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Gas residence time distribution and related flow patterns in spouted beds Lim, Choon Jim 1975

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GAS RESIDENCE TIME DISTRIBUTION AND RELATED FLOW PATTERNS IN SPOUTED BEDS by CHOON JIM LIM B . S c . , Nanyang U n i v e r s i t y , 1 9 6 8 M . A . S c . , 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 , 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the D e p a r t m e n t o f CHEMICAL ENGINEERING We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA October, 1975 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the 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 ag ree t h a t the L i b r a r y s h a l l make i t f r e e l y 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 ag ree 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 pu rposes may be g r a n t e d by the Head o f my D e p a r t -ment o r by h i s 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 p u b l i c a t i o n , i n p a r t o r i n w h o l e , or the c o p y i n g 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 not 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 . Depar tment o f C h e m i c a l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Co lumb ia Vancouve r 8 , Canada ABSTRACT A t w o - r e g i o n model o f a spou ted bed wh ich t a k e s i n t o a c c o u n t the a c t u a l pa th f o l l o w e d by the gas i n the a n n u l u s , has been d e v e l o p e d to p r e d i c t the r e s i d e n c e t ime d i s t r i b u t i o n o f gas i n the b e d . The model i s based on the a s s u m p t i o n s o f p l u g f l o w o f gas i n the spou t and d i s p e r s e d p l u g f l o w a l o n g the f l o w pa th i n the a n n u l u s . The f l o w path i n the a n n u l u s i s d e s c r i b e d by p o s t u l a t i n g t h a t a l l the gas e n t e r i n g the a n n u l u s f rom the spou t a t a g i v e n l e v e l t r a v e l s r a d i a l l y and v e r t i c a l l y a l o n g a p a r t i c u l a r f l o w path w i t h o u t r a d i a l d i s p e r s i o n or m i x i n g . T h i s p i c t u r e i s c o n s i s t e n t w i t h v i s u a l o b s e r v a t i o n s made u s i n g N 0 2 gas as t r a c e r . The hydrodynamic d a t a needed as i n p u t to the model a re gas v e l o c i t i e s i n spou t and a n n u l u s , spou t shape and spou t v o i d a g e . The r e s i d e n c e t ime d i s t r i b u t i o n o f gas t o g e t h e r w i t h the a b o v e - m e n t i o n e d hydrodynamic f e a t u r e s were measured e x p e r i -m e n t a l l y f o r a w ide range o f s p o u t i n g c o n d i t i o n s . The RTD d a t a were o b t a i n e d f rom s t i m u l u s - r e s p o n s e e x p e r i m e n t s u s i n g h e l i u m gas i n j e c t e d as a n e g a t i v e s t e p i n t o the s p o u t i n g gas downstream o f the gas i n l e t . The gas v e l o c i t y i n the spou t was d e t e r m i n e d by p i t o t t u b e , and i n the a n n u l u s by s t a t i c i i p r e s s u r e measu remen ts . H igh speed c i n e - p h o t o g r a p h y was employed to measure spou t p a r t i c l e v e l o c i t i e s ( i n h a l f - s e c t i o n a l beds) and spou t v o i d a g e d i s t r i b u t i o n was d e t e r m i n e d f rom spou t and a n n u l u s p a r t i c l e v e l o c i t i e s by s o l i d s mass b a l a n c e . The v a l u e s o f the a x i a l d i s p e r s i o n c o e f f i c i e n t f o r the a n n u l u s gas wh ich i s an a d j u s t a b l e pa rame te r o f the m o d e l , were e s t i m a t e d by compar ing p r e d i c t e d and e x p e r i m e n t a l RTD c u r v e s . The c o e f f i c i e n t s f o r spou ted beds were found to be g e n e r a l l y h i g h e r than t hose r e p o r t e d f o r packed b e d s , bu t a t l e a s t an o r d e r o f magn i tude s m a l l e r than t h o s e f o r f l u i d i z e d b e d s . The hydrodynamic d a t a o b t a i n e d were a n a l y z e d to t e s t p u b l i s h e d t h e o r i e s and c o r r e l a t i o n s and to improve upon t h e s e whereve r p o s s i b l e . The M a m u r o - H a t t o r i e q u a t i o n was found to g i v e good p r e d i c t i o n o f a n n u l u s l o n g i t u d i n a l gas v e l o c i t y p r o f i l e s f o r 15 .2 cm d i a m e t e r beds but to u n d e r - e s t i m a t e v e l o c i t i e s f o r l a r g e r c o l u m n s . The e q u a t i o n o f Yokogawa et al. p roved to be u n s a t i s f a c t o r y f o r p r e d i c t i v e pu rposes and was m o d i f i e d . The m o d i f i e d v e r s i o n can p r e d i c t the gas v e l o c i t y p r o f i l e i n the a n n u l u s c o r r e c t l y p r o v i d e d t h a t one such p r o f i l e f o r the p a r t i c u l a r s o l i d m a t e r i a l i s known. The d a t a f rom the p r e s e n t s tudy showed good agreement w i t h the e q u a t i o n o f G r b a v c i c et al. f o r gas v e l o c i t y a t the top o f the a n n u l u s . i i i A s i m p l e model was f o r m u l a t e d , based on the o b s e r v e d s o l i d s f l o w p a t t e r n in the a n n u l u s , wh ich e n a b l e s the c a l c u l a t i o n o f s o l i d s f l o w path and r e t e n t i o n t ime i n the a n n u l u s f rom ave rage p a r t i c l e v e l o c i t y d a t a . For p a r t i c l e v e l o c i t y i n the s p o u t , the f o r c e b a l a n c e model o f T h o r l e y et at. as amended by Mathur and E p s t e i n was f u r t h e r improved by i n t r o d u c i n g the t h e o r e t i c a l r e l a t i o n s h i p between spou t v o i d a g e and number o f p a r t i c l e s i n the s p o u t . The r e s u l t i n g e q u a t i o n was found to g i v e good agreement w i t h e x p e r i m e n t a l v a l u e s o f no t o n l y spou t p a r t i c l e v e l o c i t y but a l s o o f spou t v o i d a g e . TABLE OF CONTENTS Page ABSTRACT i i L IST OF TABLES . . . v i i i L IST OF FIGURES i x ACKNOWLEDGMENTS x v i C h a p t e r 1 INTRODUCTION 1 2 LITERATURE REVIEW 5 2.1 Flow P a t t e r n o f Gas 5 2 .2 Flow P a t t e r n o f S o l i d s . 23 2 .3 Bed S t r u c t u r e 40 2 .4 C o n c l u s i o n s 47 3 PROGRAMME OF PRESENT INVESTIGATION 49 4 APPARATUS AND MATERIALS 52 4.1 A p p a r a t u s 52 4 . 2 S o l i d s P r o p e r t i e s 62 5 MODELLING OF GAS FLOW IN SPOUTED BEDS 68 V 5.1 V i s u a l O b s e r v a t i o n 68 5.2 M a t h e m a t i c a l Model . . . ' 70 v C h a p t e r Page • 5 . 3 Measurement o f R e s i d e n c e Time D i s t r i b u t i o n o f Gas 8 2 5 .3 .1 E x p e r i m e n t a l P r o c e d u r e . . . . 82 5 . 3 . 2 A c c u r a c y o f Data 82 5 . 3 . 3 R e s u l t s 86 5 . 3 . 4 P r o c e s s i n g o f Data 86 5.4 Compar i son Between P r e d i c t e d and E x p e r i m e n t a l RTD Curves 91 6 GAS DISTRIBUTION BETWEEN SPOUT AND ANNULUS . . . 100 6.1 Measurement T e c h n i q u e s . . . . 100 6 .2 R e s u l t s and D i s c u s s i o n . • 103 6 .2 .1 In the Spout 103 6 . 2 . 2 In the Annu lus 108 7 FLOW PATTERN OF SOLIDS 124 7.1 In the Annul us . . . 124 7 .1 .1 E x p e r i m e n t a l R e s u l t s and D i s c u s s i o n 124 7 . 1 . 2 A Model f o r S o l i d s Move-ment i n the A n n u l u s 136 7 .2 In t he Spout 140 7 .2 .1 E x p e r i m e n t a l R e s u l t s and Di s c u s s i o n 1 40 7 . 2 . 2 A Model f o r S o l i d s Move-ment i n the Spout 150 8 BED STRUCTURE 166 8.1 Spout S h a p e . . . 166 8 .2 Spout Vo idage D i s t r i b u t i o n 172 v i C h a p t e r Page 8 . 2 . 1 E x p e r i m e n t a l R e s u l t s and D i s c u s s i o n 172 8 . 2 . 2 A Model f o r Spout V o i d a g e D i s t r i b u t i o n 179 9 CONCLUSIONS 183 10 SUGGESTIONS FOR FURTHER WORK . . 187 NOTATION 188 LITERATURE CITED 196 -APPENDICES I E x p e r i m e n t a l C o n d i t i o n s 201 I I C a l i b r a t i o n Cu rves f o r P r e s s u r e T r a n s d u c e r s . . . 203 I I I P r e s s u r e Drop-Gas V e l o c i t y E q u a t i o n s f o r L o o s e - P a c k e d Beds o f the S o l i d M a t e r i a l s Used i n RTD E x p e r i m e n t s 206 IV N u m e r i c a l Method f o r S o l v i n g the P a r t i a l D i f f e r e n t i a l E q u a t i o n f o r R e s i d e n c e Time D i s t r i b u t i o n o f Gas i n the A n n u l u s 207 V Computer Program f o r E s t i m a t i n g the Model P a r a m e t e r D 215 VI E x p e r i m e n t a l and P r e d i c t e d RTD C u r v e s 229 V I I Gas F low Data 235 V I I I S o l i d s F low Data 241 v i i LIST OF TABLES T a b l e Page 3.1 Ranges o f V a r i a b l e s S t u d i e d 50 4.1 Key to F i g u r e 4 . 4 58 4 . 2 P r o p e r t i e s o f S o l i d M a t e r i a l s Used 64 5.1 V a l u e s o f the D i s p e r s i o n C o e f f i c i e n t , D and C o r r e s p o n d i n g E x p e r i m e n t a l C o n d i t i o n s 96 6.1 Data f o r Gas T r a n s p o r t R a t i o and I n t r o d u c e d Momentum 107 6 .2 V a l u e s o f H and U u 114 m aH m 7.1 S o l i d s F low i n Annu lus 131 7.2 Compar i son Between P r e d i c t e d and Observed V a l u e s o f C r o s s - F l o w Ra tes i n the Upper P a r t o f the Bed 134 8.1 E x p e r i m e n t a l Spout V o i d a g e D i s t r i b u t i o n 174 8 . 2 Compar i son o f P r e d i c t e d and E x p e r i m e n t a l S p o u t V o i d a g e s 182 v i i i LIST OF FIGURES F i g u r e Page 1.1 Schema t i c d i ag ram of a spou ted bed 2 2.1 A i r d i s t r i b u t i o n i n spou ted wheat beds [ 2 ] , D c / D 0 = 6 7 2 .2 L o n g i t u d i n a l p r o f i l e s o f gas v e l o c i t y a t spou t a x i s [ 1 7 ] . D 0 = 2 cm, e x c e p t f o r wheat D 0 = 4 cm . . . 9 2 .3 R a d i a l p r o f i l e s o f upward gas v e l o c i t y i n the spout o f wheat bed [ 1 7 ] . D = 2 1 . 5 cm, D 0 - 4 cm, H = 80 cm, u"s = 1.3 S i /sec 11 2 .4 S k e t c h o f gas v e l o c i t y p r o f i l e s i n spou t [17 ] . . . 11 2 . 5 Mamuro and H a t t o r i ' s model f o r f o r c e s a c t i n g i n the annu lus [16 ] 14 2 .6 Yokogawa 's model f o r f o r c e s a c t i n g i n the a n n u l u s [17 ] 14 2 . 7 ' V e r t i c a l p r o f i l e s o f gas v e l o c i t y [ 1 , 1 1 , 1 2 , 2 1 , 2 4 ] . Compar i son o f e x p e r i m e n t a l r e s u l t s w i t h E q s . ( 2 . 1 3 ) and ( 2 . 3 6 ) 22 2 .8 L o n g i t u d i n a l p r o f i l e s o f p a r t i c l e v e l o c i t y a l o n g the spou t a x i s o f wheat beds [10 ] 25 2 .9 R a d i a l p r o f i l e s o f upward p a r t i c l e v e l o c i t y . . . . 26 2 .10 L o n g i t u d i n a l p r o f i l e s o f r a d i a l a v e r a g e upward p a r t i c l e v e l o c i t y i n the spou t o f g l a s s beads bed [ 3 0 ] . D = 7 .5 cm, d = 1.05 mm 28 c P i x F i gure Page 2.11 L o n g i t u d i n a l v e l o c i t y p r o f i l e s f o r i n d i v i d u a l p a r t i c l e s e n t e r i n g i n t o the spou t f rom the a n n u l u s a t bed l e v e l Z [ 1 7 ] . Wheat b e d ; D = 2 1 . 5 cm, D 0 = 4 cm, H - 80 cm, U g = 1.15 c m/sec 29 2 .12 S o l i d s f l o w i n a n n u l u s [ 2 , 1 1 ] . Wheat ; D c / D 0 = 6 . 0 , 9 = 6 0 ° , U /LI = 1 . 1 , H/D„ = 3 31 s ms c 2 .13 L o n g i t u d i n a l p r o f i l e s o f r a d i a l a v e r a g e p a r t i c l e v e l o c i t y i n the s p o u t . Wheat b e d ; D c = 15 .2 cm, H = 6 3 . 5 cm. by the M a t h u r - E p s t e i n model [ 1 0 ] , by the L e f r o y - D a v i d s o n model [24 ] and o e x p e r i m e n t a l d a t a 34 2 .14 L e f r o y and D a v i d s o n ' s c o l l i s i o n model [ 2 4 ] : the sequence o f c o l l i s i o n A , B and C l e a d i n g to the e n t r a i n m e n t o f p a r t i c l e No. 2 38 2 .15 Obse rved spou t shapes [10 ] 41 2 .16 F o r c e b a l a n c e models c o n c e r n i n g spou t s h a p e , (a) L e f r o y and D a v i d s o n [ 2 4 ] ; (b) B r i d g w a t e r and Mathur [37 ] 43 4.1 D iagram o f 15 .2 cm d i a m e t e r co lumn and gas i n l e t n o z z l e 53 4 . 2 P h o t o g r a p h s o f 24.1 cm and 2 9 . 2 cm co lumns . . . . . 54 4 . 3 P h o t o g r a p h s o f h a l f - c y l i n d r i c a l co lumns 56 4 .4 S c h e m a t i c f l o w d iag ram o f the a p p a r a t u s 57 4 . 5 S c h e m a t i c d iag ram of p i t o t tube 6 0 4 .6 S c h e m a t i c d i ag ram of s t a t i c p r e s s u r e p robe 61 5.1 S t r e a m l i n e s o f N 0 2 t r a c e r i n a h a l f c y l i n d r i c a l spou ted b e d . S o l i d m a t e r i a l ; Ot tawa s a n d , -20 + 30 mesh 69 x F i gure Page 5.2 B a c k m i x i n g o f N 0 2 t r a c e r i n the c o n i c a l p a r t o f a spou ted bed of p o l y s t y r e n e p e l l e t s : U s / U m s = 1 - 3 ' H = 4 5 - 7 c m 7 1 5 .3 C o - o r d i n a t e sys tem and g r i d p o i n t s f o r gas s t r e a m l i n e c a l c u l a t i o n s 73 5.4 C a l c u l a t e d s t r e a m l i n e s f o r a 24.1 cm d i a x 72 cm deep bed o f p o l y s t y r e n e p e l l e t s : D 0 / D c = 0 . 1 2 , U s / U m s = 1.1 74 5 .5 T r a c e r mass b a l a n c e i n the a n n u l u s 76 5.6 T r a c e r mass b a l a n c e i n the spou t 80 5 .7 Compar i son o f the r e s p o n s e s o f the two the rma l c o n d u c t i v i t y c e l l s used i n s t i m u l u s -r e s p o n s e e x p e r i m e n t s ; s c a l e 0 .2 s e c / s q u a r e . . . . 84 5 .8 R e p r o d u c i b i l i t y o f t r a c e r c o n c e n t r a t i o n c u r v e s : s c a l e 0 .2 s e c / s q u a r e , wheat b e d s , D c = 24.1 cm . . . 85 5 .9 T r a c e r c o n c e n t r a t i o n c u r v e s f o r bed o f p o l y s t y r e n e p e l l e t s . D = 24.1 cm, H = 72 cm. S c a l e 0 .2 s e c / s q u a r e 87 5 .10 T r a c e r c o n c e n t r a t i o n c u r v e s f o r bed o f p o l y s t y r e n e p e l l e t s , D = 2 9 . 2 cm, H = 91 .4 cm. S c a l e 0 .2 s e c / s q u a r e 89 5.11 N o r m a l i z e d RTD c u r v e s c o r r e s p o n d i n g to the r e s p o n s e c u r v e s o f F i g u r e s 5 . 9 , f o r a p o i n t h a l f - w a y a c r o s s the a n n u l u s s u r f a c e 92 5 .12 V a r i a t i o n of A 2 w i t h D c a l c u l a t e d by E q . ( 5 . 1 7 ) f o r the da ta o f F i g u r e 5 .12 93 5 .13 N o r m a l i z e d RTD c u r v e s c o r r e s p o n d i n g to the r e s p o n s e c u r v e s o f F i g u r e 5 .9 f o r d i f f e r e n t r a d i a l p o s i t i o n s a c r o s s the a n n u l u s s u r f a c e 95 x i 5.14 Compar i son o f a x i a l d i s p e r s i o n c o e f f i c i e n t r e s u l t s i n packed [ 5 0 ] , f l u i d i z e d [ 5 5 , 5 6 ] and spou ted beds 98 6.1 R a d i a l p r o f i l e s o f upward a i r v e l o c i t y i n the s p o u t , H / D c = 3 , U s / U m s = 1.1 . 1 0 4 6 .2 R e l a t i o n s h i p between gas t r a n s p o r t r a t i o , TR, and i n t r o d u c e d momentum, MI, i n 15 .2 cm d i a column 106 6 . 3 E f f e c t o f bed dep th on a i r v e l o c i t y i n t he a n n u l u s . D c = 15 .2 cm, D„ = 1.95 cm 109 6 .4 E f f e c t o f s p o u t i n g a i r v e l o c i t y on a i r v e l c o i t y i n the a n n u l u s . D = 15 .2 cm, D 0 = 1 .95 cm c. 110 6 .5 E f f e c t o f a i r i n l e t d i a m e t e r on a i r v e l o c i t y i n t he a n n u l u s . D c = 1 5 . 2 cm, wheat beds H = 6 3 . 5 cm, p o l y s t y r e n e b e d s , H = 45 .7 cm I l l 6 . 6 R e l a t i o n s h i p between u a ^ / u a ^ a n d H / / H m ' m Compar i son o f e x p e r i m e n t a l r e s u l t s w i t h E q . ( 6 . 1 2 ) . 115 6 .7 R e l a t i o n s h i p between U a / U H and Z / H m . m Compar i son o f e x p e r i m e n t a l r e s u l t s f o r 15 .2 cm d i a beds a g a i n s t E q s . ( 6 . 1 3 ) and ( 6 . 1 4 ) . L s \ s 1 1 7 6 .8 R e l a t i o n s h i p between U a / U a H and Z / H m -m Compar ison o f e x p e r i m e n t a l r e s u l t s f o r 24.1 cm and 29 .2 cm d i a beds w i t h E q s . ( 6 . 1 3 ) and ( > 6 - 1 4 ) ' U s = ] ' ] U m s 1 1 8 6 .9 E f f e c t o f C f / k s on a i r v e l o c i t y p r o f i l e s p r e d i c t e d by E q . ( 2 . 3 4 ) : wheat b e d , D c = 15 .2 cm, H = 6 3 . 5 cm, K = 0 .0088 gm/cm 3 . . . . 120 x i i F i g u r e Page 6 .10 R e l a t i o n s h i p between U U/U „ and Z/H i n a H a ri m m wheat b e d s . Compar i son o f e x p e r i m e n t a l r e s u l t s w i t h those p r e d i c t e d by E q . ( 6 . 1 5 ) w i t h K k s / C f = 0 . 0 5 9 . U g = 1.1 U m s 122 7.1 R a d i a l s e c t i o n s o f a n n u l u s f o r p a r t i c l e v e l o c i t y measurements 125 7 .2 P a r t i c l e v e l o c i t y p r o f i l e s i n the a n n u l u s : P o l y s t y r e n e p e l l e t s , D c = 15 .2 cm, D 0 / D c = 0 .1 2 5 , U = 1 .1 II 127 s ms 7 .3 P r o f i l e s o f r a d i a l a v e r a g e p a r t i c 1 e v e l o c i t y i n the a n n u l u s : p o l y s t y r e n e p e l l e t s , D c = 15 .2 cm, D 0 / D c = 0 .125 128 7.4 V o l u m e t r i c f l o w r a t e o f s o l i d s i n a n n u l u s : P o l y s t y r e n e p e l l e t s , D c = 1 5 . 2 cm, D 0 / D c = 0 .125 . . 129 7 .5 C o - o r d i n a t e sys tem and g r i d p o i n t s f o r p a r t i c l e f l o w pa th c a l c u l a t i o n s 138 7.6 Compar i son between e x p e r i m e n t a l and c a l c u l a t e d p a r t i c l e f l o w pa ths i n the a n n u l u s o f wheat b e d s : H/ D = 3 . 0 , D 0 / D = 0 . 1 2 5 , U c = 1.1 ll . . . 139 c c s ms 7.7 D i s t r i b u t i o n o f r e t e n t i o n t ime o f s o l i d s i n the a n n u l u s . Wheat b e d s : H / D c = 3 . 0 , D 0 / D c = 0 . 1 2 5 , U s = .1 .1 U m s 141 7 .8 R a d i a l s e c t i o n s o f spou t f o r p a r t i c l e v e l o c i t y measurements 143 7 .9 R a d i a l p r o f i l e s o f p a r t i c l e v e l o c i t y i n the spou t 145 7 .10 E f f e c t o f s p o u t i n g v e l o c i t y and p a r t i c l e d i a m e t e r on spou t p a r t i c l e v e l o c i t y : G l a s s b e a d s , D c = 15 .2 cm, H / D c = 3 . 0 , D 0 / D c •= 0 .125 . . . 146 x i i i F i gure Page 7.11 E f f e c t o f bed dep th on spou t p a r t i c l e v e l c o i t y : Wheat , D c = 15 .2 cm, D 0 / D c = 0 . 1 2 5 , V U m s " , 4 7 7.12 E f f e c t o f co lumn d i a m e t e r on spou t p a r t i c l e v e l c o i t y : Wheat , H / D c = 3..0, U s / U m s = 1.1 148 7 .13 E f f e c t o f gas i n l e t d i a m e t e r on spou t p a r t i c l e v e l o c i t y : P o l y s t y r e n e p e l l e t s , D = 1 5 . 2 cm, H / D c = 3 . 0 , U s / U m s = 1.1 . . . . . 149 7 .14 Compar i son between e x p e r i m e n t a l spou t p a r t i c l e v e l o c i t i e s and p r e d i c t i o n by the amended f o r c e b a l a n c e model o f T h o r l e y et al. [ 1 0 ] . P o l y s t y r e n e p e l l e t s , D c = 15 .2 cm, H / D r = 3 . 0 , D 0 / D o = 0 . 1 2 5 , U / U = 1 . 1 . 1 5 1 s ms 7 .15 Compar i son between e x p e r i m e n t a l spou t p a r t i c l e v e l o c i t i e s and p r e d i c t i o n by the L e f r o y and D a v i d s o n ' s model [ 2 4 ] , u s i n g e m p i r i c a l e q u a t i o n s f o r d e t e r m i n i n g u s and e^. P o l y s t y r e n e p e l l e t s , D c = 15 .2 cm, H / D c = 3 . 0 , D 0 / D = 0 . 1 2 5 , V u . » - i . i . . . : 1 5 4 7.16 Compar i son between e x p e r i m e n t a l spou t p a r t i c l e v e l o c i t i e s and p r e d i c t i o n by L e f r o y and D a v i d s o n ' s model [24 ] u s i n g e x p e r i m e n t a l l y d e t e r m i n e d v a l u e s o f u s and e . P o l y s t y r e n e p e l l e t s , D = 15 .2 cm, H/D = 3 . 0 , D 0 / D - = 0 . 1 2 5 , V U m s " 1-1 • • • • • 1 5 6 7.17 C o m p a r i s o n between e x p e r i m e n t a l spou t p a r t i c l e v e l o c i t i e s and p r e d i c t i o n by m o d i f i e d f o r c e b a l a n c e model o f T h o r l e y et al., E q . ( 7 . 2 6 ) . P o l y s t y r e n e p e l l e t s , D c = 15 .2 cm, H / D c = 3 . 0 , D 0 / D c = 0 . 1 2 5 , U s / U m s = 1.1 162 7 .18 Compar i son between e x p e r i m e n t a l spou t p a r t i c l e v e l o c i t i e s and p r e d i c t i o n by m o d i f i e d f o r c e b a l a n c e model o f T h o r l e y et al., E q . ( 7 . 2 6 ) . Wheat , D c = 15 .2 cm, H / D c = 3 . 0 , D 0 / D c = 0 . 1 2 5 , XI V F i g u r e Page 7 .19 Compar i son between e x p e r i m e n t a l spou t p a r t i c l e v e l o c i t i e s and p r e d i c t i o n by m o d i f i e d f o r c e b a l a n c e model of T h o r l e y et a l . , E q . ( 7 . 2 6 ) . M i l l e t , D = 15 .2 cm, H/D = 3 . 0 , D 0 / D c = 0 . 1 2 5 , V U m s = ' - 1 6 4 8.1 Compar i son o f spou t shape i n f u l l and h a l f c o l umns 168 8 .2 Observed spou t shapes 169 8 . 3 E f f e c t o f s p o u t i n g v e l o c i t y and bed dep th on spou t shape : P o l y s t y r e n e p e l l e t s , D = 1 5 . 2 cm, D / D 0 = 0 . 1 2 5 . 170 c 8 .4 E f f e c t o f column d i a m e t e r and p a r t i c l e s i z e on spou t s h a p e : P o l y s t y r e n e p e l l e t s , D / D 0 = 0 . 1 2 5 , H / D c = 3 .0 C 171 8 . 5 E f f e c t o f U s / U m s on spou t v o i d a g e : P o l y s t y r e n e p e l l e t s , D = 15 .2 cm, H/D = 3 . 0 , D 0 / D = 0 .125 . . 175 8 .6 E f f e c t o f bed dep th on spou t v o i d a g e : D = 15 .2 cm, D 0 / D c = 0 . 1 2 5 , U s / U m s = 1.1 . 176 8 .7 E f f e c t o f column d i a m e t e r on spou t v o i d a g e : Wheat , H / D c = 3 ^ 0 , D 0 / D c = 0 . 1 2 5 , U s / U m s = 1.1 . . . 177 8 .8 E f f e c t o f gas i n l e t d i a m e t e r on spou t v o i d a g e : P o l y s t y r e n e p e l l e t s , D = 15 .2 cm, H/D = 3 . 0 , U / U = 1 . 1 c 1 78 s ms 8 .9 Compar i son o f p r e d i c t e d and e x p e r i m e n t a l spou t v o i d a g e p r o f i l e s : P o l y s t y r e n e p e l l e t s , D = 15 .2 cm, H/D = 3 . 0 , D 0 / D „ = 0 . 1 2 5 , c V .s • • 1 8 1 11.1 C a l i b r a t i o n c u r v e s f o r p r e s s u r e t r a n s d u c e r A • ( range ± 2 .5 p s i d ) . . . . . 204 11.2 C a l i b r a t i o n c u r v e s f o r p r e s s u r e t r a n s d u c e r B ( range ± 0 .3 p s i d ) . 205 IV.1 Mesh s p a c i n g f o r s o l v i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n , E q . ( 5 . 5 ) 209 xv ACKNOWLEDGMENTS I w i s h to e x p r e s s my thanks to the f a c u l t y and s t a f f o f the Chem ica l E n g i n e e r i n g Depa r tmen 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 . S p e c i a l t hanks a re ex tended to D r . K i s h a n B. M a t h u r , under whose d i r e c t i o n t h i s work was c o n d u c t e d , f o r h i s h e l p f u l g u i d a n c e and encouragement and a l s o to D r . N. E p s t e i n f o r h i s a d v i c e and comments t h r o u g h o u t the c o u r s e o f t h i s i n v e s t i g a t i o n . I am i n d e b t e d to the gen t lemen o f the Chem ica l E n g i n e e r i n g Depar tment workshop and s t o r e s f o r t h e i r i n v a l u a b l e h e l p . I w i sh to thank Mr . Paddy J a r v i s i n p a r t i c u l a r f o r h i s i n f i n i t e p a t i e n c e and s u p e r i o r s k i l l i n c o n s t r u c t i n g the e x p e r i m e n t a l a p p a r a t u s . I am g r a t e f u l to the N a t i o n a l R e s e a r c h C o u n c i l o f Canada f o r s c h o l a r s h i p and r e s e a r c h g r a n t s , and 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 a G radua te F e l l o w s h i p . F i n a l l y , I would l i k e to e x p r e s s my g r a t i t u d e to my w i f e He len f o r her c o n t i n u a l s u p p o r t and f o r e b e a r a n c e t h r o u g h o u t t h i s work . x v i C h a p t e r 1 INTRODUCTION The u s e f u l n e s s o f f 1 u i d i z a t i o n as a f l u i d - s o l i d c o n -t a c t i n g t e c h n i q u e i s w e l l e s t a b l i s h e d . I t s a p p l i c a t i o n i n the case o f g a s - s o l i d sys tem h a s , however , been l a r g e l y l i m i t e d to r e l a t i v e l y f i n e p a r t i c l e s (< 1 mm d i a ) because c o a r s e p a r t i c l e s , when s u b j e c t e d to f 1 u i d i z a t i o n , g i v e r i s e to the f o r m a t i o n o f l a r g e gas bubb les and a tendency towards s l u g g i n g . I t was t h i s l i m i t a t i o n of the f l u i d i z a t i o n phenomenon wh ich l e d Mathur and G i s h l e r [ 1 , 2 ] to d e v e l o p the spou ted bed t e c h -n i q u e f o r c o n t a c t i n g f l u i d s w i t h c o a r s e p a r t i c l e s wh ich a re d i f f i c u l t to f l u i d i z e . F i g u r e 1.1 shows a t y p i c a l spou ted bed i n a c y l i n -d r i c a l c o l u m n . The bed c o n s i s t s o f two d i s t i n c t z o n e s , the spou t and the a n n u l u s . The spou t i s a f a s t moving s t ream of f l u i d c a r r y i n g p a r t i c l e s upwards i n a d i l u t e p h a s e , w h i l e i n the s u r r o u n d i n g d e n s e - p h a s e a n n u l u s , the p a r t i c l e s move s l o w l y downwards*and to some e x t e n t r a d i a l l y i n w a r d s . The f l u i d p e r c o l a t e s th rough t h e s e a n n u l a r p a r t i c l e s , f l a r i n g out r a d i a l l y as i t t r a v e l s upwards . 1 / / s s / / / / / / „ — F O U N T A I N B E D S U R F A C E S P O U T A N N U L U S S P O U T - A N N U L U S I N T E R F A C E C O N I C A L B A S E F L U I D I N L E T Figure 1.1. Schematic diagram of a spouted bed. 3 A l t h o u g h the s p o u t e d bed was o r i g i n a l l y c o n s i d e r e d as a m o d i f i e d v e r s i o n o f a f l u i d i z e d b e d , deve lopmen ts ove r the l a s t f i f t e e n y e a r s have d e m o n s t r a t e d t h a t i t p o s s e s s e s s e v e r a l un ique hydrodynamic f e a t u r e s wh ich make i t more s u i t a b l e f o r c e r t a i n t y p e s o f a p p l i c a t i o n s than more c o n v e n t i o n a l f l u i d -s o l i d s y s t e m s . Spou ted beds have thus p roved to be o f i n t e r e s t f o r c a r r y i n g ou t a v a r i e t y o f p r o c e s s e s i n v o l v i n g c o a r s e s o l i d s , such as d r y i n g , c o o l i n g and c o a t i n g o f v a r i o u s m a t e r i a l s , g r a n u l a t i o n , c o a l c a r b o n i z a t i o n , s h a l e p y r o l y s i s , t he rma l c r a c k i n g o f p e t r o l e u m , e t c . [3], D e t a i l e d knowledge o f f l u i d f l o w b e h a v i o u r , wh ich d e t e r m i n e s the e f f e c t i v e n e s s o f f l u i d - s o l i d s c o n t a c t , i s impo r -t a n t f o r the d e s i g n and s c a l e - u p o f any c o n t a c t i n g s y s t e m . For spou ted b e d s , c o n s i d e r a b l e i n f o r m a t i o n e x i s t s on the b u l k d i s t r i b u t i o n o f gas between the spou t and the a n n u l u s but no work has been r e p o r t e d , e i t h e r e x p e r i m e n t a l or t h e o r e t i c a l , on gas r e s i d e n c e t ime d i s t r i b u t i o n . A knowledge o f gas r e s i -dence t ime d i s t r i b u t i o n , though o f m inor consequence f o r a s o l i d t r e a t m e n t p r o c e s s such as g r a i n d r y i n g , c o u l d have an i m p o r t a n t e f f e c t on c e r t a i n o t h e r t y p e s o f p r o c e s s e s l i k e t h e r m o - c h e m i c a l d e p o s i t i o n [ 4 , 5 , 6 , 7 ] , vapour phase c h e m i c a l r e a c t i o n [8 ] and a e r o s o l c o l l e c t i o n [ 9 ] , f o r wh ich the use o f s p o u t e d beds has r e c e n t l y a t t r a c t e d a t t e n t i o n . The p r i m a r y o b j e c t i v e o f the p r e s e n t r e s e a r c h i s to d e v e l o p a m a t h e m a t i c a l m o d e l , based on e x p e r i m e n t a l d a t a , to d e s c r i b e the r e s i d e n c e t ime d i s t r i b u t i o n o f gas i n spou ted b e d s . Such a model would have to be based on a d e t a i l e d knowledge o f the movement o f g a s , as w e l l as o f o t h e r h y d r o -dynamic f e a t u r e s , v i z . s o l i d s f l o w p a t t e r n , spou t s h a p e , and spou t v o i d a g e . The e x p e r i m e n t a l da ta r e q u i r e d , t h e r e f o r e , a re RTD measurements t o g e t h e r w i t h p a r a l l e l da ta on the a b o v e -men t i oned hydrodynamic f e a t u r e s . A s e c o n d a r y o b j e c t i v e o f the work i s to use the hyd rodynamic da ta o b t a i n e d to t e s t e x i s t i n g t h e o r i e s and c o r r e l a t i o n s , and to improve upon t h e s e whe reve r p o s s i b l e . C h a p t e r 2 LITERATURE REVIEW S i n c e the deve lopmen t o f the spou ted bed t e c h n i q u e by G i s h l e r and Mathur [1 ] i n 1954, a l a r g e volume of i n f o r m a -t i o n on t h i s t o p i c has appeared i n the t e c h n i c a l l i t e r a t u r e . In the r e c e n t l y p u b l i s h e d book e n t i t l e d Spouted Beds, Mathur and E p s t e i n [10 ] have c o m p r e h e n s i v e l y r e v i e w e d most o f the i n f o r m a t i o n a v a i l a b l e on the s u b j e c t . The p r e s e n t r e v i e w i s t h e r e f o r e c o n f i n e d to t hose a s p e c t s o f spou ted bed b e h a v i o u r wh ich have a d i r e c t b e a r i n g on the p r e s e n t r e s e a r c h . 2.1 Flow P a t t e r n o f Gas Experimental Work Gas d i s t r i b u t i o n i n spou ted beds has been d e t e r m i n e d by two me thods ; f rom s t a t i c p r e s s u r e measurements i n the a n n u l u s , and f rom p i t o t tube measurements of l o c a l gas v e l o c i t i e s i n the s p o u t . v Mathur and c o - w o r k e r [ 2 , 1 1 , 1 2 ] , A b d e l r a z e k [ 4 ] , Aggor et al. [ 5 ] , and r e c e n t l y van V e l z e n et al. [ 6 , 7 ] have 5 6 used s t a t i c p r e s s u r e measurements to d e t e r m i n e gas v e l o c i t i e s i n the a n n u l u s . T h i s method i n v o l v e s the a s s u m p t i o n t h a t the spou ted bed a n n u l u s i s s i m i l a r i n v o i d a g e to a l o o s e l y packed b e d , hence any v e r t i c a l p r e s s u r e g r a d i e n t caused by the f l o w o f gas th rough the a n n u l u s i s s i m i l a r to t h a t f o r a l o o s e l y - p a c k e d b e d . The a n n u l u s v o i d a g e i n the l ower p a r t o f the bed has been o b s e r v e d to be somewhat s m a l l e r than near the top o f the a n n u l u s [ 1 2 , 1 4 ] , but t h i s s m a l l v a r i a t i o n was n e g l e c t e d by a l l the i n v e s t i g a t o r s . The method i s a p p l i c a b l e f o r o n l y the c y l i n d r i c a l p a r t o f the bed where the p r e s s u r e a c r o s s a h o r i z o n t a l s e c t i o n has been found to be a l m o s t un i f o r m . T y p i c a l r e s u l t s o f gas f l o w p r o f i l e s i n wheat beds o b t a i n e d by Mathur and G i s h l e r [2 ] a re p r e s e n t e d i n F i g u r e . 2 . 1 , wh ich shows t h a t a s u b s t a n t i a l p r o p o r t i o n o f the i n l e t gas f l a r e s ou t i n t o the a n n u l u s as i t t r a v e l s upwards . R a d i a l f l o w f rom spou t to a n n u l u s i s more p ronounced i n the l ower h a l f o f the b e d . B e c k e r [ 1 5 ] , Mamuro and H a t t o r i [ 1 6 ] , van V e l z e n et al. [ 6 , 7 ] , and Yokogawa et al [ 1 7 , 1 8 , 1 9 ] used p i t o t tubes to measure l o c a l gas v e l o c i t i e s . T h i s method i s s u i t a b l e f o r gas v e l o c i t y measurement i n the spou t o n l y . In the a n n u l u s , the low gas v e l o c i t i e s i n v o l v e d t o g e t h e r w i t h the d i s t u r b a n c e of l o c a l p o r o s i t y caused by the i n s e r t i o n o f the p i t o t tube make t h i s method u n r e l i a b l e [ 1 0 , 1 5 , 1 7 ] . 7 1.0 0.8 o O o 0.6 0.4 0.2 0 0 o 0.2 D c ,cm H/D c A 15 4 o 30 4 © 61 3 0.4 0.6 Z/H 0.8 1.0 F i g u r e 2,1 A i r d i s t r i b u t i o n D c / D 0 = 6 . i n spou ted wheat beds [ 2 ] , B e c k e r [15 ] p r e s e n t e d the f o l l o w i n g e m p i r i c a l c o r -r e l a t i o n f o r v e r t i c a l a i r v e l o c i t y p r o f i l e s measured a l o n g the spou t a x i s i n wheat b e d s , a t bed dep ths a p p r o a c h i n g the maximum s p o u t a b l e bed d e p t h : u p „ - 9 Z / H s c = 1 + 7.6 e m ( 2 . 1 ) u t where u i s the l o c a l gas v e l o c i t y a t the spou t a x i s and Uj. i s the p a r t i c l e f r e e f a l l t e r m i n a l v e l c o i t y Yokogawa et al. [ 1 7 , 1 8 ] a l s o measured v e r t i c a l gas v e l o c i t y p r o f i l e s a l o n g the spou t a x i s . T h e i r r e s u l t s p l o t t e d i n F i g u r e 2 .2 show t h a t the r a t i o u s c / u 0 i n c r e a s e s w i t h d e c r e a s i n g Z / D 0 , r e a c h i n g a maximum v a l u e o f u n i t y a t a f i n i t e v a l u e o f Z / D 0 ( = Z 0 / D 0 ) r a t h e r than a t Z / D 0 = 0. The v a l u e o f Zo/Do was found to be d i f f e r e n t f o r the d i f f e r e n t s o l i d m a t e r i a l s s t u d i e d and c o u l d be e x p r e s s e d e m p i r i c a l l y as 7 fn i ° - 3 ^ = 0 .80 1^- (2.2) Yokogawa et al. c o r r e l a t e d t h e i r d a t a u s i n g the form o f B e c k e r ' s e m p i r i c a l e q u a t i o n ( E q . 2 . 1 ) . u s c = b i + b 2 e n Z / H ( 2 . 3 ) 1.0 0.8 o 1 3 0 .6 o CO 3 0.4 0 .2 0 D 0 r — o •u s c /u 0 = |.0 \ \ o i o * A. 1 r D c,cm A Red bean I 2.8 A Sesame 21.5 • Rice 3 0 . 0 o Wheat 3 0 . 0 2 0 3 0 4 0 Z / D o Figure 2.2. Longitudinal p ro f i l es of gas ve loc i ty at spout axis [17]. except for wheat D0 = 4 cm. D0 = 2 cm, 10 and assuming t h a t a t Z = Z 0 , u sc u 0 and a t Z = H, u sc u „ . The gas v e l o c i t y a t the spou t a x i s i s g i v e n by SCrl SC Uo = 1 - 4 1 scH u o J - n Z 0 / H _ e - n Z / H J - n Z 0 / H _ e - n ( 2 . 4 ) They gave the f o l l o w i n g e m p i r i c a l e x p r e s s i o n s f o r the v a l u e s o f n and u s c H i n the above e q u a t i o n n = 2 5 . 0 0 .5 ( 2 . 5 ) and u s c H = U 2 U s H ( 2 . 6 ) where 'sH i s the ave rage gas v e l o c i t y ove r the c r o s s s e c t i o n o f the spou t a t Z - H. R a d i a l p r o f i l e s o f upward gas v e l o c i t y i n the s p o u t , measured by Yokogawa et a l . , a re i l l u s t r a t e d i n F i g u r e 2 . 3 . By a n a l o g y w i t h v e l o c i t y p r o f i l e s i n a f r e e j e t , t hey d e v e l o p e d the f o l l o w i n g e q u a t i o n s f o r the r a d i a l p r o f i l e s o f gas v e l o c i t y i n the s p o u t : R e f e r r i n g to the n o m e n c l a t u r e i n F i g u r e 2 . 4 , f o r a t Z = 0 to Z 0 r = 0 to r i Figure 2 .3 . Radial p ro f i l es of upward gas ve loc i ty in the spout of wheat bed [17], D = 21.5 cm, Do = 4 cm, H = 80 cm, U g = 1.3 m/sec. 1 2 s z U 0 ( 2 . 7 ) a t T i to r sz / 3 /2 ' Uo -- u 1 1 - r - r i + s s j r 0 - r i + u. ss ( 2 . 8 ) a t r = r 0 to r. u s z = u s s ( 2 . 9 ) f o r Z = Z 0 , to H sz u - u sc s s 3/2 + u s s (2.10) u s s b e i n g t n e upward gas v e l o c i t y a t the s p o u t - a n n u l us i n t e r f a c e . Gas f l o w i n the l o w e r c o n i c a l p a r t o f the a n n u l u s i s more complex and d i f f i c u l t to measure by the s t a t i c p r e s s u r e method because o f the sudden e x p a n s i o n o f the gas wh ich o c c u r s a t the i n l e t p o i n t and the d i m i n i s h i n g w i d t h o f the annu lus i n the c o n i c a l b a s e . N e v e r t h e l e s s , A b d e l r a z e k [4 ] and van V e l z e n et al. [ 6 , 7 ] made s t a t i c p r e s s u r e measurements i n the l ower c o n i c a l s e c t i o n o f s m a l l beds (up to 12 .5 cm d i a x 20 cm d e e p ) , u s i n g a probe i n s e r t e d f rom the t o p . A b d e l r a z e k [4 ] found t h a t s t a t i c p r e s s u r e s a t the spou t a x i s were s l i g h t l y 1 3 lower than a t the s p o u t - a n n u l us i n t e r f a c e w h i l e van V e l z e n et al. [ 6 , 7 ] o b t a i n e d n e g a t i v e v e r t i c a l p r e s s u r e g r a d i e n t s under c e r t a i n o p e r a t i n g c o n d i t i o n s . Both r e s u l t s s u g g e s t r e c i r c u l a t i o n o f gas i n the l o w e r c o n i c a l s e c t i o n of the b e d , a l t h o u g h the q u a n t i t a t i v e da ta a re open to q u e s t i o n . Back f l o w o f gas i n the lower p a r t o f the a n n u l u s was o b s e r v e d d i r e c t l y i n t h i s work [20 ] under c e r t a i n s p o u t i n g c o n d i t i o n s , u s i n g c o l o u r e d N 0 2 gas as t r a c e r (see F i g u r e 5 . 2 ) . Theoretical Analysis A t t e m p t s to a n a l y s e the gas f l o w p a t t e r n t h e o r e t i -c a l l y have been made by Mamuro and H a t t o r i [16 ] and by Yokogawa et al. [ 1 7 , 1 9 ] . Both g roups o f w o r k e r s s t a r t the a n a l y s i s w i t h a f o r c e b a l a n c e on d i f f e r e n t i a l h e i g h t , d Z , o f the a n n u l u s , assuming the v e r t i c a l s t a t i c p r e s s u r e g r a d i e n t to f o l l o w D a r c y ' s Law, dP - i n r = K U a ( 2 - 1 ] ) Mamuro and H a t t o r i [16 ] n e g l e c t e d the f r i c t i o n a l f o r c e a c t i n g on both v e r t i c a l b o u n d a r i e s o f the annu lus and assumed t h a t the k i n e t i c p r e s s u r e due to c r o s s - f l o w o f gas f rom the spou t to the annu lus i s i n b a l a n c e w i t h the r a d i a l component o f the annu lus p r e s s u r e (see F i g u r e 2 . 