s i n g i n c l i n a t i o n . The d i g i t i z e d bed i n c l i n a t i o n s f o r t h i s same s l u m p , the second phase o f t h i s a n a l y s i s , i s shown i n F i g u r e 6 . 3 , and the p o s i t i o n s o f the t r a c e d bed i n c l i n a t i o n s o f F i g u r e s 6.1 and 6 .2 are c l e a r l y i n d i c a t e d . The o b s e r v a t i o n s made f rom the t r a c e d r e s u l t s , d e s c r i b e d a b o v e , are i n agreement w i t h the d i g i t i z e d d a t a . In F i g u r e 6 . 3 , t h e r e i s a s t eady i n c r e a s e i n bed i n c l i n a t i o n u n t i l about 0 .7 to 1.1 seconds 179 0.5 ~ r Time (s) 1.5 2 Position TT 2 3 2.5 Time to maximum bed inclination , tj (/) a> CD T J 48 T 56 7 Time to minimum bed inclination,^ No motion on bed surface - e -c o 44 Motion on bed^surface o o oo o o o ° o o o c "o c T J CD CD 40 o o o o a 9 ^ u 0 36 1 ° o _ J o 0 < f i L 0 30 60 Frame number 90 120 Figure 6.3 D ig i t i zed bed i nc l i na t i ons for the same slump shown in Figures 6.1 and 6.2. The times of bed motion and no motion as recorded from the bed surface are also shown. 180 where t h i s i n c r e a s e s lows down c o n s i d e r a b l y . The bed then a t t a i n s the maximum bed i n c l i n a t i o n a f t e r 1.6 seconds f rom the r e f e r e n c e f r a m e . The c o n t i n u e d r o t a t i o n o f the c y l i n d e r r e s u l t s i n a sudden d e c r e a s e i n i n c l i n a t i o n u n t i l the i n i t i a l p o s i t i o n i s a g a i n r e a c h e d , t h e r e b y c o m p l e t i n g the s l u m p i n g c y c l e . For the f i n a l phase o f t h i s a n a l y s i s o f the same s l u m p , t ime d u r a t i o n s were r e c o r d e d f o r the bu l k s o l i d s , \.when; s t a -t i o n a r y and i n r e l a t i v e mo t ion to the c y l i n d e r w a l l . These measurements were made w i t h the a i d o f an i n c l i n e d m i r r o r p l a c e d i n s i d e the c y l i n d e r wh ich r e f l e c t e d the bed s u r f a c e and was then s i m u l t a n e o u s l y p h o t o g r a p h e d . w i t h the end w a l l and r e f e r e n c e g r i d s . S t a r t i n g f rom the r e f e r e n c e f r a m e , the frame numbers at wh ich mo t ion on the bed s u r f a c e s t a r t e d and s t o p p e d were r e c o r d e d . The r e s u l t s , when supe r imposed on t he d i g i t i z e d da ta i n F i g u r e 6 . 3 and compared to F i g u r e s 6.1 and 6 . 2 , r e v e a l the f o l l o w i n g c h r o n o l o g y of e v e n t s f o r a s l u m p . The b u l k s o l i d s r o t a t e w i t h the c y l i n d e r w a l l f rom t ime ze ro to 0 .9 s e c o n d s . S u b s e q u e n t l y , some mot ion i s seen at the apex o f the b e d , but the i n c l i n a t i o n o f the bed i n c r e a s e s at a l ower r a t e u n t i l a t 1.6 seconds the maximum bed i n c l i n a t i o n i s a t t a i n e d . Hence the p r o c e s s o f d i l a t a n c y o c c u r s between 0 .9 and 1.6 seconds and i s c h a r a c t e r i z e d by a s h r i n k a g e i n volume as seen f rom the t r a c e d and d i g i t i z e d 181 i n c l i n a t i o n s o f F i g u r e s 6 . 1 and 6 . 3 . T h i s s h r i n k a g e has been a s s o c i a t e d w i t h the d i l a t a n c y o f l o o s e l y packed bu l k 1 0 8 s o l i d s . F r i c t i o n i s seen t o have reached a l i m i t i n g v a l u e a t the p o i n t o f maximum i n c l i n a t i o n wh ich by d e f i n i -t i o n i s the upper a n g l e o f r e p o s e . The s h e a r p l ane i s formed at t h i s p o i n t and those s o l i d s i n the s h e a r wedge t h a t had undergone d i l a t a n c y w i l l now f a l l towards the c h o r d a l base o f the b e d ; hence the d r a s t i c d e c r e a s e i n bed i n c l i n a t i o n p a s t the 1 . 6 second mark. When mot ion c e a s e s , t he bed i n -c l i n a t i o n i s equa l to i t s i n i t i a l s t a r t i n g p o s i t i o n , wh ich i s b e l i e v e d to be the s h e a r a n g l e . 6 . 2 . 1 . 2 The A p p l i c a t i o n o f the TR Senso r R e c a l l t ha t the aim of t h i s e m p i r i c a l approach i s to i d e n t i f y the p o s i t i o n o f the s h e a r p l ane and the s i z e o f the s h e a r wedge. I t was shown in S e c t i o n 4 . 5 . 1 t h a t t he upper a n g l e o f repose i s a f u n c t i o n o f r o t a t i o n a l s p e e d . H e n c e , f rom the a n a l y s i s i n the p r e v i o u s s e c t i o n , the s i z e o f the s h e a r wedge w i l l a l s o be a f u n c t i o n o f the r o t a t i o n a l speed and c o u l d be c a l c u l a t e d f r o m , Y = - <(>[_ ( 6 . 1 ) However , due to the poor a c c u r a c y o f the method used i n measu r i ng <j)y ( S e c t i o n 4 . 5 . 1 ) , the shea r wedge was no t c a l c u l a t e d u s i n g E q u a t i o n ( 6 . 1 ) . A n o t h e r approach t h a t was 182 c o n s i d e r e d , was to o b t a i n the da ta p h o t o g r a p h i c a l l y , as seen i n F i g u r e 6 . 3 . T h i s method i s ve ry t e d i o u s and t ime c o n s u m i n g , t h e r e f o r e the f e a s i b i l i t y o f a p p l y i n g the s e n s o r to t h i s t ask was i n v e s t i g a t e d . One o f the f e a t u r e s o f the s e n s o r , d i s c u s s e d i n S e c t i o n 4 . 3 , i s t h a t i t s mi 1.1 i vo l tag'e o u t p u t i s a f u n c t i o n o f the d i s t a n c e f rom the r e f l e c t i n g s u r f a c e . T h e r e f o r e , f o r a s l u m p , the change i n t he magni tude o f the s e n s o r o u t -put i s r e l a t e d to the s i z e o f the s h e a r wedge. The a p p l i c a -t i o n o f the s e n s o r i n t h i s manner p r e s e n t s two d i f f i c u l t i e s ; name ly , t h a t the s e n s o r o u t p u t i s a l s o r e l a t e d to the n a t u r e o f the r e f l e c t i n g s u r f a c e and the i n i t i a l d i s t a n c e f rom wh ich i t i s p l a c e d . However , even i f t h i s e f f e c t was e l i m i -n a t e d by s t a n d a r d i z a t i o n t e c h n i q u e s , a second and more s e r i o u s drawback would s t i l l be p r e s e n t i . e . , the n o n - l i n e a r r e l a t i o n s h i p between the s e n s o r o u t p u t and the r e f l e c t i n g d i s t a n c e . The c o n v e r s i o n o f the s e n s o r o u t p u t to the s h e a r wedge s i z e would thus be f u r t h e r c o m p l i c a t e d , so t h a t an i n d i r e c t approach appeared n e c e s s a r y . The t imes to maximum, and minimum i n c l i n a t i o n s , t-j and t2 r e s p e c t i v e l y , were measured f rom the s e n s o r o u t p u t , and are compared i n Tab le XI I to the t imes to maximum bed i n c l i n a t i o n wh ich were measured p h o t o g r a p h i c a l l y . E x c e l l e n t agreement i s o b s e r v e d . The da ta u s i n g the s e n s o r were o b t a i n e d w h i l e r e c o r d i n g the 183 s l u m p i n g f r e q u e n c y by i n c r e a s i n g the c h a r t speed on the c h a r t r e c o r d e r to about 0 .2 Sl/mm (5 s / i n ) . S u b s e q u e n t l y , t he d i s t a n c e on t he c h a r t o u t p u t between the maxima and minima o f these expanded s l ump ing s i g n a l s was measured d i r e c t l y and c o n v e r t e d to t i m e s . From the response c h a r a c t e r i s t i c s o f the s e n s o r a l r e a d y d i s c u s s e d above and from the e x p e r i m e n t a l v e r i f i c a t i o n ( T a b l e X I I ) , i t i s c o n -c l u d e d t h a t t he maxima and minima o f the s l u m p i n g s i g n a l s r e p r e s e n t the upper a n g l e o f repose and the s h e a r a n g l e o f the bed r e s p e c t i v e l y . The s i z e o f the s h e a r wedge c o u l d thus be e a s i l y c a l c u l a t e d f r o m , Y-| = 6 . t . j . n ( 6 . 2 ) where the 6 i s a c o n v e r s i o n f a c t o r . The s i z e and posi t ion of the shear...wedge, w e r e ; i d e n t i f i e d u s i n g t he s e n s o r and the s h e a r a n g l e measurements r e s p e c t i v e l y . T h e r e f o r e , the m a t h e m a t i c a l e x p r e s s i o n r e l a t i n g the s h e a r wedge to the sys tem v a r i a b l e s rema ins to be d e r i v e d f o r subsequen t mode 11 ing u s a g e . 6 . 2 . 1 . 3 Times to Maximum and Minimum  Bed I n c l i n a t i o n s The t imes to maximum and minimum bed i n c l i n a t i o n s , t-j and t^ r e s p e c t i v e l y , were measured f o r d i f -f e r e n t bed depths and r o t a t i o n a l speeds f o r the v a r i o u s TABLE XI I Time to Maximum Bed I n c l i n a t i o n Method of Measurement Rotational Speed (r/min) 0.54 Fi lm 0.85 0.96 1.82 Average Time (s) 2.17 1.59 1.19 0.86 Standard Deviation (s) 0.18 0.28 0.18 0.14 Number of Observations 14 22 Sensor Average Time (s) 1.92 1.60 1.33 0.86 Standard Deviation (s) 0.18 0.13 0.14 0.08 Number of Observations 30 16 27 : 27 185 m a t e r i a l s used i n t h i s s t u d y . A t y p i c a l s e t o f r e s u l t s i s p r e s e n t e d g r a p h i c a l l y f o r g r a v e l i n F i g u r e 6 . 4 . Not a l l the e x p e r i m e n t a l da ta are i n c l u d e d on the d iagram f o r the sake of c l a r i t y i n p r e s e n t a t i o n . F u r t h e r m o r e , each o f the p o i n t s p l o t t e d r e p r e s e n t s the average o f a t l e a s t 25 measurements . The r e s u l t s o f the o t h e r m a t e r i a l s t e s t e d d i s p l a y e d the same c h a r a c t e r i s t i c s as the g r a v e l and are shown i n Append ix C. V i s u a l i n s p e c t i o n o f the r e s u l t s r e v e a l s t h a t the t imes t-j and t 2 a re both i ndependen t o f bed d e p t h . At low and h igh r o t a t i o n a l s p e e d s , they appear to be a s y m p t o t i c to the t ime a x i s and to a l i n e p a r a l l e l to the r o t a t i o n a l speed a x i s r e s p e c t i v e l y . The t ime t^ i s g e n e r a l l y g r e a t e r than t ime and t h e i r d i f f e r e n c e d e c r e a s e s wi th i ncreasii nxj r o t a t i ona l s p e e d s . The form o f the e q u a t i o n used f o r the r e g r e s s i o n o f the t-j da ta was , t 1 = . Y o 1 . + C ] ( 6 . 3 ) 6 . n The form o f t h i s e q u a t i o n r e f l e c t s the p r i n c i p l e s of d i l a t i o n and o f the m o b i l i z a t i o n o f f r i c t i o n as w e l l as a d e q u a t e l y p r o v i d i n g t h e i r i n t e r p r e t a t i o n i n the s lump ing p r o c e s s . T h i s i s e v i d e n t when both s i d e s o f E q u a t i o n ( 6 . 3 ) are m u l t i p l i e d by ( 6 . n ) y i e l d i n g , 8 to cu E 0 o 0 0.5 G r a v e l *| B e d depth % F i l l (m ) O © 0.089 16 A 0.065 1 0 • 0.054 7.8 O 0.042 5.5 o 0.0 31 3.5 R e g r e s s i o n curves 1.5 Rotational speed (r/min) Figure 6 . 4 The t 1 and t 2 measured resul ts as well as the regression curves for gravel in cy l inder A (0 .4 m ID x 0 .46 m L ) . CO CTl 187 Y ] = YO1 + 6 . C r n (6.: 4) The s i z e o f the s h e a r wedge i s t h e r e f o r e equa l to a c o n s t a n t , t he minimum s h e a r wedge, YQi > p l u s the r o t a t i o n a l speed ti;mes a t ime c o n s t a n t , C| , when d i l a t a n c y i s b e l i e v e d to o c c u r and w i l l be r e f e r r e d to as the d i l a t a n c y c o e f f i c i e n t . Re-f e r r i n g a g a i n to the p h o t o g r a p h i c r e s u l t s i n F i g u r e ( 6 . 3 ) , Y o l wou ld r e p r e s e n t the p e r i o d o f t ime d u r i n g wh ich no 6 . n mot ion was o b s e r v e d on t he bed s u r f a c e , w h i l e wou ld r e -p r e s e n t the t ime i nc remen t between the s t a r t o f mot ion on the bed and the p o i n t a t wh ich the upper ang le o f repose i s a t t a i n e d . For t ime t 2 > the f o l l o w i n g e q u a t i o n was u s e d : t2 = _2o^_ + C 2 + * C 3 n ( 6 - 5 ) 6 . n A l t h o u g h , i t i s i n agreement w i t h the e x p e r i m e n t a l r e s u l t s , the f i r s t term i n e q u a t i o n ( 6 r 5 ) has no p h y s i c a l s i g n i f i ^ cance w i t h r e s p e c t to the mot ion o f the s h e a r wedge. To i l l u s t r a t e t h i s , c o n s i d e r a c y l i n d e r r o t a t i n g a t a ve ry low r o t a t i o n a l speed w i t h a slump about to o c c u r . The bu l k s o l i d s w i l l take a f i n i t e t ime to slump f rom the apex to the c h o r d a ! base o f the b e d . M e a n w h i l e , the c y l i n d e r has r o t a -t e d th rough a f i n i t e a n g l e . As t he r o t a t i o n a l speed a p - ' p roaches z e r o , the s l u m p i n g t ime w i l l s t i l l be f i n i t e 188 but the ang le t r a v e l l e d by the c y l i n d e r w i l l approach z e r o . T h e r e f o r e , the f i r s t term i n E q u a t i o n ( 6 . 5 ) s h o u l d be z e r o . The a s y m p t o t i c b e h a v i o u r e x p e r i m e n t a l l y o b s e r v e d f o r a l l the da ta a t the low r o t a t i o n a l speeds has t h e r e f o r e been a t t r i b u t e d to the f o l l o w i n g two e f f e c t s . A t low r o t a t i o n a l s p e e d s , when t i m e s t^ and were measured f rom the r e c o r d i n g c h a r t , i t was more d i f f i c u l t to p i n p o i n t the p r e c i s e l o c a -t i o n o f the maxima and minima o f the s e n s o r o u t p u t . T h i s was due to the s e n s i t i v i t y o f the s e n s o r a t both p o s i t i o n s o f bed i n c l i n a t i o n s . A c o n s i s t e n t method o f l o c a t i n g the maxima and min ima o f the s e n s o r o u t p u t was t h e r e f o r e a d o p t e d . The second and more i m p o r t a n t cause was t h a t the sys tem has a l o n g e r response t ime a t the low r o t a t i o n a l s p e e d s , hence more t ime f o r the i n t e r n a l d i s t r i b u t i o n o f s t r e s s . However , s i n c e these ' b u l k s o l i d s are n o t composed o f i d e a l l y u n i f o r m shapes and s i z e s , a s c a t t e r i n the measured l i m i t i n g s h e a r wedge wou ld thus be e x p e c t e d . In f a c t the e x p e r i m e n t a l measurements o f the t imes t^ and had h i g h e r s t a n d a r d d e v i a t i o n s a t low r o t a t i o n a l speeds than a t the h i g h e r o n e s ; thus t he f i r s t te rm i n E q u a t i o n ( 6 ' . 5 ) i s due to bo th a measu r i ng and a random e x p e r i m e n t a l e r r o r . The minimum s h e a r wedge s h o u l d t h e r e f o r e be equa l t o , Y Q = Y Q L + Y Q 2 ( 6 . 6 ) and the t ime to the l i m i t i n g f r i c t i o n c o n d i t i o n i s g i v e n b y , t 1 = y 0 + C 1 ( 6 . 7 ) 6 . n 189 The t h i r d term i n E q u a t i o n ( 6 . 5 ) i s i n c l u d e d because i t wou ld be e x p e c t e d t h a t the t ime o f bed c o l l a p s e s h o u l d be a f u n c t i o n o f r o t a t i o n a l s p e e d . In f a c t i t s h o u l d d e c r e a s e w i t h i n c r e a s i n g s p e e d , thus w i l l be n e g a t i v e . Now add ing E q u a t i o n ( 6 . 3 ) and ( 6 . 5 ) y i e l d s , t T = Y o l * Y 0 2 + ( C 1 + Z 2 ) + C 3 , n ( 6 , 8 ) 6 . n B u t , t h e s l u m p i n g f r e q u e n c y i s g i v e n b y , 60 ( 6 . 9 ) V Th e r e f o r e , 1 = 1 S 60 Y o l + Y o 2 + <C1 + V + C 3 ' n 6 . n ( 6 . 1 0 ) To v e r i f y the r e g r e s s i o n c o e f f i c i e n t s o b t a i n e d f rom the t^ and t 2 d a t a , the s l u m p i n g f r e q u e n c y da ta a l r e a d y p r e s e n t e d i n S e c t i o n 4 . 5 . 1 . were r e g r e s s e d u s i n g E q u a t i o n (6110) th us ,y i e l d i ng an i ndependen t s e t o f c o e f f i c i e n t s . A s i m p l e r e g r e s s i o n (SIMREG) was pe r fo rmed on the t^ da ta u s i n g E q u a t i o n ( 6 . 3 ) and a s t a n d a r d m u l t i p l e r e g r e s s i o n (STPREG) on the t^ and the s l u m p i n g f r e q u e n c y d a t a u s i n g E q u a t i o n s ( 6 . 5 ) and (6110) r e s p e c t i v e l y . The UBC T r i a n g u l a r 14 3 R e g r e s s i o n Package . was used to o b t a i n these cu rve f i t s 190 on the UBC Amdahl 470 V6 Model II compute r . The r e g r e s s i o n c o e f f i c i e n t s o b t a i n e d f o r each o f the s e t s o f da ta f o r the v a r i o u s m a t e r i a l s on hand are l i s t e d i n T a b l e s X I I I to XV a l o n g w i t h the s t a n d a r d e r r o r s f o r each o f the c o e f f i c i e n t s , based on a 95% c o n f i d e n c e l i m i t . The s q u a r e d m u l t i p l e c o r -r e l a t i o n c o e f f i c i e n t s a re i n the m a j o r i t y o f cases g r e a t e r than 0 .95 i n d i c a t i n g a good f i t . T h i s i s f u r t h e r i l l u s t r a t e d i n F i g u r e s 6.4 and C . l to C. IO where the f i t t e d e q u a t i o n s o f t.j and t^ are .compared w i t h the e x p e r i m e n t a l d a t a . The c o e f f i c i e n t s f rom the t^ and da ta are now compared i n Tab le XVI w i t h those from the s l u m p i n g f r e q u e n c y d a t a . Very good agreement i s o b s e r v e d . H e n c e , the l i m i t i n g s h e a r wedge s i z e and the s h e a r wedge s i z e as a f u n c t i o n o f r o t a t i o n a l speed are m a t h e m a t i c a l l y i d e n t i f i e d i n a form t h a t a l l o w s t h e i r a p p l i c a t i o n to s tudy the c h o r d a l t r a j e c t o r y o f the s i ump . 6 . 2 . 2 The C h o r d a l T r a j e c t o r y o f a Slump When the f r i c t i o n f o r c e i s m o b i l i z e d to i t s maximum v a l u e , the g r a n u l a r s o l i d s c o n t e n t o f the s h e a r wedge i s u n s t a b l e a t the apex and must f a l l to the c h o r d a l base o f the b e d . The c h o r d a l t r a j e c t o r y o f each o f the g r a n u l e s i n the s h e a r wedge (wedge ABC, F i g u r e 6 .5 ) wou ld be g i v e n by the t h i r d p r i n c i p l e o f s o l i d s f l o w , the m i n i m i z a t i o n of e n e r g y . However , s i n c e t h e r e i s no i n f o r m a t i o n on the manner TABLE X I I I Regression Results for the t^ Measurements (t^= Y o 1 , +• C|) Material Cyl inder Diameter (m) Y o l (degrees) Standard Error on Y 0 ] (degrees) C l (s) Standard Error on G-j (s) Squared Mul t ip le Corre lat ion ^ Coef f ic ient R Number of Observations* Gravel 0.4 3.67 0.0463 0.536 0.0219 0.9911 58 Gravel 1.0 . 3.75 0.0371 0.699 0.0251 0.9970 33 Iron Oxide : ; 0.4 4.13 0.147 0.722 0.0941 0.9839 15 Limestone B 0.4 4.41 0.0624 0.491 0.0410 0.9974 15 Limestone B 1.0 4.79 0.0588 0.487 0.0624 0.9955 32 Limestone C 0.4 5.53 0.124 0.158 0.0830 0.9793 44 Limestone C 1.0 4.49 0.0807 0.153 0.0716 0.9917 28 Limestone D 0.4 4.30 0.0661 0.103 0.0445 0.9858 63 Limestone F 1.0 4.48 0.105 0.770 0.175 0.9882 24 Nickel Oxide. : 0.4 2.70 0.0345 0.614 0.0699 0.9955 30 Sand B 0.4 1.78 0.0764 0.364 0.111 0.9347 40 * - each observation is an average of at least 25 readings. TABLE XIV R e g r e s s i o n R e s u l t s f o r t h e t 7 M e a s u r e m e n t s ( t , = Y o 2 + C , + C , n ) c d 6 . n M a t e r i a l C y l i n d e r D i a m e t e r (m) Y o 2 ( d e g r e e s ) S t a n d a r d E r r o r on ( d e g r e e s ) C 2 (s) S t a n d a r d E r r o r on C, , ( s ) C 3 (6?,-ii, r e v S t a n d a r d E r r o r on , 6 0 . s 2 , r e v S q u a r e d M u l t i p l e R e g r e s s i o n , C o e f f i c i e n t R Number o f O b s e r v a t i o n s * G r a v e l 0 . 4 1 . 3 3 0 . 0 5 4 3 1.11 0 . 0 5 0 2 - 0 . 2 4 3 0 . 0 3 4 9 . 0 . 9 7 1 2 58 G r a v e l 1 .0 0 . 8 8 0 . 0 3 8 5 1 . 3 5 0 . 0 4 8 7 - 0 . 3 8 7 0 . 0 4 3 0 0 . 9 8 3 5 33 I r o n O x i d e " 0 . 4 0 . 6 0 0 . 0 6 2 2 . 2 5 0 . 8 0 6 - 1 . 6 7 1 .26 0 . 6 6 5 0 15 L i m e s t o n e B 0 . 4 0 . 7 8 0 . 0 4 9 1 .07 0 . 0 6 5 2 - 0 . 2 2 0 0 . 0 4 0 4 0 . 9 8 8 6 15 L i m e s t o n e B 1.0 0 . 8 8 0 . 0 6 6 1 .66 0 . 1 3 5 - 0 . 5 8 1 0 . 1 8 5 0 . 9 4 3 4 32 L i m e s t o n e C 0 . 4 . 0 .94 0 . 0 4 2 1 .09 0 . 0 4 8 9 - 0 . 2 3 1 0 . 0 3 5 3 0 . 9 6 9 5 44 L i m e s t o n e C 1.0 1 .09 0 . 0 3 1 6 1 .27 0 . 0 5 1 7 - 0 . 2 9 6 0 . 0 6 0 6 0 . 9 9 1 7 28 L i m e s t o n e D 0 .4 0 . 3 8 0 . 0 4 3 1 .18 0 . 0 5 3 - 0 . 3 5 3 0 . 0 6 9 0 0 . 8 6 9 9 63 L i m e s t o n e F 1 .0 1.02 0 . 0 4 6 1 .66 0 . 1 6 3 - 0 . 6 3 4 0 .381 0 . 9 7 9 6 24 N i c k e l O x i d e 0 . 4 0 .24 0 . 0 2 4 2 . 0 8 0 . 1 0 0 - 2 . 4 9 0 . 3 2 2 0 . 9 4 8 4 30 Sand B 0 .4 0 . 3 8 0 . 0 3 0 1 .44 0 . 0 8 6 9 - 1 . 9 6 0 . 2 9 5 0 .9711 40 * - E a c h o b s e r v a t i o n i s an a v e r a g e o f a t l e a s t 25 r e a d i n g s . TABLE XV R e g r e s s i o n R e s u l t s f o r t h e S l u m p i n g F r e q u e n c y Measurements ( t T = yo + C 4 + C j n) 6 . n M a t e r i a l C y l i n d e r D i a m e t e r ( « ) Y o ( d e g r e e s ) S t a n d a r d E r r o r on y 0 ( d e g r e e s ) C 4 ( s ) S t a n d a r d E r r o r on C^ ( s ) C 5 ( 6 0 . s 2 , r e v S t a n d a r d E r r o r on Cc 2 ( 6 0 . s<, r e v S q u a r e d M u l t i p l e R e g r e s s i o n , C o e f f i c i e n t R Number o f O b s e r v a t i o n s * G r a v e l 0 . 4 4 89 0 0729 1.80 0 0639 - 0 364 0 0396 0 . 9 9 3 3 82 G r a v e l 1 .0 4 95 0 0753 1 .82 0 0594 - 0 234 0 0665 0 9980 33 I r o n O x i d e 0 . 4 4 91 0 491 2 .98 0 623 -1 49 0 931 0 9775 16 L i m e s t o n e B 0 .4 5 14 0 0747 1.71 0 0940 - 0 308 0 0627 0 9949 57 L i m e s t o n e B 1.0 5 87 0 0706 1 .89 0 128 - 0 330 0 149 0 9966 48 L i m e s t o n e C 0 .4 6 62 0 156 0 . 