Correlations Predicting Frictional Pressure

8/19/2019 Correlations Predicting Frictional Pressure

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Int. J. Multiphase Flow

Vol. 15, No. 6, pp. 947-964, 1989 0301-9322/89 $3.00 + 0.00

Printed in Great Britain. All rights reserved Copy right ~ 1989 Pergam on Press/Elsevier

C O R R E L A T I O N S P R E D I C T I N G F R I C T I O N A L P R E S S U R E

D R O P A N D L I Q U I D H O L D U P D U R I N G H O R I Z O N T A L

G A S - L I Q U I D P I P E F L O W W I T H A S M A L L

L I Q U I D H O L D U P

J . H A R T

P. J . HAMERSMA'[ a n d J . M . H . FORTUIN

L a b o r a t o r i u m v o o r C h e m i s c h e T e c h n o lo g i e , U n i v e r si t e i t v a n A m s t e r d a m , N i e u w e A c h t e r g r a c h t 1 6 6 ,

1 0 1 8 W V A m s t e r d a m , T h e N e t h e r l a n d s

Received

15

February

1 9 8 8 ;

in revised for m

22

March

1989)

A b s t r a c t - - E x p e r i m e n t a l d a t a a n d c o r r e l at i o n s a v a i l a b le i n t h e l i t e r at u r e f o r t h e l i q u id h o l d u p ~ L a n d t h e

p r e s s u r e g r a d ie n t

A P T p / L

f o r g a s - l iq u id p ip e f lo w, g e n e r a l ly , d o n o t c o v e r th e d o m a in 0 < ~L < 0 .0 6 .

Re l i a b le p r e s s u r e - d r o p c o r r e l a t io n s f o r th i s h o ld u p r a n g e a r e imp o r ta n t f o r c a lc u la t in g f lo w r a te s o f

n a tu r a l g a s , c o n ta in in g t r a c e s o f c o n d e n s a te . I n th e p r e s e n t p a p e r a t t e n t io n i s f o c u s e d o n r e l i a b le

m e a s u r e m e n t s o f ~L a n d

A P T p / L

v a l u e s a n d o n t h e d e v e l o p m e n t o f a p h e n o m e n o l o g i c a l m o d e l f o r t h e

l iq u id - h o ld u p r a n g e 0 < ~L < 0 . 0 6. Th i s mo d e l i s c a l l e d th e a p p a r e n t r o u g h s u r f a c e mo d e l a n d i s r e f e r r e d

t o a s t h e A R S m o d e l . T h e e x p e r i m e n t a l r e s u l t s p r e s e n t e d i n t h i s p a p e r r e f e r t o a i r - w a t e r a n d

a i r - wa te r + e th y le n e g ly c o l s y s te ms wi th v a r y in g t r a n s p o r t p r o p e r t i e s in h o r i z o n ta l s t r a ig h t s mo o th g la s s

t u b e s u n d e r s t e a d y - s t a t e c o n d i t i o n s . T h e h o l d u p a n d p r e s s u r e g r a d i e n t v a l u e s p r e d i c t e d w i t h t h e A R S

m o d e l a g r e e s a t i s f a c t o r i l y w i t h b o t h o u r e x p e r i m e n t a l r e s u l t s a n d d a t a o b t a i n e d f r o m t h e l i t e r a t u r e

r e f e r r in g to s ma l l l i q u id - h o ld u p v a lu e s 0 < ~L < 0 .0 6 . F u r th e r , i t h a s b e e n s h o w n th a t in th e d o m a in

3 8 < o < 7 2 mP a m th e in t e r r a c ia l t e n s io n o f th e g a s - l iq u id s y s te m h a s n o s ig n i f i c an t e f fe c t o n th e l iq u id

h o ld u p . Th e p r e s s u r e g r a d ie n t , h o we v e r , i n c r e a s e s s l ig h t ly w i th d e c r e a s in g s u r f a c e t e n s io n v a lu e s .

