Two-phase turbulent mixing for air-water flows in adjacent triangular array subchannels.

University of Windsor University of Windsor

Scholarship at UWindsor

Scholarship at UWindsor

Electronic Theses and Dissertations Theses, Dissertations, and Major Papers

1-1-1970

Two-phase turbulent mixing for air-water flows in adjacent

Two-phase turbulent mixing for air-water flows in adjacent

triangular array subchannels.

triangular array subchannels.

Kaziu F. Rudzinski

University of Windsor

Follow this and additional works at: https://scholar.uwindsor.ca/etd

Recommended Citation Recommended Citation

Rudzinski, Kaziu F., "Two-phase turbulent mixing for air-water flows in adjacent triangular array subchannels." (1970). Electronic Theses and Dissertations. 6626.

https://scholar.uwindsor.ca/etd/6626

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TWO-PHASE TÜBBÜLEÎIT MIXING FOR AIR-¥ATER FLOWS IN ADJACENT TRIANGULAR ARRAY SUBCHANNELS

by

K azlu F . R u d z in s k i

A T h e s is

S u b m itte d t o t h e F a c u lty o f G ra d u a te S t u d i e s th r o u g h th e D ep artm en t o f C hem ical E n g in e e r in g i n P a r t i a l F u l f i l m e n t

o f th e R e q u ire m e n ts f o r t h e D eg ree o f M a s te r o f A p p lie d S c ie n c e a t t h e

U n i v e r s i t y o f W indsor

W indsor, O n ta r io

1970

UMI Number: EC52833

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ERRATA

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[ 1 6 ] f o u n d

1 0 D = h y d r a u l i c d i a m e t e r ( f t ) 1 2 H e t s r o n i e_t [ 2 4 ]

1 2 Rowe a n d A n g l e [ 1 5 , 2 0 ] h a v e f o u n d

14 R o g e r s [ 2 2 ] h a s

16 t h i s t r a n s i t i o n p o i n t . T h e d a t a o f

P e t r u n i k [ 1 8 ] a n d U a l t o n [ 1 ] s h o w e d t h a t

2 2 b u b b l e f l o w c a n o c c u r [ 4 3 ]

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26 F r o m a v a i l a b l e e x p e r i m e n t a l e v i d e n c e [ 2 9 ]

26 S t e e n ' s [ 3 5 ] c r i t e r i a ^

3 7 T h e a i r p r e s s u r e w a s

4 8 p r e v i o u s i n v e s t i g a t o r s [ l y 1 5 , 1 8 , 2 0 , “ 4 6 ]

54 T h e f l o w s a n d q u a l i t i e s w e r e c a r e f u l l y 66 4 . S h e r w o o d , T . K . , H e a t T r a n s f e r , M a s s

T r a n s f e r a n d F l u i d F r i c t i o n R e l a t i o n s h i p s i n T u r b u l e n t F l o w , I n d . & E n g . Ch e m. 4 2 , N o . 1 0 , pp 2 0 7 7 - 2 0 8 4 , ( 1 9 5 0 ) .

66 1 3 . R o g e r s , J . T . , a n d T a r a s u k , W . R . , A G e n e r a l i z e d C o r r e l a t i o n f o r N a t u r a l

T u r b u l e n t M i x i n g o f C o o l a n t i n F u e l B u n d l e s , P r e s e n t e d A . N . S . M e e t i n g , T o r o n t o , O n t a r i o ,

( 1 9 6 8 ) .

6 7 1 4 . B o w r i n g , R . W. , HAMBO, A C o m p u t e r P r o g r a m f o r t h e S u b c h a n n e l A n a l y s i s o f t h e H y d r a u l i c a n d B u r n o u t C h a r a c t e r i s t i c s o f Rod C l u s t e r s ,

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A t o m i c E n e r g y E s t a b l i s h m e n t , W i n f r i t h , D o r s e t , A E E H - R - 5 8 2 ( 1 9 6 8 ) .

6 7 . . 2 6 . Va n D e r R o s , T . , a n d B o g a a r d t , M . , On t h e M a s s a n d E n e r g y E x c h a n g e B e t w e e n I n t e r a c t i n g W a t e r C h a n n e l s , T e c h n o l o g i c a l U n i v e r s i t y o f E i n d h o v e n , R e p o r t R e q ' d ( 1 9 6 9 ) .

.

6 7 2 7 . I n g e r s o n , L . , H e a t T r a n s f e r B e t w e e n

S u b c h a n n e l s i n a Rod B u n d l e , A E - R L - 1 1 2 5 , ,

S w e d e n , ( 1 9 6 9 ) . . . . . .

. . . 6 8 . . ; . . . 40 . . Owe n s , W. C . , T w o - P h a s e P r e s s u r e G r a d i e n t , I n t e r n a t i o n a l D e v e l o p m e n t i n H e a t T r a n s f e r , P T - I I , pp 3 6 3 - 3 6 8 , ASME ( 1 9 6 1 ) .

6 7 4 7 . H e w i t t , G . E . , K i n g , R . D . , a n d L o v e g r o v e , P..C '. , T e c h n i q u e s f o r L i q u i d F i l m a n d P r e s s u r e D r o p S t u d i e s i n A n n u l a r T w o - P h a s e F l o w , ,. A E R E - R - 3 9 2 1 , H a r w e l l , E n g l a n d ( 1 9 6 2 ) .

83 - T r a c e r C o n c e n t r a t i o n ( p p m ) ' '

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S 2 1 9 1 5

ABSTRACT

T u r b u le n t in te r c h a n g e r a t e s b etw ee n two a d j a c e n t t r i a n g u l a r

s u b c h a n n e ls w ere m easured f o r tw o -p h a se a i r - w a t e r f lo w s . The su b c h a n n e ls w ere p h y s i c a l l y s e p a r a te d b y an 0 .0 0 6 i n s t a i n l e s s s t e e l s t r i p

e x c e p t i n t h e m ix in g s e c t i o n , w here a s l o t I S i n b y 0.04- i n w ide (L/Dg=115) was c u t . M ethane and p o ta s s iu m n i t r a t e w ere em ployed a s t r a c e r s

f o r a i r and w a te r r e s p e c t i v e l y . A g as chrom ato g ra p h and an a to m ic a b s o r p t i o n u n i t w ere u s e d f o r g as and l i q u i d a n a ly s e s r e s p e c t i v e l y .

Tw o-phase a .ir - w a te r t u r b u l e n t i n t e r c h a n g e r a t e s w ere d e te rm in e d a t a sy stem p r e s s u r e o f 50 p s i a o v e r a m ass f l u x r a n g e o f 0 .5 z 10^ t o 1 .5 X 10^ ( l b m ) / ( h r ) ( f t ) an d a q u a l i t y * ra n g e o f 0.01% t o 10%. I n a d d i t i o n t h e tw o -p h a se p r e s s u r e d ro p was r e c o r d e d a s w e ll a s v o id f r a c t i o n d a t a .

The a i r and w a te r m ix in g r a t e s showed maximas a t 6% q u a l i t y f o r t h e a i r an d 0.7% q u a l i t y f o r th e w a te r . The a i r and w a te r m ix in g r a t e s in c r e a s e d a s th e q u a l i t y and m ass f l u x i n c r e a s e d . The m ix in g r a t e s w ere fou nd t o b e a f u n c t i o n o f t h e s u b c h a n n e l q u a l i t y and w ere flo w reg im e d e p e n d e n t. T h is was i n q u a l i t a t i v e a g re e m e n t w ith p r e v io u s w o rk s. At low q u a l i t i e s t h e l i q u i d in t e r c h a n g e was en h an ced , w ith a g r a d u a l change t o t h e o p p o s ite e f f e c t i n t h e in t e r m e d i a t e q u a l i t i e s , and th e n e q u a l m ix in g a s homogeneous flo w c o n d i tio n s w ere a p p ro a c h e d .

e x i s t e d i n t h e a d j a c e n t f lo w a r e a s .

I t was n o te d t h a t f o r t h e above c o n d i tio n s t h e r e was a n e t flo w o f a i r from t h e h ig h t o t h e lo w q u a l i t y s u b c h a n n e l. S i m i l a r l y t h e r e was a n e t flo w o f w a te r from t h e low q u a l i t y t o t h e h ig h q u a l i t y

s u b c h a n n e l. T h is t r a n s f e r o f m ass (g a s o r l i q u i d ) was te rm e d t h e t u r b u l e n t m ass flo w w ith t h e sym bol, j ^ (ib m /h r ) o r t h e t u r b u l e n t m ass

flu x ;, (ib m /h r f t ^ ) u s i n g t h e gap i n t e r c o n n e c t i o n s p a c e a s t h e

c o n t r o l a r e a . The w a te r t u r b u l e n t m ass f l u x was fo u n d t o b e p r o p o r t i o n a l t o a mean d r i v i n g f o r c e , ^ ( 1 - ( X ) , t h a t e x i s t e d b etw ee n t h e s u b c h a n n e ls . T h is was a l s o t h e c a s e f o r a i r u p t o a A o f 8%. Any f u r t h e r i n c r e a s e i n t h e mean d r i v i n g f o r c e d i d n o t r e s u l t i n h ig h e r a i r t u r b u l e n t

m ass f l u x e s .