9 5 0 171 n 003 0 116 0 9826 59 L i m e s t o n e C 1 .0 5 77 0 0860 0.81 0 138 0 519 0 157 0 9969 29 L i m e s t o n e D 0 . 4 4 68 0 109 1.24 0 147 - 0 211 0 209 0 9890 67 L i m e s t o n e F 1 .0 4 79 0 0427 2 . 7 6 0 138 - 0 716 0 280 0 9988 29 N i c k e l O x i d e 0 . 4 3 36 0 0327 2 . 0 3 0 115 - l 15 0 358 0 9966 79 Sand B 0 . 4 2 33 0 162 1.54 0 473 - l 43 1 64 0 9588 39 * - E a c h o b s e r v a t i o n was t a k e n o v e r a p e r i o d o f a t l e a s t 5 m i n u t e s . O J TABLE XVI Compar ison of R e g r e s s i o n C o e f f i c i e n t s O b t a i n e d f rom t- j , t^ Data and  f rom the S lumping Frequency Data Cylinder Data from t , , t? Regressions Data from Slumping Material Diameter Frequency Regressions (m) V (deg) C l , + C 2. (s) C 3 (60 . s ) rev ' •yo. (deg) C 4 (s) C 5 (60. s ) rev ' Gravel 0.4 5.00 1.64 -0.243 4.89 1.80 : -0.364 Gravel 1.0 4.63 2.05 -0.387 4.95 1.82 -0.234 Iron Oxide 0.4 4.73 2.97 -1.67 4.91 2.98 -1.49 Limestone B 0.4 5.19 1.56 -0.220 5.14 : 1.71 -0.308 Limestone B 1.0 5.67 2.15 -0.581 5.87 1.89 -0.330 Limestone C 0.4 6.47 1.25 -0.231 6.62 0.95 0.003 Limestone C 1.0 6.58 1.43 -0.296 5.77 0.81 0.519 Limestone D 0.4 4.68 1.28 -0.353 4.68 1.24 -0.211 Limestone F 1.0 5.50 2.43 -0.634 4.79 2.76 -0.716 Nickel Oxide " 0.4 2.94 2.69 -2.49 3.36 2.03 -1.15 Sand B 0.4 2.16 1.80 -1.96 2.33 1.54 -1.43 195 i n wh ich the p o t e n t i a l energy o f the s o l i d s i s d i s s i p a t e d , a t r a j e c t o r y model w i l l be d e v e l o p e d based on the c o n v e r s i o n o f the p o t e n t i a l energy o f the g r a n u l e s i n the shear wedge s o l e l y to k i n e t i c e n e r g y . F u r t h e r m o r e , i t w i l l be assumed t h a t t he g r a n u l e s below the s h e a r wedge w i l l not be d i s t u r b e d by the t r a j e c t o r y o f t hose i n i t ; hence the g r a n u l e s i n the s h e a r wedge w i l l f a l l i n t o wedge PSC ( F i g u r e 6 . 5 ) . T h e i r average t r a j e c t o r y w i l l be g i v e n by the s t r a i g h t l i n e j o i n -i n g the c e n t r o i d s o f the two wedges and the r e s u l t a n t f o r c e on t he s h e a r wedge w i l l a c t th rough i t s c e n t r o i d and w i l l be g i v e n b y , Ma = F G s i n n - F p ( 6 . 1 1 ) where the g r a v i t a t i o n a l f o r c e i s g i v e n b y , F Q = Mg ( 6 . 1 2 ) and the f r i c t i o n a l f o r c e b y , Fp = y L F N ( 6 . 1 3 ) The f o r c e b a l a n c e normal to the l i n e o f the average c h o r d a l t r a j e c t o r y y i e l d s , F N = Mg cos n ( 6 . 1 4 ) Now s u b s t i t u t i n g E q u a t i o n s • ( 6 .12) to ( 6 . 1 4 ) i n t o ( 6 . 1 1 ) , a = g ( s i n n - y , c o s n ) ( 6 . 1 5 ) 196 Figure 6.5 Force balance on a shear wedge. 197 S i n c e the l i m i t i n g f r i c t i o n c o n d i t i o n o c c u r s a t the s h e a r a n g l e , i t f o l l o w s t h a t , u L = t an c|>^ ( 6 . 1 6 ) I n t e g r a t i n g E q u a t i o n • (6 .1 5 )• w i t h the i n i t i a l c o n d i t i o n s , ( i ) t = 6 V q T = 0 ( 6 . 1 7 ) and ( i i ) t = 0 S q = 0 ( 6 . 1 8 ) and s u b s t i t u t i n g e q u a t i o n ( 6 . 1 6 ) , y i e l d s , 2 s = 1 gt ( s i n v q -. tan ^ cos n) ( 6 . 1 9 ) 2 I t wou ld have been more p r e c i s e to e x p r e s s the i n i t i a l c o n -d i t i o n i n ( 6 . 1 7 ) as f o l l o w s , when t = 0 v T = <4y~A B C cos n t x A B C s i n n) ( 6 . 2 0 ) However , due to the s m a l l a n g u l a r v e l o c i t i e s o f s l u m p i n g beds the c o n t r i b u t i o n o f the i n i t i a l c o n d i t i o n ( 6 . 2 0 ) wou ld have a n e g l i g i b l e e f f e c t on the average t r a j e c t o r y t i m e . H e n c e , i n i t i a l c o n d i t i o n ( 6 . 1 7 ) was used i n the m o d e l . F i n a l l y , s i n c e the s h e a r wedge i s bounded by the upper and l owe r a n g l e s o f repose the d i s t a n c e between the wedge c e n t r o i d s , ABC and P S C , and the s l o p e o f the average c h o r d a l t r a j e c t o r y may be c a l c u l a t e d f r o m , S = ( ( * A B C ~ ^ P S C ) 2 + ^ A B C " VPSC)2)k ( 6 ' 2 1 ) and „ n = tan 198 ^1 y A B C " y P S C  X ABC " X P S C ( 6 . 2 2 ) where the c a l c u l a t i o n o f the c e n t r o i d c o - o r d i n a t e s ( x ^ B C ' y A B C ^ and ( X p c j r j , y p s c ^ 1 S P r e s e n t e c ' i n Append ix D. H e n c e , the ave rage s l u m p i n g t ime o f the c o n t e n t s o f the s h e a r wedge may be c a l c u l a t e d f rom E q u a t i o n (6 .19) . 6 . 2 . 3 The C r i t e r i o n o f the S l u m p i n g - R o l l i n g Boundary As a s h e a r wedge f a i l s and i t s c o n t e n t s f a l l i n t o the c h o r d a l base o f the bed more g r a n u l e s c r o s s the shea r p l ane due to the c o n t i n u o u s r o t a t i o n o f the c y l i n d e r . Thus i n the s l u m p i n g t ime o f the shea r wedge, c a l c u l a t e d f rom E q u a t i o n ( 6 . 1 9 ) , the a n g u l a r r o t a t i o n o f the c y l i n d e r ( i . e . the q u a n t i t y o f s o l i d s c r o s s i n g the s h e a r p l a n e ) , may be c a l c u l a t e d f r o m , Y* = 6 . n . t ( 6 . 2 3 ) I f t he ang le t r a v e l l e d by the c y l i n d e r , y* , d u r i n g the s l u m p i n g t ime i s l e s s than the minimum s h e a r wedge (j*<y^) t h e n i t i s p r o p o s e d t h a t the bed w i l l c o n t i n u e to s l u m p . I f the a n g l e t r a v e l l e d by the c y l i n d e r i s g r e a t e r than the minimum s h e a r wedge .(Y*>YQ) then the bed w i l l r o l l . I t f o l l o w s t h a t the c o n d i t i o n when the ang le t r a v e l l e d e q u a l s the minimum s h e a r wedge (y* = Yq) r e p r e s e n t s the s l u m p i n g -r o l l i n g boundary and t h a t the c h o r d a l t r a j e c t o r y t ime i s 199 g i v e n b y , t Q = 30 f y n \ ( 6 . 2 4 ) o r ^ 1 M n ( 6 . 2 5 ) S u b s t i t u t i n g E q u a t i o n ( 6 . 2 4 ) i n (6 .19 ) , y i e l d s , 2 s = 450 9 / Y q \ ( s i n n - t an <]>L cos.n) ( 6 . 2 6 ) 2 A l s o s u b s t i t u t i n g , S = ( 6 . 2 7 ) R and d i v i d i n g by s , E q u a t i o n . (6 .26) becomes, 1 - / 4 5 0 \ / g R \ / R \ y n 2 ( s i n n - t an o). cos n) = 0 ( 6 . 2 8 ) T h i s e q u a t i o n i s t h e r e f o r e the d i m e n s i o n l e s s form of the c r i t e r i o n f o r the s i u m p i n g - r o l 1 i n g boundary and p r e s e n t s the s c a l e - u p c r i t e r i a f o r the m a t e r i a l , the c y l i n d e r and the o p e r a t i n g v a r i a b l e s . The f i r s t d i m e n s i o n l e s s pa rame te r gR i s r e c o g n i z e d as the i n v e r s e o f the Froude number wh ich i s a r a t i o o f the i n e r t i a l to g r a v i t a t i o n a l f o r c e s . The second s c a l e - u p f a c t o r , ( R / s ) r e f l e c t s the e f f e c t o f f i l l i n g degree as w e l l as the s i z e o f the shea r wedge, s i n c e the c a l c u l a t i o n o f s i s based on the l o c a t i o n o f the wedge 200 c e n t r o i d s . Thus i t i s an o p e r a t i n g and a m a t e r i a l s c a l e -up f a c t o r . The t h i r d , Y » r e p r e s e n t s a m a t e r i a l v a r i a b l e , the minimum shea r wedge; w h i l e the l a s t one ( s i n n -tan <|>^  cos •..n) i s dependent on the f i l l r a t i o , the s i z e o f the s h e a r wedge, the s h e a r a n g l e and the c y l i n d e r d i a m e t e r . For a g i v e n m a t e r i a l and c y l i n d e r , o n l y the f i r s t two d i m e n s i o n l e s s pa rame te rs (;g R/v ) and ( R / 5 ) c o u l d v a r y , and t h e i r v a l u e wou ld be s o l e l y a f u n c t i o n o f r o t a t i o n a l speed and bed d e p t h . H e n c e , t h e i r use i s m a t h e m a t i c a l l y j u s t i f i e d f o r the axes o f the B e d - B e h a v i o u r D iag ram. In c o n c l u s i o n t he m a t h e m a t i c a l model f o r p r e d i c t i n g the s i u m p i n g - r o l l i n g boundary o f a b u l k s o l i d r e q u i r e s the f o l l o w i n g i n p u t v a r i a b l e s ; the f i r s t t h r e e o f wh ich must be e x p e r i m e n t a l l y d e t e r m i n e d : 1) the minimum s h e a r wedge, 2) the s h e a r wedge as a f u n c t i o n o f r o t a t i o n a l s p e e d , 3) the s h e a r a n g l e , and 4) the c y l i n d e r i n n e r r a d i u s . The e q u a t i o n i n Tab le D . l to D.3 (Append ix D) and E q u a t i o n ( 6 . 2 8 ) were s o l v e d by t r i a l and e r r o r on the UBC Amdahl compu te r . The sou rce l i s t i n g o f the program i s i n c l u d e d i n Append i x E and the model p r e d i c t i o n s w i l l be p r e s e n t e d i n Chap te r 7. 201 6 . 3 The S l i p p i n g Model A model w i l l now be p r e s e n t e d to p r e d i c t t hose c o n -d i t i o n s when s l i p p i n g r a t h e r than s l u m p i n g , r o l l i n g or c a s c a d i n g w i l l o c c u r . The model i s based on the p r i n c i p l e s o f mechan ics f o r r i g i d b o d i e s . Assume t h a t the bed of bu l k s o l i d s i n F i g u r e 6 .6 i s r o t a t e d w i t h the c y l i n d e r w a l l .arid has the a n g l e o f i n c l i n a t i o n <f>1 . The g r a v i t y f o r c e and the c e n t r i f u g a l f o r c e w i l l a c t th rough the c e n t r e o f g r a v i t y o f the bed whose l o c a t i o n i s g i v e n b y , R n •= 2 R 3 s i n 3 A ( 6 . 2 9 ) 8 3A where the a rea o f the bed may be c a l c u l a t e d u s i n g E q u a t i o n ( 5 . 6 ) o r , A = R^ (2 X - s i n 2 X ) ( 6 . 3 0 ) 2 The g r a v i t y f o r c e i s g i v e n by E q u a t i o n ( 6 . 1 2 ) and the c e n t r i -f u g a l f o r c e b y , F c = M V R b ( 6 . 3 1 ) The r e s u l t a n t f o r c e i s t h e r e f o r e , F R = ( F * + 2 F C F G cos <},' + F*)h ( 6 . 3 2 ) The moment b a l a n c e about the c e n t r e o f r o t a t i o n of the c y 1 i nder y i e l d s , F F = R B F G s i n <j>' ( 6 . 3 3 ) X 202 203 In o r d e r f o r the f o r c e s to be i n e q u i l i b r i u m ( i . e . no s l i p -p i ng a t the w a l l ) the normal and f r i c t i o n a l f o r c e s must a l s o have a r e s u l t a n t , F^ , wh ich i s equa l i n magn i tude but o p p o s i t e i n d i r e c t i o n to F^ . T h e r e f o r e , F p = F^ s i n and N R F i cos 3 ( 6 . 3 4 ) ( 6 . 3 5 ) I t i s known t h a t the maximum f r i c t i o n a l f o r c e i s g i v e n b y , F F " y W/S F N ( 6 . 3 6 ) and t h a t . y W / S = t a n * S ( 6 . 3 7 ) T h e r e f o r e s u b s t i t u t i n g E q u a t i o n s ( 6 . 3 2 ) to ( 6 . 3 5 ) and ( 6 . 3 7 ) i n ( 6 . 3 6 ) y i e l d s , tan <|> = B r G F r s in cb ( F C + 2 F C F G C 0 S * ' + F G ) J S c o s { s i n _ 1 (6.38) } ( F C + 2 F C F G C 0 S * ' + F G ) % E q u a t i o n ( 6 . 3 8 ) was s o l v e d by t r i a l and e r r o r f o r the r e -s u l t a n t bed i n c l i n a t i o n , <i>1 , u s i n g the UBC Amdahl computer ( t h e program l i s t i n g i s shown i n Append ix F ) . I t was then compared to the upper a n g l e o f r e p o s e , <j>y, and the 204 r e s u l t a n t dynamic a n g l e o f i n c l i n a t i o n <(>p, to be d e s c r i b e d i n the f o l l o w i n g s e c t i o n ) . Three s i t u a t i o n s may a r i s e , (i) ' . *" > 4>u > * • ( i i ) 4>y > 4>' > <J>D and ( i i i ) > *D > * ' In the f i r s t s i t u a t i o n , no s l i p p i n g wou ld o c c u r ; w h i l e i n t he second s l i p p i n g wou ld o c c u r r a t h e r than s l u m p i n g . F i n a l l y , s l i p p i n g wou ld r e p l a c e a l l o t h e r modes o f bed b e h a v i o u r s i n the l a s t c o n d i t i o n . For r o l l i n g beds the a n a l y s i s c o u l d be s i m p l i f i e d by compar ing $ 1 to <j>D. F u r t h e r -more , when the c e n t r i f u g a l f o r c e i s n e g l i g i b l e , as i n r o t a r y k i l n s f o r e x a m p l e , E q u a t i o n ( 6 . 3 8 ) s i m p l i f i e s t o : tan (j)' = 3 tan <(><. ( A - cos •A s i n.x)1 ( 6 . 3 9 ) 2 s i n 3 A S l i p p i n g i n r o t a r y k i l n s i s t h e r e f o r e a f u n c t i o n o f the s l i p a n g l e <j>s and the degree o f f i l l . The r e s u l t s o f the s l i p -p i n g model c o u l d t h e r e f o r e be p l o t t e d on a B e d - B e h a v i o u r Diagram and w i l l be p r e s e n t e d i n C h a p t e r 7. 6 .4 The C a s c a d i n g and C a t a r a c t i n g Models The c a l c u l a t i o n o f the o n s e t o f c a s c a d i n g and c a t a -r a c t i n g have thus f a r r e c e i v e d l i t t l e a t t e n t i o n i n the 205 l i t e r a t u r e . C a s c a d i n g has been i d e n t i f i e d by the c r e s c e n t shape o f the b e d ; w h i l e c a t a r a c t i n g has been c h a r a c t e r i z e d by a s i g n i f i c a n t p o r t i o n o f g r a n u l a r s o l i d s b e i n g p r o j e c t e d i n t o the f r e e space o f the c y l i n d e r . These s o l i d s shower back i n t o the b e d . S e v e r a l a t t emp ts have been made to m a t h e m a t i c a l l y a n a l y z e a c a t a r a c t i n g b e d , but most o f them a re c o n c e r n e d w i t h the t r a j e c t o r y o f the s o l i d s above the . . 4 4 , 4 9 , 5 1 , 5 3 , 5 6 , 5 8 , 7 9 , 1 2 0 , 1 4 8 n , • . . + „ b e d . Only two i n v e s t i g a t o r s have a t t emp ted to c a l c u l a t e the ang le o f detachment o f p a r t i c l e s from the s o l i d s b e d : D a v i s ^ and U g g l a . ^ T h e i r c a l c u l a t i o n o f the onse t o f c a t a r a c t i n g was based on the f o r c e b a l a n c e on a s i n g l e p a r t i c l e r e s t i n g on the s u r f a c e o f the b e d . When the c y l i n d e r has ze ro r o t a t i o n a l s p e e d , Dav i s p l a c e d h i s p a r t i c l e a t the c y l i n d e r w a l l , i n l i n e w i t h the c e n t r e o f r o t a t i o n o f the c y l i n d e r . T h i s c o r -responds to a f i l l r a t i o o f 0 .5 and a h o r i z o n t a l bed s u r f a c e . Dav i s a l s o d i d not t a k e the f r i c t i o n f o r c e i n t o a c c o u n t . Ugg la on the o t h e r hand i n c l u d e d i t and c a r r i e d ou t a f o r c e b a l a n c e on a s i n g l e p a r t i c l e w i t h the f r i c t i o n f o r c e a p p l i e d as an i n t e r p a r t i c l e f r i c t i o n on the bed s u r f a c e . Improved e s t i m a t e s o f the a n g l e o f de tachment were o b t a i n e d . The s t a t i c ang le o f f r i c t i o n , d>^, a p p l i e d by Ugg la as an i n t e r p a r t i c l e f r i c t i o n on the bed s u r f a c e i s an e m p i r i c a l measure o f the s t a t i c f r i c t i o n c o e f f i c i e n t o f the 206 bed bu t t he dynamic ang le o f f r i c t i o n , <)>D, s h o u l d have been a p p l i e d . T h e r e f o r e , f o r the c u r r e n t model the f o r c e b a l a n c e was c a r r i e d ou t at the c e n t r o i d o f the bed w i t h the ang le o f k i n e t i c f r i c t i o n e q u a l to the dynamic a n g l e o f r epose o f the bu l k s o l i d s . The f o r c e po l ygon i s now s o l v e d f o r a g i v e n bed d e p t h , c y l i n d e r r o t a t i o n a l speed and m a t e r i a l -dynamic ang le o f f r i c t i o n ( F i g u r e 6 . 7 a ) . The f o r c e b a l a n c e y i e l d s , F G s i n * D - F f = o F i n a l l y , F F = y D F N (6 .40) ( 6 . 4 1 ) ( 6 . 4 2 ) S u b s t i t u t i n g f o r the a p p r o p r i a t e terms and s i m p l i f y i n g , ( S i n cb' - y COS cbp)- y n to Rp ( 6 . 4 3 ) w h e r e , y n = tan cbD ( 6 . 4 4 ) Hence , the bed i n c l i n a t i o n , f o r wh ich the f o r c e s a c t i n g on the bed are i n e q u i l i b r i u m , can be c a l c u l a t e d by t r i a l and e r r o r u s i n g E q u a t i o n ( 6 . 4 3 ) wh ich can a l s o be s i m p l i f i e d t o , s i n - <j>D) = to Rg s i n cbn (6 .45) o r s i n (cb-j - <bD)" = .u_Rs: 2 s i n X 3( x -cos?x s i nx) s 1 n <j> D ( 6 . 4 6 ) 207 Y X F i g u r e 6 .7 (a) F o r c e b a l a n c e on t h e bed o f b u l k s o l i d s f o r the c a s c a d i n g model w i t h t h e apex in t h e f i r s t q u a d r a n t . (b) Bed c o n f i g u r a t i o n w i t h the apex i n the f o u r t h q u a d r a n t . " (c) Bed c o n f i g u r a t i o n d e f i n i n g the rolling-cascading boundary. 208 The r e s u l t a n t bed i n c l i n a t i o n i s t h e r e f o r e a f u n c t i o n o f the Froude number, the degree o f f i l l and the dynamic a n g l e o f r e p o s e . When the r e s u l t a n t bed i n c l i n a t i o n , cb^, p l a c e s the apex o f the bed i n the f o u r t h q u a d r a n t , the ou te rmos t p a r t i -c l e a t the c y l i n d e r w a l l w i l l have an i n i t i a l v e l o c i t y i n t o the w a l l ( F i g u r e 6 .7b ) and the ang le 0 w i l l be n e g a t i v e . However , once the r e s u l t a n t bed i n c l i n a t i o n p l a c e s the apex i n t h e f i r s t q u a d r a n t , the v e l o c i t y o f the ou te rmos t p a r t i -c l e w i l l be away f rom the w a l l and w i l l be p r o j e c t e d i n t o the open space o f the c y l i n d e r ( F i g u r e 6 . 7 a ) . T h i s has been d e f i n e d as c a s c a d i n g . Note t h a t i n t h i s c o n f i g u r a t i o n the ang le 6 i s p o s i t i v e . The r o l l i n g - c a s c a d i n g boundary i s t h e r e f o r e d e f i n e d by the r o t a t i o n a l speed wh ich y i e l d s a r e s u l t a n t bed i n c l i n a t i o n ^ wh ich p l a c e s the apex o f the bed on the a b s c i s s a ; as i l l u s t r a t e d in F i g u r e 6 . 7 c . In t h i s c o n f i g u r a t i o n , e = 0 ( 6 . 4 7 ) and A = 90 - cbp ( 6 . 4 8 ) S u b s t i t u t i n g E q u a t i o n ( 6 . 4 8 ) i n t o ( 6 . 4 6 ) and s i m p l i f y i n g y i e l d s , U R / C R = 3 ftan *D " ^J'2 " ^ l " t a n *' 9 2 \ y D ( 6 . 4 9 ) 209 Equation: ( 6 . 4 9 ) r e v e a l s t ha t the r o l l i n g - c a s c a d i n g boundary i s a f u n c t i o n o f the c y l i n d e r s i z e , t he dynamic a n g l e o f f r i c t i o n o f the m a t e r i a l and the degree o f f i l l . Hence , the r o l l i n g - c a s c a d i n g boundary may be d e s c r i b e d on a B e d -B e h a v i o u r D iag ram. The p r e d i c t e d b o u n d a r i e s u s i n g t h i s model w i l l be p r e s e n t e d i n C h a p t e r 7 . Once a bed i s c a s c a d i n g i t s s u r f a c e w i l l no l o n g e r be p l a n a r due to the s o l i d s a t the apex b e i n g p r o j e c t e d i n t o the c y l i n d e r f r e e s p a c e . In t h i s a n a l y s i s , a p a r a b o l i c t r a j e c t o r y w i l l be a p p l i e d to a s i n g l e p a r t i c l e at the apex o f t he b e d . No i n t e r a c t i o n s w i t h o t h e r p a r t i c l e t r a j e c t o r i e s w i l l be a c c o u n t e d f o r as i t wou ld g r e a t l y c o m p l i c a t e the s o l u t i o n o f the p r o b l e m . Wh i l e t h i s t r e a t m e n t may seem 32 49 u n r e a l i s t i c , i t i s a good a p p r o x i m a t i o n ' and w i l l i l -l u s t r a t e the b a s i s o f u s i n g the B e d - B e h a v i o u r Diagram to d e s c r i b e c a t a r a c t i n g . The c a s c a d i n g - c a t a r a c t i n g boundary has been d e f i n e d as the l o c u s o f r o t a t i o n a l speeds a t v a r i o u s bed depths f o r wh ich the p a r a b o l i c t r a j e c t o r y o f a p a r t i c l e p r o j e c t e d f rom the apex w i l l i n t e r s e c t the m i d -p o i n t o f the bed s u r f a c e i n c l i n e d at the r e s u l t a n t ang le o f i n c l i n a t i o n . T h i s a n g l e i s c a l c u l a t e d u s i n g E q u a t i o n ( 6 . 4 6 ) . The p a r a b o l i c t r a j e c t o r y o f the uppermost p a r t i c l e on the bed s u r f a c e i s g i v e n b y , 210 Y = R s i n (R cos e - X) + g (R cos 9 - X)' (6.50) ta n e 2 . 2 2 v s i n ee o w h e r e X = R cos 2v s i n 0 o cos::0 +:/Rg s i n . 2 s i n f (6.51) a n d v = CJR o ( 6 . 5 2 ) T h e c o - o r d i n a t e o f t h e m i d - p o i n t o f t h e b e d s u r f a c e ( F i g u r e 6 . 7 a ) i s g i v e n b y , X c = ( R - H ) c o s ( 0 - x ) a n d Y c = ( R - H ) s i n ( e - x ) ( 6 . 5 3 ) (6 .54) F r o m t h e a b o v e e q u a t i o n s i t i s s e e n t h a t a s w i t h t h e r o l l i n g -c a s c a d i n g b o u n d a r y , t h e c a s c a d i n g - c a t a r a c t i n g b o u n d a r y i s d e p e n d e n t o n t h e c y l i n d e r d i a m e t e r , t h e d y n a m i c a n g l e o f r e p o s e o f t h e m a t e r i a l a n d t h e f i l l r a t i o . A c o m p u t e r p r o -gram was w r i t t e n t o p r e d i c t a c o m p l e t e B e d - B e h a v i o u r D i a g r a m f o r a g i v e n m a t e r i a l , s h o w i n g t h e v a r i o u s a r e a s o f p r e -d o m i n a n c e o f s l i p p i n g , s l u m p i n g , r o l l i n g , c a s c a d i n g , c a t a r a c t i n g a n d c e n t r i f u g i n g . T h e p r o g r a m , l i s t e d i n A p p e n d i x G r e q u i r e s t h e f o l l o w i n g i n p u t v a r i a b l e s : 1. t h e r a d i u s o f t h e c y l i n d e r , R, 2. t h e a n g l e c o n t a i n i n g t h e s h e a r w e d g e , Y 0 > 211 3 . the shear wedge as a f u n c t i o n of r o t a t i o n a l s p e e d , Y 0 and C 1 . 