Key Words:

g a s - l iq u id p ip e flow, l iq u id h o ld u p , p r e s s u r e d r o p , s m a l l li q u id h o ld u p , s t r a t i f i e d - w a v y f lo w

r e g ime , a n n u la r f lo w

I . I N T R O D U C T I O N

F o r a q u a n t i t a t iv e d e s c r i p t i o n o f g a s - l i q u i d f l o w in p i p e li n e s , i t is o f i m p o r t a n c e t o u n d e r s t a n d t h e

p h e n o m e n a i n t h is k i n d o f fl ow . D e s p i t e t h e n u m e r o u s th e o r e t i c a l a n d e x p e r i m e n t a l i n v e s ti g a t io n s

o n g a s - l i q u i d p i p e f lo w , n o r e li a b le f r i c t io n a l p r e s s u r e g r a d i e n t a n d l i q u i d - h o l d u p c o r r e l a t i o n s a r e

a v a i l a b l e , m a i n l y a s a r e s u l t o f t h e l a r g e n u m b e r o f v a r i a b l e s i n v o l v e d .

S o m e a u t h o r s (e .g . S t o r e k B r a u e r 1 9 80 ; O l u j ic 1 9 85 ; G r e g o r y e t a i 1 9 8 5 ; B a t t a r r a e t a l 1985;

M i il l e r- S t e in h a g e n H e c k 1 98 6) h a v e a t t e m p t e d t o f in d g e n e r a l m o d e l s f o r t h e fr i c ti o n a l p r e s s u r e

g r a d i e n t a n d l i q u id h o l d u p i n t w o - p h a s e p i p e fl o w b y c u r v e f i tt in g o n e x i st in g d a t a . A d i s a d v a n t a g e

o f t h i s a p p r o a c h i s t h a t t h e c o r r e l a t i o n s o b t a i n e d i n t h i s w a y a r e s t r o n g l y d e p e n d e n t o n t h e

c o m p o s i t i o n o f t h e d a t a b a n k s u s e d .

G e n e r a ll y , b e t t er a g r e e m e n t b e tw e e n e x p e r i m e n t a n d t h e o r y i s f o u n d w h e n p h e n o m e n o l o g i c a l

m o d e l s a r e u s e d f o r t h e d e s c r i p t i o n o f g a s - l i q u i d p i p e f lo w . M o s t o f th e s e m o d e l s h a v e b e e n a p p l i e d

t o s t r a t i f i e d f lo w ( e.g . C h e r e m i s i n o f f D a v i s 19 7 9 ; C h e n S p e d d i n g 1 9 8 1; K a d a m b i 1 98 1) , a n n u l a r

f l ow ( e. g. H o o g e n d o o r n 1 9 6 5; W a U is 1 9 7 0 ; H a s h i z u m e 1 98 5) a n d s l u g f lo w ( e.g . D u k l e r H u b b a r d

1 9 7 5 ; K o r d b y a n 1 9 8 5 ) .

I n a p r e v i o u s p a p e r H a m e r s m a H a r t (1 98 7) p r e s e n t e d a p h e n o m e n o l o g i c a l m o d e l t o c a l c u la t e

t h e v a l u e o f t h e f r i c t i o n a l p r e s s u r e g r a d i e n t i n g a s - l i q u i d p i p e f l o w w i t h a s m a l l l i q u i d h o l d u p

(EL < 0 .0 4 ) c o v e r i n g t h e s tr a ti fi e d , w a v y a n d a n n u l a r f l o w r e g im e s . A d i s a d v a n t a g e o f a p h e n o m e n o -

l o g ic a l a p p r o a c h i s t h a t i n p r a c ti c a l s i t u a t io n s , w h e r e , f o r e x a m p l e , h ig h p r e s s u r e s a n d / o r l a rg e p i p e

d i a m e t e r s o c c u r , o f t e n t h e p r e s e n t f l o w r e g i m e i s u n k n o w n ( e . g . S i m p s o n e t a l 1987).