The t u r b u l e n t v o lu m e tr i c f l u x e s , ( f t ^ / h r f t ^ ) , gave

i d e n t i c a l sh ap ed p l o t s a s t h e t u r b u l e n t m ass f l u x e s , J ^ ( l b m /h r f t ^ ) , b u t w ith t h e a i r t u r b u l e n t v o lu m e tr i c f l u x a p p r o x im a te ly a n o r d e r o f m a g n itu d e g r e a t e r . P ic k s F i r s t la w o f d i f f u s i o n was u s e d t o d e f i n e a m ass t r a n s f e r c o e f f i c i e n t , k ^ .

U sin g a s i m p l i f i e d t r a c e r m ass b a l a n c e t h e t u r b u l e n t m ix in g + 4

-r a t e s , ¥ g and w ere p r e d i c t e d when a i r an d w a te r t u r b u l e n t m ass flo w s w ere t a k i n g p l a c e . The a i r and w a te r t u r b u l e n t m ix in g r a t e s w ere fo u n d t o b e s u r p r e s s e d u s i n g t h i s m o d e l.

I V

ACKNOWLEDGEMENTS

The a u t h o r w is h e s t o e x p r e s s h i s g r a t i t u d e t o D r. G.G. S t . P i e r r e f o r h i s a s s i s t a n c e an d a d v ic e i n t h i s s t u d y .

The e x p e r im e n ta l work f o r t h i s t h e s i s was p e rfo rm e d a t th e U n i v e r s i t y o f W indsor, C hem ical E n g in e e r in g L a b o r a t o r i e s . Thanks a l s o g o es t o K. P e t r u n i k an d K. S in g h f o r t h e i r g u id a n c e i n e x p e r im e n ta l te c h n iq u e . S p e c i a l th a n k s g o es t o G . Eyan f o r h i s t e c h n i c a l a s s i s t a n c e i n b u i l d i n g n eed e d e q u ip m e n t f o r t h i s s tu d y .

TABLE OF CONTENTS

Page

ABSTRACT i ü

ACKNOWLEDGEMENTS ?

TABLE OF CONTENTS "vl

LIST OF FIGURES i x

LIST OF TABLES %

I INTRODUCTION 1

I I LITERATURE SURVEï 6

A. I n t r o d u c t i o n an d D e f i n i t i o n o f Terms 6

1 . N a tu r a l M ixin g E f f e c t s 6

2 . F o rced M ixing E f f e c t s 7

B. S in g le C hannel E x p e rim e n ts 7

C. M u lti-C h a n n e l (Rod B u n d le ) E x p e rim e n ts 8

D. Two C hannel E x p e rim e n ts 10

E. S in g le P h ase T u r b u le n t M ixing C o r r e l a t i o n s 12

1 . S t . P i e r r e C o r r e l a t i o n 12

2 . Bow ring C o r r e l a t i o n 13

3 . R og ers and T a ra su k C o r r e l a t i o n 14

4 . Rowe an d A ngle C o r r e l a t i o n 14

5. Moyer C o r r e l a t i o n 14

6 . S k in n e r e;b a l C o r r e l a t i o n 15

7 . I n g e r s o n C o r r e l a t i o n 15

F . Two-Phase T u r b u le n t I n te r c h a n g e D a ta 16

I I I THEORY 18

A. D e r iv a t io n o f M ixing E q u a tio n s 18

V I

Page

1 . B a s ic A ssu m p tio n s 18

2 . S im ple M ix ing Model 19

3 . D i f f e r e n t i a l M ix in g M odel 20

B . I d e n t i f i c a t i o n o f Flow Regim es 21

1 . B u b b ly -S lu g T r a n s i t i o n 22

2 . S lu g -A n n u la r T r a n s i t i o n 24

IV AIR-WATER TEST LOOP AND ASSOCIATED EQUIPMENT 27

A. E x p e rim e n ta l E quipm ent 27

1 . A ir-W a te r T e s t Loop 2 7

2 . T e s t S e c tio n A ssem bly 29

(a ) I n j e c t i o n - E n t r a n c e S e c t io n 31

(b ) Flow D evelopm ent S e c t io n 31

(c ) S e p a r a te d S u b c h a n n e ls S e c t io n 31

(d ) M ixin g S e c t io n 31

(e ) S e p a r a te d S u b c h a n n e l S e c t i o n 31

( f ) E x i t S e c t io n 33

B. M easurem ent o f E x p e r im e n ta l V a r ia b l e s 33

1 . P r e s s u r e M easurem ents 33

2 . A ir an d W ater Flow s 35

(a ) A ir Flow s 35

(b) W ater Flow s 36

3 . V oid F r a c t i o n 37

A. T em p eratu re 3 7

5. T r a c e r C o n c e n tr a tio n s 37

Page

V EXPEKOyiEKTAL PROCEDURE 39

VI RESULTS AND DISCUSSION 41

A. A ir-W a te r T u r b u le n t M ix in g D ata 41

1 . V a r i a t i o n o f Tw o-Phase M ix in g R a te s w ith

Q u a li ty an d V oid F r a c t i o n 41

2 . V a r i a t i o n o f Tw o-Phase M ix in g R a te s w ith

Mass F lu x 45

3 . C om parison o f S u b c h a n n e l Q u a l i t y w ith th e

T u r b u le n t M ixin g Q u a li ty 45

4 . D is c u s s io n o f T u r b u le n t M ix in g D ata 48 B. C om parison o f P r e s s u r e Drop P r e d i c t i o n s w ith

E x p e rim e n ta l D ata 51

C. V oid F r a c t i o n ' 54

D. U n eq u al Mass F lu x a n d Q u a li ty 54

1 . V a r i a t i o n o f T u r b u le n t Mass F lu x w ith I n t e r -

S u b ch an n el Mean Volume V oid F r a c t i o n D if f e r e n c e 56

2 . T u r b u le n t M ix in g R a te s w ith U neq ual Mass

F lu x an d Q u a li ty 58

V II CONCLUSIONS 61

A. E q u a l Mass F lu x and Q u a li ty 61

B. U n eq u al Mass F lu x and Q u a li ty 61

NOMENCLATURE 63

REFERENCES 66

APPENDIX I 70

APPENDIX I I 74

APPENDIX I I I 81

VITA AUCTOEIS 85

v x i i

LIST OF FIGURES

F ig u r e Page

1 .1 R e a c to r F u e l Rod B u n d le S u b c h a n n e ls 2

1 .2 D e t a i l o f Flow C h an n els 3

3 .1 M ixing Flow D iagram 19

3 . 2 Flow Regime D iagram s 21

4 . 1 S ch em atic Flow D iagram o f T e s t Loop 28

4 . 2 T e s t S e c tio n A ssem bly 30

4 .3 M ixing S l o t 32

4 . 4 P r e s s u r e Tap L o c a tio n 34

6 .1 P l o t o f A ir an d W ater M ix in g R a te s v e r s u s Q u a lity 42 6 .2 P l o t o f A ir M ix in g R a te s v e r s u s V oid F r a c t i o n 44

6 .3 P l o t o f W ater M ixing R a te s v e r s u s V oid F r a c t i o n 46

6 . 4 P l o t o f T u r b u le n t M ix in g Q u a l i t y v e r s u s S u b ch an n el

Q u a lity 47

6 .5 P l o t o f T u r b u le n t M ix in g V o lu m e tric Q u a l i t y v e r s u s

S u bchan nel V o lu m e tric Q u a l i t y 49

6 .6 Com parison o f M a r t i n e l l i - L o c k h a r t Model w ith O b serv ed

P r e s s u r e G r a d ie n t 52

6 . 7 Com parison o f Owens Homogeneous Model w ith O bserved

P r e s s u r e G r a d ie n t 53

6 . 8 P l o t o f V oid F r a c t i o n v e r s u s S u b ch an n el Q u a li ty 55

6 .9 P l o t o f T u r b u le n t Mass Flu:-: v e r s u s Mean Volume V oid

F r a c ti o n D if f e r e n c e 57

6 .1 0 M ixing Flow D iagram 59

LIST OF TABLES

T able Page

2 .1 D e s c r i p t i o n o f D a ta A v a il a b le on Two S u b ch an n el

E x p e rim e n ts 11

4 .1 A ir R o ta m e te r - Flow Range 35

4 .2 W ater R o ta m e te r - Flow Range 36

4 .3 A ir R o ta m e te r - Flow Range 36

5 .1 Tw o-Phase A ir-W a te r E q u a l Mass F lu x P a r a m e te r

Range 40

5 .2 Two-Phase A ir - V a te r U n eq u al M ass F lu x P a ra m e te r

Range 40

6 .1 C om parison o f Tw o-Phase T u r b u le n t M ix in g D a ta

w ith Two-Phase T u r b u le n t M ix in g D a ta when T u r b u le n t

Mass F lu x was P r e s e n t 60

1 1 . 1 Two-Phase A ir-W a te r M ix in g D ata (L/Dg = 115) 76 1 1 .2 Tteo-Phase A ir-W a te r M ix in g D ata (L/Dg = 115) 77 1 1 .3 C om parison o f T h e o r e t i c a l P r e s s u r e D rop M odels w i t h