5 ) . The r a d i a l component was then taken to be p r o p o r t i o n a l to the ? + d ? C T Q + d c r U a + d U a Spout —m ±_ Kine t i c pressure due to gas cross flow Radia l component of pressure due to weight of bed T A n n u l u s T Figure 2.5. Mamuro and Ha t to r i ' s Model for forces act ing in the annulus [16]. ? + d P a O T Q + d c T a U a + d U G k-Spout —>< A Drag fo rce due to gas cross flow Rad ia l component of pressure due to weight of bed Annulus A dZ Figure 2.6. Yokogawa's model for forces act ing in the annulus [17]. 1 5 v e r t i c a l component o f the a n n u l a r bed p r e s s u r e . The g e n e r a l s o l u t i o n o f t h e i r d i f f e r e n t i a l f o r c e - b a l a n c e e q u a t i o n i s TTD S U a = 3A~ C l a (z + c 2 ) 3 - c ( 2 . 1 2 ) where C i and C 2 a re c o n s t a n t s . Wi th the boundary c o n d i t i o n s . U a = 0 a t Z = 0 , and U a = U f (minimum f l u i d i z a t i o n v e l o c i t y ) a t Z = H , they o b t a i n e d the f o l l o w i n g r e s u l t f rom E q . ( 2 . 1 2 ) U a - 1 - (1 - Z / H J 3 ( 2 . 1 3 ) U mf V ' m E q . ( 2 . 1 3 ) i s r e s t r i c t e d to beds o f maximum s p o u t a b l e depth because the upper boundary c o n d i t i o n a p p l i e s o n l y when H = H . Mamuro and H a t t o r i [16 ] and van V e l z e n et al. [7 ] have a r b i t r a r i l y m o d i f i e d E q . ( 2 . 1 3 ) f o r the s i t u a t i o n where H < H , but both o f t h e s e e m p i r i c a l m o d i f i c a t i o n s have no g e n e r a l v a l i d i t y . On the o t h e r hand , G r b a v c i c et al. [21 ] have r e c e n t l y r e p o r t e d the e x p e r i m e n t a l f i n d i n g t h a t ll a t a any g i v e n bed l e v e l i s i n d e p e n d e n t of bed d e p t h , and t h i s i s c o r r o b o r a t e d by our e x p e r i m e n t a l measurements . Hence , E q . ( 2 . 1 3 ) s h o u l d a p p l y f o r any bed d e p t h . W r i t i n g E q . ( 2 . 1 3 ) f o r Z = H„ G r b a v c i c e t al. [ 21 ] o b t a i n e d the f o l l o w i n g r e l a -t i o n s h i p between U ,. and U c \ a H mf U a_H = 1 U mf 1 - -iL m 3 ( 2 . 1 4 ) . 16 Yokogawa et al. [ 1 7 , 1 9 ] i n c l u d e d the f r a c t i o n a l f o r c e a c t i n g on bo th s i d e s o f the a n n u l u s i n t o t h e i r f o r c e b a l a n c e e q u a t i o n s . S i n c e the f r i c t i o n a l f o r c e a c t i n g on the o u t e r boundary o f the a n n u l u s i s d i f f e r e n t f o r the upper p a r t o f the bed ( c y l i n d r i c a l s e c t i o n ) f rom t h a t f o r the l ower p a r t ( c o n i c a l s e c t i o n ) , they d i v i d e d the a n n u l u s i n t o two r e g i o n s a c c o r d i n g l y . T h e o r e t i c a l a n a l y s i s was c a r r i e d ou t f o r each r e g i o n s e p a r a t e l y . For the upper p a r t o f the a n n u l u s , t h e i r f o r c e b a l a n c e e q u a t i o n i s as f o l l o w s (see F i g u r e 2 . 6 ) : dP d a ' 4 x D 4 T D I T + "df" + D 2 - D C * + D 2 - D S * = p b ( 2 ' 1 5 ) c s c s where x c i s the f r i c t i o n a l f o r c e between the a n n u l a r s o l i d s and the column w a l l and i s g i v e n by x c = U C c r c ( 2 . 1 6 ) u c b e i n g the c o e f f i c i e n t o f f r i c t i o n between the s o l i d s and the co lumn w a l l and a ^ c the r a d i a l component o f p r e s s u r e due to w e i g h t o f s o l i d s a t the co lumn w a l l . The f r i c t i o n a l f o r c e between spou t and a n n u l u s , x $ , was e v a l u a t e d by c o n s i d e r i n g the momentum b a l a n c e ove r a d i f f e r e n t i a l h e i g h t , d Z , o f the s p o u t . dZ + f d T <Vs"> • f " « * <PS - P f ) (1 - e s ) (1 - e s ) v s ( 2 . 1 7 ) 1 7 On the b a s i s o f t h e i r p r e s s u r e drop s tudy [22] Yokogawa et al. assumed t h a t the changes w i t h r e s p e c t to bed l e v e l i n f l u i d and s o l i d s momentum i n the spou t and i n the e f f e c t i v e w e i g h t o f the s o l i d s were n e g l i g i b l y s m a l l i n c o m p a r i s o n w i t h the p r e s s u r e drop d P a - Hence E q . ( 2 . 1 7 ) becomes D dP T s = - T " - d f < 2 - 1 8 > Assuming t h a t the r a d i a l p r e s s u r e due to w e i g h t o f the a n n u l a r s o l i d s , or, i s a f u n c t i o n o f r and i s p r o p o r t i o n a l to a a , the v e r t i c a l p r e s s u r e due to we igh t o f the a n n u l a r s o l i d s , they o b t a i n e d and o r c = k c a a ( 2 . 1 9 ) a . r s = k s a a ( 2 . 2 0 ) where a „ and a „ a re the v a l u e s o f a a t the i n t e r -r c r s r . f a c e w i t h the co lumn w a l l and the i n t e r f a c e w i t h the s p o u t , r e s p e c t i v e l y , and k c > a c c o r d i n g to R a n k i n e ' s t h e o r y [ 2 3 , p. 8 ] , i s g i v e n by . = 1 - s i n (j> , 2 2 ] v K c 1 + s i n cf> K C ' c 1 ; <J) b e i n g the a n g l e o f i n t e r n a l f r i c t i o n o f the s o l i d m a t e r i a l . To d e t e r m i n e k g Yokogawa et al. a rgued t h a t s i n c e the i n t e r f a c e between the spou t and the a n n u l u s i s m a i n t a i n e d 18 s t e a d y , a r s , the r a d i a l p r e s s u r e due to w e i g h t o f s o l i d s a t the s p o u t - a n n u l us i n t e r f a c e , must be i n b a l a n c e w i t h the r a d i a l f o r c e due to gas f l o w f rom the spou t to the a n n u l u s . They presumed t h a t the c r o s s - f l o w i n g gas i s i n the l a m i n a r f l o w r e g i o n a l ( D a r c y ' s r e g i o n ) , hence r s C . U e f r s ( 2 . 2 2 ) U r s b e i n g the r a d i a l v e l o c i t y o f gas a t the s p o u t - a n n u l us i n t e r f a c e and C f the d rag c o e f f i c i e n t . By s u b s t i t u t i n g Eq ( 2 . 2 2 ) i n t o E q . ( 2 . 2 0 ) , they o b t a i n e d k s = C f U r s / ° c ( 2 . 2 3 ) where U r s i s g i v e n by the f o l l o w i n g gas mass b a l a n c e e q u a t i o n : r s 1 4 21 dU a dZ ( 2 . 2 4 ) The f i n a l d i f f e r e n t i a l e q u a t i o n r e l a t i n g U a and Z i s o b t a i n e d by s u b s t i t u t i n g E q . ( 2 . 1 1 ) ( D a r c y ' s L a w ) , and E q s . ( 2 . 1 6 ) to ( 2 . 2 4 ) , i n t o E q . ( 2 . 1 5 ) , and r e a r r a n g i n g d 2 U_ dU -2- + A a , « ~ ~ a , B dZ' + — u dZ E a 1 ( 2 . 2 5 ) where 19 -4 k u D A - c c c rt D z - D z ( 2 . 2 6 ) K D c 2 ( 2 . 2 7 ) E = -D 2 - D 2 f c s 4k Pu D s K b s ( 2 . 2 8 ) and where K i s the Darcy c o n s t a n t , and k c and k g the c o n s t a n t s d e f i n e d by E q s . ( 2 . 2 1 ) and ( 2 . 2 3 ) r e s p e c t i v e l y . The g e n e r a l s o l u t i o n o f E q . ( 2 . 2 5 ) i s ll = a i e X l Z + a ( 2 . 2 9 ) i n wh ich 2 x " E A 2 = A 2 B T " E ( 2 . 3 0 ) ( 2 . 3 1 ) and a i and a 2 a re c o n s t a n t s . Wi th the boundary c o n d i t i o n s U = 0 a t Z = 0 , and a dU / d Z = 0 a t Z = H, Yokogawa et al'. d e r i v e d the f o l l o w i n g e x p r e s s i o n f o r ll f rom E q . ( 2 . 2 9 ) a 20 ii - X i e A l H (1 - e X z Z ) - A 2 e X z H (1 - e X l Z ) ( 9 B ( x i e ^ H - X 2 e A 2 H ) ( } For Z = H, U g = U g H , t h e r e f o r e E q . ( 2 . 3 2 ) becomes U a H = A * 6 ^ - \ > - X>1 * ( 1 - e ) ( 2 . 3 3 ) a H B ( x i e X l H - X 2 e X 2 H ) To ge t the f i n a l wo rk i ng e q u a t i o n f o r the c y l i n d r i c a l p a r t o f the b e d , Yokogawa et at. d i v i d e d E q . ( 2 . 3 2 ) by E q . ( 2 . 3 3 ) , w i t h the r e s u l t t h a t U a = X i e X l H (1 - e X z Z ) - X 2 e X z H (1 - e X l Z )  U aH A i e X l H (1 - e X 2 H ) - X 2 e X 2 H (1 - e X l H ) For the l ower p a r t of the a n n u l u s , t h e i r a n a l y s i s i s s i m i l a r to t h a t f o r the upper p a r t e x c e p t t h a t the e f f e c t o f the a n g l e of the c o n i c a l w a l l on x c was taken i n t o a c c o u n t . The r e s u l t i n g e q u a t i o n s a re much more c o m p l i c a t e d than t hose f o r the upper c y l i n d r i c a l p a r t o f the a n n u l u s , but Yokogawa et al. found t h a t the gas f l o w p a t t e r n was no t ve ry s e n s i t i v e to the a n g l e o f the v e s s e l w a l l . A s e m i e m p i r i c a l method f o r p r e d i c t i n g gas f l o w d i s t r i b u t i o n between spou t and annu lus has been p roposed by L e f r o y and D a v i d s o n [ 2 4 ] . These w o r k e r s e x p e r i m e n t a l l y e s t a b l i s h e d t h a t a t minimum s p o u t i n g v e l o c i t y the p r e s s u r e v a r i a t i o n w i t h bed l e v e l Z i n t h e i r f l a t - b a s e d column c o u l d b e d e s c r i b e d b y 21 • y -P a = AP cos ~ (2 where AP i s the t o t a l p r e s s u r e drop a c r o s s the bed Assuming t h a t gas f l o w i n the a n n u l u s f o l l o w s D a r c y ' s Law, they o b t a i n e d the f o l l o w i n g e x p r e s s i o n f o r U f o r H = H : a m a. mf 2 Hm  (2 A c o m p a r i s o n o f E q . ( 2 . 1 3 ) a g a i n s t a v a i l a b l e e x p e r i -menta l da ta has been made by G r b a v c i c et al. [21 ] and i s shown i n F i g u r e 2 .7 a l o n g w i t h a p l o t o f E q . ( 2 . 3 6 ) . F i g u r e 2 .7 i n d i c a t e s t h a t E q . ( 2 . 3 6 ) i s not i n good agreement w i t h the e x p e r i m e n t a l d a t a , p r o b a b l y because the b a s i c a s s u m p t i o n r e p r e s e n t e d by E q . ( 2 . 3 5 ) o v e r - s i m p l i f i e s the gas f l o w p a t t e r n . The M a m u r o - H a t t o r i e q u a t i o n , E q . ( 2 . 1 3 ) , i s i n b e t t e r ^ a g r e e m e n t w i t h the da ta though the da ta show c o n s i d e r a b l e s c a t t e r a round the t h e o r e t i c a l c u r v e , e s p e c i a l l y f o r Z < 0 .5 H . r J - m Yokogawa et al. [ 1 7 , 1 9 ] showed t h a t E q . ( 2 . 3 4 ) f i t t e d t h e i r e x p e r i m e n t a l da ta ve ry w e l l . An a t t emp t was made to compare E q . ( 2 . 3 4 ) a g a i n s t the e x p e r i m e n t a l da ta i n F i g u r e 2 . 7 . I t was f o u n d , however , t h a t p r e d i c t i o n o f U /U u by a an E q . ( 2 . 3 4 ) i s v e r y s e n s i t i v e to the v a l u e o f C f / k s i n E q . ( 2 . 2 8 ) (see F i g u r e 6 . 9 ) , and the p r o c e d u r e f o r e s t i m a t i n g C ^ / k s used by Yokogawa et al. i n v o l v e s a g r e a t dea l of u n c e r t a i n t y (see F i g u r e 6 .9) . 22 . 0 0.8 1 0 . 6 o 0.4 0.2 0 Eq. (2 . i3 ) Qy y E q . ( 2 . 3 6 ) D c Solids dp c m mm © 6 1 . 0 W h e a t 3 . 6 o 15.2 Whea t 3 . 6 " * 3 0 . 5 Kale seeds I . 7 A I 1.0 G l a s s 2 . 5 0 0.2 0.4 0.6 Z/Hm 0.8 .0 Figure 2.7. Ver t i ca l p ro f i l es of gas ve loc i t y [1,11,12,21,24] . Compari-son of experimental resu l ts with Eqs. (2.13) and (2.36). 23 Summary E x p e r i m e n t a l t e c h n i q u e s d e v e l o p e d to d e t e r m i n e gas v e l o c i t i e s i n the spou ted bed a re dynamic p r e s s u r e measu re -ment u s i n g a p i t o t t u b e , and s t a t i c - p r e s s u r e measurement . The f o rmer method has been found a p p l i c a b l e o n l y to the spou t and the l a t t e r o n l y to the c y l i n d r i c a l p a r t o f the a n n u l u s . No s a t i s f a c t o r y method has been e s t a b l i s h e d to measure gas v e l o c i t y i n the l ower c o n i c a l s e c t i o n o f the a n n u l u s . The t h e o r e t i c a l e q u a t i o n d e r i v e d by Mamuro and H a t t o r i [ 1 6 ] , E q . ( 2 . 1 3 ) , has been found to be the b e s t amongst the a v a i l a b l e e q u a t i o n s to d e s c r i b e the l o n g i t u d i n a l gas v e l o c i t y - p r o f i l e i n the c y l i n d r i c a l p a r t o f the a n n u l u s , f rom wh ich the bu l k d i s t r i b u t i o n o f gas between spou t and a n n u l u s can be d e d u c e d . The e x p e r i m e n t a l da ta s u p p o r t i n g the e q u a t i o n , however , c o v e r o n l y a nar row range o f v a r i a b l e s . S m a l l - s c a l e a s p e c t s o f gas f l o w p a t t e r n such as a x i a l or r a d i a l m i x i n g , and any l o c a l i z e d r e c i r c u l a t i o n o f g a s , have r e c e i v e d l i t t l e a t t e n t i o n . No work , e i t h e r e x p e r i m e n t a l o r t h e o r e t i c a l , has been r e p o r t e d on r e s i d e n c e t ime d i s t r i b u t i o n o f gas i n spou ted b e d s . 2 .2 Flow P a t t e r n o f So l i d s Experimental Work The movement o f s o l i d s i n the spou t and the a n n u l u s has been the s u b j e c t o f s e v e r a l s t u d i e s , e x p e r i m e n t a l [ 2 , 1 7 , 2 4 , 2 5 , 2 6 , 2 7 , 2 8 , 2 9 , 3 0 ] as w e l l as t h e o r e t i c a l [ 1 1 , 1 2 , 1 7 , 2 4 , 2 8 , 2 9 ] . 24 * Mathur and G i s h l e r [2 ] used h a l f - s e c t i o n a l co lumns to s tudy the s o l i d s f l o w p a t t e r n and to measure p a r t i c l e v e l o c i t i e s i n both the spou t and the a n n u l u s f o r wheat b e d s . H igh speed c i n e pho tog raphy (2000 f r a m e s / s e c o n d ) was used to r e c o r d p a r t i c l e v e l o c i t i e s i n the s p o u t . F i g u r e 2 .8 shows the l o n g i t u d i n a l p r o f i l e s o f p a r t i c l e v e l o c i t y a l o n g the spou t a x i s r e p o r t e d by t h e s e w o r k e r s . I t i s seen t h a t p a r t i c l e s s t a r t i n g f rom the base o f the bed a c c e l e r a t e to a maximum v e l o c i t y and then d e c e l e r a t e u n t i l they r e a c h z e r o v e l o c i t y a t the top o f the f o u n t a i n . G o r s h t e i n and Soroko [31 ] d e v e l o p e d the p i e z o e l e c t r i t e c h n i q u e to measure spou t p a r t i c l e v e l o c i t i e s i n a f u l l co lumn T h i s t e c h n i q u e was s u b s e q u e n t l y used by G o r s h t e i n and Mukh lenov [25 ] to measure p a r t i c l e v e l o c i t i e s o f c a t a l y s t beads i n a c o n i c a l bed and by M i k h a i l i k and A n t a n i s h i n [27 ] to measure p a r t i c l e v e l o c i t i e s o f m i l l e t , s i l i c a ge l and p o l y s t y r e n e i n c o n i c a l - c y l i n d r i c a l b e d s . R a d i a l p r o f i l e s o f p a r t i c l e v e l o c i t i e s a t any l e v e l i n the spou t were found to be p a r a b o l i c by both groups o f w o r k e r s . P a r a b o l i c r a d i a l v e l o c i t y p r o f i l e s were a l s o o b t a i n e d i n h a l f - s e c t i o n a l c y l i n d r i c a l co lumns by L e f r o y and Dav idson [24 ] and Yokogawa et al. [ 1 7 ] . T y p i c a l r e s u l t s , shown i n F i g u r e 2 . 9 , can be r e p r e s e n t e d by the e q u a t i o n 25 Z / H Figure 2.8. Longitudinal p ro f i l es of pa r t i c l e ve loc i t y along the spout axis of wheat beds [10]. 26 Figure 2.9. Radial p ro f i l es of upward pa r t i c l e ve loc i t y [10]. o Conical vesse l ; ca ta lys t marbles, d = 1.5 mm; 8 = 20° - 60°, H/D0 = 1.5-6.0, Z = p 5 cm and 10 cm. • Cy l i nd r i ca l vessel ( h a l f - s e c t i o n a l ) , D c = 30.5 cm; polyethylene ch ips , d = 3.5 mm; e = 180°, H/D 0 = 39, Z not g iven. P 27 where v i s the maximum v e l o c i t y a t the a x i s , sc J Van V e l z e n et al. [30 ] i n v e s t i g a t e d p a r t i c l e move-ment i n t h e spou t of s m a l l beds (up to 12 .5 cm d i a x 30 cm deep) of g l a s s beads by i n t r o d u c i n g a r a d i o a c t i v e 1 y marked p a r t i c l e i n t o the bed and r e c o r d i n g i t s movement u s i n g a s c i n t i l l a t i o n c o u n t e r . I n d i v i d u a l p a r t i c l e v e l o c i t i e s a l o n g the spou t w e r e d e t e r m i n e d and the a v e r a g e p a r t i c l e v e l o c i t y a t any g i v e n l e v e l was t aken as the a r i t h m e t i c mean v a l u e o f a l l i n d i v i d u a l p a r t i c l e v e l o c i t i e s a t t h a t l e v e l . F i g u r e 2 .10 shows l o n g i t u d i n a l p r o f i l e s o f r a d i a l ave rage p a r t i c l e v e l o c i t y o b t a i n e d by t h e s e w o r k e r s . An e m p i r i c a l e q u a t i o n o f p a r a b o l i c type was p roposed to d e s c r i b e the v e r t i c a l v e l o c i t y p r o f i l e s , but t h i s e q u a t i o n has been found to be i n a p p l i c a b l e to e x p e r i -menta l r e s u l t s o b t a i n e d in the p r e s e n t i n v e s t i g a t i o n . Yokogawa et al. [ 1 7 , 2 8 , 2 9 ] a l s o s t u d i e d the movement o f i n d i v i d u a l p a r t i c l e s i n the spou t u s i n g c i n e p h o t o g r a p h y . A p a r t i c l e was found to move v e r t i c a l l y upward and r a d i a l l y toward the spou t a x i s as soon as i t e n t e r e d the spou t f rom the a n n u l u s . V e r t i c a l v e l o c i t y p r o f i l e s f o r i n d i v i d u a l p a r t i c l e s e n t e r i n g the spou t f rom the a n n u l u s a t d i f f e r e n t l e v e l s i n the bed have been r e p o r t e d by t h e s e wo rke rs ( F i g u r e 2 . 1 1 ) . The p a r t i c l e s a re seen to a c c e l e r a t e r a p i d l y to a maximum v e l o c i t y as i n F i g u r e 2 .8 and F i g u r e 2 . 1 0 , though the d e c e l e r a t i o n a c c o r d i n g to t h e s e r e s u l t s ( F i g u r e 2 . 1 1 ) o c c u r s more s l o w l y . 28 Figure 2.10. Longitudinal p ro f i l es of rad ia l average upward pa r t i c l e ve loc i t y in the spout of glass beads bed [30] D = 7.5 cm, d„ = 1.05 mm. c p 29 Figure 2.11. Longitudinal ve loc i t y p ro f i l es for ind iv idual pa r t i c les entering into the spout from the annulus at bed level Z [17]. Wheat bed; D = 21.5 cm, D0 = 4 cm, H = 80 cm, U = 1.15 m/sec. 30 Mathur and c o - w o r k e r s [ 2 , 1 1 ] used both f u l l and h a l f - c y l i n d r i c a l co lumns to s tudy downward p a r t i c l e v e l o c i t i e s i n thp a n n u l u s by d i r e c t o b s e r v a t i o n a g a i n s t the column w a l l , and a g a i n s t the f l a t w a l l o f ha 1 f - c y 1 i n d r i c a 1 c o l u m n s . P a r t i c l e s were o b s e r v e d to move v e r t i c a l l y downward and r a d i a l l y i nward toward the s p o u t - a n n u l us i n t e r f a c e . They found t h a t f o r the upper p a r t o f the a n n u l u s , the downward p a r t i c l e v e l o c i t y near the column w a l l was o n l y s l i g h t l y s m a l l e r than t h a t near the s p o u t w a l l . A l s o , the p a r t i c l e c r o s s - f l o w r a t e f rom the a n n u l u s i n t o the spou t per u n i t bed h e i g h t was found to rema in c o n s t a n t w i t h bed l e v e l f o r the upper p a r t o f the b e d , as shown i n F i g u r e 2 . 1 2 . The c r o s s - f l o w r a t e per u n i t h e i g h t was found to 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 p o u t i n g gas v e l o c i t y and gas i n l e t d i a m e t e r and to rema in more o r l e s s c o n s t a n t w i t h i n c r e a s i n g column d i a m e t e r . I n c r e a s i n g the cone a n g l e . 9, was found to i n c r e a s e the c r o s s - f l o w r a t e i n the l a r g e co lumn ( D c = 61 cm) but had l i t t l e e f f e c t i n the s m a l l co lumn (D = 15 .2 cm) . Yokogawa [17 ] a l s o measured p a r t i c l e v e l o c i t i e s i n the a n n u l u s , and found t h a t p a r t i c l e c r o s s - f l o w r a t e per u n i t h e i g h t rema ins c o n s t a n t f o r bed l e v e l s g r e a t e r than D c / H . Theoretical Analysis T h o r l e y and c o - w o r k e r s [ 1 1 , 1 2 ] , based on t h e i r o b s e r v a t i o n s , p roposed a t h e o r e t i c a l model to p r e d i c t the l o n g i -t u d i n a l p a r t i c l e v e l o c i t y p r o f i l e i n the s p o u t . The model was 31 5.0 0 0.2 0.4 0 6 0-8 1.0 Z / H Figure 2.12. Sol ids flow in annulus [2,11]: wheat; D c /D 0 = 6 .0 , 9 = 60°, U s / U m s = 1 . 1 , H/D c = 3. 32 l a t e r m o d i f i e d by Mathur and E p s t e i n [10] to a c c o u n t f o r the c r o s s - f l o w of p a r t i c l e s f rom a n n u l u s to s p o u t . The m o d i f i e d model s t a r t s w i t h the f o l l o w i n g g e n e r a l f o r c e b a l a n c e equa -t i o n f o r the spou t p a r t i c l e s , wh ich assumes t h a t the o n l y a c c e l e r a t i n g f o r c e on the p a r t i c l e s i s the f r i c t i o n a l d rag o f the f l u i d , and t h a t the d e c e l e r a t i n g f o r c e i s g r a v i t y : v s 2 d n z , v s d v s 3 P f K - v s ) 2 V C D <P5 - Pf> ( 0 n z dZ dZ 4 d p ' p s p s where n z i s the c u m u l a t i v e number o f r i s i n g p a r t i c l e s a t a g i v e n l e v e l and Cp i s the d rag c o e f f i c i e n t . The spou t was d i v i d e d i n t o t h r e e v e r t i c a l r e g i o n s and s p e c i f i c a s s u m p t i o n s were made f o r each r e g i o n to o b t a i n c o r r e s p o n d i n g e q u a t i o n s f o r p a r t i c l e v e l o c i t y f rom E q . ( 2 . 3 8 ) . In the r e g i o n between the top o f bed s u r f a c e and the top o f f o u n t a i n , i t was assumed t h a t drag f o r c e due to a s c e n d i n g f l u i < i s n e g l i g i b l e . S i n c e t h e r e i s no c r o s s f l o w of p a r t i c l e s i n t o the spou t i n t h i s r e g i o n , E q . ( 2 . 3 8 ) r e d u c e s to 2 g ( p . - p f ) V 2 = s f_ ( H . _ z ) ( 2 p s H 1 b e i n g the h e i g h t o f f o u n t a i n . In the nex t r e g i o n , f rom the l e v e l o f maximum p a r t i c l e v e l o c i t y to the bed s u r f a c e , the a s s u m p t i o n o f n e g l i g i b l e d rag f o r c e was r e t a i n e d , w h i l e the 33 e f f e c t o f c r o s s - f l o w i n g p a r t i c l e s was t aken i n t o a c c o u n t by assuming t h a t n z i s d i r e c t l y p r o p o r t i o n a l to Z . I n t e g r a t i o n o f the f o r c e b a l a n c e e q u a t i o n s , w i t h the upper boundary c o n d i -t i o n g i v e n by E q . ( 2 . 3 9 ) a t Z = H, g i v e s the f o l l o w i n g r e s u l t : 2 g ( p c - p f ) v - s s p M s 3H H 2 - 2 H 3 - Z , 3 T 2 r ( 2 In the l o w e s t r e g i o n , between the f l u i d i n l e t and the l owe r boundary of the p r e v i o u s r e g i o n , c r o s s - f l o w o f p a r t i c l e s to the annu lus was assumed to be c o n c e n t r a t e d a t the base o f the spou t ( i . e . d n z / d Z = 0 ) . E q . ( 2 . 3 8 ) t h e r e f o r e s i m p l i f i e s to „ 3 p f ( u s - v s > 2 d v 2 C D <PS - P f ) v s = - 4 d p 3 p s ^ 9 ( 2 The l ower boundary c o n d i t i o n f o r i n t e g r a t i o n o f E q . ( 2 . 4 1 ) i s u = u 0 and v = 0 a t Z = 0 , w h i l e the upper boundary i s l o c a t e d by i n t e r s e c t i o n w i t h E q . ( 2 . 4 0 ) . To t e s t the v a l i d i t y o f t h e i r m o d e l , Mathur and E p s t e i n [10 ] c a l c u l a t e d p a r t i c l e v e l o c i t y p r o f i l e s f o r wheat b e d s , and o b t a i n e d -good agreement w i t h e x p e r i m e n t a l d a t a (o f F i g u r e 2 . 8 ) , as shown i n F i g u r e 2 . 1 3 . I t s h o u l d be no ted t h a t the measured v e l o c i t i e s i n F i g u r e 2 . 1 3 were b i a s e d toward v e l o c i t i e s a t the spou t a x i s w h i c h , a c c o r d i n g to E q . ( 2 . 3 7 ) , a re about t w i c e the r a d i a l a v e r a g e v e l o c i t i e s . The c o m p a r i s o n between e x p e r i m e n t a l and p r e d i c t e d p a r t i c l e v e l o c i t i e s i n 34 Z / H Figure 2.13. Longitudinal p ro f i l es of rad ia l average pa r t i c l e ve loc i t y in the spout. Wheat bed: D c = 15.2 cm, H = 63.5 cm. by the Mathur-Epstein model [10] ; by the Lefroy-Davidson model [24]: Curve 1 using average spout diameter by McNab's equation, curve 2 assuming D s var ies l i n e a r l y from Do at Z = 0 to the value given by McNab's equation at Z = H/3 and then re -mains constant between H/3 < Z < H; o experimental data. 35 F i g u r e 2 . 1 3 , t h e r e f o r e , i n d i c a t e s t h a t the m o d e l , i n f a c t , o v e r - e s t i m a t e s the r a d i a l ave rage p a r t i c l e v e l o c t i e s ove r the e n t i r e h e i g h t o f the s p o u t . L e f r o y and Dav idson [24 ] a t t e m p t e d to p r e d i c t spou t p a r t i c l e v e l o c i t i e s f rom mass and momentum b a l a n c e s f o r both gas and s o l i d s f l o w i n the s p o u t . The momentum b a l a n c e f o r gas i s d K " s 2 > dZ dP £ s ST " B ( u s v s > 2 ( 2 . 4 2 ) w h i l e t h a t f o r s o l i d s i s (1 - , s ) v s > dZ dP = " ( 1 " e s } " d T " ( p s ' p f ) ( 1 - e s } 9 + 3 (u s - v s ) * ( 2 . 4 3 ) where B i s the g a s - s o l i d i n t e r a c t i o n f a c t o r , and i s g i v e n by the f o l l o w i n g e x p r e s s i o n B = 0 .33 p f ( l - e s ) dp e s 1 .78 ( 2 . 4 4 ) L e f r o y and Dav idson s o l v e d the above e q u a t i o n s v n u m e r i c a l l y f o r u , v , and e , w i t h the a s s i s t a n c e o f the mass b a l a n c e e q u a t i o n f o r spou t g a s , and the f o l l o w i n g e m p i r i c a l e q u a t i o n f o r l o n g i t u d i n a l p r e s s u r e d i s t r i b u t i o n i n the s p o u t : 36 dP " ~dT = B L ( p s - p f ) ( 1 " e m f ) 9 s i n [2HJ frrZl ( 2 . 4 5 ) where Mathur and E p s t e i n [10 ] found t h a t n u m e r i c a l s o l u -t i o n o f the above e q u a t i o n s f o r v g was ve ry s e n s i t i v e to the spou t c o n t o u r , as shown i n F i g u r e 2 . 1 3 . Curve 1 i n F i g u r e 2 .13 was computed u s i n g McNab 's e q u a t i o n [32] f o r ave rage spou t d i a m e t e r , w h i l e c u r v e 2 was based on the a s s u m p t i o n t h a t McNab 's e q u a t i o n a t Z = H / 3 , the l a t t e r v a l u e r e m a i n i n g c o n -s t a n t f o r the upper t w o - t h i r d s o f the b e d . The l a r g e d i f f e r e n c e between the two c u r v e s i n d i c a t e s t h a t an a c c u r a t e knowledge o f spou t d i a m e t e r and i t s v a r i a t i o n w i t h bed l e v e l , p a r t i c u -l a r l y near the j e t e n t r a n c e , i s e s s e n t i a l f o r g e t t i n g c o r r e c t answers f rom the L e f r o y - D a v i d s o n M o d e l . T h o r l e y et al. [ 11 ] and L e f r o y and Dav idson [24 ] examined h igh speed movie f i l m s ( 2 , 0 0 0 - 3 , 0 0 0 f r a m e s / s e c ) o f spou ted beds i n h a l f - c y l i n d r i c a l c o l u m n s . They o b s e r v e d t h a t near the a i r i n l e t n o z z l e , the spou t w a l l s were c o n t i n u a l l y c o l l a p s i n g , and p a r t i c l e s were b e i n g swept i n t o the h i g h v e l o c i t y j e t , but i n the upper p a r t o f the bed p a r t i c l e s near the i n t e r f a c e bounced i n t o the spou t by c o l l i s i o n w i t h the f a s t a s c e n d i n g p a r t i c l e s . Based on t h i s o b s e r v a t i o n L e f r o y and Dav idson [24 ] p roposed a c o l l i s i o n model to p r e d i c t the c r o s s - f l o w r a t e (o r e n t r a i n m e n t r a t e ) o f p a r t i c l e s f rom the D $ v a r i e d l i n e a r l y f rom D 0 a t Z = 0 to the v a l u e g i v e n by 37 a n n u l u s i n t o the s p o u t . They r e l a t e d the e n t r a i n m e n t v e l o c i t y , v ^ * of a p a r t i c l e ( no . 2) l e a v i n g the a n n u l u s to the v e l o c i t y o f the r i s i n g spou t p a r t i c l e ( no . 1 ) , V i , by c o n s i d e r i n g the sequence o f c o l l i s i o n s shown i n F i g u r e 2 . 1 4 . The model p o s t u -l a t e s t h a t p a r t i c l e no . 1 c o l l i d e s w i t h the s t a t i o n a r y p a r t i c l e no . 2 near the spou t w a l l . The l a t t e r p a r t i c l e i s t h e r e b y thrown a g a i n s t the w a l l and then rebounds back i n t o the spout a t v e l o c i t y . They a rgued t h a t p a r t i c l e no . 2 would e i t h e r become e n t r a i n e d i n the spou t gas or i t m igh t rebound back i n t o the spou t w a l l i f i t impac t s w i t h p a r t i c l e s a l r e a d y i n the s p o u t . The p r o b a b i l i t y o f s u c c e s s f u l e n t r a i n m e n t o f n o . ' 2 t ype p a r t i c l e s t h e r e f o r e depends on the v o i d a g e i n the spou t and i n the a n n u l u s ( e c and e , r e s p e c t i v e l y ) , and i s g i v e n by (e - e ) / ( l - e 3 ) . -» a s a a Assuming t h a t the r a d i a l p a r t i c l e v e l o c i t y p r o f i l e i n the spou t i s p a r a b o l i c ( E q . ( 2 . 3 7 ) ) , and p a r t i c l e no . 1 on the ave rage i s one p a r t i c l e d i a m e t e r away f rom the spou t w a l l , they a r r i v e d a t the f o l l o w i n g e x p r e s s i o n f o r c r o s s - f l o w r a t e per u n i t bed h e i g h t : 7 T 3 e ( l + e) d v (1 - e_)(e_ - e ) S - ) * - ( 2 . 4 6 ) a where e i s the c o e f f i c i e n t o f r e s t i t u t i o n . L e f r o y and Dav idson [24 ] d i d not make any d i r e c t v c o m p a r i s o n between p r e d i c t i o n by E q . ( 2 . 4 6 ) and e x p e r i m e n t a l d a t a but showed t h a t c r o s s - f l o w r a t e s c a l c u l a t e d by E q . ( 2 . 4 6 ) ag reed w e l l w i t h t h o s e o b t a i n e d f rom t h e i r momentum b a l a n c e e q u a t i o n s ( E q s . ( 2 . 4 2 ) and ( 2 . 4 3 ) ) . However , Mathur and Figure 2.14. Lefroy and Davidson's c o l l i s i o n model [24]: the sequence of c o l l i s i o n A, B and C leading to the entrainment of p a r t i c l e No. 2. CO CO 39 E p s t e i n [10] found t h a t c r o s s - f l o w r a t e s c a l c u l a t e d by E q . ( 2 . 4 6 ) were a l m o s t an o r d e r of magn i tude g r e a t e r than e x p e r i -menta l v a l u e s f o r wheat beds i n 15 .2 cm and 61 cm d i a m e t e r co lumns ( F i g u r e 2 . 1 2 ) . Summary High speed pho tog raphy and p i e z o e l e c t r i c t e c h n i q u e have been found to g i v e r e l i a b l e r e s u l t s o f p a r t i c l e v e l o c i t i e s i n the s p o u t . P a r t i c l e v e l o c i t i e s i n the a n n u l u s have been d e t e r m i n e d by d i r e c t measurement a g a i n s t the column w a l l . R a d i a l p r o f i l e s o f upward p a r t i c l e v e l o c i t y i n the spou t have been found to be g e n e r a l l y p a r a b o l i c . Data on l o n g i t u d i n a l p a r t i c l e v e l o c i t y p r o f i l e s i n the spou t a re ve ry l i m i t e d . R a d i a l ave rage v e l o c i t i e s a t d i f f e r e n t bed l e v e l s have been r e p o r t e d f o r g l a s s beads i n s m a l l co lumns ( D c = 7 .5 and 12 .5 cm) , and v e l o c i t i e s a l o n g the spou t a x i s i n l a r g e r . co lumns ( D £ = 15 cm and 61 cm) but f o r wheat o n l y . P a r t i c l e s i n the a n n u l u s have been o b s e r v e d to move v e r t i c a l l y downward and r a d i a l l y i nward toward the s p o u t - a n n u l us i n t e r f a c e . The c r o s s - f l o w r a t e o f p a r t i c l e s f rom the a n n u l u s i n t o the spou t has been found to be c o n s t a n t i n the upper p a r t o f the b e d . v T h e o r e t i c a l e q u a t i o n s have been d e v e l o p e d f o r p a r t i c l e v e l o c i t i e s both i n the spou t and in the a n n u l u s , but the agreement between p r e d i c t i o n f rom t h e s e e q u a t i o n s and the a v a i l a b l e e x p e r i m e n t a l da ta has p roved to be u n s a t i s f a c t o r y . 40 2 . 3 Bed S t r u c t u r e Spout Shape I t i s c l e a r f rom the d i s c u s s i o n i n the p r e c e d i n g s e c t i o n s t h a t an a c c u r a t e knowledge of spou t shape i s i m p o r t a n t f o r e s t i m a t i n g gas and s o l i d s f l o w no t o n l y i n the spou t but a l s o i n the a n n u l u s . Most o f the a v a i l a b l e i n f o r m a t i o n on spou t shape comes f rom o b s e r v a t i o n s made i n h a l f - c y l i n d r i c a l co lumns [ 1 0 ] . The use of a ha 1 f - c y l i n d r i c a 1 column r a i s e s the q u e s t i o n o f p o s s i b l e d i s t o r t i o n o f the shape o f the spou t by the f l a t f a c e o f the c o l u m n . The p o i n t has been i n v e s t i g a t e d by M i k h a i l i k [ 3 3 ] , who c a r r i e d out p a r a l l e l measurements o f spou t d i a m e t e r a t d i f f e r e n t bed l e v e l s i n h a l f and f u l l - c o l u m n s (9 .4 cm d i a ) , u s i n g a p i e z o e l e c t r i c probe f o r the l a t t e r . He c o n c l u d e d t h a t spou t shapes o b t a i n e d i n the two c o l u m n s , f o r a v a r i e t y o f s o l i d m a t e r i a l s , were q u i t e s i m i l a r . C o n f i r m a t i o n o f the above c o n c l u s i o n has been o b t a i n e d i n the p r e s e n t work [20 ] f o r a l a r g e bed o f r e c t a n g u l a r c r o s s - s e c t i o n (29 cm x 14 cm x 36 cm d e e p ) , u s i n g x - r a y pho tog raphy to d e t e r m i n e the spou t shape i n the f u l l column ( F i g u r e 8 . 1 ) . In h a l f - c y l i n d r i c a l c o l u m n s , the spou t has been o b s e r v e d to assume a v a r i e t y o f shapes under d i f f e r e n t e x p e r i -menta l c o n d i t i o n s ( F i g u r e 2 . 1 5 ) . The spou t d i a m e t e r can i n c r e a s e c o n t i n u o u s l y a l o n g t he h e i g h t o f the bed [ 2 , 1 1 , 3 4 ] , i t can i n c r e a s e in the l ower p a r t o f the bed and then t a p e r down s l i g h t l y i n the upper p a r t [ 1 1 , 2 4 ] , or i t can d e c r e a s e 41 DIVERGES CONTINUOUSLY NECKS,EXPANDS EXPANDS, THEN TAPERS THEN TAPERS SLIGHTLY v Figure 2.15. Observed spout shapes [10]. 42 s u d d e n l y near the gas e n t r a n c e , then i n c r e a s e and f i n a l l y rema in s u b s t a n t i a l l y unchanged f u r t h e r up [ 5 , 3 5 ] . Malek et al. [ 36 ] found t h a t the spou t d i a m e t e r d i d no t change much above a l e v e l o f about 2 .5 cm f rom the a i r i n l e t , d i a m e t e r s measured above t h a t l e v e l b e i n g w i t h i n abou t 10% o f the ave rage v a l u e . S e v e r a l e m p i r i c a l e q u a t i o n s [ 3 0 , 3 1 , 3 2 , 3 3 , 3 4 ] have been p roposed f o r the a v e r a g e spou t d i a m e t e r . Of t h e s e , the f o l l o w i n g e q u a t i o n due to McNab [ 3 2 ] , wh ich i s based on a s t a t i s t i c a l r e g r e s s i o n a n a l y s i s o f d a t a c o v e r i n g a w ide range o f v a r i a b l e s , has been found to be the most r e l i a b l e : 0 . 1 1 8 G 0 - 4 9 D ° ' 6 8 D s = 074T <2'47> p b T h e o r e t i c a l a n a l y s e s o f the b a l a n c e o f f o r c e s a c t i n g a t the s p o u t - a n n u l us i n t e r f a c e (see F i g u r e 2 . 1 6 ) have been c a r r i e d out by L e f r o y and Dav idson [24 ] and by B r i d g w a t e r and Mathur [ 3 7 ] . The outcome o f bo th a n a l y s e s a re s e m i - e m p i r i c a 1 e q u a t i o n s f o r ave rage spou t d i a m e t e r s , g i v e n b e l o w : L e f r o y and D a v i d s o n D s = 1 .07 D 2 7 3 d p / 3 ( 2 . 4 8 ) B r i d g w a t e r and M a t h u r Figure 2.16. Force balance models concerning spout shape, (a) Lefroy and Davidson [24]; (b) Bridgwater and Mathur [37]. 44 N e i t h e r e q u a t i o n g i v e s q u a n t i t a t i v e l y c o r r e c t p r e d i c t i o n o f spou t d i a m e t e r but E q . ( 2 . 4 9 ) i n c l u d e s a l l the i m p o r t a n t v a r i a b l e s i n v o l v e d and i t s f u n c t i o n a l form a g r e e s w i t h McNab 's e m p i r i c a l e q u a t i o n ( E q . ( 2 . 4 7 ) ) , wh ich i s based on a s t a t i s t i c a l r e g r e s s i o n a n a l y s i s o f a wide range o f e x p e r i m e n t a l d a t a . Voidage Distribution A n n u l u s : The a s s u m p t i o n t h a t the a n n u l u s of the spou ted bed i s e s s e n t i a l l y a l o o s e - p a c k e d bed [2 ] has been a c c e p t e d by a l l the wo rke rs in the f i e l d so f a r . T h i s i s a key a s s u m p t i o n f o r e s t i m a t i n g gas v e l o c i t y i n the a n n u l u s f rom p r e s s u r e g r a d i e n t d a t a , and f o r any t h e o r e t i c a l d e s c r i p -t i o n of annu lus gas f l o w . S l i g h t v a r i a t i o n o f v o i d a g e i n d i f f e r e n t p a r t s o f the a n n u l u s , e s p e c i a l l y i n the l o w e r p a r t , a re o b s e r v a b l e i n h a l f c y l i n d r i c a l c o l u m n s . E l j a s [14 ] r e c e n t l y measured the a n n u l u s v o i d a g e in a t w o - d i m e n s i o n a l bed (2 cm x 20 cm x 30 cm h e i g h t ) o f g l a s s beads by r e c o r d i n g the a b s o r p -t i o n of a y - r a y beam th rough the b e d . He found t h a t v a r i a t i o n o f v o i d a g e a t d i f f e r e n t l o c a t i o n s i n the a n n u l u s c o u l d be as h i g h as 0 . 1 . V a r i a t i o n s of t h i s o r d e r migh t be caused by the e f f e c t o f column w a l l s i n a t w o - d i m e n s i o n a l b e d . No work has v been done to measure the annu lus v o i d a g e d i s t r i b u t i o n i n t h r e e -d i m e n s i o n a l c o l u m n s . S p o u t The spou t i s l i k e a r i s e r i n wh i ch p a r t i c l e s a re be ing t r a n s p o r t e d in a d i l u t e p h a s e . Spout v o i d a g e 45 t h e r e f o r e depends on the i n t e r a c t i o n between gas and s o l i d f l o w p a t t e r n s i n the s p o u t . S e v e r a l methods have been used to s t udy the r a d i a l a v e r a g e v o i d a g e a l o n g the h e i g h t o f the s p o u t . Mathur and c o - w o r k e r [ 2 , 1 1 ] and Yokogawa e t <aZ.