4. the shear a n g l e , <j>^, 5 . the dynamic angle of r e p o s e , <j>g,' and 6. the angle of s l i p p i n g f r i c t i o n , <t>j,. The p r e d i c t i o n s of the model w i l l be p resen ted in the f o l l o w -ing chapte r f o r the m a t e r i a l s used in t h i s s t u d y . C h a p t e r 7 212 MODEL PREDICTIONS AND DISCUSSION 7 .1 I n t r o d u c t i on The p r e d i c t e d b o u n d a r i e s f o r the v a r i o u s modes o f bed b e h a v i o u r w i l l be p r e s e n t e d i n t h i s c h a p t e r and w i l l be com-pared both to the e x p e r i m e n t a l r e s u l t s d i s c u s s e d i n c h a p t e r 4 and to e x p e r i m e n t a l and i n d u s t r i a l o b s e r v a t i o n s r e p o r t e d i n the l i t e r a t u r e . The s c a l e - u p c r i t e r i a s u g g e s t e d by the model e q u a t i o n s and by the e x p e r i m e n t a l r e s u l t s w i l l a l s o be t e s t e d and the r e s u l t s w i l l be a p p l i e d to the i n d u s t r i a l o p e r a t i o n s o f r o t a r y k i l n s . The e f f e c t of t empera tu re on the bed b e h a v i o u r o f bu l k s o l i d s w i l l a l s o be b r i e f l y d i s c u s s e d , a l o n g w i t h some a p p l i c a t i o n s o f the B e d - B e h a v i o u r Diagram i n o t h e r s i t u a t i o n s . A compar i son between the Phase Diagram of f l u i d - s o l i d sys tems and the B e d - B e h a v i o u r Diagram w i l l then be made. A d e r i v a t i o n o f the i n t e r n a l ang le o f f r i c t i o n o f c o h e s i o n l e s s bu l k s o l i d s w i l l be p roposed based on an a n a l y s i s o f s l u m p i n g . 7.2 The S l u m p i n g - R o l l i n g Boundary Model p r e d i c t i o n s showing the e f f e c t s o f p a r t i c l e s i z e , p a r t i c l e s h a p e , combined e f f e c t s bed d e p t h , of s i z e and 213 shape and o f c y l i n d e r d i a m e t e r w i l l be d i s c u s s e d . The s c a l e -up c r i t e r i a o f the s i u m p i n g - r o l 1 i n g boundary w i l l be i n -v e s t i g a t e d u s i n g e x p e r i m e n t a l r e s u l t s to v e r i f y the model p r e d i c t i o n s . 7 .2 .1 P r e d i c t e d v e r s u s Measured R e s u l t s Model p r e d i c t i o n s o f the s i u m p i n g - r o l 1 i n g boundary a re p r e s e n t e d i n F i g u r e s ' 7'. l ' . ' tb 7 . 6 . The p r e d i c t e d and e x p e r i m e n t a l b o u n d a r i e s f o r l i m e s t o n e B t e s t e d i n c y l i n d e r A are shown i n F i g u r e 7 . 1 . Us ing the s t a n d a r d r • e r r o r on each of the r e g r e s s i o n c o e f f i c i e n t s . ( T a b ! e 6 .2 and 6 . 3 ) , the maximum and minimum v a l u e s o f Yq and were c a l c u -l a t e d f o r a 95% c o n f i d e n c e l i m i t and were used to o b t a i n the upper and l o w e r bounds o f the s i u m p i n g - r o l 1 i n g b o u n d a r y , a l s o p l o t t e d i n F i g u r e 7 . 1 . For l i m e s t o n e B, the v a r i a t i o n s i n Yq o f + 0 . 3 degrees and i n Cj o f + 0.1 seconds r e s u l t i n a change i n the p r e d i c t e d boundary o f about + 0 .15 r /m in and + 0 .25 r / m i n f o r the upper and l owe r c o n f i d e n c e l i m i t s of the c o e f f i c i e n t s r e s p e c t i v e l y . T h i s i s o f the same o r d e r as the w i d t h o f the e x p e r i m e n t a l l y d e t e r m i n e d t r a n s i t i o n z o n e . Good agreement i s o b s e r v e d between p r e d i c t e d and e x p e r i m e n t a l b o u n d a r i e s , p a r t i c u l a r l y w i t h r e s p e c t to the bed depth dependency . Compar ing t he p r e d i c t e d and the e x p e r i m e n t a l bed Ixib 6 IxlO 5 E Q. CO "O TD CO CO 0.12 0.09 0.06 Q.03 IXIO Froude number (F r=GJR/g ) 4 214 Slumping 0.5 IXIO 3 Limestone B Best fit data 9 5 % confidence limits Experimental 25 2 0 15 10 Rolling I 1.5 Rotational speed (r/min) 2.5 Figure 7.1 Predicted and experimental boundaries for limestone B tested in cylinders A and B (0.4 m ID x 0.46 m L and 0.86 m L respectively). - 6 -5 IXIO 1X10 Froude number ( F r * G J R / g ) IXIO C4 0.1 2 h-- 0.09 E r. Q. CO T= 0J06 •o CO CD 0.03 ! 0.5 I 1.5 Rotational speed (r/min) 1 1 1 — ; . Nickel oxide i 1 Best fit dota \ 1 \ • 9 5 % confidence limits — — 1 ' \ ; \ \ \\ \ Experimental \ • \ \ ', \ \ '. v - \ Rolling \ \ Slumping ^ » \ \Transit ion \ \ \ 1 > \ * \ \ ^ 1 1 1 IxlO 25 2 0 15 10 n-Figure 7.2 Predicted and experimental boundaries for nickel oxide tested in cylinder A (0.4 m ID x 0.46 m L). 215 b e h a v i o u r b o u n d a r i e s f o r n i c k e l o x i d e ( F i g u r e 7 . 2 ) , i t i s o b s e r v e d t h a t the form of the p r e d i c t e d bed depth dependency i s s i m i l a r to t ha t f o r the measured s i u m p i n g - t r a n s i t i o n b o u n d a r y , i n d i c a t i n g t h a t a t low degrees o f f i l l the enhanced c u r v a t u r e o f the t r a n s i t i o n - r o l l i n g boundary may be due to c i r c u m f e r e n t i a l w a l l e f f e c t s . However , t h i s c o u l d no t be e x p e r i m e n t a l l y c o n f i r m e d . The 95% c o n f i d e n c e l i m i t on the r e g r e s s i o n c o e f f i c i e n t s has y i e l d e d a much s m a l l e r range in the p r e d i c t e d boundary than had been e x p e r i m e n t a l l y o b t a i n e d . The model p r e d i c t i o n s f o r o t h e r m a t e r i a l s t e s t e d are a l s o i n good agreement w i t h the e x p e r i m e n t a l r e s u l t s and are shown i n F i g u r e s 7 .3 to 7 . 6 . 7 . 2 . 2 The Bed Depth The e f f e c t o f bed depth on the boundary was not r e v e a l e d u s i n g any o f the s l ump ing or r o l l i n g c h a r a c t e r -i z a t i o n t e c h n i q u e s d e s c r i b e d i n S e c t i o n 4 . 5 . 1 and w i l l now be e x p l a i n e d i n the l i g h t o f the m a t h e m a t i c a l model d e v e l o p e d . I t was shown i n S e c t i o n 6 . 2 . 1 . 3 t h a t the t imes to maximum and minimum i n c l i n a t i o n s a re i ndependen t o f bed d e p t h , as i s the s i z e o f t h ' e s h e a r wedge. R e c a l l t h a t the s h e a r a n g l e was a l s o i n d e p e n d e n t o f bed d e p t h . Thus f o r v a r y i n g bed depths o f a g i v e n m a t e r i a l , the shea r p l a n e s and bed s u r f a c e s w i l l be p a r a l l e l i n a g i v e n c y l i n d e r . Hence , 216 the a n g l e s d e f i n i n g the geometry o f the sys tem w i l l be s i m i l a r ; however , the d i s t a n c e between the upper and l owe r c e n t r o i d s w i l l i n c r e a s e w i t h i n c r e a s i n g bed d e p t h . S i n c e the c o n t e n t s o f the shea r wedge have a l o n g e r d i s t a n c e to t r a v e l a t the h i g h e r bed d e p t h s , t h e i r ave rage t r a v e l l i n g t ime w i l l c o r r e s p o n d i n g l y i n c r e a s e . At equa l r o t a t i o n a l s p e e d s , t he a n g u l a r d i s t a n c e t r a v e l l e d by the c y l i n d e r d u r i n g the c o l l a p s e o f the shea r wedge w i l l t hus be g r e a t e r f o r the h i g h e r bed d e p t h s . S i n c e the minimum s h e a r wedge f o r a l l bed dep ths i s a c o n s t a n t , the deeper beds would r e q u i r e s m a l l e r r o t a t i o n a l speeds to s a t i s f y the s i u m p i n g - r o l 1 i n g boundary c r i t e r i o n , hence the bed depth dependency o f the bounda ry . T h i s can a l s o be m a t h e m a t i c a l l y i l l u s t r a t e d u s i n g E q u a t i o n ( 6 . 2 8 ) whose r a t i o f o r two bed dep ths y i e l d s , M W P ( 7 . 1 ) If H P < H M f rom geometry i t f o l l o w s t h a t S P < S M In o r d e r to m a i n t a i n the e q u a l i t y o f E q u a t i o n ( 7 . 1 ) , the a n g u l a r v e l o c i t i e s must have the f o l l o w i n g r e l a t i o n s h i p , 217 T h e r e f o r e , the s i u m p i n g - r o l 1 i n g boundary model c o r r e c t l y p r e d i c t s the bed depth dependency o b s e r v e d i n the B e d -B e h a v i o u r D i a g r a m s . 7 . 2 . 3 P a r t i c l e Shape Compar ing now the r e s u l t s o f F i g u r e s 7.1 and 7 . 2 , i t i s c l e a r t h a t the model has a c c o u n t e d f o r the d i f -f e r e n c e i n p a r t i c l e shape between the i r r e g u l a r l i m e t s t o n e B and the s p h e r i c a l n i c k e l o x i d e . A s u r v e y o f model i n p u t s ( T a b l e s V I I , X I I I and XIV) f o r both m a t e r i a l s r e v e a l s t h a t the s h e a r a n g l e and the minimum s h e a r wedge are s m a l l e r f o r the s p h e r i c a l s o l i d s ; w h i l e t h e i r d i l a t a n c y c o e f f i c i e n t i s l a r g e r . The s h e a r a n g l e s a re r e p o r t e d to be 3 7 . 7 ° and 2 9 . 9 ° f o r l i m e s t o n e B and n i c k e l o x i d e r e s p e c t i v e l y , w h i l e the r e s p e c t i v e minimum s h e a r wedges a re 5 .2 ° and 2 . 9 ° . A l t h o u g h the d i l a t a n c y c o e f f i c i e n t f o r l i m e s t o n e B, 0 .49 s e c o n d s , i s l ower than t h a t f o r n i c k e l o x i d e , 0.61 s e c o n d s , the shea r wedge o f the n i c k e l o x i d e w i l l be s m a l l e r u n t i l a r o t a t i o n a l speed o f 3 .2 r / m i n , where t h e i r s h e a r wedges would be e q u a l . Hence , o v e r the speeds o f i n t e r e s t the s l o p e o f the ave rage c h o r d a l t r a j e c t o r y w i l l be s t e e p e r f o r l i m e s t o n e B than f o r n i c k e l o x i d e . T h i s , combined w i t h a s m a l l e r minimum s h e a r wedge r e s u l t s i n a s i u m p i n g - r o l 1 i n g boundary o c c u r r i n g a t l ower r o t a t i o n a l speeds f o r the s p h e r i c a l bu l k s o l i d s ( F i g u r e s 7.1 and 7 . 2 ) . The boundary f o r the i r r e g u l a r 218 l i m e s t o n e i s a l s o more bed -dep th dependent than the s p h e r i -c a l n i c k e l o x i d e . The e f f e c t o f p a r t i c l e shape on bed b e h a v i o u r i s a l s o i l l u s t r a t e d i n F i g u r e s 7 .3 and 7 . 4 , f o r the i r r e g u l a r l i m e -s tone D and the n o d u l a r sand B. A compar i son o f the model i n p u t s f o r both m a t e r i a l s r e v e a l s t h a t a l t h o u g h the shea r a n g l e s o f both t hese bu l k s o l i d s a re n e a r l y e q u a l , 3 3 . 5 ° f o r l i m e s t o n e D and 32 .2 ° f o r sand B, the minimum shea r wedge f o r l i m e s t o n e D i s a t l e a s t doub le t h a t f o r sand B, 4 . 7 ° and 2 . 2 ° r e s p e c t i v e l y . On the o t h e r h a n d , the d i l a t a n c y c o e f f i c i e n t o f the i r r e g u l a r s o l i d s i s l ower than t h a t of the n o d u l a r s o l i d s , 0 .10 and 0 .36 s e c o n d s , b u t , as i n the p r e v i o u s i l l u s t r a t i o n w i t h l i m e s t o n e B and n i c k e l o x i d e , t h i s d i f f e r e n c e i s no t s u f f i c i e n t to y i e l d a s t e e p e r t r a -j e c t o r y f o r sand B. Hence , the s 1 u m p i n g - r o l 1 i n g boundary f o r sand B o c c u r s a t l ower r o t a t i o n a l speeds than f o r l i m e s t o n e D, wh ich a l s o d i s p l a y s a more a c c e n t u a t e d bed depth depen-dency . The e f f e c t o f p a r t i c l e s h a p e , can be m a t h e m a t i c a l l y e x p r e s s e d f rom E q u a t i o n ( 6 . 2 8 ) as f o l l o w s , w M = ( s i n n - cos n tan <j>L) 2 ] M a>p ( 7 . 2 ) rj [ Y ( s i n n - cos n tan <f>L)2]p -6 -5 IXIO IXIO Froude number (Fr =OJ R/g) 219 0.12 0.09 •D 00 0.06 0,0 3 \— i r Limestone D Best fit doto 95% confidence limits Experimental Slumping IxlO 25 20 15 10 £ 0.5 I 15 Rotational speed (r/min) Figure 7.3 Predicted and experimental boundaries for limestone D tested in cy l inder A (0.4 m ID x 0.46 m L ) . ixio6 ixio5 0.12 - 0.09 _£ c f 0.06 "D CO CO 0.03 Slumping Froude number ( F r = O J R / g ) IXIO Sand B Best fit dalo 95% confidence Experimental Rolling 0.5 I I.5 Rotational speed (r/min) IXIO 3 •25 •20 • 15 • 10 Figure 7.4 Predicted and experimental boundaries for sand B tested in cy l inder A (0.4 m ID x 0.46 m L) . 220 and w i l l be f u r t h e r d i s c u s s e d i n the f o l l o w i n g s e c t i o n s . 7.2.4 P a r t i c l e S i z e A compar i son o f the p r e d i c t e d b o u n d a r i e s f o r l i m e s t o n e s B and D ( F i g u r e 7.1 and 7 .3 ) r e v e a l s t h a t r o l l i n g s e t s i n a t l ower r o t a t i o n a l speeds f o r the s m a l l e r s i z e d bu lk s o l i d s . The r e s p e c t i v e model i n p u t s f o r l i m e s t o n e s B and D a r e : the s h e a r a n g l e , 3 7 . 7 ° and 3 3 . 5 ° , the minimum s h e a r wedge, 5 .2 ° and 4 . 7 ° and the d i l a t a n c y c o e f f i c i e n t , 0 .49 and 0 .10 s e c o n d s ; w h i l e t h e i r r e s p e c t i v e ave rage p a r t i -c l e s i z e s a re 4 . 3 mm and 0 .58 mm. T h u s , as the s i z e d e -c r e a s e s the s l o p e o f the c h o r d a l t r a j e c t o r y d e c r e a s e s and the minimum s h e a r wedge d e c r e a s e s . Both t h e s e f a c t o r s w i l l r e s u l t : ; as o b s e r v e d , i n the s 1 umpi n g - r o l 1 i n g boundary o c c u r r i n g at l owe r r o t a t i o n a l s p e e d s . T h i s i s f u r t h e r c o n -f i r m e d '"in F i g u r e 7 .5 f o r l i m e s t o n e C whose model i n p u t s a r e : <j>L = 3 3 . 6 ° , Y q = 6 . 5 ° and = 0 .16 s e c o n d s . Excep t f o r the minimum s h e a r wedge, the magn i tude o f the r e m a i n i n g i n p u t v a r i a b l e s ' T ' i e between t h o s e f o r l i m e s t o n e s B and D. The d e v i a t i o n o f Y f o r l i m e s t o n e C i s a t t r i b u t e d to a g r e a t e r o e x p e r i m e n t a l s c a t t e r i n the maximum and minimum bed i n c l i n a -t i o n t imes wh ich i s e v i d e n t by the l a r g e 95% c o n f i d e n c e range o f the p r e d i c t e d b o u n d a r i e s . Hence , the e f f e c t o f p a r t i c l e s i z e i s r e f l e c t e d i n E q u a t i o n ( 6 . 2 8 ) i n the same manner as p a r t i c l e shape and may a l s o be d e s c r i b e d by E q u a t i o n ( 7 . 2 ) . Figure 7.5 Predicted and experimental boundaries fo r limestone C tested in cy l inder A (0.4 m ID x 0.46 m L ) . Froude number (Fr=oJR/g) IXI06!XI05 IXIO4 IxlO3 0.0 3 h 0.5 I Rotational speed (r/min) 1.5 Figure 7.6 Predicted and experimental boundaries fo r limestone B tested in cy l inder C (1.06 m ID x 0.4 m L) . 222 7 . 2 . 5 Combined E f f e c t o f P a r t i c l e S i z e and Shape I t was shown i n the p r e v i o u s s e c t i o n s t h a t both p a r t i c l e s i z e and shape a f f e c t the s l o p e o f the c h o r d a l t r a j e c t o r y of the s o l i d s as w e l l as the s i z e o f the minimum s h e a r wedge, and t h a t E q u a t i o n ( 7 . 2 ) p r e s e n t s the s c a l e - u p c r i t e r i a f o r each o f t h e s e m a t e r i a l v a r i a b l e s . I t t h e r e -f o r e f o l l o w s t h a t when compar ing two b u l k s o l i d s h a v i n g d i f -f e r e n t ave rage s i z e s and s h a p e s , E q u a t i o n ( 7 . 2 ) s h o u l d s t i l l a p p l y i n p r e d i c t i n g the boundary o f one o f the m a t e r i a l s f rom t h a t o f the o t h e r . F u r t h e r m o r e , t h e s e d i f f e r e n c e s i n p a r t i c l e s i z e and shape would a l s o be r e f l e c t e d i n the i n p u t v a r i a b l e s o f t h e model f o r both m a t e r i a l s as w i l l be i l l u s t r a -ted f o r the s p h e r i c a l n i c k e l o x i d e and the n o d u l a r sand B ( r e c a l l t h a t t h e i r p a r t i c l e s i z e s a re 4 .9 mm and 0 .50 mm r e s p e c t i v e l y ) . T h e i r r e s p e c t i v e model i n p u t s a r e : the s h e a r a n g l e , 2 9 . 9 ° and 3 2 . 2 ° , . the minimum s h e a r wedge, 2 . 9 ° and 2 . 2 ° , and the d i l a t a n c y c o e f f i c i e n t 0.61 and 0 .36 s e c o n d s . As i t was d i f f i c u l t i n Chap te r 4 to e s t i m a t e - ' q u a T i t a t i v e l y whether the p a r t i c l e s i z e o r the p a r t i c l e shape would d o m i n a t e , so i t i s e q u a l l y d i f f i c u l t a t t h i s p o i n t to e s t i m a t e f rom the above i n p u t da ta wh ich v a r i a b l e w i l l have a dominant e f f e c t on the model p r e d i c t i o n o f the b o u n d a r i e s . The s h e a r a n g l e i n d i c a t e s t h a t the boundary f o r n i c k e l o x i d e w i l l be to the l e f t o f t h a t f o r sand B; w h i l e the minimum s h e a r wedge and the d i l a t a n c y c o e f f i c i e n t i n d i c a t e 223 the r e v e r s e . The model p r e d i c t i o n s c l e a r l y show the l a t t e r v a r i a b l e s to have domina ted ( F i g u r e s 7.2 and 7 . 4 ) . Hence , the model does i ndeed accoun t f o r the e f f e c t s o f both p a r t i c l e s i z e and shape th rough a p p r o p r i a t e changes i n the shear a n g l e , <|>^ , the ave rage s l o p e o f the c h o r d a l t r a -j e c t o r y , n» and the minimum s h e a r wedge, y . F u r t h e r m o r e , i n v iew of the r e s u l t s p r e s e n t e d thus f a r , the minimum s h e a r wedge appears to be the dominant m a t e r i a l v a r i a b l e i n d e t e r -m i n i n g the r e l a t i v e p o s i t i o n o f the boundary . 7 . 2 . 6 C y l i n d e r D iamete r The minimum shear wedge, the d i l a t a n c y and the shea r a n g l e were de te rm ined f o r l i m e s t o n e B i n c y l i n d e r C (1 .06 m ID) to be 5 . 7 ° , 0 .49 seconds and 3 4 . 5 ° r e s p e c t i v e l y . Us ing t h i s da ta the s i u m p i n g - r o l 1 i n g boundary was p r e d i c t e d and compared i n F i g u r e 7.6 to the e x p e r i m e n t a l r e s u l t s . At the h i g h e r degrees o f f i l l the model o v e r p r e d i c t e d the boundary by about 0 .3 r /m in and the e x p e r i m e n t a l boundary d i s p l a y s a more a c c e n t u a t e d bed depth dependency . N e v e r -t h e l e s s , i n v iew o f the v e r y nar row e x p e r i m e n t a l t r a n s i t i o n zone and the many s i m p l i f y i n g a s s u m p t i o n s o f the m o d e l , the agreement i s r e a s o n a b l y good . The i n p u t da ta f o r t h i s same m a t e r i a l , t e s t e d i n c y l i n d e r s A and B, r e v e a l s t h a t o n l y the c y l i n d e r d i a m e t e r 224 and the shea r a n g l e d i f f e r e d f rom t h o s e l i s t e d above f o r the l a r g e r c y l i n d e r . That i s , the minimum s h e a r wedge and the d i l a t a n c y c o e f f i c i e n t were e q u a l . T h e r e f o r e , the r a t i o o f the c h o r d a l t r a j e c t o r y e q u a t i o n f o r both c y l i n d e r s r educes t o , / s \ - = ( F r ) p ( s i n n - cos n tan c& L) M / s \ ( 7 . 3 ) WM ^ F r ^ M ( s i n n - cos n tan <j>L)p \ R / p In o r d e r to i d e n t i f y the r e l a t i v e c o n t r i b u t i o n o f the a n g u l a r f u n c t i o n i n E q u a t i o n ( 7 . 3 ) , the s h e a r a n g l e f o r c y l i n d e r s A and B was used to p r e d i c t a boundary f o r c y l i n d e r C ; thus e n s u r i n g t h a t the a n g u l a r r e l a t i o n s i n both c y l i n d e r s were e q u a l . N e g l i g i b l e d i f f e r e n c e i n the boundary r e s u l t e d , i n d i c a t i n g t h a t E q u a t i o n ( 7 . 3 ) may be f u r t h e r s i m p l i f i e d t o , E q u a t i o n ( 7 . 4 ) t h e r e f o r e p o i n t s to two d i m e n s i o n l e s s v a r i a b l e s f o r the s c a l e - u p o f the s 1 u m p i n g - r o l 1 i n g boundary w i t h r e s p e c t to the c y l i n d e r d i a m e t e r , the degree o f f i l l and the Froude number. 7 . 2 . 7 S e a l e - u p An a t t emp t w i l l be made i n t h i s s e c t i o n to c o n -s o l i d a t e and s i m p l i f y the s c a l e - u p c r i t e r i a t h a t have been ( 7 . 4 ) 225 p r e s e n t e d f o r m a t e r i a l and c y l i n d e r v a r i a b l e s . The a n a l y s i s w i l l be aimed at r e l a t i n g the r e g r e s s i o n c o e f f i c i e n t s to m a t e r i a l and c y l i n d e r v a r i a b l e s . The minimum shea r wedges and the d i l a t a n c y c o e f f i -c i e n t s f o r a l l m a t e r i a l s and c y l i n d e r s used i n t h i s s tudy a re p l o t t e d as a f u n c t i o n o f p a r t i c l e s i z e i n F i g u r e s 7.7 and 7 . 8 . The bars on t hese graphs r e p r e s e n t the 95% c o n f i d e n c e l i m i t s f o r the r e s p e c t i v e c o e f f i c i e n t s . The e f f e c t s o f p a r t i c l e s h a p e , p a r t i c l e s i z e and c y l i n d e r d i a m e t e r a re e v i d e n t . S p h e r i c a l and n o d u l a r s o l i d s have l ower minimum s h e a r wedges than i r r e g u l a r and a n g u l a r s o l i d s . F u r t h e r -more , a dependency o f p a r t i c l e s i z e on y Q i s appa ren t f o r s p h e r i c a l and n o d u l a r s o l i d s whereas f o r the i r r e g u l a r and a n g u l a r s o l i d s the minimum s h e a r wedge appears to be i n -dependent o f p a r t i c l e s i z e . However , e x c l u d i n g the r e s u l t s f o r l i m e s t o n e G, whose da ta e x h i b i t e d a g r e a t e r s c a t t e r , the minimum s h e a r wedge does appear to d i s p l a y a s l i g h t d e -pendency on p a r t i c l e s i z e . The c y l i n d e r d i a m e t e r i s o b s e r v e d to have n e g l i g i b l e e f f e c t on both Yq and . S i n c e the minimum s h e a r wedge and the s h e a r a n g l e a re r e l a t e d by the e q u a t i o n •R = Y0 + * L ( 7 . 