E x p e rim e n ta l D a ta (L/Dg = 115) 78

1 1 .4 Two-Phase T u r b u le n t M ix in g F a c t o r s 79 1 1 . 5 Tbro-Phase T u r b u le n t M ix in g F a c t o r s 80

1 1 1 .1 U nequal Mass F lu x an d Q u a li ty D ata 82

1 1 1 .2 U nequal Mass F lu x an d Q u a l i t y D ata 83 1 1 1 .3 U nequal Mass F lu x an d Q u a l i t y D ata 84

1 . INTRODUCTION

The h e a r t o f a n u c l e a r pow ered r e a c t o r i s i t s f u e l b u n d le s w h ic h c o n s i s t o f a p a r a l l e l m a t r i x o f r o d s fo rm in g i n t e r c o n n e c t e d s u b c h a n n e ls ’' a s shown i n F i g . 1 . 1 . The p r e d i c t i o n o f c r i t i c a l h e a t f l u x i n o r d e r t o d e s ig n r e a c t o r s o f h ig h e r th e r m a l e f f i c i e n c y h a s n e c e s s i t a t e d t h e d e t e r m i n a t i o n o f th e e f f e c t s w hich a c c o u n t f o r t h e in t e r c h a n g e o r m ix in g o f c o o l a n t b e tw e e n b u n d le s u b c h a n n e ls . W ith t h e a d v e n t o f b o i l i n g w a te r r e a c t o r s t h e s tu d y o f c o o l a n t m ix in g h a s expan ded t o i n c l u d e tw o -p h a se l i q u i d v a p o u r s y s te m s .

T h is s tu d y c o n s i s t e d o f two m a jo r p a r t s . The f i r s t was c o n c e rn e d w ith t h e m easu rem en t o f t u r b u l e n t m ix in g r a t e s b etw ee n two i d e n t i c a l a d j a c e n t f lo w c h a n n e ls i n a t r i a n g u l a r a r r a y when d i v e r s i o n c r o s s - f l o w and f o r c e d m ix in g e f f e c t s w ere a b s e n t . The sec o n d p a r t w as c o n c e rn e d w i t h t h e m easurem ent o f t r a n s p o r t r a t e s i n t h e same t e s t s e c t i o n when t h e a d j a c e n t s u b c h a n n e ls i n i t i a l l y c a r r i e d u n e q u a l m ass f l u x an d q u a l i t y .

The t e s t s e c t i o n u s e d f o r t h i s s tu d y was d e s ig n e d and c o n s t r u c t e d b y W a l t o n [ l ] .

T y p ic a l f u e l r o d b u n d le m a tr ix

/ \

/ \

/

V

T r i a n g u l a r s u b c h a n n e l a r r a y

F ig u r e 1 . 1 . R e a c to r F u e l Rod B u nd le S u b c h a n n e ls

DETAIL OF FLOW CHANNELS

( S C A L E : 5 X )

%

CROSS SECTIONAL VIEW

( S C A L E ; F U L L S I Z E )

STAINLESS STEEL DIVIDING STRIP

SUBCHANNEL DIMENSIONS c / s AREA 3 - 6 5 0 x

W E T T E D PE RIMETER I I I 6 x I0” f1. e q u i v a l e n t h y d r a u l i c

DIAMETER 1 - 3 0 8 x 10'" ft. PITCH T O D IA M E TE R RATIO 1 - 0 5

%6 'O'-RING GROOVE /a ‘O'-RING GROOVES

A. Two a d j a c e n t i d e n t i c a l flo w a r e a s , 8 0 - in lo n g (s e e F ig , 1 .2 )

w ere s e p a r a t e d , e x c e p t f o r an i n t e r c o n n e c t i o n l e n g t h , b y an 0 .0 0 6 - i n s t a i n l e s s s t e e l s t r i p . The m ix in g s e c t i o n , w here t h e s u b c h a n n e ls w ere i n t e r c o n n e c t e d , was 1 8 - i n lo n g (L/Dg = 115) and 0.04- i n w id e . A tw o -p h a s e tw o-com ponent sy stem was em ployed u s i n g p o ta s s iu m n i t r a t e an d m ethane a s w a te r an d a i r t r a c e r s r e s p e c t i v e l y . I n i t i a l l y , t u r b u l e n t in t e r c h a n g e r a t e s w ere m e asu re d w h ile m a in ta i n in g an

o p e r a t i n g p r e s s u r e o f 50 p s i a a t t h e m id - p o in t o f t h e i n t e r c o n n e c t i o n l e n g t h . The t o t a l m ass f l u x was v a r i e d from 0 .5 % 10^ t o 1 .5 x 10°

( l b f f l ) / ( h r ) ( f t ^ ) and t h e q u a l i t y ra n g e d from 10^ t o

0.002%

w ith c o r r e s p o n d in g volum e v o id f r a c t i o n s o f a p p r o x im a te ly 80^ t o 9 ^ . The tw o -p h a se flo w re g im e s e n c o u n te re d e x te n d e d from t h e b u b b le - s l u g t r a n s i t i o n a t t h e lo w q u a l i t i e s t o t h e s l u g - a n n u l a r t r a n s i t i o n a t t h e h ig h q u a l i t i e s .

I n t h e seco n d p a r t o f t h i s i n v e s t i g a t i o n t u r b u l e n t t r a n s p o r t r a t e s b e tw e e n s u b c h a n n e ls w ere m e asu re d a t o p e r a t in g p r e s s u r e s J u s t above a tm o s p h e ric p r e s s u r e w ith u n e q u a l q u a l i t y and m ass f l u x e x i s t i n g i n i t i a l l y i n e a c h s u b c h a n n e l. The m ass f l u x and q u a l i t y w ere s e l e c t e d to g iv e th e same p r e s s u r e d ro p p e r

u n i t a x i a l l e n g t h , so t h a t t h e p r e s s u r e i n e a c h flo w c h a n n e l a t th e b e g in n in g o f th e m ix in g r e g io n w ere e q u a l . T hese c a r e f u l l y

ch o sen m ass f l u x e s and q u a l i t i e s gave a t t h e s t a r t o f t h e i n t e r c o n n e c t i o n l e n g t h a t l e a s t a 10^ volum e v o id f r a c t i o n d i f f e r e n c e b etw ee n

s u b c h a n n e ls . The m ass f l u x was v a r i e d from 0 . 6 x 10^ t o 1 . 4 x 10& ( l b m ) / ( h r ) ( f t ^ ) w h ile t h e v o id f r a c t i o n v a r i e d from 80$ t o 20 $.

V oid f r a c t i o n d a ta o v e r t h e q u a l i t y and m ass f l u x r a n g e s o f i n t e r e s t w ere o b ta in e d u s i n g a q u ic k c l o s i n g v a lv e t e c h n i q u e .

I n b o th p h a s e s o f t h i s s tu d y a x i a l p r e s s u r e d ro p d a ta

I I . LITERATURE SURVEY

A. I n t r o d u c t i o n an d D e f i n i t i o n o f Terms

S in c e t h i s s tu d y c o n tin u e d and e x te n d e d a p r e v io u s p r o j e c t t h e l i t e r a t u r e s u r v e y i s an u p d a te d v e r s i o n o f t h a t w r i t t e n

b y W alton [ I ] . The s u r v e y i s i n c lu d e d to f a c i l i t a t e t h e r e a d i n g a n d com p rehen sio n o f t h i s t h e s i s .

R og ers and T o d r e a s [ 2 ] h av e p ro p o s e d f o u r

m ixing

p r o c e s s e s w h ich a r e c a t e g o r iz e d d e p e n d in g on w h e th e r t h e m ix in g i s n a t u r a l o r f o r c e d and d i r e c t i o n a l o r n o n d i r e c t i o n a l . The f o llo w in g i s a b r i e f d e s c r i p t i o n o f t h e s e p r o c e s s e s .

1 . N a tu r a l M ix in g E f f e c t s

T hese in c lu d e p r o c e s s e s w hich o p e r a t e i n t h e a b s e n c e o f p r o tu b e r a n c e s i n t h e ro d b u n d le s , i . e . , a b u n d le o f sm ooth, b a r e

r o d s .

a . T u r b u le n t in te r c h a n g e r e s u l t s from n a t u r a l ed dy d i f f u s i o n b etw ee n s u b c h a n n e ls . T u r b u le n t i n t e r c h a n g e i s u s u a l l y m e asu re d b y i n j e c t i o n o f t r a c e r s . T h is r e s u l t s i n t h e m easurem en t o f t h e eddy d i f f u s i v i t y o f m a s s , 5 ^ , w h ich i n g e n e r a l i s n o t t h e same a s th e eddy d i f f u s i v i t y o f h e a t , F o r t u n a t e l y , i t h a s b een fo u n d t h a t t h e r e i s no s i g n i f i c a n t d i f f e r e n c e b e tw e e n 6!^ and

C 2 , 3, 4 3.

b . D iv e r s io n c r o s s - f l o w i s t h e d i r e c t e d flo w c a u s e d b y r a d i a l p r e s s u r e g r a d i e n t s b etw ee n a d j a c e n t s u b c h a n n e ls . T h ese g r a d i e n t s may be in d u c e d by g r o s s d i f f e r e n c e s b etw ee n t h e s u b c h a n n e l h e a t - f l u x

7.

d i s t r i b u t i o n s and d i f f e r e n c e s i n s u b c h a n n e l e q u i v a le n t d ia m e te r s o r l o c a l g eo m etry .