[19] deduced spout v o i d a g e f rom e x p e r i m e n t a l da ta on upward p a r t i c l e v e l o c i t y i n the spou t and downward v e l o c i t y i n the a n n u l u s . The f o l l o w i n g mass b a l a n c e e q u a t i o n was then used to e s t i m a t e , e s , the r a d i a l a v e r a g e spou t v o i d a g e a t any g i v e n l e v e l i n the b e d : D , * ( l - c , ) v , - 0 - E , ) ( D C « - D s » ) v a ( 2 . Mathur and c o - w o r k e r [ 2 , 1 1 ] measured v g f rom c i n e f i l m w i t h -ou t p a y i n g a t t e n t i o n to r a d i a l v a r i a t i o n s , and t h e i r spou t v o i d a g e r e s u l t s c a l c u l a t e d f rom E q . ( 2 . 5 0 ) , wh ich r e q u i r e s a r a d i a l ave rage v a l u e o f v , a re to t h a t e x t e n t a p p r o x i m a t e . L e f r o y [38 ] e s t i m a t e d spou t v o i d a g e i n a h a l f -c y l i n d r i c a l column by p h o t o g r a p h i n g the s p o u t and compar i ng the number of p a r t i c l e s per u n i t a rea w i t h the number i n a s i m i l a r a rea of known v o i d a g e ( a n n u l u s ) . T h i s me thod , as p o i n t e d out by L e f r o y , i s s u b j e c t to e r r o r , because the dep th o f f o c u s o f the camera was o n l y about one p a r t i c l e d i a m e t e r , V so t h a t p a r t i c 1 e s s l i g h t l y away f rom the column f a c e would not g i v e a c l e a r image on the pho tog raph and may not have been c o u n t e d . S e v e r a l S o v i e t wo rke rs [ 2 6 , 2 7 , 3 9 , 4 0 , 4 1 ] have used the p i e z o e l e c t r i c t e c h n i q u e to measure spou t v o i d a g e m o s t l y 46 f o r c o n i c a l b e d s , a l t h o u g h some measurements made i n s m a l l c y l i n d r i c a l beds ( 9 .4 cm d i a ) have a l s o been r e p o r t e d . Yokogawa et al. [ 1 9 ] , based on t h r e e s e t s o f da ta on v e r t i c a l p r o f i l e s o f r a d i a l ave rage spou t v o i d a g e , c o n -c l u d e d t h a t spou t v o i d a g e d e c r e a s e s l i n e a r l y w i t h i n c r e a s i n g bed l e v e l and r e a c h e s a v a l u e o f 0 .7 a t the bed s u r f a c e . T h i s c o n c l u s i o n i s not borne out by o t h e r e x p e r i m e n t a l r e s u l t s (see F i g u r e 5 .3 o f R e f . 1.0), nor by the da ta o b t a i n e d i n the p r e s e n t work ( C h a p t e r 8 . 2 ) . The o n l y t h e o r e t i c a l a n a l y s i s i n v o l v i n g spou t v o i d a g e has been made by L e f r o y and Dav idson [ 2 4 ] , spou t v o i d a g e b e i n g a pa ramete r i n t h e i r momentum b a l a n c e e q u a t i o n s f o r gas and s o l i d s f l o w i n the spou t ( E q s . ( 2 . 4 2 ) and ( 2 . 4 3 ) ) . These e q u a t i o n s , however , have been found to be ve ry s e n s i t i v e to spou t shape (see F i g u r e s 2 .13 and 7 . 1 5 ) , and d i d not p r e d i c t r e l i a b l e p a r t i c l e v e l o c i t i e s i n the s p o u t . C o n s e q u e n t l y , the e q u a t i o n s canno t be used to e s t i m a t e spou t v o i d a g e s , s i n c e i n a spou ted bed the upward f l o w r a t e o f s o l i d s i n the spou t must equa l the downward f l o w r a t e i n the a n n u l u s a t any g i v e n bed l e v e l ( E q . ( 2 . 5 0 ) ) . Summary No c o r r e l a t i o n s , e i t h e r e m p i r i c a l o r t h e o r e t i c a l , a re a v a i l a b l e f o r p r e d i c t i n g the shape o f the spout wh ich can undergo a s u b s t a n t i a l change i n i t s d i a m e t e r i n the lower p a r t o f the b e d . For e s t i m a t i n g ave rage spout d i a m e t e r s , s e v e r a l 47 e q u a t i o n s have been p r o p o s e d . Of t h e s e , McNab 's e m p i r i c a l e q u a t i o n ( E q . ( 2 . 4 7 ) ) has been found to be the most r e l i a b l e . In a l l spou ted bed work , whether e x p e r i m e n t a l o r t h e o r e t i c a l , the annu lus has a lways been c o n s i d e r e d e s s e n t i a l l y as a l o o s e - p a c k e d bed w i t h c o n s t a n t v o i d a g e . S l i g h t v a r i a t i o n s i n annu lus v o i d a g e have been o b s e r v e d i n h a l f - c y l i n d r i c a l c o l u m n s , the maximum v a r i a t i o n measured i n a t w o - d i m e n s i o n a l column b e i n g 0 . 1 . . No s t u d y has been made to i n v e s t i g a t e annu lus v o i d a g e i n f u l 1 - c y l i n d r i c a l c o l u m n s . Spout v o i d a g e has been measured d i r e c t l y u s i n g a p i e z o e l e c t r i c probe and i n d i r e c t l y f rom upward p a r t i c l e v e l o c i t y i n the spou t and downward p a r t i c l e v e l o c i t y i n the a n n u l u s . The e x p e r i m e n t a l da ta a r e , however , l i m i t e d and g i v e no c l e a r i n d i c a t i o n of the e f f e c t o f p a r t i c l e p r o p e r t i e s , column geometry and s p o u t i n g c o n d i t i o n on spou t v o i d a g e d i s t r i b u t i o n . 2 .4 C o n c l u s i o n s From the p r e c e d i n g l i t e r a t u r e r e v i e w , the f o l l o w i n g c o n c l u s i o n s can be drawn on the p r e s e n t s t a t u s o f t h e o r e t i c a l and e x p e r i m e n t a l work on gas and s o l i d s f l o w , and bed s t r u c -t u r e , i n spou ted b e d s . 1 (1) E x p e r i m e n t a l t e c h n i q u e s f o r measu r i ng gas v e l o c i t i e s i n the spou t and i n the c y l i n d r i c a l p a r t o f the a n n u l u s , p a r t i c l e v e l o c i t i e s i n the spou t and i n the a n n u l u s , spou t v o i d a g e and spou t shape have been e s t a b l i s h e d . 48 (2) The i n f o r m a t i o n a v a i l a b l e on gas f l o w p a t t e r n i s r e s t r i c t e d to v e r y nar row ranges o f the i n d e p e n d e n t v a r i a b l e s , and m o s t l y c o n s i s t s o f v e r t i c a l gas v e l o c i t y p r o f i l e s i n the c y l i n d r i c a l p a r t o f the a n n u l u s and r a d i a l p r o f i l e s i n the s p o u t . F low p a t t e r n o f gas i n the l owe r c o n i c a l p a r t o f the annu lus 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 , and no work has been done to measure gas r e s i d e n c e t ime d i s t r i b u t i o n i n spou ted b e d s . (3) The e q u a t i o n o f Mamuro and H a t t o r i , E q . ( 2 . 1 3 ) , has been f o u n d , on the b a s i s o f the a v a i l a b l e d a t a , to p r e d i c t the v e r t i c a l gas v e l o c i t y p r o f i l e i n the c y l i n d r i c a l p a r t o f the a n n u l u s r e a s o n a b l y w e l l . (4) C o n s i d e r a b l e e x p e r i m e n t a l i n f o r m a t i o n i s a v a i l -a b l e on r a d i a l p r o f i l e s o f upward p a r t i c l e v e l o c i t y , and on s i n g l e p a r t i c l e t r a j e c t o r i e s , i n the s p o u t . However , da ta on l o n g i t u d i n a l p r o f i l e s o f p a r t i c l e v e l o c i t y , whe the r f o r d i f f e r e n t r a d i a l p o s i t i o n s or r a d i a l a v e r a g e v a l u e s , a re r e l a -t i v e l y few . None o f the t h e o r e t i c a l e q u a t i o n s f o r p r e d i c t i n g l o n g i t u d i n a l p a r t i c l e v e l o c i t y p r o f i l e s g i v e s r e l i a b l e r e s u l t s . (5) No s y s t e m a t i c s t udy has been made o f spou t v o i d a g e d i s t r i b u t i o n and spou t s h a p e , and t h e r e a re no s a t i s -f a c t o r y methods f o r p r e d i c t i n g t h e s e c h a r a c t e r i s t i c s . C h a p t e r 3 PROGRAMME OF PRESENT INVESTIGATION The programme o f the p r e s e n t i n v e s t i g a t i o n i s o u t -l i n e d b e i o w : (1) V i s u a l o b s e r v a t i o n o f g a s f l o w p a t t e r n i n s p o u t e d b e d s u s i n g N0 2 g a s a s t r a c e r . ( 2 ) F o r m u l a t i o n o f a m a t h e m a t i c a l m o d e l , b a s e d o n t h e a b o v e v i s u a l o b s e r v a t i o n s , t o d e s c r i b e g a s m o v e m e n t a n d t o p r e d i c t g a s r e s i d e n c e t i m e d i s t r i b u t i o n i n t h e s p o u t e d b e d . (3) M e a s u r e m e n t o f r e s i d e n c e t i m e d i s t r i -b u t i o n o f g a s i n t h e s p o u t a n d i n t h e a n n u l u s , u s i n g h e l i u m g a s a s t r a c e r , u n d e r a w i d e r a n g e o f e x p e r i m e n t a l c o n d i t i o n s . (4) M e a s u r e m e n t o f b u l k d i s t r i b u t i o n o f g a s b e t w e e n t h e s p o u t a n d t h e a n n u l u s , s p o u t s h a p e a n d s p o u t v o i d a g e u n d e r c o n d i t i o n s c o r r e s p o n d i n g t o e a c h r e s i -d e n c e t i m e d i s t r i b u t i o n e x p e r i m e n t . (5) V e r i f i c a t i o n a n d p o s s i b l y m o d i f i c a t i o n o f t h e m a t h e m a t i c a l m o d e l by c o m p a r i n g e x p e r i m e n t a l RTD c u r v e s a g a i n s t m o d e l p r e d i c t i o n . (6) A n a l y s i s o f t h e h y d r o d y n a m i c d a t a m e n -t i o n e d u n d e r (4) a b o v e t o m a k e c r i t i c a l a s s e s s m e n t o f p u b l i s h e d t h e o r i e s a n d c o r r e l a t i o n s , a n d t o i m p r o v e u p o n t h e s e whe r e v e r p o s s i b l e . 49 50 The ranges o f i n d e p e n d e n t v a r i a b l e s s t u d i e d i n t h i s work a re shown i n T a b l e 3 . 1 . T a b l e 3.1 Ranges o f V a r i a b l e s S t u d i e d M a t e r i a l s d P mm Ps gm/cm 3 cm Do cm. H cm U / U s ms Ammonium n i t r a t e G l a s s beads 1 .1 1 .05 15 .2 1 .27 15 .2 1 .1 M i l l e t to to to to to to P o l y s t y r e n e p e l l e t s 3 .5 2 .96 2 9 . 2 3.86 91 .4 1 .3 Wheat The methods used to o b t a i n the e x p e r i m e n t a l da ta a r e b r i e f l y summar ized b e l o w : (1) V i s u a l o b s e r v a t i o n : by i n j e c t i n g c o l o u r e d gas ( N 0 2 ) c o n t i n u o u s l y a t v a r i o u s p o s i t i o n s i n t o the s p o u t e d bed a t p o i n t s c l o s e to the f l a t f a c e o f ha 1 f - c y l i n d r i c a 1 c o l u m n s . (2 ) R e s i d e n c e t i m e d i s t r i b u t i o n : by i n j e c t i n g h e l i u m gas t r a c e r i n t o the bed a t the gas i n l e t n o z z l e as a n e g a t i v e s t e p f u n c t i o n and m e a s u r i n g the t r a c e r c o n c e n t r a t i o n by the rma l c o n d u c t i v i t y c e l l s . (3) Gas v e l o c i t y i n t h e s p o u t : by the dynamic p r e s s u r e t e c h n i q u e u s i n g a p i t o t t u b e . 51 (4) Gas v e l o c i t y i n t h e u p p e r s e c t i o n o f t h e  a n n u l u s : by the s t a t i c - p r e s s u r e t e c h n i q u e d e v e l o p e d by Mathur and c o - w o r k e r s . ( 5 ) Gas v e l o c i t y i n t h e - lower s e c t i o n o f t h e  a n n u l u s : by c a l c u l a t i o n f rom t o t a l gas f l o w r a t e and spou t gas v e l o c i t y measured by p i t o t t u b e , and knowledge o f spou t d i a m e t e r . (6) S p o u t d i a m e t e r and s h a p e : measured d i r e c t l y a g a i n s t the f l a t f a c e o f h a l f - c y l i n d r i c a l c o l u m n s . (7 ) S p o u t v o i d a g e d i s t r i b u t i o n : o b t a i n e d i n d i r e c t l y f rom p a r t i c l e v e l o c i t i e s i n the spou t and the a n n u l u s u s i n g the s o l i d s mass b a l a n c e g i v e n by E q . ( 2 . 5 0 ) . (8) S o l i d s v e l o c i t y i n t h e a n n u l u s : measured d i r e c t l y i n h a l f - c y l i n d r i c a l c o l u m n s . (9) S o l i d s v e l o c i t y i n t h e s p o u t : by. t a k i n g h i g h speed movie f i l m (2000-3000 f r a m e s / s e c o n d ) o f the p a r t i c l e s i n the spou t i n h a l f - c y l i n d r i c a l co lumns and then a n a l y s i n g the f i l m f rame by f r a m e . C h a p t e r 4 APPARATUS AMD MATERIALS 4 .1 Appara tus Three c y l i n d r i c a l co lumns o f 1 5 . 2 , 2 4 . 1 , and 29 .2 cm i n s i d e d i a m e t e r s were used i n the e x p e r i m e n t s . The columns were made o f 4 . 8 mm t h i c k P I e x i g l a s s , and each one was f i t t e d w i t h a con - i ca l base made o f coppe r s h e e t , as shown i n F i g u r e 4 . 1 . D e t a i l s o f the gas i n l e t o r i f i c e , wh ich c o u l d be c o n -v e n i e n t l y changed to d i f f e r e n t d i a m e t e r s , a re a l s o shown i n the f i g u r e ( F i g u r e 4 . 1 ) . Both the c y l i n d r i c a l and the c o n i c a l s e c t i o n s were d r i l l e d and tapped w i t h 3 mm p i p e t h r e a d h o l e s t h rough the w a l l , 5.1 cm a p a r t , to a l l o w i n s e r t i o n o f p i t o t tube and s a m p l i n g p robes f o r t r a c e r c o n c e n t r a t i o n measu remen ts . A d raw ing of the 15 cm d i a m e t e r column i s shown i n F i g u r e 4.1 and pho tog raphs o f the two l a r g e r u n i t s i n F i g u r e 4 , 2 . Three h a l f - c y l i n d r i c a l P l e x i g l a s s co lumns h a v i n g the same d i a m e t e r s as the f u l l co lumns were a l s o c o n s t r u c t e d , and were used to o b s e r v e p a r t i c l e s m o t i o n , spou t s h a p e , and gas f l o w p a t t e r n ( w i t h N 0 2 t r a c e r ) . The h a l f - c o l u m n c o n s i s t e d of a ha 1 f - c y 1 i n d e r and a h a l f - c o n e ( s e m i - c i r c u l a r c r o s s - s e c t i o n ) , 52 53 Conica base, copper I. 5mm thick Solids discharge 1 2.5cm I. D. Copper pipe 5 cm I. D. P r o b e ho lder Dr i l l & t a p i /e" N.P.T. C o l u m n , P l e x i g l a s s 4 .8 m m th i ck Dr i l l 8 t a p 1/8" N .P .T . D o V7777, \*- 3 mm S ^ 7 7 7 - 7 7 - ^ --5.5cm • I 6 mm Opening for gas inlet nozzle G a s in let n o z z l e S p o u t i n g g a s Figure 4 . 1 . Diagram of 15.2 cm diameter column and gas i n l e t nozzle. 55 w i t h a s e m i - c i r c u l a r a i r - i n l e t o r i f i c e and a f l a t f a c e o f 4 . 76 mm, t h i c k g l a s s s h e e t , as shown i n F i g u r e 4 . 3 . A s c h e m a t i c d iag ram o f the a p p a r a t u s used i s shown i n F i g u r e 4 . 4 , and a key to the f i g u r e i s g i v e n i n T a b l e 4 . 1 . S p o u t i n g a i r f rom the mains f l o w e d th rough the r o t a m e t e r and the s t r a i g h t e n i n g s e c t i o n E , e n t e r i n g the bed th rough the i n l e t o r i f i c e . Compressed h e l i u m from a c y l i n d e r , H, was reduced to about 5 p s i g by p a s s i n g i t t h rough the p r e s s u r e r e g u l a t o r , G. I t then f l o w e d th rough a sys tem o f b u f f e r v e s s e l s , a f l o w m e t e r , and s o l e n o i d v a l v e s b e f o r e e n t e r i n g the s p o u t i n g a i r p i p e a t a p o i n t 5 cm below the a i r - i n l e t o r i f i c e . A s e p a r a t e s u p p l y o f mete red a i r , c o n t r o l l e d by the s o l e n o i d v a l v e S 3 , was c o n n e c t e d to the h e l i u m l i n e j u s t b e f o r e i t s e n t r y i n t o the s p o u t i n g a i r p i p e . S o l e n o i d v a l v e s S I , S2 and S3 were c o n n e c t e d i n such a way t h a t when SI and S2 were o f f , S3 was o n , and v i c e v e r s a . The a i r s u p p l y c o n t r o l l e d by S3 was used to sweep away the r e s i d u a l h e l i u m gas i n the i n j e c t i o n l i n e when t r a c e r v a l v e s SI and S2 were c l o s e d , so as to s h o r t e n the t r a i l i n g p a r t o f the h e l i u m c o n c e n t r a t i o n v s . t ime c u r v e . The c o n c e n t r a t i o n o f h e l i u m i n the s p o u t i n g a i r a t the p o i n t o f i t s e n t r y i n t o the b e d , and above the b e d , was measured s i m u l t a n e o u s l y by two the rma l c o n d u c t i v i t y c e l l s ( C a r l e Model 100 m i c r o d e t e c t o r s y s t e m s ) , TI and T2 i n F i g u r e 4 . 4 . C o n t i n u o u s measurement o f h e l i u m c o n c e n t r a t i o n was a c c o m p l i s h e d by s a m p l i n g a s m a l l volume of a i r - h e l i u m m i x t u r e Spouting Figure 4 .4 . Schematic flow diagram of the apparatus (see Table 4.1 for descr ip t ion) T a b l e 4.1 Key to F i g u r e 4 .4 A A n n u l u s B Spout B l , B 2 , B 3 B u f f e r v e s s e l s C Column, P I e x i g l a s s (see F i g u r e 4.1 and 4 . 2 ) E S t r a i g h t e n i n g s e c t i o n 5 cm I .D . x 60 cm, coppe r p i p e F Ro tame te rs G P r e s s u r e r e g u l a t o r H Compressed h e l i u m c y l i n d e r L I , L 2 Samp l i ng 1i nes 0 O s c i 1 1 o s c o p e P 1 , P 2 P r e s s u r e t r a n s d u c e r s PT P i t o t tube (see F i g u r e 4 . 5 ) R R e c o r d e r SP S t a t i c p r e s s u r e p robe (see F i g u r e 4 . 6 ) S 1 , S 2 , S 3 S o l e n o i d v a l u e s T l , T 2 Thermal c o n d u c t i v i t y c e l l s Q S t a t i c p r e s s u r e probe h o l d e r (see F i g u r e 4 . V Vacuum pump X V a l v e s C a l b e P r e s s u r e t r a n s m i s s i o n tube — ^ — Gas l i n e 59 c o n t i n u o u s l y i n t o the t h e r t n i s t e r d e t e c t o r s t h rough the s a m p l i n g l i n e s L l and L 2 . The s u c t i o n f o r c e needed f o r s a m p l i n g was p roduced by the vacuum pump, V , a t t a c h e d to the s a m p l i n g sys tem as shown i n F i g u r e 4 . 4 . F low r a t e s o f the r e f e r e n c e a i r and the sample gas p a s s i n g t h rough the d e t e c t o r s were c o n t r o l l e d by r o t a m e t e r s . B u f f e r v e s s e l , B 3 , s e r v e d to m i n i m i z e f l u c -t u a t i o n s i n f l o w r a t e caused by the vacuum pump. The o u t p u t s of the d e t e c t o r sys tem were f ed i n t o a dua l -beam o s c i l l o s c o p e , and t r a c e r c o n c e n t r a t i o n c u r v e s a p p e a r i n g on the o s c i l l o s c o p e s c r e e n were pho tog raphed u s i n g a p o l a r o i d camera s p e c i a l l y d e s i g n e d f o r t h i s p u r p o s e . A i r v e l o c i t y i n the spout was measured w i t h two p i t o t tube p robes made of 1.58 mm. O . D . tube w i t h tube l e n g t h o f 20 cm and 30 cm (see F i g u r e 4 . 5 f o r d e t a i l s ) . The p i t o t tube was i n s e r t e d i n t o the spou t t h rough o p e n i n g s i n the column w a l l and was h e l d i n p o s i t i o n by a 1/8 i n c h p i pe t h r e a d and c o m p r e s s i n g f i t t i n g , as shown i n the f i g u r e . A s c a l e a t t a c h e d to the tube i n d i c a t e d the e x a c t r a d i a l p o s i t i o n o f the p r o b e . Measurements were made a t 0 .2 to 1 cm i n t e r v a l s a c r o s s the d i a m e t e r o f the s p o u t , and a t s e v e r a l l o c a t i o n s a l o n g the h e i g h t . S t a t i c p r e s s u r e g r a d i e n t s i n the annu lus were measured w i t h the p r e s s u r e probe shown i n F i g u r e 4 . 6 . The probe c o n s i s t s o f two c o n c e n t r i c s t a i n l e s s s t e e l tubes w i t h hypodermic n e e d l e s s i l v e r - s o l d e r e d a t the e n d , and bent a t an a n g l e o f 9 0 ° , so t h a t the open ends o f the n e e d l e s a re Figure 4 .5 . Schematic diagram of p i to t tube. 61 S e a l Mount ing chuck ' / / / / / / / / / a m ^ z . Probe holder 4 .8 mm O.D. 120 cm 2 2 Gauge hypodermic needle Figure 4 .6 . Schematic diagram of s ta t i c pressure probe. 62 normal to the f l o w when the probe i s i n s e r t e d v e r t i c a l l y i n t o the annu lus f rom the t o p . The probe was h e l d i n p o s i t i o n by b o l t i n g i t to a P l e x i g l a s s d i s c , w i t h a 1.27 cm wide s l o t a c r o s s i t s d i a m e t e r , wh ich was used to cove r the top o f the column (see F i g u r e 4 . 1 ) . Thus the probe c o u l d be moved l a t e r a l l y a l s o i n the s l o t , to any d e s i r e d r a d i a l p o s i t i o n i n the column . The p i t o t tubes and the s t a t i c p r e s s u r e probe were c o n n e c t e d to p r e s s u r e t r a n s d u c e r s (PI and P2 i n F i g u r e 4 . 4 ) . Two Sta tham d i f f e r e n t i a l p r e s s u r e t r a n s d u c e r s w i t h ranges o f ±0 .3 p s i g and ±2 .5 p s i g were u s e d . The t r a n s d u c e r s c o n -v e r t e d p r e s s u r e to an e l e c t r i c a l s i g n a l wh ich was r e c o r d e d on a c h a r t r e c o r d e r . The p r e s s u r e t r a n s d u c e r s were c a l i b r a t e d a g a i n s t mic romanometer (Model MM3) wh ich has an a c c u r a c y o f 0 .005 m i l l i m e t e r s o f N - b u t y l a l c o h o l c o l u m n . The c a l i b r a t i o n c u r v e s and o t h e r d e t a i l s of the t r a n s d u c e r s a re g i v e n v i n A p p e n d i x I I . 4 . 2 S o l i d s P r o p e r t i e s The s o l i d m a t e r i a l s used i n t h i s work were whea t , m i l l e t , O t t a w a s a n d , two s i z e s o f p o l y s t y r e n e p e l l e t s , two s i z e s o f g l a s s beads and two s i z e s o f ammonium n i t r a t e . Each m a t e r i a l was s c r e e n e d to a r e l a t i v e l y narrow s i z e r a n g e . P a r t i c l e s i z e , p a r t i c l e d e n s i t y , bu l k d e n s i t y , l o o s e l y - p a c k e d v o i d a g e , a n g l e of r e p o s e , a n g l e of i n t e r n a l f r i c t i o n and the 63 a n g l e o f f r i c t i o n between a P l e x i g l a s s s h e e t and the p a r t i c l e s were measured f o r a l l the m a t e r i a l s u s e d . The r e s u l t s a r e p r e s e n t e d i n T a b l e 4 . 2 . Particle Size The g l a s s beads used were a l m o s t p e r f e c t s p h e r e s o f u n i f o r m s i z e and t h e i r d i a m e t e r was measured w i t h a m i c r o -meter (20 p a r t i c l e s o f each s i z e ) . The d i a m e t e r o f Ot tawa s a n d , wh ich i s a l s o n e a r l y s p h e r i c a l i n s h a p e , was t aken as the a r i t h m e t i c mean o f s c r e e n a p e r t u r e s . For whea t , m i l l e t , p o l y s t y r e n e and ammonium n i t r a t e , wh ich a r e a l l n o n - s p h e r i c a l i n s h a p e , the p a r t i c l e d i a m e t e r i s d e f i n e d as the d i a m e t e r o f a sphe re o f the same volume as the p a r t i c l e . I t was d e t e r -mined by c o u n t i n g a l a r g e number o f p a r t i c l e s ( e . g . 6000 p a r t i c l e s f o r p o l y s t y r e n e , 2000 p a r t i c l e s f o r w h e a t ) , w e i g h i n g them and m e a s u r i n g t h e i r , volume by wa te r d i s p l a c e m e n t , u s i n g a 100 c . c . v o l u m e t r i c f l a s k . Benzene was used as the l i q u i d f o r wa te r s o l u b l e ammonium n i t r a t e w h i l e f o r wheat and m i l l e t , wh ich a re pe rmeab le to w a t e r , the p a r t i c l e s were c o a t e d w i t h a l i q u i d p l a s t i c b e f o r e the volume measurement . The s i z e o f w h e a t , m i l l e t and p o l y s t y r e n e p a r t i c l e s was a l s o o b t a i n e d by m e a s u r i n g the s i z e o f about 20 randomly p i c k e d p a r t i c l e s f rom wh ich the equ i vo lume sphe re d i a m e t e r s were c a l c u l a t e d , assuming wheat to be a p r o l a t e s p h e r o i d , m i l l e t an o b l a t e s p h e r o i d and p o l y s t y r e n e an e l l i p t i c a l c y l i n d e r . The ave rage 64 Table 4.2 Propert ies of So l id Mater ia ls Used Mater ia ls V mm p s ' gm/cm3 gm/cm3 e a a Degree Degree Y Degree 5 Ammonium n i t ra te 1.45 1 .74 0.98 0.44 34.8 60.2 20.3 1.99 1.75 0.94 0.46 34.8 58.2 22.2 Glass beads 1.10 2.96 1 .74 0.41 24.1 51.0 15.2 1.00 2.93 2.94 1 .72 0.42 25.1 52.2 17.0 1.00 * M i l l e t 2.34 mm x 1.77 mm 2.15 1 .18 0.68 0.42 28.3 62.4 20.5 0.98 Ottawa sand 0.718 2.32 1.35 0.42 63.5 22.0 * Polystyrene p e l l e t s , 3.33 nun height x 2.66 mm x 1.97 mm 1 .61 1 .05 0.53 0.49 33.4 65.5 25.4 2.93 1 .05 0.57 0.41 35.5 62.2 22.5 0.85 *Wheat 5.14 mm x 2.82 mm 3.50 1 .24 0.70 0.44 28.0 53.8 22.3 0.94 * Assume: m i l l e t is oblate spheroid wheat is prolate spheroid polystyrene (d = 2.93 mm) is e l l i p t i c a l cy l inder 65 p a r t i c l e d i a m e t e r s o b t a i n e d by t h i s method ag reed c l o s e l y (± 1.5%) w i t h t h o s e o b t a i n e d f rom the l i q u i d d i s p l a c e m e n t me thod . Density P a r t i c l e d e n s i t y was o b t a i n e d by m e a s u r i n g the volume o f l i q u i d d i s p l a c e d by a known w e i g h t o f p a r t i c l e s , as d e s c r i b e d a b o v e . Bu l k d e n s i t y o f l o o s e l y packed s o l i d s was measured by p a r t i a l l y f i l l i n g a 500 c . c . g r a d u a t e d c y l i n d e r w i t h a known w e i g h t o f s o l i d m a t e r i a l . The c y l i n d e r was i n v e r t e d w i t h i t s mouth c o v e r e d and was then q u i c k l y i n v e r t e d a g a i n to i t s o r i g i n a l p o s i t i o n . The volume o c c u p i e d by the s o l i d s a f t e r t h i s l o o s e n i n g p r o c e d u r e was r e c o r d e d . The ave rage r e s u l t ^ o f f i v e r e p l i c a t e t e s t s was used to c a l c u l a t e the b u l k d e n s i t y . The v a r i a t i o n between t e s t s was ±1 .0%. T h i s p r o c e d u r e has been used p r e v i o u s l y by Oman and Watson [ 42 ] and Eastwood et al. [ 4 3 ] . The p o r o s i t y o f the s o l i d s i n the l o o s e - p a c k e d c o n d i t i o n s was c a l c u l a t e d f rom p a r t i c l e and bu l k d e n s i t i e s as f o l l o w s : Angle of Repose, a The a n g l e of r e p o s e , a, i s the a n g l e formed between the s l o p i n g s i d e o f a f r e e l y formed heap o f the s o l i d 66 m a t e r i a l and the h o r i z o n t a l base [ 5 3 ] . A heap o f about 0 .02 m 3 of the p a r t i c l e s was formed by p o u r i n g the m a t e r i a l t h rough a s m a l l f u n n e l , h e l d i n p l a c e s l i g h t l y above the apex o f the c o n i c a l heap . Angle of Internal Friction, § The a n g l e o f i n t e r n a l f r i c t i o n o f the s o l i d s , cj), was d e t e r m i n e d u s i n g the method s u g g e s t e d by Zenz and Othmer [ 4 4 ] . The a p p a r a t u s used was a c y l i n d r i c a l P l e x i g l a s s c o l u m n , 9 .53 cm I .D . x 45 cm h e i g h t , open a t the top and c l o s e d a t the b o t t o m . A s m a l l ho l e ( 1 .27 cm d i a m e t e r ) was d r i l l e d a t the c e n t r e o f the f l a t - b o t t o m e d c l o s u r e . The column was f i l l e d w i t h the m a t e r i a l under t e s t , wh i ch was then a l l o w e d to d r a i n f r e e l y t h rough the open ing i n the b a s e . The s u r f a c e o f the p a r t i c l e s i n the column i n i t i a l l y moved down i n a h o r i z o n t a l p l a n e , and a d e p r e s s i o n s t a r t e d to d e v e l o p a t the c e n t r e o f the s u r f a c e when the bed h e i g h t d e c r e a s e d to a c e r t a i n v a l u e , H^ . The t a n g e n t o f the a n g l e o f i n t e r n a l f r i c t i o n o f the p a r t i c l e s was taken as WJ\).. Angle of Friction, y The a n g l e o f f r i c t i o n between the p a r t i c l e s and P l e x i g l a s s s u r f a c e , y> was measured by J a m i e s o n ' s method [ 4 5 ] . A r e c t a n g u l a r wooden t r a y (25 cm x 20 cm x 5 cm h i g h ) was f i l l e d w i t h the p a r t i c l e s under t e s t and c o v e r e d w i t h a f l a t 67 P l e x i g l a s s p l a t e (80 cm x 30 cm) . The t r a y was then i n v e r t e d so t h a t the s o l i d m a t e r i a l r e s t e d on the h o r i z o n t a l s u r f a c e o f the P l e x i g l a s s p l a t e . The p l a t e was then r a i s e d s l o w l y a t one end u n t i l the t r a y c o n f i n i n g the p a r t i c l e s began to move down. The a n g l e formed between the P l e x i g l a s s p l a t e and the h o r i z o n t a l , when the t r a y s t a r t e d to move was t aken as the a n g l e o f f r i c t i o n , y> f ° r the p a r t i c l e s and P l e x i g l a s s s u r f a c e . The c o e f f i c i e n t o f f r i c t i o n , u , between p a r t i c l e s and a s u r f a c e i s d e f i n e d as the t a n g e n t o f y. C h a p t e r 5 MODELLING OF GAS FLOW IN SPOUTED BEDS 5 .1 V i s u a l O b s e r v a t i o n N 0 2 g a s , wh ich has a deep red c o l o u r , was chosen as a t r a c e r to o b s e r v e the gas f l o w b e h a v i o u r i n the spou ted b e d . In o r d e r to a c c e n t u a t e the c o l o u r o f N 0 2 g a s , both the gas and the s p o u t i n g a i r were p r e h e a t e d to about 50°C [46 ] b e f o r e i n t r o d u c t i o n i n t o the b e d . Ot tawa s a n d , g l a s s beads and p o l y s t y r e n e p e l l e t s were used as the bed m a t e r i a l s . N 0 2 gas was i n j e c t e d c o n t i n u o u s l y a t v a r i o u s pos , i n t o the spou ted bed a t p o i n t s c l o s e to the f l a t f a c e o f a ha 1 f - c y 1 i n d r i c a l co lumn (15 cm d i a m e t e r ) so t h a t the movement o f t r a c e r gas c o u l d be seen c l e a r l y a g a i n s t the f l a t f a c e . I t was o b s e r v e d t h a t the gas e n t e r i n g the annu lus f rom the spout a t a g i v e n l e v e l t r a v e l l e d r a d i a l l y ou tward and v e r -t i c a l l y upward a l o n g a p a r t i c u l a r s t r e a m l i n e w i t h l i t t l e r a d i a l m i x i n g between s t r e a m l i n e s . C r o s s - f l o w i n g gas a t the bot tom of the spou t was seen to p e n e t r a t e deeper towards the column w a l l and then to f l o w upward a l o n g the w a l l to the s u r f a c e o f the b e d . A t y p i c a l s e t o f gas s t r e a m l i n e s f o r an Ot tawa sand bed i s shown i n F i g u r e 5 . 1 . 68 70 No gas b a c k m i x i n g was o b s e r v e d i n beds o f Ot tawa s a n d , p o l y s t y r e n e p e l l e t s and g l a s s beads a t low s p o u t i n g v e l o c i t i e s . However , a t h i g h v e l o c i t i e s ( U s ~ 1.3 ^ m s ) b a c k m i x i n g o f gas i n the c o n i c a l s e c t i o n o f the annu lus was o b s e r v e d i n beds o f p o l y s t y r e n e p e l l e t s and g l a s s b e a d s , as i l l u s t r a t e d i n F i g u r e 5 . 2 . T h i s o b s e r v a t i o n i s c o n s i s t e n t w i t h p i t o t tube measurements wh ich a l s o show e v i d e n c e o f b a c k m i x i n g a t h i gh gas v e l o c i t i e s i n beds o f p o l y s t y r e n e and g l a s s beads (see T a b l e 6 . 1 ) . In the h i g h v e l o c i t y s p o u t , t r a c e r gas moved too s w i f t l y to be o b s e r v e d but i t i s s a f e to assume t h a t i n t h i s h i g h l y t u r b u l e n t r e g i o n (Rep > 1 0 3 ) , t h e r e w i l l be good r a d i a l m i x i n g . o f gas a t any g i v e n bed l e v e l g i v i n g r i s e to s u b s t a n t i a l l y p l ug f l o w b e h a v i o u r [ 5 9 ] . 5 . 2 M a t h e m a t i c a l Model Based on the q u a l i t a t i v e p i c t u r e o f gas f l o w p a t t e r n as r e v e a l e d by the N 0 2 t r a c e r e x p e r i m e n t s , a m a t h e m a t i c a l model to d e s c r i b e gas movement i s f o r m u l a t e d . The model e n a b l e s the r e s i d e n c e t ime d i s t r i b u t i o n o f gas p a s s i n g t h rough the spou ted bed to be p r e d i c t e d f rom da ta on c e r t a i n h y d r o -dynamic f e a t u r e s o f the b e d . The v a l i d i t y o f the model i s l a t e r v e r i f i e d by compar ing the p r e d i c t e d gas r e s i d e n c e t ime d i s t r i b u t i o n a g a i n s t e x p e r i m e n t a l l y measured r e s u l t s . The model i s based on the f o l l o w i n g p o s t u l a t e s : 72 ( a ) S p o u t g a s i s i n p l u g f l o w . ( b ) A n n u l u s g a s f o l l o w s t h e p a t h d e s c r i b e d b y E q . ( 5 . 1 ) i n p l u g f l o w w i t h d i s p e r s i o n a l o n g t h e f l o w p a t h . N o d i s p e r s i o n o r m i x i n g o c c u r s i n t h e d i r e c t i o n n o r m a l t o t h e s t r e a m l i n e w h i l e d i s p e r s i o n a l o n g t h e s t r e a m l i n e i s r e p r e s e n t e d b y a d i s p e r s i o n c o e f f i c i e n t , D. ( c ) C r o s s - f l o w o f g a s f r o m t h e s p o u t i n t o t h e a n n u l u s o c c u r s a t a l l l e v e l s i n t h e t h e b e d . For c a l c u l a t i n g the gas f l o w path i n the a n n u l u s , we d i v i d e the v e r t i c a l h e i g h t o f the a n n u l u s i n t o M equa l i n t e r v a l s , and the w i d t h o f the a n n u l u s a t the top i n t o N equa l i n t e r v a l s . L e t each i n t e r v a l a t the a n n u l u s top r e p r e s e n t one pa th o f gas f l o w (see F i g u r e 5 . 3 ) , and l e t Q(J - 1) be the v o l u m e t r i c f l o w r a t e between s t r e a m l i n e s J - 1 and J . Gas mass b a l a n c e a t any l e v e l I g i v e s Q(J - 1 ) = u a ( i ) T T J J U J - 1 , I ) 2 - R ( J , I ) ( 5 . 1 ) where u" a( I) i s the upward s u p e r f i c i a l gas v e l o c i t y a t l e v e l I and R ( J , I ) the r a d i a l d i s t a n c e f rom the co lumn a x i s to the i n t e r s e c t i o n between the J t h s t r e a m l i n e and I. E q . ( 5 . 1 ) d e s c r i b e s the gas s t r e a m l i n e s wh ich can be c a l c u l a t e d u s i n g e x p e r i m e n t a l v a l u e s o f IL ( I ) o b t a i n e d f rom s t a t i c p r e s s u r e measure-a m e n t s . V e r t i c a l p r e s s u r e g r a d i e n t measured a t d i f f e r e n t r a d i a l p o s i t i o n i n the a n n u l u s d i d not show any s i g n i f i c a n t v a r i a t i o n , hence same v a l u e o f U n ( I ) was t a k e n f o r c a l c u l a t i n g Q ( J ) . The a outcome o f a s t r e a m l i n e c a l c u l a t i o n , s t a r t i n g f rom the top o f the a n n u l u s and end ing a t the s p o u t - a n n u l us i n t e r f a c e , i s shown i n F i g u r e 5 . 4 . I t was assumed t h a t the 73 J <j Q(J-I) N+l N j f j - l 2 I Figure 5 .3 . Co-ordinate system and gr id points for gas stream-l i ne ca lcu la t ions (note: z is the l i nea r distance along a st reaml ine, s ta r t ing from spout-annulus in te r face . The d i rec t ion of z changes with bed l e v e l , fo l lowing the s t reaml ine) . 74 75 f i r s t s t r e a m l i n e ( J = 1) c o i n c i d e s w i t h the column w a l l , as s u g g e s t e d by v i s u a l o b s e r v a t i o n o f N 0 2 t r a c e r movement ( F i g u r e 5 . 1 ) . The computed f l o w p a t t e r n i n F i g u r e 5.4 i s seen to be g e n e r a l l y s i m i l a r to t h a t o b s e r v e d v i s u a l l y . In o r d e r to d e t e r m i n e the gas v e l o c i t y a l o n g each s t r e a m l i n e , the a n g l e 9 formed between the s t r e a m l i n e and the v e r t i c a l a x i s was a measured f rom F i g u r e 5 .3 and u, ( I ) was c a l c u l a t e d u s i n g the a z f o l l o w i n g r e l a t i o n s h i p : u a z ( D = U a ( I ) / { e a cos 6 a > (5 To c a l c u l a t e the r e s i d e n c e t ime d i s t r i b u t i o n o f t r a c e r gas p a s s i n g t h rough the a n n u l u s , c o n s i d e r the c o n t r o l zone dz i n F i g u r e 5 .5 wh ich r e p r e s e n t s a v e r t i c a l s e c t i o n o f the a n n u l u s between two s t r e a m l i n e s . From c o n s e r v a t i o n o f t r a c e r f o r the c o n t r o l z o n e , we ge t A e a ~JT d z [ C a " <Ca + "jT d z ) Q(J)" 3C_ 9C (5 where A i s the ave rage c r o s s - s e c t i o n a l a r e a between l e v e l s I and I + 1 . E q u a t i o n ( 5 . 3 ) can be s i m p l i f i e d to 76 Figure 5.5. Tracer mass balance in the annulus. 77 3C a n M 1 3C n 3 2 C _a _ Q( J ) a . . D _ " a A £ a 3t — + A L 2" £ a s T ^ ( 5 ' 4 ) where z ' = z / L , L b e i n g the l e n g t h o f the gas f l o w p a t h . S i n c e Q ( J ) / e A e q u a l s u . , , the ave rage i n t e r s t i t i a l v e l o c i t y a a z between the s t r e a m l i n e s , E q . ( 5 . 4 ) may be r e w r i t t e n as f o l l o w s : 9C u 3C 3 2 C 3t L 3 z ' X7 JT^ [b-*> For n e g a t i v e s t e p f u n c t i o n i n p u t o f t r a c e r (used i n the e x p e r i -menta l wo rk , see p. 8 2 ) , the i n i t i a l and boundary c o n d i t i o n s a r e : f o r t < 0 , C = 1.0 ( 5 . 6 ) — a f o r t > 0 , 3 C a t z * = ° ' f I F " = u a z ( C a " C s > < 5 - 7 > 3 C a t z ' = 1 . 0 , j^- = 0 ( 5 . 8 ) The boundary c o n d i t i o n s r e p r e s e n t e d by E q s . ( 5 . 7 ) and ( 5 . 8 ) , have been d i s c u s s e d a t l e n g t h by Danckwer ts [ 4 7 ] , Wehner and W i l h e l m [48 ] and Gunn [ 4 9 ] , and have been w i d e l y used f o r d i s p e r s i o n models o f f l u i d f l o w i n packed and f l u i d i z e d beds [ 4 9 , 5 0 , 5 1 , 5 2 , 5 3 e t c . ) . E q . ( 5 . 7 ) s t a t e s t h a t the r a t e a t wh ich t r a c e r i s f ed to the bed ( i n t h i s case the a n n u l u s ) i s equa l to the r a t e a t wh ich i t c r o s s e s p l a n e z = 0 by 78 combined f l o w and d i f f u s i o n . E q . ( 5 . 8 ) r e p r e s e n t s a c o n c e p t u a l boundary c o n d i t i o n wh ich f o l l o w s f rom the i n t u i t i v e a rgumen t , o r i g i n a l l y advanced by Danckwer ts [ 4 7 ] , t h a t f o r f l o w th rough a packed v e s s e l , the t r a c e r c o n c e n t r a t i o n i m m e d i a t e l y below and above the bed s u r f a c e s h o u l d rema in the same. E q u a t i o n s ( 5 . 5 ) to ( 5 . 8 ) a p p l y to any p a r t i c u l a r gas pa th i n the a n n u l u s . For N p a t h s , we have N s e t o f t h e s e e q u a t i o n s , f o r each v a l u e o f J between 1 and N: 3 C a ( J ) u a z ( J ) 8 C a ( 0 ) D 3 2 C a ( J ) 3t + L(J) ' 3 z ' " LTTP 3 Z ' 2 = 0 { 5 ^ 9 ) f o r t < 0 , C a ( J ) - 1 ,0 = 0 ( 5 . 