A s t r i k i n g p h e n o m e n o n i n t h e e x i s t i n g l i t e r a t u r e c o n c e r n i n g t w o - p h a s e f l o w i s t h e l a c k o f

t h e o r e t i c a l a n d e x p e r i m e n t a l s t u d i e s o n h o r i z o n t a l g a s - l i q u i d p i p e f l o w w i t h s m a l l v a l u e s o f t h e

l i q u i d h o l d u p ( e.g . EL ~< 0 .0 6 ). T h i s t y p e o f f l o w m a y o c c u r , w h e n n a t u r a l g a s o f h i g h p r e s s u r e i s

t r a n s p o r t e d t h r o u g h p i p e li n e s . A t c e r t a i n p o s i t i o n s i n th e p i p e t r a c es o f li q u id c a n b e f o r m e d b y

t T o w h o m a l l c o r r e s p o n d e n c e s h o u l d b e a d d r e s s e d .

9 7


2/18

9 8 J HART

et a

retrograde condensation as a result of pressure decrease. The presence of these traces of liquid may

have a marked effect on the gas-liquid frictional pressure gradient. It can be shown (Hamersma

& Hart 1987) that in a pipe a liquid holdup of 0.005 may result in a rise of the pressure gradient

of up to 30%, as compared with single-phase gas flow.

In spite of this large effect of the presence of liquid on the pressure gradient in gas-l iquid pipe

flow, this kind of flow has not been studied extensively before. For example, Dukler e t a l . (1964)

stated that one source of experimental errors in two-phase flow data, in general, is the low accuracy

of low liquid-holdup data and that comparing these data (0.01


3/18

PRESSURE DROP AND LIQUID HOLDUP IN GAS LIQUID FLOW 949

2 . M O D E L

2.1. The fr ict ional pressure drop d Pr e

H a m e r s m a & H a r t 0 9 8 7 ) h a v e d e r iv e d a m o d e l f o r th e f r ic t io n a l p r e s s u re g r a d i e n t i n t h e g a s

p h a s e o f i s o t h e r m a l , s t e a d y - s t a t e g a s - l i q u i d f l o w t h r o u g h h o r i z o n t a l p i p e s w i t h s m a l l l iq u i d -

h o l d u p v a l u e s (EL < 0 . 0 4) in th e s t r a t i f i e d - w a v y , w a v y a n d a n n u l a r f l o w r e g i m e s . F o r t h i s k i n d o f

f l o w i t w a s s h o w n t h a t t h e l i q u i d , f lo w i n g a l o n g t h e t u b e w a l l , m a y b e c o n s i d e r e d a s a lo c a l

r o u g h n e s s o v e r a f r a c t i o n 0 = ~ t/2 1 t o f t h e t u b e w a l l , se e f ig u r e 1 . T h e i n t e r f a c i a l s h e a r s t r e s s zi

e x e r t e d b y t h e g a s p h a s e o n t h e l i q u i d f i l m m a y b e c o n s i d e r e d a s s h e a r s t r e s s e x e r t e d b y t h e g a s

p h a s e o n a r o u g h w a l l. T h e s e a s s u m p t i o n s l e a d t o 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 t h e t w o - p h a s e

p r e s s u r e d r o p A P T p :

L i 2

[ ]

w h e r e f TP i s t h e t w o - p h a s e f r i c ti o n f a c t o r a n d VG i s t h e a v e r a g e a x i al v e l o c i t y o f t h e g a s p h a s e ;

VG = UG /(I -- EL).

I f , h o w e v e r , t h e i n t e r f a c e v e l o c i t y vi m a y n o t b e n e g l e c t e d w i t h r e g a r d t o V G , [3 ] c h a n g e s i n t o

( s e e [A . 5 ] i n a p p e n d i x A w i t h f l = 0 ) :

~PGVG

~pG (2t~G ~ i- Vi ), [4]

w h e r e

0 -- ¢t/ 21r)

i s t h e w e t t e d f r a c t i o n o f t h e t u b e w a l l a n d f i is th e i n t e r f a c e f r i c t i o n f a c t o r . I n