5 ) and s i n c e lower v a l u e s o f the s t a t i c a n g l e o f r epose i n d i c a t e 226 CD OJ cn CU •o 3 8| cu cn CU 5 C y l i n d e r internal <b ( m ) 0 .4 1.0 © S p h e r i c a l & n o d u l a r o A I r regular 8 angu la r o cu x : to £ E 0 lm C o loo o _ CO • o • « "5 I E CO CO I CO > o O m a> c o 07 E CO o "a o co c o to CO £ CO "x o c o 0 8 0 Average particle size dp(mm) F i g u r e 7 . 7 Minimum s h e a r wedge v e r s u s a v e r a g e p a r t i c l e s i z e f o r a l l m a t e r i a l s t e s t e d . 227 (JT c OJ o .0 0.8 0.6 C y l i n d e r interna <|> (m) 0.4 1.0 © S p h e r i c a l & nodular O A I r regular & angular CD O o o c o 5 0.4 0.2 0 6 m -o c, D CO c o CO > c o O i -- 0 . 2 A 0 ca CO c o In co £ co 'x o o CO c o w cu E CO T» X O c o CO CO E E Li 1^ 1 2 4 6 8 Average particle size d. 10 12 (m m) Figure 7.8 Di latancy coe f f i c i en t versus average p a r t i c l e s i ze for a l l mater ials tested. 228 e a s i e r f"! owabi 1 i t y o f s o l i d s , t h e r e f o r e the l owe r the v a l u e s o f Y q the e a s i e r the f l o w a b i l i t y o f the bu l k s o l i d . Hence , the l ower the r o t a t i o n a l speed a t wh ich a bu l k s o l i d w i l l r o l l , as has been e x p e r i m e n t a l l y o b s e r v e d and m a t h e m a t i c a l l y p r e d i c t e d . F i n a l l y , the d i l a t a n c y c o e f f i c i e n t s shown i n F i g u r e 7 .8 seem to be o n l y a f u n c t i o n o f p a r t i c l e s i z e . T h e r e f o r e , i t i s appa ren t t h a t ove r the range o f v a r i a b l e s i n v e s t i g a t e d , the minimum s h e a r wedge i s the most i m p o r t a n t m a t e r i a l v a r i a b l e wh ich i s a f f e c t e d by shape and s i z e and wh ich r e f l e c t s the l o c a t i o n o f the s i u m p i n g - r o l 1 i n g b o u n d a r y . The m a t e r i a l s c a l e - u p c r i t e r i a c o u l d t h e r e f o r e be s i m p l i f i e d i n terms o f Y q and c o u l d be e x p r e s s e d i n terms o f more p r i m a r y m a t e r i a l v a r i a b l e s , namely the p a r t i c l e s i z e and s h a p e . S i n c e the l a t t e r was not q u a n t i f i e d i n t h i s s tudy and i s i n h e r e n t l y ve ry d i f f i c u l t to q u a n t i f y , the fo rmer was used i n the f o l l o w i n g a n a l y s i s . The e x p e r i m e n t a l b o u n d a r i e s f o r l i m e s t o n e B and C de te rm ined i n c y l i n d e r s A and B are p r e s e n t e d on a d i m e n s i o n l e s s p l o t o f % f i l l v e r s u s Froude number i n F i g u r e 7 . 9 a . No agreement i n the b o u n d a r i e s i s o b s e r v e d . I f the Froude numbers o f the l i m e s t o n e C b o u n d a r i e s a re m o d i f i e d as f o l l o w s w i t h r e s p e c t to p a r t i c l e s i z e but f o r equa l c y l i n d e r d i a m e t e r , [ F r ] = dp % [ F r ] p ( 7 . 6 ) 229 20-1 6 -12-Slumping Rolling 0.4 m 1 D cylinder dp(mm) A Limestone C 1.5 O B 4.3 2x10 r4 4X10 6 X 1 0 8 X 1 0 10X10 Froude number (Fr»0JR/g) (a) F i gure 7 .9 20 I 6 Slumping 0.4m ID cylinder Prototype Model Limestone C B A O " - 4 2 X l O 4 x 1 0 " 6 X 1 0 ' 8 X 1 0 I 1/2 10X10* I2XI0 4 - 4 - 4 5 8xH [ F r ] M = H P k M / d " p ] ' (b) D i m e n s i o n l e s s P l o t o f the e x p e r i m e n t a l b o u n d a r i e s o f l i m e s t o n e s B and C t e s t e d i n c y l i n d e r s A and B (0 .4 m ID x 0 .46 m L and 0 .86 m L r e s p e c t i v e l y ) . (a) u s i n g Froude number and % F i l l (b) u s i n g Froude number , % F i l l and s i z e r a t i o I 230 where the s u b s c r i p t ^ r e p r e s e n t s the m o d e l , l i m e s t o n e B, and the s u b s c r i p t p the p r o t o t y p e , l i m e s t o n e C , t h e t r a n s i t i on zones o f both m a t e r i a l s o v e r l a p as shown i n F i g u r e 7 . 9 b . T h e r e f o r e , even though the p a r t i c l e s i z e i s not e x p l i c i t l y e x p r e s s e d i n E q u a t i o n ( 6 . 2 8 ) i t i s i m p l i c i t l y accoun ted f o r i n the minimum s h e a r wedge s i z e , y - ( F i g u r e 7 .7 ) and i n the s h e a r a n g l e , <f>L ( T a b l e V I I ) . That i t cannot be d i r e c t l y i n t r o d u c e d i n the o r i g i n a l t r a j e c t o r y e q u a t i o n i s one o f the main drawbacks i n a p p l y i n g the p r i n c i p l e s o f mechan ics f o r r i g i d b o d i e s to bu l k s o l i d s wh ich has been e n c o u n t e r e d i n 1 gg o t h e r models o f bu l k s o l i d s f l o w . For e x a m p l e , i n m o d e l l i n g the d i s c h a r g e r a t e o f bu l k s o l i d s f rom b i n s and h o p p e r s , the p a r t i c l e s i z e does not appear i n the e q u a t i o n s o f " k i n e m a t i c s but i s known to a f f e c t the d i s c h a r g e r a t e . In f a c t , i t was found i n t hose a p p l i c a t i o n s t h a t the r a t i o o f p a r t i c l e d i a m e t e r to o r i f i c e d i a m e t e r i s a s c a l e - u p f a c t o r . In s o l i d - f l u i d r e a c t o r s such as spou ted beds and f l u i d i z e d beds the (D/dp) r a t i o has a l s o been found to 100 149-151 a p p l y as one o f the s c a l i n g c r i t e r i a . ' Rose and B l u n t ^ have e x p e r i m e n t a l l y d e t e r m i n e d t h a t (D/dp) i s one o f the s c a l e - u p c r i t e r i a f o r s l i p p i n g i n r o t a r y c y l i n d e r s and i n d i c a t e t h a t i t s pu rpose i s to m a i n t a i n the e q u a l i t y of the number o f c o n t a c t p o i n t s between the bu lk s o l i d s and the c y l i n d e r w a l l . A p p l y i n g t h i s o b s e r v a t i o n to the s i u m p i n g - r o l 1 i n g boundary and the p a r t i c l e s i z e r e l a t i o n s h i p 231 i n E q u a t i o n ( 7 . 6 ) would i n d i c a t e t h a t the f u n c t i o n a l r e l a t i o n -s h i p o f t h i s s c a l e - u p f a c t o r i s ( D / d p ) 2 . T h e r e f o r e a t equa l degrees of f i l l the s c a l e - u p c r i t e r i o n would be g i v e n b y , [ F r ] , d h ~D " P D d M P [ F r ] , ( 7 . 7 ) The e x p e r i m e n t a l b o u n d a r i e s o f l i m e s t o n e s B, C and D i n c y l i n d e r s A and B were s c a l e d - u p to the p a r t i c l e s i z e o f l i m e s t o n e B i n c y l i n d e r C u s i n g E q u a t i o n 7.7 and were com-pared to the e x p e r i m e n t a l r e s u l t s o f the l a t t e r , r e f e r r e d to as the m o d e l . The r e s u l t s , p l o t t e d i n F i g u r e 7 . 1 0 , i n d i c a t e t h a t the t r a n s i t i o n zones o f a l l t h r e e s c a l e d l i m e s t o n e s o v e r l a p . The t r a n s i t i o n zone o f l i m e s t o n e D a l s o o v e r l a p s w i t h the model r e s u l t s . The d i s c r e p a n c y between the b o u n d a r i e s o f l i m e s t o n e s B and C p r o t o t y p e s and t hose o f the model i s v e r y s m a l l , o f the o r d e r o f 0 .3 r /m in i n c y l i n d e r s A and B (0 .4 m ID)and 0 .15 r /m in i n c y l i n d e r C (1 .06 m D) . T h e r e f o r e f o r bu l k s o l i d s o f equa l shape the s c a l e - u p f a c t o r s o f the s i u m p i n g - r o l 1 i n g boundary may be s i m p l i f i e d t o : the degree o f f i l l and the Froude I , number m u l t i p l i e d by ( D / d ) 2 as g i v e n by E q u a t i o n ( 7 . 7 ) . As has been d i s c u s s e d i n S e c t i o n 3 . 2 . 2 , the p a r t i c l e shape o f the m a t e r i a l s used i n t h i s s t udy has not been q u a n t i f i e d , hence a s c a l i n g c r i t e r i o n based on p a r t i c l e shape c o u l d not be d e v e l o p e d but i s recommended f o r f u t u r e Equivalent rotational speed in 0.4 m I.D. c y l i n d e r ( r / m i n ) 1.0 1.25 1.5 1.75 2 . 0 Im I.D. c y l i n d e r 0.5 0 .7 0.9 I.I 1.3 HM=HP[DP/DJ/%,/<W]1'2 Figure 7.10 Dimensionless Plot of experimental boundaries of limestones B, C and D tested in cyl inders A and B, scale-up, using the Froude number, the % F i l l and the p a r t i c l e s ize to cy l inder diameter r a t i o , and compared to the boundaries of limestone B tested in cy l inder C. 233 work . Such an i n v e s t i g a t i o n , c a r r i e d ou t a l o n g the same l i n e s as the c u r r e n t o n e , would a l s o i d e n t i f y the shape f a c t o r wh ich • i n f l u e n c e s the f l o w a b i l i t y o f bu l k s o l i d s ( the s p h e r i c i t y o r the Heywood r a t i o s ) . S i n c e s e v e r a l i n v e s t i g a t o r s have s u g -g e s t e d t h a t the number o f i n t e r p a r t i c l e c o n t a c t p o i n t s , the c o - o r d i n a t i o n number, i s r e l a t e d to s o l i d s m i x i n g and f l o w -a b i l i t y 1 ^ ' 1 ^ 2 and s i n c e i t i s a more fundamenta l p r o p e r t y o f bu l k s o l i d s , i t wou ld t h e r e f o r e be f r u i t f u l to a l s o q u a n t i f y i t . A s t udy o f i t s r e l a t i o n s h i p to the shape f a c t o r , p a r t i c l e s i z e and minimum s h e a r wedge, w i t h the a i d o f B e d - B e h a v i o u r D i a g r a m s , s h o u l d be c o n c u r r e n t l y c a r r i e d out u s i n g i d e a l l y s i z e d and shaped m a t e r i a l s . 7 . 2 . 8 Gas E v o l u t i o n and the Tempera tu re o f ^ e S o l i d s In a c o n t i n u o u s l y o p e r a t e d r o t a r y k i l n the e v o l u t i o n o f gases f rom the bed o f the s o l i d s and the t e m p e r a t u r e o f the s o l i d s may enhance or r e t a r d the s o l i d s m o t i o n . Some o f t hese a s p e c t s have a l r e a d y been a d d r e s s e d i n Chap te r 5 i n c o n n e c t i o n w i t h s e g r e g a t i o n . Two e f f e c t s of t e m p e r a t u r e and of gas e v o l u t i o n t h a t r e f l e c t on the p r e d i c t e d s i u m p i n g - r o l 1 i n g boundary a re o f i n t e r e s t ; the f i r s t i s conce rned w i t h normal k i l n o p e r a t i o n and the second touches on c e r t a i n a s p e c t s o f o v e r h e a t i n g the s o l i d s . 234 In some r o t a r y k i l n o p e r a t i o n s , such as l i m e s t o n e c a l c i n a t i o n gases may be g e n e r a t e d i n the bed . In r i s i n g th rough the cha rge to the f r e e b o a r d , t hey e x e r t a drag f o r c e on the bu l k s o l i d s wh ich a l s o undergo changes i n t h e i r p h y s i c a l and c h e m i c a l n a t u r e as they t r a v e l t h rough the k i l n . The drag f o r c e e x e r t e d on the bed by the r i s i n g gases may be a c c o u n t e d f o r i n the - s i u m p i n g - r o l 1 i n g model by q u a n t i -f y i n g the mass o f gas e v o l v i n g a t a g i v e n l o c a t i o n a l o n g the k i l n and m o d i f y i n g the g r a v i t y f o r c e o f the s h e a r wedge a c c o r d i n g l y . T h i s would r e s u l t i n a s 1 u m p i n g - r o l 1 i n g boun-dary wh ich w i l l o c c u r a t l ower r o t a t i o n a l speeds than i n the absence o f t hese g a s e s . To c o n f i r m t h a t t hese gases have an e f f e c t on bed b e h a v i o u r , the s l u m p i n g f r e q u e n c y o f l i m e s t o n e C, measured i n the h o r i z o n t a l c y l i n d e r , i s compared to the measurements made d u r i n g c o l d and hot runs i n the UBC p i l o t k i l n ( F i g u r e 7 . 1 1 ) . The d i f f e r e n c e between the s l u m p i n g f r e q u e n c y o f the c o l d run and t hose f o r the h o r i z o n t a l c y l i n d e r has been a t t r i b u t e d to the a x i a l s l o p e o f the k i l n . The r e s u l t s f rom the hot run a re h i g h e r than t h o s e o f the c o l d run and are up to 4 s l u m p s / m i n g r e a t e r than the f r e -q u e n c i e s i n the h o r i z o n t a l c y l i n d e r . In the hot r u n , the s o l i d s a t t a i n e d a t e m p e r a t u r e o f 1107°K a t 0 .4 m f rom the s o l i d s d i s c h a r g e end . C a l c i n a t i o n t e s t s c a r r i e d out by 15 3 Brimacombe and Watk inson have showed t h a t a t 1107°K, t h e r e i s a 2.5% c o n v e r s i o n to l i m e and t h a t on comp le te 4 4 r — 4 0 3 6 O o ° o ° o CP O 8 3 2 o o o © p _ 2 8 | c £ E to >> o c CO C T CO c '5L E CO 2 0 16 1 6? • i s A ° O o 8 -<9 Hor i zon ta l cy l inder O Pi lot kiln cold test O Pi lot kiln hot test • 8 IO 0 l 1 0 4 0.8 1.2 1.6 2.0 Rota t iona l s p e e d ( r / m i n ) 2.4 ure 7.11 Slumping frequencies of limestone C in cy l inder A (0.4 m ID x 0.46 m L) compared with those measured in p i l o t k i l n during hot and cold runs. ; 236 c a l c i n a t i o n the change i n p a r t i c l e s i z e and the f i n e s g e n e r a t i o n were m i n i m a l . V i s u a l i n s p e c t i o n of the s tone i n d i c a t e d t h a t no change r e s u l t e d i n p a r t i c l e shape and t h a t c a l c i n a t i o n had s t a r t e d , hence the carbon d i o x i d e wh ich e v o l v e d f rom the cha rge may have been r e s p o n s i b l e f o r the i n c r e a s e i n s l u m p i n g f r e q u e n c y . As d i s c u s s e d i n Chap te r 4 , a h i g h e r s l u m p i n g f r e q u e n c y c u r v e i s i n d i c a t i v e o f a s l u m p i n g -r o l l i n g boundary o c c u r r i n g at l ower r o t a t i o n a l s p e e d s . To f u r t h e r v e r i f y t h i s c o n c l u s i o n , the c o n t r i b u t i o n o f the drag f o r c e was e s t i m a t e d . I f a l l the gases g e n e r a t e d f rom the c a l c i n a t i o n r e a c i o n i n the bed were e v o l v e d t h rough the shea r p l a n e , the g r a v i t y f o r c e o f the shea r wedge would have been reduced by a p p r o x i m a t e l y 10% f o r the sys tem de-s c r i b e d above . From E q u a t i o n ( 6 . 2 8 ) t h i s would r e s u l t i n the s 1 u m p i n g - r o l 1 i n g boundary o c c u r r i n g a t a l ower r o t a t i o n a l speed and hence the e v o l u t i o n o f gases would a f f e c t bed b e h a v i o u r . The p h y s i c a l changes undergone by the s o l i d s , w i l l a f f e c t t h e i r shape and s i z e as they t r a v e l t h rough the k i l n . Fo r e x a m p l e , o b s e r v a t i o n s o f i n d u s t r i a l c a l c i n i n g k i l n s have r e v e a l e d t h a t a n g u l a r l i m e s t o n e fed to a r o t a r y k i l n w i l l d i s c h a r g e as i r r e g u l a r s o l i d s . Coa l lumps fed w i t h i r o n o x i d e p e l l e t s w i l l be q u i c k l y ground to a powder and w i l l be g a s i f i e d . The i r o n o x i d e p e l l e t s i n t u r n may undergo 237 s w e l l i n g a t the e a r l y s t a g e s o f r e d u c t i o n , then p r i o r to d i s c h a r g i n g they may d e c r e a s e a g a i n i n s i z e . Under t h e s e c o n d i t i o n s i t i s not p o s s i b l e to p r o v i d e a s i n g l e Bed -B e h a v i o u r Diagram to d e s c r i b e the t r a n s v e r s e mot ion o f s o l i d s ove r the e n t i r e l e n g t h o f the k i l n . R a t h e r , s e v e r a l d iag rams c o u l d be d e t e r m i n e d f o r each c h a r a c t e r -i s t i c zone i n the f u r n a c e wh ich c o u l d be used to e s t a b l i s h e f f e c t i v e o p e r a t i n g s t r a t e g i e s and wh ich c o u l d be i n c o r p o r a -t ed i n t o f u t u r e k i l n d e s i g n s . O v e r h e a t i n g the s o l i d s c o u l d have a s e r i o u s d e t r i -menta l e f f e c t on bed b e h a v i o u r due to the a g g l o m e r a t i o n o f s o l i d s i n the bed . T h i s would i n c r e a s e the i n t e r n a l f r i c t i o n o f the bu l k s o l i d s , p r o v i d e poor g a s - s o l i d c o n t a c t as w e l l as poor s o l i d s m i x i n g . To a n t i c i p a t e t h i s o c c u r r e n c e a s e r i e s o f Bed B e h a v i o u r Diagrams c o u l d be e x p e r i m e n t a l l y de te rm ined ove r a range o f o p e r a t i n g t e m p e r a t u r e s . The s h i f t i n the s 1 u m p i n g - r o l 1 i n g b o u n d a r y , c o r r e s p o n d i n g to v a r i o u s o p e r a t i n g s o l i d s t e m p e r a t u r e s , c o u l d be de te rm ined and the c o n d i t i o n s o f s o l i d s o v e r h e a t i n g i d e n t i f i e d . T h i s i n f o r m a t i o n c o u l d be u s e f u l i n d e v e l o p i n g an o p e r a t i n g p r a c t i c e f o r i n d u s t r i a l k i l n s as w e l l as i n p r o v i d i n g u s e -f u l i n f o r m a t i o n to i n d i c a t e some d e s i g n and o p e r a t i n g c o n -s t r a i n t s . 238 7 .3 S l i p p i n g The p r e d i c t i o n s o f the s l i p p i n g m o d e l , p r e s e n t e d i n C h a p t e r 6 , a re shown i n F i g u r e 7.12 as a p l o t o f the r e -s u l t a n t bed i n c l i n a t i o n , wh i ch i s c a l c u l a t e d f o r a g i v e n f r a c t i o n o f c r i t i c a l speed and degree o f f i l l v e r s u s the s l i p p i n g a n g l e . These r e s u l t s were o b t a i n e d f o r t h r e e k i l n d i a m e t e r s : 0 .4 m, 1.0 m and 2 .7 m; as w e l l as f o r t h r e e Froude numbers : 0 , 0 .02 and 0 . 0 7 5 . For equa l s l i p a n g l e s and Froude numbers the model p r e d i c t i o n s f o r the t h r e e k i l n d i a m e t e r s a re i d e n t i c a l a t equa l f i l l r a t i o s . I t i s f u r t h e r o b s e r v e d i n F i g u r e 7.12 t h a t the e f f e c t o f r o t a -t i o n a l speed i s n e g l i g i b l e . For e x a m p l e , f o r a s l i p p i n g a n g l e o f 40° and a 10% f i l l the e q u i l i b r i u m a n g l e o f bed i n c l i n a t i o n f o r a Froude number o f 0 and 0 .075 a re 5 2 . 5 ° and 55° r e s p e c t i v e l y . The n e g l i g i b l e e f f e c t o f r o t a t i o n a l speed i s f u r t h e r s u b s t a n t i a t e d by i n d u s t r i a l o b s e r v a t i o n s o f b a l l m i l l s where changes i n r o t a t i o n a l speed have had a 2 4 55 n e g l i g i b l e e f f e c t i n e l i m i n a t i n g s l i p p i n g . ' ' T h i s w i l l be f u r t h e r d i s c u s s e d i n the f o l l o w i n g s e c t i o n . T h e r e -f o r e the e q u i l i b r i u m a n g l e o f bed i n c l i n a t i o n need o n l y be compared to the s t a t i c and dynamic a n g l e s o f r epose o f a m a t e r i a l r a t h e r than cp^ and cf)p as d e s c r i b e d i n S e c t i o n 6 . 3 . The a p p l i c a t i o n o f F i g u r e 7.12 f o r p r e d i c t i n g the o c c u r r e n c e o f s l i p p i n g w i l l now be i l l u s t r a t e d u s i n g a 239 \ Figure 7.12 Predic t ions of the s l i pp ing model for k i l n diameters of 0.4 m, 1.0m and 2.7 m. 240 b e d / w a l l a n g l e o f s l i p o f 30° and s t a t i c and dynamic a n g l e s o f repose o f 40° and 35° r e s p e c t i v e l y . When the bed i s n o r m a l l y s l u m p i n g the s l i p p i n g a n g l e o f e q u i l i b r i u m i s com-pared to the s t a t i c a n g l e o f r e p o s e . For e x a m p l e , l o c a t e i n F i g u r e 7.12 the c o - o r d i n a t e s ( 3 0 ° , 4 0 ° ) . I f the degree o f f i l l i n the k i l n i s 5%, then the e q u i l i b r i u m bed i n c l i n a -t i o n f o r the s l i p p i n g f o r c e and moment b a l a n c e i s 3 5 ° . Hence , the bed w i l l s l i p p r i o r to r e a c h i n g the s l u m p i n g a n g l e o f 4 0 ° . I f on the o t h e r hand the degree o f f i l l i s 20%, the e q u i l i b r i u m a n g l e o f s l i p i s then 45° and the bed w i l l slump b e f o r e r e a c h i n g the e q u i l i b r i u m bed i n c l i n a t i o n o f s l i p . T h e r e f o r e , i t appears i n t h i s case t h a t a l l degrees of f i l l l e s s than 12.5% w i l l r e s u l t i n the bed s l i p p i n g w h i l e t hose g r e a t e r than 12.5% w i l l r e s u l t i n the o c c u r r e n c e o f s l u m p i n g . I f the bed were r o l l i n g then the s l i p p i n g boundary would o c c u r a t the 10% f i l l . Hence , the upper l e f t of F i g u r e 7.12 r e p r e s e n t s the no s l i p s i t u a t i o n and the bot tom r i g h t , the s l i p p i n g s i t u a t i o n . From t h i s a n a l y s i s i t i s c l e a r t h a t to a v o i d s l i p p i n g i n k i l n s , o n l y two v a r i a b l e s a re o f i n t e r e s t : the b e d / w a l l f r i c t i o n a n g l e and the degree o f f i l l . In k i l n o p e r a t i o n s , the fo rmer may be i n c r e a s e d by r oughen ing the w a l l wh ich may be e f f e c t e d i n a number o f ways . In the l i m e i n d u s t r y the k i l n w a l l may be roughened by f e e d i n g s a l t i n t o the k i l n ; 241 however , t h i s has the accompany ing d e t r i m e n t a l e f f e c t of d e c r e a s i n g the r e a c t i v i t y o f the l i m e s u b s e q u e n t l y p r o -d u c e d ^ ' ^ and i s t h e r e f o r e not w i d e l y a p p l i e d . A l t e r n a t e l y , o p e r a t o r s a re f a c e d w i t h a t o t a l p r o d u c t i o n shu t down of up to one week to m a n u a l l y roughen the r e f r a c t o r y w a l l . The p lacement o f k i l n i n t e r n a l s has a l s o the e f f e c t o f i n -c r e a s i n g the s o l i d s / w a l l f r i c t i o n a n g l e and i s r e p o r t e d to 47 be s u c c e s s f u l l y a p p l i e d . O ther methods f o r a v o i d i n g s l i p p i n g i n c l u d e i n c r e a s i n g the f e e d r a t e to the k i l n , thus i n c r e a s i n g the f i l l r a t i o . C a u t i o n would be a d v i s e d i n t a k i n g t h i s app roach as an u n a c c e p t a b l e p r o d u c t c o u l d be o b t a i n e d due to the p o s s i b l e i n c r e a s e i n s o l i d s r e -s i d e n c e t ime a t t e m p e r a t u r e wh ich may be compensated f o r by a c o r r e s p o n d i n g i n c r e a s e i n r o t a t i o n a l s p e e d , f o r examp le . A more nove l app roach would be to c o n t r o l the a c c r e t i o n b u i l d - u p on the k i l n w a l l t hus r o u g h e n i n g the w a l l o r i n -c r e a s i n g the h o l d - u p o f the k i l n . A g a i n , to ensu re t h a t the s o l i d s r e s i d e n c e t ime i s not e x c e s s i v e l y i n c r e a s e d due to the p r e s e n c e o f a c c r e t i o n dams the r o t a t i o n a l speed o f the k i l n may a l s o be i n c r e a s e d . A f u r t h e r c o m p l i c a t i o n i n c h a r a c t e r i z i n g s l i p p i n g i n k i l n s i s t h a t the w a l l roughness and the p h y s i c a l p r o p e r t i e s o f the s o l i d s may va ry a l o n g the k i l n l e n g t h . Under t h e s e c o n d i t i o n s , the s o l i d s and the s o l i d s / w a l l f r i c t i o n can be 242 c h a r a c t e r i z e d u s i n g the methods d e s c r i b e d i n t h i s t h e s i s ( S e c t i o n 3 .4 ) and an o p e r a t i n g s t r a t e g y d e v e l o p e d f o r each s e c t i o n or zone i n the k i l n . T h i s app roach c o u l d a l s o be used to h e l p i d e n t i f y t h o s e o p e r a t i n g c o n d i t i o n s when s l i p p i n g might be e n c o u n t e r e d and a p p r o p r i a t e p r e v e n t a t i v e a c t i o n t a k e n . 