2 . F o rc e d M ixing E f f e c t s

T h ese in c lu d e e f f e c t s in d u c e d b y t h e p r e s e n c e o f p i n s p a c e r s o r g r i d s i n t h e f lo w c h a n n e l.

a . Flow s c a t t e r i n g r e f e r s t o t h e n o n d i r e c t i o n a l m ix in g e f f e c t s a s s o c i a t e d w ith a x i a l o r c i r c u m f e r e n t i a l f i n s , w a r t- ty p e s p a c e r s ,

g r i d s p a c e r s , and end p l a t e s , w h ich b r e a k up s t r e a m lin e s and in d u c e tu r b u l e n c e , b u t do n o t f a v o r a s i n g l e d i r e c t i o n .

b . Flow sw eep in g i s d i r e c t i o n a l and r e f e r s t o t h e c r o s s - f l o w e f f e c t s a s s o c i a t e d w i t h w ire 'v/rap s p a c e r s , h e l i c a l f i n s , c o n to u re d g r i d s , an d m ix in g v a n e s w h ich g iv e a n e t c r o s s - f l o w i n a p r e f e r r e d d i r e c t i o n .

B. S in g le C hannel E x p e rim e n ts

Eddy d i f f u s i v i t i e s h av e b e e n m e asu re d e x p e r im e n ta lly b y i n j e c t i o n o f a t r a c e r ( o r h e a t ) from a p o i n t s o u r c e i n a m oving f l u i d s tre a m and m e a su rin g th e s u b s e q u e n t c o n c e n t r a t i o n ( o r te m p e r a tu r e ) p r o f i l e s dow nstream 0 - 8 ] . By r e p la c e m e n t o f t h e th e rm a l d i f f u s i v i t y w ith t h e c o n c e p t o f an eddy d i f f u s i v i t y i n W ils o n ' s C 9 ] i n t e g r a t e d F o u r ie r e q u a tio n , t h e c o n c e n t r a t i o n ( o r te m p e r a tu r e ) p r o f i l e s w ere a n a ly s e d t o g iv e t h e eddy d i f f u s i v i t i e s .

An a l t e r n a t e a p p ro a c h o f d e te r m in in g eddy d i f f u s i v i t i e s in v o lv e d th e m easurem ent o f t h e w a te r - v a p o r g r a d i e n t a c r o s s a t u r b u l e n t g as

s tre a m i n a w e tt e d - w a ll co lum n. By d e te r m in in g th e r a t e o f v a p o r iz a ­

-, 8. t u r b u l e n t p o r t i o n o f t h e g a s s tre a m [10,111 .

S t . P i e r r e [ l 2 ] h a s c o r r e l a t e d m o st o f th e a v a i l a b l e d a t a w hich a r e p l o t t e d a s th e i n v e r s e o f t h e P e c l e t n u m b e r ,'|/, v e r s u s t h e R e;/nolds num ber. The i n v e r s e o f P e c l e t number r e p r e s e n t s t h e r a t i o o f th e t u r b u l e n t c o n d u c tiv e h e a t (m ass) t r a n s f e r t o th e b u lk h e a t (m ass) t r a n s f e r .

C. M u lti-C h a n n e l (Rod B u n d le ) E x p e rim e n ts

E x p erim en ts t o d e te r m in e t o t a l flo w r e d i s t r i b u t i o n from a l l m ix in g e f f e c t s have b e e n c a r r i e d o u t i n m u ltir o d b u n d le s . A t r a c e r was i n j e c t e d i n one s u b c h a n n e l and dow nstream c o n c e n tr a ti o n s w ere m easu red i n a d j a c e n t s u b c h a n n e ls t o d e te r m in e t h e t u r b u l e n t m ix in g r a t e . M ixing c o e f f i c i e n t s w ere d e te rm in e d from a t r a c e r m ass b a la n c e .

R o gers [13] h a s p r o p o s e d t h e f o llo w in g s i n g l e p h a se m ix in g c o r r e l a t i o n b a s e d on ro d b u n d le m ix in g d a t a . T a k in g an e n e rg y b a la n c e betw een t h e h e a t t r a n s p o r t r a t e b y th e p o s t u l a t e d m ix in g flo w and t h e t r a n s v e r s e h e a t t r a n s p o r t b y eddy d i f f u s i v i t y betw een an y two s u b c h a n n e ls , i an d j .

0 ^ ^ ( 2 -1 )

The a v e ra g e te m p e ra tu re g r a d i e n t may b e a p p ro x im a te d by

( f ) ^ ^

w here i s an in t e r s u b c h a n n e l d i s t a n c e . U sing t h i s a p p ro x im a tio n

E q u a tio n ( 2 .1 ) may be s i m p l i f i e d t o g iv e t h e b a s i c e q u a tio n f o r m ix in g flo w r a t e :

^ Î j ^ ^ 2 i j ( 2 .3 ) i j

A nalogous form s o f t h i s e q u a t io n , d i f f e r i n g o n ly i n d e f i n i t i o n o f i n t e r sub c h a n n e l d i s t a n c e , h av e b een p ro p o s e d b y S t . P i e r r e [ l ^ .

Bownring [1 4] and Rowe and A ngle [_15] . S t . P i e r r e

wp =

(2 .4 )

1/2 (Re^i+Re

,)

Bovnring

w h = L b e . . L f

(2 .5 )

' S . a Rowe and A ngle

.

e h c s .

L

^

(

2

.

6

)

A y

R o g ers [13] h a s p o s t u l a t e d t h e i s a f u n c t i o n o f b / d and w as a b l e t o c o r r e l a t e e x i s t i n g d a t a ( s e e T a b le I I , r e f e r e n c e C13] ) u s i n g t h i s ty p e o f f o r m u l a tio n f o r .0 5 < b /d < .5 , T h is p o s t u l a t e d d ep en dence on b / d ivas fo u n d t o y i e l d n e a r l y c o n s t a n t m ix in g r a t e s o v e r t h e ra n g e o f b / d s i n c e £ i n c r e a s e d a s b / d d e c r e a s e d . W hereas t h i s c o r r e l a t i o n may a d e q u a te l y p r e d i c t m ix in g o v e r a l i m i t e d ra n g e o f b / d i t i s o b v io u s t h a t i f i s a l s o a f u n c t i o n o f b / d th e n

10.

S k in n e r e t a l f46~| fo u n d t h a t t h e r a t e o f t r a n s f e r (mass o r e n e rg y ) v a s h ig h e r th a n c o u ld b e a c c o u n te d f o r b y t u r b u l e n t d i f f u s i o n . They p ro p o s e d t h a t s e c o n d a ry flo w s e x i s t i n t h e gap a r e a (up t o

2%

f o r sm ooth r o d s , 6$ f o r ro u g h e n e d o n e s ) , w hich a c c o u n ts f o r t h e h ig h e r m ix in g .

Van D er Ros and B o g a a rd t [ 1 7 ] fo u n d t h a t h e a t f l u x i n th e gap had no e f f e c t on m ix in g . The te m p e r a tu r e o f t h e i n n e r c h a n n e l in c r e a s e d l i n e a r l y w ith d i s t a n c e dcw nstream show ing t h a t th e

o v e r a l l e f f e c t i v e h e a t ex ch an g e a lo n g t h e c h a n n e l was a p p r o x im a te ly c o n s t a n t . U sin g an eddy d i f f u s i v i t y f o r d a ta a n a l y s i s th e y fo u n d

t h a t :

£

= 0 .0 0 9 (7/D e) ( 2 .7 ) w here V= v e l o c i t y ( f t / s e c )

w here De= h y d r o l i c d i a m e te r ( f t )

w h ich was t h e b e s t f i t o f t h e d a t a . The P r a n d t l m ix in g l e n g t h th e o r y was u s e d t o e x p l a i n t h e f a c t t h a t m ix in g c o e f f i c i e n t was i n v e r s e l y p r o p o r t i o n a l t o t h e gap w ûdth.

D. Two S u b ch an n el E x p e rim e n ts

A t p r e s e n t l i m i t e d d a t a have b e e n o b ta in e d f o r s i n g l e p h a s e t u r b u l e n t in te r c h a n g e b e tw e e n two a d j a c e n t s u b c h a n n e ls . T a b le 2 .1 l i s t s th e a v a i l a b l e d a t a s o u r c e s .

T ab le 2 .1

D e s c r ip ti o n o f D ata -A v ailab le on Two S u b ch an n el E x p erim en ts

11.

S o u rc e and r e f . n o .