1 0 ) f o r t > 0 , a t z' = 0 , D 9 C a ( J ) L W ~Zp " u a z ( J ) c a ( J ) - c s ( j ) I s 0 ( 5 , 1 1 ) a t z' = 1 . 0 , 3C ( J ) a  3z ' = 0 ( 5 , 1 2 ) 79 In o r d e r to s o l v e the above s e t o f e q u a t i o n s an e x p r e s s i o n f o r C s , wh ich appears i n E q . ( 5 . 1 1 ) , i s r e q u i r e d . T r a c e r mass b a l a n c e ove r a d i f f e r e n t i a l h e i g h t o f the spou t dZ (see F i g u r e 5 . 6 ) , assuming p l ug f l o w of spou t gas and r a d i a l u n i f o r m i t y o f t r a c e r c o n c e n t r a t i o n , g i v e s 9C u 9C s s s 9t H 3Z ' ( 5 . 1 3 ) u s be i ng the i n t e r s t i t i a l upward gas v e l o c i t y i n the spou t and Z ' the d i m e n s i o n l e s s bed l e v e l Z / H . The r e l e v a n t i n i t i a l and boundary c o n d i t i o n s a r e , f o r t < 0 , C = 1 .0 f o r t > 0 , C s = C ! ( 5 . 1 4 ) ( 5 . 1 5 ) The s o l u t i o n o f E q . ( 5 . 1 3 ) u s i n g L a p l a c e t r a n s f o r m a t i o n i s as f o l l o w s : C s = 6 ( t ) - 6 V JL o U S dZ + C: rZ dZ ' •r x 6 d Z ' ( 5 . 1 6 ) where <5(t) i s the u n i t 6 ( t > 0) .= 1 . s t e p f u n c t i o n , 6 ( t < 0) = 0 , and 80 d Z C s + ^ s d Z dz dz A Spou t 6353 •> Annu lus Figure 5.6. Tracer mass balance in the spout. 81 E q u a t i o n s ( 5 . 1 ) and ( 5 . 2 ) t o g e t h e r w i t h e q u a t i o n s ( 5 . 9 ) , ( 5 . 1 0 ) , ( 5 . 1 1 ) , ( 5 . 1 2 ) and ( 5 . 1 6 ) c o n s t i t u t e the t h e o r e t i c a l model and can be s o l v e d n u m e r i c a l l y to y i e l d C , a as a f u n c t i o n of t ime ( i . e . the RTD c u r v e ) f o r any l o c a t i o n i n the b e d . The hydrodynamic da ta r e q u i r e d a re l o n g i t u d i n a l p r o f i l e s o f U" a, u s , e g and D g . The d i s p e r s i o n c o e f f i c i e n t i n E q s . ( 5 . 9 ) and ( 5 . 1 1 ) i s the o n l y a d j u s t a b l e pa ramete r o f the m o d e l , and t h i s has been e v a l u a t e d f o r each e x p e r i m e n t f rom a c o m p a r i s o n between p r e d i c t e d and o b s e r v e d RTD c u r v e s (see s e c t i o n 5 . 4 ) . The hydrodynamic da ta men t i oned were d e t e r m i n e d e x p e r i m e n t a l l y (see C h a p t e r s 6 and 7 ) , i n p a r a l l e l w i t h t r a c e r r e s p o n s e measurements . The n u m e r i c a l p r o c e d u r e s used f o r s o l v i n g the model e q u a t i o n s f o r C and f o r e v a l u a t i n g the a p a r a m e t e r D, t o g e t h e r w i t h the computer programmes u s e d , a r e p r e s e n t e d i n A p p e n d i c e s IV and V . I t s h o u l d be no ted t h a t a s i m p l e r model assuming v e r t i c a l l y upward p l u g - f l o w o f gas i n the a n n u l u s , w i t h s u p e r -imposed a x i a l d i s p e r s i o n but w i t h o u t any r a d i a l f l o w , was f i r s t p roposed [ 2 0 ] . E x p e r i m e n t s however showed t h a t r e s p o n s e c u r v e s measured a t d i f f e r e n t r a d i a l p o s i t i o n s i n the a n n u l u s were n o t i c e a b l y d i f f e r e n t , e s p e c i a l l y i n l a r g e r d i a m e t e r (> 15 cm) beds . ^ T h i s r a d i a l v a r i a t i o n o b v i o u s l y a r i s e s f rom the f a c t t h a t the l e n g t h of the f l o w path i n the a n n u l u s becomes s m a l l e r w i t h i n c r e a s i n g r a d i a l d i s t a n c e f rom the column w a l l , and was not a c c o u n t e d f o r i n the s i m p l e r model s i n c e the a s s u m p t i o n o f v e r t i c a l l y upward f l o w i m p l i e s r a d i a l u n i f o r m i t y o f gas r e s i d e n c e t i m e ! 82 5 .3 Measurement o f R e s i d e n c e Time D i s t r i b u t i o n o f Gas 5 .3 .1 E x p e r i m e n t a l P r o c e d u r e He l i um gas was i n j e c t e d i n t o the a i r app roach p i p e as a n e g a t i v e s t e p f u n c t i o n , u s i n g the sys tem o f s o l e n o i d v a l v e s shown i n F i g u r e 4 . 4 . P u l s e i n j e c t i o n was found to be u n s a t i s f a c t o r y s i n c e i t caused n o t i c e a b l e d i s t u r b a n c e and somet imes even c o l l a p s e o f the s p o u t i n g a c t i o n . The c o n c e n t r a t i o n o f t r a c e r i n the i n l e t gas ( s t i m u l u s ) and i n the e x i t gas ( r e s p o n s e ) , a t t he l o c a t i o n s shown i n F i g u r e 4 . 4 , were measured s i m u l t a n e o u s l y by d raw ing gas samples a t a p r e c i s e l y c o n t r o l l e d r a t e th rough a p a i r o f t he rma l c o n -d u c t i v i t y c e l l s . A t l e a s t t h r e e s e t s o f measurements were made f o r each s e t o f e x p e r i m e n t a l c o n d i t i o n s , k e e p i n g the s t i m u l u s m e a s u r i n g probe a t the same l o c a t i o n , and m e a s u r i n g the r e s p o n s e a t (1) the top c e n t r e o f the s p o u t , (2) the c e n t r e o f the a n n u l u s s u r f a c e , and (3) the a n n u l u s s u r f a c e , h a l f - w a y between the a n n u l u s c e n t r e and column w a l l . For some runs i n 15 .2 cm d i a m e t e r column and a l l the runs i n 24.1 cm and 2 9 . 2 cm d i a m e t e r c o l u m n s , a d d i t i o n a l r e s p o n s e measurements were made a t o t h e r r a d i a l l o c a t i o n s a c r o s s the bed s u r f a c e . 5 . 3 . 2 A c c u r a c y o f Data The main f a c t o r s wh ich have a b e a r i n g on the a c c u r a c y o f the RTD measurement a r e s a m p l i n g r a t e o f gas and any d i f f -e r e n c e i n the c h a r a c t e r i s t i c s o f the two c o n d u c t i v i t y c e l l s 83 used a t the gas i n l e t and o u t l e t p o i n t s . The e f f e c t o f both t h e s e f a c t o r s was c o r r e c t e d e x p e r i m e n t a l l y as f o l l o w s : ( 1 ) The s a m p l i n g r a t e s f o r b o t h s t i m u l u s and r e s p o n s e m e a s u r e m e n t s f o r e a c h e x p e r i m e n t were so c o n t r o l l e d t h a t when b o t h p r o b e s were p l a c e d a t t h e same l o c a t i o n , t h e i r r e s p o n s e t o a c h a n g e i n t r a c e r c o n c e n t r a t i o n o c c u r r e d a t p r e c i s e l y t h e same i n s t a n t . (2) The d i f f e r e n c e i n c h a r a c t e r i s t i c s o f t h e two t h e r m a l c o n d u c t i v i t y c e l l s was c h e c k e d by a g a i n p l a c i n g two sampl i n g p r o b e s a t t h e same l o c a t i o n , and r e c o r d i n g t h e r e s p o n s e s o f b o t h e e l Is t o i n j e c t i o n o f t r a c e r . F i g u r e 5 .7 shows a pho tog raph o f the r e s p o n s e c u r v e s o b t a i n e d u s i n g the two c o n d u c t i v i t y c e l l s , bo th p l a c e d a t the c e n t r e o f the gas i n l e t o r i f i c e . The t r a c e r was i n t r o d u c e d both as a p u l s e and as a n e g a t i v e s t e p i n j e c t i o n , the fo rmer f o r d e t e r m i n -i n g any s a m p l i n g r a t e a d j u s t m e n t r e q u i r e d and the l a t t e r f o r c o r r e c t i n g the s t i m u l u s t r a c e r c o n c e n t r a t i o n c u r v e s f o r the s l i g h t d i f f e r e n c e wh ich i s seen to e x i s t between the r e s p o n s e s o f the two the rma l c o n d u c t i v i t y c e l l s ( see F i g u r e 5 . 7 ) . S e v e r a l r e p l i c a t e measurements o f the s t i m u l u s and r e s p o n s e c u r v e s were c a r r i e d out to t e s t the r e p r o d u c i b i l i t y o f the r e s u l t s . R e p l i c a t e t e s t s showed t h a t bo th the s t i m u l u s and r e s p o n s e t r a c e r c o n c e n t r a t i o n c u r v e s c o u l d be r e p r o d u c e d v e r y c l o s e l y , as i l l u s t r a t e d by the r e s u l t s i n F i g u r e 5 . 8 . Response to tracer inject ion by cells TI S T 2 for sampl-ing rate adjustment Response to tracer stimulation (negative step) for stimulus curve correct ion. by cell T 2 by cel l TI Figure 5.7. Comparison of the responses of the two thermal conduct iv i ty c e l l s used in stimulus-response experiments ( ce l l s TI and 12 in Figure 4 .4 ) . Scales: 0.2 sec/square. 00 85 F i r s t tes t Repl icate test Figure 5.8. Reproduc ib i l i t y of t racer concentration curves: Scale 0.2 sec/ square, wheat bed, D c = 24.1 cm. Stimulus curves measured at gas i n l e t nozzle and response curves at bed sur face. For re-sponse curves: Al and Bl at spout axis (r = 0 ) , A2 and B2 at annulus centre (r = 7.6 cm) and A3 and B3 at r = 10 cm. 86 5 . 3 . 3 R e s u l t s F i g u r e s 5 .9 and 5 .10 show a s e r i e s o f t r a c e r c o n c e n -t r a t i o n c u r v e s measured i n two d i f f e r e n t d i a m e t e r beds a t s e v e r a l r a d i a l p o s i t i o n s a c r o s s the bed s u r f a c e , i n the spou t as w e l l as i n the a n n u l u s . These r e s u l t s , and a l l the o t h e r t r a c e r c o n c e n t r a t i o n da ta o b t a i n e d i n t h i s wo rk , show the f o l l o w i n g t r e n d s : ( i ) gas r e s i d e n c e t i m e i n t h e s p o u t i s much s h o r t e r t h a n i n t h e a n n u l u s ; ( i i ) t r a c e r c o n c e n t r a t i o n c u r v e s m e a s u r e d a t d i f f e r e n t r a d i a l p o s i t i o n s i n t h e s p o u t a r e s i m i l a r i n shape t o e a c h o t h e r , and a l s o t o t h e s t i m u l u s c u r v e a f t e r t h e l a t t e r i s c o r r e c t e d f o r t h e d i f f e r e n c e between t h e two t h e r m a l c o n d u c t i v i t y c e l l s ( s e e F i g u re 5 . 7 ) ; ( i i i ) t h e t i m e l a p s e b e t ween t h e i n s t a n t o f s t i m u l a t i o n and of r e s p o n s e f o r t h e s p o u t r e g i o n i n c r e a s e s s l i g h t l y w i t h i n c r e a s i n g r a d i a l d i s t a n c e f r o m t h e s p o u t a x i s ; ( i v ) t r a c e r c o n c e n t r a t i o n c u r v e s m e a s u r e d a t d i f f e r e n t p o i n t s a c r o s s t h e a n n u l u s s u r f a c e a r e n o t i c e a b l y d i f f e r e n t i n shape f r o m e a c h o t h e r ; and (v ) t h e t i m e l a p s e between s t i m u l a t i o n and r e s p o n s e f o r t h e a n n u l u s r e g i o n i n -c r e a s e s c o n s i d e r a b l y w i t h i n c r e a s i n g d i s t a n c e f r o m t h e s p o u t - a n n u I u s i n t e r f a c e . 5 . 3 . 4 P r o c e s s i n g o f Data A l l the t r a c e r c o n c e n t r a t i o n v s . t ime c u r v e s were d i g i t i z e d a t 0 .012 second i n t e r v a l s by a INSTRONICS GRADICON d i g i t e r i z e r h a v i n g an a c c u r a c y o f ±0 .001 i n c h . A computer 87 • A f 4-4-4 . i 1 B +•) 4 4- | A ) (||) M M • 1 1 1 tiit 4-M-f- 4 flf H •4-4 4 .4. i i i ) 1 M 1 M M -4-4 m c 4-f-4-f , 4 If .4-L 4-4.4-4 - M - M • M M L , \ ••f f 4-t -\ t 1 Figure 5.9. Tracer concentration curves for bed of polystyrene p e l l e t s , D c = 24.1 cm, H = 72 cm; scale 0.2 sec/square. Stimulus curves measured at gas i n l e t nozzle and response curves at CONTINUED 88 bed surface. For response curves: A - spout axis at r = 0, B - in spout at r = 1 cm, C - in spout at r = 2 cm, D - in annulus at r = 3.5 cm, E - in annulus at r = 5 cm, F - in annulus at r = 6.5 cm, G - in annulus at r = 8.4 cm, H = in annulus at r = 9 cm and I - in annulus at r = 10 cm. 89 Figure 5.10. Tracer concentration curves for bed of polystyrene p e l l e t s , D c = 29.2 cm, H = 91.4 cm. Scale 0.2 sec/square. Stimulus curves measured at gas i n l e t nozzle and response curves at bed surface, For response curves: A - spout axis at r = 0, B - in spout at r = 2 cm, C - in annulus of r = 6 cm, D - in annulus centre, E - in annulus at r = 10 cm and F - in annulus at r = 12 cm. 90 programme was then w r i t t e n to read i n the d i g i t i z e d da ta ( i n i n c h e s ) f o r each c u r v e and to n o r m a l i z e the c u r v e . N o r m a l i z a -t i o n o f the t r a c e r c o n c e n t r a t i o n cu rve i n v o l v e s c a l c u l a t i n g the d i m e n s i o n l e s s t r a c e r c o n c e n t r a t i o n , C / C 0 a t each p o i n t a l o n g the cu rve ove r the whole range o f t ime r e c o r d e d . The i n i t i a l t r a c e r c o n c e n t r a t i o n C 0 was t aken as the h e i g h t o f the t r a c e r c o n c e n t r a t i o n c u r v e above the base l i n e ( i . e . no o r n e g l i g i b l e t r a c e r p r e s e n t i n the gas s t r eam) a t t < 0 ( i . e . b e f o r e t r a c e r i n j e c t i o n was shu t o f f ) , and C was t aken as the h e i g h t o f the c u r v e above the base l i n e a t t > 0 . The n o r -m a l i z e d t r a c e r r e s i d e n c e t ime d i s t r i b u t i o n (RTD) c u r v e s were used to t e s t the v a l i d i t y o f the m a t h e m a t i c a l model p r e s e n t e d i n C h a p t e r 5 . 2 . The n o r m a l i z e d t r a c e r s t i m u l a t i o n c u r v e s were a l l found to be p r a c t i c a l l y o f the same s h a p e , and c o u l d be s u p e r -imposed on each o t h e r . A l s o the d i f f e r e n c e between the r e s p o n s e g i v e n by t he rma l c o n d u c t i v i t y c e l l T i and c e l l T 2 (see F i g u r e 4 . 4 ) was found to remain the same i n a l l e x p e r i m e n t s . Because o f the above f i n d i n g s , a s i n g l e common s t i m u l a t i o n c u r v e has been used f o r the purpose o f t e s t i n g the t h e o r e t i c a l m o d e l . T h i s s t i m u l a t i o n c u r v e was a r r i v e d a t by a v e r a g i n g a l l the n o r m a l i z e d s t i m u l a t i o n c u r v e s o b t a i n e d w i t h the the rma l c o n -d u c t i v i t y c e l l , T 2 , wh i ch was used to measure a l l the r e s p o n s e c u r v e s a t the bed s u r f a c e . 91 5.4 Compar i son Between P r e d i c t e d and E x p e r i m e n t a l RTD Curves The p r e d i c t e d and e x p e r i m e n t a l RTD c u r v e s have been compared to e v a l u a t e the a d j u s t a b l e model pa rame te r D by s e a r c h i n g f o r the v a l u e o f D r e q u i r e d to g i v e a c l o s e match between the p r e d i c t e d and measured c u r v e s . The c r i t e r i o n used f o r a " c l o s e match" was to m i n i m i z e the sum of the squa red d e v i a t i o n , A 2 , between p r e d i c t e d and o b s e r v e d v a l u e s o f C , a o v e r the e n t i r e range o f v a l u e s o f C g t aken a t 0 .012 seconds t ime i n t e r v a l s . That i s ( 5 . 1 7 ) where the s u b s c r i p t s p and e r e f e r to p r e d i c t e d and e x p e r i m e n t a l v a l u e s r e s p e c t i v e l y ( see A p p e n d i x V f o r computer programmes used) . A s e r i e s o f p r e d i c t e d RTD c u r v e s f o r f o u r d i f f e r e n t a s s i g n e d v a l u e s o f D i s shown i n F i g u r e 5 . 1 1 , wh ich a l s o i n -c l u d e s the c o r r e s p o n d i n g e x p e r i m e n t a l RTD c u r v e . A l l the c u r v e s shown r e f e r to a l o c a t i o n h a l f way a c r o s s the w i d t h o f the a n n u l u s , i m m e d i a t e l y above the a n n u l u s s u r f a c e . The v a r i a -t i o n o f A 2 , the sum o f the squa red d e v i a t i o n between the e x p e r i m e n t a l and p r e d i c t e d c u r v e s i n F i g u r e 5 . 1 1 , a c c o r d i n g to E q . ( 5 . 1 7 ) i s shown i n F i g u r e 5 . 1 2 . The minimum v a l u e o f A 2 i s seen to o c c u r a t D = 0 .016 m 2 / s e c . W i th t h i s v a l u e of D, the p r e d i c t e d c u r v e i n F i g u r e 5.11 g i v e s a v e r y c l o s e match 0- 0 . 2 0 . 4 0 . 6 0 . 8 1.0 1.2 1.4 T I M E , S E C . Figure 5.11. Normalized RTD curves corresponding to the response curves of Figure 5.9, for a point half-way across the annulus sur face. 93 Figure 5.12. Var ia t ion of A 2 with D ca lcu lated by Eq. (5.17) for the data of Figure 5.11. 94 w i t h the e x p e r i m e n t a l c u r v e o v e r the e n t i r e range o f r e s i d e n c e t i m e . A s e r i e s o f r e s p o n s e c u r v e s r e c o r d e d a t d i f f e r e n t r a d i a l p o s i t i o n s a c r o s s the annu lus s u r f a c e i s shown i n F i g u r e 5 .13 ( n o r m a l i z e d c u r v e s o f F i g u r e 5 . 9 ) . These c u r v e s i l l u -s t r a t e the l a r g e r a d i a l v a r i a t i o n i n r e s i d e n c e t ime wh ich o c c u r s i n the spou ted bed a n n u l u s , and wh ich i s c o r r e c t l y p r e d i c t e d by the t h e o r e t i c a l model as shown i n F i g u r e 5 . 1 3 . A l l the p r e d i c t e d c u r v e s i n the f i g u r e a re based on a common v a l u e o f D ( 0 . 0 1 6 m 2 / s e c ) wh ich was a r r i v e d a t by m a t c h i n g the p r e d i c t e d and o b s e r v e d c u r v e s f o r a s i n g l e l o c a t i o n ( m i d - p o i n t o f a n n u l u s s u r f a c e ) . F i g u r e 5 .13 shows t h a t r e s p o n s e c u r v e s f o r o t h e r r a d i a l l o c a t i o n s a re a l s o c o r r e c t l y p r e d i c t e d by the m o d e l , w i t h -ou t any a l t e r a t i o n o f the D v a l u e . T h i s agreement p r o v i d e s s t r o n g s u p p o r t f o r the v a l i d i t y o f the t h e o r e t i c a l m o d e l , and i n p a r -t i c u l a r f o r the p r o c e d u r e f o r t a k i n g the gas f l o w pa th i n t o a c c o u n t . A c o m p i l a t i o n o f a l l the D v a l u e s o b t a i n e d i n t h i s work i s p r e s e n t e d i n T a b l e 5 . 1 , t o g e t h e r w i t h the main e x p e r i m e n t a l c o n d i t i o n s f o r each run ( a l l the e x p e r i m e n t a l and p r e d i c t e d RTD c u r v e s appear i n A p p e n d i x V I ) . The d a t a show t h a t the v a l u e o f D i n c r e a s e s w i t h i n c r e a s i n g p a r t i c l e s i z e ( run n o s . 18 and 1 ; 17 and 15) and bed depth ( run n o s . 4 and 1; 10 , 9 and 8 ; 31 and 3 9 ) , w h i l e i t r ema ins s u b s t a n t i a l l y i n d e p e n d e n t o f co lumn d i a m e t e r ( run n o s . 1, 29 and 3 2 ; 8 , 28 and 3 0 ) , o r i f i c e d i a m e t e r ( run n o s . 1, 23 and 2 5 ; 13 and 2 1 ; 32 and 3 3 ) , and s p o u t i n g v e l o c i t y ( run n o s . 1, 2 and 3 ; 4 , 5 and 6 ; 8 and 1 2 ; 0. 0.2 0.4 0.6 0.8 1.0 l'.2 1.4 T I M E , S E C . Figure 5.13. Normalized RTD curves corresponding to the response curves of Figure 5.9 for d i f fe ren t rad ia l pos i t ions across the annulus surface. Curve no. 3 i s fo r a point half-way across the annulus surface. 96 Table 5.1 Values of the Dispersion Coef f i c ien t D and Corresponding Experimental Condit ions Run No. Mater ia l D c m Do /D c H/D c s ms dp. mm D, m 2 /sec 1 2 3 4 5 6 Polystyrene pe l le ts P s - 1.05 gm/cm3 0.152 0.152 0.152 0.152 0.152 0.152 0.125 0.125 0.125 0.125 0.125 0.125 3.00 3.00 3.00 2.00 , 2.00 2.00 1.1 1.2 1.3 1.1 1.2 1.3 2.93 2.93 2.93 2.93 2.93 2.93 0.024 0.024 0.025 0.008 0.007 0.008 23 25 0.152 0.152 0.084 0.167 3.00 3.00 1.1 1.1 2.93 2.93 0.017 0.018 29 32 33 0.241 0.292 0.292 0.120 0.125 0.087 3.00 3.10 3.10 1.1 1.1 1.1 2.93 2.93 2.93 0.016 0.017 0.022 18 Polystyrene p e l l e t s , small 0.152 0.125 3.00 1.1 1.61 0.011 8 10 9 12 Wheat P s = 1.24 gm/cm3 0.152 0.152 0.152 0.152 0.125 0.125 0.125 0.125 3.00 1 .00 2.00 3.00 1.1 1.1 1.1 1.3 3.50 3.50 3.50 3.50 0.021 0.015 0.017 0.020 28 30 31 0.241 0.292 0.292 0.120 0.125 0.125 3.00 3.10 1.57 1.1 1 .1 1 .1 3.50 3.50 3.50 0.029 0.017 0.010 13 14 21 M i l l e t P s = 1.18 gm/cm3 0.152 0.152 0.152 0.125 0.125 1 .084 3.00 3.00 3.00 1.1 1.3 1.3 2.15 2.15 2.15 0.013 0.014 0.014 15 16 Glass beads P s = 2.96 gm/cm3 0.152 0.152 0.125 0.125 2.60 2.60 1.1 1.3 2.93 2.93 0.044 0.044 17 Glass beads, P s = 2.94 cm/cm3 0.152 0.125 2.60 1.1 1.10 0.003 20 19 Ammonium n i t ra te P s = 1.75 gm/cm3 0.152 0.152 0.125 0.125 3.00 3.00 1 .1 1 .1 1.99 1.45 0.007 0.005 97 13 and 14 ; 15 and 1 6 ) . A l l t h e s e t r e n d s can be e x p l a i n e d , though o n l y q u a l i t a t i v e l y , i n terms o f the gas v e l o c i t y i n the a n n u l u s on the p rem ise t h a t i n packed b e d s , a x i a l d i s p e r s i o n c o e f f i c i e n t i s more p ronounced a t h i g h e r gas v e l o c i t i e s [ 5 0 , 5 4 ] . S i n c e IJ , and hence the ave rage v a l u e o f IJ , would i n c r e a s e m s a w i t h i n c r e a s i n g d p as w e l l as w i t h i n c r e a s i n g H, D wou ld a l s o be e x p e c t e d to i n c r e a s e . The r e l a t i v e i ndependence o f D f rom D c and D 0 f o r g e o m e t r i c a l l y s i m i l a r beds can be a t t r i b u t e d to the weak e f f e c t o f both t hese v a r i a b l e s on U" [ 1 0 ; t h i s wo rk , a C h a p t e r 6 ] , w h i l e i n c r e a s i n g the s p o u t i n g v e l o c i t y above U m s would not a f f e c t U, and t h e r e f o r e £>, s i n c e the e x c e s s g a s , a U g - U m s i s known to t r a v e l p r e f e r e n t i a l l y t h rough the spou t r e g i o n [ 1 0 ; t h i s wo rk , C h a p t e r 6 ] . In a d d i t i o n to the v a r i a b l e s d i s c u s s e d , the c h a r a c t e r i s t i c s of the s o l i d m a t e r i a l , e . g . shape and s u r f a c e p r o p e r t i e s , p r o b a b l y have an i n f l u e n c e on D [54 ] but t h i s i s d i f f i c u l t to. i s o l a t e f rom the d a t a i n T a b l e 5 . 1 . An a t t emp t to see how a x i a l d i s p e r s i o n i n spou ted beds compares w i t h t h a t i n packed beds and f l u i d i z e d beds i s p r e s e n t e d i n F i g u r e 5 . 1 4 . An a c c u r a t e c o m p a r i s o n i s d i f f i c u l t to make s i n c e the gas v e l o c i t y i n the spou ted bed a n n u l u s , u n l i k e t h a t i n packed and f l u i d i z e d b e d s , changes a l o n g the h e i g h t o f the b e d . The v e l o c i t i e s used f o r p l o t t i n g the spou ted bed da ta i n F i g u r e 5 .14 a re l o n g i t u d i n a l i n t e g r a t e d a v e r a g e v a l u e s f o r the a n n u l u s . The c o m p a r i s o n shows t h a t v a l u e s o f D f o r s p o u t e d beds a re g e n e r a l l y h i g h e r than f o r packed beds [ 5 0 ] , the o 98 o LU CO CVJ o o o o o o 0.0) T T T / 4 3 / N I J '2 ft ! £ • Spouted beds A A A A A Polystyrene _ • Wheat o G lass b e a d s ' • Millet A Ammonium nitrate _ J I I L 0.1 1.0 U , M . / S E C . 10. Figure 5.14. Comparison of ax ia l d ispers ion coe f f i c i en t resu l ts in packed [50], f l u i d i z e d [55,56] and spouted beds (d D for (1): 0.1-0.16 mm, (2) : 0.11 mm, (3): 0.25 mm, w (4) : 0.5 mm, (5): 6 mm, spouted bed data, d p = 1.1-3.5 mm). d i f f e r e n c e b e i n g more p ronounced f o r l a r g e r spou ted bed p a r t i c l e s . When compared w i t h the d a t a f o r f l u i d i z e d beds [ 5 5 , 5 6 ] , the spou ted bed r e s u l t s a re seen to be a t l e a s t o r d e r o f magn i tude s m a l l e r . C h a p t e r 6 GAS DISTRIBUTION BETWEEN SPOUT AND ANNULUS 6 .1 Measurement T e c h n i q u e s C y l i n d r i c a l co lumns o f 15 .2 cm, 24.1 cm and 29 .2 cm d i a m e t e r were used to s t u d y the gas d i s t r i b u t i o n between the spou t and the a n n u l u s o f beds composed of s e v e r a l d i f f e r e n t m a t e r i a l s (see T a b l e 4 . 2 ) . Gas f l o w i n the c y l i n d r i c a l p a r t o f the a n n u l u s was d e t e r m i n e d f rom measurements o f v e r t i c a l s t a t i c p r e s s u r e g r a d i e n t s a l o n g the c e n t r e ( h a l f - w a y between the co lumn w a l l and the s p o u t - a n n u l us i n t e r f a c e ) o f the a n n u l u s . S t a t i c p r e s s u r e r e a d i n g s were taken a t 2 cm i n t e r v a l s a l o n g the h e i g h t u s i n g the s t a t i c p r e s s u r e probe shown i n F i g u r e 4 . 6 . R a d i a l p r e s s u r e g r a d i e n t s i n the c y l i n d r i c a l p a r t o f the a n n u l u s were found to be i n s i g n i f i c a n t . T h i s method i s , how-e v e r , u n s u i t a b l e f o r the l ower c o n i c a l s e c t i o n ' o f the bed where s i g n i f i c a n t r a d i a l s t a t i c p r e s s u r e g r a d i e n t s e x i s t [ 2 , 1 1 , 1 2 ] . In the c o n i c a l p a r t , t h e r e f o r e , t he spou t a i r v e l o c i t y was measured by the p i t o t tube shown i n F i g u r e 4 . 5 , wh ich r e a c h e d as low as 1.4 cm above the a i r - i n l e t o r i f i c e . 100 101 The b a s i c a s s u m p t i o n o f the s t a t i c p r e s s u r e t e c h -n ique i s t h a t the p r e s s u r e g r a d i e n t caused by the f l o w of gas t h rough the a n n u l u s a t a g i v e n f l o w r a t e i s the same as f o r a 1 o o s e l y - p a c k e d b e d . C a l i b r a t i o n c u r v e s f o r the r e l a t i o n s h i p between gas v e l o c i t y and p r e s s u r e drop f o r the bed p a r t i c l e s i n the s t a t i c l o o s e l y - p a c k e d c o n d i t i o n a r e , t h e r e f o r e , needed f o r c o n v e r t i n g the measured p r e s s u r e g r a d i e n t i n the a n n u l u s i n t o v e r t i c a l gas v e l o c i t y . These c a l i b r a t i o n c u r v e s f o r a l l the m a t e r i a l s used i n t h i s work were d e t e r m i n e d i n a 15 .2 cm d i a m e t e r c o l u m n , w i t h the same s t a t i c p r e s s u r e probe ( F i g u r e 4 . 6 ) as used f o r a n n u l u s measu remen ts , t h e da ta e x p r e s s e d i n terms o f a i r s u p e r f i c i a l v e l c o i t y ( c m / s e c ) v e r s u s m i l l i v o l t o u t p u t o f the p r e s s u r e t r a n s d u c e r per 2 cm o f bed h e i g h t were f i t t e d by p o l y n o m i a l s , u s i n g UBC OLQF s u b r o u t i n e . The r e s u l t s a re summar ized i n A p p e n d i x I I . In the c o n i c a l s e c t i o n , the p i t o t tube was i n s e r t e d i n t o the spou t h o r i z o n t a l l y t h rough the v e s s e l w a l l . A i r v e l o c i t i e s were measured a t s e v e r a l r a d i a l p o s i t i o n s , 0 .2 to 0 .5 cm a p a r t , a c r o s s the s p o u t . The s p o u t - a n n u l us i n t e r f a c e c o u l d not be l o c a t e d a c c u r a t e l y f rom the p i t o t tube p r e s s u r e r e a d i n g s , t h e r e f o r e , spou t d i a m e t e r s measured i n the c o r r e s p o n d -i n g ha 1 f - c y 1 i n d r i c a 1 co lumn were used f o r c o n v e r t i n g the measured v e l o c i t i e s i n t o v o l u m e t r i c f l o w r a t e s . R a d i a l p r o f i l e s o f a i r v e l o c i t y i n the spou t a t any g i v e n bed l e v e l were found to be symmet r i c w i t h a maximum a t the spou t a x i s . Hence , a i r 102 v e l o c i t i e s measured a t p o s i t i o n s between the a x i s and the p e r -i p h e r y o f the spou t were used to c a l c u l a t e the i n t e g r a t e d a v e r a g e spou t gas v e l o c i t y . The v o l u m e t r i c gas f l o w r a t e i n the s p o u t , Q s , a t any g i v e n l e v e l was then c a l c u l a t e d as f o l 1 o w s : ' s - S \ u s t 6 - 1 ' where e s i s the r a d i a l ave rage spou t v o i d a g e o b t a i n e d from p a r t i c l e v e l o c i t y measurements ( C h a p t e r s 7 and 8) and u g i s the i n t e g r a t e d ave rage i n t e r s t i t i a l a i r v e l o c i t y i n the spou t o b t a i n e d f rom p i t o t tube measuremen ts . P i t o t tube measurements were made o n l y i n the l ower h i g h - v o i d a g e s e c t i o n o f the spou t s i n c e any e r r o r caused by the p r e s e n c e of p a r t i c l e s wou ld no t be s i g n i f i c a n t i n t h i s r e g i o n . Spout a i r f l o w r a t e s i n the upper c y l i n d r i c a l p a r t o f the bed were o b t a i n e d by d i f f e r e n c e s between the t o t a l a i r f l o w r a t e and the a n n u l u s f l o w r a t e as d e t e r m i n e d by the s t a t i c p r e s s u r e t e c h n i q u e . Annu lus f l o w r a t e s f o r the c o n i c a l s e c t i o n were a l s o o b t a i n e d by d i f f e r e n c e s , i n t h i s case between the t o t a l f l o w r a t e ' a n d the spou t f l o w r a t e as d e t e r m i n e d f rom p i t o t tube measurements . In o r d e r to check the r e l i a b i l i t y o f the measurement t e c h n i q u e s u s e d , a d d i t i o n a l p i t o t tube measurements were made 103 a c r o s s the spout a t a l e v e l j u s t above the c o n i c a l b a s e . The a n n u l u s f l o w a t the same l e v e l was d e t e r m i n e d by the s t a t i c p r e s s u r e t e c h n i q u e . The sum of the spou t and a n n u l u s f l o w r a t e s measured by the two d i f f e r e n t methods was found to ag ree w i t h i n ±3.0% of the t o t a l s p o u t i n g a i r f l o w r a t e i n s e v e r a l t e s t s c a r r i e d out under d i f f e r e n t e x p e r i m e n t a l c o n d i t i o n s . 6 .2 R e s u l t s and D i s c u s s i o n 6 . 2 . 1 In the Spout T y p i c a l r a d i a l p r o f i l e s o f v e r t i c a l gas v e l o c i t y i n the spou t f o r d i f f e r e n t bed l e v e l s a re shown i n F i g u r e s 6 . 1 ( a ) and 6 . 1 ( b ) , f o r wheat and p o l y s t y r e n e beds r e s p e c t i v e l y . These p r o f i l e s a re s i m i l a r i n shape to t hose r e p o r t e d i n the l i t e r a t u r e [ 6 , 7 , 1 5 , 1 7 , 1 8 ] . van V e l z e n et al. [ 6 , 7 ] , on the b a s i s o f t h e i r e x p e r i m e n t a l da ta ( D c = 1 2 . 5 cm) , have s u g g e s t e d t h a t the f r a c t i o n o f t o t a l gas f l o w i n g t h rough the spou t (gas t r a n s p o r t r a t i o ) , i . e . (6 i s a f u n c t i o n of the i n t r o d u c e d momentum a t the gas i n l e t n o z z l e , p f Q F LI0 MI = — — 5  g u 0 be i ng the gas v e l o c i t y t h rough the i n l e t o r i f i c e . (a) Polystyrene Pellets 6 0 10 0 Z , c m -j D c = 15.2 cm g ^ . \ * I I I 1 o~—A 0 0 .2 0-4 0 .6 0.8 1.0 r/r s • Figure 6 .1 . Radial p ro f i l es of upward H / D c = 3 ' V Ums = (b) Wheat a i r ve loc i t y in the spout. 105 T h i s h y p o t h e s i s has been t e s t e d u s i n g some o f the da ta o b t a i n e d i n the p r e s e n t work . V a l u e s o f TR and MI c a l -c u l a t e d by E q s . ( 6 . 2 ) and ( 6 . 3 ) and p l o t t e d i n F i g u r e 6 .2 show t h a t i n g e n e r a l , f o r a g i v e n co lumn s i z e and s o l i d m a t e r i a l , the t r a n s p o r t r a t i o a t a l l bed l e v e l s does i n c r e a s e w i t h i n c r e a s i n g v a l u e s o f i n t r o d u c e d momentum. The i n d e p e n d e n t v a r i a b l e s i n v o l v e d a re U / U , H and D 0 , the momentum i n t r o d u c e d b e i n g d i r e c t l y p r o p o r t i o n a l to U s / U m s and H, and i n v e r s e l y to D 0 . However , MI i s a l s o a f u n c t i o n o f m a t e r i a l p r o p e r t i e s and co lumn d i a m e t e r ( t h r o u g h Q j ) , but the e f f e c t o f t h e s e v a r i a b l e s on TR i s not c o r r e c t l y a c c o u n t e d f o r by the momentum term s i n c e da ta f o r d i f f e r e n t m a t e r i a l s and column s i z e s f a i l to come t o g e t h e r on a p l o t o f TR v s . M I . V o l u m e t r i c gas f l o w th rough the a n n u l u s i s r e l a t e d to the t r a n s p o r t r a t i o TR as f o l l o w s : Q a = Q T (1 - TR) ( 6 . 4 ) T h i s e q u a t i o n i m p l i e s t h a t when TR i s g r e a t e r than u n i t y , the d i r e c t i o n o f gas f l o w i n the a n n u l u s i s not upward but down-w a r d , wh ich can o c c u r o n l y i f t h e r e i s r e c i r c u l a t i o n o f gas i n the a n n u l u s . The da ta i n T a b l e 6.1 show t h a t r e c i r c u l a t i o n o f gas does o c c u r i n the l owe r p a r t o f the a n n u l u s under c e r t a i n c o n d i t i o n s , v i z . a t u^ /U^s > 1.2 ( run n o s . 2 , 3 , 6 , 12 and 1 4 ) , H/D > 4 ( run n o s . 7 and 11) and D 0 / D _ < 0.1 ( run n o s . 2 3 , 21 and 3 3 ) . The o c c u r r e n c e o f gas r e c i r c u l a t i o n [0) po lystyrene penets 6 0 1 0 0 1 4 0 180 2 2 0 M l , gm r i n n » > o fi 9 D o l a t i n n c h i n h o t u i o o n n a Q ^D; w n e a i 140 180 2 2 0 2 6 0 3 0 0 M l , gm r a t i o . TR. and introduced momentum, MI in 1 0 7 Table 6 . 1 Data for Gas Transport Ratio and Introduced Momentum Run Mater ial Dc> H/D c D 0 /D c U /U s' ms MI TR at Z, cm No. cm gm 1 . 6 4 . 4 9 . 8 1 Polystyrene 1 5 . 2 3 . 0 0 0 . 1 2 5 1 . 1 1 2 0 1 . 0 0 0 . 9 6 6 2 M 1 . 2 1 4 7 1 . 0 1 3 " H 1 . 3 1 7 4 1 . 0 6 4 2 . 0 1 . 1 1 0 8 . 9 4 9 . 7 8 4 5 II 1 . 2 1 3 6 . 9 9 3 . 9 6 8 . 8 5 9 6 " 1 . 3 1 5 5 1 . 0 1 4 . 9 9 5 . 8 6 5 7 5 . 9 2 1 . 1 1 8 1 1 . 0 6 6 1 . 0 1 7 . 8 8 0 2 3 3 . 0 0 . 0 8 4 1 . 1 2 3 6 1 . 1 1 4 1 . 0 9 1 1 . 0 2 2 2 5 0 M 0 . 1 6 7 1 . 1 7 5 . 9 6 6 . 9 1 0 . 7 3 9 8 Wheat 1 5 . 2 3 . 0 0 . 1 2 5 1 . 1 . 2 0 1 . 9 8 2 . 9 2 3 . 6 3 2 1 2 II II 1 . 3 2 6 9 1 . 0 4 . 9 9 8 . 7 3 3 11 II 4 . 1 7 II 1 .1 2 4 1 1 . 0 2 . 9 5 8 . 7 1 3 9 n 2 . 0 n 1 . 1 1 7 4 . 9 8 0 . 8 9 5 . 6 2 5 1 3 M i l l e t 1 5 . 2 3 . 0 . 1 2 5 1 .1 7 8 . 9 9 5 . 9 0 4 . 6 4 7 1 4 H II II 1 . 3 1 0 2 1 . 0 3 . 9 6 4 . 7 2 9 21 n . 0 8 4 1 . 1 1 4 9 1 . 0 4 . 9 8 4 . 6 7 5 Z = 3 . 8 8 . 6 1 5 . 0 3 0 Wheat 2 9 . 2 3 . 0 . 1 2 5 1 .1 4 5 0 . 9 2 4 . 7 2 7 . 6 9 3 3 2 polystyrene n n II II 2 5 2 0 . 9 8 5 3 3 n' II M . 0 8 7 1 .1 5 1 0 1 . 0 6 0 . 9 5 2 . 8 1 5 1 08 under some of the above c o n d i t i o n s was c o n f i r m e d by v i s u a l o b s e r v a t i o n u s i n g N 0 2 t r a c e r ( e . g . F i g u r e 5 . 2 ) . The gas momentum pa rame te r however does not show any s i m p l e c o r r e l a t i o n w i t h e i t h e r the o n s e t o r the e x t e n t o f r e c i r c u l a t i o n , e x c e p t inasmuch as f o r the same column s i z e and s o l i d m a t e r i a l , r e c i r c u l a t i o n i s a s s o c i a t e d w i t h h i g h v a l u e s o f i n t r o d u c e d momentum (see F i g u r e 6 . 2 ) . 6 . 2 . 2 In the Annu lus In g e n e r a l , the v e r t i c a l gas v e l o c i t y i n the a n n u l u s o f a spou ted bed i n c r e a s e s w i t h i n c r e a s i n g bed l e v e l . The i n c r e a s e i s more r a p i d i n the l ower s e c t i o n of the annu lus than i n the upper s e c t i o n , the v e l o c i t y g r a d i e n t becoming ze ro near the top o f a deep bed (H ~ H m ) . No m e a s u r a b l e r a d i a l g r a d i e n t s e x i s t a c r o s s the a n n u l u s i n the c y l i n d r i c a l p a r t o f the b e d . The above t r e n d s were noted i n t h i s work and a re c o n s i s t e n t w i t h p r e v i o u s l y r e p o r t e d o b s e r v a t i o n s [ 1 0 ] . T y p i c a l a i r v e l o c i t y p r o f i l e s i n the annu lus above the c o n i c a l s e c t i o n of the b e d , o b t a i n e d f rom p r e s s u r e g r a d i e n t measuremen ts , a re p r e s e n t e d i n F i g u r e s 6 . 3 , 6 .4 and 6 . 5 . In F i g u r e 6 . 5 a v e l o c i t y p r o f i l e f rom the work o f Mathur and G i s h l e r [2 ] i s a l s o i n c l u d e d . These p l o t s i11 u s t r a t e • the f o l l o w i n g a d d i t i o n a l f e a t u r e s of gas f l o w i n the a n n u l u s . ( i ) Gas v e l o c i t y p r o f i l e s o b t a i n e d i n beds o f d i f f e r e n t dep th w i t h o t h e r c o n d i t i o n s ( m a t e r i a l , D „ , D 0 and U /U _) h e l d c o n -v c ' u s ms ' s t a n t c o i n c i d e w i t h each o t h e r ( F i g u r e 6 . 3 ) . T h i s means t h a t o ® o © o • o A A A A A A A A i 0 A A ^ A A A A A @ ^ Sol ids H , c m O • A | ^ 0 Wheat 1 5 - 2 0 S A • 3 0 . 5 Q A © 4 5 . 7 • Polystyrene 3 0 . 5 A 4 5 . 7 A 9 0 . 1 • 0 10 2 0 3 0 4 0 5 0 6 0 7 0 8 0 Z , cm Figure 6 .3 . Ef fect of bed depth on a i r ve loc i ty in the annulus. D c = 15.2 cm, D0 = 1.95 cm. 160 140 120 §100 E o 80 6 0 4 0 20 0 no IS / // ® / If 3 • / •o-.o-o •A' "A / A' A' S o l i d s U s / U m s A'tr s G lass beads I • ( d p = 2.93mm)l © Wheat o I A P o l y s t y r ene I A I l . l -A ! 0 20 30 4 0 50 0 Figure 6.4. Ef fect of spouting a i r ve loc i t y on a i r ve loc i t y in the Z c m annulus. D c = 15.2 cm, D0 = 1.95 cm. 80 O CD if) e 60 o 40 20 0 0 So l i d s D 0 , cm ,91 0 . 9 5 Wheat (by Mathur et al. [27j) Polystyrene 2 . 5 4 1.91 1.27 10 20 30 Z 7 cm 40 50 60 Figure 6.5 . Effect of a i r i n l e t diameter on a i r ve loc i t y in the annulus. D = 15.2 cm, wheat beds H = 63.5 cm, polystyrene beds H = 45.7 cm, U s / U m s = 7.10. 1 1 2 the a i r v e l o c i t y a t a g i v e n l e v e l rema ins the same r e g a r d l e s s o f bed d e p t h . ( i i ) For a g i v e n s o l i d m a t e r i a l and column d i a m e t e r , the a i r v e l o c i t y a t any bed l e v e l d e c r e a s e s w i t h i n l e t d i a m e t e r ( F i g u r e 6 . 5 ) . The e f f e c t o f gas i n l e t d i a m e t e r , however , becomes p r o g r e s s i v e l y l e s s p ronounced a t h i g h e r bed l e v e l s ( F i g u r e 6 . 5 ) , the c u r v e s f o r d i f f e r e n t v a l u e s o f D 0 a p p r o a c h i n g each o t h e r near the top o f the b e d . i n the annu lus w i t h t hose p r e d i c t e d by E q s . ( 2 . 1 3 ) , ( 2 . 1 4 ) and ( 2 . 3 6 ) , the maximum s p o u t a b l e bed d e p t h , H , and the minimum f l u i d i z a t i o n v e l o c i t y , must be known. V a l u e s o f H m f o r the 15 .2 cm d i a m e t e r column were measured e x p e r i m e n t a l l y , w h i l e t hose f o r 24.1 cm and 29 .2 cm d i a m e t e r c o l u m n s , b e i n g t oo l a r g e f o r e x p e r i m e n t a l measurement , were c a l c u l a t e d by B e c k e r ' s e q u a t i o n [ 1 5 ] . i n c r e a s i n g U / U m s ( F i g u r e 6 . 