[ 3] a n d [ 4] t h e t w o - p h a s e f r i c t i o n f a c t o r f TP i s g i v e n b y

f ~ , = (1 - 0 ) ' f ~ + 0 . f . [5 ]

T h e s i n g l e -p h a s e f r i c ti o n f a c t o r f G , d u e t o d r a g e x e r t e d b y t h e g a s p h a s e o n t h e n o n - w e t t e d p a r t

o f t h e t u b e w a l l, c a n b e o b t a i n e d f r o m ( E c k 1 9 73 )

0 . 0 7 7 2 5

fG = [1Ogl0 Re G /7)] 2 ' [6 ]

w h i c h i s v a l i d f o r 2 1 0 0 < R e G < l 0 s , w h e r e

DVGPG

R e G

= - - [7 ]

T h e i n te r r a c ia l f r i c ti o n f a c t o r f r e s u l ts f r o m d r a g e x e r t e d b y t h e g a s p h a s e o n a r o u g h s u r f a c e , i. e.

t h e r ip p l i n g l i q u id p h a s e , a n d i s o b t a i n e d f r o m ( E c k 1 9 7 3)

0 . 0 6 2 5

, ,

l O g l0 k -~eG '~ 3 . 7 1 5 D J _ ]

w h e r e

k / D

i s t h e r e l a t iv e s a n d r o u g h n e s s o f t h e in n e r t u b e w a l l.

E x p e r i m e n t a l r e su l ts ( H a m e r s m a H a r t 1 9 8 7 ) s h o w e d t h a t t h e v a l u e o f t h e a p p a r e n t r o u g h n e s s

k

o f t h e l i q u i d f i lm ,

i . e . k - - 6 M ^ x -

6 M I ~ ( s e e a l s o f i g u r e I ) , i s e q u a l t o

~

2 . 3 . 6 , [ 9 ]

i n w h i c h 6 is t h e a v e r a g e t h i c k n e s s o f th e l i q u i d f il m o n t h e

wet ted

p a r t 0 o f t h e t u b e w a l l ; 6 i s

a s s u m e d t o b e c o n s t a n t i n t h e t a n g e n t i a l d i r e c t i o n . R e f e r r i n g t o f ig u r e l i t c a n e a s i l y b e d e r i v e d

t h a t

E L D i f 6

~ 4--'0-' D ~ 1. [10 ]

I t m u s t b e n o t e d t h a t t h e c o r r e l a t i o n o f [8 ] i s b a s e d o n t h e so - c a l le d s a n d r o u g h n e s s . T h i s t y p e

o f r o u g h n e s s t e n d s t o b e d i f f e re n t f r o m t h e r o u g h n e s s i n g a s - l i q u i d f l o w . I n t h e l a t te r c a s e


4/18

950 J. HART

e t a t

F i g u r e I . S c h e m a t i c r e p r e s e n t a t i o n o f g a s - l i q u i d f l o w w i th a s m a l l l i q u id h o l d u p i n s t r a i g h t s m o o t h t u b e s .

I n t h e A R S m o d e l i t i s a s s u m e d t h a t i n a c r o s s - s e c t i o n o f t h e t u b e t h e w a l l i s c o v e r e d o v e r a f r a c t i o n

0 = a / 2 n ) b y a l i q u i d l a y e r o f c o n s t a n t a v e r a g e t h i c k n e s s ~ in t h e t a n g e n t i a l d i r e c t i o n . T h e v a l u e o f 0

i s a s s u m e d t o b e d e p e n d e n t o n t h e k i n e t i c e n e r g y o f t h e l i q u i d s e e [ B .7 ]) . I n t h e a x i a l d i r e c t i o n t h e t h i c k n e s s

o f t h e l i q u i d l a y e r v a r i e s i n t h e d o m a i n 6 m tn < 6 < d im ax .