G e o m e tric a l

d e s c r i p t i o n b / d

M ass f l u x

( I b m / h r . f t ) F lu id

E x p e rim e n ta l te c h n iq u e

S i n g l e t o n [ I9J s q u a re f i l l e r s ro u n d f i l l e r s

.453 .84 0 .2 0 7 .400

6 .3 4 2 -1 .8 3 x 1 0 .9 4 5 -1 .0 7 x 1 0 ^

w a te r w a te r

dye i n j e c t i o n dye i n j e c t i o n

Rowe and A ngle [15,2(3

s q u a r e - t r i a n g l e * s q u a r e - s q u a r e *

.0355

.149 .1 4 2

.1 5 0

.1 5 6

1 .0 - 3 .0 x 1 0 ^ .9 5 0 -2 .9 0 x 1 0 ^

w a te r w a te r

e n th a lp y b a la n c e LiOH, DpO

T2O t r a c e r s

P e t r u n i k [ i S l r e c t a n g l e -r e c t a n g l e

.3 2 0

.80 0

.3 2 0

.80 0

0 .2 - 2 .0 x 1 0 ^

0 .1 - 1 .0 x 1 0 ^

w a te r

a i r

KNO^ t r a c e r

CH^ t r a c e r

H e ts r o n i e± a l [22]

s q u a r e - s q u a r e * .2 5 1 .5 3 -2 .2 2 x 1 0 ^ w a te r e n t h a lp y b a la n c e

W alto n L ll t r i a n g l e - t r i a n g l e* . 05 .0 5

3 . 8 - 1 .7 x 1 0 ^ .1 6 -2 .7 x 1 0 5

w a te r a i r

ERO? t r a c e r CH^ t r a c e r

¥ a n d e r Ros an d

B o g a a r t [ I ' d s q u a r e - s q u a r e ..1 .2

3 .6 -1 2 .8 x 1 0 ^ 3 i 6 -1 2 .8 x 1 0 ^

w a te r e n th a lp y b a la n c e

^ s im u la te d ro d b u n d le a r r a y s

P e tr u n i k [ I S ] and Rowe and A ngle [17,2Q] have fou nd t h e m ix in g r a t e t o be d e p e n d e n t on gap w id th (r o d s p a c i n g ) . S in g le to n [19] fo u n d th e m ix in g r a t e i n c r e a s e t o b e a p p r o x im a te ly 1.4. tim e s t h e r a t i o o f th e gap w id th s f o r t h e s q u a r e f i l l e r s and a p p r o x im a te ly p r o p o r t i o n a l t o t h e r a t i o o f gap w id th s f o r t h e ro un d f i l l e r s .

1 2 .

a r r a y was l e s s th a n f o r th e s q n a r e - t r i a n g u l a r a r r a y . Hence m ix in g d e c r e a s e d a s t h e c e n t r o i d a l d i s t a n c e b e tw e e n s u b c h a n n e ls i n c r e a s e d . T h is o b s e r v a t i o n h a s b e e n s u p p o r te d by t h e w ork o f H e ts r o n i e t. a l .

[24] , ( s e e A ppendix I o f R e f . [ 1 ] ) and t h e t h e o r e t i c a l work o f Lowe [ 2 $ ] . Van D er Ros a n d B o g a a rd t [ 1 7 ] u s in g a s im p le s q u a r e - s q u a r e

s u b c h a n n e l g eo m etry d e te r m in e d m ix in g f o r s e v e r a l gap w id th s i n c l u d i n g an a s y m m e tr ic a l o n e . I t was fo u n d t h a t b etw ee n a R ey n o ld s num ber o f 6000— 1 7 ,0 0 0 t h e r e was no e f f e c t o f gap w id th on m ix in g . At R ey n o ld s num ber l e s s th a n 6OOO t h e m ix in g d e c r e a s e d r a p i d l y f o r t h e 0 .0 3 8 i n . g a p . F o r R eyn old s num ber g r e a t e r th a n

1 7 ,0 0 0 m ix in g i n c r e a s e d f o r th e 0 .0 7 8 8 i n . g a p . I n s t a b i l i t i e s a s s o c i a t e d w ith t h e t r a n s i t i o n fro m a la m in a r t o t u r b u l e n t b o u n d a ry l a y e r i n t h e gap was b e l i e v e d t o b e t h e c a u s e o f t h i s i n c r e a s e d m ix in g , ^’o r t h e a s y m m e tr ic a l gap g eo m etry i t was fo u n d t h a t th e gap form i n f l u e n c e d m ix in g . The w id e r p a r t o f

t h e gap i n c r e a s e s m ix in g . From a b e s t f i t o f t h e d a t a t h e f o ll o w i n g c o r r e l a t i o n ws,s o b ta in e d

£ = 0 .0 0 4 (V/De) (2.1 4)

F u r t h e r work i s b e i n g done b y t h e s e a u t h o r s i n tw o -p h a s e flo w . [ 2 7 ] . The d a t a p r e s e n t l y a v a i l a b l e i s i n s u f f i c i e n t t o p r e d i c t s i n g l e ­ p h a s e t u r b u l e n t in t e r c h a n g e r a t e s f o r a l l ty p e s s u b c h a n n e l g e o m e tr ie s . H ence, f u r t h e r e x p e r im e n ta tio n i s n e e d e d i n t h i s a r e a .

E . S in g le P h ase T u r b u le n t M ix in g C o r r e l a t i o n s 1 . S t . P i e r r e C o r r e l a t i o n Cl2]

S t . P i e r r e h a s p ro p o s e d a c o r r e l a t i o n f o r t u r b u l e n t in t e r c h a n g e r a t e s b a s e d on v a l u e s o f t u r b u l e n t d i f f u s i v i t i e s o b ta in e d from s i n g l e

1 3 .

c h a n n e l e x p e r im e n ts . W ith t h e a v a i l a b i l i t y o f m ore d a t a a t lo w e r R ey n o ld s num bers S t . P i e r r e [ 2 Ï ] h a s t e n t a t i v e l y m o d if ie d t h e c o r r e l a t i o n f o r f o r < 5 0 0 0.

= 2V ( 2 .8 )

w here

'IjJ

= < 5000

a n d 1}/ = 0.0 20 5 5000 < N ^ < 2 0 0 ,0 0 0 .

2 . Bow ring C o r r e l a t i o n [ I4]

B ow ring h a s u s e d t h e f o llo w in g e q u a t io n , i n a s u b c h a n n e l code HAMBO t o p r e d i c t s i n g l e p h a s e t u r b u l e n t i n t e r c h a n g e .

¥ t = Fm b V A + _fj (De i Gi + D ^ j G j ) ( 2 .9 )

L S^ d SO

I f t h e s u b c h a n n e ls a r e i d e n t i c a l th e n e q u a tio n ( 2 .8 ) can b e r e a r r a n g e d t o g iv e :

N gt = E l 'V zf

,

8% a 40 (2.10)

F o r a t r i a n g u l a r - t r i a n g u l a r a r r a y : 8^ = 0 .3 9 ( r e f e r e n c e 1 4 )

F* = 1.117^^ and

-

0

.

28

*

f = 0 .0 9 4 Hi_Be

1 4 .

3 . R o gers and T a ra su k C o r r e l a t i o n [131

From m u l ti- c h a n n e l e x p e rim e n ts R o g ers and T a ra su k have d e v e lo p e d t h e f o llo w in g e q u a tio n f o r t u r b u l e n t in t e r c h a n g e

1 _ T — a \ 3m

% t = 5 s Nae

2 ''d/

m-1

1 + _{_&uL} ^ e , i ;

D

(2.

11)

R ogers [20]] h a s t e n t a t i v e l y p ro p o s e d t h e f o llo w in g v a l u e s f o r th e c o n s ta n t s Kg, m, n andcC .

Kg = 0 .0 0 4 0

m = 0 .9 0 n = 0 .2 0

cC ij = 0 .5 7 7

4* Rowe and A ngle C o r r e l a t i o n P -5 ,2(3

The Rowe and A ngle c o r r e l a t i o n was d e v e lo p e d b y f i n d i n g a v a lu e o f th e e q u i v a l e n t S ta n to n num ber w h ich m o st c l o s e l y m atch ed t h e i r

e x p e r im e n ta l d a t a .

% t “ &r

w here = 0 .0 0 6 2 = 0 .0 0 3 6

N-_Re- n / 2

t r i a n g u l a r - s q u a r e a r r a y s q u a r e - s q u a r e a r r a y n = 0 .2

Z .j = b (g ap w id th ) 5 . Moyer C o r r e l a t i o n |23]

Moyer h a s p ro p o s e d th e f o ll o w i n g c o r r e l a t i o n f o r t u r b u l e n t m ix in g :

1 5 .

= \ Ü ^ ( 2 .1 3 )

S t ^ Z ij

w here Z = c e n t r o i d a l d i s t a n c e b e tw e e n s u b c h a n n e ls i j

f i s d e f in e d a s i n e q u a t io n ( 2 . 9 ) . 6 . S k i n n e r q t a l C o rrela tio n Ol6l

S k in n e r e t a l [ l 6 ]

proposed

t h e f o l l o w i n g c o r r e l a t i o n f o r s i n g l e p h a s e a i r m ix in g .

Ngt = - 0 . 1 l 6 ( ^ 4 x i n (N ) ) (2 .1 4 )

w here j / = - 0 .1 8 4 f o r sm ooth r o d c l u s t e r s = 0 .4 6 2 f o r ro u g h ro d c l u s t e r s an d fet = 0 .0 1 3 f o r sm ooth r o d c l u s t e r s

= - 0 .0 5 4 f o r ro u g h r o d c l u s t e r s

F o r a co m p ariso n w ith e x p e r im e n ta l d a t a s e e A ppendix I .