4 ) , and w i t h d e c r e a s i n g gas In o r d e r to compare the measured a i r v e l o c i t i e s H m = 42 d - 0 . 7 6 P ( 1 2 . 2 D 0 / D c ) / cu ( 6 . 5 ) where oi = - 1 . 6 e x p ( - 0 . 0 0 7 2 R e ) ( 6 . 6 ) Re m p m t ( 6 . 7 ) U = - a 2 + /otf + 2a 3 ( 6 . 8 ) ( 6 . 9 ) 113 a 3 = d p ( p s - p f ) g X / ( 3 3 p f ) ( 6 . 1 0 ) a , = (22 + 2 6 0 0 / R e j R e J 7 3 X 2 / 3 ( 6 . 1 1 ) The term U f appea rs i n E q s . ( 2 . 1 3 ) , ( 2 . 1 4 ) and ( 2 . 3 6 ) , because of the a s s u m p t i o n t h a t the a i r v e l o c i t y a t the top o f the a n n u l u s f o r a bed o f maximum s p o u t a b l e d e p t h , U „ , e q u a l s m ^ m f T h i s i s an a p p r o x i m a t i o n wh ich was a v o i d e d by u s i n g measured v a l u e s o f U u ( a t U = 1.1 U ) i n s t e a d o f U , . Wi th t h i s a n m s ms mf s u b s t i t u t i o n , E q s . ( 2 . 1 4 ) , ( 2 . 1 3 ) and ( 2 . 3 6 ) become U aH = 1 m i - J i -m 3 ( 6 . 1 2 ) 'aH. 1 - Z/H m ( 6 . 1 3 ) m and s l n aH TT Z / 2 H m ( 6 . 1 4 ) m The r e s u l t s o f H m and u" a H a re l i s t e d i n T a b l e m 6 . 2 , w h i l e a c o m p a r i s o n o f the e x p e r i m e n t a l v a l u e s o f u a H / u a H m a g a i n s t t h o s e p r e d i c t e d by E q . ( 6 . 1 2 ) i s shown i n F i g u r e 6 . 6 . A l s o i .nc luded i n the f i g u r e a re the d a t a f o r a 61 cm d i a m e t e r wheat b e d , c a l c u l a t e d by Mathur and E p s t e i n [10 ] f rom the e x p e r i m e n t a l r e s u l t s of T h o r l e y et al. [ 2 ] . The e x p e r i m e n t a l p o i n t s i n F i g u r e 6 .6 a re i n q u i t e good agreement w i t h E q . ( 6 . 1 2 ) 114 Table 6.2 Value of H and U Material cm D 0 , cm V mm H m cm U a H m ' E x P K cm/sec Polystyrene pe l le ts 15.2 1.91 2.93 90.2 58.5 15.2 1.27 2.93 95.0 58.5 15.2 2.54 2.93 83.0 53.5 24.1 2.86 2.93 131.7* 58.5 29.2 2.54 2.93 213.1* 58.5 Wheat 15.2 1 .91 3.50 63.5 82.0 24.1 2.81 3.50 * 119.2 82.0 29.2 3.81 3.50 163.4* 8 2 . 0 M i l l e t 15.2 1.91 2.15 48.0 56.4 15.2 1 .27 2.14 69.8 56.4 Glass beads 15.2 1.91 2.93 42.0 146.0 Ammonium n i t ra te 15.2 1.91 1.99 53.0 67.5 15.2 1.91 1 .45 49.0 52.0 * Note: Calculated by Becker's equation [15]. Other resu l ts of H are experimental va lues. m 115 A Eq- 6.12 S o l i d s o Wheat M i l l e t G lass beads Po lys t rene -A m m o n i u m n i t ra te Wheat by Mathur et al (10) 0 . 4 0 - 6 H/H 0 . 8 1 . 0 m Figure 6.6. Relat ionship between U ^ / U . ^ and H/H m . Comparison m of experimental resu l ts with Eq. (6.12) (a l l data for U s = 1.1 U m s , except those with / across the s i gn , in th is case U s > 1.1 U ). 116 E x p e r i m e n t a l a i r v e l o c i t y p r o f i l e s a re compared a g a i n s t E q . ( 6 . 1 3 ) and E q . ( 6 . 1 4 ) i n F i g u r e 6.7 f o r 15 .2 cm d i a m e t e r b e d s , and i n F i g u r e 6 .8 f o r 24.1 cm and 2 9 . 2 cm d i a m e t e r b e d s . A l l the e x p e r i m e n t a l v a l u e s o f U a H and u"aH i n the f i g u r e were measured a t U s = 1.1 U m s . The c o m p a r i -m son shows t h a t ( i ) the L e f r o y and D a v i d s o n [24 ] e q u a t i o n , E q . ( 6 . 1 4 ) , c o n s i s t e n t l y u n d e r - p r e d i c t e s II , the d i s c r e p a n c y b e i n g as l a r g e as 100% f o r some of the d a t a , ( i i ) the Mamuro and H a t t o r i e q u a t i o n , E q . ( 6 . 1 3 ) , i s i n b e t t e r agreement w i t h the r e s u l t s f o r 15 .2 cm d i a m e t e r b e d s , though the da ta show c o n s i d e r a b l e s c a t t e r (± 30%) around the t h e o r e t i c a l c u r v e . For the l a r g e r d i a m e t e r beds ( F i g u r e 6 . 8 ) , t h i s e q u a t i o n a l s o u n d e r - p r e d i c t s LI , though no t as s e r i o u s l y as E q . ( 6 . 8 ) . a T h e o r e t i c a l e q u a t i o n s d e r i v e d by Yokogawa and c o -w o r k e r s [ 1 7 , 1 9 ] f o r p r e d i c t i n g gas v e l o c i t y p r o f i l e s i n the a n n u l u s ( E q s . ( 2 . 3 2 ) , ( 2 . 3 3 ) and ( 2 . 3 4 ) ) i n c l u d e the a n g l e o f i n t e r n a l f r i c t i o n o f the p a r t i c l e s , <f>, the a n g l e o f f r i c -t i o n between the p a r t i c l e s and the v e s s e l w a l l , y ( t a n y = u c , c o e f f i c i e n t o f f r i c t i o n ) , the Darcy law c o n s t a n t , K, and the r a t i o C f / k s , where i s the d rag c o e f f i c i e n t and k $ r e p r e -s e n t s the r a t i o between r a d i a l and v e r t i c a l p r e s s u r e s o f a n n u l a r s o l i d s a t the spout b o u n d a r y . A l l the above q u a n t i t i e s can be measured e x p e r i m e n t a l l y e x c e p t C ^ / k s wh i ch t h e r e f o r e becomes an a d j u s t a b l e pa rame te r i n the e q u a t i o n s . Yokogawa et al. e v a l u a t e d C ^ / k s f rom e x p e r i m e n t a l v a l u e s o f U a H u s i n g E q . ( 2 . 3 3 ) , and used t h i s v a l u e o f C f / k s i n E q . ( 2 . 3 4 ) to p r e d i c t the v e r t i c a l gas v e l o c i t y p r o f i l e i n 1.0 0 . 8 0 . 6 0 . 4 0 . 2 0 Eq. 6.13 Eq.6.14 Sol ids Wheat Mil let Glass beads Polystyrene Ammonium ni t rate 0 0 . 2 0 . 4 0 . 6 Z/H, 0 . 8 1.0 m Figure 6.7. Relat ionship between U /U „ and Z/H . Comparison a an_ m m of experimental resu l ts for 15.2 cm diameter beds against Eqs. (6.13) and (6.14). U = 1 1 U s ms' 118 .0 Sol ids D c , cm Wheat 24.1 2 9 . 2 _ A Polystyrene 24.1 • 2 9 . 2 0 . 4 0 . 6 Z/H, 1.0 m Figure 6.8. Relat ionship between Ug/U.^ and Z / H m . Comparison of m m experimental resu l ts for 24.1 cm and 29.2 cm diameter beds with Eqs. (6.13) and (6.14). II = 1.1 U ms 119 the a n n u l u s . The v a l u e o f K used i n E q . ( 2 . 3 3 ) was d e t e r m i n e d e x p e r i m e n t a l l y and assumed to be i n v a r i a n t w i t h gas v e l o c i t y . T h i s i s a q u e s t i o n a b l e a s s u m p t i o n s i n c e a t the h i g h e s t v e l o c i t i e s e n c o u n t e r e d near the top o f the a n n u l u s , the p a r t i c l e Reyno lds numbers r each as h igh as ~ 100 , wh ich i s o u t s i d e the range o f a p p l i c a b i l i t y o f D a r c y ' s Law. The v a l u e o f K, a c c o r d i n g to E q . ( 2 . 3 3 ) , has a s t r o n g i n f l u e n c e on II ,, and any e r r o r a r i s i n g f rom the a s s u m p t i o n o f c o n s t a n t K i s r e f l e c t e d i n the v a l u e o f the l o o s e pa rame te r C ^ / k s . In c o n s e q u e n c e , the p r e d i c t i o n o f v e l o c i t y p r o f i l e by E q . ( 2 . 3 4 ) becomes c o r r e -s p o n d i n g l y u n r e l i a b l e , as i l l u s t r a t e d i n F i g u r e 6 . 9 . Hence, E q . ( 2 . 3 3 ) canno t be e x p e c t e d to g i v e a r e l i a b l e e s t i m a t e o f C ^ / k s f o r use i n E q . ( 2 . 3 4 ) to p r e d i c t the a i r v e l o c i t y p r o f i l e i n the a n n u l u s . A method f o r ove rcom ing the above d i f f i c u l t y i n -v o l v i n g m o d i f i c a t i o n of Yokogawa 's e q u a t i o n s , i s now p r o p o s e d . The m o d i f i c a t i o n i s based on the e m p i r i c a l o b s e r v a t i o n t h a t f o r a g i v e n s o l i d m a t e r i a l , co lumn geometry and U s / U ' m s r a t i o , the a n n u l u s gas v e l o c i t y II a t any l e v e l Z i s i n d e p e n d e n t o f a bed depth H (see page 1 0 9 , F i g u r e 6 . 3 ) . E q . ( 2 . 3 4 ) , t h e r e f o r e , can be g e n e r a l i z e d to a p p l y to a l l bed dep ths by w r i t i n g t h i s e q u a t i o n f o r H = H„ when II u = LI u and D„ = D „ . 3 ^ m aH aH„, s sm m II A 2 Z X 2 H A j Z U a _ A x e m (1 - e ) - X 2 e m (1 - e ) ( 6 1 5 ) U a H X l H m X 2 H m *2H m X i H m A i e , ( l - e ) - A 2 e ( 1 - e ) where 1 20 0 0 .2 0 . 4 0 .6 0-8 1.0 Z/H Figure 6 .9 . Ef fect of C f / k s on a i r ve loc i t y p ro f i l es predicted by Eq. (2.34): wheat bed, D c = 15.2 cm, H = 63.5 cm, K = 0.0088 gm/cm3. 1 21 + / A " w ( 6 . 1 6 ) 9- - /A" 2" 2 -jp - w ( 6 . 1 7 ) and A = -4k u D / c c c D 2 - D 2 c sm ( 6 . 1 8 ) w Kk D 2 D s c sm 4Cx D 2 - D 2 c sm ( 6 . 1 9 ) The g r o u p , K k s / C f i n E q . ( 6 . 1 9 ) can be c o n s i d e r e d as an a d j u s t a b l e pa rame te r wh ich i s dependent on the s o l i d s p r o p e r t i e s o n l y . The v a l u e o f Kk / C y f o r a g i v e n s o l i d m a t e r i a l can be e s t i m a t e d by f i t t i n g a s i n g l e e x p e r i m e n t a l a n n u l u s a i r v e l o c i t y p r o f i l e to E q . ( 6 . 1 5 ) , and t h i s v a l u e o f K k $ / C f can then be used f o r p r e d i c t i n g v e l o c i t y p r o f i l e s f o r o t h e r bed dep ths and d i a m e t e r s by the same e q u a t i o n . For a wheat b e d , 15 .2 cm d i a m e t e r x 6 3 . 5 cm deep the v a l u e o f K k s / C ^ e s t i m a t e d as above was found to be 0 .059 (see F i g u r e 6 . 1 0 ) . The a i r v e l o c i t y p r o f i l e f o r a 29 .2 cm d i a m e t e r x 91 .4 cm deep bed of w h e a t , c a l c u l a t e d by E q . ( 6 . 1 5 ) w i t h K k s / C f = 0 .059 i s seen to ag ree w e l l w i t h the measured p r o f i l e ( F i g u r e 6 . 1 0 ) . C a l c u l a t e d r e s u l t s f o r 29 .2 cm d i a m e t e r beds o f p o l y s t y r e n e p e l l e t s , u s i n g K k s / C f (= 0 .023 ) e s t i m a t e d f rom da ta f o r a 15 .2 cm d i a m e t e r 1 22 1.0 Exptl. P red ic ted D c , c m H , c m 5 . 2 6 3 . 5 2 9 . 2 9 1 . 4 0 0 . 2 0 . 4 0 . 6 Z/H 0 . 8 1.0 Figure 6 J 0 . Relat ionship between U a H / U a H and Z/H in wheat beds. m Comparison of experimental resu l ts with those pre-dicted by Eq. (6.15) with K k / c , = 0.059. U_ = ms 1 23 p o l y s t y r e n e b e d , showed s i m i l a r agreement w i t h e x p e r i m e n t a l v e l o c i t y p r o f i l e s . C h a p t e r 7 FLOW PATTERN OF SOLIDS 7.1 In the Annu lus 7 .1 .1 E x p e r i m e n t a l R e s u l t s and D i s c u s s i o n Annu lus p a r t i c l e v e l o c i t i e s were measured a g a i n s t the f l a t f a c e o f h a l f - c o l u m n s by measu r i ng w i t h a s top watch the t ime taken f o r a p a r t i c l e to t r a v e l down a s m a l l v e r t i c a l d i s t a n c e (4 cm) . An ave rage o f f o u r r e a d i n g s was taken as the v e l o c i t y a t each l o c a t i o n , w h i l e r a d i a l a v e r a g i n g was done by m e a s u r i n g v e l o c i t i e s at. t h r e e r a d i a l p o s i t i o n s as shown i n F i g u r e 7 . 1 , and u s i n g the f o l l o w i n g e q u a t i o n to o b t a i n the ave rage v e l o c i t y : v A + v A + v A v = aw aw ac ac as as { 1 ,s v a A \i >\) In some runs the a n n u l u s p a r t i c l e v e l o c i t i e s , p a r t i c u l a r l y i n the top p o r t i o n o f the b e d , were too f a s t to be measured a c c u r a t e l y by a s t op w a t c h , and the v e l o c i t i e s i n t hese c a s e s were d e t e r m i n e d f rom movie f i l m s taken a t h i gh s p e e d . 124 1 25 Spout v as> A A S v a c A A C AQW Annulus Figure 7 .1 . Radial sect ions of annulus for pa r t i c l e ve loc i t y measurements. v , r , v . and v, : pa r t i c l e v e l o c i t i e s ; as ac aw A a s ' A a c ' a n d A aw : cross-sect iona l areas. 1 26 The v o l u m e t r i c f l o w r a t e o f s o l i d s i n the a n n u l u s was c a l c u l a t e d f rom v as f o l l o w s : ( 7 . where A i s the annu lus c r o s s s e c t i o n a l a rea and a e a the annu lus v o i d a g e Measured v a l u e s o f v aw' v and v as ' t o g e t h e r w i t h ac v a l u e s o f v a c a l c u l a t e d by E q . ( 7 . ) , f o r a l1 the runs a re p r e s e n t e d i n Append i x V I I I ( a ) , w h i l e t y p i c a l r e s u l t s a re shown i n F i g u r e s 7.2 to 7 . 4 . F i g u r e 7.2 i l l u s t r a t e s the v e r t i c a l v e l o c i t y p r o f i l e s a t t h r e e r a d i a l p o s i t i o n s a c r o s s the a n n u l u s , F i g u r e 7 .3 shows some examp les o f r a d i a l ave rage v e l o c i t y p r o f i l e s c a l c u l a t e d by E q . ( 7 . 1 ) , and F i g u r e 7.4 p r e s e n t s v o l u m e t r i c s o l i d s f l o w r a t e s c a l c u l a t e d f rom the da ta o f F i g u r e 7 .3 u s i n g E q . ( 7 . 2 ) . R a d i a l v a r i a t i o n i n v e l o c i t y was c o n s i d e r a b l y l e s s f o r l a r g e r d i a m e t e r beds than shown i n F i g u r e 7 . 2 . In a l l c a s e s , the a v e r a g e p a r t i c l e v e l o c i t y and v o l u m e t r i c f l o w r a t e were found to d e c r e a s e l i n e a r l y f rom the top o f the a n n u l u s to Z = D c , a l e v e l somewhat above the top o f the c o n i c a l p a r t o f the bed (see F i g u r e s 7 .3 and 7 . 4 ) . T h i s l i n e a r d e c r e a s e o f v , and G w i t h bed l e v e l was a l s o a a o b s e r v e d by T h o r l e y et al. [11 ] and Yokogawa et al. [ 1 7 ] ; The r e l a t i o n s h i p o f G^ v s . Z can be d e s c r i b e d by the f o l l o w i n g a e x p r e s s i o n : 1 27 Figure 7.2. P a r t i c l e ve loc i t y p ro f i l es in the annulus: polystyrene pe l le ts (see Figure 7.1 for the locat ion of measurements): D c = 15.2 cm, D 0 /D c = 0.125, U $ = 1.1 U m s . 1 28 H/D, U s / U m s o to E o >° 0 0 0 .4 0.6 Z/H Figure 7.3. P ro f i l es of rad ia l average pa r t i c l e ve loc i t y in the annulus: Polystyrene p e l l e t s , D = 15.2 cm, D 0 /D c = 0.125 c 1 29 Figure 7.4. Volumetric flow rate of so l ids in the annulus: Polystyrene p e l l e t s , D c = 15.2 cm, D 0 /D c = 0.125. 1 30 G a = G a H - S(H - Z) ( 7 . 3 ) where G a ^ i s the v o l u m e t r i c f l o w r a t e of s o l i d s a t the top of the a n n u l u s and S (= AG /AZ ) i s the c r o s s - f l o w r a t e of s o l i d s f rom a the annu lus i n t o the spou t per u n i t o f bed h e i g h t ( c m 3 / ( s e c ) ( c m ) ) . A c o m p i l a t i o n o f a l l the r e s u l t s o f G u and S a n o b t a i n e d by f i t t i n g a s t r a i g h t l i n e th rough the e x p e r i m e n t a l v a l u e s o f G f o r the p a r t o f bed between D < Z < H i s p r e -d C ~ ~ s e n t e d i n T a b l e 7 . 1 , t o g e t h e r w i t h the main e x p e r i m e n t a l c o n d i -t i o n s f o r each r u n . These d a t a show t h a t both G a ^ and S i n c r e a s e w i t h i n c r e a s i n g U / U ( run n o s . 1, 2 and 3 ; 4 , 5 and 6 ; 8 3 s ms and 1 2 ; 13 and 1 4 ; 15 and 1 6 ) , bed d e p t h , H ( run n o s . 1 and 4 ; 2 and 5 ; 3 and 6 ; 8 , 9 and 1 1 ) , and p a r t i c l e d i a m e t e r , dp ( run n o s . 1 and 18 ; 15 and 17 ; 19 and 2 0 ) , w h i l e they rema in s u b s t a n t i a l l y i n d e p e n d e n t o f cone a n g l e , 8 ( run n o s . 1 and 2 7 ; 13 and 2 6 ) . I n c r e a s i n g the o r i f i c e d i a m e t e r has no s i g n i f i c a n t e f f e c t on G a ^ but i t c a u s e s the c r o s s - f l o w r a t e S to i n c r e a s e ( run n o s . 1, 23 and 2 5 ; 8 and 24 ; 13 and 2 1 ) . For g e o m e t r i c a l l y s i m i l a r b e d s , column d i a m e t e r has a marked i n f l u e n c e on G „ u a n wh ich i n c r e a s e s w i t h i n c r e a s i n g D c ( run n o s . 1, 29 and 3 2 ; 8 , 28 and 3 0 ) , but the e f f e c t o f D c on S appears to be more c o m p l e x . For wheat b e d s , S was found to i n c r e a s e w i t h i n c r e a s -i ng D c ( run n o s . 8 , 28 and 30) and f o r p o l y s t y r e n e beds to 1 3 1 Table 7 . 1 Sol ids Flow in Annulus Run No. Mater ial cm Do/D c H/D c U /U s ms G a H ' cm 3 /sec s, cm 2 /sec 1 Polystyrene 1 5 . 2 0 . 1 2 5 3 . 0 1 .1 2 7 7 . 7 4 . 4 0 2 pe l le ts n n 1 . 2 3 1 7 . 8 4 . 7 3 3 d p = 2 . 9 3 mm II n 1 . 3 3 3 8 . 4 5 . 0 0 4 P s = 1 . 0 5 gm/cm3 H 2 . 0 1 . 1 1 6 9 . 2 2 . 4 4 5 II n 1 . 2 1 9 0 . 1 2 . 4 1 6 n M 1 . 3 2 1 8 . 6 2 . 8 8 2 3 0 . 0 8 7 3 . 0 0 1 .1 2 8 7 . 5 3 . 1 5 2 5 " 0 . 1 6 7 3 . 0 0 1 . 1 3 0 8 . 8 4 . 8 8 * 2 7 " 0 . 1 2 5 3 . 0 0 1 .1 2 7 7 . 5 4 . 1 4 2 9 2 4 . 1 0 . 1 2 0 3 . 0 1 . 1 5 2 4 . 9 4 . 1 3 3 2 2 9 . 2 0 . 1 2 5 3 . 1 1 . 1 6 5 6 . 1 3 . 2 5 3 3 2 9 . 2 0 . 0 8 7 3 . 1 1 .1 6 3 7 . 6 2 . 5 0 1 8 Polystyrene pe l le ts d = 1 . 6 1 mm P 1 5 . 2 0 . 1 2 5 3 . 0 1 .1 1 4 3 . 3 2 . 2 5 8 Wheat 1 5 . 2 0 . 1 2 5 3 . 0 0 1 . 1 4 3 3 . 8 6 . 9 8 1 2 d p = 3 . 5 0 mm II H 1 . 3 5 5 6 . 7 6 . 9 1 9 P s = 1 . 2 4 gm/cm 3 n II 2 . 0 1 .1 2 9 2 . 7 6 . 0 1 1 0 1 . 0 0 1 .1 1 5 0 . 0 --11 n 4 . 1 7 7 6 4 . 5 8 . 6 7 CONTINUED 1 3 2 Table 7 . 1 (Continued) Run No. Material cm Do/Dc H/Dc u /u s' ms G ah" cm 3/sec s, cm 2/sec 2 4 1 5 . 2 0 . 1 6 7 3 . 0 0 1 .1 4 4 6 . 7 8 7 . 7 9 2 8 2 4 . 1 0 . 1 2 0 3 . 0 II 1 0 6 5 . 5 11 . 6 8 3 0 2 9 . 2 0 . 1 2 5 3 . 1 n 1 4 3 5 . 2 1 3 . 9 1 1 3 m i l l e t 1 5 . 2 0 . 1 2 5 3 . 0 1 . 1 3 0 6 . 6 5 . 6 5 1 4 d = 2 . 1 5 mm P II n II 1 . 3 3 8 6 . 0 6 . 3 8 21 P S = 1 . 1 8 gm/cm3 II 0 . 0 8 4 II 1 .1 3 1 0 . 0 5 . 2 8 * 2 6 II 0 . 1 2 5 2 . 6 7 1 . 1 2 9 6 . 1 6 . 1 2 1 5 Glass beads 1 5 . 2 0 . 1 2 5 2 . 6 0 1 . 1 4 0 0 . 3 5 . 8 7 1 6 d p = 2 . 9 3 mm P S = 2 . 9 6 gm/cm3 II n n 1 . 3 4 8 8 . 9 6 . 4 6 1 7 Glass beads, dp = 1 . 1 0 mm 1 5 . 2 0 . 1 2 5 2 . 6 0 1 . 1 9 6 . 8 1 . 4 9 Ammonium n i t ra te P S = 1 . 7 4 gm/cm3 1 9 dp = 1 . 4 5 mm 1 5 . 2 0 . 1 2 5 3 . 0 1 . 1 1 9 7 . 3 3 . 7 5 2 0 d p = 1 . 9 9 mm n n 2 7 1 . 2 4 . 5 4 Cone angle 9 = 1 8 0 ° , and for a l l the other runs 9 = 6 0 ° 133 d e c r e a s e ( run n o s . 1, 29 and 3 2 ) . Most o f the above t r e n d s o f G a ^ and S a re c o n s i s t e n t w i t h those o b s e r v e d by T h o r l e y et al. [ 1 1 , 1 2 ] f o r wheat b e d s . In the l o w e r c o n i c a l p a r t o f the b e d , the v e r t i c a l g r a d i e n t o f downward p a r t i c l e v e l o c i t y became s t e e p e r than i n the upper p a r t o f the bed (see F i g u r e 7 . 2 ) , p o s s i b l y due to the c o n v e r g i n g c r o s s - s e c t i o n a rea of the a n n u l u s . No s i m p l e l i n e a r r e l a t i o n s h i p e x i s t s between v and Z , o r between G, a a and Z , f o r the l ower p a r t o f the bed (see F i g u r e s 7 .3 and 7 .4 ) However , a l l the s o l i d s f l o w r a t e da ta o b t a i n e d i n t h i s work shows t h a t G c o n t i n u e s to d e c r e a s e w i t h d e c r e a s i n g bed l e v e l a 3 i n the c o n i c a l p a r t o f the bed a l s o . The o n l y a t t emp t to f o r m u l a t e a t h e o r y to d e s c r i b e c r o s s - f l o w o f s o l i d s f rom annu lus to spou t was made by L e f r o y and D a v i d s o n ' s [24 ] whose c o l l i s i o n model l e d to the f o l l o w i n g e q u a t i o n : T T 3 e(1 + e ) d v s ( 1 - £ s ) U s - E a )  S = 8(1 - e a ) ( 2 The above e q u a t i o n p r e d i c t s h i g h e r c r o s s - f l o w r a t e s , S , than e x p e r i m e n t a l r e s u l t s , a s shown i n T a b l e 7 . 2 . The d i s c r e p a n c y may be due to the f a c t t h a t v o i d a g e shows a c o n s i d e r a b l e r a d i a l v a r i a t i o n , w i t h a minimum near the spou t w a l l and a maximum a t the spou t a x i s [ 1 0 ] . The f r a c t i o n o f p a r t i c l e s p e r m a n e n t l y e n t r a i n e d i s r e p r e s e n t e d i n E q . ( 2 . 4 6 ) by the v o i d a g e group ( e g - e a ) / ( l - e a ) ; however , e g was assumed by 1 34 Table 7.2 Comparison Between Predicted and Observed Values of Cross-Flow Rates in the Upper Part of the Bed Run No. Mater ia ls U /U s' ms * S , by Eq. (2.46) cm 2 /sec S, exp l . cm 2 /sec + e \ s O - S ) ( V e a ) cm/sec 1 Polystyrene pel l e t s 1.1 7.00 4.40 0.4 6.5 ± 0.5 15 Glass beads (dp=2.93 mm) 1.1 19.05 5.87 0.8 6.6 ± 0.3 16 Glass beads (dp=2.93 mm) 1 .3 19.94 6.46 0.8 6.9 ± 0.3 NOTE: it S i s average value between 0.4 <.Z/H < 1.0. The coe f f i c i en t of r e s t i t u t i o n , e, was determined by dropping a pa r t i c l e on a heavy plate of same material as the pa r t i c l e and measuring the ra t io of height of r i s e to height of f a l l . The experiment was repeated many times (~30) with d i f fe ren t pa r t i c l es and the highest ra t i o taken as e [38], V a r i a t i o n of v c ( l - 0( £<- - e j between 0.4 < Z/H < 1.0. 5 S S a - -135 L e f r o y and Dav idson to be the r a d i a l mean v o i d a g e r a t h e r than a l owe r v a l u e wh ich would o c c u r near the s p o u t - a n n u l us i n t e r f a c e . A l t h o u g h E q . ( 2 . 4 6 ) does not p r e d i c t c o r r e c t v a l u e s o f c r o s s - f l o w r a t e , S , i t i s q u a l i t a t i v e l y c o n s i s t e n t w i t h the o b s e r v e d t r e n d s o f S w i t h the main v a r i a b l e s i n v o l v e d , as d i s c u s s e d b e l o w : ( i ) E f f e c t o f bed l e v e l ( f o r Z > D ): Fo r a g i v e n r u n , both v s and e s d e c r e a s e w i t h i n c r e a s i n g bed l e v e l ( s e c t i o n s 7.2 and 8 . 2 ) . A c c o r d i n g to E q . ( 2 . 4 6 ) , S 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 bed l e v e l due to the e f f e c t o f v s , w h i l e i t s h o u l d i n c r e a s e w i t h i n c r e a s i n g bed l e v e l due to d e c r e a s i n g e . These two o p p o s i t e e f f e c t s tend to c a n c e l out l e a v i n g S sub -s t a n t i a l l y i n d e p e n d e n t o f bed l e v e l , as i n d i c a t e d by the v a l u e s o f v s ( l - e s ) U s - e a ) i n T a b l e 7 . 2 . ( i i ) E f f e c t o f bed d e p t h : A t a g i v e n bed l e v e l v s i n c r e a s e s w i t h i n c r e a s i n g H ( F i g u r e 7 . 1 1 ) , w h i l e £ s r ema ins more or l e s s the same ( F i g u r e 8 . 6 ) . S s h o u l d t h e r e f o r e i n c r e a s e w i t h i n c r e a s i n g H, as i t i n d e e d does ( run n o s . 4 and 1,, 5 and 2 , 6 and 3 , 9 , 8 and 11 i n T a b l e 7 . 1 ) . ( i i i ) E f f e c t o f U /U : A t a g i v e n bed l e v e l , bo th s—ms— 3 v and e i n c r e a s e w i t h i n c r e a s i n g U /U ( F i g u r e s 7 .10 and 8 . 5 ) . The p r o d u c t v _ ( l - £ c ) ( e r - £ ) a l s o i n c r e a s e s though s s s a 1 36 s l i g h t l y , w i t h i n c r e a s i n g U s / U m s (see T a b l e 7.2 da ta f o r g l a s s b e a d s ) , and so does as e x p e c t e d . ( i v ) E f f e c t o f d p : E q . ( 2 . 4 6 ) shows t h a t S i s d i r e c t l y p r o p o r t i o n a l to dp . P a r t i c l e d i a m e t e r e f f e c t a l s o e n t e r s i n d i r e c t l y s i n c e a t a g i v e n bed l e v e l , v g was found to i n c r e a s e w i t h i n c r e a s i n g d ( F i g u r e 7 . 1 0 ) , w h i l e e showed P S l i t t l e v a r i a t i o n w i t h c h a n g i n g p a r t i c l e s i z e ( run n o s . 1 and 1 8 , 15 and 17 i n T a b l e 8 . 1 ) . S would t h e r e f o r e be e x p e c t e d to i n c r e a s e w i t h i n c r e a s i n g d . The da ta i n T a b l e 7.1 a re i n p agreement w i t h the e x p e c t e d t r e n d . (v) E f f e c t o f D c: A t a g i v e n bed l e v e l v g i n c r e a s e s w i t h i n c r e a s i n g D c ( F i g u r e 7 .12 ) w h i l e e s rema ins s u b s t a n t i a l l y the same ( F i g u r e 8 . 7 ) . T h e r e f o r e a c c o r d i n g to E q . ( 2 . 4 6 ) , S s h o u l d i n c r e a s e w i t h i n c r e a s i n g D c . T h i s i s a g a i n c o n s i s t e n t w i t h the o b s e r v e d t r e n d f o r wheat beds ( run n o s . 8 , 28 and 30 i n T a b l e 7 . 1 ) . However f o r beds o f p o l y s t y r e n e p e l l e t s , o b s e r v e d v a l u e s o f S show a d e c r e a s e w i t h i n c r e a s e D c ( run n o s . 1, 29 and 32 i n T a b l e 7 .1 ) and no s a t i s f a c t o r y e x p l a n a t i o n c o u l d be found to e x p l a i n t h i s r e v e r s e d t r e n d . 7 . 1 . 2 A Model f o r S o l i d s Movement i n the Annu lus A p a r t i c l e i n the a n n u l u s moves v e r t i c a l l y downward and r a d i a l l y i nward towards the spou t a l o n g a p a r t i c u l a r f l o w 1 37 path w i t h o u t r a d i a l m i x i n g . P a r t i c l e s c l o s e to the column w a l l tend to t r a v e l down a l o n g the w a l l and to e n t e r the spou t i n the lower p a r t o f the b e d . On the b a s i s o f t h e s e o b s e r v a -t i o n s , a s i m p l e model i s p roposed wh ich e n a b l e s the c a l c u l a t i o n o f p a r t i c l e f l o w pa th and r e t e n t i o n t ime i n the a n n u l u s . L e t us d i v i d e the v e r t i c a l h e i g h t o f the a n n u l u s i n t o M equa l i n t e r v a l s , and the w i d t h of the annu lus a t the top i n t o N equa l i n t e r v a l s . L e t each i n t e r v a l a t the a n n u l u s top r e p r e s e n t one f l o w p a t h , so t h a t the t o t a l number o f f l o w p a t h s e q u a l s N, as shown i n F i g u r e 7 . 5 . L e t G y ( J - 1) be the f l o w r a t e o f s o l i d s between f l o w pa ths J - 1 and J . S o l i d s mass b a l a n c e a t any bed l e v e l I g i v e s , G v ( J - l ) = T r v a ( I ) ( l - e a ) { R ( I , J - 1 ) 2 - R ( I , J ) 4 ( 7 . 4 ) where v ( I ) i s the downward p a r t i c l e v e l o c i t y a t bed l e v e l I, between r a d i a l p o s i t i o n s R ( I , J - l ) and R ( I , J ) . The f l o w pa ths d e s c r i b e d by E q . ( 7 . 4 ) have been c a l c u l a t e d u s i n g e x p e r i m e n t a l v a l u e s o f v ( I ) w i t h the assump-a t i o n t h a t the f i r s t f l o w pa th c o i n c i d e s w i t h the column w a l l , as s u g g e s t e d by v i s u a l o b s e r v a t i o n . Observed and p r e d i c t e d f l o w pa ths a re compared i n F i g u r e 7 . 6 , wh ich shows t h a t E q . ( 7 . 4 ) g i v e s a f a i r l y a c c u r a t e d e s c r i p t i o n o f p a r t i c l e f l o w p a t h s . 1 38 Figure 7.5. Co-ordinate system and gr id points for pa r t i c l e flow path ca l cu la t i ons . (b) D r = 2 9 . 2 c m 140 The r e t e n t i o n t i m e , T ( R ) , o f s o l i d s i n each f l o w pa th o f the annu lus i s d e f i n e d as fL cos(e ) T(R) = n - 9 - dz ( 7 . 5 ) J 0 a where L i s the l e n g t h o f the p a r t i c l e f l o w p a t h , 6 g i s the a n g l e between the f l o w pa th and the v e r t i c a l a x i s , and v . i s the downward p a r t i c l e v e l o c i t y (see F i g u r e 7 . 5 ) . a The r e t e n t i o n t ime f o r any f l o w pa th can be c a l c u l a t e d f rom E q . ( 7 . 5 ) , u s i n g e x p e r i m e n t a l r e s u l t s o f v . C a l c u l a t e d v a l u e s a o f T(R) f o r p a r t i c l e s s t a r t i n g a t d i f f e r e n t r a d i a l p o s i t i o n s a c r o s s the annu lus s u r f a c e , t o g e t h e r w i t h measured r e s i d e n c e t i m e s , a re shown i n F i g u r e 7 . 7 . The c a l c u l a t e d and measured r e s u l t s a re seen to be i n e x c e l l e n t ag reemen t . 7 .2 In the Spout 7 .2 .1 E x p e r i m e n t a l R e s u l t s and D i s c u s s i o n In the s p o u t , p a r t i c l e s move too s w i f t l y to be o b s e r v e d by the naked e y e . P a r t i c l e v e l o c i t i e s i n the spou t w e r e , t h e r e f o r e , measured by t a k i n g h i g h speed mot ion p i c t u r e s o f the spou t a t 2000-3000 f r a m e s / s e c , u s i n g a 16 mm Hycam r o t a t i n g p r i s m type (model K20S4E) c a m e r a . A t i m i n g l i g h t g e n e r a t o r was used to g e n e r a t e l i g h t s p o t s a t 1 m i l l i s e c o n d 141 Figure 7.7. D is t r ibu t ion of retent ion time of so l ids in the annulus. Wheat beds H/D = 3.0, D 0/D = 0.125, U = 1.1 U . c c s ms 142 i n t e r v a l s on the edge o f the f i l m wh ich e n a b l e d the e x a c t f i l m speed to be d e t e r m i n e d . In o r d e r to o b t a i n c l e a r p i c t u r e s o f t he p a r t i c l e s i n the s p o u t , a s m a l l s e c t i o n o f the s p o u t , about 15 cm h i g h , was f i l m e d a t a t i m e . In the case o f c o l o u r l e s s m a t e r i a l s , such as g l a s s b e a d s , ammonium n i t r a t e and p o l y -s t y r e n e p e l l e t s , a s m a l l p r o p o r t i o n o f the p a r t i c l e s was c o a t e d w i t h a red (o r b l u e ) p a i n t to make them c l e a r l y v i s i b l e i n the p i c t u r e s . P a r t i c l e v e l o c i t i e s were measured by p r o j e c t i n g the f i l m w i t h a 16 mm s t o p - a c t i o n p r o j e c t o r wh ich p e r m i t t e d f rame by f rame a n a l y s i s to be c a r r i e d o u t . S i n c e t h e r e e x i s t s a p ronounced r a d i a l g r a d i e n t o f p a r t i c l e v e l o c i t y i n the s p o u t , the v e l o c i t y was measured a t s i x e q u a l l y spaced l o c a t i o n s a c r o s s the d i a m e t e r o f the spou t (see F i g u r e 7 . 8 ) . For each l o c a t i o n , f i v e r e p l i c a t e measurements were made f rom the c i n e f i l m , and an ave rage v a l u e was t aken to be the v e l o c i t y a t t h a t l o c a t i o n . The r e s u l t s o f s e v e r a l e a r l y t e s t s showed the r a d i a l v e l o c i t y p r o f i l e s to be f u l l y symmet r i c a round the v e r t i c a l a x i s o f the s p o u t . T h e r e f o r e , i n s u b s e q u e n t e x p e r i -m e n t s , v e l o c i t i e s were measured o n l y a t t h r e e l o c a t i o n s a c r o s s the spou t r a d i u s a t any g i v e n bed l e v e l . P a r t i c l e v e l o c i t i e s a t D and D 1 ( F i g u r e 7 .8 ) were t aken to be z e r o and the f u l l r a d i a l p r o f i l e was o b t a i n e d by i n t e r p o l a t i o n between the v e l o c i t i e s a t A , B, C and D. The i n t e r p o l a t i o n was done by U . B . C . SPLINE s u b r o u t i n e i n wh ich c u b i c s p l i n e f u n c t i o n s a re used to i n t e r p o l a t e the da ta between known q u a n t i t i e s . The 143 Annulus Spout Annulus i D' C' B' A B C D Figure 7.8. Radial sect ions of spout for pa r t i c l e ve loc i t y measurements. 144 r a d i a l l y ave raged p a r t i c l e v e l o c i t y , v , a t any g i v e n bed l e v e l was c a l c u l a t e d by i n t e g r a t i n g the r a d i a l v e l o c i t y p r o f i l e as f o l l o w s : v s . 2 v r ' d r ' ( 7 . 6 ) J O 3 i-where v s z i s the v e r t i c a l p a r t i c l e v e l o c i t y a t a d i s t a n c e r f rom the spou t a x i s and r 1 the d i m e n s i o n l e s s d i s t a n c e f rom the spou t a x i s ( r / r ) . T y p i c a l r a d i a l p r o f i l e s f o r s e v e r a l bed l e v e l s a r e shown i n F i g u r e 7 . 9 , w h i l e v e r t i c a l p r o f i l e s o f the r a d i a l a v e r a g e p a r t i c l e v e l o c i t y , v , a r e p r e s e n t e d i n F i g u r e s 7 . 1 0 , 7 . 1 1 , 7 .12 and 7 . 1 3 . P a r t i c l e v e l o c i t y da ta f o r a l l t he runs appea r i n A p p e n d i x V I I I ( b ) . The da ta i n F i g u r e 7.9 and f rom a l l the o t h e r runs show t h a t the r a d i a l p r o f i l e s a r e not a lways p a r a b o l i c as s u g g e s t e d by G o r t h s a i n and Makh lenov [25 ] and o t h e r s [ 1 7 , 2 4 ] . Most o f the da ta o b t a i n e d i n the p r e s e n t work can be d e s c r i b e d by the e q u a t i o n v ^ = 1 r i v s c r r s n ( 7 . 7 ) w i t h the v a l u e o f n v a r y i n g between 1.3 and 2 . 2 . Fo r a g i v e n r u n , the v a l u e o f n i n g e n e r a l i n c r e a s e s w i t h i n c r e a s -i ng bed l e v e l ( F i g u r e 7 . 9 ) . (a) Millet (b) Wheat o CD t o e 4 . N CO > 3 0 0.2 0.4 0-6 0.8 1.0 r / r c 0 0.2 0.4 0-6 0.8 1.0 r/r. Figure 7.9. Radial p ro f i l es of pa r t i c l e ve loc i t y in the spout. 1 46 Figure 7.10. Ef fect of spouting ve loc i t y and pa r t i c l e diameter on spout pa r t i c l e ve loc i t y . Glass beads, D = 15.2 cm, H/D c = 3.0, D 0 /D c = 0.125. C 1 48 4 o cu CO - 2 0 1 1 1 1 D c , c m D 0 / D c o 15.2 0.125 r "/V \ A N @X ^ A , © 24.1 0 .120 * 2 9 . 2 0.125 ^ A ^ 1 1 1 1 0 2 0 4 0 6 0 Z, cm 8 0 100 Fiqure 7.12. Ef fect of column diameter on spout pa r t i c l e ve loc i t y : Wheat, H/D c = 3.0, U s / U m s = 1 . 1 . 149 150 Examples of l o n g i t u d i n a l p r o f i l e s o f the r a d i a l a v e r a g e p a r t i c l e v e l o c i t y , v s , f o r d i f f e r e n t s o l i d m a t e r i a l s and o p e r a t i n g c o n d i t i o n s , a re shown i n F i g u r e 7 .10 to F i g u r e 7 . 1 3 . I t i s seen t h a t a p a r t i c l e s t a r t i n g f rom the base o f the bed a c c e l e r a t e s f rom r e s t to a maximum v e l o c i t y , then d e c e l e r a t e s , r a p i d l y a t f i r s t and then more s l o w l y i n the upper p a r t o f the s p o u t . P a r t i c l e v e l o c i t i e s a t a l l l e v e l s i n the spou t were g e n e r a l l y found to i n c r e a s e w i t h i n c r e a s i n g U s / U m s ( F i g u r e 7 . 1 0 ) , p a r t i c l e s i z e ( F i g u r e 7 . 1 0 ) , bed dep th ( F i g u r e 7 . 1 1 ) , co lumn d i a m e t e r ( F i g u r e 7 .12 ) and w i t h d e c r e a s i n g gas i n l e t d i a m e t e r ( F i g u r e 7 . 1 3 ) . 7 . 2 . 2 A Model f o r P a r t i c l e Movement i n the Spout Two t h e o r e t i c a l models d e s c r i b i n g the l o n g i t u d i n a l p r o f i l e o f spou t p a r t i c l e v e l o c i t y have been p r e v i o u s l y p r o p o s e d and a r e r e v i e w e d i n C h a p t e r 2 . The v a l i d i t y o f bo th t h e s e models was t e s t e d a g a i n s t e x p e r i m e n t a l r e s u l t s o b t a i n e d i n the p r e s e n t work . F i g u r e 7.14 shows t h a t p a r t i c l e v e l o c i t i e s p r e -d i c t e d by the f o r c e b a l a n c e model o f T h o r l e y et al. [11 ] as amended by Mathur and E p s t e i n [10 ] a re a t l e a s t t w i c e the e x p e r i m e n t a l v a l u e s a t any g i v e n bed l e v e l . In s o l v i n g the t h e o r e t i c a l e q u a t i o n s ( E q . ( 2 . 4 1 ) , ( 2 , 4 0 ) and ( 2 . 3 9 ) ) e x p e r i m e n t a l v a l u e s o f u $ , e s and H 1 were u s e d . The i n i t i a l c o n d i t i o n s f o r E q . ( 2 . 4 1 ) were taken as v c = 0 , u c = u 0 151 O CD C O 0 •o— Exper imenta l_ ™.~ Calculated 'Eq.2.41 Eq .2 .40 0 Figure 7.14. 0.2 0.4 0.6 Z / H 0.8 .0 Comparison between experimental spout pa r t i c l e ve loc i t i es and predic t ion by the amended force balance model of Thorley et al. [10]. Polystyrene p e l l e t s , D = 15.2 cm, H/D = 3.0, Do/D, 0.125, U s / U m s 1.1 152 and e $ = 1 . 0 0 a t Z = 0 , u 0 b e i n g the gas v e l o c i t y t h r o u g h t h e i n l e t o r i f i c e . P r e d i c t i o n s f rom the same model were found by Mathur and E p s t e i n [ 1 0 ] to be i n r e a s o n a b l y good agreement w i t h two s e t s o f p r e v i o u s l y p u b l i s h e d d a t a f o r wheat beds [ 1 1 , 1 2 ] . The p a r t i c l e v e l o c i t i e s i n t h e s e da ta w e r e , how-e v e r , b i a s e d towards v e l o c i t i e s a t the spou t a x i s , and wou ld t h e r e f o r e be l a r g e r than r a d i a l a v e r a g e v e l o c i t i e s wh ich a re p r e d i c t e d by the m o d e l . In a d d i t i o n , the da ta d i d not i n c l u d e d i r e c t l y measured v a l u e s o f U s - These were e s t i m a t e d by Mathur and E p s t e i n f o r use i n E q . ( 2 . 4 1 ) , but w i t h r e s e r v a t i o n c o n c e r n i n g the a c c u r a c y o f the e s t i m a t -i n g p r o c e d u r e , p a r t i c u l a r l y f o r the l o w e r p a r t o f t he s p o u t . The p r e s e n t d a t a a re f r e e f rom the above u n c e r t a i n t i e s and p r o v i d e a more r e l i a b l e b a s i s f o r e v a l u a t i o n o f the t h e o r e t i c a l model than the d a t a used by Mathur and E p s t e i n . The model o f L e f r o y and D a v i d s o n [ 2 4 ] was t e s t e d i n two ways . F i r s t , the same p r o c e d u r e as used by L e f r o y and D a v i d s o n [ 2 4 ] and a l s o by Mathur and E p s t e i n [ 1 0 ] , was f o l l o w e d . T h i s i n v o l v e s s o l v i n g the momentum b a l a n c e e q u a t i o n s , e q s . ( 2 . 4 2 ) and ( 2 . 4 3 ) , t o g e t h e r w i t h the e m p i r i c a l e q u a t i o n s f o r d P / d Z , E q . ( 2 . 4 5 ) , f o r u , E q . ( 2 . 3 6 ) , and the gas mass b a l a n c e e q u a t i o n , E q . ( 7 . 9 ) . C a l c u l a t i o n s were c a r r i e d out u s i n g measured v a l u e s o f spou t d i a m e t e r a t v a r i o u s bed l e v e l s to e s t i m a t e u s > and a l s o w i t h a s i n g l e 1.53 v a l u e o f spou t d i a m e t e r , o b t a i n e d by a v e r a g i n g the measured v a l u e s ove r the h e i g h t o f the b e d . The c o m p a r i s o n i n F i g u r e 7 .15 shows t h a t t h e r e i s poor agreement between e x p e r i m e n t a l da ta and model p r e d i c t i o n s , r e g a r d l e s s o f whe ther the p r e -d i c t i o n i s based on the measured spou t c o n t o u r or an ave rage v a l u e of d i a m e t e r . Mathur and E p s t e i n a l s o came to the same c o n c l u s i o n (see F i g u r e 2 . 1 3 ) . The second method used f o r e v a l u a t i n g tffe L e f r o y and D a v i d s o n model c o n s i s t e d o f c o m b i n i n g the momentum b a l a n c e e q u a t i o n s ( e q s . ( 2 . 