r i p p l i n g o c c u r s . I t w a s r e c o g n i z e d ( e .g . S c h l i c h t i n g 1 9 80 ) t h a t r i p p l e s , p e r p e n d i c u l a r t o t h e f l o w

d i r e c t i o n , a r e t h e c a u s e o f a l a r g e r f r ic t i o n t h a n t h a t w h i c h c a n b e e x p e c t e d f r o m t h e i r g e o m e t r y ,

b a s e d o n s a n d r o u g h n e s s . T h e r e f o r e , t h e v a l u e o f 2. 3 • 6 f o r t h e r o u g h n e s s k o f t h e l i q u id f il m is

p r o b a b l y l a r g e r t h a n t h e r e a l r o u g h n e s s o f t h e l iq u i d f il m . B u t s in c e t h e r e is n o c o r r e l a t i o n

a v a i l a b l e , w h i c h a c c u r a t e l y d e s c r i b e s t h e fr i c ti o n f a c t o r in p i p e s w i t h r i p p l i n g r o u g h n e s s , [8 ] i n

c o m b i n a t i o n w i t h [ 9] a n d [ 10 ] i s u s e d . T h e r e i s n o d o u b t t h a t t h e f a c t o r 2 . 3 i s d e p e n d e n t o n t h e

a m p l i t u d e a n d f r e q u e n c y o f t h e w a v e s o f t h e l iq u i d f il m , a n d , t h e r e f o r e , a l s o o n t h e f lo w r e g im e

a n d t r a n s p o r t p r o p e r t ie s o f th e g a s - l iq u i d s y s te m .

2 2 The w et t ed wal l f rac t ion 0

F o r t h e w e t t e d f r a c t i o n 0 o f t h e t u b e w a l l d u r i n g g a s - l i q u i d f l o w w i t h a s m a l l li q u id h o l d u p i n

t h e s tr a t i fi e d - w a v y f lo w r e gi m e , t h e f o ll o w i n g e q u a t i o n w a s d e r i v e d in a p r e v i o u s p a p e r ( H a m e r s m a

& H a r t 1 9 8 7 ) :

0 = ( 1 ) a r c c o s ( l - C , F r ), [11 ]

i n w h i c h t h e c o n s t a n t C t w a s d e t e r m i n e d e m p i r i c a ll y a s C ~ = 0 . 66 . E q u a t i o n [1 1] is v a l i d f o r

F r ~< 2 / C ~ . i f F r > 2 /C ~ t h e n 0 = I , i. e. a n n u l a r f l o w o c c u r s . I n [ 1 1] t h e m o d i f i e d F r i s d e f i n e d b y

Fr = fP L ~ / V ~ ~ [1 2]

\av \gD

w h e r e

Ap = PL

P G a n d g i s th e a c c e l e r a t i o n d u e t o g r a v i t y .

A n i m p l i c a t i o n o f [ i 1 ] i s t h a t t h e v a l u e o f t h e w e t t e d w a l l f r a c t io n 0 s h o u l d a p p r o a c h z e r o f o r

t h e h y p o t h e t i c a l s i tu a t i o n t h a t F r - - ,0 . F r o m o u r m e a s u r e m e n t s , h o w e v e r , i t a p p e a r e d t h a t f o r s m a l l

v a l u e s o f F r ( F r < 0 . 2 5 ) a c e r t a i n m i n i m u m v a l u e o f t h e w e t t e d w a l l f r a c t io n 0 = 0 0 i s h e l d b y t h e

l iq u i d p h a s e . S i n c e f o r s m a l l v a l u e s o f F r t h e d i s c r e p a n c y b e t w e e n t h e 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 t h e w e t t e d w a l l f r a c t i o n a n d v a l u e s c a l c u l a t e d w i t h [1 i ] c o u l d b e > 3 0 0 % , a m o r e a c c u r a t e

d e s c r i p t i o n o f 0 i s r e q u i r e d . I n a p p e n d i x B a m o d e l h a s b e e n d e r iv e d , r e s u l t i n g i n t h e f o l lo w i n g

c o r r e l a t i o n :