7 . I n g e r s o n C o r r e l a t i o n [2 ? ]

I n g e r s o n [ 2 ? ! p ro p o s e d t h e f o ll o w i n g c o r r e l a t i o n s :

= ( 6 .1 2 p/D e - 3 .0 3 ) l / 2 0 _ A _ v / l L (2 .1 5 i j V

S t Z i i P

w h ere Z . = c e n t r o i d a l d i s t a n c e P,, = w e tte d p e r i m e t e r ._w

T h is c o r r e l a t i o n i s f o r an y r o d c o n f i g u r a t i o n w ith a p i t c h t o d ia m e te r r a t i o g r e a t e r th a n 1 .1 5 .

F o r an y ro d c o n f i g u r a t i o n w ith a n y p i t c h to d ia m e te r r a t i o

h e p r o p o s e s :

N gt = 0 .9 5 / E . i V z f " ( 2 .1 6 ) ( 5 7 ^ / ( d De) z o Z ij

16 .

F . Two-Phase T u r b u le n t I n te r c h a n g e Data

Rowe and A ngle [15,20] h ave o b ta in e d e x p e r im e n ta l d a ta on tw o -p h a s e s te a m -w a te r t u r b u l e n t in t e r c h a n g e a t e l e v a t e d p r e s s u r e s i n s q u a r e - t r i a n g l e an d s q u a r e - s q u a r e a r r a y s . T h e i r r e s u l t s

showed t h a t a maximum v a lu e o f m ix in g was r e a c h e d a t low q u a l i t i e s an d t h a t m ix in g was g tr o n g l y a f f e c t e d by ro d s p a c in g d u r in g b o i l i n g . The r a t i o o f steam t o w a te r m ix in g was fo u n d t o b e a b o u t th e same a s t h a t o c c u r r i n g i n t h e b u lk f lo w .

W alton [ 1 ] m e asu re d tw o -p h a s e a i r - w a t e r t u r b u l e n t in te r c h a n g e r a t e s b etw een two a d j a c e n t t r i a n g u l a r s u b c h a n n e ls . I t was

fo u n d t h a t t h e l i q u i d - p h a s e m ix in g d e c r e a s e d >d.th q u a l i t y beyond t h e s l u g - a n n u l a r t r a n s i t i o n , w h ile t h e g a s - p h a s e ( a i r ) m ix in g

w as e s s e n t i a l l y c o n s t a n t . I n a d d i t i o n f o r b o th p h a s e s th e r e l a t i v e m ix in g ,

{ %/ f t ) ,

was fo u n d t o a lw a y s d e c r e a s e w ith i n c r e a s i n g

q u a l i t y beyond t h i s t r a n s i t i o n p o i n t . B o th P e t r u n i k [ 1 8 ] and W alton [1 ] d a ta showed t h a t tw o -ph s.se m ix in g was g r e a t e r th a n t h e s in g le - p h a s e m ix in g ta k e n a t th e same t o t a l m ass flo w r a t e . I t was a l s o n o te d t h a t th e h i g h e s t m ix in g o c c u re d i n a r e g io n o f lo w q u a l i t y and low m ass f l u x ; t h a t i s i n t h e t r a n s i t i o n r e g io n b e tw e e n s lu g flo w and a n n u la r f lo w .

Lahey and Sehr.aub [ 2 8 ] r e c e n t l y re v ie w e d a l l a v a i l a b l e d a t a on tw o -p h a se m ix in g , v o id f r a c t i o n , and flo w re g im e s . They h av e l i s t e d th e f o llo w in g t r e n d s w hich h av e b e e n o b s e rv e d f o r tw o -p h a s e su b c h a n n e l m ix in g ;

(1 ) Tw o-phase m ix in g i s a f u n c t i o n o f r o d / r o d s p a c in g " b " .

(2 ) Tî-ra-phase m ix in g i s a f u n c t i o n o f flo w re g im e .

1 7 .

(3) S u b ch an n el shap e and s i z e i s an im p o r ta n t p a r a m e te r when c o n s id e r i n g t h e m ix in g .

(4 ) The sh ap e o f th e p a s s a g e c o n n e c tin g s u b c h a n n e ls i s a p p a r e n tly

im p o r ta n t .

(5 ) The e n th a lp y o f flo w d i v e r s i o n d o e s n o t , i n g e n e r a l, a p p e a r t o b e th e a v e ra g e e n th a lp y o f t h e d o n o r s u b c h a n n e l.

I I I . THEORY A. D e r iv a t io n o f M ix in g E q u a tio n s

The m ix in g e q u a tio n s d e v e lo p e d h e r e have b e e n d e r iv e d p r e v i o u s l y b y W alton [ 1 ] . C o n s id e r t h e t u r b u l e n t flo w o f a f l u i d i n two i d e n t i c a l , a d j a c e n t s u b c h a n n e ls . B o th s u b c h a n n e ls a r e c o n n e c te d b y a s l o t o f w id th "b" a n d l e n g t h " I " , th r o u g h w hich t h e t u r b u l e n t t r a n s f e r o f l i q u i d ta k e s p l a c e . I n one s u b c h a n n e l a t r a c e r i s i n j e c t e d u p s tr e a m o f t h e m ix in g z o n e . The am ount o f l i q u i d t r a n s f e r r e d from one s u b c h a n n e l t o a n o t h e r i s d e te rm in e d b y em p lo y in g a m ass b a l a n c e u s i n g m e asu re d t r a c e r e x i t c o n c e n tr a ti o n s i n e a c h s u b c h a n n e l.

1 . B a s ic A ssu m p tio n s:

The f o llo i- d n g b a s i c a s s u m p tio n s a r e made i n t h i s a n a l y s i s : ( i ) The t o t a l m ass flo w r a t e i s t h e same i n e a c h s u b c h a n n e l. ( i . e . , l i q u i d p l u s t r a c e r ) .

( i i ) S u b ch an n el a x i a l p r e s s u r e g r a d i e n t s a r e i d e n t i c a l , th u s e l i m i n a t i n g r a d i a l p r e s s u r e g r a d i e n t s and an y n e t t r a n s f e r o f f l u i d from one s u b c h a n n e l t o a n o t h e r .

( i i i ) T r a c e r c o n c e n t r a t i o n s a r e s m a ll h e n c e t h e t r a c e r h a s n e g l i g i b l e e f f e c t on p h y s i c a l p r o p e r t i e s .

( i v ) The f l u i d l e a v i n g o n e s u b c h a n n e l h a s t h e a v e r a g e p r o p e r t i e s ( t r a c e r c o n c e n tr a ti o n ) o f t h a t c h a n n e l.

(v ) A f t e r f l u i d h a s l e f t t h e d o n o r s u b c h a n n e l i t m ix es im m e d ia te ly i n t h e r e c e i v i n g s u b c h a n n e l.

( v i ) T h ere i s no r e l a t i v e v e l o c i t y b e tw e e n f l u i d and t r a c e r .

I S .

1 9 .

2 . S im p le M ixing M odel

A m ass b a la n c e on t h e t r a c e r o v e r t h e e n t i r e m ix in g l e n g t h L , f o r s u b c h a n n e l 1 r e s u l t s i n E q u a tio n 3 .1 ( s e e F i g . 3 . 1 ) .

F ig u r e I Cl

I 2I

2 . 1

_{'Î .2}

2

¥2

C’ = e x i t co n cen tra tio n flbm tr a cer) VLbm f l u i d /

(ibffl tracer] VLbm f l u i d / C = i n l e t c o n c e n t r a t i o n flbm t r a c e r

Lbm f l u i d

¥ = s u b c h a n n e l m ass flo w r a t e (ib m /h r)

¥ ^ = m ix in g r a t e (ib m /h r) L = s l o t l e n g t h ( f t )

3.1. M ixing Flow D iagram rt

¥ , + ¥2 2 ( C2 +0?) = ¥ ^0^ + ¥2 p (C2+C2)

^ • 2 ' - T "

(3.1)

F o r t r a c e r i n j e c t i o n i n s u b c h a n n e l 1 , C2 e q u a ls z e r o . N o tin g t h a t W^=¥2=¥2=¥2= ¥ an d

2~^2

1 " ^ ^ E q u a tio n 3 .1 r e d u c e s t o :

¥(C^-C{) =

\^/2

(C

2

+C

2

-Cg)

(3.2)

An o v e r a l l t r a c e r b a la n c e f o r t h e m ix in g l e n g t h L, r e s u l t s i n :

Cq = C2 +C2

From e q u a tio n ( 3 .3 ) an d ( 3 .2 ) we o b t a i n

¥02 = ¥ t / 2 (02 +O2 +O2 -Og)

w h ich r e d u c e s t o

. 0’

¥ ^ = ¥ - 2

(3 .3 )

( 3 .4 )

( 3 .5 )

I I t

20.