7.4 The Comple te B e d - B e h a v i o u r Diagram A l l s i x p r e d i c t e d modes o f bed b e h a v i o u r a re p l o t t e d i n F i g u r e 7 .13 f o r g r a v e l i n c y l i n d e r A. O the r p r e d i c t i o n s a re shown i n F i g u r e s 7 .14 to 7 . 1 6 . The s o l i d l i n e s on the d iagrams denote the b o u n d a r i e s wh ich were c a l c u l a t e d u s i n g the r e s p e c t i v e models p r e s e n t e d i n Chap te r 6. In the case o f g r a v e l ( F i g u r e 7 .13) s i n c e the b e d / w a l l f r i c t i o n a n g l e , <}><, f o r t h i s case was 4 8 . 8 ° and the s t a t i c and dynamic a n g l e s o f repose were 4 0 . 7 ° and 3 7 . 5 ° r e s p e c t i v e l y , no s l i p p i n g took p l a c e . Had the c y l i n d e r w a l l been s m o o t h e r , however , w i t h a •([><. = 3 5 ° ; the s l i p p i n g boundary would have been g i v e n as i n d i c a t e d i n F i g u r e 7 . 1 3 , by the <{><. = 35° l i n e s , below wh ich o n l y s l i p p i n g would have o c c u r r e d . As the b e d / w a l l f r i c t i o n a n g l e d e c r e a s e s ( i . e . smoother w a l l ) the a rea on the d iag ram domina ted by s l i p p i n g i n c r e a s e s . As shown i n the p r e v i o u s s e c t i o n the r o t a t i o n a l speed has a n e g l i g i b l e e f f e c t on the s l i p p i n g boundary . F u r t h e r m o r e , f o r the same the s l i p p i n g boundary i n the s l u m p i n g zone F r o u d e number ( F r = G J R / g ) 0.1 I 10 Rot a t i o n a l speed (r/min) Figure 7.13 Complete Bed-Behaviour Diagram of gravel in cy l inder A (0.4 m ID x 0.46 m L ) . Froude number (Fr =OJR/g) Figure 7.14 0.12 0.09 £• 0.06 "O CD 0.03 0 0.1 IXIO5 i x i o 4 IXIO3 i x i o 2 ixio' 1 1 Slumping ^ Cascading ^ Full cotaracting'T/ Cataraeting-^t Centrifuging —t-i i 1 j i t i 1 1 I 1 1 I Rolling 1 "~ " ~ i 1 V 1 1 I \ —4 \ | \ \ t \ \ I \ \ \ I \ \ \ Nickel oxide Experimental _ _ Slipping 1 ^ ; = 3 0 ° | I 10 Rot a t i o n a l speed (r/min) • Complete Bed-Behaviour Diagram of n ickel oxide in cy l inder A (0.4 m ID x 0.46 m L ) . 2 0 IO -100 244 o c c u r s a t a h i g h e r degree o f f i l l than t h a t i n the r o l l i n g or c a s c a d i n g z o n e s . T h i s i s due to the dynamic a n g l e o f r epose b e i n g l ower than the s t a t i c . Hence , the s l i p p i n g boundary i s a f f e c t e d by the b e d / w a l l f r i c t i o n a n g l e , <f> , and the degree o f f i l l , as e a r l i e r d i s c u s s e d . I t i s b e l i e v e d t h a t the (D/d ) s c a l e - u p r a t i o p roposed by Rose 54 and B l u n t i s i n c o r p o r a t e d i n the f^.measurement. The c r i t e r i a p roposed i n t h i s t h e s i s f o r s l i p p i n g a re i n agreement w i t h the e x p e r i m e n t a l and i n d u s t r i a l o b s e r v a t i o n s o f o t h e r w o r k e r s . The e x p e r i m e n t s o f Rose 54 and B l u n t on b a l l m i l l s showed t h a t the Froude number was not an i m p o r t a n t s c a l i n g c r i t e r i o n f o r s l i p p i n g . A l -though t h e i r r e s u l t s a re p r e s e n t e d on a p l o t s i m i l a r to the Bed B e h a v i o u r D i a g r a m , they canno t be v e r i f i e d u s i n g the c u r r e n t model as the <j>R and q> o f the m a t e r i a l s used were not r e p o r t e d . The n e g l i g i b l e e f f e c t o f r o t a t i o n a l speed 51 on s l i p p i n g was a l s o o b s e r v e d by K o r o t i c h u s i n g a c y l i n d e r , 0 .3 m D, r o t a t i n g between 10-65 r /m in (Froude numbers f rom 0.02 to 0 . 7 1 ) . The cha rge i n the c y l i n d e r a lways s l i p p e d ove r t h a t range o f r o t a t i o n a l speeds a t a l l degrees o f f i l l up to 20% when s l i p p i n g c e a s e d . The o b s e r v a t i o n s o f Gow e t 5 5 4 a l . and Duda on b a l l m i l l s a re a l s o s i m i l a r . In r o t a r y k i l n s , R o n c o ^ o b s e r v e d s l i p p i n g to cease w i t h i n c r e a s i n g 3 6 f i l l r a t i o . F i n a l l y the o b s e r v a t i o n s o f Lehmberg e t a l . 245 seem at f i r s t s i g h t to c o n t r a d i c t the o b s e r v a t i o n s o f a l l the above wo rke rs w i t h r e s p e c t to the e f f e c t o f r o t a t i o n a l speed on s l i p p i n g . On c l o s e r e x a m i n a t i o n t h e i r r e s u l t s c o u l d be e a s i l y e x p l a i n e d u s i n g the Bed B e h a v i o u r Diagram and the s l i p p i n g m o d e l . They had obse rved s l i p p i n g a t 2 r / m i n ; by i n c r e a s i n g the r o t a t i o n a l speed to 5 r / m i n , s l i p p i n g ceased and r o l l i n g s e t i n . Wi th a rougher c y l i n d e r w a l l or a h i g h e r s t a t i c a n g l e o f repose o f the c h a r g e , t h e s e wo rke rs may have o t h e r w i s e o b s e r v e d s l ump ing a t 2 r / m i n . By i n c r e a s i n g the r o t a t i o n a l speed to 5 r /m in the ( s i u m p i n g ) - r o l 1 i n g boundary was c r o s s e d and the s o l i d s were r o l l i n g , s i n c e the r o l l i n g - s l i p p i n g boundary p r o b a b l y o c c u r -red a t l owe r f i l l r a t i o s . T h e r e f o r e , the s l i p p i n g model p r e d i c t i o n s a re i n agreement w i t h the o b s e r v a t i o n s r e p o r t e d i n the l i t e r a t u r e . At the h i g h e r deg rees o f f i l l , i n F i g u r e 7 . 1 3 , r o l l i n g i s absen t and s l ump ing changes d i r e c t l y to c a s c a d i n g . T h i s f o l l o w s f rom the d e f i n i t i o n o f c a s c a d i n g g i v e n i n Chap te r 6 and r e s u l t s i n the o c c u r r e n c e o f a t r i p T e p o i n t between s l ump-i n g , r o l l i n g and c a s c a d i n g wh ich w i l l be a f u n c t i o n o f the dynamic ang le o f r e p o s e , c p ^ The lower cpp, the h i g h e r the degree o f f i l l r e q u i r e d f o r t h i s t r i p l e p o i n t wh ich i s i l -l u s t r a t e d f o r n i c k e l o x i d e (cp^ = 3 0 . 2 ° ) i n F i g u r e 7 . 1 4 . Compar ing the c a s c a d i n g - c a t a r a c t i n g b o u n d a r i e s o f both 246 F i g u r e s 7 .13 and 7 . 1 4 , i t i s e v i d e n t t h a t the lower cpD, the h i g h e r the Froude number r e q u i r e d f o r c a t a r a c t i n g a t equa l degrees o f f i l l . The f u l l c a t a r a c t i n g l i n e on both t h e s e d iagrams r e p r e s e n t s the r o t a t i o n a l speed f o r wh ich the p a r a b o l i c t r a j e c t o r y o f the ou te rmos t g r a n u l e on the apex i n t e r s e c t s the c y l i n d e r w a l l a t the c h o r d a l base o f the c h a r g e . As e x p e c t e d , i t behaves s i m i l a r l y to the c a s c a d i n g -c a t a r a c t i n g bounda ry . I t i s a l s o obse rved t h a t w i t h i n -c r e a s i n g bed depth the r o t a t i o n a l speed r e q u i r e d to get c a t a r a c t i n g d e c r e a s e s . T h i s i s due to the s o l i d s a t the apex b e i n g p r o j e c t e d i n t o the f r e e b o a r d a t l ower r o t a t i o n a l s p e e d s . Hence , c a t a r a c t i n g i s r e a l l y an e x t e n s i o n o f c a s c a d i n g . F i n a l l y , the e f f e c t o f c y l i n d e r d i a m e t e r i s shown i n F i g u r e s 7 .15 and 7 . 1 6 . The dynamic a n g l e o f repose o f l i m e -s tone C i n c y l i n d e r ' A i s 3 6 ° ; w h i l e t h a t - f o r l i m e s t o n e B i n c y l i n d e r C i s 3 6 . 5 ° . Both m a t e r i a l s n e a r l y have equa l (D/d ) r a t i o s as shown e a r l i e r i n t h i s c h a p t e r . Compar ing t h e i r comp le te B e d - B e h a v i o u r D i a g r a m s , i t i s o b s e r v e d t h a t the b o u n d a r i e s f o r s l u m p i n g , r o l l i n g , c a s c a d i n g , c a t a r a c t i n g and c e n t r i f u g i n g o c c u r a t equa l Froude numbers and degree o f f i l l s . The minor d e v i a t i o n s o b s e r v e d are due to the s m a l l d i f f e r e n c e s i n (D /dp) and c p ^ . Hence , the s c a l e - u p c r i t e r i a f o r c a s c a d i n g and c a t a r a c t i n g a re the dynamic Froude number (Fr=CUR/g) 10 Rotational speed (r/min) Figure 7.15 Complete Bed-Behaviour Diagram of limestone C in cy l inder A (0.4 m ID x 0.46 m L) . 2 Froude number ( F r - C U R / g ) IXIO 5 IXIO 4 IXIO3 IXIO 2 i x i o ' 0.1 I 10 Rotational speed (r/min) Figure 7.16 Complete Bed-Behaviour Diagram of limestone B in cy l inder C (1.06 m ID x 0.4 m L ) . 248 a n g l e o f r e p o s e , the Froude number and the degree o f f i l l r a t h e r than the a n g l e o f detachment wh ich has o f t e n been 56 79 s u g g e s t e d i n the l i t e r a t u r e . ' I t i s v e r y d i f f i c u l t to d i r e c t l y compare e x p e r i m e n t a l o b s e r v a t i o n s o f c a s c a d i n g and c a t a r a c t i n g r e p o r t e d i n the l i t e r a t u r e w i t h the model p r e d i c t i o n s because the m a j o r i t y o f the fo rmer were made a t degrees o f f i l l o f 30% and g r e a t e r . Hence , o n l y t r e n d s i n the o b s e r v a t i o n s c o u l d be compared. C a t a r a c t i n g has been o b s e r v e d to o c c u r a t Froude 2 6 1 1 5 4 numbers i n the range o f 0 .25 to 0 .72 ' ' wh ich i s i n agreement w i t h the p r e d i c t e d r e s u l t s . The speed o f the p a r t i c l e s f a l l i n g down the s l o p e o f the bed i s r e p o r t e d by ; 1 5 2 M u l l e r to be a f u n c t i o n o f the s l o p e o f the b e d , hence c a s c a d i n g i s a f u n c t i o n o f the dynamic a n g l e o f f r i c t i o n 72 as shown i n t h i s s t u d y . F i n a l l y , Oyama p r e s e n t s the f o l l o w i n g e m p i r i c a l r e l a t i o n s h i p f o r p r e d i c t i n g the < c a s c a d i n g - c a t a r a c t i n g and the f u l l c a t a r a c t i n g b o u n d a r i e s , n = K ( 7 . 8 ) D 0 . 4 7 -j0.14 where K i s a c o n s t a n t equa l to 54 f o r the c a s c a d i n g -c a t a r a c t i n g boundary and 72 f o r the f u l l c a t a r a c t i n g c o n -d i t i o n . A l t h o u g h Oyama has not a c c o u n t e d f o r the dynamic a n g l e o f r e p o s e , the p r e d i c t e d b o u n d a r i e s u s i n g E q u a t i o n ( 7 . 8 ) agree w i t h i n + 15% of the r o t a t i o n a l speeds p r e d i c t e d 249 w i t h the models p r e s e n t e d i n Chap te r 6. The c a s c a d i n g and c a t a r a c t i n g p r e d i c t i o n s a re t h e r e f o r e r e l i a b l e . 7 .5 Other A s p e c t s 7 .5 .1 I n t e r n a l F r i c t i o n J e n i c k e and Johanson have d e v e l o p e d a s h e a r c e l l f o r measu r i ng the i n t e r n a l f r i c t i o n o f c o h e s i v e powders which has not been w i d e l y a p p l i e d to c o h e s i o n l e s s bu l k s o l i d s . S i n c e t h e i r t e s t i n g method and da ta a n a l y s i s i s 155-158 f u l l y d e s c r i b e d and d i s c u s s e d i n the l i t e r a t u r e , o n l y i t s r e l e v a n c e to the B e d - B e h a v i o u r Diagram w i l l be d i s c u s s e d . In a r o t a r y c y l i n d e r , when the l i m i t i n g f r i c t i o n f o r c e i s r eached the bu l k s o l i d s f a i l a t the shea r p l a n e , wh ich r e p r e s e n t s the maximum s h e a r s t r e s s . I t s magn i tude may be e s t i m a t e d f rom the component o f the g r a v i t a t i o n a l f o r c e on the s h e a r wedge p a r a l l e l to the s h e a r p l a n e . The normal s t r e s s a s s o c i a t e d w i t h i t would be g i v e n by the s t r e s s n o r -mal to the s h e a r p l a n e i n the s o l i d s b e d . M o h r ' s c i r c l e may t h e r e f o r e be drawn f o r each t ype o f bu lk s o l i d i n each c y l i n d e r t e s t e d . S i n c e c o h e s i o n l e s s bu l k s o l i d s have t h e i r y i e l d l o c u s p a s s i n g th rough the o r i g i n o f the s t r e s s a x e s , the i n t e r n a l a n g l e o f f r i c t i o n may thus be i d e n t i f i e d . Such p l o t s have been d e r i v e d by J e n i c k e and Johanson and a re c a l l e d Y i e l d L o c i Diagrams and as d e s c r i b e d above may r e l a t e 250 the i n t e r n a l a n g l e o f f r i c t i o n o f a bu l k s o l i d to the s h e a r a n g l e and the minimum s h e a r wedge s i z e . S i n c e the Y i e l d L o c i i Diagrams have been a p p l i e d to p r e d i c t the c o n d i t i o n s of f l o w and no f l o w i n b i n s and hoppers and s i n c e i t may a l s o be used to d e s c r i b e a s l u m p , i t s r e l a t i o n s h i p to the B e d - B e h a v i o u r D iagram s h o u l d be t h o r o u g h l y i n v e s t i g a t e d i n f u t u r e r e s e a r c h p rog rams . 7 . 5 . 2 The Phase Ru le From the d i s c u s s i o n i n the above s e c t i o n , the v a r i a b l e s a f f e c t i n g the s i u m p i n g - r o l 1 i n g boundary may be reduced to f o u r : the degree o f f i l l , the i n t e r n a l a n g l e o f f r i c t i o n , the g r a v i t y and the c e n t r i f u g a l f o r c e s . For the r e m a i n i n g b o u n d a r i e s on a Bed B e h a v i o u r Diagram the i n t e r n a l a n g l e o f f r i c t i o n may b e r e p l a c e d by the dynamic a n g l e o f repose or the b e d - w a l l f r i c t i o n a n g l e . Hence , f o r each e q u i l i b r i u m s t a t e o f s o l i d s m o t i o n , s l i p p i n g , s l u m p i n g e t c . , f o u r i ndependen t pa rame te rs d e f i n e a s y s t e m : a m a t e r i a l v a r i a b l e , the f r i c t i o n a l b e h a v i o u r o f the s o l i d s ; a sys tem v a r i a b l e , the degree o f f i l l ; and two i ndependen t e x t e r n a l f o r c e s a c t i n g on the s y s t e m , the g r a v i t y and the c e n t r i -f u g a l f o r c e s . The l a t t e r two may be r e g a r d e d as f o r c e componen ts , C , wh ich a re r e q u i r e d t o d e s c r i b e an e q u i l i b r i u m s t a t e o f m o t i o n , P. I f the o n l y i ndependen t pa ramete r h e l d c o n s t a n t i s the g r a v i t y f o r c e , the sys tem w i l l have t h r e e 251 degrees of f r eedom. T h i s c o r r e s p o n d s to o n l y one e q u i l i -b r i um s t a t e o f s o l i d s m o t i o n , as seen on a B e d - B e h a v i o u r D i a g r a m , f o r example s l u m p i n g , i . e . P = 1. I f two i n -dependent pa rame te rs a re h e l d c o n s t a n t , f o r example the g r a v i t y f o r c e and the m a t e r i a l , then o n l y two deg rees o f f reedom e x i s t , F = 2 , the c e n t r i f u g a l f o r c e and the degree of f i l l . T h i s c o n d i t i o n c o r r e s p o n d s to a. boundary between two modes o f bed b e h a v i o u r , P = 2 . For the case when t h r e e i ndependen t pa rame te rs a re h e l d c o n s t a n t , o n l y one degree of f reedom e x i s t s , F = 1. I t f o l l o w s t h a t the e q u i l i b r i u m s t a t e s o f mot ion w i l l be t h r e e , P = 3 , a t r i p l e p o i n t . These e q u i l i b r i u m s t a t e s o f mot ion may be r e g a r d e d as phases in a c o n t e x t s i m i l a r to t h a t used to d e s c r i b e S o l i d - F l u i d 1 4 9 - 1 5 1 1 5 9 Phase D iag rams . ' The r e l a t i o n s h i p o f the p a r a -meters d e s c r i b e d above c o u l d then be termed a phase r u l e and may be e x p r e s s e d as f o l l o w s , P + F = C + 2 ( 7 . 9 ) where the 2 r e p r e s e n t s the degree o f f i l l and the f r i c t i o n a l p r o p e r t i e s o f a bu l k s o l i d . T h i s same approach when a p p l i e d to the phase d iag rams 149-151 157 o f f l u i d - s o l i d sys tems ' would r e s u l t . i n - ' t he s u p e r -f i c i a l v e l o c i t y , p l o t t e d on the a b s c i s s a , to be i n d i c a t i v e o f the r a t i o o f the buoyancy f o r c e to the g r a v i t y f o r c e 252 by means o f a m o d i f i e d Froude number and i s t h e r e f o r e a b a s i c a l l y s i m i l a r r e p r e s e n t a t i o n to the B e d - B e h a v i o u r D iag ram. I t f o l l o w s , t h e r e f o r e , t h a t when gases a re e v o l v e d f rom r o t a r y k i l n beds due to the o c c u r r e n c e o f c h e m i c a l r e a c t i o n s or d r y i n g t h e r e would be t h r e e components i n the s y s t e m , the g r a v i t y f o r c e , the c e n t r i f u g a l f o r c e and the buoyancy f o r c e . Hence , a t e r n a r y B e d - B e h a v i o u r Diagram o r a m o d i f i e d b i n a r y d iag ram c o u l d be d e v e l o p e d . S i n c e the B e d - B e h a v i o u r D i a g r a m , the S o l i d s - F l u i d s Phase Diagram and the Y i e l d L o c i i Diagram each r e p r e s e n t an e q u i l i b r i u m s t a t e o f the bu l k s o l i d s and s i n c e i t appears t h a t they may be r e l a t e d as d e s c r i b e d a b o v e , i t i s p roposed t h a t f u t u r e work be d i r e c t e d to s tudy t h e i r i n t e r r e l a t i o n -s h i p s and to i n v e s t i g a t e w h e t h e r , ,,as has been s u g g e s t : ' t e d / ^ ' ^ ^ ' ^ ^ a thermodynamic b a s i s does e x i s t f o r bu l k s o l i d s m o t i o n . I t was not f e a s i b l e i n t h i s s t udy to d e v e l o p t hese a s p e c t s o f the B e d - B e h a v i o u r Diagram and i t i s recom-mended t h a t they be pu rsued i n f u t u r e s t u d i e s . C h a p t e r 8 253 SUMMARY AND CONCLUSIONS 8.1 Summary and C o n c l u s i o n s The v a r i o u s modes o f bed b e h a v i o u r e n c o u n t e r e d i n r o t a r y c y l i n d e r s have been q u a l i t a t i v e l y d e s c r i b e d based on the o b s e r v a t i o n s r e p o r t e d i n the l i t e r a t u r e and i n t h i s s t u d y . A B e d - B e h a v i o u r D i a g r a m , wh ich i s a p l o t o f bed depth v e r s u s r o t a t i o n a l s p e e d , was p roposed to i l l u s t r a t e the r e g i o n s o f dominance o f each o f t hese modes. An i n depth e x p e r i m e n t a l a n a T y s i s o f the s i u m p i n g - r o l 1 i n g boundary was c a r r i e d out to i l l u s t r a t e the e f f e c t s o f m a t e r i a l , c y l i n d e r and o p e r a t i n g v a r i a b l e s . I t was shown t h a t a d e c r e a s e i n p a r t i c l e s i z e and a change i n p a r t i c l e shape f rom a n g u l a r to s p h e r i c a l r e s u l t e d i n the boundary o c c u r -r i n g at l owe r r o t a t i o n a l s p e e d s ; i n c r e a s i n g the d i a m e t e r o f the c y l i n d e r a l s o d i s p l a y e d the same e f f e c t . S lump ing beds were e x p e r i m e n t a l l y c h a r a c t e r i z e d u s i n g the s t a t i c and the upper a n g l e s o f r e p o s e , the s h e a r a n g l e , the s l u m p i n g f r e q u e n c y ; w h i l e the d y n a m i c • a n g l e o f repose and the a c t i v e l a y e r t h i c k n e s s were used to c h a r a c t e r i z e r o l l i n g . Wi th the a i d o f the B e d - B e h a v i o