4 2 ) and ( 2 . 4 3 ) ) to e l i m i n a t e d P / d Z , d [ ( l s s ) v s > ] r l d Z ' e s H p f d K u s 2 ) dZ 6 (u, v ) 2 ( p , p f ) g H ( 7 . 3 ) and s o l v i n g E q . ( 7 . 8 ) f o r v s , e g and u g t o g e t h e r w i t h the f o l l o w i n g gas and s o l i d s mass b a l a n c e e q u a t i o n s . Gas mass b a l a n c e : d ^ A s e s u s d Z ' dQ. dZ 1 (7.9.) S o l i d s mass b a l a n c e : ^ V 1 - £s>vs dZ ' 5 l d Z ' ( 7 . 1 0 ) 1 54 O if) 0 — E x p e r i m e n t a l C a l c u l a t e d L I I / / / / 1 o ' \ o Figure 7.15. 0.2 0.4 0.6 0.8 0 Z / H « / I I Comparison between experimental spout p a r t i c l e ve l oc i t i e s and predict ion by the Lefroy and Davidson's model [24] using empirical equations for determining u s and e s . Polystyrene p e l l e t s , D g = 15.2 cm, H/D c = 3 .0, D 0 / D c = 0.125, Us/Ums = 1 . 1 . using measured values of spout diameter, using average spout diameter. 155 The above e q u a t i o n s were s o l v e d u s i n g e x p e r i m e n t a l l y d e t e r m i n e d v e r t i c a l p r o f i l e s o f a i r and s o l i d s f l o w r a t e s i n the a n n u l u s (d Q / d Z ' and dG / d Z 1 ) and a l s o measured v a l u e s a a o f spou t d i a m e t e r a t v a r i o u s bed l e v e l s . The v a l u e s o f v g o b t a i n e d by t h i s me thod , w i t h the i n i t i a l c o n d i t i o n s v g = 0 , u s = u 0 and = 0 a t Z ' = 0 , were found to be much h i g h e r than the e x p e r i m e n t a l r e s u l t s (see F i g u r e 7 . 1 6 ) . However , i f e x p e r i m e n t a l v a l u e s o f v s , u s and e g a t a bed l e v e l some-what removed f rom the gas e n t r a n c e r e g i o n ( Z ' > 0 . 1 0 ) a re used as the i n i t i a l c o n d i t i o n s f o r s o l v i n g E q s . ( 7 . 8 ) , ( 7 . 9 ) and ( 7 . 1 0 ) , the c a l c u l a t e d v a l u e s o f v g ag ree w e l l w i t h the e x p e r i m e n t a l r e s u l t s (see F i g u r e 7 . 1 6 ) . I t would a p p e a r , t h e r e f o r e , t h a t the momentum b a l a n c e e q u a t i o n s o f L e f r o y and D a v i d s o n a re not a p p l i c a b l e i n the v i c i n i t y o f the gas i n l e t ( f o r Z 1 < 0 . 1 ) , p o s s i b l y because o f the a b r u p t change i n spou t d i a m e t e r wh ich o c c u r s i n t h i s r e g i o n ( s e c t i o n 8 . 1 ) . I t has been found p o s s i b l e to f u r t h e r mod i f y the f o r c e - b a l a n c e model o f T h o r l e y et a l . , as amended by Mathur and E p s t e i n , so as to o b t a i n b e t t e r agreement between p r e -d i c t i o n and e x p e r i m e n t a l r e s u l t s . The m o d i f i c a t i o n s t a r t s f rom E q . ( 2 . 3 8 ) , wh ich was d e r i v e d by T h o r l e y et a l . f rom a f o r c e b a l a n c e on the spou t p a r t i c l e s . v 2 d n , v„ d v r 3 p , ( u „ - v J 2 (Pc. - p.) s " " z . v s u " s " M f v u s s ' . 2 r v p s ^f n n z dZ dZ " 4 d p d p s d v D P s ( 2 . 1 56 4 o CD CO 0 m a x i m u m vQ of \ 125.0 at Z/H = . 025 Exper imenta l Ca l cu la ted -A* ^ s , o 0.2 0.4 0.6 0.8 .0 Z / H Figure 7.16. Comparison between experimental spout pa r t i c l e v e l oc i t i e s and predict ion by Lefroy and Davidson's model [24] using exper i -mentally determined values of u s and e s. Polystyrene p e l l e t s , D, 15.2 cm, H/D„ = 3.0, D 0/D = 0.125 U /U s' ms 1.1. c "c solv ing s ta r t at Z/H = 0, solv ing s ta r t at Z/H = 0 .1 . 1 57 In g e n e r a l , the r e l a t i o n s h i p between spou t v o i d a g e and number o f p a r t i c l e s n z i n the spou t a t any g i v e n bed l e v e l can be e x p r e s s e d as I V <' " s s> " "z \ V ( 7 ' , 1 ) which on r e a r r a n g i n g g i v e s D 2 n z = ^ (1 - c s ) ( 7 . 1 2 ) D i f f e r e n t i a t i o n o f E q . ( 7 . 1 2 ) w i t h r e s p e c t to bed l e v e l Z g i v e s d n z D s 2 d e s (1 - z%) d D g 2 ~dl = " d - 2 " ~dl + ~d~^ d T " ( 7 - 1 3 ) P P R e p l a c i n g n z and d n z / d Z i n E q . ( 2 . 3 8 ) a c c o r d i n g to E q s . ( 7 . 1 2 ) and (7 . 1 3 ) , we ge t v de v 2 dD 2 v „ dv s s , s s . s s + —n r=i— + 1-e dZ D 2 d Z - dZ s s 3 p f ( u - v ) 2 d 2 C (p - p f ) r s s v D s f _ g ( 7 . 1 4 ) The mass b a l a n c e o f s o l i d s , e x p r e s s e d by E q . ( 7 . 1 0 ) , can be r e w r i t t e n as f o l l o w s : 1 58 dZ = < (1 -dv v (1 - E.) dD : _ ^ s , _ S s s e s J "cTT + — T J 7 2 dT A . dZ d G a l - d f } / v : ( 7 . 1 5 ) E q s . ( 7 . 1 4 ) and ( 7 . 1 5 ) a r e combined and r e a r r a n g e d to o b t a i n the f i n a l form of the w o r k i n g e q u a t i o n = 1.0 1 3 P f ( u s - v s P d / C p 4d ' p_ { p s " p f } ) d G a The e x p r e s s i o n f o r i n E q . ( 7 . 1 6 ) can be o b t a i n e d f rom the R i c h a r d s o n - Z a k i [57 ] e q u a t i o n w h i c h , f o r R e p > 500 , i s as f o l l o w s : e t 2 .39 ; = e l t ( 7 . 1 7 ) where v i t i s the t e r m i n a l v e l o c i t y o f p a r t i c l e s a t i n f i n i t e d i l u t i o n and V t i s the s u p e r f i c i a l t e r m i n a l v e l o c i t y o f a swarm o f p a r t i c l e s w i t h v o i d a g e e. In the s p o u t , V £ t i s equa l to ( u g - v s ) e s , so E q . ( 7 . 1 7 ) becomes u s - V s 1.39 v . = e s 1 1 ( 7 . 1 8 ) At the t e r m i n a l v e l o c i t y , the d rag c o e f f i c i e n t e q u a l s 0 .44 f o r R e p > 500 [ 5 7 ] . T h e r e f o r e , d rag f o r c e , F , on each p a r t i c l e a t v l t i s 159 F = 0 .44 8 V ^ V L T 2 <7-19) S u b s t i t u t i n g E q . ( 7 . 1 8 ) i n t o E q . ( 7 . 1 9 ) to e l i m i n a t e v 1 t , , g i v e s 0 .44 2 .78 e s "J (f]dp2 P f ( u s " v s ^ 2 < 7 - 2 0 > From the d e f i n i t i o n o f Cp, we o b t a i n e s A l t e r n a t i v e l y , C Q can be o b t a i n e d f rom the e q u a -t i o n o f Wen and Yu [58 ] who found t h a t f o r any f l o w reg ime F T " ~T77 (7.22) e where F x i s the drag f o r c e o f a f l u i d a c t i n g on a s i n g l e sphe re a t i n f i t e d i l u t i o n (e = 1) and F £ i s the drag f o r c e o f a f l u i d a c t i n g on a sphe re i n an assemb lage w i t h v o i d a g e e. Both F x and F £ a r e a t the same r e l a t i v e s u p e r f i c i a l v e l o c i t y between the f l u i d and the p a r t i c l e s . In the s p o u t , the r e l a -t i v e i n t e r s t i t i a l v e l o c i t y between p a r t i c l e s and f l u i d i s u s " v s * T h e r e f o r e , drag f o r c e a t e = 1, F i , i s 1 60 F i 'Dl V p f ( u s v )2 ( 7 . 2 3 ) and at e = e s , the d rag f o r c e i s d 2 P P F (u v s ) 2 ( 7 . 2 4 ) CQ-J , the drag c o e f f i c i e n t a t z - 1 , e q u a l s 0 .44 f o r R e p > 500 [ 5 7 ] . S u b s t i t u t i n g E q s . ( 7 . 2 3 ) , ( 7 . 2 4 ) and 0.44 f o r C i n t o E q . ( 7 . 2 2 ) and r e a r r a n g i n g y i e l d s Dl 'D 0 .44 2 .70 ( 7 . 2 5 ) wh ich i s a l m o s t i d e n t i c a l to E q . ( 7 . 2 1 ) d e r i v e d f rom the R i c h a r d s o n - Z a k i e q u a t i o n . S u b s t i t u t i o n o f Cp from E q . ( 7 . 2 1 ) i n t o E q . ( 7 . 1 6 ) g i v e s 1 .0 A c " 0 . 3 3 P f ( u c - v ) z d : s f s S V pf) dG a ST p s e s 2 . 78 9J ( 7 . 2 6 ) For s o l v i n g the above e q u a t i o n , the a i r and s o l i d s mass b a l a n c e e q u a t i o n s , E q s . ( 7 . 9 ) and ( 7 . 1 0 ) , and e x p e r i m e n t a l v a l u e s o f D . G, and G\ a re needed . E q . ( 7 . 2 6 ) would then S a a y i e l d not o n l y the v e r t i c a l p r o f i l e of v $ i n the s p o u t , but a l s o o f u s and e $ 161 I t s h o u l d be no ted t h a t a c c o r d i n g to the above t h e o r y , v g ove r the e n t i r e h e i g h t o f the spou t i s d e s c r i b e d by the same e q u a t i o n v i z . E q . ( 7 . 2 6 ) r a t h e r than by t h r e e d i f f e r e n t e q u a -t i o n s a p p l i c a b l e to the l o w e r , m i d d l e and upper zones o f the spou t as i n the T h o r l e y m o d e l . The g e n e r a l i z e d c h a r a c t e r o f E q . ( 7 . 2 6 ) f o l l o w s from e x p r e s s i n g the p a r t i c l e c r o s s - f l o w d n z / d Z by E q . ( 7 . 1 3 ) and the d rag c o e f f i c i e n t C Q by E q . ( 7 . 2 1 ) . In the T h o r l e y m o d e l , d n z / d Z was assumed to be ze ro i n the l owe r and upper zones and c o n s t a n t i n the m i d d l e z o n e , w h i l e the d rag f o r c e was t a k e n i n t o a c c o u n t o n l y i n the l owe r z o n e . C a l c u l a t i o n s by E q . ( 7 . 2 6 ) have been c a r r i e d out s t e p by s t e p w i t h the s t ep l e n g t h Z ' = 0 . 0 2 5 and the i n i t i a l c o n d i t i o n s v $ = 0 , e s = 1 and u s = u 0 a t Z = 0 , u s i n g a s u c c e s s i v e a p p r o x i -ma t i on p r o c e d u r e . A t each s t e p , a v a l u e o f e c a l l e d E was assumed to e v a l u a t e u $ and v g f rom E q s . ( 7 . 9 ) and ( 7 . 1 0 ) . These v a l u e s o f u g , v s and E z were s u b s t i t u t e d i n t o the r i g h t hand s i d e o f E q . ( 7 . 2 6 ) and the v a l u e o f e g t hus o b t a i n e d was compared w i t h the assumed v a l u e E z - I f the d i f f e r e n c e between e s and E z was g r e a t e r than 0 . 0 0 0 5 , a new v a l u e o f E z was chosen and the c a l c u l a t i o n p r o c e d u r e was r e p e a t e d . Some examples o f the outcome o f t h e s e c a l c u l a t i o n s a r e shown i n F i g u r e s 7 . 1 7 , 7 . 1 8 , and 7 . 1 9 , f o r t h r e e d i f f e r e n t s o l i d m a t e r i a l s . The agreement between e x p e r i m e n t a l and p r e -d i c t e d r e s u l t o f v $ i s r e m a r k a b l y good and v e r y much b e t t e r than t h a t o b t a i n e d w i t h e i t h e r the T h o r l e y - M a t h u r - E p s t e i n model 162 O If) £ 2 > o o 0.2 — o — Exper imenta l Ca l cu la ted 0.4 0.6 0.8 1.0 Z/H Figure 7.17, Comparison between experimental spout pa r t i c l e ve l oc i t i e s and predic t ion by modified force balance model of Thorley et al., Eq. (7.26). Polystyrene p e l l e t s , D c = 15.2 cm, H/D c = 3.0, D 0 /D c = 0.125, l y i J ^ = 1.10. 163 Figure 7.18. Comparison between experimental spout pa r t i c l e v e l o c i t i e s and predict ion by modified force balance model of Thorley et al.t Eq. (7.26). Wheat, D = 15.2 cm, H/D = 3.0, D 0 /D c = 0.125, U s / U m s = 1.1. c c 164 O CO CO > 4 Experimental Calculated P Q Q 0.2 0 . 4 0 . 6 0 . 8 1.0 Z / H Figure 7.19. Comparison between experimental spout pa r t i c l e ve l oc i t i e s and predic t ion by modified force balance model of Thorley et al., Eq. (7.26). M i l l e t , D = 15.2 cm, H/D = 3.0, D 0 / D c = 0.125, U s / U m s = 1.1. C 165 ( F i g u r e 7 .14 ) or the L e f r o y - D a v i d s o n model ( F i g u r e s 7 .15 and 7 . 1 6 ) . The main improvement ove r the fo rmer model i s c o n -s i d e r e d to be the i n t r o d u c t i o n o f a g e n e r a l i z e d e x p r e s s i o n f o r s o l i d s ~ c r o s s - f 1 o w ( E q . ( 7 . 1 3 ) ) , and the i ndependence o f s u c c e s s i v e v a l u e s o f v $ c a l c u l a t e d by E q . ( 7 . 2 6 ) f rom the r e s u l t o f the p r e v i o u s c a l c u l a t i o n s t e p and t h e r e f o r e f rom the e f f e c t o f u n c e r t a i n t i e s in the d i f f i c u l t g a s - e n t r a n c e r e g i o n . C h a p t e r 8 BED STRUCTURE 8.1 Spout Shape H a l f - c y l i n d r i c a l co lumns made o f P l e x i g l a s s were used f o r measu r i ng the spou t shape as v iewed a g a i n s t the f l a t t r a n s p a r e n t f a c e o f the c o l u m n . The gas i n l e t o r i f i c e s used were a l s o s e m i c i r c u l a r i n s h a p e . Spout d i a m e t e r s were measured a t s e v e r a l bed l e v e l s so t h a t the f u l l c o n t o u r o f the spou t c o u l d be d rawn . The spou t boundary or s p o u t - a n n u l us i n t e r f a c e was c l e a r l y i d e n t i f i a b l e as the p o i n t where c r o s s -f l o w i n g p a r t i c l e s from the a n n u l u s r e v e r s e d t h e i r d i r e c t i o n o f movement f rom downward ( i n the a n n u l u s ) to upward ( i n the s p o u t ) . In o r d e r to f i n d out i f the f l a t f a c e o f the h a l f -co lumn has any d i s t o r t i n g e f f e c t on spou t s h a p e , a spou ted bed o f mus ta rd seed c o n t a i n e d i n a 29 x 14 cm r e c t a n g u l a r co lumn was x - r a y p h o t o g r a p h e d . The pho tog raphs were taken (by D r . K . B . Ma thu r ) w i t h the s p e c i a l equ ipment i n P r o f . P . N . Rowe's l a b o r a t o r y a t U n i v e r s i t y C o l l e g e , London [ 4 6 ] , 1 66 167 wh ich i s d e s i g n e d f o r t a k i n g x - r a y pho tog raphs o f f l u i d i z e d beds f o r s t u d y o f bubb le b e h a v i o u r . The spou t shape was a l s o d e t e r m i n e d i n a bed of mus ta rd seed c o n t a i n e d i n a h a l f -s e c t i o n a l v e r s i o n o f the r e c t a n g u l a r column (29 x 7 cm) where the spou t c o u l d be o b s e r v e d a g a i n s t the column f a c e , under the same o p e r a t i n g c o n d i t i o n s as used f o r the f u l l c o l u m n . A c o m p a r i s o n o f the spou t c o n t o u r s o b t a i n e d i n the f u l l and h a l f - r e c t a n g u l a r c o l u m n s , shown i n F i g u r e 8 . 1 , i n d i c a t e s t h a t no s i g n i f i c a n t d i s t o r t i o n o f spou t shape o c c u r s due to the p r e s e n c e o f the f l a t w a l l i n the h a l f - c o l u m n . T h i s c o n f i r m s M i k h a i l i k ' s [33 ] c o n c l u s i o n , based on spou t shape measurements i n f u l l and h a l f co lumns ( 9 . 4 cm d i a m e t e r ) u s i n g a p i e z o e l e c t r i c probe f o r the f u l l c o l u m n s . The x - r a y p h o t o g r a D h i c method i s c o n s i d e r e d to be more r e l i a b l e than t h a t employed by M i k h a i l i k , s i n c e i t e l i m i n a t e s the p o s s i b i l i t y o f any d i s t o r t i o n due to i n s e r t i o n o f a p r o b e . The spou t shapes o b s e r v e d d u r i n g t h i s work can be c l a s s i f i e d i n t o t h r e e g e n e r a l t y p e s i l l u s t r a t e d i n F i g u r e 8 . 2 . Types A and B i n F i g u r e 8 .2 c o r r e s p o n d to t ypes (a) and (b) i n F i g u r e 2 .15 [ 1 0 ] , but shape type ( c ) o f F i g u r e ( 2 . 1 5 ) i n wh ich the n e c k i n g o f the spou t o c c u r s i m m e d i a t e l y above the gas i n l e t was not o b s e r v e d i n the p r e s e n t wo rk . I t has not p roved p o s s i b l e to d e v e l o p any q u a n t i t a -t i v e c o r r e l a t i o n s f o r spou t s h a p e , but the f o l l o w i n g q u a l i t a -t i v e t r e n d s can be seen f rom the da ta p r e s e n t e d i n F i g u r e 8 . 3 and 8 . 4 : (1) Bed dep th and s p o u t i n g v e l o c i t y ' (U /U ) have e ° 3 CO Q 0-0-0 - 0 -^-o-0-OT)-0°0-0-0 o •oo-o-o-^rcTcT'0 ° 'O •o o S o l i d s i M u s t a r d s e e d Fu l l co l umn , 2 9 x 14 c m X - ray r e s u l t s H a l f c o l u m n , 2 9 x 7 c m 0 8 12 16 2 0 2 4 Z , cm 2 8 3 2 3 6 Figure 8 .1 . Comparison of spout shape in f u l l and hal f columns. CTl CD b c Observed spout shapes. a . Diverges cont inuously, b. Expands, then tapers or remains constant in diameter, c . Expands, necks and then diverges. 1 70 8 Figure 8 .3 . Ef fect of spouting ve loc i t y and bed depth on spout shape: polystyrene p e l l e t s , D = 15.2 cm, D / D 0 = 15.2 cm, D c /D 0 = 0.125. c C 171 Figure 8.4. Ef fect of column diameter and pa r t i c l e s ize on spout shape: Polystyrene.pel l e t s , D c /D 0 = 0.125, H/D c = 3.0. 1 72 l i t t l e e f f e c t on spou t s h a p e , a l l t h r e e shapes i n F i g u r e 8 . 3 b e i n g o f t ype A ( F i g u r e 8 . 2 ) . (2) Spout shape i s s t r o n g l y dependent on co lumn d i a m e t e r and p a r t i c l e s i z e . The da ta i n F i g u r e 8 .4 show t h a t the spou t shape changes f rom type A to type B w i t h i n c r e a s e i n column d i a m e t e r ( f rom 15 .2 cm to 24.1 cm) , and f rom t ype A to type C w i t h d e c r e a s e i n p a r t i c l e s i z e ( f rom 2 .93 mm to 1.61 mm) . V a l u e s o f spou t d i a m e t e r ave raged o v e r the f u l l h e i g h t o f the bed were found to be s m a l l e r than t h o s e p r e -d i c t e d by the McNab e q u a t i o n , E q . ( 2 . 4 6 ) [ 3 2 ] . Ave rage v a l u e s o v e r the h e i g h t 0 .2 < Z ' < 1 . 0 , i . e . e x c l u d i n g the gas e n t r a n c e r e g i o n a t the b o t t o m , however , showed good agreement w i t h the e q u a t i o n . 8 . 2 Spout V o i d a g e D i s t r i b u t i o n 8 .2 .1 E x p e r i m e n t a l R e s u l t s and D i s c u s s i o n C o n s i d e r a c r o s s - s e c t i o n a l a rea of the spou ted bed a t a d i s t a n c e Z f rom the gas i n l e t and l e t e s be the r a d i a l a v e r a g e v o i d a g e a c r o s s the s p o u t . The v o l u m e t r i c s o l i d s f l o w r a t e i n the s p o u t , 6 S , i s g i v e n by Gs " J V Cl - e s] v s ( 8 . 1 ) and i n the a n n u l u s , G . i s g i v e n by a 1 73 H 4 D 2 - D 2 c s ( 8 . 2 ) In the above e q u a t i o n s , v $ and v Q a re the r a d i a l l y a v e r a g e d p a r t i c l e v e l o c i t i e s i n the spou t and the a n n u l u s r e s p e c t i v e l y , and e g i s the a n n u l u s v o i d a g e wh ich i s equa l to the v o i d a g e i n a l o o s e l y packed b e d . By mass b a l a n c e o f s o l i d s f l o w i n the s p o u t and annu l us , we have ( 8 . 3 ) S u b s t i t u t i n g E q s . ( 8 . 1 ) and ( 8 . 2 ) i n t o E q . ( 8 . 3 ) , and r e a r r a n g ^ i n g y i e 1 d s 1 - ^ ( 1 - e . J . ( 8 . 4 ) In t h i s w o r k , a l l the q u a n t i t i e s on the r i g h t hand s i d e o f E q . ( 8 . 4 ) were m e a s u r e d , and the r a d i a l ave rage spou t v o i d a g e , e , was then c a l c u l a t e d . A l l the r e s u l t s o f e a re p r e s e n t e d i n T a b l e 8 . 1 , and some t y p i c a l da ta a re shown i n F i g u r e s 8 . 5 to 8 . 8 . The r e s u l t s i n d i c a t e t h a t : ( i ) spou t v o i d a g e a t any bed l e v e l i n c r e a s e s w i t h i n c r e a s i n g s p o u t i n g v e l o c i t y above the minimum s p o u t i n g v e l o c i t y , i . e . u s / u m s ( F i g u r e 8 .5 and T a b l e 8 . 1 ; run n o s . 1, 2 and 3 ; 4 , 5 and 6 ; 8 and 1 2 ; 13 and 1 74 Table 8.1 Experimental Spout Voidage D is t r ibu t ion (for experimental condit ions see Appendix I) Run Ma "ho v i a 1 Z/H l t d L c i l a i IN 0 . 0 .0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0 1 Polystyrene 1 .0 .971 .946 .921 .896 .858 .829 .804 .788 .769 .758 2 pe l le ts 1 .0 .974 .957 .939 .917 .888 .857 .835 .815 .798 .783 3 1 .0 .972 .958 .944 .927 .902 .876 .854 .835 .818 .808 4 1 .0 .989 .968 .946 .920 .898 .881 .861 .844 .826 .804 5 1 .0 .985 .963 .945 .926 .904 .887 .869 .854 .839 .830 6 1 .0 .985 .967 .953 .934 .912 .893 .878 .862 .847 .834 23 1 .0 .959 .948 .936 .916 .887 .845 .810 .768 .735 .708 25 1 .0 .951 .909 .867 .828 .788 .773 .748 .708 .673 .650 27 1 .0 .980 .952 .915 .880 .843 .802 .764 .728 .693 .660 29 1 .0 .991 .980 .955 .919 .885 .852 .820 .793 .777 .761 32 1 .0 .996 .990 .970 .928 .890 .861 .834 .815 .792 .778 33 1 .0 .991 .983 .962 .937 .910 .879 .849 .820 .795 .775 18 1 .0 .981 .962 .931 .898 .874 .847 .825 .799 .796 .785 Wheat 8 1 .0 .944 .915 .881 .855 .827 .795 .757 .724 .691 .668 12 1 .0 .934 .909 .877 .850 .824 .793 .767 .742 .717 .688 9 1 .0 .965 .941 .914 .888 .361 .840 .814 .787 .765 .733 11 1 .0 .921 .879 .844 .800 .755 .716 .689 .672 .673 .680 24 1 .0 .930 .902 .855 .815 .780 .748 .716 .687 .661 .644 28 1 .0 .962 .935 .899 .861 .828 .786 .740 .680 .644 .588 30 1 .0 .965 .940 .903 .868 .820 .778 .747 .723 .713 .706 31 1 .0 .983 .981 .974 .958 .938 .907 .871 .833 .801 .773 13 M i l l e t 1 .0 .951 .925 .870 .826 .764 .723 .661 .653 .640 .630 14 1 .0 .957 .949 .922 .883 .844 .808 .771 .766 .771 .781 21 1 .0 .956 .928 .906 .868 .823 .775 .726 .679 .627 .610 26 1 .0 .980 .958 .902 .848 .791 .725 .652 .587 .535 .496 15 Glass beads 1 .0 .960 .943 .931 .911 .883 .855 .834 .808 .789 .762 16 1 .0 .964 .959 .952 .936 .912 .889 .873 .859 .849 .836 17 1 .0 .989 .974 .945 .905 .875 .844 .810 .788 .760 .742 Ammonium 19 ni t rate 1 .0 .889 .969 .930 .890 .859 .831 .805 .781 .759 .738 20 1 .0 .972 .951 .921 .896 .870 .846 .810 .781 .756 .729 175 Figure 8 .5 . Ef fect of U s / U m S on spout voidage: Polystyrene p e l l e t s , D_ = 15.2 cm, H/D = 3.0, D 0/D„ = 0.125 (run nos. 1, 2 L c c and 3) . 1.0 0.9 0.8 0.7 0.6 • « • • • • • — • © « — O -i . . A-—A-S o l i d s H , c m Wheat 3 0 . 5 4 5 . 7 6 3 . 5 Polystyrene 3 0 . 5 4 5 7 0.5 'A. 0 10 2 0 3 0 Z , cm 4 0 5 0 60 Figure 8.6. Ef fect of bed depth on spout voidage: D = 15.2 cm, D 0/D = 0.125, U s / U m s = 1 , 1 ^ r u n n o s " ] ' 4 ' 8 ' 9 a n d 0.4 0.6 Z / H .0 Figure 8 .7 . Ef fect of column diameter on spout voidage: wheat, H/D c = 3.0, D 0 / D c = 0.125, U s / U m s = 1.1 (run nos. 8, 28 and 30). 0 0.9 0.8 0.7 0.6 0.5 'A. 0 ~ s X ' O ' . . . , •o a M M A* • 0 . 2 'Q ^ A . ' 4 D 0 , cm 1.27 1.91 2 . 5 4 0 . 4 0 . 6 0 . 8 1.0 Z/H Figure 8.8. Ef fect of gas i n l e t diameter on spout voidage: Polystyrene p e l l e t s , D c = 15.2 cm, H/D c = 3.0, U s / U m s = 1 , 1 ( r u n n o s ' ] ' 2 3 a n d 2 5 ^ -179 1 4 ; 15 and 1 6 ) ; ( i i ) spou t v o i d a g e a t a g i v e n bed l e v e l r ema ins more o r l e s s the same r e g a r d l e s s o f bed depth ( F i g u r e 8 .6 and T a b l e 8 . 1 ; run n o s . 1 and 4 ; 8 , 9 and 1 1 ; 30 and 3 1 ) ; ( i i i ) spou t v o i d a g e at a g i v e n reduced bed l e v e l , Z ' , r ema ins s u b -s t a n t i a l l y the same f o r g e o m e t r i c a l l y s i m i l a r beds ( c o n s t a n t H / D c and D c / D o ) o f the same m a t e r i a l , o p e r a t i n g a t the same r a t i o o f I) / U „ ( F i g u r e 8 .7 and T a b l e 8 . 1 ; run n o s . 8 , 28 and s ms 3 3 0 ; 1, 29 and 3 2 ) . ( i v ) Spout v o i d a g e i n most c a s e s was found to i n c r e a s e w i t h d e c r e a s i n g gas i n l e t d i a m e t e r ( F i g u r e 8 .8 and T a b l e 8 . 1 ; run n o s . 1 and 2 5 ; 13 and 2 1 ; 8 and 2 4 ) . In some c a s e s , however , t h i s t r e n d was not c l e a r , and v o i d a g e p r o f i l e s f o r d i f f e r e n t o r i f i c e d i a m e t e r s were found to c r o s s each o t h e r ( F i g u r e 8 . 8 , and T a b l e 8 . 1 ; run n o s . 1 and 2 3 ; 32 a nd 3 3 ) . 8 . 2 . 2 A Model f o r Spout V o i d a g e D i s t r i b u t i o n The f o l l o w i n g t h e o r e t i c a l e q u a t i o n = 1.0 A 0 .33 p f ( u s - v s ) < d / P s - p f P o K S S 2 .78 dG dz d e v e l o p e d i n C h a p t e r 7 f o r s o l i d s movement i n the spou t can a l s o be used to d e s c r i b e the spou t v o i d a g e d i s t r i b u t i o n . As d i s c u s s e d i n C h a p t e r 7 , t h i s e q u a t i o n when s o l v e d i n c o n -j u n c t i o n w i t h E q s . ( 7 . 9 ) and ( 7 . 1 0 ) , u s i n g e x p e r i m e n t a l v a l u e s o f 180 D , and Q . y i e l d s v e r t i c a l p r o f i l e s not o n l y o f gas and s a a p a r t i c l e v e l o c i t i e s i n the spou t but a l s o of spou t v o i d a g e . A few examples of the agreement between spout v o i d a g e p r o f i l e s c a l c u l a t e d by E q . ( 7 . 2 6 ) and e x p e r i m e n t a l p r o f i l e s o b t a i n e d by the method d e s c r i b e d i n s e c t i o n 8 .2 .1 a re p r e - ' s e n t e d i n F i g u r e 8 .9 and T a b l e 8 . 2 . E x p e r i m e n t a l da ta f o r a l l the o t h e r runs showed s i m i l a r agreement ( w i t h i n ~ 10%) w i t h c a l c u l a t i o n by E q . ( 7 . 2 6 ) . 181 0 0 - 9 0 . 8 0 . 7 0 . 6 0 . 5 0 o. Experimental Predicted by Eq-7.26 0 . 2 0 . 4 0 . 6 Z/H 0 . 8 LO Figure 8.9. Comparison of predicted and experimental spout voidage p r o f i l e s : Polystyrene p e l l e t s , = 15.2 cm, H/D c = 3.0, D 0 / D c = 0.125, U s / U m s = 1.1 (run no. 1) . Table 8.2 Comparison of Predicted and Experimental Spout Voidages (H/D = 3.0, D 0 /D c = 0.125, l y U ^ . = 1.1) Polystyrene M i l l e t Wheat Z/H D = 15.2 cm c 15.2 cm D = 15.2 cm c D = 24.1 cm c D = 29.2 cm c Run No. 1 Run No. 13 Run No. 8 Run No. 28 Run No. 30 e s 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0 1.00 0.97 0.95 0.92 0.90 0.86 0.83 0.80 0.79 0.77 0.76 1 .00 0.95 0.94 0.92 0.90 0.87 0.84 0.81 0.78 0.75 0.73 1.00 0.95 0.92 0.87 0.83 0.76 0.72 0.66 0.65 0.76 0.63 1.00 0.93 0.89 0.88 0.86 0.82 0.80 0.75 0.71 0.67 0.63 1.00 0.94 0.91 0.88 0.86 0.83 0.79 0.76 0.72 0.69 0.67 1 .00 0.94 0.90 0.86 0.82 0.78 0.74 0.71 0.68 0.66 0.63 1.00 0.96 0.93 0.90 0.86 0.83 0.79 0.74 0.68 0.64 0.59 1 .00 0.94 0.91 0.85 0.82 0.76 0.72 0.70 0.68 0.67 0.65 1 .00 0.96 0.94 0.90 0.87 0.82 0.78 0.75 0.72 0.71 0.70 1.00 0.94 0.89 0.85 0.80 0.74 0.70 0.68 0.66 0.64 0.61 Note 1 Experimental spout voidage Predicted spout voidage by Eq. (7.26) CO r o C h a p t e r 9 C O N C L U S I O N S R e s i d e n c e Time D i s t r i b u t i o n (1) E x p e r i m e n t a l e v i d e n c e i n d i c a t e s t h a t spou t gas t r a v e l s upward i n p l ug f l o w , w h i l e gas e n t e r i n g the a n n u l u s f rom the spou t moves r a d i a l l y ou tward and v e r t i c a l l y upward a l o n g i d e n t i f i a b l e s t r e a m l i n e s , w i t h l i t t l e r a d i a l m i x i n g . N o t i c e a b l e d e v i a t i o n f rom p lug f l o w t h e r e f o r e o c c u r s i n the a n n u l u s . (2) A t w o - r e g i o n model wh ich t a k e s i n t o a c c o u n t the a c t u a l path f o l l o w e d by the gas i n the a n n u l u s , has been s u c c e s s f u l l y d e v e l o p e d to p r e d i c t the r e s i d e n c e t ime d i s t r i -b u t i o n o f gas i n the spou ted b e d . The model i s based on the a s s u m p t i o n s o f p l ug f l o w o f gas i n the spou t and d i s p e r s e d p l u g f l o w a l o n g the f l o w pa th i n the a n n u l u s . The model pa rame te r D ( d i s p e r s i o n c o e f f i c i e n t o f gas a l o n g the f l o w pa th i n the a n n u l u s ) was e v a l u a t e d by compar i ng p r e d i c t e d and e x p e r i m e n t a l r e s i d e n c e t ime d i s t r i b u t i o n c u r v e s , under a w ide range o f e x p e r i m e n t a l c o n d i t i o n s . 183 184 The v a l u e o f D was found to i n c r e a s e w i t h i n c r e a s i n g p a r t i c l e s i z e and bed d e p t h , and to rema in s u b s t a n t i a l l y i n d e p e n d e n t o f column d i a m e t e r , o r i f i c e d i a m e t e r and s p o u t i n g v e l o c i t y . The v a l u e s a re g e n e r a l l y h i g h e r than a x i a l d i s -p e r s i o n c o e f f i c i e n t s r e p o r t e d f o r packed b e d s , but a t l e a s t an o r d e r o f magn i tude s m a l l e r than t hose f o r f l u i d i z e d b e d s . Gas F low P a t t e r n (3) The gas t r a n s p o r t r a t i o ( Q s / Q T ) was found to d e c r e a s e w i t h i n c r e a s i n g bed l e v e l , w h i l e a t a g i v e n bed l e v e l the t r a n s p o r t r a t i o i n c r e a s e d l i n e a r l y w i t h i n c r e a s i n g momentum of the s p o u t i n g g a s . (4) R e c i r c u l a t i o n o f gas was o b s e r v e d to o c c u r i n the l owe r p a r t of the a n n u l u s , g e n e r a l l y a t h i g h s p o u t i n g v e l o c i t i e s ( U s / U m s > 1 . 2 ) , w i t h deep beds ( H / D c > 4 . 0 ) and s m a l l o r i f i c e s i z e s ( D 0 / D c < 0 . 1 ) . (5) A t a g i v e n bed l e v e l , the gas v e l o c i t y i n the a n n u l u s was found to d e c r e a s e w i t h i n c r e a s i n g s p o u t i n g v e l o c i t y and d e c r e a s i n g gas i n l e t d i a m e t e r , and to rema in i n d e p e n d e n t o f bed d e p t h . (6) The M a m u r o - H a t t o r i e q u a t i o n ( E q . ( 6 . 1 3 ) ) gave good p r e d i c t i o n o f a n n u l u s gas v e l o c i t y p r o f i l e s ( l o n g i t u d i n a l ) 185 f o r 15 .2 cm d i a m e t e r beds but u n d e r - e s t i m a t e d v e l o c i t i e s f o r . l a r g e r c o l u m n s . The e q u a t i o n o f Yokogawa et al. ( E q . ( 6 . 1 5 ) ) , p roved to be u n s a t i s f a c t o r y f o r p r e d i c t i v e p u r p o s e . The e q u a t i o n was m o d i f i e d and c o u l d then p r e d i c t the gas v e l o c i t y p r o f i l e i n the a n n u l u s c o r r e c t l y p r o v i d e d t h a t one such p r o f i l e f o r the p a r t i c u l a r s o l i d m a t e r i a l was known. (7) The da ta o b t a i n e d i n the p r e s e n t i n v e s t i g a t i o n s u p p o r t the e q u a t i o n r e c e n t l y p roposed by G r b a v c i c et al. [21 ] ( E q . ( 6 . 1 2 ) ) to d e s c r i b e the r e l a t i o n s h i p between U a H / U a u m and H/H . m So l i d s F low P a t t e r n (8) The r a t e o f c r o s s - f l o w o f s o l i d s f rom the a n n u l u s i n t o the spou t was found to be i n v a r i a n t w i t h bed l e v e l ove r most o f the c y l i n d r i c a l s e c t i o n o f the bed ( D c < Z < H ) . The c r o s s - f l o w r a t e i n c r e a s e d w i t h i n c r e a s i n g U / U . - 3 s ms bed d e p t h , p a r t i c l e d i a m e t e r and o r i f i c e d i a m e t e r , and rema ined s u b s t a n t i a l l y i n d e p e n d e n t o f cone a n g l e . The L e f r o y - D a v i d s o n c o l l i s o n model ( E q . ( 2 . 4 6 ) ) o v e r - e s t i m a t e d c r o s s - f l o w r a t e s but c o u l d be used to e x p l a i n most o f the o b s e r v e d t r e n d s o f c r o s s - f l o w r a t e q u a l i t a t i v e l y . (9) A s i m p l e model to d e s c r i b e the movement o f a n n u l a r s o l i d s has been f o r m u l a t e d , wh ich e n a b l e s the 186 c a l c u l a t i o n o f s o l i d s f l o w pa th and r e t e n t i o n t ime i n the a n n u l u s f rom a v e r a g e p a r t i c l e v e l o c i t y d a t a . (10) The r a d i a l a v e r a g e v a l u e o f spou t p a r t i c l e v e l o c i t y a t a g i v e n bed l e v e l was found to i n c r e a s e w i t h i n c r e a s i n g U s / U m s » p a r t i c l e s i z e , bed d e p t h , column d i a m e t e r and w i t h d e c r e a s i n g gas i n l e t d i a m e t e r . (11) The f o r c e b a l a n c e model o f T h o r l e y et al. f o r p a r t i c l e v e l o c i t y i n the spou t as amended by Mathur and E p s t e i n has been f u r t h e r improved by i n t r o d u c i n g the t h e o -r e t i c a l r e l a t i o n s h i p between spout v o i d a g e and number o f p a r t i c l e s i n the s p o u t . The r e s u l t i n g e q u a t i o n ( E q . ( 7 . 2 6 ) ) showed good agreement w i t h e x p e r i m e n t a l v a l u e s o f not o n l y spou t p a r t i c l e v e l o c i t y but a l s o o f spou t v o i d a g e . Bed S t r u c t u r e (12) Spout shape and d i a m e t e r s measured i n h a l f -and f u l l - c o l u m n s under s i m i l a r o p e r a t i n g c o n d i t i o n s were found to be no d i f f e r e n t . (13) A t a g i v e n bed l e v e l , spou t v o i d a g e i n c r e a s e d w i t h i n c r e a s i n g U / l l and i n most c a s e s w i t h d e c r e a s i n g 3 s ms gas i n l e t d i a m e t e r . For g e o m e t r i c a 1 1 y s im i1 a r b e d s , spou t v o i d a g e s were s u b s t a n t i a l l y the same a t any g i v e n reduced bed l e v e l ( Z/H) . C h a p t e r 10 SUGGESTIONS FOR FURTHER WORK The f l o w model f o r gas movement i n spou ted beds d e v e l o p e d i n t h i s work i s c o n s i d e r e d to be s u f f i c i e n t l y c o m p l e t e to p r o v i d e a b a s i s f o r a n a l y s i n g and p r e d i c t i n g the pe r f o rmance o f spou ted bed p r o c e s s e s i n v o l v i n g g a s - p a r t i c l e i n t e r a c t i o n . A c h a l l e n g i n g programme f o r f u r t h e r work would be to a p p l y the gas f l o w model to g a s - s o l i d s heat a n d / o r mass t r a n s f e r , and vapor phase c h e m i c a l r e a c t i o n i n the p r e s e n c e o f c a t a l y s t o r hea t c a r r i e r p a r t i c l e s . E x p e r i m e n t a l work w i l l be r e q u i r e d to t e s t t he v a l i d i t y o f the m o d e l , and to mod i f y and improve i t where n e c e s s a r y . 187 NOTATION A C r o s s - s e c t i o n a r e a o f gas f l o w pa th i n the a n n u l u s ( F i g u r e 5 . 3 ) . Or d e f i n e d by E q . ( 2 . 2 6 ) A 3 C r o s s - s e c t i o n a l a r e a o f a n n u l u s a A C r o s s - s e c t i o n a l a rea o f a n n u l u s between a c R = 1/3 (2R + r ) and R = 1/3 (R + 2 r ) C 5 C S ( F i g u r e 7 .1 ) A C r o s s - s e c t i o n a l a rea o f a n n u l u s between R = as 1 / 3(R C + 2)V a n d R = r s ( p i 9 u r e 7 - 1 ) -A C r o s s - s e c t i o n a l a rea o f a n n u l u s between a R = R c and R = l / 3 ( 2 R c + r g ) ( F i g u r e 7 .1 ) A g C r o s s - s e c t i o n a l a rea o f spou t B D e f i n e d by E q . ( 2 . 2 7 ) B L C o n s t a n t i n E q . ( 2 . 4 5 ) C T r a c e r c o n c e n t r a t i o n of t > 0 Co T r a c e r c o n c e n t r a t i o n a t t < 0 Ci C / C 0 a t z = 0 , d i m e n s i o n l e s s 188 189 C / C 0 i n a n n u l u s , d i m e n s i o n l e s s Drag c o e f f i c i e n t d e f i n e d by E q s . ( 7 . 2 1 ) and ( 7 . 2 5 ) Drag c o e f f i c i e n t d e f i n e d by E q . ( 2 . 2 2 ) C / C 0 i n s p o u t , d i m e n s i o n l e s s D i s p e r s i o n c o e f f i c i e n t o f gas a l o n g the f l o w path i n the a n n u l u s , m 2 / s e c Column d i a m e t e r , m Gas i n l e t d i a m e t e r , m Spout d i a m e t e r , m D c a t H = H , m s m D iame te r o f sphe re of same volume as p a r t i c l e , mm C h a r a c t e r i s t i c 1 i n e a r d i m e n s i o n o f p a r t i c l e , mm D e f i n e d by E q . ( 2 . 2 8 ) C o e f f i c i e n t o f r e s t i t u t i o n ( E q . ( 2 . 4 6 ) ) Drag f o r c e o f a f l u i d a c t i n g on a s i n g l e sphere ( E q . ( 7 . 1 9 ) ) Drag f o r c e o f a f l u i d a c t i n g on a s i n g l e sphe re ( E q . ( 7 . 2 2 ) ) Drag f o r c e o f a f l u i d a c t i n g on a c o n s t i t u e n t sphe re i n a m u l t i p a r t i c l e sys tem ( E q . ( 7 . 2 2 ) ) 1 90 F ,F F o r c e s a c t i n g on an e lement o f p a r t i c l e s i n the spou t w a l l ( F i g u r e 2 .16 ) G Mass f l o w r a t e o f s p o u t i n g gas ( E q . ( 2 . 4 7 ) ) , K g / m 2 sec G, V o l u m e t r i c f l o w r a t e o f s o l i d s i n a n n u l u s , c m 3 / s e c a G a H G a a t Z = H, c m 3 / s e c G s V o l u m e t r i c f l o w r a t e o f s o l i d s i n s p o u t , c m 3 / s e c G V o l u m e t r i c f l o w r a t e o f s o l i d s between s t r e a m l i n e s J - l and J i n a n n u l u s ( F i g u r e 7 . 5 ) , c m 3 / s e c H Bed d e p t h , m H' V a l u e o f Z a t top o f f o u n t a i n , m H m Maximum s p o u t a b l e bed d e p t h , m I H o r i z o n t a l g r i d l i n e s i n F i g u r e s 5 .3 and 7 .5 J Gas s t r e a m l i n e s i n F i g u r e 5 .3 S o l i d s s t r e a m l i n e s in F i g u r e 7 .5 K C o n s t a n t i n D a r c y ' s e q u a t i o n ( E q . ( 2 . 1 1 ) ) k c D e f i n e d by E q . ( 2 . 1 9 ) k $ D e f i n e d by E q . ( 2 . 2 0 ) L Leng th o f gas or s o l i d s f l o w pa th i n a n n u l u s , m M Number o f d i v i s i o n i n H ( F i g u r e s 5 .3 and 7 .5 ) MI Gas i n p u t momentum ( E q . ( 6 . 3 ) ) , gm Number o f gas or s o l i d s s t r e a m l i n e s i n a n n u l u s ( F i g u r e s 5 .3 and 7 .5 ) C u m u l a t i v e number of r i s i n g p a r t i c l e s i n spou t ( E q . ( 2 . 3 8 ) ) S t a t i c p r e s s u r e i n the a n n u l u s T o t a l p r e s s u r e drop a c r o s s the bed Gas f l o w r a t e between s t r e a m l i n e s J - l and J i n the annu lus ( F i g u r e 5 . 3 ) , m 3 / s e c Gas f l o w r a t e i n a n n u l u s , m 3 / s e c Minimum s p o u t i n g gas f l o w r a t e , m 3 / s e c T o t a l s p o u t i n g gas f l o w r a t e , m 3 / s e c Gas f l o w r a t e i n spout , m 3 / s e c R a d i a l d i s t a n c e f rom spou t a x i s , r s < R < R £ R / R c Column r a d i u s R a d i a l d i s t a n c e f rom spou t a x i s , 0 < r < r r / r See R i g u r e 2 .4 See F i g u r e 2 .4 Spout r a d i u s s s 1 92 S o l i d s c r o s s - f l o w r a t e f rom a n n u l u s to spou t ( E q . ( 7 . 3 ) ) , c m 2 / s e c S o l i d s r e t e n t i o n t ime i n the a n n u l u s ( E q . ( 7 . 