3 . D i f f e r e n t i a l M ixing Model

I t i s c o n v e n ie n t h e r e t o d e f i n e r e l a t i v e t r a c e r c o n c e n tr a ti o n s : •K

-C j = 1 and Cg = 2 so t h a t + C2 = 1

C2+C2 C2+C2

(3 .6 )

A

t r a c e r b a la n c e o v e r a d i f f e r e n t i a l m ix in g l e n g t h dx, f o r s u b c h a n n e l

two g iv e s :

t

+ \ . 2 ^1 dx = ^ 2 .1

^2

dx + ¥2(0% + dC%)

L L

T h is re d u c e s to

f 2 = v L ( ( ^

-dx ¥L

( 3 .7 )

( 3 .8 )

S in c e + Cg = 1 E q u a tio n ( 3 .8 ) becom es *

^^2 _ (1 - 2CÏ) dx Tr/L

(3.9) R e a rra n g in g and i n c o r p o r a t i n g t h e a p p r o p r i a t e b o u n d a ry c o n d i tio n s

(3 .1 0 )

I n t e g r a t i o n y i e l d s

*1 +

- i I n ( I -2C2 )= ¥ 2 ¥ #1

S in c e Cg ?

CÎ+C2

( 3 .1 1 )

(3 .1 2 )

E q u a tio n ( 3 .1 1 ) becom es

¥^ = - ¥ / 2 I n

I

1

-Cl + C2

(3.13)

Comparing E q u a tio n s ( 3 .5 ) and (3 .1 3 ) and d e f i n i n g t h e r e l a t i v e e r r o r t o b e t h e d i f f e r e n c e o f th e m ix in g r a t e s o b ta in e d u s i n g

E q u a tio n s (3*5) an d ( 3 .1 3 ) d iv i d e d b y t h a t o b ta in e d u s i n g E q u a tio n (3 .1 3 ) t h e f o ll o w i n g e x p r e s s io n i s o b ta in e d .

2 1 .

R e l a t i v e e r r o r = 1 + 20 ( 3 .1 4 )

G iln

T h is e r r o r i s l e s s th a n

1%

f o r l e s s th a n 0 . 1 . B. I d e n t i f i c a t i o n o f Flow Regim es

The tw o -p h a se a i r - w a t e r d a t a w ere o b ta in e d h e r e i n t h e flo w re g im e s n o rm a lly e n c o u n te r e d from t h e b u b b l e - s l u g t r a n s i t i o n t o th e s l u g - a n n u l a r t r a n s i t i o n . Thus t h e a b i l i t y t o p r e d i c t t h e t r a n s i t i o n from t h e b u b b le t o s l u g flo w and s l u g to a n n u l a r f lo w i s o f im p o r ta n c e .

0

,

0 0 0

s ^ >

ooo

,

0 o ,

( 9 0 0

b ■© 0

0 o

11 Î

#

1 . B u b b ly -S lu g T r a n s i t i o n

One d e f i n i t i o n o f a n " I d e a l B ubbly" flo w regim e [4 5 ] i s , "The b u b b ly flo w reg im e o c c u r s when t h e g a s i s i n th e form o f d i s c r e t e s m a ll b u b b le s d i s p e r s e d u n if o r m ly i n a l i q u i d , an d u n a b le t o a g g lo m e ra te w ith e a c h o t h e r " . Thus i t w i l l b e fou nd t h a t

t h e b u b b ly reg im e i s u n s t a b l e u n d e r a l l c o n d i t i o n s s in c e t h e " i d e a l " o f no a g g lo m e r a tio n n e v e r o c c u r s . T h is i s due t o th e v a r i a t i o n o f b u b b le v e l o c i t y "sdith d i f f e r e n t b u b b le s i z e .

C o n s e q u e n tly l a r g e r b u b b le s t e n d to o v e r ta k e th o s e o f s m a l le r d ia m e te r and a g g lo m e r a tio n r e s u l t s .

T here i s no lo w e r l i m i t t o t h e v o id f r a c t i o n a t w h ich

b u b b le flo w can o c c u r C38J . Thus t o o b t a i n a b u b b le flo w re g im e , i t m ust b e c r e a t e d . G r i f f i t h an d W a llis C 44 j p e rfo rm e d a n

e x p e r im e n ta l s tu d y on tw o -p h a s e s l u g flo w , i n w hich t h e y d is c o v e r e d t h a t i t was q u i t e p o s s i b l e t o h ave a b u b b ly flo w a t t h e tu b e

e n t r a n c e w h ile a few f e e t dow nstream ch an ce e n c o u n te r s w ould c r e a t e a b u b b le s u f f i c i e n t l y l a r g e to f i l l t h e tu b e and to s t a r t g ro w th i n t o a t y p i c a l s l u g flo w b u b b le b y a b s o r b in g i t s n e ig h b o u r s . The b u b b le g ro w th i s more in f l u e n c e d b y t h e e n t r a n c e r e g io n e f f e c t s th a n b y t h e d ro p i n h y d r o s t a t i c p r e s s u r e . As a r e s u l t , when th e t r a n s i t i o n o c c u r s and how i t i s d e f in e d i s a f f e c t e d b y th e p a r t i c u l a r a p p a r a tu s u s e d , t h e m ethod o f g as i n j e c t i o n , t h e l e n g t h and tim e from t h e i n j e c t i o n l o c a t i o n to p o i n t o f o b s e r v a ti o n

The t r a n s i t i o n from b u b b ly flo w to s l u g flo w o c c u r s s p o n ta n e o u s ly due t o :

2 3 .

( i ) C o a le sc e n c e o f b u b b le s , w hich can o c c u r when b u b b le s a r e j u s t form ed, o r b y c o l l i s i o n a s th e y flo w a lo n g th e flo w c h a n n e l.

( i i ) The b o u n d ary c o n d i t i o n s , m ethod o f g as i n j e c t i o n o r fo r m a tio n , o r t h e flo w dynam ics o f t h e c h a n n e l a r e in c o m p a tib le f o r b u b b le flo w .

A cco rd in g t o ¥ a l l i s [43"} b o th t h e abo ve p r o c e s s e s a r e r a t h e r

w h im s ic a l, s i n c e t h e s p e e d

\rlth.

w h ich c o a le s c e n c e o c c u r s i s p a r t i c u l a r l y s e n s i t i v e t o i m p u r i t i e s . V e lo c i ty g r a d i e n t s an d t u r b u l e n c e te n d

t o i n c r e a s e t h e r a t e a t w h ic h th e s m a ll b u b b le s c o l l i d e , th u s p ro m o tin g a g g lo m e r a tio n , b u t a l s o h ave th e e f f e c t o f t e a r i n g a p a r t th e

b i g g e r b u b b le s . A v e r y a p p ro x im a te " r u l e o f thum b" som etim es u s e d p l a c e s th e t r a n s i t i o n from b u b b ly to s l u g flo w a t 10 ^ v o id f r a c t i o n i n ta p w a te r .

The seco n d mode o f b re a k d o ™ o f t h e b u b b ly flo w m ust b e u s e d i n c o n ju n c tio n m t h t h e e n t i r e sy stem c h a r a c t e r i s t i c s . H ere th e i n i t i a l f o r m a tio n o f t h e b u b b le i s im p o r ta n t . I f t h e b u b b le s when fo rm ed a r e l a r g e enough t o form s lu g s t h a t n e a r l y f i l l s th e c h a n n e l t h e n t h e r e i s e s s e n t i a l l y no b u b b ly flo w re g im e . G r i f f i t h

[_U5~\

s u g g e s ts t h e fo llo w d n g c r i t e r i o n f o r d e te r m in in g i f b u b b le flo w e x i t s .

- 0 - 4 / 1 .1 (3 .1 5 )

2 4 .

F o r t h i s s tu d y t h e c r i t e r i o n o f v o id f r a c t i o n l e s s th a n 10% was u s e d t o d e te r m in e t h e b u b h ly - s lu g t r a n s i t i o n . Due t o t h e m ethod o f g a s i n j e c t i o n t h e h ig h m ass f l u x e s u s e d , an d a i d e d b y v i s u a l o b s e r v a ti o n t h i s was b e l i e v e d t o b e a v a l i d c r i t e r i o n .

2 . S lu g -A n n u la r T r a n s i t i o n

W a l lis an d C o l l i e r [ 2 9 j p r e s e n t e d a c o m p re h e n siv e d i s c u s s i o n o f t h e a v a i l a b l e m eth o d s f o r p r e d i c t i n g t h e t r a n s i t i o n from

s lu g t o a n n u l a r f lo w . The m etho ds w ere p r e d o m in a te ly e m p i r i c a l . G r i f f i t h and H a b e r s tr a h

[

3 0

J

a n d B re n n e r [31] u s e d an e l e c t r i c a l c o n d u c ta n c e p ro b e t o d e te r m in e t h e o n s e t o f l i q u i d b r i d g i n g a c r o s s t h e flo w c r o s s - s e c t i o n . L iq u id b r i d g e s w ere d e t e c t e d b y d i f f e r e n c e s i n r e s i s t a n c e s f o r l i q u i d a n d g a s b etw ee n th e p ro b e and th e w a ll o f t h e t e s t s e c t i o n . T h is m ethod o f d e f i n i n g t r a n s i t i o n i s r a t h e r a r b i t r a r y s i n c e a r a t h e r b ro a d r e g i o n e x i s t s i n w h ich s l u g s become few an d f a r b e tw e e n b u t do n o t d is a p p e a r a l t o g e t h e r . The t r a n s i t i o n was c o r r e l a t e d b y th e d im e n s io n le s s p a r a m e te r s ;

= 0 .9 + 0 .6 jg ( j f < 1 ) (3 .1 6 )

j g = (7 + 0 . 0 6 ( & ) ) & ) ^ j^ ( j ^ > l ) (3 .1 7 )

w here j g = (Wg/ljCg^ [gOg (<?«- - (^ )] ^ (3 .1 8 ) = t h e d im e n s io n le s s g as v e l o c i t y

an d j j ( 3 .1 9 )

= t h e d im e n s io n le s s l i q u i d v e l o c i t y

2 5 .