u r D i a g r a m , the e f f e c t o f the a d d i t i o n o f f i n e p a r t i c l e s on the f l o w c h a r a c t e r i s t i c s 254 and s e g r e g a t i o n p a t t e r n s o f b u l k s o l i d s was i n v e s t i g a t e d . When compared to the c o a r s e r m a t e r i a l s , the p r e s e n c e o f f i n e s d i d not a f f e c t the a n g l e s of r e p o s e , the s l u m p i n g f r e q u e n c i e s , the s h e a r a n g l e s or the a c t i v e l a y e r t h i c k n e s s e s o f the t h r e e m i x t u r e s t e s t e d . S lump ing was o b s e r v e d to be enhanced f o r the l i m e s t o n e m i x t u r e and r o l l i n g f o r the two sand m i x t u r e s . The mechanism o f t r a n s v e r s e s e g r e g a t i o n o f f i n e s o l i d s i n r o t a r y c y l i n d e r s and the c o m p o s i t i o n and l o c a t i o n o f the c e n t r a l s e g r e g a t e d c o r e , the ' k i d n e y ' , were a l s o i d e n t i f i e d . I t was found t h a t the f i n e s s e g r e g a t e d i n the cha rge by the p e r c o l a t i o n mechanism and o c c u p i e d a c o r e o f the same shape as the b e d . I t s c o m p o s i t i o n c o r -responded to the maximum bed c o n t r a c t i o n , A e m a x > o f the components m i x e d . A second s e g r e g a t i o n co re was a l s o i d e n t i -f i e d to be i n the top p o r t i o n o f the bed near the w a l l . I t o c c u r r e d f o r s l ump ing and r o l l i n g beds and i s b e l i e v e d to be r e s p o n s i b l e f o r s l i p p i n g i n i n d u s t r i a l r o t a r y k i l n s . L o n g i -t u d i n a l s e g r e g a t i o n was a l s o o b s e r v e d to take p l a c e and was found to be dependent on bed b e h a v i o u r . A s e m i - e m p i r i c a l model was d e v e l o p e d to p r e d i c t the s i u m p i n g - r o l l i n g boundary o f bu l k s o l i d s . The model was s u c c e s s f u l l y a p p l i e d to p r e d i c t the e f f e c t s o f bed d e p t h , p a r t i c l e s i z e , p a r t i c l e s h a p e , combined e f f e c t s o f s i z e and shape and of c y l i n d e r d i a m e t e r . Based on the 255 exper imenta l and t h e o r e t i c a l a n a l y s e s , s c a l e - u p c r i t e r i a have been proposed f o r m a t e r i a l , c y l i n d e r and o p e r a t i n g v a r i a b l e s . These were the Froude number, the degree of f i l l and the minimum shear wedge or the ( D / d p ) r a t i o f o r s o l i d s of equal shape. S l i p p i n g , c a s c a d i n g and c a t a r a c t i n g have a l s o been m a t h e m a t i c a l l y model led in t h i s study and t h e i r p r e d i c t i o n s were in agreement with exper imenta l o b s e r v a t i o n s r e p o r t e d in the l i t e r a t u r e . A comprehensive a n a l y s i s of the s c a l e - u p c r i t e r i a , has been c a r r i e d out f o r each of the bed behav iours and has been r e l a t e d to the Bed -Behav iour Diagram. These a r e , f o r s l i p p i n g , the bed /wa l l s l i p angle and the degree of f i l l , and f o r c a s c a d i n g and c a t a r a c t i n g , the Froude number, the degree of f i l l and the dynamic angle of r e p o s e . 8.2 Recommendations f o r Future Work It i s recommended that f u t u r e s t u d i e s be c a r r i e d out in the f o l l o w i n g a r e a s : 1. A more d e t a i l e d v e r i f i c a t i o n of the width of the c e n t r a l s e g r e g a t i o n core model should be undertaken f o r v a r y i n g coarse to f i n e s i z e r a t i o s , v a r i o u s bed d e p t h s , p a r t i c l e s h a p e s , d e n s i t i e s , and c y l i n d e r d i a m e t e r s . The model shou ld a l s o be r e f i n e d to 256 p r e d i c t the d i s t a n c e the s u r f a c e o f the c o r e s i s r e c e s s e d f rom the bed s u r f a c e . 2 . The cause o f the s h i f t i n the s i u m p i n g - r o l 1 i n g boundary w i t h f i n e s a d d i t i o n s h o u l d be i n v e s t i g a t e d . 3 . The s i z e o f p a r t i c l e s a s s o c i a t e d w i t h the second s e g r e g a t i o n zone s h o u l d be de te rm ined and a s i z e c r i t e r i o n d e v e l o p e d . 4 . The e f f e c t s o f s e g r e g a t i o n on i n c l i n e d c o n t i n u o u s k i l n o p e r a t i o n s h o u l d a l s o be i n v e s t i g a t e d . 5. U s i n g e q u a l l y s i z e d s o l i d s o f i d e a l s h a p e s , the a p p l i c a b i l i t y o f the "Heywood r a t i o s " o r the : " s p h e r i c i t y " to q u a n t i f y p a r t i c l e shape and i t s e f f e c t on bed b e h a v i o u r c o u l d be e s t a b l i s h e d . 6. The c h a r a c t e r i z a t i o n o f s l ump ing u s i n g the s l ump ing f r e q u e n c y and o f r o l l i n g u s i n g the a c t i v e l a y e r t h i c k n e s s a l l o w s a new approach to m o d e l l i n g the r e s i d e n c e t ime of s o l i d s i n r o t a r y k i l n s . P a r t i -c u l a r l y o f i n t e r e s t wou ld be t h o s e c a s e s where two o f t h r e e t y p e s o f bed b e h a v i o u r o c c u r i n a g i v e n k i l n . 7. T h i s : . c h a r a c t e r i z a t i o n ' o f the s o l i d s mot ion may a l so be a p p l i e d to f o r m u l a t e a more a c c u r a t e d e s c r i p t i o n o f the g a s - s o l i d s heat f l o w s tep i n a k i l n . A n o t h e r a rea recommended f o r f u t u r e work i s the q u a n t i t a t i v e s tudy o f the e f f e c t o f gas e v o l u t i o n f rom the charge on i t s m o t i o n . F i n a l l y , r e s e a r c h i n b u l k s o l i d s f l o w p r o p e r t i e s as d e s c r i b e d by F l u i d - S o l i d s Phase D i a g r a m s , B e d -B e h a v i o u r Diagrams and Y i e l d L o c i i D iagrams w i l l p rove most u s e f u l i n h e l p i n g to i n c r e a s e our f u n d a -menta l u n d e r s t a n d i n g o f the f l o w o f bu l k s o l i d s and o f t h e i r m i x i n g c h a r a c t e r i s t i c s . 258 BIBLIOGRAPHY 1 Z a b l o t n y , W.W., "The Movement of the Charge in R o t a r y K i l n s " , I n t e r n a t i o n a l Chemica l E n g i n e e r i n g , v o l . 5 , no . 2 , 1965 , pp . " .360-366. 2 R u t g e r s , R. , "Longitudinal M i x i n g o f Granu l ia r : M a t e r i a l F l o w i n g th rough a R o t a t i n g C y l i n d e r ; P a r t I: D e s c r i p t i v e and T h e o r e t i c a l " , Chemica l E n g i n e e r i n g S c i e n c e , v o l . 20 , 1965 , pp . 1079 -1087 . 3 H e j j a , A . A . , R o t a r y K i l n s in the M e t a l l u r g i c a l  Indus t r y , Resea rch Repo r t no. 1, Department o f M e t a l l u r g y , U n i v e r s i t y o f the W i t w a t e r s r a n d , J o h a n n e s b u r g , 1976. 4 Duda, W . H . , Cement Data Book , I n t e r n a t i o n a l P r o c e s s E n g i n e e r i n g i n the Cement I n d u s t r y , 2nd e d . , London , Macdona ld and E v a n s , 1977. 5 W a t k i n s o n , A . P . and Br imacombe, J . K . , "Heat T r a n s f e r i n a D i r e c t - F i r e d R o t a r y K i l n : I T H e a t Flow R e s u l t s and T h e i r I n t e r p r e t a t i o n " , M e t a l l u r g i c a l T r a n s a c t i o n s B, v o l . 9 B , June 1978 , pp . 2 0 9 - 2 1 9 . 6 T h e m e l i s , N . J . , D o n a l d s o n , J .W. and Udy, M . C . , "Use o f the S i m i l a r i t y P r i n c i p l e i n P r e d i c t i n g the Optimum Pe r fo rmance o f I ron R e d u c t i o n K i l n s " , C o n f e r e n c e o f M e t a l l u r g i s t s T r a n s a c t i o n s , v o l . 6 7 , 1964 , pp. 7 4 - 8 2 . 7 C r o s s , M. and Young , R .W. , " M a t h e m a t i c a l Model o f R o t a r y K i l n s used i n the P r o d u c t i o n o f I ron Ore P e l l e t s " . I ronmak ing and S t e e l ma k i n g , no . 3 , 1976 , pp. 1 2 9 - 1 3 7 . 8 M a t t h e e , H . , " S e g r e g a t i o n Phenomena R e l a t i n g to B u n k e r i n g o f B u l k M a t e r i a l s : T h e o r e t i c a l C o n s i d e r a -t i o n s and E x p e r i m e n t a l I n v e s t i g a t i o n s " , Powder T e c h n o l o g y , v o l . 1 , 1 9 6 7 / 6 8 , pp. 2 6 5 - 2 7 1 . 9- S i n g e r , R. and A h i e r , S . , "Cheap S t e e l f o r Hous ing the W o r l d " , New S c i e n t i s t , v o l . 2 3 , 1977 , pp. 7 2 1 - 7 2 3 . 10 W i l s o n , K . , "The SL/RN P r o c e s s at the G r i f f i t h M i n e " , Canad ian M e t a l l u r g i c a l Q u a r t e r l y , v o l . 1 8 , 1979 , pp. 1 0 5 - 1 0 9 . 259 11 B o l d , D.A. and E v a n s , N . T . , " D i r e c t R e d u c t i o n Down Under : the New Z e a l a n d S t o r y " , I ron and S t e e l I n t e r n a t i o n a l , June 1977 , pp. 1 4 5 - 1 5 2 . 12 Kuoppamak i , R . , K u u s i , J . and B l o m q v i s t , S . , "Use o f A c t i v a t e d P r o c e s s M a t e r i a l s as T r a c e r s in S m e l t e r P r o c e s s S t u d i e s " , P r o c e e d i n g s of a Symposium on N u c l e a r Techn iques in the B a s i c Me ta l I n d u s t r i e s , H e l s i n k i , J u l y - Augus t 1972 , 1973 , pp . 2 2 7 - 2 3 6 . 13 Akerman, K. , Ho f fman , P . , and Z a b l o t n y , W. , "Mechan ism Rate of Passage .o f M a t e r i a l s i n R o t a r y K i l n s " , B r i t i s h Chemica l E n g i n e e r i n g , v o l . 1 1 , no . 1 , 1966 , pp. 2 6 - 2 9 . 14 C r o s s , M . , P e l l e t Movement th rough a R o t a r y K i l n , Repo r t no . 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Appendix A Results of Screen Analyses 275 TABLE A.1 S c r e e n A n a l y s i s R e s u l t s f o r G r a v e l S i e v e D e s i g n a t i o n s ( U . S . S i e v e S e r i e s ) F r a t i o n a l Weight ( P e r c e n t ) > + 5 0 . 5 - 5 +6 32 .8 -6 +7 28 .6 -7 +8 25 .8 -8 +10 9 .5 -10 +12 1 .2 -12 1 . 5 T o t a l . 9 9 . 9 Average P a r t i c l e S i z e (mm) 3.0 276 TABLE A . 2 Sc reen A n a l y s i s R e s u l t s f o r I r on Ox ide S i e v e D e s i g n a t i o n s ( U . S . S i e v e S e r i e s ) F r a c t i o n a l Weight ( P e r c e n t ) +0.530 i n 12.1 - 0 . 5 3 0 i n + 1/2 i n 9 .2 - 1/2 i n + 7 /16 i n 43.1 - 7 /16 i n + 3 .8 i n 32 .0 - 3 /8 i n + 5 /16 i n 2 .5 - 5 /16 i n +0.265 i n 0 .2 - 0 . 2 6 5 i n 0 .9 T o t a l 100 .0 Average P a r t i c l e S i z e (mm) 11 .6 TABLE A . 3 Sc reen A n a l y s i s R e s u l t s f o r L imes tone B ! S i e v e D e s i g n a t i o n s ( U . S . S i e v e S e r i e s ) F r a c t i o n a l Weight ( P e r c e n t ) + 1/4 i n 0 .05 - 1/4 i n + 3 1/2 4 . 5 - 3 1/2 + 4 28 .2 - 4 + 5 26 .9 - 5 + 6 2 5 . 3 - 6 + 7 12 .6 - 7 + 8 1 .6 - 8 + 10 10 0 .05 0 .9 T o t a l 100.1 Average P a r t i c l e S i z e (mm) 4 . 3 TABLE A . 4 Sc reen A n a l y s i s R e s u l t s f o r L imes tone C S i eve ( U . S . Des i g n a t i ons S i e v e S e r i e s ) F r a c t i o n a l Weight ( P e r c e n t ) + 7 0 .2 - 7 + 8 2.6 - 8 + 10 8 .8 - 10 + 12 17 .4 - 12 + 14 22 .2 - 14 + 16 21 .2 - 16 + 18 15 .0 - 18 + 20 . 7 . 7 - 20 + 25 2 .5 - 25 + 30 1.1 - 30 + 35 0 .7 - 35 + 40 0 .3 - 40 0 .5 T o t a l 100 .2 Average P a r t i c l e S i z e (mm) 1 .5 TABLE A . 5 Sc reen A n a l y s i s R e s u l t s f o r L imes tone D S i eve ( U . S . D e s i g n a t i o n s S i e v e S e r i e s ) F r a c t i ona l Weight ( P e r c e n t ) + 18 3 .6 - 18 + 20 9.2 - 20 + 25 12 .2 - 25 + 30 13 .3 - 30 + 35 2 0 . 3 - 35 + 40 18.1 - 40 + 45 16 .9 - 45 + 50 5.7 - 50 0 .8 T o t a l 100.1 Average P a r t i c l e S i z e (mm) 0 .58 TABLE A . 6 Sc reen A n a l y s i s R e s u l t s f o r L imes tone E S i eve ( U . S . Des i g n a t i ons S i e v e S e r i e s ) F r a c t i o n a l Weight ( P e r c e n t ) + 18 2 .0 - 18 + 20 1 1 . 4 - 20 + 25 13 .7 - 25 + 30 12 .0 - 30 + 35 14 .7 - 35 + 40 11 .5 - 40 + 45 8.7 - 45 + 50 6 .9 - 50 + 60 4 .6 - 60 + 70 1 .2 - 70 13 .6 T o t a l 100 .3 Average P a r t i c l e S i z e (mm) 0. 54 TABLE A . 7 Sc reen A n a l y s i s R e s u l t s f o r L imes tone F S i e v e D e s i g n a t i o n s ( U . S . S i e v e S e r i e s ) F r a c t i o n a l Weight ( P e r c e n t ) + 7 /16 i n 9 .8 - 7 /16 i n + 3 / 8 i n 16 .8 - 3 /8 i n + 5/16 i n 19.1 - 5 /16 i n +0.265 i n 25.1 - 0 . 2 6 5 i n + 1/4 i n 7.9 - 1/4 i n + 3 1/2 i n 11 .6 - 3 1/2 9.9 T o t a l 100 .2 Average P a r t i c l e S i z e (mm) 8.1 232 TABLE A . 8 Sc reen A n a l y s i s R e s u l t s f o r N i c k e l Ox ide S i e v e D e s i g n a t i o n s ( U . S . S i e v e S e r i e s ) F r a c t i o n a l Weight ( P e r c e n t ) + 1/4 i n 0 .4 - 1/4 i n + 3 1/2 7.1 - 3 1/2 + 4 46 .4 - 4 + 5 39 .9 - 5 + 6 6.1 - 6 0 .2 T o t a l 100.1 Average P a r t i c l e S i z e (mm) 4 .9 283 TABLE A . 9 Sc reen A n a l y s i s R e s u l t s f o r Sand B S i e v e ( U . S . Des i g n a t i ons S i e v e S e r i e s ) F r a c t i o n a l Weight ( P e r c e n t ) + 20 0 .4 - 20 ••+ 25 3.1 - 25 + 30 11 .2 - 30 + 35 35.4 - 35 + 40 28 . 2 - 40 + 45 12 .0 - 45 + 50 5.1 - 50 + 60 2 .6 - 60 + 70 0 .8 - 70 1 .1 T o t a l 99 .9 Average P a r t i c l e S i z e (mm) 0.50 284 TABLE A . 1 0 Sc reen A n a l y s i s R e s u l t s f o r Sand C S i e v e ( U . S . D e s i g n a t i o n s S i e v e S e r i e s ) F r a c t i o n a l Weight ( P e r c e n t ) + 30 - 30 + 35 4 .6 - 35 + 40 6.9 - 40 + 45 4 . 5 - 45 + 50 4 . 5 - 50 + 60 7.6 - 60 + 70 8 .6 - 70 + 80 10 .4 - 80 + 100 14 .2 - 100 + 1 20 1 7 . 5 - 120 + 140 10 .9 - 140 + - 170 170 5 .3 3.1 T o t a l 9 9 . 8 Average P a r t i c l e S i z e (mm) 0 . 23 APPENDIX B Iden t i f i ca t ion of the Run Numbers Corresponding to the Bed Behaviour Observati ons. 286 TABLE B . l I d e n t i f i c a t i o n o f the Run Numbers C o r r e s p o n d i n g to the Bed B e h a v i o u r O b s e r v a t i ons F i gure M a t e r i a l C y l i nder Bed Depth Run (mm) 4 . 5 Sand A A 32 . 56 20 . 57 4 2 . 58 52 . 59 60 . 60 71 . 61 8 2 . 62 94 . 63 4 . 8 L imes tone B A 9 2 . 5 65 8 2 . 5 66 ,66A 72 . 5 67 6 2 . 5 68 52 . 5 69 42 . 5 70 4 .9 N i c k e l Ox ide A 9 2 . 5 78 8 2 . 5 79 72 . 5 80 62 . 5 81 52 .5 82 4 2 . 5 83 4 . 1 0 L imes tone B A 8 7 : 5 117 35 . 118 46 . 119 66 . 120 4.11 N i c k e l Ox ide A 8 8 . 183 77. 184 68. 185 56 . 186 4 5 . 187 ,188 4 .12 L imes tone B B 4 3 . 175 70 . 179 8 5 . 180 55 . 181 30. 182 287 TABLE B . l I d e n t i f i c a t i o n o f the Run Numbers ( C o n t ' d ) C o r r e s p o n d i n g to the Bed B e h a v i o u r O b s e r v a t i o n s F i gure M a t e r i a l C y l i n d e r Bed Depth Run (mm) 4 . 1 3 L imes tone D A 87 . 5 199 71 . 200 56. 201 4 2 . 202 31 . 203 4 .14 Sand B A 9 2 . 194 73 . 195 60 . 1 96 4 5 . 197 27 . 198 4 . 1 5 L imes tone C A 9 4 . 221 81 . 222 68 . 223 52 . 224 39 . 225 29 . 226 4 .16 G r a v e l A 89 . 204 6 5 . 205 54. 206 42 . 207 31 . 208 72 . 209 84. 210 97 . 211 4 .17 L imes tone B C 78 . 216 106. 21 7 132 . 218 165 . 219 288 TABLE B.2 I d e n t i f i c a t i o n o f the Run Numbers C o r r e s p o n d i n g to the Bed B e h a v i o u r O b s e r v a t i o n s  ( S e g r e g a t i o n T e s t s ) F i gure M a t e r i a l C y l i n d e r Bed Depth Run (mm) 5 .2 Sand Mix A A 9 2 . 5 1 21 72 . 1 22 54. 123 34. 124 5 .3 Sand Mix B A 99 . 125 62 . 127 38. 128 77 .5 129 52 . 1 30 5.4 •Limestone Mix A A 9 3 . 1 32 71 . 5 133 53 . 1 34 38. 5 1 35 52 . 145 Append ix C E x p e r i m e n t a l and R e g r e s s i o n R e s u l t s  o f the Times to Maximum and Minimum Bed I n c l i n a t i o n s . 290 10 £ i-8h-6 r - O 0 0 G ravel t, B e d d e p t h % Fi 11 (m) O © 0 - 0 4 5 1.5 R e g r e s s i o n c u r v e s \ o \ a , :0~ F i g u r e C l The t , and t ? f o r g r a v e l i n 0.5 I 1.5 R o t a t i o n a l speed ( r / m i n ) measured r e s u l t s as w e l l as the r e g r e s s i o n c u r v e s c y l i n d e r C (1.06 m ID x 0.4 m L) 7 £ 3 O Iron oxide tl t2 Bed depth % Fi 11 O © 0.100 19 A A 0.084 1 5 • • 0.070 1 2 Regression curve 0.2 R o t a t i o n a l 0.4 speed ( r /m in ) 0.6 F i g u r e C.2 The t , and C measured r e s u l t s as w e l l as the r e g r e s s i o n c u r v e s f o r i r o n o x i a e i n c y l i n d e r A (0.4 m ID x 0.46 m L ) . 291 Limestone B <| *2 Bed depth • / . Fill (m) O • 0.04 3 5.7 Regression curves 0.5 I 1.5 Rotational speed (r/min) Figure C.3 The t-j and tp measured resu l ts as wel l as the regression curves for limestone B in cy l inder B (0.4 m ID x 0.86 m L), 63 .1 20 16 £ 1 2 E O S °fa \ Limestone B <l Bed depth % F i l l (m ) O • Q 0 7 8 3.3 A • 0.1 0 6 5.2 a • 0.1 3 2 7.1 o • 0.165 9.9 Regression curves 0.2 0.4 0.6 0.8 1.0 Rotational speed (r/min) 1.2 Figure C.4 The t] and t~ measured resu l ts as well as the regression curves for limestone B in cy l inder C (1.06 m ID x 0.4 m L) 16 12 4D V ft 292 Limestone C »l Bed depth % Fill (m) O • 0.094 18 A A 0.08 1 14 D • 0.06 8 1 1 o • 0.05 2 7.5 o • 0.0 39 4.9 + X 0.0 2 9 3.2 Regression curves 0.5 I 1.5 2 Rotational speed (r/min) Figure C.5 The t, and t 2 measured resu l ts as well as the regression curves for lfmestone C in cy l inder A (0.4 m ID x 0.46 m L) E Limestone C '2 Bed depth (m) % F i l l o • 0.030 0.8 degression curves 0.2 0.4 0.6 0.8 Rotational speed (r/min) 1.0 1.2 Figure C.6 The t-\ and to_ measured resu l ts as well as the regression curves for limestone C in cy l inder C (1.06 mID x0.4mL). 2 9 3 Limestone D Bed depth % F i l l 1 (tn) o o 0.088 16 A • 0.0 71 12 • D 0.056 8.3 O • 0.0 42 5.5 o 0.0 31 1 3"5 Regression curves 0.2 0.4 0.6 0.8 R o t a t i o n a l s p e e d ( r / m i n ) 1.0 1.2 FiaureC 7 The t i and t , measured resu l ts as wel l as the regression ¥-^^L Curves for limestone D in cy l inder A (0.4 m ID x 0.46 m L ) 294 3 0 25 20 I 5 § i -10 L i m e s t o n e F u *2 Bed depth % F i l l (m) o O 0.050 1.7. A • 0.075 3.1 - Regress ion cu rves 0 0.2 0.4 Rotational speed ( r /min) 0.6 Figure C.8 The t-| and t ? measured resu l ts as wel l as the regression curves for limestone F in cy l i nce r C (1 .06 m ID x 0.4 m L) 295 • 20 — N i c k e l ox ide *l B e d depth % Fill ( m ) o © 0.045 6 A A 0.0 6 8 1 1 • B 0.0 8 8 1 6 — R e g r e s s i o n c u r v e s 1 6 -A 1 2 -to Rotational speed (r/min) Figure C.9 The t-j and t? measured resu l ts as well as the regression curves for n icke l oxide in cy l inder A ( 0 . 4 m ID x 0 .46 m L ) . 296 5 K to E 3 h-0 2 h -o S a n d B f| f2 B e d d e p t h ( m ) % F i l l O A • O o © A • 0.092 0.073 0.0 60 0.04 5 0.027 17 1 2 9.2 6 2.8 R e g r e s s i o n c u r v e s 0 0.1 0.2 0.3 0.4 Rotat iona l speed ( r /m in ) F i g u r e C . 10 The t-j and t ? measu red r e s u l t s as w e l l as t he r e g r e s s i o n c u r v e s f o r sand B i n c y l i n d e r A ( 0 . 4 m ID x 0 . 4 6 m L ) . 297 Appendix D  The Co-Ordinates of the Wedge Centroids 2 9 8 The c a l c u l a t i o n o f the c e n t r o i d s o f the upper and l owe r wedges o f a s l ump ing bed w i l l be p r e s e n t e d i n t h i s a p p e n d i x . These wedges are bounded by the i n i t i a l (<j>^ ) and f i n a l (<p^ ) s u r f a c e i n c l i n a t i o n s f o r a s l ump ing bed o f u n i t c y l i n d e r l e n g t h ( F i g u r e D . l ) . A p p l y i n g the method o f compos i t e shapes to the upper wedge ABC y i e l d s the f o l l o w i n g e q u a t i o n s f o r the c o - o r d i n a t e s o f the c e n t r o i d ABC, XABC = X0ABA0AB + x0DBA0DB - X0EAA0EA - XQDEAQDE - XDECADEC (D. l ) AABC a n d , yABC = ^0ABA0AB + ^ 0DBA0DB - % A A 0 E A - ^0DEA0DE - ^DECADEC (D.2) A ABC whe re , AABC = A0AB + A0DB - A0EA - A0DE - ADEC (D.3) The c e n t r o i d o f wedge PCS w i l l be c a l c u l a t e d u s i n g the g e o m e t r i c a l symmetry o f the s y s t e m . B e f o r e c a l c u l a t i n g the terms i n e q u a t i o n s ( D . l ) to (D .3 ) the geometry o f the sys tem must f i r s t be d e f i n e d . Assume t h a t the apexes o f the bed b e f o r e and a f t e r s l ump ing l i e i n the f i r s t quadran t ( F i g u r e D . l ) . T h e r e f o r e , 299 Figure D.l Geometrical construct ion of the bed for the apex in the f i r s t quadrant for both the cb,, and bed i n c l i n a t i o n s . 300 / ATU = / OGC = <J,U L BQR = L GHC = c?L i ACB = Y = *y - * L (D.4) C o n s t r u c t , OE 1 AS a n d , OD | BP In AOEA and AODB , " g i v e n OA = OB = R L OEA = l_ ODB = 90° OE = OD = (R-H) .'. AOEA = AODB ( s a s ) J_ OAE = l_ OBC = 6 a n d , (D .5 ) From AOGA, L AGH = l_ GOA + i_ OAG .'. 4>n = e + 6 (D .6 ) A l s o i n AOBH, l_ BHI = j_ BOH + J_ OBH •'• <|>L = (e- L AOB) + 6 S u b s t i t u t i n g e q u a t i o n (D .4 ) and (D .6 ) and s i m p l i f y i n g y i e l d s , J_ AOB = y (D .7 ) 301 Now, j_ AOE = l_ AOB + L BOG + /_ GOE a n d , L BOD = L BOG + L GOE + J_ EOD 4 V A OEA = A ODB L AOE = I BOD .*. / . AOB = L EOD = Y (D .8 ) F u r t h e r m o r e , i n AEOC and ACOD, OC i s common OE = OD and / OEC = J_ ODC = 90° A EOC = A COD ( s a s ) I EOC = L COD = Y (D .9 ) 2 and l_ ECO = J_ DCO = 90 - Y 2 I t f o l l o w s t h a t , A EOF = A OD F ( s a s ) and A EFC = A DFC ( s a s ) DF = EF = (R-H) s i n x 2 Hence , the geometry of wedge ABC i s d e f i n e d QEF. 302 The c e n t r o i d s and a r e a s o f the components l i s t e d i n e q u a t i o n s ( D . l ) to (D .3 ) may now be c a l c u l a t e d . They c o n -s i s t o f two b a s i c g e o m e t r i c a l s h a p e s : the c i r c u l a r s e c t o r and the t r i a n g l e . The c e n t r o i d o f the s e c t o r i s c a l c u l a t e d f rom (see F i g u r e D . 