0 = 00 + C 2 F r ° 'sS, [13]

w h e r e t h e c o n s t a n t C 2 h a s b e e n o b t a i n e d f r o m e x p e r i m e n t s (C 2 = 0 . 26 ; s e e s e c t i o n 4 .1 ) . T h i s

e q u a t i o n g i v e s a b e t t e r d e s c r i p t i o n o f t h e e x p e r i m e n t a l r e s u l ts ( s e e s e c t i o n 4 . 1 ) t h a n [ I 1] a n d i s v a l i d

fo r 0 ~< 1 o r F r ~< [ (1 - 0 0 ) / C2 ] 1 TI. I f F r > [ ( I - 0 0 )/ C2 ] I ra , an n u l a r f l o w o c cu r s an d 0 = 1 . T h e v a l u e

o f 00 , w h i c h c a n b e r e g a r d e d a s th e m i n i m u m v a l u e o f 0 i f F r ~ 0 , h a s a ls o b e e n d e r iv e d i n a p p e n d i x

B a n d c a n b e a p p r o x im a t e d b y

0 0 ~ 0 . 5 2 ~ 374.

[14]


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P R E SS U R E D R O P A N D L I Q U ID H O L D U P I N G A S L I Q U I D F L O W

951

2 .3 . Th e l i q u id h o ld u p e t

B u t t e r w o r t h ( 19 75 ) f o u n d t h a t m o s t o f t h e l i q ui d - o r g a s - h o l d u p c o r r e l a ti o n s i n t h e l i te r a t u r e

c a n b e r e w r i t te n i n t o t h e f o l l o w i n g e q u a t i o n :

e L a ( U L ~ b ( p L ~ C ( L ~,

[ 1 5 ]

w h e r e % a n d ~?L a r e t h e d y n a m i c v i sc o si t ie s f g a s a n d l i q ui d , r e sp e ct i v el y . h e v a l u e s o f a, b , c

a n d d i n [ 1 5] a p p e a r t o b e c o n s t a n t f o r a l i m i t i n g r a n g e o f v a l u e s o f t h e s up e rf i ci a l e l oc i ti e s G

a n d U L a n d t h e t r a n s p o r t p r o p e r ti e s . E q u a t i o n s [ I] a n d [ 2] a r e s p e c i a l c a s e s o f [ 15 ].

F r o m t h e f o r c e b a l a n c e u n d e r s t ea d y - s ta t e c o n d i t i o n s w e d e r i v e d t h e f o l l o w i n g e q u a t i o n

d e s c r i bi n g t h e l i qu i d h o l d u p i n th e st ra ti fi ed , a v y a n d a n n u l a r f l o w r e g i m e s ( s e e a p p e n d i x C ) :

e L

U L [

1 - e L u-~ \ f P G / J fo r { L ~ 0.06 [16]

w h e r e f L is t h e f r i c t i o n f ac t o r r e f e r r in g t o t h e s h e a r s t r e ss b e t w e en t h e l i q u i d f i lm an d t u b e w a l l .

I n s ec t i o n 4 . 2 it w il l b e s h o w n o n t h e b a s i s o f ex p e r i m en t a l r e s u l t s t h a t t h e r a t i o f L / f i s a f u n c t i o n

o f t h e s u p e r f i c i a l R e y n o l d s n u m b e r ResL only Resg

= D U L P L / q L .

3 . T E S T F A C I L I T Y

3 .1 . Te s t se c t i o n a n d a c c e sso r ie s

A s ch em a t i c r ep r e s e n t a t i o n o f t h e t e s t f a c i l it y i s g i v en i n fi g u r e 2 . T h e ex p e r i m en t s w e r e ca r -

r i ed o u t i n a s tr a i g h t , h o r i z o n t a l co p p e r t u b e ( 1 2 ) w i t h i .d . 51 m m an d l en g t h ap p r o x ~ - 1 7 m , i n c l u d -

i n g a 1 .4 m g l a s s s ec t i o n (1 3) . T h e l a r g e s t d ev i a t i o n o f t h e t u b e f r o m t h e h o r i zo n t a l w as a b o u t 0 . 0 5 ° .