E n tra in m e n t m easu rem en ts [2 4 ] u s in g a c o l l e c t i o n ty p e p ro b e w ere made t o d e t e c t l i q u i d b r i d g e s an d e n t r a i n e d d r o p l e t s . By p l o t t i n g p e r c e n ta g e e n tr a in m e n t v e r s u s f o r v a r i o u s v a lu e s o f and e x t r a p o l a t i n g t h e s l u g flo w p o r t i o n t o t h e a b s c i s s a i t i s p o s s i b l e to d e f i n e a lo w e r c r i t i c a l g as v e l o c i t y a t w hich t h e s l u g - a n n u l a r t r a n s i t i o n o c c u r s . The t r a n s i t i o n l i n e i s c o r r e l a t e d b y t h e e q u a tio n

jg = 0 .4 + 0 .6

(3.20)

w h ich i s s i g n i f i c a n t l y b e lo w t h e e l e c t r i c a l p ro b e t r a n s i t i o n . An a t te m p t was made b y G o v ie r ^ a l Q 2 , 3 3 , 3 4 ] t o c o r r e l a t e re g im e b o u n d a r ie s an d p r e s s u r e d ro p maxima and m in im a. The p r e d i c t i o n s u s i n g t h i s m ethod a r e n o t v e r y a c c u r a t e s i n c e some o f th e maxima an d m inim a a r e r e l a t i v e l y f l a t . However, t h e o r d e r o f m a g n itu d e p r e d i c t e d i s c o r r e c t , show ing t h a t p r e s s u r e d ro p b e h a v i o r i s q u i t e c l o s e l y r e l a t e d t o t r a n s i t i o n s i n flo w re g im e .

The o n ly n o n - e m p ir ic a l m ethod f o r p r e d i c t i n g t h e a n n u l a r -

s l u g t r a n s i t i o n i s one b a s e d on v o id f r a c t i o n p r e d i c t i o n . T h is m ethod h a s p e rh a p s t h e m o st m ean ing from an e n g i n e e r in g s ta n d p o i n t s in c e i t c o n s id e r s t h e p o i n t w here s l u g v o id f r a c t i o n t h e o r y m u st be r e p l a c e d b y a n n u l a r v o id f r a r t i o n th e o r y [2*3 • T h is o c c u r s a t a V alu e o f = 0 . 8 an d g iv e s r i s e t o t h e e q u a tio n

= 0 .3 3 + 0 .7 5 j j (3 .2 1 )

T h is e q u a tio n a g r e e s c l o s e l y w ith e q u a tio n (3 .2 0 ) w h ich was o b ta in e d fro m t h e e n tr a in m e n t p ro b e r e s u l t s .

26.

e q u a tio n s (3 .1 7 ) and (3 .1 8 ) may b e i n t e r p r e t e d a s a r e g io n o f " s l u g - a n n u la r " flo w i n which, a r a t h e r g r a d u a l t r a n s i t i o n b e h a v i o r o c c u r s .

At p r e s e n t no u n i v e r s a l e q u a tio n e x i s t s f o r p r e d i c t i n g

e n tr a in m e n t. From a v a i l a b l e e x p e r im e n ta l e v id e n c e

£393

t h e ra n g e o f c o n d i tio n s f o r w hich a n n u l a r flo w , w ith no e n tr a in m e n t, e x i s t s i s v e r y s m a ll. S t e e n 's [1361] c r i t e r i a f o r t h e o n s e t o f e n tr a in m e n t d o es n o t a g r e e p a r t i c u l a r l y w e l l w ith e x p e r im e n ta l e v id e n c e . Up

to 30% e n tra in m e n t was e x p e r ie n c e d b e f o r e S t e e n 's c r i t e r i a was f u l f i l l e d . E n tra in m e n t a t th e u p p e r ra n g e o f i s e x p e c te d t o be l a r g e and

a p p ro a c h in g m i s t flo w .

IV . AIR-WATER TEST LOOP AND ASSOCIATED EQUIPMENT A. E x p e r im e n ta l E quipm ent

1 . A ir-W a te r T e s t Loop

The a i r - w a t e r t e s t lo o p a n d a s s o c i a t e d e q u ip m en t u s e d i n t h i s s tu d y w ere l o c a t e d i n t h e C h em ical E n g in e e r in g L a b o r a t o r i e s , E sse x H a ll , a t t h e U n i v e r s i t y o f W in d so r. The lo o p was d e s ig n e d and b u i l t s p e c i f i c a l l y f o r m ix in g e x p e r im e n ts em p lo y in g a i r - w a t e r f lo w s . The m a jo r lo o p com ponents c o n s i s t e d o f a r e g u l a t e d 100 p s i g a i r s u p p ly , a 100 p s i g w a te r s u p p ly , tw o r o ta m e te r b a n k s , a

t e s t s e c t i o n a s s e m b ly an d tw o c y c lo n e s e p a r a t o r s . The lo o p i s d ia g ra m e d s c h e m a t i c a l l y i n F i g . 4 . 1 .

A ir from t h e u n i v e r s i t y 100 p s i g a i r s e r v i c e l i n e was

f i l t e r e d , s p l i t i n t o two s e p a r a t e s tr e a m s and m e te re d s e p a r a t e l y u s i n g m a tch ed r o t a m e t e r s . W ater was s u p p l i e d b y a G oulds

(m odel 3775) c e n t r i f u g a l pump w h ic h c o u ld p r o v id e 125 I P ® a t a h ead o f 130 p s i a . The w a te r s u p p ly was d iv i d e d a l s o i n t o two s e p a r a t e s tr e a m s an d m e te r e d s e p a r a t e l y u s i n g m a tch ed r o t a m e t e r s .

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The v a t e r sam p les w ere c o l l e c t e d from e a c h s e p a r a t o r f o r l a t e r a n a l y s i s . The e x c e s s w a te r was d is c h a r g e d i n t o a d r a i n .

S e v e r a l lo o p m o d i f i c a t i o n s w ere r e q u i r e d f o r t h e seco n d p a r t o f t h i s s tu d y . The v a l v e s f o r t h e dow nstream r e s i s t a n c e w ere removed and t h e w a t e r - a i r m ix tu r e from e a c h s u b c h a n n e l w ent d i r e c t l y t o t h e c y c lo n e s e p a r a t o r s . The s e p a r a t o r s w ere r e d e s ig n e d t o h a n d le 600 I b m /h r o f w a te r an d 125 Ib m /h r a i r . They w ere a l s o c o m p le te ly s e a l e d . The a i r from e a c h s e p a r a t o r was m e te re d s e p a r a t e l y b y m a tch ed r o ta m e te r s and th e n v e n te d t o t h e a tm o s p h e re . The w a te r flo w r a t e was d e te rm in e d u s i n g t h e d i r e c t w e ig h in g te c h n iq u e . E x ce ss w a te r was d is c h a r g e d i n t o t h e d r a i n . A ll w a te r s a m p le s w ere ta k e n d i r e c t l y from th e

s e p a r a t o r s .

2 . T e s t S e c tio n A ssem bly

The t e s t s e c t i o n a s s e m b ly u s e d i n t h i s s tu d y was d e s ig n e d a n d c o n s tr u c t e d b y W alton T l J • The t e s t s e c t i o n c o n s i s t e d o f s i x s e p a r a t e u n i t s : I - an i n j e c t i o n e n t r a n c e s e c t i o n , I I - flo w d ev elo p m en t s e c t i o n , I I I - s e p a r a t e d s u b c h a n n e ls , IV - m ix in g s e c t i o n , V- s e p a r a te d s u b c h a n n e ls , V I- e x i t s e c t i o n ( s e e F ig . 4 -.2 ).

S e c t io n s I I and V w ere i d e n t i c a l i n c o n s t r u c t i o n , and I and VI w ere i d e n t i c a l e x c e p t f o r t h e i n j e c t i o n p o r t i o n o f s e c t i o n I .

A l l u n i t s w ere c o n s tr u c t e d o f a c r y l i c t o a llo w v i s u a l i d e n t i f i c a t i o n o f flo w re g im e s . The v a r i o u s u n i t s w ere b u t t e d t o g e t h e r u s i n g 0 .0 0 5 i u t h i c k n a t u r a l r u b b e r g a s k e ts hand c u t t o f i t t h e c r o s s -

3 0 .

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F ig u r e 4.«2 T e s t S e c t i o n A ssem bly