2 ) , x0AB. = / x df l 1 (D. 1 0 ) •' ' / d A ' where x 2^  R cos ^ ( D . l l ) 3 and dA' = 1 R 2 d ij, (D .12 ) 2 S u b s t i t u t i n g e q u a t i o n ( D . l l ) and (D .12) i n t o ( D . 1 0 ) , i n t e -g r a t i n g and s i m p l i f y i n g y i e l d s , *0'A'B": = 120 R [ s i n e - s i n ( e - y ) ] ( D . l 3) Tf Y S i m i l a r l y , OAB = 120 R [cos ( 0 - Y ) - c o s e ] ( D . 1 4 ) IT Y and A 0AB = _L_ R 2 Y (D .15 ) '3.60 303 Figure D.2 Construction i l l u s t r a t i n g the der ivat ion of the centroid of sector OAB. 304 The c e n t r o i d o f a t r i a n g l e i s g i v e n by the p o i n t o f i n t e r s e c t i o n o f the l i n e s c o n n e c t i n g the v e r t e c e s and the m i d -p o i n t s o f t h e i r o p p o s i t e s i d e s . I t w i l l be s u f f i c i e n t to s o l v e f o r t h i s p o i n t o f i n t e r s e c t i o n u s i n g two s t r a i g h t l i n e s . The r e q u i r e d c o - o r d i n a t e s o f the v e r t e c e s o f the t r i a n g l e s are l i s t e d i n T a b l e D. l w h i l e i n T a b l e D.2 to D.3 the d e r i v e d c e n t r o i d c o - o r d i n a t e s and a r e a s o f the r e q u i r e d t r i a n g l e s a re l i s t e d . S u b s t i t u t i n g the a r e a s f rom T a b l e D.3 i n e q u a t i o n (D .3 ) y i e l d s , A ABC = _ [ r _ R 2 Y - ( R - H ) 2 tan Y ( D . l 6) 360 2 F i n a l l y , the c e n t r o i d o f wedge ABC may be c a l c u l a t e d . I f e i t h e r o r both a p e x e s , A and B, l i e i n the f o u r t h quad ran t the above e q u a t i o n s w i l l s t i l l h o l d i f t he d e f i n i t i o n s o f (pyS <f>|_ J Y > 6 and 0 a re m a i n t a i n e d as shown in F i g u r e D.3 and D.4 . The c e n t r o i d o f the l ower wedge PSC must now be c a l c u l a t e d ( F i g u r e D . 5 ) . In A OCA a n d A O C P , OC i s common OA = OP and PC = AC .\ A OCA = A OCP L AOC = j_ POC 305 Figure D.3 Geometrical construct ion of the bed fo r the apex in the f i r s t and fourth quadrants at both the cby and ^ bed i nc l i na t i ons respec t ive ly . Figure D.4 Geometrical construct ion of the bed for the apex in the fourth quadrant at both the cb.. and *. bed i nc l i na t i ons . 307 Figure D.5 Geometrical construct ion of the bed i l l u s t r a t i n g the ca lcu la t ion of the centroid of PSC. 308 TABLE D. l V e r t e x C o - O r d i n a t e s V e r t e x C o - O r d i n a t e s (x,y) 0 ( 0 , 0 ) A ( R c o s e , R s i n e ) B (R cos ( e - y) , - R s i n ( e - Y ) ) D ( ( R - H ) s i n <J>L, - : (R-H) cos <PL) E ( ( R - H ) s i n * u , - (R-H) COS TABLE D.2 C e n t r o i d C o - O r d i n a t e of Component T r i a n g l e Areas Tr iangle Abscissa (x) Ordinate (y) ODB 1 (R cos (0-Y) 3 + (R-H) s in <PL) 1 (R s in (6-Y) - (R-H) cos *, ) 3 L OEA 1 ((R-H) s in cp.. 3 u + R cos e) 1 (R sin e - (R-H) cos cp..) 3 u ODE 2 (R-H) cos Y 3 2 s in ( c p u -• Y) 2 - 2 (R-H) COS Y COS (cp - Y ) 3 2 u 2 DEC (R-H) s in (cby - y ) 2 - (R-H) cos (cp - Y ) u 2 (cos Y + 1 s in 2 3 Y tan Y 2 2 ) (cos Y + 1 s in Y tan Y) 2 3 2 2 TABLE D.3 Areas of Tr iangular Components Tr iangle Area ODB 1 (R 2 - ( f r - H ) 2 ) % (R-H) 2 OEA 1 (R 2 - es-H ) 2 )^ (R-H) 2 ODE (R-H) 2 cos Y_ s i n y 2 2 DEC ; ( ( R - H ) sin •2 Y ) tan Y 2 2 310 and 0C i s a T ine o f symmetry o f the upper and l ower wedges In "A HOD, / HOD = 90 - cbL Us ing e q u a t i o n (D .9 ) , [_ HOC = 90-cf>- - Y . L 2 I_ JOC = 90-cbL-j_+ t a n " 1 y A B C  2 X ABC I KOC = L J 0 C and l_ HOK = a = 1 80-2<f>L-Y ' + t a n - 1 y A B C (D.17) X ABC S u b s t i t u t i n g e q u a t i o n (D .4 ) i n t o ( D . 1 7 ) , a = 1 8 0 - 2<j>|J + y + t a n " 1 y A B C (D.18) X ABC a 1 s o R ABC = ( X A B C 2 + y A B C 2 ^ 2 ( D ' l 9 ) •'• X P S C = R ABC C 0 S a ( D ' 2 0 ) and y p s c = ^/\BC s i n a ( D ' 2 1 ) T h e r e f o r e the r e q u i r e d e q u a t i o n s to c a l c u l a t e the c o - o r d i n a t e s o f the c e n t r o i d s o f the upper and l ower wedges have been d e r i v e d . APPENDIX E Program L i s t i n g for Slumping-Rol l ing Boundary. 312 M I C H I G i N T E R M I N A L S Y S T E M F O R T R A N G I 2 1 . 8 I MA ! N 1 2 - 1 9 - 8 0 1 0 : 3 2 : 5 3 0 C 0 1 O C 0 2 0 0 0 3 C 0 C 4 OC 0 5 0 0 0 6 c o o ? 0 0 C 3 CUO-5 0 0 1 0 0 0 1 1 o o i : 0 0 1 3 JO i<t O O i 5 0 C 1 6 0 0 1 7 0 0 1 3 0 C 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 C 2 t B E D B E H A V I O U R M O D E L O F S L U M P I N G / R O L L I N G B O U N D A R Y T U S P R O G R A M C A L C U L A T E S T H E R O T A T I O N A L S P E E D A T W H I C H A G I V E N B E D O F S O L I D S W I L L C H A N G E F R C M A S L U M P I N G T O A R O L L I N G M O O E O F 9 E D P ? h A V ! C U R . T H : I N P U T S T O T H I S M O D E L A R E : R - R A C I U S C F R O T A R Y K I L N ( C M ) . G M A O 1 - A N G U L A R C H A N G E I N B E D I N C L I N A T I O N B E T W E E N MAX. £ M I N . 3 E D I N C L I N A T I O N A S R O T A T I O N A L S P E E D A ° R O A C H E S Z E R O F O R A B E D M O V I N G F R O M T H E M I N . T O M A X . B E D I N C L I N A T I O N ( D E G R E E S ! . G M A 0 2 - S A M E A S G M A 0 1 B U T F C P A B E D G O I N G F R O M M A X . T O M I N . B E D I N C L I N A T I O N ( D E G R E E S ) . C 2 - S L O P E O F G M A 1 O R G M A 2 V S R P M C U R V E S ( S E C S ) . P H l R - A L S O C A L L E D L A M C A I N L O G - L O W E S T B E D I N C L I N A T I O N ( C E G R E E S I R E A L N . N K D I M E N S I O N D F ( 1 0 0 ) . D F C I 1 0 0 1 , H I 1 0 0 ) . D E L T A ( 1 0 0 1 . C E L T A R I 1 0 0 ) I N T E G E « C H A R G E ( 2 0 ) R E « D ( 5 , 1 ) C H A R G E F O R M A T t 2 0 A A I ~ R t 1 C ( 5 , 2 ) R , G M A 0 1 , G M A C 2 , C 2 , P M R F L ) R M A T | F 7 . 2 , 2 X , F 5 . 2 , 2 X , F 5 . 2 , 2 X , F 6 . 3 , 2 X , F 5 . 1 ) « R ! T C ( 0 t l 5 J C H A R G F F O R M A T t ' 1 ' , 2 X , • B E D B E H A V I C U R D I A G R A M F O R : • , 2 X , 2 0 4 4 I W R I T E ( 6 , 2 1 ) R . G M A 0 1 , G V A 0 2 , C 2 , P H I R F O R M A T C , , ' R = , . F 7 . 2 . 2 X , ' G " A 0 1 = , . F 5 . 2 . 2 X , , G M 5 0 2 = , , F 5 . 2 , 2 X , ' C 2 = , , F 6 C . 3 . 2 X , • P H l R i ' , F 5 . 1 ) W R I T E ( 6 , 1 6 I F 0 = M A T ( • - ' , 2 X , • T H E B O U N D A R Y B E T W c f - ' S S L U M P I N G A N D R O L L I N G O C C U R S A T C I W S ! T F ( 6 , 1 7 ) 1 7 C C R M A T ( > > , 3 X , ' 3 E D D E P T H ' , 2 X D E G R E E O F F I L L • . 2 X , • R O T A T I DMAL S ° E E O C • I W - : I T F ( £ , 1 E > 1 8 c 7 « « 4 T ( i " , 1 0 X , ' ( C M ) • , 7 X , ' I P E R C E U ~ I ' , 7 X , • ( R - V / M I N I•I C A L C U L A T E B E D D E P T H S T O B E L S E D F O R M O D = L C A L C U L A T I O N S . B E D D E P T H S W I L L B E O B T A I N E D C O R R E S P C N D I N G TO 2 5 , 1 5 , 1 0 , £ 5 P E R C E N T F I L L . C R O S S S E C T I O N A L A R E A O F K I L N . A = 3 . 1 4 1 5 9 3 » < R * « 2 I . C A L C U L A T I O N O F B E D D E P T H S . 1 = 1 D F ( I ) = 1 . 0 ' H ( I I = 1 . 0 4 R 1 = R - H ( ! ) R 2 = A R C C S ( R 1 / R ) R 3 = S C R T < ( R * « 2 I - ( P l - * 2 I I C F C ( ! ! = ( ( < ( P . * * 2 > * R i ) - < R l * R 3 ) ) / A 1 * 1 0 0 . 0 I F ( D F C U ) . E C . O F ( I I ) G C T C 5 1 2 1 5 2 1 16 1 . 0 0 0 2 . 0 0 0 3 . 0 0 0 A • 0 0 c 5 . 0 0 0 6 . 0 0 0 7 . O O C 8 . 0 0 0 9 . 0 C C 1 0 . 0 0 0 1 1 . 0 0 0 1 2 . 0 0 0 1 3 . O O C 1 4 . 0 0 0 1 5 . 0 0 0 1 6 . 0 0 0 1 7 . 0 0 0 1 8 . O O C 1 9 . 0 0 0 2 C . D 0 C 2 1 . O O C 2 2 . 0 0 0 2 3 . O O C 2 A . O O C 2 5 . O O C 2 6 . 0 0 0 2 7 . O O C 2 3 . 0 0 0 2 9 . 0 0 0 3 0 . 0 0 0 3 1 . 0 0 0 3 2 . O O C 3 3 . 0 0 0 3 4 . O O C 3 5 . 0 0 0 3 6 . 0 0 0 3 7 . 0 C C 3 8 . 0 0 0 3 9 . 0 0 0 4 0 . 0 0 0 4 1 . 0 0 0 4 2 . 0 0 0 4 3 . O O C 4 4 . 0 0 0 4 5 . 0 0 0 4 6 . 0 0 0 4 7 . 0 0 0 4 8 . O O C 4 9 . O O C 5 0 . 0 0 0 5 1 . O O C 5 2 . 0 0 0 5 3 . O O C 5 4 . 0 0 0 5 5 . 0 0 0 5 6 . O O C 5 7 . 0 0 0 5 8 . O O C 313 M.'OIGS'M T-rRM I NAL S V S T F M FORTRAN G ( 2 1 . 8 ) M A I N 1 2 - 1 9 - 8 0 1 0 : 3 2 : 5 3 0C2 7 002-3 0C29 0C3J 0C31 0C32 0033 003- . 0035 0C36 0037 00 3 8 0039 n o v o 0041 0042 0043 0044 0043 004c 0047 004 6 0C4 9 00 3 0 0051 0C52 0"0 5 3 0054 0055 0056 0057 0C5H 0C59 0060 O C t l 0 06 2 0063 0C64 0065 0066 00= 7 0 01 ^ OCti 0C70 0 0 7 i 0072 I F ( D F C ( I ) . L T . O F ( I ) ) G C TO 3 ! F ( ( D F C ( ! I - D F ( I 1 1 . L E . C . l I G O TO 5 H 1 I ) = M I 1 -0 .02 GO T O 4 3 H ( I ) = M I > * 1 . 0 G O TC 4 5 1 F ( I . F C . 2 5 1 G O TO 6 1 = 1 + 1 D F l I l = C F ( 1 - 1 1 + 1 . 0 M I I - M I - 1 1 GO TO 4 C F INISH I N I T I A L I Z I N G A L L CTHFB V A R I A B L E S : C DELTA - FUNCTION OF CEGREE OF F I L L ( D E G R E E S ) . C THETA - DEFINES U°PER POSITION OF THE B E D SURFACE (DEGREE S I . C NOTE: WHEN ANY ANGULAR VARIABLE ENDS WITH AN R (EG CELT AR) THAT C ANGLE IS IN RACIANS ( E X E PT PHIR - PHIRR IN R A D I A N S ) . r 6 1 = 1 8 DELTAR(I ) = A O S ! N ( ( R - H ( I I 1/R) DELTA( I ! = ( C E L T A R ( I l * ( 1 8 C . 0 / 2 . 1 4 1 5 9 3 I) I F I I . F C . 2 5 I GO TC 19 1 = 1*1 GO T O 3 1 9 1 = 1 7 J = 0 R0*! = C.1O 11 G M i ] = G * A 0 1 M C 2 * 6 . 0 * R D V . I+GMAC2 ALPHA = 0HIR + G>141 T H - f A = A L P H A - D D L T A ( I ) G . * ' - 1 » = ( G * 6 1 * 3 . 1 4 1 5 9 3 1 / 1 6 0.0 ALPHAP = ( A L P H A * 3 . 141593 ) /18C.0 T H j T 4 p i ( T H c T A * 3 . 1 4 1 5 ? 3 l / 1 8 C . O P H ! R R = ( P H I R « 3 . 1 4 1 5 9 3 ) / 1 8 0 . C GMA0=GKAC1+GMA02 G f A 0 R=(GMA0 * 3 . 1 4 1 5 9 3 ) / 1 6 0.0 C CALCULATE CENTRCID OF UPPER WEDGE - X A B C , Y A B C . C X 0 A 3 = ( ( 2 . C » R ) / ( 3 . 0 * G M A 1 R ) ) » ( S I N ( T H E T A R ) - S ! N ( T H F T A R - G M A 1 1 ' ) ) X C D B = ( ( R * C a S ( T H E T A R - G H A l R | | * ( ( R - H ( t | ) * S ! N ( P H l R R I I ) / 3 . 0 X 0 E A = ( ( R * C 0 S ( T H E T A R I ) + ( ( R - H ( I ) ) * S ! N ( A L P H A R I I ) / 3 . 0 X 0 O E = ( 2 . 0 * ( R - H ( I ) l » C C S ( G ^ A l P / 2 . 0 l * S I N ( A L P H A R - ( G M A l R / 2 . 0 ) ) l / 3 . 0 XD2C= (R-H( I ) l * S I N ( A L P H A F . — ( G M A 1 R / 2 . 0) » * ( C 0 S ( G M A l R / 2 . 0 ) » ( SIN (GMA1R/2 C . 0 ) * T A N ( G H A l R / 2 . 0 ) / 3 . 0 ) l Y 0 A B = ( ( 2 . 0 * R I / i 3 . 0 * G ^ A l P | ) * ( C 0 S ( T H E T A R - G M A 1 R l - C O S ( T H E T A R ) I Y O O B = ( ( R « S I N ( T H E T A R - G M A 1 R ) ) - ( ( R - H ( I ) ) » C O S ( P H I R S ) I ) / 3 . 0 Y C E A = ! ( R * S I N ( T H E T A R I l - ( ( R - H ( I ) ) * C O S ( A L P H A R | ) 1 / 3 . 0 Y 0 D F = - ( 2 . 0 « ( R - H ( I ) I » C 0 S ( G M A 1 R / 2 . 0 ) * C 0 S ( A L P H A R - ( G M A I R 1 1 . 0 1 1 1 / 3 . 0 YD-C = - ( R - H ( I I ) * C C S ( A L P H « R - ( G ' ' A 1 R / 2 . 0 1 I * ( C O S ( G M A 1 R / 2 . 0 I » ( S I N ( G M A 1 R / C2.0 1 * T A N ( G M A 1 R / 2 . 0 ) / 3 . C I I A O ' - e = ( R » * 2 ) * G M A l P / 2 . 0 AO;T> = ( S Q R T ( ( P * * 2 ) - { (R-H ( t 1 1 **2I 1 I « ( R-H( I I I 11 .0 A O F A = A C C B 4 0 n " = ( ( R - H ( I ) ) * « 2 l « S I N ( G M A l R / 2 . 0 l * C O S ( G 1 « A l R / 2 . 0 ) A C E C = ( ( ( R - H ( I ) ) * S I N ( G M A 1 R / 2 . 0 ) ) * * 2 1 * T A N ( G M 4 1 R / 2 . 0 ) l l H C = A C A e - ( ( ( R - H ( I I ) * * 2 1 * T A N ( G M A l R / 2 . 0 1 l X « 9 C = ( ( X 0 A B « A 0 A 3 > • ( X C C B * A 0 C B 1 - ( X O E A * A O F « ) - < X C D ? * A 3 0 E I - ( X 0 E C * A D f c C > ) 59 .OOC i 0 .000 6 1 . 0 0 0 6 2 .000 63.OOC 6 4 . 0 0 0 6 5 . 0 0 0 6 6 . 0 0 0 67 .0DC 68 .OOC 6 9 . 0 0 0 7 C . U 0 0 7 1 . 0 0 0 72 .000 73.OOC 7 4 . 0 0 0 75.OOC 76 .000 7 7 . 0 JO 78.OOC 79 .00D SC.OOC 8 1 . 0 0 0 6 2 .000 83 .OCC 3 4 . 0 0 0 e E . o o : E6.000 87 . 68 . 89 . 90 . o c : OOC o c : 9 1 . 0 0 0 9 2 . 0 0 0 93 .000 9 4 . 0 3 0 9 5 . 0 0 0 9 6 . 0 0 0 9 7 .C0C 93 .000 9 9 . 0 0 0 i o c . o : o 101 .OCC 1 0 2 . 0 0 0 1 0 3 . 0 0 0 104 .000 1 0 5 . 0 0 0 106.OCC 107.OOC 103 .OCC 1 0 9 . 0 0 0 110.OOC 1 1 1 . 0 0 c 112 .030 113.OCC 114 .030 1 15.OOC 1 1 6 . 000 314 MICHIGAN TERMINAL SYSTEV FCRTRAN G ( 2 1 . 8 ) MAIN 1 2 - 1 9 - 3 0 10:32:53 0073 0C74 0075 0076 0077 0078 0C79 0030 0031 coe2 0033 0 C S 4 CC65 0086 0 C 8 7 008B 0C89 0090 0091 0092 0093 C094 009 5 0096 0C97 0098 "OPTIONS IN • u P T I C N S IN « S T A T ! S T I C S • S T A T I S T ICS iC ERRORS IN *A C /AABC Y A R C = l ( Y 0 » S * A ? A 3 I H Y C C E » A C C E I - ( YOE A* AOF A 1 - ( Y QCE*AOOE I - ( YO = C*AO=CI I C / A A B C C C CALCULATE CENTROID CF L CK FR wc CGF - X 0 1Q2C . YOI 02C r R A B C = J C R T I ( Y A B C * * 2 I • ( X A E C » « 2 I ) A C l Q 2 C = < - 3 . 1 4 1 5 9 3 + 1 2 . C * A L P H A R ) - G * A I R - A T A N ( Y A 8 C IX i 3 C I ( XC1C2C=RABC*COS(A0102CI YQ1C2C=RABC*SIN( AQ1C2C.I C C CALCULATF DISTANCE BETWEEN CENTROIDS - S ( C I ) f. ITS SLOPE - N O E G R E c S l i C NR(RACIANS I. C S = SORT II ( X A B C - X Q 1 C 2 C I * * 2)M ( Y A B C - Y C 1 0 2 C I * « 2 I 1 N P - A T A M ( Y Q 1 Q 2 C - Y 4 B C I / O C K 2 C - X A 9 C I I N = N P * 1 3 C . 0 / 3 . 1 4 1 5 9 3 C C EVALLATF EGUATTCN FOR INTERSECTION CF T<= £ T T * CJR.VES. C FCUN= ( ( ( ( 9 0 0 . * 9 e 0 . i l / ( 2 . 0 * l 3 . 1 4 1 5 9 3 " 2 l * ( r . P y * » 2 ) - ^ i - ? : ) l * I R A B C / S ) * G C V A O R » ( S I M N R ) - < C O S ( N R >»TAN< PHIRR) t ) ) -< Y A B C / S ) ) * G " 1 0 3 IF< J . F C . O t G j TO 20 I F( r .OUN.GE. 1 . 0 ) G 0 T 0 •= GC TC- 12 20 I F I E Q L N . G T . l . O I G O TC IC I F ( E C U N . L 7 . 1 . 0 1 GO TO 12 10 R P M - R F V t O . 1 0 GC TC 11 12 J = l cpw = F ,pv -0 .001 GC TC 11 9 WRITE i t , 131 H| T ) ,DFC < I I , » » 13 F a R M A T ( « 0 , , 1 0 X , F 5 . 1 . E X , F 5 . 1 , 1 2 X , F 6 . 3 l I F U .EC.251C-C TO 14 1 = 1*1 GC TC 7 14 STOP E \ 0 e F ^ E C T * I C , E 5 C 0 I C t S C U P C E , N C L l S T , N 0 O E : K , L 0 A 0 , N ? - . A P E F F E C T * NA« = = VAIN , LINECNT = 60 * SOURCE STATEMENTS = 93.P3CG.R.AM SIZE = 6014 * NQ DIAGNOSTICS GENERATES IN 117 .OOC ne.ooc 1 1 9 . 0 0 0 12C.O0O 12 i .-oo; 122 .000 123.OOC 1 2 4 . 0 0 0 125 .OOC <26.000 127.OOC 128.OOC 129, I 3C, 3.31. GOO OOC 000 13 2 .00C 1 3 3 . 0 0 0 134.OOC 135.OOC 1 3 6 . 0 0 0 137 .000 1 3 3 . OOC 139.OOC 140.OOC 141 , 142, OOC OOC 1 4 3 . 0 0 0 144 .000 145.oo: 14 6 .000 1 4 7 . 0 0 0 1 4 3 . 0 0 0 149 .000 1 5 0 . 0 0 0 151.OOC 152 .000 152.OOC i 5 4 . 0 0 0 155 .O00 >10 STATEM^-vJTS FLAGGEC IN Tt-E A30VE C O M 0 ' LAT I C N S . EXECUTION TtRMINATEC 10: 32: 54 T = 0 .524 RC = C $ .23 315 APPENDIX F Program L i s t i n g for S l ipp ing . 316 M I C H GAN TORMINA!. SYSTEM FORTRAN 0 (21.81 M A I N 1 2 - 1 9 - 8 0 1 0 : 3 2 : 5 5 C G E N F R A L I 2 C C S L I ° P I N G B= D S E h A V I C U R . 0 0 0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0 8 0 C C 9 0 O 1 0 0 0 1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 0 2 6 0 0 2 7 0 C 2 8 0 0 2 9 0 0 3 0 0 0 3 1 0 0 3 2 0 0 3 2 0 0 5 4 0 0 3 5 GENERALIZE: PLOT C F EQUILIBRIUM BED INCLINATION F O R A GIVEN BED/WALL FRICTION ANGLE AT A GIVEN KILN RPM £ BED D E P T K . I N P U T I N N E = R A D I U S O F K I L N IN C » . 0 ! M = N S I 0 N H<25 I , D C { 2 5 I , C F C ( 251 R f A C ( 5 , 1 ) R 1 C 0 R M A T ! F 7 . 2 > W R I T S ( 6 , 1 4 ) 1 4 FC'MIT (• 1 • , ' G E N E R A L I Z E C P L C T O F E Q U I L I B R I U M S E C I N C L ! N ' T I TN C ^ I ) W R I T F ( 6 , 1 5 ) 15 F O R M A T ! • • , ' G I V E N 6 E C / W A L L F R I C T I O N A N G L E AT A G I V E N . K I L N R P « £. B= C O O F . P T i - . • | W R I T F ( 6 , 1 6 I 16 F C R X A T I ' - ' ) W R I T E l i , 1 7 1 17 F C R M t T ( • 0 * , 5 X , ' 3 E 0 D E P T H 1 » 3X » ' D P M • f 2 X t ' B E D / W A L L F R I C T I O N A N G L £ ' , 3 X C , * 3 E C ! N C L I N A T T g « j . ) W P I T 7 ( 6 . 1 S I R 0 R M a T (' o * I ' 18 C = D S S S E C T I O N A L A » E « O F K I L N . P I " = 3 . 1 4 1 5 9 - 3 s = ( P I ? » ( R » * 2 > 1 C A L C U L A T E B E D D E P T H S . 1 = 1 0 E ( I 1 = 1 . 0 H ( I » = 1 . 0 4 R . 4 = ' . - H . ( I ) R 2 = S R C C S ( R 4 / R ) R ? = S O S T ( ( = * * 2 ) - ( R 4 * * 2 I I D F C ( ! l = ( < ( ( R * * 2 ) * R 2 ) - < R 4 * = 3 l ) / A ) * l C 0 . O I F [ D F C ( I ) . E C . D F I I ) I G C T C 5 I F ( D F C ( I ) . L T . D F ( I ) ) G C T C 3 I F U C F C I I I - O F I I I I . L E . C . 1 I G 0 T O 5 H ( I l = H ( I 1 - 0 . 0 2 G C T r A 3 H I I l = H < I I » 1 . 0 C O T C 4 5 I F I I . F C . 2 5 I G - 0 T O 2 1 = ! * l D F ( I l = C F < 1 - 1 1 *1 . 0 H ( I l = h ( 1 - 1 ) G C T C 4 C A L C U L A T E C E N T R I F U G I N G R E M . 2 R . P M C = ( ( 3 0 . 0 / o i E I * ( S 0 R T ( ( 9 8 C . 6 l / P I I I I N I T I A L I Z E V A R I A B L E S . . O O C . 0 0 0 • O O C . 0 0 0 • O O C . 0 0 0 . 0 0 0 e.ooc 9 . 0 0 0 1 C 0 0 C 1 1 . 0 0 0 1 2 . 0 0 0 1 3 . 0 C C 1 4 . 0 0 0 1 5 . O O C 1 6 . 0 0 0 1 7 . O O C 1 8 . 0 0 0 1 9 . 0 0 0 2 0 . 0 0 0 2 1 . 0 0 0 2 2 . O O C 2 2 . 0 0 0 2 4 . 0 0 0 2 5 . O O C 2 6 . U 0 0 2 7 . O O C 2 8 . 0 0 0 2 9 . 0 0 0 3 0 . 0 0 0 3 1 . O O C 3 2 . O O C 3 3 . 0 0 0 3 4 . 0 0 0 3 5 . 0 0 0 2 6 . O O C 3 7 . 0 0 0 3 8 . 0 0 0 3 9 . 0 0 0 4 0 . 0 0 0 4 1 . O O C 4 2 . 0 0 0 4 3 . 0 0 0 4 4 . 0 0 0 4 5 . 0 0 0 4 6 . O O C 4 7 . 0 0 0 4 3 . O O C 4 9 . 0 0 0 5 C . 0 0 C 5 1 . 0 0 0 5 2 . 0 0 0 5 3 . O O C 5 4 . 0 0 0 5 5 . O O C 5 6 . 0 0 0 5 7 . O O C 5 9 . 0 0 C 317 io.:ci\' T :RMI \AL SYSTPM. FCRTRAN G(21 .S I MAIN 1 2 - 1 9 - 8 0 10:32:55 0036 0037 0038 0C39 0C4C 004 1 0042 0043 004-. 0045 0 041: 0047 0043 004 9 0050 0051 005 2 0 0 53 0034 0C55 005a 0057 3d 0 0 C 5 9 00 c J 006 1 00c2 00 b 2 0 J-'.,4 CCt 5 OOo-i C Cc 7 00 6 S 0069 C070 GO J I CC72 007? C074 00 7 5 u J76 0077 0U7.3 0 C 7 9 "OPT - C P T »STA » S T » f- D P Or 19 13 10 1-JNS I CNS T I ST T 1ST S 1U 21 IN EPFE T M r ? c -I C S * I C S " NC M A T - : J=l A 1 = A R C C S ( ( R - H ( J l l / R ) A e = D = ( ( R * * 2 ) * < ( 2 . 0 * A l l - ( S I N ( 2 . 0 * 4 1 1 1 1 1 / 2 . 0 R1=<2 .C»(R* *3 I«<<S!N IA1) ) * *3 I ) / ( 3 .0 *ABE0 I U E E 0 = 9 £ 0 . 6 0 F ! S = 1 .0 P H I S C = P H I S M P I E / 1 3 0 . C ) B=TAN<PHISR> R P V = C.C A L P H A P = ( 0 . 0 1 » P I E ) / i e C . C C 5 : C = ( R 1 * ( ( ( R P " " P I E 1 /30 .01* *21) L S O S l = ( SCRT( I CBE0««2 I • ( 2 . 0 * C B E D ' W B E D * COS ( ALPH AR 1 I * t •*3~0*•* 2 1*1 S 2 = ( R l * w e E 0 * S I N ( A L P H « R I ) / R S 3 = ( ( S 2 ) / ( S 1 * C 0 S ( A R S I N ( S 2 / S 1 I ) ) ) S4=3/S2 I H L S - E Q . C I O O TO 7 I F ( S 4 . G E . 1 .CIGC TO 3 CC T 0 9 I F t S 4 . L E . 1 . 0 ' l G C TO 9 1 c ( A L P r A R . G E . ( P I E / 2 . C l I G O TC 20 <l5u ( P = A L P H t R t ( o ! - / I S O . C ) G" TO 10 L S = 1 A L=H.AC = A L P H A R - ( I C . 1 * F 1 ' ; ) / 1 3 C . 0 1 C-C TC 10 A L P H A = » L " h J R * < 1 8 C . 0 / P I E I W 3 ! T E ( t , l l l H ( J I , a P ^ , P H I S . ALPHA F G » M A T ( • • , 7 X , F 5 . 1 , 3 X , F 6 . 1 , 6 X , F 5 . 1 , ! 8 X . F 5 . 1 ) ! C ( S P M . E O . I C . O ) G C TC 12 5 CM=:R F 1 .0 GO TC 13 I>=IP U IS.C-E.50.0IGD TO 2<t P H I S = P H S » 5 . 0 CC TO 19 IF<J .EC .25100 TO 23 J = J+ 1 GC TO t I F ( J . E C . 2 5 I G O TG 23 W R ! T E ( 6 , 2 1 ) H I J ) , R O M , P H I S FORMAT(• • , 7 X , F 5 . 1 , 2 X . F 6 . 1 , 6 X , F 5 . 1 , 8 X , • B E D IS UNSTABLE* I GO T 0 22 STOP END CT» IC . E B C D I C SOURCE .NOL I ST , NOOECK , LOAD , N 3 MA P C T * NA»E = XAIN . LINECNT = 60 59.OOC 6 C . 0 0 C 61.OOC 6 2 . 0 0 0 63 .00C 6 4 . 0 0 0 65.OOC 6 6 . 0 0 0 67.OOC 6 S . 0 0 C 6 9 .00 0 7 0 . 0 0 0 71.OOC 7 2 . C 0 C 7 3 . 0 0 0 74 .000 7 ; . o : c 7 6 . 0 0 0 77.OCC 7 8 .000 79.OCC 79 .100 S C . O C C 3 1.030 82 .000 8 3 . J 0 C 84 .000 e 5. o o c 6 6 .000 87, p o , 39, 9C. OOC OOC 0 0 C OCC 9 1 . 0 0 0 c i ( OOC 000 94.0 00 94.01C 94 .020 94.IOC 9 4.203 94.300 94.400 95.OOC 96.OOC SOURCE STATEMENTS • D ! A G N C S T I C S GE N E c A T E C 79,PROGRAM S I Z ; 2592 STATr.McNTS FLAGCFC CUTION TERMIV A T P C IN T (• C ( ABOVE COMPILATIONS. 10:32:56 T=0.357 =C=C i. 19 APPENDIX G Program L i s t i n g for Predict ing Ful l Bed-Behaviour Diagram. 319 MICHIGAN TfcRMINM SYSTEM FORTRAN G I 2 1 . 8 ) MAIN 12 -19 -BO 1 0 : 3 2 : 5 7 C 1 .000 C BED REI-AVIOUR MODEL 2 .000 r # » » « » • » « » * « * * * « » » * * 3 . 0 C C C 4.OOC C 5.OOC C THIS PROGS!" CALCULATES T H C ROTATIONAL SPEED AT WHICH A GIVEN BED 6 .000 C CE SOLIDS wILL CHANGE F c C w : 7.OOC C 1 . SLUMPING TC ROLLING. 5.OOC C 2 . S L I P P I N G , 9.OOC C 3. ROLLING (CASCADING) TO C A T AP. i C T ! NG , 10 .000 C 4 . CATARACTING TC C E N T R I E U G I N G . 11 . 0 0 0 C 12.OOC C T H c INPUTS TO THIS MODEL A R E : 1 3 . 0 0 0 C R - R A C K S CF ROTARY CYLINCER ( C M ) . 14.OOC C GMA01 - ANGULAR CHANGE IN BED INCLINATION BETWEEN MAX. S - I N . SEC • 1 5 . 0 0 0 C INCLINATION AS ROTATIONAL SPEED APR 0 AC HES ZERO FOR A 3 C D 16 .000 C MOVING FROM THF MIN. TO MAX. BED INCLINATION (DEGR=E5>. 17.OOC C GMA02 - SAME AS GMA01 BUT FOR A BcC GOING FROM Max. TO MIN. 3ED 13 .000 ' C INCLINATION ( D E G R E E S ) . 19.OOC C CI - SLOPE 0= GMAI QR G"A2 VS RPM CURVES ( S E C S ) . 2 0 . 0 0 0 C PHIR - ALSO CALLFO L A y C A IN LOG - LOWEST BIO ! N C L I M * T ! 0 N (DEGR-ESI 21 .000 C CAVG - AVERAGE PARTICLE S IZE OF MATERIAL (CM). 2 2 . J 0 C C ° H i S - BEC/WALL FRICTICN ANGL C ( D E G R E E S ) . 