5 ) ) , Gas t r a n s p o r t r a t i o , Q S / Q T Time, sec S u p e r f i c i a l gas v e l o c i t y i n a n n u l u s U a t Z = H a U a t Z = H = H a m Minimum s u p e r i f i c a l gas v e l o c i t y f o r f l u i d i z a t i o n Minimum s u p e r f i c i a l gas v e l o c i t y f o r s p o u t i n g V o l u m e t r i c r a t e o f r a d i a l gas p e r c o l a t i o n per u n i t a rea o f s p o u t - a n n u l us i n t e r f a c e S u p e r f i c i a l s p o u t i n g gas v e l o c i t y Gas v e l o c i t y a l o n g the s t r e a m l i n e i n the a n n u l u s ( F i g u r e 5 .3 ) Gas v e l o c i t y t h rough i n l e t o r i f i c e R a d i a l a v e r a g e upward gas v e l o c i t y i n s p o u t , i n t e r s t i t i a l Gas v e l o c i t y a l o n g spou t a x i s , u g z a t r = 0 u s c a t Z = H 193 u a t Z = H s ii a t Z = r sz s L o c a l upward gas v e l o c i t y i n the spou t a t bed l e v e l Z P a r t i c l e f r e e f a l l t e r m i n a l v e l o c i t y S u p e r f i c i a l p a r t i c l e t e r m i n a l v e l o c i t y a t v o i d a g e e V e r t i c a l p a r t i c l e v e l o c i t y i n the spou t ( F i g u r e 2 .14 ) R a d i a l p a r t i c l e v e l o c i t y i n the spou t ( F i g u r e 2 .14 ) P a r t i c l e t e r m i n a l v e l o c i t y a t i n f i n i t e d i l u t i o n P a r t i c l e e n t r a i n m e n t v e l o c i t y ( F i g u r e 2 .14 ) R a d i a l ave rage downward p a r t i c l e v e l o c i t y i n the a n n u l u s R a d i a l ave rage o f v between R = 1/3(2R + r ) and R = 1/3(R + 2r|5 ( F i g u r e 7 .1 ) c s R a d i a l a v e r a g e o f v between R = 1/3(R + 2 r c ) a z c s and R = r $ ( F i gure 7 .1 ) R a d i a l a v e r a g e o f v between R = R and R = l / 3 ( 2 R c ,+ r s ) a z ( F i g u r e 7.1 ) c Downward p a r t i c l e v e l o c i t y i n the a n n u l u s R a d i a l a v e r a g e upward p a r t i c l e v e l o c i t y i n the spou t 194 v „ „ v , a t r = 0 sc s z v . Upward v e l o c i t y o f an i n d i v i d u a l p a r t i c 1 e i n the spou t v g z Upward p a r t i c l e v e l o c i t y i n the spou t w D e f i n e d by E q . ( 6 . 1 9 ) Z V e r t i c a l d i s t a n c e f rom f l u i d i n l e t , m Z ' Z /H Z 0 V a l u e o f Z a t wh ich u s c / u 0 r e a c h e s u n i t y , see F i g . 2 .2 z L i n e a r d i s t a n c e a l o n g a s t r e a m l i n e , s t a r t i n g f rom s p o u t - a n n u l us i n t e r f a c e , m z ' z / L a A n g l e of r e p o s e 3 Gas s o l i d s i n t e r a c t i o n f a c t o r ( E q . ( 2 . 4 4 ) y A n g l e o f f r i c t i o n between s o l i d s and P l e x i g l a s s w a l l 6 ( t ) U n i t s t e p f u n c t i o n ( E q . ( 5 . 1 6 ) ) e Vo idage ( E q . ( 7 . 1 7 ) ) e g A n n u l u s v o i d a g e e s R a d i a l a v e r a g e spou t v o i d a g e S p h e r i c i t y = [ s u r f a c e o f s p h e r e ^ v J [ s u r f a c e o f p a r t i c l e ^ both o f same volume 1 9 5 I n c l u d e d a n g l e o f c o n i c a l base A n g l e between the gas or s o l i d s f l o w pa th i n a n n u l u s and v e r t i c l e a x i s ( F i g u r e 5 .3 and 7 .5 ) P a r t i c l e shape f a c t o r ( E q . ( 6 . 1 0 ) ) D e f i n e d by E q . ( 2 . 3 0 ) D e f i n e d by E q . (2 .31 ) V i s c o s i t y o f a i r , gm/cm sec C o e f f i c i e n t o f f r i c t i o n , t a n ( y ) S o l i d s bu l k d e n s i t y i n l o o s e packed c o n d i t i o n , gm/cm 3 Gas d e n s i t y , gm/cm 3 P a r t i c l e d e n s i t y , gm/cm 3 V e r t i c a l p r e s s u r e due to w e i g h t o f a n n u l a r s o l i d s , o r downward force due to w e i g h t o f s o l i d s per u n i t c r o s s - s e c t i o n a l a r e a o f a n n u l u s R a d i a l p r e s s u r e due to w e i g h t o f a n n u l a r s o l i d s a a t R = R r c a r a t R = r s F r i c t i o n a l f o r c e between a n n u l u s and co lumn w a l l F r i c t i o n a l f o r c e between a n n u l u s and spou t A n g l e of i n t e r n a l f r i c t i o n o f p a r t i c l e s Sum of squa red d e v i a t i o n s between e x p e r i m e n t a l and model p r e d i c t e d RTD c u r v e s , d e f i n e d by E q . ( 5 . 1 7 ) LITERATURE CITED 1. G i s h l e r , P . E . , and M a t h u r , K . B . , U . 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O s t e r g a a r d , K. and M i c h e l s e n , M . L . , T r a n s a c t i o n s No. 1 1 , Dansk I n g e n i o r f o r e i n i n g , Copenhagen , 1 967 . 54 . U r b a n , J . C . , and G o m e z p l a t a , A . , C a n . J . Chem. E n g . , 4 7 , 353 ( 1 9 6 9 ) . 5 5 . G i l l i l a n d , E . R . , and Mason , E . A . , I n d . E n g . C h e m . , 4 4 , 21 8 ( 1 952) . 56 . S c h u g e r l , K . , P r o c . I n t . Symp. on F l u i d i z a t i o n , 'Ne the r 1 ands U n i v . P r e s s , 782 ( 1 9 6 7 ) . 57 . R i c h a r d s o n , J . F . and Z a k i , W . N . , T r a n s . I n s t n . Chem. E n g r s . , 3 2 , 35 ( 1 9 5 4 ) . 200 58 . Wen, C . Y . , and Y u , Y . H . , Chem. Eng. P r o g . Symp. S e r i e s , 6 2 , 101 ( 1 9 6 6 ) . 59 . E p s t e i n , N. , C a n . J . Chem. E n g . , 3_6 , 21 0 ( 1 9 5 8 ) . APPENDIX I EXPERIMENTAL CONDITIONS ( S p o u t i n g F l u i d : A i r ) Run No . M a t e r i a l mm D c m D 0 / D c H / D c e u /u s ' ms 1 P o l y s t y r e n e p e l l e t s 2 . 9 3 0 . 1 5 2 0 . 1 2 5 3 . 0 0 6 0 1 . 1 2 •I II II II n 1 . 2 3 II ti M 1 . 3 4 n II II 2 . 0 0 n 1 . 1 5 II n II II 1 . 2 6 n II II II 1 . 3 7 n H it 5 . 9 2 n 1 . 1 2 3 II n 0 . 0 8 4 3 . 0 0 II 1 . 1 2 5 0 . 1 6 7 II II 1 . 1 2 7 II n 0 . 1 2 5 II 1 8 0 1 . 1 2 9 II 0 . 2 4 1 0 . 1 2 0 II 6 0 1 . 1 3 2 II 0 . 2 9 2 0 . 1 2 5 . n M 1 . 1 3 3 II 0 . 0 8 7 n n 1 . 1 1 8 1 . 6 1 0 . 1 5 2 0 . 1 2 5 II 1 . 1 8 Wheat 3 . 5 0 0 . 1 5 2 0 . 1 2 5 3 . 0 0 6 0 1 • 1. 1 2 n II M n II 1 . 3 9 II n n 2 . 0 0 H 1 . 1 1 0 II H n 1 . 0 0 II 1 . 1 1 1 II II II 4 . 1 7 n 1 . 1 2 4 M M 0 . 1 6 7 3 . 0 0 H 1 . 1 2 8 it 0 . 2 4 1 0 . 1 2 0 H 1 . 1 3 0 n 0 . 2 9 2 0 . 1 2 5 M II 1 . 1 3 1 n II II 1 . 5 7 II 1 . 1 CONTINUED 2 0 1 2 0 2 Run No . M a t e r i a l mm D c m D 0 / D c H / D c 0 Us/U ms 1 3 Mi 1 l e t 2 . 1 5 0 . 1 5 2 0 . 1 2 5 3 . 0 0 6 0 1 . 1 1 4 H n n 1 . 3 2 1 II n 0 . 0 8 4 n n 1 . 1 2 6 II II 0 . 1 2 5 H 1 8 0 1 . 1 1 5 G l a s s beads 2 . 9 3 0 . 1 5 2 0 . 1 2 5 2 . 6 0 6 0 1 . 1 1 6 II II M II 1 . 3 1 7 1 . 1 0 n II II II 1 . 1 1 9 Ammon i um n i t r a t e 1 . 4 5 0 . 1 5 2 0 . 1 2 5 3 . 0 0 6 0 1 . 1 2 0 T . 9 9 II n n II 1 . 1 APPENDIX II CALIBRATION CURVES FOR PRESSURE TRANSDUCERS 203 204 Figure 1 1 . 1 . Ca l ib ra t ion curves for pressure transducer A (range ± 2 . 5 ps id ) . 205 Figure I I .2 . Ca l ib ra t ion curve for pressure transducer B (range ± 0 . 3 ps id ) . APPENDIX I I I PRESSURE DROP-GAS VELOCITY EQUATIONS FOR LOOSE-PACKED BEDS OF THE SOLID MATERIALS USED IN RTD E X P E R I M E N T S M a t e r i a l s a o a i a 2 a 3 a 5 Polystyrene pe l le ts (d p = 1.61 mm) 0.0283 52.7872 -9.6028 0.0 0.0 0.0 Polystyrene pe l le ts (d = 2.93 mm) P 0.0143 115.786 -273.0747 607.8411 -608.4216 212.5646 Glass beads (d p = 1.10 mm) 0.0139 22.3337 17.2931 -13.3795 3.4867 -0.2939 Glass beads (d p = 2.93 mm) 1.2059 115.0895 -70.2980 33.3312 -8.0545 0.7406 Ammonium n i t ra te (dp = 1.45 mm) 0.0240 41.4799 -13.5580 5.3725 -0.8862 0.0 Ammonium n i t ra te (dp = 1.99 mm) 0.0085 59.1293 -3.8764 -27.4734 22.6112 -5.0834 Wheat 0.0209 117.1497 -144.3286 133.7972 -58.0697 9.1062 M i l l e t 0.0026 46.6431 72.5961 -136.1061 61.4883 0.0 Equation: U (cm/sec) = a 0 + a i x + a 2 x 2 + a 3 x 3 + a ^ x " + a 5 x 5 a where x i s the output of pressure transducer B in m i l l i v o l t s (see Figure I I .2 for ca l i b ra t i on cu rve ) . 206 APPENDIX IV NUMERICAL METHOD FOR SOLVING THE PARTIAL DIFFERENTIAL EQUATIONS FOR RESIDENCE TIME DISTRIBUTION OF GAS IN THE ANNULUS P a r t i a l d i f f e r e n t i a l e q u a t i o n s t o g e t h e r w i t h the i n i t i a l and boundary c o n d i t i o n s , E q s . ( 5 . 5 ) to ( 5 . 8 ) , f o r d e s c r i b i n g the r e s i d e n c e t ime d i s t r i b u t i o n o f gas i n the a n n u l u s a re r e w r i t t e n as f o l l o w s : 3C u a , 3C n 3 2 C -sf • - r - £ - n * " 0 < I V - " f o r t < 0 , C = 1 .0 ( I V . 2 ) a f o r t > 0 a t z ' = 0 , D d C I J T = u a z ( C a " C s> ( I V . 3 ) a t z 1 = 1.0 8 C g f r = 0 ( I V . 4 ) 207 208 N u m e r i c a l s o l u t i o n o f the above d i f f e r e n t i a l e q u a t i o n s was o b t a i n e d by e v a l u a t i n g the d e r i v a t i v e i n terms o f f i n i t e d i f f e r e n c e s and i n t e g r a t i n g the r e s u l t i n g d i f f e r e n c e e q u a t i o n s n u m e r i c a l l y . To f o r m u l a t e a d i f f e r e n c e e q u a t i o n f o r E q . ( I V . 1 ) , a net o f m e s h - s i z e A z 1 x At was e s t a b l i s h e d , as shown i n F i g u r e I V . 1 . S u b s c r i p t s i and k denote d i s t a n c e and t ime p o s i t i o n s r e s p e c t i v e l y f o r t r a c e r c o n c e n t r a t i o n C . The d i s t a n c e L ( i n a E q . ( I V . 1 ) ) was d i v i d e d i n t o M-l i n t e r v a l s , so t h a t s u b s c r i p t i = 1 , 2 , 3 , « ' « , M . The C r a n k - N i c h o l s o n d i f f e r e n c e scheme was used to r e p l a c e the d e r i v a t i v e s w i t h f i n i t e d i f f e r e n c e s . T h u s , 3 C C . i i i — C . a a _ a , i , k+1 a , i , k 3 t At ( I V . 5 ) 9 C a - C a , i + l , k + l " C a , i - l , k + 1 + C a , i + 1,k " C a , i - l , k , T , , ^ jzrr ( I V . 6 ) 3 2 C g dT^ = 1 C a , i + l , k + l " 2 C a , i , k + l + C a , i - l , k + l " C a , i + l , k " 2 C a , i , k + Ca,i-l,k}A2 Az'2) (IV-7) S u b s t i t u t i n g E q s . ( I V . 5 ) to ( I V . 7 ) i n t o E q . ( I V . 1 ) , and l e t t i n g . A = 1 /A t B = D / ( 2 L 2 A Z ' 2 ) E. = u z i / ( 4 L A z ' ) ( I V . 8 ) ( I V . 9 ) ( I V . 1 0 ) 209 k + l £ f- k Ca,i-I,k+I Ca,i,k+I Ca,i + I,k+I 1 C a, i-I,k C a , i , k r Ca,i+I,k <—Az'—• <»—- Az'—• 0 i + l M Figure IV .1 . Mesh spacing for solv ing par t ia l d i f f e ren t i a l equation, Eq. (5 .5) . 210 g i v e s , a f t e r r e a r r a n g e m e n t A z i C a , i - l , k + l + A a C a, i ,k+1 - B z i C a, i+ l ,k+ l F i for 2 < i < M-l (IV.11) w h e r e F. = -A . C . , . + B. C . , - B . C , (IV.12) i z i a , i - l , k b a , i , k z i a, i+ l ,k A z i = - ( E . + B) (IV.13) A = A + 2B (IV.14) a B z i = E i - B (IV.15) B f a = A - 2B (IV.16) The boundary c o n d i t i o n a t z ' = 0 , g i v e n by E q . ( I V . 3 ) , w r i t t e n i n terms o f f i n i t e d i f f e r e n c e s , becomes A a l C a , l , k + 1 + B z l C a ,2 , k + 1 = F ! < I V - , 7 > in which A a l =D/(l Az 1 ) + U z l (IV.18) 211 B z l = -D/{1 A z ' ) ( I V . 1 9 ) F l = U z l C s , l , k + 1 < I V - 2 0 > The boundary c o n d i t i o n a t z 1 = 1, i . e . E q . ( I V . 4 ) , s t a t e s t h a t the c o n c e n t r a t i o n g r a d i e n t a t the top s u r f a c e o f the annu lus e q u a l s z e r o . To a p p r o x i m a t e t h i s e q u a t i o n by the f i n i t e d i f f e r e n c e method, we ex tend the C domain on z ' p o s i -a t i o n beyond the top to i = M+l and a p p l y the method i l l u s t r a t e d b e l o w . S u b s t i t u t e E q . ( I V . 4 ) i n t o E q . ( I V . 1 ) , g i v e s 9C _ 3 2 C • W ' H W * ( I V - 2 1 ) The d i f f e r e n c e scheme f o r E q . ( I V . 4 ) a t t ime = k i s C..H + l.k 1 C « . M - l . k ( I V ' 2 2 > a t t ime = k+1, i s C a , M + l , k + l C a , M - l , k + l ( I V . 2 3 ) Hence E q . ( I V . 7 ) a t i = M becomes 3 2 C 3z T T [ C a , M - l , k + l " C a , M , k + l + C a , m - l , k " C a , M , k / A z1 2 ( I V . 2 4 ) 212 Now r e p l a c i n g the d e r i v a t i v e s i n E q . ( I V . 2 1 ) by the f i n i t e d i f f e r e n c e s g i v e n by E q s . ( I V . 5 ) and ( I V . 2 4 ) , we ge t A z M C a , M - l , k + l + A a C a , M , k + l = F M < I V - 2 5 > where A z M = -2B ( V . 2 6 ) F M = -A M C M , , + B, C M . ( I V . 2 7 ) M z M a , M - l , k b a , M , k \ » . <-/ , E q s . ( I V . 1 1 ) , ( I V . 1 7 ) and ( I V . 2 5 ) form a s e t o f M s i m u l t a n e o u s e q u a t i o n s wh ich have a t r i d i a g o n a l c o e f f i c i e n t m e t r i c f o r each t ime i n t e r v a l A t . The method s u g g e s t e d by Thomas was used to s o l v e the e q u a t i o n s f o r C f o r each t ime a i n t e r v a l , s t a r t i n g w i t h k = 2 ( f o r k = 1 , C , = 1 a c c o r d i n g to a E q . ( I V . 2 ) ) . T h i s method i s e q u i v a l e n t to p l a i n G a u s s i a n e l i m i n a t i o n , but i t a v o i d s the e r r o r growth a s s o c i a t e d w i t h t he back s o l u t i o n o f the e l i m i n a t i o n method and a l s o m i n i m i z e s the s t o r a g e i n the c o m p u t e r . Our sys tem of f i n i t e d i f f e r e n c e e q u a t i o n s to be s o l v e d i s as f o l 1 o w s : L e t The s o l u t i o n i s 21 3 A a l C a , l , k + l + B z l C a , 2 , k + 1 = F l A z i C a , i - l , k + l + A a C a , i , k + 1 + B z i C a , i + l , k + l " F i f o r 2 < i < M-l { A z M C a , M - l , k + l + A a C a , M , k + l " F M ( I V . 1 7 ) ( I V . 1 1 ) (IV .25) w, = A , 1 a 1 ( I V . 2 8 ) w i = A a " A z i b i - l > f o r 2 5 i 5 ( I V . 2 9 ) b i = B z i / w i , f o r 1 < i < M-l ( I V . 3 0 ) 9 l = F l / W l ( IV .31 ) g i = ( F i " A z i 9 i - l ) / w i ' f o r 2• - 1 - M ( I V . 3 2 ) ' a , M , k + l y M ( I V . 3 3 ) ' a , i ,k + l " 9 i " b i C a , i + 1 ,k+l f o r 1 < i < M-l I V . 3 4 ) 214 The c a l c u l a t i o n p r o c e d u r e used was to compute w, b and g i n o r d e r of i n c r e a s i n g i , and then to e v a l u a t e C i n o r d e r of a d e c r e a s i n g ! ' . APPENDIX V COMPUTER PROGRAM FOR ESTIMATING THE MODEL PARAMETER D 215 216 C Q M * «-A: v. * * * if. -is A • * A a ( C ! j i • v? V ; J;.«« •:: s : i : A * ; i ; « - 1 : »>• '• j>. * r e : - * v.: c NAME J I M c P E A 0 IM ( 1) COLUMN GEOMETRICAL INFORMATIONS c * :i: :%\ ( 2 ) BED HYDRODYNAMIC DATA ; Q A, OS» EA c # ' « £ E S 0 c * * * (3) EXPERIMENTAL MEASURED STIMULUS c AND RESPONSE TRACER CONCo CURVES c COMPUTE ( I ) GAS FLOW PATHS IN THE ANNULUS c (2) RTD CURVE OF GAS IN THE SPOUT c (3) RTD CURVE'OF GAS AT THE TOP CENTRE OF c * * * THE ANNULUS WITH ASSUMED VALUES OF D c. (4) ESTIMATE D WHICH GIV CS- THE MINIMUM c c OF THE SUM OF S Q U A R E DEVIATION 6ETW 0 EXPo AND THEORETICAL RTD CURVES c PLOT THE ME ASURED STIMULUS AND RESPONSE TRACER c CONCo CURVES 7OGETHER WITH PREDICTED RTD c CURVE WITH D ESTIMATED IN ( 4 ) ABOVE c * £ * c x"c & # & jjr x : # c OOOl DIMENSION 1(50),OA(50),ES(50),DS(50),QQ{5) 1 , M P A R T ( 8 ) , R ( 5 0 , 2 5 ) , S A ( 5 0 ) , U A { 5 9 ) , A O ( 5 0 ) 2 , Q 3 ( 5 0 ) , Z 3 ( 5 0 ) , S 2 ( 5 0 ) , 12 (50) ,QZZ(50) ,RS(25) 3 , S Z I ( 2 5 ) , Z L ( 5 0 , 5 ) , P L ( 5 0 , 5 ) ,ANGLE(50,5) ,U(50,5) 4 , DEG(50,5) , T L ( 5 ) , C i ( 1 5 0 ) ,0(24-) ,CIP( 150) 5 , C S ( 5 0 , 1 5 0 ) , T T A M ( 5 ) , T { 1 5 0 ) , U S ( 5 0 ) , N I < 2 5 ) 6 , C S l ( 1 5 0 ) , C A l ( i 5 0 ) , C A 2 < 1 5 0 ) , S S Z I ( 5 ) , N N I ( 5 ) 7 , Z Z L ( 5 0 ) , U U ( 5 0 ) , S 4 ( 5 0 ) , U U A ( 5 0 ) , D U U A ( 5 0 ) 8 , V ( 5 0 , 5 ) , C A ( 5 0 , 2 ) , C M { 5 , 1 5 0 ) , X E ( 5 0 ) , A X ( 5 0 > 9 , CX( 50) , SLH50 ) , D( 50 ) , SG( 50 ) , TP ( 150) , 01 F( 30) 1 , DEL W(30) » DE LC(30)» CAW(150)» C A C ( 1 5 0 ) , R R{50,5) 0002 CALL PLOTS C *** READ FORM F I L E MEWDEQ 0003 DO 2000 KKK=1,2 0004 READ (5 , 1 ) NU?, 0005 READ(5,2) (NPART(I ) , I = 1 , 8} 0006 READ(5,3) DC » H,QT » E A,D3,ANG 0007 READ(5,4) ( Z ( I ) , Q A ( I ) , E S ( I ) , D S ( I ) , I = 1 , 4 1 ) C *** READ FROM F I L E INRTO 0008 READ(3,1) IT 0009 READ(3,6) ( C l ( I ) , 1 = 1, I T I 0010 REWIND 3 C **# READ FROM F I L E EXRTD 0011 READ(4, 21) L 00 12 I F ( K K K o E Q . l ) R E A D ( 4 , 2 2 ) ( T ( K ) , C AC ( K ) ,K=1,L) 0013 I F ( K K K o E 0 o 2 ) R E A D ( 4 , 2 4 ) ( T ( K ) , C A C ( K ) ,K= 1, L) 0014 WRITE(6, 7) NUR,(NPART( I ),I=1 , 8 ) 0015 WRITE(6,0) H , QT,DC,EA 0016 M=41 0017 MM=M-1 0018 N=21 0019 NN=N-1 00 20 NNN=NN-1 0021 LL=L-1 00 22 ZI=1»/MM 0023 T!=0.012 0024 R ( M , l ) = D C / 2 . 0025 R I = ( D C - 0 S < M ) ) / ( 2 o * N N ) 0026 DO 20 J=1,NN 0027 20 R(M,J+1)=R<M,J)-RI C *** C *** COMPUTE US c 0028 DO 2 5 1 = 1 ,M 00 29 2 5 US( I )=4. *( QT-QA{ I ) ) / ( 3. 1416*.:S (I )* i)S ( I )**2 ) C C *** COMPUTE SAt'JA AND R C *** 0030 ANG2-3ol416*ANG/180«/2o 0031 HC=(DC-DB)/(TAN(ANG2)*2.) 0032 HC=HC/H 0033 0 0 30 1=1,M 0034 I F( Z ( I )oL5oHC) P. ( I , 1) = H*Z ( I ) *T AN ( ANG2 ) +DA/2. 0035 I F {Z < I ) o G T o H C ) R ( I , l ) = D C / 2 o 0036 SA( I )=3ol416 * ( R (I,1)**2-DS(I)**2/4« 1 0037 30 UA( I)=QA{ I ) / ( S A ( I )*EA) 0038 R ( l , l ) - D S ( l ) / 2 o C *** C *** COMPUTE 0 AND Z2 0039 AQ'(1)=0.00 0040 DO 40 J=1,NN 0041 Q ( J ) = 3 . 1 4 1 6 * ( R ( M , J ) 2 -R ( M , J +1 ) * * 2 ) *U A ( M ) •• EA 0042 40 A O ( J + 1) = A Q ( J ) + Q ( J ) 0043 DO 50 1=1,M 0044 IF{QAU).LE.OoOO.AND.OM I + 1)„GT.0.U0) GO TO 60 0045 50 CONTINUE 0046 60 11= I 0047 DO 7 0 1 = 1 I ,MM 0048 I F ( 0 A { I ) o G H o Q A ( 1 + 1 ) ) GO TO 30 0049 70 CONTINUE 0050 • IM=M 0051 GO TO 90 0052 80 IM=I 0053 90 DO 100 I=11,IM 0054 Q3 ( I - I I *• 1 ) = QA { I ) 0055 100 Z 3 ( I - I I + 1 ) = Z ( I ) 00 56 IMM= I M-I I + 1 0057 I F ( A Q ( N ) 0 G E o Q 3 ( IMM) ) A Q ( N ) = Q 3 { IMM) 0058 CALL SPLINE(03,Z3,S2,IMM,AO,Z2.DZZ,N,£999) 0059 RSI 1),= 0S( l ) / 2 o 0060 Z 2 ( l ) = 0 o 0 0 0061 Z2(N)=1«00 0062 DO 110 J=2,N 0063 DO 120 1=1., MM 0064 I F ( Z ( I ) o L E o Z 2 ( J ) o A N D c Z(1 + l ).GT„Z2(J) ) GO TO 125 0065 120 CONTINUE 0066 125 N I ( J ) = I + 1 0067 S Z H J ) = Z ( H - 1 ) - Z 2 ( J ) 218 0068 0069 0070 0071 0072 00 73 00 74 00 75 00 76 0077 0078 0079 0030 0081 0032 0033 0084 0035 0086 0087 0038 0089 0090 0091 0092 0093 0094 0095 0096 0097 0098 0099 0100 0101 0102 0103 0104 0105 0106 0107 0108 0109 0110 0111 0112 0113 RS(J)=D5( H-l )/2, R U » J ) = R S U ) - ( 0 S ( I +1 ) - 0 S I I ) ) * .3 Z I ( J ) / {?.o l l ) 110 c C * * * COMPUTE STREAMLINE R C DO 130 J=1,NNN i i =.M i < j + i ) DO 130 1 = 11 ,M W=R{ I , J ) * * 2 - Q { J ) / ( 3 , R (I,J+1)=SQRT(W) 1 4 1 6 * E A * U A ( I ) ) 130 C * * # C Q »: sc:k P. ( M » ;'•!) =RS (M) 132 C C C 150 140 CHOSEN J = l l FOR CURVE FITTING JJ= 1 J = l l J J J = J NNI {J J ) = N I ( J ) S S Z I U J ) = S Z I U ) 11= N NI U J ) - 1 DO 132 I=II,M RR I I , J J ) = R( I ,J) 0 0 ( J J ) = Q ( J - l > COMPUTE P L i T L » ANGLE,H AND U J = l I I = N N I ( J ) - 1 SUM = Oo 0 ZL{ I I , J ) = 0 o 0 0 DO 150 1 = 1 I,MM WI=ZI I F ( I o c Q . I I ) WI=SSZI(J) Y = { WI*H )**2+( RR{ I +1, J )-P.R { I, J ) )**2 P L ( I , J ) = SQRT(Y) SUM=SUM+PL(I,J) ZL(I+1 , J ) = Z L ( I , J ) + P L ( I , J ) WX=(RR{I+1,J)-RR(I,J))/{WI*H) ANGLE( I , J ) =AT AN(WX) LK I , J) =UA { I ) /COS ( ANGLE ( I , J) ) DEG(I,J)=180 oWANGLE(I,J)/3o i 4 1 6 CONTINUE ANGLE(M, J )=0 o 00 DEG(M,J)=0o00 U ( M , J ) = U A ( M ) TL ( J) = SUM J = l I I = N N I ( J ) - 1 T T A N U )=OoO DO 155 1 = 1 I,M T T A N ( J ) = 2 . * P L ( I » J ) / ( U ( I » J ) + U ( I + 1 , J ) ) + T T A N U ) WRITEt6,9) WRITE (6,11) J J J , TL( J ) , (JQ( J ) , RR ( M, J ) , T T A N U ) J = l I I = N N I ( J ) - l 2 1 9 0 1 1 4 0 1 1 5 0 1 1 6 0 1 1 7 0 1 1 3 0 1 1 9 0 1 2 0 0 1 2 1 0 1 2 2 0 1 2 3 0 1 2 4 0 1 2 5 0 1 2 6 0 1 2 7 0 1 2 8 0 1 2 9 0 1 3 0 0 1 3 1 0 1 3 2 0 1 3 3 0 1 3 4 0 1 3 5 0 1 3 6 0 1 3 7 0 1 3 8 0 1 3 9 0 1 4 0 0 1 4 1 0 1 4 2 0 1 4 3 0 1 4 4 0 1 4 5 0 1 4 6 0 1 4 7 0 1 4 3 0 1 4 9 0 1 5 0 0 1 5 1 0 1 5 2 0 1 5 3 0 1 5 4 0 1 5 5 0 1 5 6 0 1 5 7 0 1 5 8 01 59 0 1 6 0 0 1 6 1 0 1 6 2 1 6 5 1 6 0 C * * * c c 1 7 0 1 8 0 2 0 0 2 1 0 2 2 0 2 3 0 1 9 0 C c * * * c * * * 6 0 0 2 4 1 2 4 2 2 4 3 DO 1 6 5 I = I I , M Z ZL ( I- I I + 1 ) = ZL ( I , J ) / T L ( J ) UU( I — I I + 1 ) = U ( I f J ) I M = M - I 1 + 1 C A L L S P L IMF ( Z Z L , ' J U , S4 , I M , Z • UU A , DUU A , H , £9 9 9 ) DO 1 6 0 1=1 ,M V( I , J ) = UUA (' I ) C O M P U T E CS L L T = I T + 1 DO 1 7 0 K = L L T , L C1IK )=0 ,00 DO 1 3 0 K = 1 , L C S < 1 , K ) = C 1 ( K ) A T U = 0 o 0 0 DO 1 9 0 I = 2 , M A U T = 2 « * Z I * H / I U S ( I ) + U S ( I - 1 ) ) A T U = A T U + A U T Z L L = A T U / T I L L Z = Z L L I F ( L L Z 8 L T 0 1 ) GO TO 2 1 0 DO 2 0 0 K = 1 , L L Z C S U , K ) = 1 . 0 0 DO 2 2 0 K = 1 , L L T I K = L L Z + K C S ( I , I K ) = C 1 ( K ) IK = IK + 1 DO 2 3 0 K = I K , L C S ( I , K ) = 0 „ 0 0 C O N T I N U E C O M P U T E RTD C U R V E FOR THE A N N U L U S I F G = 1 D A = 0 o 0 0 6 DO 1 0 0 0 M K = 1 , 2 0 D E L W { M K ) = 0 o 0 0 0 £ L C ( M K)=OoOO D I F ( M K ) = 0 A J = l 0 0 2 4 1 1 = 1 , M C A ( I , 1 ) = 1 « 0 0 C M ( J , I ) = 1 . 0 0 11 = N N I ( J ) - 1 I F ( I F G „ E Q o 2 ) GO TO 2 4 2 0A= D I F ( MK ) GO TO 2 4 3 D A = D I F { I C ) C O N T I N U E X l = l o / T I X 2 = D A / ( 2 o * ( T L ( J ) * Z I ) * * 2 ) X4 = D A / ( T L { J ) * Z I ) 8 X 1 = X 4 + V ( 1 , J ) C X ( 1 ) = - X 4 220 0163 01 64 0165 0 166 0167 0160 0169 0170 0171 0172 0173 0174 0175 0176 0177 0178 0179 0180 0131 0132 0183 0134 0135 01 86 0137 0188 0189 0190 0191 0192 0193 0194 0195 0196 0197 0193 0199 0200 0201 0202 0203 02 04 0205 0206 0207 0208 0209 0210 0211 0212 0213 250 260 232 280 290 300 270 240 C c * c 350 1000 400 1100 700 8X=Xl+2o*X2 0X=Xl-2o*X2 DO 250 I=2,M X3( I ) = V( I , J ) / ( 4o *TL ( J ) *ZI ) AX( I ) = - X 3 ( I ) - X 2 C X ( I ) = X 3 ( I}-X2 AX(M) = -2o*X2 CX(M )=0.00 SB( 1)=CX( 1)/BX1 00 260 !=2,M Sfi( I ) = CX{ I ) /(3X-AX ( I )*!S3( I - l ) ) 00 2 70 K=L,LL Q ( 1 ) = V ( 1 , J ) * C S ( II.K + l ) S G ( 1 ) = U ( i ) / B X 1 00 2 80 I = 2,M IF(IoEQoM) GO TO 282 D( I ) = - A X { I ) * C A ( I - l , L ) + O X * C \ i I , 1 ) - C X ( I ) * C A ( 1 + 1 , 1 ) GO TO 230 D(M ) = - A X < M l * C A ( M - l , 1 ) + D X * C A ( M , 1) SG(I ) = ( 0 ( I ) - A X ( I M ; S G ( I - l ) )/{3 X - A X ( I ) * S B ( I - l ) ) CA<M,2)=SG(M) 00 290 1=1 , MM IJ=M-I CA ( I J , 2 ) = S G ( IJ ) - S B ( I J ) * C A ( I J + 1 , 2 ) C M ( J , K + 1) = C A { M , 2 ) 00 300 1=1,M C A C I , 1 ) = C A ( I , 2 ) CONTINUE CONTINUE 1 F{ I FG.EQ.2) GO TO 700 COMPUTE SUM OF SQUARED DEVIATIONS BTW EXP. RTD AND PREDICTED RTD CURVES DO 350 K=1,L OELC(M K)={C M{1,K)-CAC(K) )**2 + D £ L C ( M K ) CONT INUE DA=DA+0oQ01 CONTINUE WRITE(6,12) WRITE(6,13) (DIF(MK),DELC(MK),MK=1,20) DO 400 K=l,19 I F { D E L C ( K + l ) o G E 0 D E L C ( K ) ) GO TO 1100 CONTINUE IC = K WRITE(6,14) D A = D I F ( I C ) I FG=2 GO TO 600 WRITEt6,15) W RI T E ( 6 , 16 ) WRITE(6,18) WRIT EM 6, 16) WRITE(6,23) J = l D I F ( I C ) , D E L C ( I C ) ( C i ( K ) ,K=1,L ) RR(M,1 ) (CACIK) ,K = 1,L) 221 0214 1=1 0 2 1 5 I I = N N I ( J ) - l 0216 W R I T £ ( 6 , 1 7 ) 0 2 1 7 W R I T E ( 6 , 1 6 ) ( C S ( I I , K ) , K = l , L) 0218 WRIT E ( 6 , 1 9 ) R R ( M , I ) ,DA 0219 W R I T E ( 6 , 1 6 ) (CM( I , K ) , K = 1 , L ) 0220 650 r •*• x- A CQ'NTINUE U • • c : CALL PLOTEP. 0221 DO 3 20 K = 1 , L 0222 T P ( K ) = 5 . 0 * T ( K ) 0223 C l P ( K ) = 2 o O + 5 o O * C K K ) 0224 CS1 IK) = 2 « 0 + 5 o 0" , ;CS I M, K ) 0225 C A K K I =2o0 + 5o 0«CM( 1 , K ) 02 26 C A C ( K ) = 2 o 0 + 5 o 0 * C ' \ C ( K ) 0227 3 2 0 CONTINUE 0228 C A L L P L O T t O o O , 2 o 0 , 3 ) 02 29 C A L L A X I 3{0 o 0 » 2 o 0 » * T t T IM E IN S~C',-13 , 8» 0 , O o 0 , 0 . 0 , 0 . 2 0230 CALL A XI 5 ( Oo 0 , 2o 0 , ' CI - C S - C A l-C A2-C A3 ' , 1 7 , 5 „ , 9 0 o , O o , 0 „ 0 2 3 1 CALL L I N E ( T P , C 1 P , L L , 1 ) 0232 CALL L I N E ( T P , C S 1 , L L , 1 ) 0233 C A L L L I N E ( T P , C A I , L L , 1 ) 0 2 3 4 DO 750 K = 1 , L , 2 0 2 3 5 750 CALL S Y M 5 0 L ( T P ( K ) , C A C ( K ) , 0 . 1 0 , 3 0 , 0 . 0 , - 1 ) 0 2 3 6 CALL S Y M B O L ( 4 o , 5 o 5 , 0 1 4 , ' R U N NO = ' , O o O , 8 ) 0237 CALL S Y M B O L ( 4 o , 5 o , o 1 4 , « D = ' , 0 o 0 , 3 ) 0238 S P = 0 o l 4 * 6 „ / 7 o 02 39 XQ=4c+9o^SP 0240 Y Q = 4 o + 4 0 * S P 0241 RUN=NUR 0 2 4 2 CALL N U M 8 E R ( X Q , 5 o 5 , 0 o 1 4 , R U N , 0 o , - 1 ) 0243 CALL N U M B E R ! Y 0 , 5 » , 0 o 1 4 , D A , 0 o , 3) 0 2 4 4 C A L L P L O T ( 1 3 o , 0 « 0 , - 3 ) 0 2 4 5 2000 CONTINUE 0246 1 F O R M A T ( 1 5 ) 0247 2 F O R M A T { 3 X , 8 A 4 ) 0248 3 FORMAT(6 F l O o 5) 0249 4 FORMAT(4E12« 5) 0250 6 FORMAT!12X,E12» 5) 0251 7 F 0 RM AT(1 H I , 9 X , ' R U N = • , I 5 / / 1 2 X , 3 A 4 ) 0 2 5 2 8 F O R M A T ( 1 3 X , • H E I G H T OF THE BED =• , F 1 0 o 4 , ' M»/ 0253 0254 02 55 0256 0 2 5 7 1 3 X , • T O T A L AIR FLOW RA • , F 1 0 o 4 , 1 2 3 1 3 X , ' D I A M E T E R OF TH^ COLUMN = ' , F 1 0 o 4 , ' . 4 1 3 X , ' V O I D A G E OF ANNULUS = ' , F 1 0 « 4 ) FORMAT(/10X,«CALCULATED S TRE AM L I N E • / 1 4 X , 5 1 ( • . • )/ •1' / 1 2 15X, PATH LENGo FLOW RATE RR 11 12 13 14 , ' T I M E ' / 1 4 X , 5 1 ( • o ' ) ) F O R M A T ( 1 3 X , I 3 , 2 X , 4 E 1 2 » 5 ) F O R M A T ( / / 1 0 X i ' D A USE FOR 1 2 0 X , ' D A ' , 1 0 X , ' D E L C ' / 1 7 X , 2 3 ( ' , FORMAT( 1 6 X , 2 E 1 2 o 5) F O R M A T ( / / / 1 0 X , ' R E S U L T DF CURVI 1 1 5 X , ' D A A V E ' )/ ) ) F I T T I N G ' / / SQM/S ' / 1 5 X , ' MIN DEL C 0258 0259 0260 0261 0262 15 16 17 13 19 0263 0264 0265 0266 0267 0268 0 2 69 * OPTIONS • OPTIONS • S T A T I S T I C S * • S T A T I S T I C S * 21 22 24 23 999 2 ,£12.5/) FORMAT(IH1/10X,'STIMULUS 1 11X,'TI = 0 o 0 1 2 SEC. '//) TRACER CJNC< 222 CURVE*,/ FORMAT(6E12.4) FORMAT(//10X, •COMPUTED TRACER COMC< SEC , THE FORMAT(//10X, •EXP RTD CURVE 1 ,'THE ANNULUS'/10X,' 1=1 2 11X,'TI = 0.012 SEC. ' / ) FORMAT(//10X,•PREDICTED RTD CURV* « / ) TOP C IN THE SPOU AT & R =' ,F3o 4, :NTI: •JF :u: DF THE i VES A" 'INNULUS'/lOX, • Z = 1 6 R S O . M / S ' / l l X , ' T I TH-M * / TOP = ' r F 3 o 4 , ' = 0.012 S 5) 5) IN IN 1 , 2 10X,» 0 =• , FORMAT(4X,14) cORM AT(E12o5,24X,E12 FORMAT(E12o 5,60X,E 12 FORMAT(1H1 ) CALL PLOTNO STOP • END EFFECT* ID,EBCDIC,SOURCE,NOLI ST,NODECK,LOAD,NOMAP EFFECT -* NAME = MAIN , LI NEC NT = 57 SOURCE STATEMENTS = 269,PROGRAM SIZE = NO DIAGNOSTICS GENERATED CENTRE M ' / EC . ' / ) 66316 NO ERRORS IN MAIN NO STATEMENTS FLAGGED IN THE EXECUTION TERMINATED ABOVE COMPILATIONS $RUN -LOAD 3=INRTD 4=EXRT03 5=NEW0EQ3 EXECUTION BEGINS 223 N o t a t i o n ANG I n c l u d e d a n g l e o f the c o n i c a l base Cl S t i m u l u s t r a c e r c o n c e n t r a t i o n c u r v e CA P r e d i c t e d RTD c u r v e i n the a n n u l u s CAC E x p t l . RTD cu rve a t the top c e n t r e o f the a n n u l u s CM P r e d i c t e d RTD c u r v e a t the top of the a n n u l u s a t any r a d i a l p o s i t i o n CS P r e d i c t e d RTD c u r v e i n the spou t DA D i s p e r s i o n c o e f f i c i e n t , m 2 / s e c DB D iamete r of the bed a t Z = 0 , m DC D iame te r o f the c o l u m n , m DEG A n g l e between the s t r e a m l i n e and the v e r t i c a l a x i s , degree DELC The sum o f squa red d e v i a t i o n s between p r e d i c t e d and E x p t l . RTD c u r v e s a t top c e n t r e o f a n n u l u s DS Spout d i a m e t e r , m EA A n n u l u s v o i d a g e ES Spout v o i d a g e H Bed d e p t h , m ( f rom gas i n l e t ) 224 HC H e i g h t o f the c o n i c a l ' s e c t i o n o f the b e d , m L No. of da ta p o i n t s read i n f o r CAC M-1 No. of d i v i s i o n s o f H N- l No. o f l o n g i t u d i n a l d i v i s i o n s of the annu lus NUR Run No. Q F low r a t e a l o n g each f l o w p a t h , m 3 / s e c QA Flow r a t e i n the a n n u l u s , m 3 / s e c QT T o t a l v o l u m e t r i c gas f l o w r a t e , m 3 / s e c R,RR R a d i a l d i s t a n c e f rom column a x i s to gas s t r e a m l i n e s i n the a n n u l u s , m RS Spout r a d i u s , m SA Annu lus c r o s s - s e c t i o n a 1 a r e a , m 2 T T i m e , s e c TI = A t , mesh w i d t h o f t ime used to s o l v e E q . ( 5 . 5 ) , see F i g u r e IV.1 TL Length o f gas f l o w path i n the a n n u l u s , m TTAN R e s i d e n c e t ime of gas i n any f l o w path i n the a n n u l u s , sec U,UU,V Gas v e l o c i t y a l o n g the f l o w p a t h , m/sec 225 UA Upward gas v e l o c i t y i n the a n n u l u s , m/sec US Upward gas v e l o c i t y i n the s p o u t , m/sec Z Bed l e v e l , d i m e n s i o n 1 ess ZI = A z , mesh -w id th o f bed l e v e l used to s o l v e the E q . ( 5 . 5 ) ZL V e r t i c a l d i s t a n c e o f any p o i n t a l o n g the gas f l o w path f rom the p o i n t o f o r i g i n o f the f l o w pa th 226 Sample Output RUN = 29 POLYSTYRENE (LARGE) HEIGHT OF THE 3E0 = 0 o 7 2 4 0 M TOTAL AIR FLOW RATE = 0 „ 0 3 7 6 CU 0M/SEC DIAMETER OF THE COLUMN = 0.2413 M VOIDAGE OF ANNULUS . = 0 „ 4 5 6 0 CALCULATED STREAMLINE ooo 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 oooooooooooooooo 0 0 0 0 0 0 J PATH LENGo FLOW RATE RR AVETIME « o o « e e o e o o « o o « e « * o o * o o e o o o o o o o o o o o o o c o « o o « o o o o o o o o « 11 0 o 4 8 3 0 7 E + 00 0 o 1 1 1 6 2 E - 0 2 0 o 7 ? 5 7 9 E - 0 1 Oo48330E+00 DA USE FOR CURVE FITTING o o a o o o o o o o o o i O O O O d O O O O DA DELC O O O O O O O O 0 O O O O O O O O O O O O O O 0o60000E 0. 70000E 0 o 8 0 0 0 0 c 0 o 9 0 0 0 0 E 0.10000E 11000E 12000E 13000E 14000E 15000E 16000E 0 o l 7 0 0 0 E Oo18000E 0.19000E 0.20000E 0.21000E 0o22000E 0.23000E 0 „ 2 4 0 0 0 E 0o25000E -02 0ol5150E+00 -02 0 . U 9 6 2 E + 00 -02 0 o 9 3 7 1 9 E - 0 1 -02 0.72801E-01 -01 0o56085E-01 -01 0o42947E-01 -01 0.32837E-01 -01 0.25490E-01 -01 0o20414E-01 -01 0.17370E-01 -01 0.16115E-01 -01 0 o l 6 4 4 4 E - 0 1 -01 0.18175E-01 -01 0.21I50G-01 -01 0.25239E-01 -01 0o30324E-01 -01 0o36299E-01 -01 0.43076E-01 -01 0.50568E-01 -01 0.58711E-01 RESULT OF CURVE F ITTING OA MIN DELC = O 0 I 6 O O O E-OI SQM/S = 0.16115E-01 o o o o o o o o o o o o o o o o o O o o s e o o 0 o 0 0 a 0 e e a o e 0 0 e o e o O o 1—1 (NJ OJ - J . r— ro O- - J >£> vO I-" K - H-< !-» t\> rj i— oo cr J> - J ro o -si O oo o o o o O o c O o o CO ro ro Ul oo U J o o o o o o o O o o 4- o o- 0 0 ro 4~- 4> o o o o o r n r n :71 I'll m ITi r n : n tn (VI r."i :T) m 111 !YI l l i I I 1 1 I + + + + + + 4 4 + + 4 4 o c o o o O o o o o o c O o o o O l \J rj i—1 1—' I--1 c o o o o o o r— 1-" V— 1— 1-4 "D II o o o o O o o o o o c o a o a c O o o II > 0 0 c 0 0 o e 0 6 o 9 s 0 s 0 9 9 0 s r—• H o o 4- O r— U) o I—1 r o 1 o un -0. CO -o t—' t~* I—1 O O c 4s- - J . - f co UJ 4- 0^ ->l 4- N O O o O O o 0 Co c o 0 0 o 4- O J Ul U) - J . o c r o c o O o O o o o o o o C r 4> r o c r 0 0 C r o o o o o o \— 7 3 t n 111 I T : r n rr: m ITI r n •it :':l r n r n r o V l 1 I 1 + + + 4 + + + 4 + + + + ll < o o o o a o O o o a o o o o O o o C o m UJ r o r— i -j o o o o o o o t— rn o e O « AT c a o o o o o o o o o o o o o o o o o o e 0 e 0 0 0 0 D o 0 9 0 0 o o s 9 0 e -vl O o O Ul t— UJ O I—1 r o UJ Ul Co •£> r— i—1 r— I-* f— O J X 4- c r Ul C r o r o - J 4- I-" - • J . C - o o o o O C r n ; o o CO O- i— c r o ui c o {••' Ul o o o o o o o o a J > r o c r Cr C o c o a o o o o —I m rn IT. i i i r n r n m i r i m r n n i m TTT m (Ti :Tl l l 1 i 4 4 + + + + + + + 4 4 + o o o o o a o o o o o o O o O o M 1— 1—1 t— c o o o o o o i— r— I-1 *— o rn o O o o c c c o o o o o o o a o o o o —1 e 0 e e o e 0 0 a e 0 0 e 0 o 0 0 m O O O UJ 1 J U l U1 i - 4 PJ Ui un co »— (—' cr J> I-1 J « — c 4> r - Ul vO O O o o O CO cr o o O r o ro o -P- - J . o o o a o C o c CO O CO co co J> cr o o o T I I'V: n l , ii I'.'l i i i !Tl i n Pi** f l 1 r n 1 1 1 I + + + + + + + + +' +' + + H o o o o o o o o O O o o o o o o X ro I-1 f o o o o o o o o o i fl c o o c o o o o O o o o o o c a o o o 0 0 e o 0 0 0 e 0 o o 0 e 0 0 0 0 0 " ' o o o ro 1—' ro ^> *^!) J> cr CO >—• 1—• CT o ro a- CO CO i-* co cr IxJ -f- >o O vO o o I— o o cr J> Cr IU ro Ul O 4> o o o C c o o c O o cr CO o CO ru o CJ o CO n i ! ' l r n r n n i r n rr. rv i ."fl I ' M i n I'Ti r n l 1 l I i ' + + + + + + + + + + + c C c o o o o o o o o o o o o o fNJ 1 J t— o o o o o o o I—1 o It— o o o c o o c c o o o o o o o o o o o e 0 e » 0 e 0 0 o 9 « o o a 0 0 0 0 o O O O 1-0 CO 4- CO (-• iU 4- c r CO (•-* •—• c 4s- 4- U) - J o - o Ul UJ o r o c o o o o o c O 4- r j o c r 1—• o r o 4-- 4- vO o o o o a O O o o c r r o r o r o CO 0 0 o c o o m ;;i r n n i r n r n rn i T i ITI i'ii r . i r n t i l I 1 1 I | + + + + + + + + 4 + + o O o c o o o o o o o c o O o o r o r o 1— i—• t— o o o o o o c h-1 O o o o o o o O o o o o O O o o o o o 0 0 o 9 0 9 e 9 0 0 9 0 0 9 0 a 9 o 9 O o o o O o o o o a o C r j r r o Ul f— r o C r t— - J CO U! c r r o CO r o o i-o - 0 Ol oo 4- 4- U l o O 4- Ul t— 4- 4> I - J o t VI r n f l l .'YI rn n i • i • ! i ' l 1 r 1 1 + + + + o o o c a o o o Ol U) !-J t— I-1 o o o r— H 1—4 -4 O o o o o o o o c o o o o o o o o o O II C 0 o e o e 9 0 9 9 0 9 0 o 9 c 9 9 0 9 r -O o O o O o o o o O c O 4r 1—' 4- 1— r o Ul vO o 4* 4- ~J c 0 0 4- UJ OJ CO CO O o 4- -^ 1 UJ 1—' 4- o 4- - I i . 1 .'"1*1 ; a r n 1*1': rv 1 rr i r o vo 1 1 I 4 + 4 + > O o c C •o O c o r n r— r— O o o o . ' T i r n O v o O o c o o o c o o o o O O o o o o o o 0 0 a 0 o 9 . 0 o 9 0 9 0 0 o 0 e 9 0 o C_i O o a a o o O o o o o O r o 4- vO r o vO CT r o 4- t—• CC i\J O O- 1—' 4- c o U l OJ (NJ 0 -^ i Ul O- o UJ o r n >n r . i .".1 r s : O 1 1 I 1 + + + d O a o o o O o C o i—• t— o O c < o o o o o o o c c o o o o o o o o o o 0 0 9 0 0 9 0 9 9 9 0 9 0 9 0 0 9 0 9 O O O O O O O C O O O O i - i - U J C O i - 4 - C O UJ - f > U i CO i o u i J N I-J r o ~j u J >ii i—' J> - J Ul O UJ ~ J vO • i l ll"; d i . " 1 . |J ,;i I I I I + + 4 o o o c o o o IU I— t-' I— c o o o 0 o o c o o o o o o o o o O 9 0 o o o o o o o o o c O O o 9 0 9 o o 0 0 9 o J.J r— UJ - 0 r— O) C r p— UJ ; — 0- ->l ~J i-O o - J OJ O 4- u-O Ul o 4- O o r . i f l : ; V i ; V : r n i I 1 1 + + o o O O O o c I - J I—1 I-' o o o 9 0 o o O o o o o O O o o o O o 9 0 0 9 0 O O O O O O o o o o o o o 9 0 9 0 0 o o 0 o (._. a- r o O i— UJ -0. o 0 0 CO UJ OJ r j o o ->( Ul o~ 0 0 Ui r\j oo i i i—* rr: 1 0-I :"i i 1 f 1 + Ul ;":! + u-: . 1 •t o o o o O o r j r o 1— t"1 o o o ro ro rn oo ~o X r - o o o O o o o o O o o o o c o o o o o o .71 o O a 6 o e o 0 0 0 0 0 e e 0 e 0 0 e • 0 o 0 r> o •4 VJI i-» U J t— U J !—• ro 0 J VJI -4 vD vO t-* l-» »-• I-" c rn H ro -o. 4> 4> •£> -J 4> O J 4> vO -0. O O co >£> o o o o H OO .—1 U! lo ro M ->l -si VJI C* vO ro —J o vO o CO o o o o •—i Oo T» CO 00 •—• CO CT- a* ro CO O J VJI ro r-« 4> CO 1—' 1*0 o o o o O T l CD ; i l r n i n j'Vt ffi M l : T i I T I : il rn .Ti rn :T. rn E l l :T1 IT; r n T i zz cz 1 I 1 1 i 1 1 1 ••• V + + + + + + + + + + r~ O o o o o C5 o a o o O c O c- o o o O o o ~\ HI T J ui VJI i O J ••>) ro r v j rv t—' 1—• r—• o o o o o o o o M (-* >— 70 e~ r* "•7 c~ oo o o c o o o o o o o a o Q o o O o o a i —I r n —i o 0 o 0 e 0 o o o e 0 e 0 0 0 a o 0 o e o i> 4*- ro CT* r—* U J CT* ro O J VJI -4 OO vO •D <J0 I-- r— i> v£ ro vO •«£) U l 0 J C* 1*0 ro a- 4> U> *£) -4 4 J vO o o o —( i ll I—1 1*0 CT- 4> 0 * ro 00 o* ro O J 4> o* o O J -4 - 0 o o o m o -f> CT* CX) o o CO ro O J O CO CD (— r— vO o o o o > i ;1 rn (Tl Til .ri 11 i rr; •  a ni rn ITI r " i i i t ) ,71 •Ti T J I 1 l 1 1 1 1 + + + + + + + + + + + + a o o c o o o o o o o C o o o O c O o o U J XJ U J ro ro r-" r— o o o o o o o o o VJI CTJ e a o o o c o o O o c O o o o o o o o o C J 0 e 0 o • 0 0 a 0 0 0 o o 0 o 0 0 0 0 T l 4> r—' ro o ro VJI ro O J U l co vO vO (._! r— 1—' io O VJI o O J vO vO a O J r-1 o o o vD vO o o o -H o VJI o 4- o -p- ~J 4> O J 0* ro CT* vC -p* U l ^ o c o a ro - C O J VJI o 0 J vD O lo c*- VJI O 0 J 4> CO o o o - i 11: '71 : n 11! ! i i i n ;v. f'fl .71 : Il i ."1 111 I' i i i l l i i i n i J > i 1 1 1 1 l 1 + + + + + + + + + + + r - o O O o o o o O o o o o o O o o o o ZZ O J ro ro ro o o O o o o o o o T J e I-a O c o •o o o o c o c o o o c o o o o O —i •s>J s e 0 0 0 0 0 0 0 0 0 o 0 0 o o 0 a 0 U l oo O ro VJI 1—• ro U l OJ J> O c*o vO vD vO 1—• 1— r— —i CT* c ro ro !-» o OJ o CC" h-" CO -j 4> VJI vO o O O i—i oo a VJI i—* VJI vO o OJ r*o U l i -1 o 0 * OJ OJ vO o o o .'II VJI r* , o VJI o U l CO 4> r— +•*• o CO •—' -J o o o r . i o rr, 111 1.1 r'T iD i':l r n ITI r : i iTI r n i l l rn ITS r n m m e 1 1 1 1 1 1 l + + + + + + + + + + + + i—. o o c o O O o o O O c o c c O o o o o 00 >> OJ 00 r*j to *— ^—- o o o o o o o o 2; O 0 c c 0 0 O 0 O O O O 0 0 O 0 0 c O O —1 0 e 0 0 0 0 0 0 O 0 0 0 e 0 0 e 0 0 0 X U ) - J i - 1 4> ro 4> ro -P* CT* CO vO 0 f—J r— r-1 ro 0 ~ J U l 0 10 1*0 c* 0 3 U l J> ( O J> CO 0 c O O vO CO ~ J 1— CT* vO U i 0 CO 0* 0 ro r-1 v0 vO 0 0 O rn CT* U l OJ 1*0 OJ OJ r-< vD t— i o CT r o vO 0 0 O •—1 . ' i " i rn i ^ 1 i 11 rn rn rn . 7 . H i rn rn m rn rn r"n : n . T l If! 1 I 1 1 1 l 1 1 + + + + + + + + + + + C 0 c O 0 O O 0 0 O O 0 c O O 0 c 0 0 O O TJ Oi OJ ro ro »— ,_. r— O 0 c O C 0 0 0 r— o n i 00 O O 0 O O c O c c O c; O O O O 0 0 0 0 0 0 0 • 0 • 0 0 0 0 e 0 0 0 0 0 0 0 0 T l ro Cr co v0 ro -P* 03 I- 1 ro J> CT* -0. <X* v0 I- J t—' CT* 0** O vO 0 ro O) un CT* ro O vC ro Co vD 0 0 O TJ +- ~ J 4- Ul -P- ro ro ro OJ J> O vO -P* CT- Ul 00 0 0 O 7> O CT* CT* 0 - J . 4- VJI (—• 05 CT* ->l vO 0 0 0 TJ rn r.i 1VI . ,1 • VI r n m ITI rn i l l rn rn IT. rn f n i n . n i n rn 1 I 1 1 1 1 l 1 + + + + + + + + + + + 0 O 0 O O 0 0 c O 0 O 0 0 O 0 0 0 O O e OJ OJ ro ro ro r-< 1—" (-* O 0 O O 0 O O 0 1—• >—• t-1 0 0 0 0 0 0 0 o e 0 o 0 o e O O O O O O O O O O O O O e o © 0 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 O O O O O O O O O C O O C O O 0 0 0 0 0 o O O O O O O c c o O O o o o o e o 0 O C O O O O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 I- 1 CT* ro 0 CO CD 0 - J U l 00 s i ; 0 : n f—• rr. 1 0 1 O 1 0 ro ro 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0** 4** rv -0. CO O J ro - j U l 0 4^ * Ul 1 n 1 rn 1 i n 1 0 1 0 O O J r*o r-* 0 0 0 0 0 0 0 O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4> r— i o vO 4> o> 4> -0 r I 0 ro 0 0 0 0 0 0 0 O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I O t—' CT* -J CT* [—J  -0. Ul V C i O ro 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i — . I--* O J 4> -P* I J 4> r:. ~sl 1 C f O ro 1 ••• 0 0 0 0 0 0 0 O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1—1 1— CO O 0 Ul 1.1 1" 0 1 O I O t.._i O O O O O 0 0 0 0 0 CT* I-1 '0J -0. • -** O) OJ ro C O CT* CO UJ ro CT* -4 U l OJ O Iii . n 1 n f 1 + + + + O O 0 O 0 0 O /—> 0 —1 »—« •MP O O 0 0 0 11 c 0 0 0 0 0 —1 Ul ro CT* !