A i r s a t u r a t ed w i t h w a t e r w as s u p p l i ed f r o m a w a t e r - r i n g co m p r e s s o r ( 1 ) , p r o v i d i n g a r e l a t i v e

h u m i d i t y o f t h e a i r o f > 9 0 % . T h i s i s e s s en t ia l i n m e a s u r i n g l o w l i q u i d - h o l d u p v a l u e s , b e c a u se

e v a p o r a t i o n o f p a r t o f t h e l iq u i d in t h e p i p e m a y h a v e a m a r k e d e f fe c t o n t h e m e a s u r e d

l i q u i d - h o l d u p v a l u e . T h e a i r f l o w r a t e w a s c o n t r o l l e d b y o n e o f t h e t w o r o t a m e t e r s ( 3 ) c o v e r i n g

the r an ges 2 .77-19 .4 an d 11 .1-83 .3 d m 3 s - ' , r espec t ive ly . Th e l iqu id w as suppl i ed f rom a ves se l (7) ,

u s i n g a r o t a r y d i s p l ace m e n t p u m p ( 8 ), t o o n e o f th e t h r ee r o t am e t e r s ( 11 ), w i t h d i f f e ren t m e as u r i n g

r an g es . T h e l i q u i d w as i n j ec t ed t h r o u g h an i n j ec t i o n p o r t ( 6) a t t h e b o t t o m o f th e p i p e .

A t a d i s t an ce o f ab o u t 4 m ( 8 0 p i p e d i a ) f r o m t h e i n j ec t i o n p o i n t a p r ec i s io n g l a s s t u b e ( I 3 ) w as

i n s e r t e d , w h i c h w a s p r o v i d e d w i t h a n a n g l e g a u g e t o m e a s u r e t h e a n g l e o v e r w h i c h t h e t u b e w a s

w e t t ed b y t h e l i q u i d p h as e w i t h an a ccu r a cy o f 2 ° T w o p a i r s o f p r e ss u r e t ap s ( 14 ) w e r e l o ca t ed

t3

\ / 14 • 17 ,

5 \ ;

6 1 2 1 3

i o I = E F - - :- - - ~ 7

I

18

O ,

20

T

19

18

l

_ 1

F i g u r e 2 . T e s t f a c i l i t y : 1 , w a t e r - r i n g c o m p r e s s o r ; 2 , g a s o u t l e t ; 3 , g a s f l o w m e t e r ; 4 , p r e s s u r e i n d i c a t o r ;

5 , s i e v e p l a t e ; 6 , l i q u i d i n j e c t i o n ; 7 , l i q u i d s t o r a g e t a n k ; 8 , r o t a r y d i s p l a c e m e n t p u m p ; 9 , l i q u i d b y p a s s ;

I 0 , m a g n e t i c t h r e e - w a y v a l v e ; I 1 , l iq u i d f l o w m e t e r ; 1 2 , c o p p e r t u b e L = 1 7 m , D = 0 . 0 51 m ) ; 1 3 , G l a s s

t u b e w i t h a n g l e g a u g e ; 1 4, p r e s s u r e t a p s ; 1 5 , w a t e r m a n o m e t e r : 1 6, m i c r o m a n o m e t e r ; 1 7, g a s - l i q u i d

s e p a r a t o r ; 1 8 , l i q u i d r e c e i v i n g b i n s ; 1 9 , t e m p e r a t u r e i n d i c a t o r : 2 0 , s w i t c h o f m a g n e t i c v e n t i l e .


6/18

952 J . HART et u l ,

at distances of 6 and 11 m (120 and 220 pipe dia) from the liquid-injection point. One pair o f

pressure taps, dia 1 mm, was located at the top o f the tube, to measure the pressure gradient over

the gas phase with a precision micromanometer (16), having an accuracy of 0.4--4 Pa, depending

on the slope of the manometer tube. The other pair of pressure taps, located at the bottom of the

tube, was used in combination with a water manometer (15) to measure the pressure gradient across

the liquid phase. During our experiments no significant differences were found between values of

the time-averaged static-pressure gradient obtained with the two methods.

At the end of the tube the liquid and gas were separated by means of a gas-liquid separator ( 7).