2 3 .000 C PHIO - DYNAMIC ANGLE OF R C D r j c rf CYLINDER CHARGE ( D E G R E E S ) . 2 4 . 0 0 0 C 25.OOC 0001 REAL N,NR 26.OOC 0002 CIMrNSION DF (100 ) . D FC < IC C ) , M 10 C) , DE L TA I 1 UO ) , DE LT 4= ( i 00 I , R 3 MC S < 6 ,1 2 7 .000 C C C ) » R P f H C S ( 6 » I C C ) » R P , ' K ( l C 0 ) 23.OOC 0003 INTEGER CHARGE(20) 29 .000 0004 RE AO(5,1 I CHARGE 3C.O0C 0005 1 C CRMAT(20A4 I 3 1 . 0 0 0 .0006 R F A C ( 5 . 2 ! R . C M A 0 1 , G M A C 2 , C 1 . P H I R , C A V G , O H ' S . P - i l O 32 .000 0007 2 F n R M A T ( E 7 . 2 . 2 X , F 5 . 2 . 2 X , F 5 . 2 , 2 X , E 6 . 3 . 2 X , E 5 . l , 2 X , F 7 . 4 , 2 < , = 5 . 1 , 2 X , C 5 . 33.OOC C l l ^ 3 4 . 0 00 OOCB W? I T E ( 6 . 1 5 I C H A R G E , 35.OOC 0C09 15 F O R M M ( • 1 2 X , ' B E D BEHAVIOUR DIAGRAM FOR : • . 2 X , 2 0A.4 I 36.OOC 0C10 WRITF(6 ,21 IR ,G.MA01,Cr 'AC2.Cl ,PHIR 1 'DAVG,PH !S , B HlC 37.OOC 0011 21 FORMAT 1' • , ' R = ' , F 7 . 2 , 2 X , ' G » A 0 l = • , F 5 . 2 , 2 X , • G M A 0 2 = ' , E 5 . 2 , 2 X , • C I = ' , F 6 3°.OOC C . 3 , 2 X , ' P > I R = ' , F 5 . 1 , 2 X , , C A V G = ' , F 7 . 4 I 2 X , ' P H ! S = , , < : 5 . 1 , 2 x , , P H i n = ' 1 ' : 5 . i 39.OOC C) 4C .O0C 0012 WRITEI6 .16 ) 4 1 . 0 J C 0013 16 FORM A T ( • - • , 2 X , • T H E 9 C D BEHAVIOUR BOUNDARIES OCCUR A T ' I 42.OOC 0014 W P . I T E ( 6 , 1 7 » 43 .OOC 0015 17 FORMAT (' 0 ' , 8X, ' BED CEPTI- • , 3X , • % F I LL • . 3 X , ' ROT AT I ON AL SPEED 44.OOC C ( R E V / M I N I : ' I 45.OOC 0016 W R ! T E ( 6 , 1 8 ) 4 6 . 0 C C 0017 18 F O R M A T ! ' ' , 1 0 X ( C M I • , 4 X , • (PER CENT) ' ,IX, •SLUMPING: • 47.OOC C ,4X , ' P O L L I N G ' , 2 X , ' C A T A R A C T I N G ' , 3 X , ' C E N T R I FUG^NG' ) 4 3.0 00 O C l a W .RITE(6,26I 49.OOC 0019 26 FORMAT( 1 • , 2 9 X , ' R O L L I N G ' , 3 X , • C A S C A D I N G ' . I X , ' B E G I N ' 5 0 . 0 0 0 C , . 4 X , ' F L L L , t 3 X , « W » L L , t 5 X , ' B E C , > 51.OOC C020 W R ! T E ( 6 , 2 7 ) . 52 .000 0U21 27 FCRMtT('O') 53.OOC C 54.OOC C CALCULATE BED DEPTHS TO BE L SE 0 FOR MODEL C A L C U L A T I O N S . 5 5 . 0 0 0 C BfD OERTHS WILL BE OBTAINED CORRESPONDING TO 2 5 , 1 5 , 1 0 , S 5 PER 56.OCO C CENT F I L L . 57 .000 C 5 E . 0 C C 320 M I C H I G A N T H M M L S Y S T E M F O R T R A N G I 2 1 . 8 I C C R O S S S E C T I O N A L A R E A C F K I L N . r 0022 A = 3 . 1 4 1 5 9 3 * ( R * * 2 ) C C C A L C U L A T I O N OF S E D C E P T H S . MAIN 1 2 - 1 9 - 3 0 1 0 : 3 2 : 5 7 002 3 0024 BC25 002c 0 02 7 0023 0029 002u 00 31 0032 0 02 2 00 34 003 5 CU3c 0.0 3 7 C03S CC2 9 0040 004 i 004^ 0043 0044 0045 O C 4 0 0047 0043 0C49 00 50 0051 0052 005 2 0054 0055 0056 0057 00 5 5 0 0 5 v 1 = 1 CF{ I ) = 1 .0 H< ! ) = 1.0 4 R 1 = R - M I ) R2=ARCCS(R1/P.I R 2 = S Q R " U ( R * * 2 ) - ( R 1 * » 2 > > C F C ( I ) = ( ( ( ( R * * 2 l * R 2 1 - ( R l * R 3 ) l / A I « i O O . O I F ( O F C ( ! I . E Q . D F ( I 1)GO TO 5 I F ( O F C ( I I . L T . O F { I ) 1 CO T C 3 I F ( ( O F C ( I 1 - 0 F { I ) ) . L E . O . 1 ) G C TO 5 H ( I ) = H ( I 1 - 0 . 0 2 GC TC 4 3 H< I » = M I 1 + 1 . 0 c: TC 4 5 I F1 I . E C . 2 5 I G O TO 6 1 = 1*1 C F U ) = C F ( 1 - 1 1 * 1 . 0 H I I « K I - 1 I GC TC 4 C C J M I S H IN IT IAL IZ ING ALL CTt- E R V A R I A B L E S : C CELTA - FUNCTION OF DEGREE CF F I L L ( D E G R E E S ) . C T H E * A - CFFINES UP»ER POSITION O c THE BED SURFACE ( D E G R E E S ) . C N O T E : WHEN ANY ANGULAR VARIABLE ENDS WITH A N R (EG D E L " A R ) THAT C ANGLE IS IN R AC I AN S (E Xf P T PHIR - RHIRR IN R A D I A N S ) . 1 9 7 11 CCL-O 1 = 1 D E L T A S ( I ) = A R S I N ( ( P - H ( I I ) / R ) D E L T A ! I ) = ( D . E L T A R ( I ) * ( 1 8 0 . 0 / 3 . 1 4 1 5 9 3 ) 1 I F ( I . E C . 2 5 ) GO T C 1 9 1 = 1 * 1 G : T C e 1 = 1 j = o R F M = C . 1 0 GMA t = G V A 0 1 * ( C 1 * 6 . 0 * R °"I*GMA02 A L P H A - C H I R + G M A 1 T H C T A = A L P H A - O E L T A ( I I G W A I R = ( G U A 1 * 3 . 1 4 1 5 9 2 ) / l E C . C A L P H A R = ( A L P F A « 3 . 1 4 1 5 9 3 1 / 1 8 0 . 0 ' T H E T A R = ( T H E T A * 3 . 1 4 1 5 E 3 I / 1 8 0 . 0 P H I R R = ( P H I R * 3 . 1 4 1 5 9 3 1 / 1 8 C . 0 G M A 0 = G M A C 1 * G M A 0 2 G M A 0 R = ( G M A 0 » 3 . 1 4 1 5 9 3 ) / 1 6 0 . 0 r C C A L C U L A T I O N O F S L U M P I N G / R O L L I N G B O U N D A R Y . r 0 C A L C U L A T C C E N T R O I D OF U ° P E R W E D G E - X A B C , Y A B C . r X O A 3 = ( ( 2 . 0 » = I / ( 3 . 0 « G M A 1 R ) ) « ( S I N ( T H E T A R I - S I N ( T H E T A R - G M A 1 R I I 5 9 . 0 0 0 6 C . 0 0 0 6 1 . O O C 6 2 . 0 0 0 6 3 . 0 0 0 6 4 . 0 C C 6 5 . 0 0 0 6 6 . 0 0 0 6 7 . O O C 6 8 . 0 0 0 6 9 . O O C 7 0 . 0 0 0 7 1 . O O C 7 2 . 0 0 0 7 3 . 0 0 0 7 4 . 0 0 0 7 5 . O O C 7 6 . O O C 7 7 . 0 0 C 7 8 . O O C 7 9 . 0 0 0 8 0 . 0 0 0 8 1 . 0 0 0 6 2 . 0 0 0 8 3 . O O C 8 4 . 0 0 0 e 5.0 0 c 86 . 0 0 0 9 7 . 0 0 0 6 6.000 8 9 . 0 0 0 9 C . 0 0 C 9 1 . 0 0 0 9 2 . 0 0 0 9 3 . 0 0 0 9 4 . 0 0 0 9 5 . O O C 9 6 . 0 0 0 9 7 . 0 0 C 9 = . 0 0 0 9 9 . 0 0 0 1 O C . O O C 1 0 1 . 0 0 0 1 0 2 . 0 0 0 1 0 3 . 0 0 0 1 0 4 . 0 0 0 1 0 5 . 0 0 0 1 0 6 . O C C 1 0 7 . 0 0 0 1 0 8 . 0 0 0 1 0 9 . O O C 1 1 0 . 0 0 0 111.ooc 1 1 2 . 0 0 0 1 1 3 . O O C 1 1 4 . O O C 1 1 5 . O C C 1 1 6 . O O C 321 MICHIGAN TERMINAL SYSTEM FCRTRAN G I 2 1 . 8 I MAIN 1 2 - 1 9 - 8 3 1 0 : 3 2 : 5 7 C061 00s2 0063 005' . 00o5 0 01 >, 0067 006 ? 0069 0C7C 0 0 7 i 0C72 0073 0074 OC75 0076 00 7 7 0078 0C7 i 0CB0 00 SI 0C3s OOiid CC3 7 0085 0039 C09 0 009 1 CC92 0093 0094 009 5 00-,i 00 57 c c > c GC-9 X 0 D 3 = ( ( R * C O S ( T H E T A R - G * A l R I ) + ( < R - H ( I M * S I N ( P H I 3 R I ) ) / 3 . 0 X O E A = ( ( P * C C S ( T H E T A R ) ! • ( ( R - H ( I 1 I ' S I N ( A L P H A R ) I ) / 3 . 0 XODE = ( 2 . 0 * ( R-H ( I I ) * C C S ( G M A l P / 2 . 0 l * S I N ( ALPHAR- I C A 1 R / 2 . 0 ) t 1 / 3 . 0 XDEC= <R-H(I I l * S I N ( A L P H A R - I G M A 1 R / 2 . 0 I ) * ( C O S ( G M A I R / 2 . 0 ) • ( S 1 N ( G » A l R / 2 C . 01 * T A M G M A 1 R / 2 . 0 1 / 3 . 0 ) I YCS3=( ( 2 . 0 * R ) / ( 3 . 0 * G M A 1 R ) ) * ( C O S ( T H E T A R - G M U R I - C j S ( T h ^ T S R J I YODB=( (R*SIN(THETAP.-GMA1R) I - ( I R - H ( I I ) * C O S ( D H I ° R I 1 ) / 3 . 0 Y O E A = ( ( R * S ! N ( T H E T A R I I - ( ( R - H ( I I I *CCS(AL"HARI I 1/3 .0 YCD C = - < 2 . 0 * < R - H { I ) ) * C 0 S < G M A l R / 2 . 0 ) * C 0 S ( A L P H A R - I G " A i R / 2 . 0 l l l / 3 . 0 Y D E C = - ( R - H ( 1 1 ) * C C S ( A L P H A R - I G M A 1 R / 2 . 0 1 1 * ( C O S I G M A 1 R / 2 . 0 ! • < S INI CM AIR/ C 2 . 0 ) » T A N ( G M A l R / 2 . 0 ) / 2 . O ) A 0 A B = ( R * * 2 I * G M A 1 R / 2 . C A 0 0 9 = ( S O R T ( I R * * 2 ) - ( ( R - H ( I ) 1 * * 2 ) 11 * ( R - H I I I I 11.0 ACEA=ACC8 A O Q E = ( ( R - H ( ! l l * * 2 I * S I N I G M A 1 R / 2 . 0 ) * COS I G M A i R / 2 . 0 ) AC5C= ( ( (R-H(11 1 *S IN(GMA1R/2 .0 I ) * *2 L*TAN( G".AIR/2.01 A A B C = A C A B - ( ( ( R - H ( I ) ) * » 2 ) * T 4 N ( G M A 1 R / 2 . 0 1 ) XAB:=( (XOA3*AOABI»- (XCC8*AOCB)- (XOEA*AOFAI - (XOOS-A;OE) - (XOEC*AOEC)) C / A A B C Y A BC = ( (YOAB*AOAB 1 + I Y 0 C E * A 0 C 3 I - ( YOE A* AO' A 1 - 1 Y J O t * A ] 3 - : ) - ( Y 0 : C « A G ; C l l C/A13C r C CALCULATE CENTROID OF LOWER WECGE - XO1Q2C,YO102C r R A R : = S £ R T ( ( Y A B C * * 2 ) + ( X A B C * » 2 ) 1 A C l C 2 C = ( - 3 . 1 4 1 5 9 3 » ( 2 . 0 « A L P H A R | - G M A l R - A T A N ( Y A 3 C / X i 3 ; i I XC102C = P.A3C*COS(A01C2C ) YC102C=RA8C*SIN(A01C2CI C CALCULATE CISTANCF BETWEEN CENT^OiDS : \ ; (RADIANS i . S(CM) C ITS SLO = = - N ( D E G R - - : S ) . S = S Q R T ( ( ( X A 3 C - X S 1 Q 2 C ) * » 2 I * ( ( Y A S C - Y Q 1 0 2 C I NR=ATAN( (YQ132C-Y4BCI /<>C1C2C-XABC1) N = N R » 1 3 0 . 0 / 3 . 1 4 1 5 9 3 '21 I EVAL U A T E EQUATION FOR INTERSECTION OF TF G T T * C U R V E S . C C ! J N = < ( ( ( 9 0 C . * 9 B 0 . 6 I / ( 2 . 0 * ( 2 . 1 4 1 5 9 3 * * 2 ) * ( R D M * * 2 ) * R 4 S : ) ) * [ R A B C / $ ) * G CMAOR*(SIN ( N R ) - ( C C S ( N R I * T A N (PHI F .RI I I I - ( Y A B C / S ) l *GMA OR I F I J . E C . O I G O TO 20 I F I E C L N . G F . 1 . 0 I G C T Q 9 GC TO 12 20 I C I E C U N . G T . I . O J G G TO 10 IF( ECL'N. L T . 1.01GC TC 12 10 RFM=RPf*0 .10 GO TC 11 12 J = l opu=Rpv-o .001 GC TO 11 RPM'S AT WHICH THE CENTRIFUGING FORCE IS IX , i t , 1 0 * , 254 , 5 0 J . £ 75? OF Tt-F GRAVITY F O R C E . 9 F R = ( ( ( ( P P M « 3 . 1 4 l 5 9 3 ) / 3 0 . C I * * 2 ) * R A B C l / 9 3 0 . 6 K = l ° = 0 . C 1 F M 9 B C . 6 * 9 0 0 . 0 1 / ( ( R - ( C A V G / 2 . 0 » > * < 3 . 1 4 1 5 9 3 * « 2 ) 1 1 1 7 . 0 0 0 1 1 8 . 0 0 0 119.OOC 1 2 0 . 0 0 0 12 1.000 1 2 2 . 0 0 0 123 .OOC 1 2 4 . 0 0 0 125 .OOC 126 .000 1 2 7 . 0 0 0 128.OOC 129 .OOC 1 3 C . 0 C C 121 .OOC 132.OOC 133 .OCC 1 34 .000 135 .OCC 1 3 5 . 0 0 0 137.OOC 1 2 3 . 0 0 0 139.OOC 1 4 0 . 0 0 C 1 4 1 . J O O 1 4 2 . 0 0 0 14 3 .000 1 4 4 . 0 0 0 1 4 5 . 0 0 C 14 6 .000 14 7.OOC 1 4 8 . 0 0 0 14 9 .000 1 5 0 . 0 0 0 151.OOC 152 .OCC 1 5 3 . 0 0 0 154.OOC 1 5 5 . 0 0 0 156 .OCC 1 5 7 . 0 0 0 158 .OCC 1 5 9 . 0 0 0 1 6 0 . 0 0 3 16 I.OOC 16 2 .000 1 6 3 . 0 0 3 . 1 6 4 . 0 0 0 165 .OOC 1 6 6 . J O O 167.OOC 1 6 S . 0 0 C 1 6 9 . 0 0 0 1 7 C . C C C J 71.OOC 1 7 2 . 0 0 0 173.OOC 1 7 4 . 0 0 0 322 MICHIGAN TERMINAL SYSTEM FORTRAN G ( 2 l . 6 l MAIN 12-19 -80 1 0 : 3 2 : 5 7 0100 «=(-=( 9 8 0 . 6 * 9 0 0.0) tl (R -H( I l»( DAVG/2 .01 )*( 3.141593**21) 1 75.OOC 0101 33 R P M C S I K . I l = S O " T ( F«P» 176 . 000 0102 RPMHCStK, I )=SQRT(FH*PI 177 . 0 0 0 0103 I F U . E C . e l G O TO 22 . 178.OCC 0104 K = K*1 1 7 9 . 0 0 0 0105 I F I K . E C . 2 ) P = 0 . 0 5 1 8 C . 0 0 C O l O t ! F ( K . E 0 . 3 > ° = 0 . 1 0 . 181.OCC 0107 I E I K . E C . 4 )P = C.25 182 . 0 0 0 C108 ! C ( K . E C . 5 I P = C . 5 C 183.OCC 0109 I F ( K . E C.6 ) P = 0.75 18 4 .000 C i l O GC TC 23 185.OOC C 1 36 .0 00 C CASCACING C C A T A R A C T I N G . 187.OOC c 188.OOC C 139 .000 C RISC BEO INCLINATION FOR WHICH FORCE BALANCE ON 3EC IS IN c O'J ! LIS.R! UM 190.OOO C FOR SCM C GIVEN RPM. 1 9 1 . 0 0 0 C 19".OOC 0111 22 RPMCAT=RPM 1 9 ? . 0 0 0 0112 LA 1 = C 1 9 4 . J J j CI 13 LR = 0 195.OOC 0114 L = 0 1 9 6 . 03C 011= LA=0 197.OCC 0116 R H I O R = ( P H l O * 3 . 1 4 1 5 9 3 t / 1 6 C . C 1 9 8 . O J O 0117 8 1NCR = C .003141593/180.C 1 9 9 . 003 C U E S = T A M ° H ! 0 R t 2 0 C . 0 3C 0119 A1 = ARCCS( M - H ( I ) )/R ) 201 .000 OlcO aS = 0= ( ( R * * 2 l*< (2 . 0 * A 1 I - ( S I M 2 . 3 * A 1 I I I 1/2.0 202.OOC 0121 Rl= I 2 .0*1 R * « 3 ) *( (.S!N( A l ) ) * » 3 ) ) / ( 2 . O ' A B E O ) 2 0 2 . 0 0 0 0122 weE0=9EC.6 2 0 4 . 0 0 0 0123 31 C B - 0 = ( R l * l ( ( R P V C A T * 3 . 1 4 1 5 9 3 ) / 3 0 . 0 ) * * 2 ) ) 2 0 5 . 0 ) 0 0124 . 24 B1=(WEED*(C0S(BINCR) ) *C3ED) 2 0 6 . 0 0 0 0125 B2=WBEC*S!N(3INCR) 207.OOC 0126 B3=B2/E1 2 0 S . O 0 C 0127 34=B/B2 209 .OOC 012 = I F ( L . E C . O ) 0 0 TO 23 ' 2 1 0 . 0 0 0 0129 I E ( e 4 . C E . 1 . 0 ) G C TO 29 2 1 1 . 0 3 : 0120 GC TO 23 2 1 2 . 0 0 0 • 0 1 0 . 22 I F ( 3 4 . L E.1 . 0 1 G O TO 2 6 21 3 .000 Oi32 eiNCR = 2 I N C R M 3 . 1 4 1 5 9 3 / i e C . C ) 214 .0 O C ( J 1 J 3 CC TO 24 21 5 .000 0134 28 L= l 216.OOC 0135 B!NCR=8 INCR- (0 .3141593/180.CI 2 1 7 . 0 0 0 01 36 C-C TO 24 2 1 3 . 0 0 0 C 2 1 9 . 0 0 0 C RI NO R°M FOR THE ROLLING/CASCADING BOUNDARY• 2 2 0 . 0 0 0 C 221.OCC 0137 2 9 ' T 1 R = B I N C R - 0 E L T A R ( ! I 2 2 2 . 0 0 0 0125 1 F < T 1 R . G T . 0 . 0 . A N D . R P ' < C A T . £ C . R P M I G D TO 30 222.OOO 0139 ' . " M L A . E C . l l G Q TO 30 2 2 4 . 0 0 0 0140 I F ( L R . G T . 0 ) G 0 TO 46 2 2 3 . O J C 0141 !F( T I R . L T . O . O I G O T 0 32 2 2 * . n 3 0142 46 I F ( T 1 C . L T . 0 . 0 ) G O T 0 30 22 7.OJO 0142 LR=1 226.OOC 0144 P.Ry.CAT=RRMCAT-0.1. 2 2 9 . 3 3 0 0145 GO TC 21 22C.OOC 0146 32 RPMCAT=RPMCAT*5.C 221 .000 0147 LR=0 232 .000 323 M I C H I G A N TERMINAL SYSTEM FORTRAN G ( 2 1 . 8 > MAIN 1 2 - 1 9 - 8 0 0148 0149 0150 0151 0152 0153 0154 0155 0156 0157 0158 0159 0160 0 161 0152 0163 0164 0165 0166 0167 01c8 0169 0170 0171 0172 0173 017-01 75 0176 0177 0 1 7 o 0 1 7 9 C 1 3 0 0 1 8 1 L = 0 GO TO 31 p r . J ^ . r r ? - ° R T H E C * S C * C I N G / C A T A R A C T I N G B O U N D A R Y C F O R F U ' L L L o T B K t C. T I N . 3C !F (LA .EQ.0 IRPM3=RRMCAT I F ( T 1 » . L T . 0 . 0 ) T 1 R = ( B 1 N C R * < 0 . 3141593 /1 80 .0 ) I -DEL TAR 1 I I I F I T l R . L T . O . 0 1 T 1 P = T 1 R * ( 0 . 3 1 4 1 5 9 3 / 1 3 0 . 0 ) LA=1 XA=R*CCS(T1R| Y A = R * S I N ( T \ q ( X C 1 = R * C O S ( T 1 R - ( 2 . 0 * « 1 I I YC1=R* S I N ( T 1 R - ( 2 . 0 « A I I ) X P = ( ( R « C 0 S < T 1 R ) ) - ( ( ( 2 . C » S I N ( T 1 R ) * ( ( R P M C A T « R * 3 . 1 4 1 5 9 3 / 3 0 . 0 ) « * 2 ) I / 9 3 C O . 6 ) * < C 0 S I T 1 R I • ( S O R T ( ( ( 9 e 0 . 6 « S I N ( T I R I * 9 0 0 . 0 ) / ( ( < R P ' C A ^ « 3 . 141593> * * C2 )*R I )-( I S ! N ( T 1 R H * » 2 ) ) ) ) I 1 Y F = ( ( R * S I N ( T l R I I - ( ( ( X P ) / ( T A N ( T l R ) l ) f ( ( 9 3 0 . 6 * ( X O * * 2 ) * 9 3 0 . 0 l / ( 2 . C * ( ( 0 R P M C A T * R * 3 . 1 4 1 5 9 2 ) « * 2 ) * ( ( S ! M T i R ) ) * * 2 l ) l ) l IF< LAA . E C O IGO TO 4e 49 I F ( X C 1 . G T . X P I G O T3 47 L» = 0 L = C op:' ,:AT=RPMCAT+0.1 GC TO 21 48 XF=(X0I + X A | / 2 . 0 IF( XF . C T . X P )RPMC = RPMCA T !F ( XF.C-T . XO) LAA = 1 GC TO 49 C C CENTRIFUGING. C r 47 R P M C a = ( ( 3 0 . 0 / 3 . 1 4 1 5 9 3 ) * ( S O R T l ( 9 8 0 . 5 ) / ( 3 - [ D A V G / 2 . 0 ) ) ) ) l R P M C w = I ( 3 C . C / 3 . 1 ' i l 5 9 3 l * ( S 0 R T l ( 9 8 0 . 6 ) / ( R - H ( T ) t ( C A V G / 2 . 3 ) ) I I ) r C O L ' T P U T . C PPMK ( I l = RPM W I TE ( 6 , 1 3 ) H ( ! I ,OFC (I ) , R P M , RPMB,R°MC,RPMCAT, RPMCW, R = MC C? 12 F C R M A T { 1 • , 1 0 X , F 5 . i , 5 X , F 5 . 1 . 5 X , F 6 . 3 , 2 X , F 7 . 2 , 2 X , F C 7 . 2 ,1X , F 7 . 2 ,1X , F 7. 2 , 1 X , >= 7. 2 ) C PROCEED TO THE NEXT BED DEPTH (FOP. A MAXIMUM OF 25 1. C !F ( I . FC . .25 IG0 TO 50 1 = 1*1 GO TC 7 r C S L I P P I N G . C c 5C WRITE16.55I 55 P O R M A T ( ' 1 • , ' S L I P P I N G B O U N C A P Y : ' ) w P I T r ( 6 , 5 6 1 56 FORMAT( ' ' . ' B E D DEPTH ( C M ) • , 5 X , ' B O U N D . IN SLUMP. Z O N E ' , 5 C X , ' I N I T I A L BOUND. IN " C L / C A S C Z O N E « , 5 X , ' O U T E R B O U N D . • ! 1 0 : 3 2 : 5 7 232 .OOC 2 3 4 . O C C 235 .OOC 2 3 6 . 0 0 0 236 .OOC 237 .OCC 238 .OOC 2 3 9 . 0 0 0 240.OOC 2 4 1 . 0 0 0 242 .OOC 2 4 2 . 0 30 244 .OOC 2 4 5 . 3 0 0 2 4 6 . 0 0 0 247 .OOC 2 4 8 . 0 0 0 2 4 9 . J O C OOC oc: o o o ooo occ o o c 250 2 5 ! 75 2 253 254 255 256.oo: 2 5 7 .003 2 5 6 . 0 C : 259 .OCC 2 6 3 . 0 0 0 26 1 .OOC 2 6 2 . 0 0 0 262.OOC 264 .OOC 2 6 5 .000 2 6 6 . 0 0 C 2 6 7 . 0 0 0 268.OOC 269.OOC 2 7 C . 0 C C 271.OOC 2 7 2 . J O C 273 .OOC 2 7 4 . 0 0 0 275.OOC 2 7 6 . 0 0 0 277 .OOC 2 7 8. 0 0 C 2 7 9 . 0 0 0 2 8 C . 0 C C 2 8 1 . 0 0 0 2 8 2 . 0 0 0 283 284 285 .OCC OJC o c : 2 3 6 . 0 0 0 287.OOC ? fl 9 . 0 0 C 2 8 9 . 0 0 C 324 MICHIGAN TERMINAL S Y S T i : M FORTRAN GI 21.81 MAIN 1 2 - 1 9 - 8 0 1 0 : 3 2 : 5 7 C i 3 2 0135 0184 Oi a5 O l E o C! E7 01 CE 0189 C19C 0191 0 i 9 2 0 I -> 3 019 4 019 5 019 s 015 7 01 ^ 3 * 2 T T » ; P T r = :> T A T » i T . . T I \ WRITE16.57I 57 FORMAT!• ' ^ S X . ' I R E V / K I M ' ^ I X . ' I R E V / M I N I ' . I B X . M R E V / M I N I ' I CC 51 1=1,25 CALL S L I P l P H I S . w e F D , R , R P V C 9 , P H I R , G M A 0 1 , G M A 0 2 , C l , R P M K U I , M ( t C 1, 3 ) 51 C C N T I N L E C FINAL OUTPUT. 14 V , R I T E ( £ , 2 5 I 25 F O R M A T C l ' , ' R P M S AT WHICH THE C ENTRIFUGING FORCE IS l l , 5%, 103, 2 C 5 * . 5CJ r, 75? CP THE GRAVITY F O R C E . • » KP. ! T F ( 6 , 3 4 ) FOR-MAT I • - ' , 12X , • IS' , 1 3 X , '5%' , 14X, • IOS 1 , 13X, • 2 5 ? ' , 13X, ' 501 ' , 13X, • 75 C % • ) WR. 'TE( t , 35 l F 0 R M A T { ' • , 6 X , ' W A L L , , 5 X , ' B E D ' , 5 X , ' W A L L , , 4 X , ' B E D ' , 5 X , ' W A L L ' , 4 X , ' B E D C , 5 X , ' W A L L ' , 4 X , , B C D ' , 5 X , ' W A L L ' . 4 X , ' B E D ' , 5 X , ' W A L L ' . 4 X , ' E - D ' I wR I TH ( t, ib) FORMAT 1*0 ' ) W R ! T r ( f c , 3 7 M ( R P M C S l K , ! ) , R P " H C S < K . , I ) , K = l , 6 l , I = l ,25> FORMAT! ' • , 5 X , F 7 . 2 , 1 X , F 7 . 2 , 1 X , F 7 . 2 , 1 X , F 7 . 2 , 1 X , F 7 . 2 , 1 X , C 7 . 2 , 1 X , F 7 . 2 C , 1 X , F 7 . 2 , 1 X , F 7 . 2 , 1 X , C 7 . 2 , 1 X , F 7 . 2 . 1 X , F 7 . 2 I $ T 0 ° E NO c R F EC T ' I C E 3 C 0 I C , SC URGE , NOLI S T . N O D E C K , LOAD, NOMA R EFFECT* N«"F = VAIN , LINECNT = 60 SOURCE STATEMENTS = 198,PROGRAM S IZE = 14870 » NC DIAGNOSTICS GEN'ERATEC " 34 36 37 29C.O0O 2 9 1 . 0 0 0 2 9 2 . 0 0 0 29 3,OOC 2 9 4 . 0 0 0 295 .OOC 296 .OOC 2 9 7 . 0 0 0 296 .OCC 299.OOC 30C.OOC 2 0 1 . 0 0 0 3 C 2 . 0 0 C 3 0 3 . O C C 3 0 4 . O J C 305.OOC 306 .000 30 7.OOC 308 .0 ;C 309.OOC 310. COC 3 1 1 . 0 0 0 312.OOC 3 1 3 . 0 0 0 3 14 . 3 0 C o o o o o o o o o o o o o o o o J>J>.r-.fJ>J>.t-J> OP -J CT U l J> VU lO *— o o o o o o o o o o o o o o o o o o M * ' Ul l>i U i W W Ul U ' O vO OD (7- vn 4" Ui ro O O O O O O O O O o O O O O ' O O o O O O O O O O O O O O i U» ^ M V J f\) W f\> ( V f\J W l\) IV r- I- i C D re ,x ro n r T J J> UJ tNJ r— *D - * Ii X) it II II II , i i 2 a CO LD J- — O O (1 ro ro IT ll T> II CO -s. T I CD — II » I . I to n .T l T J o -n r— • U I— • I to # # O — — —> o 2 n - i» — o — i-* lU l / l ^ •—• TO 2 ro .1 m o —• n >o O 2 t W - I I U I N JO • f—• — r— ro ~ J> o + k-» • O U l O ro sfl m UJ o — — "--UJ D _ J > C O II H T t— » *o re ( ; X J —• to T) II f— *: »— O l~ J O — • • t i i / i O i n II z *- ~ r-f~ O 1> m < ^ • w -n o I— — • 73 • J> + O CO «**• • n r f i O .o » • —• o « i n (/i t/» cn i/i u i rt > jo l ^ w M i - n u i n / ) -o - ii ii II II cu II - i r t H 1/1 - H 1/1 i r w • —• o n x - n ^ ^ m * - ' ] i / i o i > B T j II J> II i— J > o M T> IT1 O .Tl T7 </> —« r~ t— o o .w — • O — rn UJ • —1 • o J> 1— J > Ul o N m O ll 3 * l r> "J — * - I J O TJ ; » o II • w o o o — m —• * • — v i i i (/» -O C T r- * O — — O tt Irt CD —. — o Lo cn O * U J U J I-* ( i 2 a B ! - ' » - • to — * X r> cn r\j ft .o f— —• u> * - CD S — rn UJ — O O — 7" O ^ lrt ./I X i o o o o o o o o i o o o O O o o X i o o o o o o o -j a- u i J > Ikl l\> *— J > e — i pa r » o o i> 1 CO i — 1 r-1 — irt 1 T J -< r* Tt > i> •—> O <n 1 -o CO cn *— C B M II - H C 1 -4 -4 II II .11 II i— j : J I 1 3 ? m o — o J> n >D 1 o rs ro II "O S- o 1 • — O (/> <— t (rt T I o — n r-. -4 1 1— o # JO on O - « 1 CD — * — y T*. 1 t) H 7> W — . n 1 o JO x> i cr * — 1 CO <n 1 -* UJ # X r- r- 1 — D •—« 1 z O # —* -v -o 1 -Tl — 1NJ J O —> — . 1 • —> • — 1— T) cn o O X • —• * o •-- Co o cn > » ar i— i CD — cn O * —< # JO UJ — — ro 71 — * " O — » o UJ x> X o «— T3 •» —• X > j> — CO — J 3 : n CJ IvJ O — • o U » U J I O ' J J • * J - J o i— o t*J U J Ul Ul t>l o- o o cr (7-no r> J> UJ UJ UJ UJ ^ t> o -> UJ O UJ UJ '»> UJ n i o i Ui \Ji Oi V/l VJl v/1 CD -4 T' Ul .|> U' UJ Ul 'kl UJ 'kl UJ UJ »** VJiUlUlUiJ>J> JJ>J>J>4" • u> UJ UJ •>«> UJ UJ UJ U) U I U J U J U J U J U J Ul " « J U> •*> UJ U J U J Ul Ui U J U J • J> u 1 U J i ki u i u i u> uJ u> •>» u> f\i M fvj fvj I\J f\j ro IN* rvj ro H * f- -' O J i ro ^ i (> »n J> >u i\j M n •/) m ^ n> u i 1 M J N I r- o m <> ui O O C o . O n o u i O c"> O ( > I • O O O LJ f i O O O O CJ I C l (J O i l o I O CJ O O ft O O O O O O ; j O C C_- <-) LJ CJ 1*1 o o o i:> i J c i r j o r i o o O r i o O O (.3 O O O CJ O < I O O O O O O O O O O O O O O O O O O O O • I > t J O » i t J O C O O O O O O O O O O C J o i o r i CJ r> o n o o f J O ci o o o o n o o n o CO ho 326 MICHIGAN TERMINAL SYSTEM FCRTRSN G I 2 1 . 8 I S L ! P 1 2 - 1 9 - 3 0 10: 32:59 0049 I F I L . E C . C I G O TO 23 372 .000 0C5C I F I B 4 . C E . 1 . C I G O TO 29 374.OCO 0051 GO TO 23 375 .OOC 0052 23 I F I 6 4 . L E . 1 . C I G 0 TO 2 3 376.OOC 0053 8INCR = 5 ! N C R * 1 3 . l 4 l 5 9 3 / l S C C ) 377.OOC 0054 GC TC 24 378.OOC CC55 28 L = l 379 .OCC 0056 3 I N C F = B ! N C R - < 0 . 3 1 4 1 5 9 3 / 1 8 0 . 0 ) 380 .OOC 0C57 GO TC 24 38 1.OOC 0056 29 IFI B I N'C° . L T . A S L P R I I ) IGC TO 50 382.OOC OC59 !F(0PKC.EC.RPM|RPMT2=FPVC 383.OCC OOcO I F | ( R F V C - R R M C W ) , G E . 0 . l ) R P M T 3 = R P M C 284.TOO 0061 ! c l ( R P K - R F M C W ) . G E . O . D G C TO 52 3 6 5 . 0 0 C C0 t2 I = I » 1 3 8 6 .000 006 2 ROKC=PF"C+O.l 3 8 7 . 0 0 0 0C64 L = 0 388.OOC 0065 GO TO 21 3 8 9 .000 OCoo 50 I F I R p M C E O . R P M I G C TO 51 3 9 C . 0 0 C 006 7 2 9 1 . 0 0 0 COoS GO TC 52 3 9 2 .0 0 C 0C69 51 RF-MT2 = C. 0 39 3.00 C 0070 RPM T 2=C.C 2 9 4 . 0 0 0 007 1 52 K = I T E ( 6 , 5 4 ) E , R P M T 1 , R P M T 2 . F R « ' T 3 395.OOC 007 2 54 SC=MST(> • , 4 X , F 5 . i , i e x I = 5 . 1 f 2 3 X , F 5 . 1 , 1 9 X , = 5 . 1 ) 3 9 6 . 0 0 0 007 3 RCTiJSN 397.OOC 00 7 4 END 39R.000 » j ? 7 IC:»S IM = FFE C T * I C , E 3 C D ! C , S C U R C E , N 0 L I 5 T , N 0 n E C K , L 0 A 0 , N D ' M = > »uPT I 0 'JS IN - E F ~ C T * NAVE = SLIP , LINECNT = 60 • S T A T I S T I CS * SOURCE STATEMENTS = 74.PROGRAM S U E * 6216 • S T A T I S T I C S - N: DIAGNOSTICS GENERATSC ERR'J-5 IK SI IP FLA",GE C IN ThE AECV= C O M " I L A T I C N S . AM£ N U M 3 - 3 OF ERRORS/WARNINGS SEVERITY A I N . 0 C LIP 0 C xfccuT : a r c R M T N A T r o 10:22:59 T« 1.203 RC = C ».7i 

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