—' O rn 0- r 0 CO I V O 0 vj CO 4> 4.N Ul O O r—» 4-** 4> 1—' O —1 i : 1 f l" rn 1 :! ro 1 •+• + + + O 0 O O 0 00 0 0 *—' 0 m 0 0 m ;o 0 O O 0 r> O 0 0 B 0 4*- I-' ro Ul VL) 0 4> +* -4 0 U J CO co ~4 0 0 OJ (-* 4- O 4> /Ti r n .11 : n i l l 0 1 + + + + cr. O 0 O 0 O !-• 0 O C; O < in 0 O O O 0 0 O e 0 0 4-* ro 4** sQ I O 4> r— C J ro —1 4* CO u- U.I 1*0 X O 0 VO O 0 r r 1 V + + + 00 O 0 O 0 T J r j !—• c O 0 c —i O 0 C e 0 0 ft 0 J> O J CO 1—* 4> 00 —1 - J U) CO rv U l rv -4 u- 0; rvi U l O U-' ^ J O r -if: : : 1 r.i . .1 . it 1 1 + + + 11 O 0 0 O O 1—• 1—• 0 0 O 0 0 O O 0 0 0 0 U J r— 01 -4 U J t~" CT* -4 -4 ~4 U J O +*• U ) O •JC 4- s^ O , i l 1 . i rn . 1 1 1 1 + + + O O O 0 O I - 1 O 0 O APPENDIX VI EXPERIMENTAL AND PREDICTED RTD CURVES N o t a t i o n E x p e r i m e n t a l RTD c u r v e ooooo Model p r e d i c t e d RTD c u r v e u s i n g D v a l u e s i n d i c a t e d i n the p l o t s Curve 1 S t i m u l u s c u r v e Cu rve 2 Response c u r v e a t c e n t r e of a n n u l u s s u r f a c e Curve 3 Response c u r v e a t annu lus s u r f a c e h a l f - w a y between the annu lus c e n t r e and column w a l l 229 T.SEC T.SEC T.SEC 232 233 1 CO o OC o OC 0 c 0 <t I I 0 fi 0 /o Q 0 01 00 0 >^ 0/ i n 0 / CNJ . I I o 2: z ZD < ' , OC < < < —' " 1 CO o o UJ CO o o o 10 o 03/3 o o 0.0 0.4 0.8 1.2 1.6 T .SEC APPENDIX VI I GAS FLOW DATA N o t a t i o n MV Output o f p r e s s u r e t r a n s d u c e r B i n m i l l i v o l t s per 2 cm of bed h e i g h t UA = U , s u p e r f i c i a l a i r v e l o c i t y i n annu l us, cm/sec a Z/H Reduced bed l e v e l * C a l c u l a t e d f rom gas v e l o c i t y i n the spou t measured by p i t o t tube 235 236 RUN NO. = 1 Z / H MV UA 0.031* -0.086 0.0^6* 10.257 0.249* 20.977 0.175 0. 149 12.937 0. 219 0. 209 16.777 0.306 0.309 22.764 0.394 0.429 30.060 0.481 0. 544 3 7. 210 0.569 0.649 43.381 0.612 0.699 46.027 0.65-6 0.729 47.494 0.744 0.809 50.905 0. 831 0. 859 52.670 0.919 0.924 54.624 0.962 0.924 54.624 RUN MO. = 4 Z / H M V UA 0.144* 14.623 0.322* 19.706 0. 197 0. 100 9. 413 0.262 0.150 13.004 0.328 0.201 16.2 84 0.459 0. 340 24.619 0. 591 0.4 55 31.677 0.722 0.585 39.694 0.852 0.670 44.5 21 0.919 0.680 45.050 0.951 0.6 80 45.050 RUN MO. .= 2 Z / H MV UA 0.031* -5.072 0.133* 0.534 0.249* L6.314 0.3.75 0.120 10.909 0.219 0.190 15.598 0.306 0.290 21.633 0.394 0.380 27.042 0.481 0.490 33.862 0.569 0.590 39.992 0.612 0.635 42.600 0.700 • 0.710 46.576 0.787 0.760 48.903 0.875 0.320 51.316 0.919 0.820 51.316 0.962 0.820 51.316 RUM MO. = 5 Z / H V V UA 0.052* 3.152 0.144* 10.431 0.322* 14.556 0.262 0. 129 11. 555 0.328 0. 174 14.502 0.459 0.279 20.979 0. 591 0.379 26.981 0.722 0.489 33.799 0.8 53 0. 599 40.524 0.919 0.614 41.400 0.951 0.639 42.824 RUN NO. = 3 Z / H MV UA 0.031* -31.819 0.133* 5.346 0.252* 15.538 0.175 0.090 8.629 0.219 0.150 13.004 0.306 0.245 18.955 0.394 0.340 24.619 0.481 0.440 30.743 0.569 0. 550 37.578 0.612 0.595 40.200 0.700 0. 675 44.786 0.787 0.740 40.006 0.875 0. 785 49.959 0.919 0.815 51.131 0.962 0.815 51.131 RUM NO. = 6 Z / H MV UA 0.052* -6.362 0.144* 1.935 0.322* 14.935 0.197 0.030 3.258 0.262 0.090 8.629 0.323 0.1 45 12.666 0.459 0.2 40 18.655 0. 591 0. 340 24.619 0.722 0.440 30.743 0. 853 0. 530 36.343 0.919 0.555 37. 883 0. 951 0.575 39.095 R U N M O . 7 R U N N T . = 1 0 237 Z / H MV U A 0 . 0 2 0 * - 3 4 . 0 5 9 0 . 0 4 9 * - 6 . 2 8 7 0 . 1 0 9 * 1 4 . 4 7 8 0 . 0 6 7 - 0 . 0 2 0 - 2 . 4 1 6 0 . 0 ^ 9 0 . 1 3 0 1 1 . 6 2 6 0 . 1 1 1 0 . 2 3 0 1 0 . 0 5 4 0 . 1 3 3 0 . 2 7 0 2 0 . 4 4 5 0 . 1 7 7 0 . 3 6 0 2 5 . 8 2 6 0 . 2 2 2 0 . 4 8 0 3 3 . 2 3 8 0 . 2 6 6 0 . 5 8 0 3 9 . 3 9 5 0 . 3 1 1 0 . 6 7 5 4 4 . 7 8 6 0 . 3 5 5 0 . 7 5 0 4 3 . 4 6 1 0 . 3 9 9 0 . 3 1 0 5 0 . 9 4 3 0 . 4 4 4 0 . 8 7 0 5 ? c 0 2 4 0 . 4 8 8 0 . 9 1 5 5 4 . 3 7 1 0 . 5 3 2 0 . 9 3 5 5 4 . 9 2 9 0 . 5 7 7 0 . 9 6 0 5 5 . 6 1 1 0 . 6 2 1 0 . 9 8 0 5 6 . 1 5 5 0 . 6 6 5 1 . 0 0 5 5 6 . 8 5 3 0 . 7 1 0 1 . 0 2 0 5 7 . 2 9 2 0 . 7 5 4 1 . 0 3 0 5 7 . 5 9 6 0 , 7 9 8 1 . 0 3 2 5 7 . 6 5 8 0 . 8 4 3 1 . 0 3 3 5 7 . 6 ^ 0 0 . 8 6 5 1 . 0 3 5 5 7 . 7 5 3 0 . 8 8 7 1 . 0 4 0 5 7 . 9 1 4 R U N N O . 8 Z/K, MV UA 0 . 0 3 5 * 9 . 4 5 1 0 . 0 9 6 * 3 0 . 3 8 2 0 . 2 1 4 * 4 6 . 0 5 2 0 . 5 8 7 * 7 7 . 3 1 7 0 . 1 7 1 0 . 3 5 0 2 8 , 2 5 6 0 . 1 7 5 0 . 4 9 0 3 5 . 4 2 2 0 . 2 1 9 0 . 6 7 0 4 3 . 4 9 1 0 . 2 6 2 0 . 8 5 0 5 1 . 2 1 7 0 . 3 5 0 1 . 0 9 9 6 1 . 9 3 5 0 . 4 3 7 1 . 2 6 0 6 8 . 6 9 4 0 . 5 2 5 1 . 3 9 0 7 3 . 8 0 6 0 . 6 1 2 1 . 4 5 0 7 6 . 0 0 5 0 . 7 0 0 1 . 5 0 0 7 7 . 7 4 4 0 . 7 8 7 1 . 5 3 5 7 8 . 9 0 5 0 . 8 7 5 1 . 5 4 5 7 9 . 2 2 8 0 . 9 6 2 1 . 5 5 5 7 9 . 5 4 7 P U N Mn. 9 Z / H MV U A 0 . 0 5 2 * 9 . 4 9 6 0 . 1 4 4 * 3 8 . 2 1 0 0 . 3 2 2 * 4 3 . 6 4 0 0 . 1 9 7 0 . 4 1 9 3 1 . 9 3 8 0 . 2 6 2 0 . 5 1 9 3 6 . 7 7 9 0 . 3 2 8 0 . 6 5 9 4 3 . 0 1 5 0 . 3 9 4 0 . 8 4 4 5 0 . 9 5 9 0 . 4 5 9 0 . 9 6 9 5 6 . 3 3 8 0 . 5 9 1 1 . 1 2 9 6 3 . 2 1 6 0 . 7 2 2 1 . 2 8 9 6 9 . 8 6 9 0 . 8 5 3 t . 3 5 9 7 2 . 6 2 7 0 . 9 1 9 1 . 3 9 9 7 4 . 1 4 ? Z / H MV U A 0 . 1 0 5 * 5 . 3 1 6 0 . 2 8 9 * 4 6 . 3 1 0 0 . 2 6 2 0 . 3 2 5 2 6 . 8 2 3 0 . 2 9 4 0 . 4 3 5 3 2 . 7 4 6 0 . 5 2 5 0 . 5 0 5 3 6 . 1 2 3 0 . 6 5 6 0 . 6 4 5 4 2 . 4 0 7 0 , 7 3 7 0 . 7 9 5 4 3 . 8 5 9 0 . 3 5 3 0 . 8 9 5 5 3 . 1 5 0 0 . 9 1 9 1 . 0 4 5 5 9 , 6 1 5 ° U M Hla = 1 1 Z / H M V UA 0 . 0 2 8 * - 3 . 4 5 3 0 . 0 6 Q * 1 3 . 4 2 9 0 . 1 5 4 * 3 9 . 6 6 1 • 0 . 0 9 4 0 . 2 5 0 2 2 . 1 6 0 0 . 1 2 6 0 . 4 4 5 3 3 . 2 4 4 0 . 1 5 7 0 . 5 7 5 3 9 . 3 2 4 0 . 1 3 9 0 . 7 9 0 4 8 . 6 4 5 0 . 2 2 0 0 . 9 0 0 5 3 . 3 6 5 0 . 2 3 3 1 . 1 2 0 6 2 . 3 3 3 0 . 2 4 6 1 . 2 5 0 6 3 . 2 8 5 0 . 4 0 9 1 . 3 5 0 7 2 . 2 7 9 0 . 4 7 2 1 . 4 2 0 7 4 . 9 2 0 0 . 5 3 5 1 . 4 7 0 7 6 . 7 1 2 0 . 5 9 8 1 . 5 1 0 7 8 . 0 3 0 0 . 6 6 1 1 . 5 5 0 7 9 . 3 3 8 0 . 7 2 4 1 . 5 8 0 3 0 . 3 2 7 0 . 7 3 7 1 . 6 0 0 3 0 . 9 3 2 R U N N O . = 1 2 Z / H MV U 4 0 . 0 3 9 * - 2 3 . 4 8 3 0 . 0 9 6 * 0 . 7 7 0 0 . 2 1 4 * 3 9 . 0 6 3 0 . 2 6 2 0 . 3 2 0 4 9 . 9 3 0 0 . 3 0 6 0 . 3 9 0 5 2 . 9 3 5 O . ? 5 0 0 . 9 6 0 5 5 . 9 5 0 0 . 3 9 4 1 . 0 6 5 6 0 . 4 7 6 0 . 4 3 7 I. 1 4 0 6 3 . 6 8 4 0 . 4 3 1 1 . 2 0 0 6 6 . 2 1 5 0 . 5 2 5 1 . 2 4 0 6 7 . 8 7 4 0 . 5 6 9 1 . 2 7 0 6 9 . 1 0 1 0 . 6 1 2 1 . 3 4 0 7 1 . 8 9 0 0 . 6 5 6 1 . 3 2 0 7 1 . 1 0 5 0 . 7 0 0 1 . 3 3 5 7 1 . 6 9 5 0 . 7 4 4 1 . 3 5 0 7 2 . 2 7 9 0 . 7 3 7 1 . 3 3 5 7 3 . 6 1 8 0 . 8 3 1 1 . 4 1 0 7 4 . 5 5 2 0 . 0 7 5 1 . 4 4 5 7 5 . 3 2 6 0 . 9 1 9 1 . 4 6 0 7 6 . 3 6 0 0 . 9 6 2 1 . 4 3 5 7 7 . 2 3 2 RUN NO. = 13 7/H MV UA 0.039* 1. 503 0.096* 23.391 0.214* 27.321 0.175 0.425 24.496 0. 219 0.455 26.069 0.262 0. 535 29.931 0.306 0.640 34.226 0. 3 50 0. 635 35.808 0.394 0.8 15 39.635 0.437 0.875 41.260 0.481 0. 925 42.556 0.525 0.975 43.906 0. 569 1.025 45.333 0.612 1.075 47.069 0.656 1.125 49.056 0.700 1.185 51.980 0.744 1.215 5?.718 0.787 1.235 54.997 0.831 1.255 56.380 0.375 1.2 55 56.380 0. 919 1.255 56.380 RUN MO. = 14 7./H MV UA 0.039* ' -10.012 0. 0 '6* 10.099 0.214* 24.276 0.131 0.230 13.087 0.175 0.360 20.885 0.219 0.435 25.028 0.262 0.530 29.704 0.306 0.6 20 33.475 0.3 50 0.720 36.942 0.394 0.730 33.722 0.437 0.845 40.430 0.525 0.940 42.952 0.612 1. 000 44.624 0.700 • 1.060 46. 536 0.737 1.130 49.275 0.875 1. 150 50.193 RUN NO. = 15 Z/H MV UA 0 . I l l * 43.362 0.248* 74.778 0.2 53 0.930 63.742 0.304 1.350 37.057 0.^54 1.670 99.671 0.405 1 .900 103.220 0.456 2. 110 115.726 0.506 2.290 121.85? 0.557 2.430 126.339 0.608 2. 570 130.705 0.658 2.670 133.646 0. 709 2. 760 136.194 0.759 2.050 138.650 0.310 2.960 141.539 0. 361 3. 000 14?.563 0.91 1 7.010 14 7.817 0. 96 2 3.140 I t 6.05 9 238 "UN NO. = 16 Z/H MV UA 0.253 . 0.720 5 3.048 0. 3 04 1. 090 76.075 0. 354 1.340 86.650 0.405 1.610 97.331 0. ^ 56 1.790 104.157 0. 506 2.000 111.378 0. 557 2. 1 60 117.459 0. 60S 2. 310 122.514 0.658 2.450 127.020 0. 709 2. 600 131.6 00 0. 759 2. 700 134.506 0. 8 1 0 2.7 30 136.747 0. 361 2. 900 139.978 0 . 9 H 2.950 141.281 0.962 3.030 143.322 RUN NO. = 17 Z/H MV U4 0.046* 3.947 0.111* 12.909 0.248* 24.094 0.253 0.759 22.160 0. 304 0. 999 29. 't25 0.354 1.179 3 4.524 0.405 1.329 33.492 0. 456 1.429 40. 937 0. 506 1.539 43.597 0. 557 1.629 45.630 0.608 1.639 46.937 0.658 1.7 39 49.038 0.709 1.099 51.248 0.759 1.939 52.029 0, 310 1 .979 52.799 0.861 1. 959 52.4L5 RUN NO. = 18 Z/H MV UA 0.039* 1.101 0.096* 6. 341 0.175 0. 154 7.930 0.219 0. 1 30 9.219 0.762 0.202 10.299 0.306 0. 242 12.240 0. 350 0.275 13.319 0.394 0.314 15.657 0.437 0. 357 17.649 0.431 0.408 19.967 0.52 5 0.446 21.661 0. 569 0.484 23.328 O.M 2 0.519 24.333 0.656 0.551 26.199 0. 700 0. 532 27. 498 0. 744 0.604 20.409 0. 717 0.637 29.757 0.031 0.672 31.165 0. 0 75 0.702 32.353 0. 91 9 0. 71 7 3 7.940 0.962 0.747 3 4.102 239 RUN NT, • = 19 Z/H MV UA 0.039* 10.934 0.214* 14.3 30 0.131 0.390 14.437 0. 175 0.430 15.750 0.219 0.450 16.398 0. 262 0. 550 19.549 0.306 0.690 23.754 0.350 0.340 23.044 0.437 1.040 33.505 0.525 1.2 00 37.722 0.612 1.300 40.307 0.700 I. 480 44.881 0.737 1. 560 46.386 0. 831 1.620 48.377 0.375 1.640 48.873 0.919 1.690 50.105 0.96 2 1.730 51.086 RUN NO. = 20 Z/H MV UA 0.131 0.290 16.309 0. 175 0.330 18.360 0.219 0.390 21.327 0.262 0.490 25.979 0.350 0.740 36.161 0.43 7 0.920 42. 581 0.525 1.050 47.013 0.612 1. 190 51.798 0.700 1.310 56.032 0. 744 1.330 53.586 0.787 1.450 61.210 0.831 1.540 64.680 0.875 1.570 65.857 0.919 1.5 30 66.2 52 0.962 . 1.5 90 66.647 »UN NO. = 21 Z/H MV UA 0.03 9* -6.716 0.096* 3.352 0.214* 22.813 0.131 0.255 14.620 0.175 0.315 18.250 0.219 0.455 26.069 0.262 0.555 30.317 0.306 0.650 34.590 0.3^4 0.730 38.722 0.431 0 . 3 9 5 41.776 0. 569 0.965 43.629 0.656 1.035 45.701 0.7'»4 1.105 4 8.719 0.331 1.195 52.537 0.875 1.205 53.116 P UN NT. = 2 3 Z/H MV UA 0.039* -37.910 0.096* -25.464 0.214* - 1.934 0.175 -0.070 -9.653 0. 219 -0.020 -2.416 0.262 0.025 2. 748 0.306 0.0 80 7.313~ 0.3 94 0. 230 13.054 0.431 0.430 ^0.122 0. 569 0.535 39.694 0.656 0.630 45.050 0.744 0.755 43.634 0. 831 0. 810 50.943 0.919 0. 330 53.33 7 0.962 0.900 53.938 RUN NT.. = 25 ' Z/H MV UA 0.039* 13.303 0.096* 29.191 0.214* 26.707 0.131 0.300 22.223 0.175 0. 330 24.018 0.219 0.350 25.221 0.262 0.425 29.812 0.306 0. 430 33.238 0.394 0.600 40.53? 0.4 31 0.690 45.569 0. 569 0. 735 49.959 0.656 0.340 52.030 0. 744 0. 3 90 53.641 0.831 0.940 55.067 0.919 0.955 55.475 0.962 0. 960 55.611 RUN NO. = 27 Z/H MV UA 0.026* -2.076 0.136* 6. 241 0.245* 19.870 0.044 -0.010 -1.172 0.037 0. 030 3. 258 0.131 0.030 7.813 0. 175 0. 160 13.671 0. 219 0.260 19.850 0.262 0.350 25.221 0.350 0.490 33.862 0.4^7 0.600 40.58? ' 0.525 0.630 45.050 0.612 0. 770 49.335 0.700 0. 3?0 51.679 0. 737 0.8 30 53.337 0.375 0.930 54.791 0.919 0.930 56.155 0. 96? 0. 935 56.292 o.lIN NT. = 20 "UN NI. - 30 240 Z / H MV UA 0.051 * 6. 533 0. 106* 33.091 0.105* 36.692 0.033 0. 150 14.769 0. 11 0 0.3 30 29.900 0.7.38 0. 530 3 7. 2 36 0.166 0.530 39.547 0. 221 0. 700 44.735 0.276 0.920 54.226 0.331 1.070 60.691 0. 337 1.200 66.21 5 0.442 1.260 63. 694 0.407 1.330 71.499 0.552 1.400 74.131 0.608 1.425 75.103 0.663 1.450 76.005 0.718 1.460 76.360 0.773 1.465 76.537 0.829 1.470 76.712 0.884 1.480 77.059 RUN NO. = 29 Z / H MV UA 0.051* - 5.667 0.106* 12.936 0.195* 19.753 0.1! 0 0.140 12.323 0.138 0.200 16.222 0.166 0.275 20.742 0.221 0.300 22.228 0.276 0.405 28.576 0.331 0.475 32.926 0.387 0.530 36.343 0.442 0.5 90 39.902 0.497 0.640 42.380 0. 552 0.660 43.933 0.608 0.700 46.078 0.663 0.720 47.064 0.718 0.740 43.006 0.77? 0.750 48.461 0.829 0.760 48.903 0.834 0.780 49.754 RUN N\"o = 32 Z / H MV UA 0.042* 4. 382 0.15? 0.3?0 2 4.018 0.175 0.300 27.654 0.219 0.360 33.138 0.26? 0.6?0 42.217 0.3 06 0.600 45.050 0. 350 0. 730 4 7. 541 0.394 0.770 49.335 0.430 0. 795 50.362 0.4 0.815 51. m 0.525 0.030 51.679 0. 569 0. 335 51.355 0.61? 0. f"4f) S2.030 0. 656 0.340 5 2.03 0 0. 738 0. 3 40 5 2.0 30 7/H MV UA 0. 0+2* 31.5 77 0.004* 5 5.999 0.164* 31.010 0. 04 4 0. 520 3 6.826 0.033 0. 003 53.5*0 0. 109 0.975 56.507 0. 131 0.930 54.657 0. 153 0.330 52.505 0. 1 75 0. 340 50.783 0. 219 0. 910 53.796 0.26? 1.020 53.333 0. 306 1. 090 61.54 9 0.350 1.200 66.215 0. 304 1.2 30 69.506 0. 438 1.360 72.665 0.431 1.390 73.806 0. 525 1.430 75.285 0. 569 1.440 75.647 0.613 1.460 76.360 0.656 1.470 76.712 0. 700 1.475 76.386 0.744 1.430 77.059 0. 788 1.435 77.232 PUN NO. = Z / H M V UA 0.033* 13.815 0.138* 33.449 0.262 0. 520 36.826 0.350 0.670 43.491 0.4^7 0.730 43.216 0. 52 5 0.900 53.365 0. 612 1.010 53.107 0.700 1.090 61.549 0. 737 1. 170 64.955 0.875 1.240 67.874 PUN NO. = 33 Z / H MV UA 0.042* -17.662 0.094--- 7. 139 0.164* 14.212 0.066 -0.020 -2.416 0.038 0.100 9. 41? 0. 1 09 0. 140 12.323 0.131 0, I 90 15.598 0. 153 0.200 16.222 0. 175 0. 230 13.054 0.219 0.300 22.228 0.263 0.370 26.433 0. 306 0. 440 30.743 0.->50 0.400 3?.862 0. 304 0. 540 36.964 0. 433 0.595 40.288 0.431 0.640 42.330 0. 525 0.630 45.050 0.569 0. 700 46. 078 0.613 0.740 48.006 0.656 0. 760 43.903 0. 700 0. 770 49.?34 0. 744 0. 700 50.162 0. 7 3 3 0. 000 50.558 0. .3 7 5 O.O'.O 50. 0/,3 APPENDIX V I I I SOLIDS FLOW DATA No ta t i on VA R a d i a l ave rage downward p a r t i c l e v e l o c i t y i n the a n n u l u s , = v . cm /sec a VAC.VAS & VAW Downward p a r t i c l e v e l o c i t y i n the a n n u l u s = v ^ ^ » v , c & v , . , r e s p e c t i v e l y , l o c a t i o n s a C a o aw r e f e r to F i g u r e 7 . 1 . VS R a d i a l a v e r a g e upward p a r t i c l e v e l o c i t i e s i n the s p o u t , E v s , m/sec VSC.VSM & VAS Upward p a r t i c l e v e l o c i t i e s i n the spou t a t l o c a t i o n s A , B, and C r e s p e c t i v e l y as shown i n F i g u r e 7 . 8 , m/sec Z/H Reduced bed l e v e l 241 In the Annu lus 242 HUM HD. = 1 Z / H VA VAW \/AC VAS 0.377 1.15 9 3 .0 0. 978 3. 750 0.142 1.165 3 .672 1. 184 2.250 0.219 1.193 3 .909 1. 224 1.818 0.306 1.434 1 . 224 1. 500 1.818 0. 39<t 1.620 1 . 455 1. 622 2.000 0.4B I 1.89B 1 . 763 1. 935 2. 143 0.569 2.080 1 .933 2. 143 2.308 0.656 2. 293 2 . 143 2. 307 2.608 0.744 2.485 2 . 307 2. 500 2.057 0.331 2.641 2 . 503 2. 609 3.000 0.919 2. 862 2 .727 2. 857 3.158 RUM NO. = 5 Z / H V A V Aid 1 V A : VAS 0.115 0.972 0 .0 0. 882 3.000 0.213 1.261 3 .733 0.918 3.000 0.320 1.243 3 .873 1. 194 2.222 0.459 1.520 1 . 333 1. 500 2.000 0.59 1 1.707 1 . 579 1.714 2.000 0.722 1.829 1 . 765 1.818 2.000 0.353 1.922 1 .813 1. 935 2.143 RUM NO. = 2 Z / H VA VAW VAC VAS 0.377 1.250 3 .0 1. 216 3.750 0.142 1.303 3 .714 1. 364 2. 500 0.21 9 1. 322 1 .003 1. 364 2.000 0.306 1. 708 1 . 50? 1. 76 5 2.069 0.394 1.836 1 .702 1. 935 2.222 0.481 2.134 1 .951 2. 222 2.400 0.569 2. 277 2 .105 2. 308 2.509 0.556 2.611 2 . 503 2. 609 2.857 0.744 2. 861 2 . 727 2. 857 3. 158 0.331 3.007 2 .857 3. 000 ^.333 0.919 3. 167 3 .003 3. 158 3. 529 RUM NO. 3 Z / H VA VAW VAC VAS 0.377 1.507 0 .0 1. 103 5.000 0.142 1.559 3 .933 1. 500 3.000 0.219 1.609 1 .42? 1. 622 2.000 0. 306 1.886 1 . 702 1. 935 2.222 0.394 2.062 1 . 905 2. 069 2. 400 0.481 2.486 2 .303 2. 500 2. 857 0.5 69 2.680 2 .503 2. 72 7 3.000 0.656 2. 845 2 . 6u? 2. 857 3.333 0.744 3. 160 2 . 857 3. 333 3. 529 0.331 3.323 2 . 857 3. 529 4.000 0.919 3. 540 3 .153 3. 750 4.000 RUM NO. = 6 Z / H VA VAW VAC VAS 0. 115 1. 152 3. 0 0.832 3. 750 0. 213 1. 378 3. 789 1. 184 3. 000 0. 323 1. 355 3. 963 1.429 2. 143 0. 459 1. 745 1. 622 1. 765 2. 000 0. 591 1. 959 1. 813 2. 000 2. 222 0. 722 2. 101 1. 951 2. 069 2. 500 0. 353 2. 222 2 . 003 2. 222 2. 727 RUM NO. = 8 Z / H VA VAW VAC VAS 0. 077 2. 204 1.215 2. 500 3. 750 3. 142 1. 728 1 . 285 1. 875 2. 500 0. 219 1. 671 1.42? 1. 818 2. 000 0. 306 2. 105 1.933 2. 222 2. 308 0. 394 2. 452 2.222 2. 609 2. 720 0. 48 1 2. 772 2.609 2. 857 3. 000 0.569 3. 205 3.003 3. 333 3. 450 0. 556 3. 476 3.333 3. 529 3. 700 0. 744 3. 800 3. 529 3. 920 4. 180 0. 331 4. 102 3.813 4. 270 4. 443 0. 919 4. 407 4.062 4. 563 4. 862 RUM NO. = 4 Z / H VA VAW VAC VAS 0.131 0.572 3.0 0.417 2 . 000 0.213 0.917 3.529 0.882 1 .875 0. 328 1. 063 3.693 1. 053 2 .000 0.459 1.375 1.224 1. 364 1 . 765 0.591 1.472 1. 333 1. 500 1 . 765 0.722 1.585 1. 463 1.622 1 .818 0.353 1.687 1.533 1.714 2 .000 RUM NO. = 9 Z / H VA VAW VAC VAS 0.115 1. 264 3 .0 1. 800 3. 000 D.213 1. 490 1 .023 1. 607 2. 368 0.328 1. 554 1 .395 1. 622 1. 818 0.459 1. 929 1 .813 2. 000 2. 069 0.59 1 2. 266 2 .069 2. 400 2. 500 0.722 2. 52 1 2 . 303 2. 609 2. 857 0.953 2. 806 2 .60? 2. 857 3. 158 243 RUM NO. = 10 Z/H V A VAW VAC VAS 0. 197 1. 029 0. 0 1. 429 2. 500 0. 361 1. 225 3. 813 1.286 2. 045 0. 558 1. 149 3. 933 1.154 1. 667 0. 787 1. 351 1 . 253 1. 364 1 . 579 *UM NO. = 12 Z/H VA VAW VAC VAS 0.333 3. 489 2. 104 4. 025 5. 302 0.398 3. 024 2. 329 3. 362 3. 929 0.164 2. 589 2. 321 2. 630 3. 136 0.230 2. 753 2. 662 2. 799 2. 886 0.295 3. 198 3. 135 3. 213 3. 311 0. 427 4. 041 3. 929 4. 05 1 4. 263 0.558 4. 441 4. 343 4. 467 4. 593 0.589 4. 935 4. 795 5. 017 5. 095 0.820 5. 562 5. 435 5. 622 5. 721 0.930 5. 985 5. 929 5. 983 6. 095 RUM NO. = 13 Z/H VA VAW VA: VAS 0.066 1. 696 3.525 1. 739 4.000 0.109 1.437 0.714 1. 333 3.175 0. 153 1.242 3.803 1. 351 2. 105 0.197 1.264 1.212 1. 282 1.364 0.262 1.558 1.533 1. 563 1.600 0.437 1.913 1.863 1. 900 2.069 0.525 2.030 1.885 2. 128 2.222 0.512 2.133 1.923 2. 286 2.424 0.700 2. 328 2 . l'>2 2. 424 2.580 0.787 • 2. 569 2. 424 2. 667 2.759 0.375 2.878 2.857 2. 857 2.963 RUM NO. 14 Z/H VA VAW VA: VAS 0.066 2. 267 3.769 2. 500 5.000 0.109 2.032 3.909 1. 667 5.000 0.153 1.573 1.053 1. 667 2.609 0.306 1.844 1.702 1. 951 2.000 0.394 2. 107 1.863 2. 222 2.500 0.481 2.363 2.003 2. 581 2.857 0. 569 2.578 2.286 2. 759 2. 963 0.656 2.822 2.424 3. 077 3.333 0.744 3. 092 2.857 3. 158 3.529 0.B31 3.469 3. 333 3. 333 4.0U0 RUM NO. = 15 Z/H VA VAW VA: VAS 0.114 0. 190 0.291 3.44 3 0.519 0.595 0.571 0.747 0.399 0.975 2.613 2.316 2.387 2.868 3.056 3. 247 3.456 3.685 3.925 4. 202 1 .753 1 .995 2.112 2.487 2.533 2.631 3.037 3.253 3. 555 3.812 2. 767 2. 367 2.590 3. 004 3. 449 3. 710 3. 761 3. 990 4. 134 4. 464 4.014 2.902 2.640 3. 432 3. 520 3.690 3. 834 4. 085 4. 343 4.575 RUM NO. 16 Z/H VA VAW VAC VAS 3.114 0.190 0.266 0. 342 0.418 0.494 0.546 0.797 0.B73 3.975 3.016 3.231 3. 592 3.722 3.888 4. 219 4 . e i 2 5.281 2.755 2.-442 2.583 3.053 3.085 3.207 3.733 4.414 4. 622 4.606 2.977 3. 603 3. 877 4.159 4. 358 4. 51 5 5. 259 5. 942 5.219 4. 185 3.933 4.198 4.289 4. 486 4.701 4.902 5.535 RUM NO. 17 Z/H VA VAW VA: VAS 0.10 1 0.165 0.253 0.354 0.456 0.557 0.658 0.759 0.361 0. 962 0.269 0.300 0.339 0.537 0. 632 0.693 0.751 0.775 0. 840 0. 904 3.083 3. 107 0. 155 3.387 0.4e5 3. 567 3.635 3.667 3.695 3. 762 0.231 0. 344 0. 406 0.661 0. 777 0. 825 0. 879 0. 909 0. 964 0. 988 0. 727 0.714 0. 721 0.727 0.769 0.800 0.833 0.825 1.013 1.143 RUM NO. 18 Z/H VA VAW VA: VAS 0.377 0.142 0.219 0. 306 0.394 0.40 I 0. 569 0.656 0.744 0.33 1 0.919 0.626 0.417 0. 544 0.766 0. 894 1.026 1.096 1.190 1.283 1. 396 1.528 3. 121 3.134 3.259 0. 543 3.721 0. 923 1.003 1.081 1. 143 I .231 1. 355 0. 472 0. 332 0. 423 0.714 0. 879 1.000 1. 053 1.143 I. 231 1. 379 1. 455 1.935 1.250 1 .509 1.455 1. 379 1.356 1.429 1.569 1.778 1.800 2. 105 244 RUM .ND. = 19 Z/H V A VAW V A : V A S 0.377 1.127 3.0 1.04 7 3.000 0. 142 0. 802 0.265 0. 464 2.308 0.219 0.717 0.211 0. 531 2.222 0. 306 0.770 D. 403 0.769 1.778 0.394 0.960 3.703 0.952 1. 667 0.481 1. 140 1 .003 1.176 1.455 0.569 1.268 1. 159 1. 270 1 .667 0.656 1.427 1. 293 1. 455 1. 739 0.B31 1.635 1 . 569 1.702 1.951 0. 919 1.900 1 .667 2.000 2.308 RUM NO. 20 Z/H V A VAW V A : V A S 0.077 0.892 3.0 0. 652 3.214 0.142 0.719 3.234 0.577 2.143 0.219 0.973 3.633 0.988 • 1.818 0. 306 1. 343 1.026 1. 429 2.000 0.394 1.561 1. 333 1. 667 1.951 0.481 1.774 1. 633 1.818 2.051 0.569 1. 902 1 .739 1.951 2.222 0.656 2.171 2.033 2. 222 2.500 0.331 2.498 2.286 2.581 2.857 0.919 2.824 2. 503 2. 963 3.333 RUN ND. = 24 Z/H V A VAW V A : V A S 0.377 1.066 1 .093 2.143 3.000 0. 142 1.467 1 .15'-. 1.552 2.045 0.219 1.510 1 . 404 1. 509 1 . 765 3.306 1.959 1 .813 2. 000 2.222 0. 3 9'. 2.359 2 .222 2. 400 2.609 0.43 1 2. 721 2 .503 2.857 3.000 0.569 3.114 2 .857 3. 333 3.333 0.656 3.445 3 .333 3. 478 3. 636 0.744 3.801 3 .555 3.918 4. 150 0.331 4. 114 3 .843 4. 270 4.444 0.9 19 4. 444 4 . 123 4. 593 4.873 RUM NO. 25 Z/H V A V A W V A : V A S 0.377 1.505 3 .933 1.452 2. 727 0.142 1. 092 0 .714 1.184 1.800 0.219 1.271 1 .025 1. 364 1.702 0.306 1.518 1 .355 1. 569 1.818 0.394 1.861 1 . 702 1.905 2. 162 0.481 2.034 1 .905 2. 051 2.308 0.569 2.338 2 . 105 2. 500 2.609 0.656 2.520 2 . 308 2. 609 2.857 0.744 2.733 2, . 503 2.857 3.077 0. 331 2. 944 2 .727 3. 000 3.333 RUN NO. = 21 Z/H ; A V A W V / A : V A S 0.377 1.331 3.714 1. 500 2 . 5 0 0. 142 1.222 0.732 1. 364 2.250 0.219 1.144 3.889 1. 333 1.481 0. 306 1.424 1.231 1. 569 1.667 0.394 1.636 1.455 1. 765 1.875 0.48 1 1.851 1.7 73 1. 875 2.000 0.569 2.089 2.003 2. 143 2.222 0.656 2.334 2.222 2. 400 • .500 0. 744 2. 532 2.424 2. 500 2.857 0. 331 2.672 2. 50 - 2. 727 3.000 RUM NO. = 26 Z/H VA VAW VAC VAS 0.374 1.973 3 .375 1. 500 5.000 0.123 1.258 3 .476 1. 200 2. 727 0. 172 1.030 3 .577 1. 000 2, 000 0.221 1.143 3 .693 1. 15<V 2.133 0.295 1.137 3 . 882 1.250 1.579 0.394 1.396 1 .175 1. 500 1. 765 0.492 1. 738 1 .509 1.875 2.069 0.591 2. 054 1 .951 2. 105 2.222 0.689 2.171 2 .003 2.286 2.400 0.787 2.302 2 . 105 2. 424 2. 581 RUM NO. = 23 Z/H VA VAW VA: VAS 0.077 1. 859 3. 963 2. 143 3. 333 0. 142 1. 795 1. 203 2. 000 2. 812 0.219 1 . 738 1. 333 1. 935 2. 353 0.306 2. 026 1. 739 2. 222 2. 353 0. 394 2. 139 1 . 863 2. 353 2. 424 0.481 2. 322 2. 222 2. 353 2. 500 0.569 2. 448 2. 353 2. 500 2. 581 0.656 2. 561 2. 503 2. 58 1 2. 667 0. 744 2. 710 2. 581 2. 667 3. 07 7 0.919 2. 953 2. 857 2. 857 3. 333 RUM NO. = 27 Z/H v/A VAW VAC VAS 0.366 1. 352 3. 833 0. 833 2.857 0. 109 1. 192 3. 933 0. 938 2.000 0.153 1. 173 3. 811 1.111 2.000 0.197 1. 135 3. 865 1.184 1.667 0.252 1. 217 3. 963 1. 333 1.607 0.323 1. 436 1. 224 1.538 1. 765 0.416 1. 783 1. 667 1.818 2.000 0.503 1. 985 1 . 875 2. 000 2.222 0.59 1 2. 109 2. 003 2. 143 2.308 0.578 2. 322 2. 143 2. 400 2. 609 0. 766 2. 522 2. 303 2. 609 2.875 0.353 2. 752 2 . 503 2. 857 3.158 0. 94 1 2. 835 2. 503 3. 000 3.333 245 RUM N3. = 28 Z / H VA' VAW VA- VAS 0. 355 1 . 579 3.8B9 I. 333 3. 333 0. 124 1. 263 1.003 1. 17o 2. 000 0. 173 1. 252 1 . 124 1. 282 1. 515 3. 242 1. 632 1.613 1. 639 1. 667 0.311 1 . 996 1 .923 2. 048 2. 083 0. 380 2. 198 2.123 2. 222 2. 32 6 0. 449 2. 484 2.439 2. 500 2. 564 0. 518 2 . 646 2.632 2. 632 2. 703 0. 587 2. 919 2 .857 2. 941 3. 030 0. 556 3. 125 3.125 3. 125 3. 125 0. 725 3. 409 3.226 3. 571 3. 571 0. 794 3. 781 3.5 71 3. 846 4. 167 0. 363 3. 907 3.663 4. 010 4. 310 0. 932 4. 152 3.793 4. 260 4. 810 RUM NO. = 30 Z / H VA 1 V Aw VAC VAS 0. 38 2 0. 764 3 . 632 0.714 1. 163 0. 137 0. 695 3 .5 32 0. 725 1. 020 0. 191 0. 84 7 3 . 763 0. 820 1. 099 0. 246 1'. 140 1 .087 1. 124 1. 299 0. 30 1 1. 443 1 . 409 1. 408 1. 587 0. 355 1. 755 I . 695 1. 786 1. 852 0. 410 1. 860 1 .813 1. 818 2. 041 0. 465 2. 197 2 .033 2. 222 2. 439 0. 519 2. 219 2 . 12S 2. 222 2. 439 3. 574 2. 503 2 . 381 2. 564 2. 703 0. 529 2. 661 ? . 503 2. 778 2. 857 0. 584 2. 817 2 .703 2. 857 3. 030 0. 738 2. 998 2 .857 3. 125 3. 125 o. 7 93 3. 201 3 .033 3. 226 3. 571 0. 348 3. 458 3 . 261 3. 448 3. 947 0. 902 3. 851 3 .571 3. 947 4. 348 RUM NO. = 29 Z /H VA VAW VAC VAS 0. 355 0.611 0. 272 0.441 1.500 0. 124 0.495 3.313 0.417 1.000 0. 193 0. 599 0 .444 0. 588 0. 976 0. 262 0.933 3.833 1. 000 1.053 0. 325 1.264 1.203 1. 277 395 0. 380 1.42 7 1 .373 1. 47 1 1. 493 0. 449 1. 524 1.471 1.563 1.587 0. 518 1.625 1.587 1. 639 1.695 0. 587 1.715 1.667 1. 724 1.818 0. 656 1.783 1. 72V 1. 786 1.923 0. 725 1.817 1 .695 1.887 2.000 0. 794 1. 966 1.923 1. 961 2.083 0. 363 2.015 1. 961 2. 000 2. 174 0. 932 2.173 2.003 2.222 2. 500 RUM NO. = 32 Z / H VA VAW V A : VAS 0-3 4 4 0.141 3 .0 Q..13 3. 0.400 0.398 0. 211 3 .093 0. 178 0.500 0.153 0.220 3 .132 0. 191 0. 460 0.208 0.252 3 . 169 0.231 0.482 0.246 0. 410 3 .285 0. 441 0. 658 0.301 0. 619 3 . 495 0. 641 0.885 0.355 0.897 3 . 803 0. 901 1. 136 0.410 1.139 1 . 087 1. 124 1. 299 0.465 1.307 1 .235 1. 232 1.538 0.574 1.511 1 .471 1. 538 1.563 0.529 1.557 1 .533 1. 533 1. 639 0.534 1.624 1 . 582 1.613 1. 754 0.793 1.715 1 .667 1.667 1.923 0.348 1.739 1 .667 1. 695 2.000 0.902 1.793 1 .72 4 1.754 2.041 RUM NO. = 33 Z / H VA VAW V A : VAS 0.344 0. 592 3 .253 0. 364 1.579 0.366 0.515 3 .315 0. 385 1.111 0. 120 0.476 3 . 317 0. 440 0.889 0.153 0.443 3 . 267 0. 426 0. 870 0. 175 0. 509 0 .404 0. 460 0.851 0.246 0. 360 3 . 763 0. 952 1.042 0.301 1. 120 1 .064 1. 149 1.205 0.355 1.363 1 .315 1. 370 1.449 0.410 1. 425 1 .389 1. 449 1.471 0.465 1.468 1 . 429 1.471 1.563 0.574 1.568 1 .533 1.613 1. 563 0.634 1.599 1 .563 1.613 1.667 0.738 1. 607 1 .563 1.639 . 1.667 0. 793 1. 657 1 .613 1. 667 1.754 0.848 1 .700 1 .639 1. 72 4 1.818 0.902 1.761 1 .667 1.786 1.961 EXECJTI DM TERMIMATED In the Spout 246 RUM NO. = 1 RUN NO. = 5 Z/H VS VAS VSM VSC Z/H VS VAS VSM VSC 0.366 2.037 2. 110 3. 965 5. 080 0.C98 2 . 230 2. 337 4.197 6 . 012 0. L09 2.221 2. 293 4. 278 6. 021 0.164 2. 289 2. 299 4. 66 5 5. 636 0. 175 1.964 2. 139 3. 586 4. 516 0.295 2. 023 2. 077 3.99 0 5. O i l 0.263 1.621 1. 823 2. 803 3. 780 0.492 1. 503 1 . 523 3.019 3. 7t>7 0.416 1.277 1. 534 1. 992 2. 540 0.623 1. 277 1. 345 2.473 2. 86 1 0.547 0.966 1. 113 1. 600 2. 183 0.820 1. 061 1. 123 2. 023 2. 434 0.766 0. 81 3 3. 923 1. 379 1 . 897 0.95 1 0. 929 0. 996 1. 720 2. 260 0.941 0. 724 0. 806 1. 295 1. 515 RUN NO. RUN NO. Z/H VS VAS VSM VSC Z/H VS VAS VSM VSC 0.366 2.266 2.322 4.516 5.419 0.098 2.609 2 .604 4. 807 6.410 0. 109 2.633 2.784 4.977 6.502 0. 164 2.653 2.908 4.674 6.903 0. 175 2. 500 2.673 4. 726 5.606 0.295 2.485 2. 746 4.41 1 5.85 3 0.263 2.161 2.255 4. 204 5.080 0.492 1.744 1 .870 3. 228 4.273 0.416 1.657 1.812 3.033 3.629 0.623 1.363 1.402 2. 671 3.452 0.547 1.212 1 .539 1.652 2.478 0.820 1.182 1 .265 2.218 2.736 0.678 1.017 1. 137 1. 793 2.241 0.951 1.098 1 . 197 2.031 2.337 0.766 0.957 1.095 1. 666 1. 767 0.941 0.805 3.903 1. 43 I 1. 626 RUN NO. 3 RUN NO. 7 Z/H VS VAS VSM VSC Z/H VS VAS VSM VSC 0.066 2.519 2.605 4.877 6.502 0.044 2.832 3. 127 4. 978 7.027 0. 109 2. 746 2 .988 4.753 8. 1 28 0.06 7 2.892 3.253 5. 06 2 6. 195 0.175 2. 761 3.079 4. 926 5.806 c m 2.912 3. 166 5. 332 6. 665 0.263 2.444 2. 721 4. 300 5.606 0. 166 2. 417 2 .651 4. 356 5 .608 0.394 2.019 2. 549 2. 822 3.92 7 0.211 2. 174 2 . 392 3. 967 4.570 0.547 1. 391 1 .677 2.167 2.674 0.277 1.743 2.033 2. 853 3. 703 0.678 1.128 1.286 1. 935 2.367 0.344 1.439 1.601 2. 541 3.253 0.788 1.098 1.226 1.954 2. 322 0.410 1.201 1 .261 2. 297 2.952 0.941 0.982 1. 127 1. 660 2.081 0.477 1.213 1 . 310 2.259 2.714 0.543 1.114 1 . 152 2. 205 2.55 1 0.632 0.943 0.935 1.959 2. 295 0.721 0.940 0 . 984 1. 827 2. 159 0.787 0. 865 0.924 1.62 6 2 .008 0.876 0.815 0. 854 1.588 1. 648 0.920 0.801 0.. 847 1. 540 1.783 RUN NO. 4 RUN NO. 8 Z/H VS VAS VSM VSC Z/H VS VAS VSM VSC 0.098 1.905 1 .843 3. 885 5.970 0.066 1.606 1 .619 3.274 3.824 0.164 2. 114 2.049 4.462 5. 551 0. 109 2.029 2.081 3.934 5. 204 0.295 1.857 1 .972 3. 521 4. 334 0.197 1.714 1 .777 3.311 4.415 0.492 1.444 1 .553 2. 652 3.605 0. 328 1. 537 1 .656 2.802 3.903 0.623 1.211 1. 305 2. 23 5 2.877 0.635 0.961 1. 143 1. 540 1.851 0.820 0.966 1 .072 1.713 2.207 0.766 0.741 0. 836 1.29b 1. 547 0.951 0.855 0.955 1. 506 1.89 7 0.897 0.706 0.831 1.157 1.374 247 RUN NO. = 9 Z / H VS VAS VSM VSC 0.093 1.405 1 . 285 3. 080 4. 355 0. 164 1.876 2. 125 3. 095 5. 012 0.295 1.582 1. 725 2. 833 3. 963 0.492 1.278 1 . 398 2. 329 O c • 8t>5 0.689 0.989 1. 134 1. 681 2 . C73 0.885 0.798 0. 915 1 . 3b2 1. 619 RUN NO. = 10 Z / H VS VAS VSM VSC 0.197 1. 141 1 .384 1. 689 2. 651 0.329 1. 293 1.459 2. 190 3. 167 0.461 1. 258 1.338 2. 337 3. 242 0.658 C . 963 1.056 1. 718 2. 363 0.921 0. 806 0.870 1. 481 1. 951 RUN NO. = 11 Z / H VS VAS VSM VSC 0.047 2.071 2.356 3. 445 5. 167 0.079 2. 501 2. 820 . 4. 142 6. 81c> 0 . 142 2.218 2.503 3. 791 5. 230 0.236 1.690 1 .780 3. 204 4. 215 0.331 1.443 1.556 2. 670 3. 372 0.425 1. 198 1.292 2. 190 2. 980 0.551 1.064 1 . 128 2. 002 2. 615 0.677 0.903 0.942 1. 725 2. 351 0 . e 0 3 0.814 0. 829 1. 616 2. 080 0.929 0.786 0.601 1. 575 1. 89S RUN NO. = 13 Z / H VS VAS VSM VSC 0.033 1.052 1 . 124 1.986 2. 3c'3 0.066 1.4 6C 1. 398 3. 078 4.160 0.109 1. 866 1 .998 3. 405 4.937 0.153 2. 207 2. 532 3. 721 4. 7 fcC 0.197 2.045 2 . 155 3.952 4 .5B5 0.241 1. 854 1. 969 3. 524 4.279 0.284 1.643 1. 762 3. 064 3.873 0.372 1.375 1.463 2. 549 3. 5j6 0.416 1. 156 1 .155 2. 346 3.019 0.460 1.086 1. 109 2. 144 2.311 0.503 1.065 1.092 2.086 2. 761 0.591 0.981 1.013 1.912 2.468 0.722 G. 830 0.910 1.717 2. 179 0.810 C. 859 0. 8 99 1 . 666 2.005 0.897 0.772 0.812 1 . 501 1.691 0.94 1 0.751 0. 748 1.434 1 .666 RUN NO. 14 Z / H VS VAS VSM VSC 0.033 1. 503 1.746 2.434 3 . 526 0.06b 1.923 2. 021 3.676 4. 70 1 0.109 2.464 2.754 4. 340 5.438 0.153 2. 706 3 . 059 4. 653 6. 099 0.197 2. 791 2.995 5.129 7. 052 0.241 2. 561 2.799 4. 665 5.786 0.284 2. 176 2. 500 3. 701 4.425 0.328 2.024 2.291 3. 499 4.382 0.372 1. 740 1 .829 3. 353 4.078 0.460 1.551 1 . 660 2.92 7 3. 482 0.503 1.431 1.476 2.813 3. 4 82 0.547 1.288 1 .428 2.22 3 3-353 0.591 1.256 1. 337 2.336 3.188 0.722 1.185 1 .272 2. 197 2.865 0.853 1. 124 1. 171 2.188 2.663 0.897 1.085 1.121 2. 126 2.663 0.985 1.008 1.025 2. 03 9 2.358 RUN NO. = 12 Z / H VS VAS VSM VSC 0.033 2.059 2 .285 3. 629 4.814 0.066 2.242 2 .223 4. 556 6.137 0.109 2.429 2 .418 4, 942 6.421 0.197 2. 320 2 .330 4. 68 3 6. 020 0.24 1 1.999 1 .967 4. 173 4.637 0.284 1.870 1 .920 3. 718 4.440 0.372 1.683 1 . 742 3. 321 3.913 0.416 1.557 1 .685 2. 828 3. 864 0.547 1.188 1 . 295 2. 136 2.924 0.635 1.014 1 .066 1. 903 2.693 0 . 6 7 e 0.982 1 . 021 1. 904 2.437 0. 766 0.956 0 . 907 1 . 889 2.245 0.853 0. 060 0 .853 1. 783 2.066 0.985 0.791 0 . 760 1. 720 1.861 RUN NO. = 15 Z / H VS VAS VSM VSC 0 .038 1. 787 1.910 3. 381 4 . 000 0 . 0 7 6 1. 989 2 .125 3. 660 5 . 168 0 .127 2 . 137 2 . 0 72 4. 505 5 . 621 0 . 1 7 8 2 . 118 2 .264 3. 922 5. 292 0 .228 1. 968 2 . 089 3. 722 4 . 685 0 . 330 1. 743 2 . 153 2. 565 3. 432 0 .431 1. 362 1 .491 2. 427 3 . 379 0 . 5 3 3 1. 193 1.302 2 . 0C6 2. 524 0 .635 1.036 1. 208 1. 711 2. 122 0 . 7 3 6 0. 951 1 . 108 1. 556 2 . 05 3 0 .838 0. 894 0 .988 1. 61 9 1. 889 0 . 8 8 8 0 . 827 0 .922 1. 472 1. 757 0 . 9 3 9 C . 787 0 .911 1. 34 2 1. 445 0 . 9 9 0 0. 745 0 . 848 1. 30 7 i .408 248 RUN NO. = 16 Z/H VS VAS VSM VSC 0.076 2.767 3. 137 4. 751 6. 101 0.127 2.830 2.912 5.371 6. 319 0. 170 3. 190 3.531 5. 603 6. 8t>3 0.228 2. 942 3.262 5.224 6. 655 0.279 2.669 3. 050 . 552 5. 631 0.330 2. 398 2.650 4. 371 4. e a o 0.431 1.917 2.034 3. 704 4. 0o7 0.533 1.462 1.674 2.427 3. 486 0.635 1.310 1 .420 2.413 2. 908 0.736 1. 175 1. 315 2. 043 2. 745 0.036 1.048 1 .126 1.938 2. 554 0.939 0.981 1 .046 1 . 839 2. 362 0.990 0.917 0. 991 1. 704 2. 053 RUN NO. = 17 Z/H VS VAS VSM VSC 0.076 1.406 1 .618 2. 392 2.819 0.127 1.710 1 .935 3. 036 3. 184 0.228 1.550 I .707 2. 799 3. 438 0.381 1.301 1 .518 2. 124 2.814 0.584 0. 959 1 .012 1.815 2.307 0.736 0.919 0 .963 1.774 2.168 0.939 0.755 C .805 1 . 430 1.715 RUN NO. = 18 Z/H VS VAS VSM VSC 0.066 1.49 6 1 .828 2. 310 2.594 0.109 1.741 2 . 1 16 2. 680 3.260 0.153 1.645 1 . 7 04 3. 232 3. 574 0.197 1. 597 1 .711 3. 046 3. 327 0.328 1.351 1 .503 2. 403 2.913 0.460 1.151 1 . 245 2. 164 . .400 0.591 1 .006 1 . 044 1. 977 2. 354 0.722 0.955 1 . 01r- 1. 828 2.116 0.897 0. 872 0 .939 1. 628 1.976 RUN NO. = 20 Z/H VS VAS VSM VSC 0.066 1. 867 2 . 337 2. 700 3. 421 0. 109 2. 115 2. 470 3. 480 4, 32 0 0.197 2. 099 2. 302 3. 654 4. 263 0.284 1. 974 2. 189 3. 553 4. 154 0.416 1. 712 1 . 884 3. 068 3. 971 0.547 1. 483 1. 5 58 2. 867 3. 432 0.678 1. 136 1. 196 2. 139 2. 634 0.8 10 0. 970 1. 044 1. 806 2. 229 0.94 1 0. 828 0. 096 1. 514 2. 000 RUN NO. = 21 Z/H VS VAS VSM VSC 0.104 2. 714 3. 044 4. 736 6. 089 0.153 2. 709 3. 074 4. 63 3 6. 031 0.197 2. 547 2 . 860 4. 427 5. 24 0 0.328 2. 047 2. 328 3. 532 4. 262 0.460 1. 618 1. 903 2. 629 3. 284 0.591 1. 262 1. 427 2. 1 79 2. 78 0 0.722 1. 025 1 . 090 1. 942 2. 366 0.853 0. 862 0. 925 1. 598 2. 065 0.941 0. 762 0. 812 1. 41 7 1. 926 RUN NO. = 23 Z/H VS VAS VSM VSC 0.066 2. 665 2 . 665 5 . 449 6. 659 0. 109 3. 046 3. 262 5. 609 7. 733 0.197 2. 795 2. 933 C -> • 400 6. 445 0.328 2. 174 2. 316 4. 098 5. 114 0.460 1. 698 • 1. 885 3. 016 3. 805 0.591 1. 268 1. 442 2. 145 2. 933 0.722 1. 029 1. 175 1. 756 2. 185 0 . e 5 3 0. 887 0. 999 1. 537 1. 976 0.941 C. 831 0. 970 1. 368 1. 712 RUN NO. = 19 Z/H VS VAS VSM "SC 0.066 1 .734 2 . 066 2. 813 3. 076 0.109 1 .607 2 . 040 3. 115 4. 050 0.153 1 .839 2 .177 2. 935 3. 821 0.24 1 1 . 764 2 . 088 2. 817 3. 662 0.372 1 .609 1 .841 2. 733 3. 447 0.503 1 .338 1 . 546 2. 164 3. 135 0.635 1 . 184 1 .332 2. 077 2. 520 0.766 1 .064 1 .203 1. 84 1 2. 314 0.897 0 .935 1 .077 1 . 558 2. 051 RUN NO. = 24 Z/H VS VAS VSM VSC 0. 109 1.481 1 .776 2. 233 3. 212 0. 153 1.428 1.598 2. 49 7 3. 222 0. 197 1.400 1 .500 2. 595 3. 275 0. 328 1. 146 1. 223 2. 122 2. 8 95 0. 460 0.899 0. 959 1. 6 79 2. 169 0. 59 1 0.804 0. 864 1. 44 8 1. 8 37 0. 722 0.721 0.815 1. 24 9 1. 577 0. 853 0.652 0.694 1. 236 1 . 506 249 RUN NO. = 25 Z/H VS VAS VSM VSC 0.066 0. 109 0.153 0.197 0.284 0.416 0.547 0.678 0.810 0.941 1. 172 1.444 1.291 1.167 1.'080 0. 975 0. 840 0.722 0.692 0.588 1 . 249 1 .567 1. 362 1. 303 1 . 158 1.074 0.970 0.7 93 0.759 0.639 2. 220 2. 664 2.497 2. 060 2.017 1.737 1. 390 1. 300 1. 249 1. 056 2.664 3.240 2.854 2. 578 2. 524 2. 305 1 .885 1.648 1.582 1.440 RUN NO. 26 Z/H VS VAS VSM VSC 0.074 0.123 0. 172 0.271 0.419 0.567 0.714 0.662 1.635 1.826 1. 683 1.401 1.114 0.889 0. 733 0. 658 1.816 2. 103 1 .949 1. 586 1. 164 0. 884 0.774 0.651 2.934 3.016 2. 756 2. 350 2. 145 1.833 1. 378 1. 354 3.474 4. 206 3.842 3.513 2. 724 2.220 1.858 1.700 RUN NO. 27 Z/H VS VAS VSM VSC 0.109 0. 153 0.197 0.328 0.460 0.591 0.722 0.853 1.893 1. 751 1. 568 1.376 1. 166 0.972 0.821 0.731 1 .998 1 . 776 1.700 1.491 1.249 1.025 0.929 0. 799 3. 552 3. 526 2. 854 2. 550 2. 189 1. 83 7 1. 402 1.31 7 4.918 4.262 3.805 3.074 2.664 2.459 i . 880 1.737 RUN NO. 31 Z/H VS VAS VSM VSC 0.109 1.698 1 .985 2. 622 4.595 0.153 2.284 2 .741 3. 422 5.546 0.241 2.623 2 .726 5. 155 6.031 0.372 2.472 2 .512 5.026 5.610 0.503 2.248 2 . 173 4. 844 5. 3t>l 0.635 1. 826 1 .828 3. 740 4.467 0.766 1. 595 1 . 546 3. 386 4. 061 0.897 1.363 1 .305 2. 898 3.723 RUN NO. = 2 8 Z/H VS VAS '• VSM VSC 0.069 2.996 3 .407' •' 5 . 065 6. 955 0. 124 2. 876 3. 277 4. 825 6. 772 0.180 2.704 3. 1 32 4. 394 6. 390 0.235 2. 539 2. 893 4. 232 6. 186 0.3 18 2.245 2.533 3. 829 5. 361 0.373 1.952 2. 109 3. 541 4. 94 8 0.456 1.824 1.931 3. 342 4. 218 0.539 1.488 1.621 2. 694 3. 574 0.622 1. 364 1 .425 2. 645 3. 217 0.704 1.254 1. 314 2. 43 7 2. 852 0.787 1.074 1. 069 2. 218 2. 615 0.870 1.013 1 .018 2. 075 2. 418 0.953 0.920 0.922 1. 859 2. 400 RUN NO. = 29 Z/H VS VAS VSM VSC 0.069 3. 389 3. 923 5. 643 7. 148 0.097 3. 320 3. 740 5. 744 7. 480 0.152 3. 042 3. 323 5. 573 6. 701 0.235 2. 794 3. 242 4. 514 6. 655 0.318 2. 388 2. 680 4. 177 5. 251 0.40 1 2. 035 2. 400 3. 257 4. 361 0.483 1. 598 1 . 870 2. 553 3. 729 0.566 1.478 1. 711 2. 437 3. 282 0.649 1. 265 1. 363 2. 331 3. 063 0.732 1. 138 1. 133 2. 320 2. 978 0.815 1. 075 1 . 058 2. 203 2. 978 0.898 I. 004 1. 005 2. 010 2. 773 0.981 0. 974 0. 973 1. 96 1 2. 636 RUN NO. = 30 Z/H VS VAS VSM VSC 0.055 2. 146 2 .365 3. 971 4. 08 9 0.098 2. 686 2 .831 5. 188 6. 031 0.142 2. 649 2 . 594 5. 643 6. 127 0.208 2. 513 2 .513 5. 138 5. 956 0.274 2. 373 2 .342 4. 943 5. 840 0.339 2. 170 2 . 139 4. 530 5. 361 0.405 1. 919 1 . 870 4. 021 5. 045 0.470 1. 742 1 . 729 3. 594 4. 361 0.536 1. 573 1 .583 3. 191 4. 072 0.624 1. 520 1 .517 3. 100 3.899 0.711 1. 349 1 . 340 2. 765 3. 477 0.799 1. 291 1 .256 2. 721 3. 316 0.886 1. 269 1 .233 2. 712 3. 06 3 0.952 1. 220 1 .200 2. 561 2. 978 250 RUN NO. = 32 Z/H VS VAS VSM VSC 0.077 2. 70 4 3 .217 4. 232 5. 884 0.120 3. 040 3 . 282 5. 643 6. 932 0. 186 3. 064 3 .422 5. 45 2 6. 433 0.252 2. 590 2 . 726 5. 026 5. 744 0.317 2. 349 2 .513 4. 406 5. 475 0.383 2. 096 2 .234 3. 971 4. 786 0.470 1. 818 1 . 892 3. 574 4. 124 0.613 1. 613 1 . 658 3. 185 3. 942 0.667 1. 435 1 .4 62 3. 093 3. 74 0 0.733 1. 349 1 . 289 2. 924 3. 386 0.821 1. 266 1 .226 2. 680 3. 282 0.908 1. 187 1 . 105 2. 636 3. 093 0.974 1. 145 1 . 031 2. 493 3. 046 RUN NO. = 33 Z/H VS VAS VSM VSC 0.055 3.884 4.639 6. 12 7 7. 807 0.098 4. 149 4. 662 7. 310 8. 693 0. 142 4. 215 4.595 7. 658 9. 897 0.186 4. 138 4. 569 7. 480 &. 8 73 0.252 3. 577 4.021 6. 215 8. 041 0.317 3. 40 5 3.740 6. 186 7. 48 0 0.383 2. 740 2 .951 5. 026 6. 772 0.449 2. 286 2.474 4. 177 5. 546 0.470 2. 158 2. 365 3. 875 5. 146 0.556 1. 969 2. 127 3. 655 4. 46 7 0.624 1. 694 1 .748 3. 316 • 4. 218 0.689 1. 518 1.546 3. 006 3. 923 0.755 1. 396 1.336 2. 846 3. 740 0.842 1. 337 1.323 2. 773 3. 351 0.930 1. 252 1 .255 2. 577 2. 935 0.996 1. 130 1. 133 2. 297 2.872 EXECUTION TERMINATED 

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