After shutting off the liquid injection with a quick-closing magnetic ventile (10) and removing

the remaining liquid from the tube with the saturated air flow, the liquid holdup was determined

by weighing. The repeatabili ty and reproducibility of this method (>99 % for EL > 0.01) enabled

us to measure minimum liquid-holdup values of 0.0012 with an accuracy of about 96%.

3.2. Fluid systems and test condi t ions

Measurements were carried out at superficial air velocities ranging from about 5 to 30 m s- ',

superficial liquid velocities ranging from 0.25.

1 0 3 t O

80" 10-3ms -' and temperatures in the

range 15-25°C. Flow regimes observed during the experiments were stratified, stratified-wavy

and annular flow, as has been indicated in the flow pattern map of Mandhane et al. (1974) given

in figure 3.

For the determinat ion of the effect of the interfacial tension a on the wetted wall frac tion 0, the

liquid holdup EL and the frictional pressure gradient APTp/L , water was used in combination with

a surface active agent (Tween 80, obtained from J. T. Baker Chemicals B. V.) to diminish the

interfacial tension without changing other transport properties. Care was taken not to exceed

the concentration where foaming started to occur.

The effect of the liquid viscosity on the liquid holdup ~C, the wetted wall fraction 0 and the

two-phase pressure gradient

A P r p / L

was investigated for several air-water+ethyleneglycol

mixtures. The transport properties of the fluid systems used are listed in table 1.

4. EXPERIMENTAL RESULTS

4.1. The w et ted f rac t ion 0 of the tube wal l

Figure 4 shows the experimentally determined values of the wetted wall fraction 0 as a

function of the modified Fr [Fr =

p L / A p ) e L / g D ]

for the air-water and an a ir-water + Tween

i0 0

%

m s - 1

O O1

O OO1

O I 1 0 I0 0 I00 0

u G / m s l)

F i g u r e 3 . M a n d h a n e e t a l . s ( 1 9 7 4 ) f l o w p a t t e r n m a p f o r

h o r i z o n t a l g a s - l i q u i d p i p e f l o w . T h e h a t c h e d a r e a

c o v e r s t h e

r e g i o n t r e a t e d i n t h is p a p e r .

1 . 00 ~ O E X 9 ~ D O 9

s t r a t i f i e d - i w a v y -

w a v y a n n u l a r

0 . 7 5

+

O ~

0 . 5 0 -

0 . 2 5

F r

i I

0 I 2 3 5

F i g u r e 4 . 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 t h e w e t te d

w a l l f r a c t io n 0 a s a f u n c t i o n o f t h e m o d i f i e d F r : + , a i r -

w a t e r s y s t e m ( a = 7 2 m N m - i) ; ( 3 ,

a i r - w a t e r

+ 0 .11

w t

T w e e n 8 0 s y s t e m ( a = 3 8 m N m - ) .

I

a n n u l a r

+ 1 +

++


7/18

PRESSURE DROP AND LIQUID HOLDUP IN GAS-LIQUID FLOW 953

Table 1. Transport properties of the fluid systems used at 20°C

Dynamic Surface

viscosity tension Density

Fluid system (mPa s) (raN m- t) (kg m-3)

Humid air 0.0178 - - 1.23

Water 1.00 72 998

Water + 0.008 wt Tween 80 1.00 60 998

Water + 0.020 wt Tween 80 1.00 47 998

Water + 0.110 wt Tween 80 1.00 38 998

Water + 11.5 wt glycol 1.13 65 1011

Water + 25 wt glycol 1.76 63 1025

Water + 35 wt glycol 2.47 61 1039

Water + 80 wt glycol 8.50 56 1091

a = 3 8 m N m - ~ ) s y s t e m . F r o m t h is f i g ur e it a p p e a r s t h a t n o s i g n i fi c a n t e f fe c t o f th e s u r f a c e t e n s i o n

a o n t h e v a l u e o f th e w e t t e d w a l l f ra c t i o n 0 c a n b e o b s e r v e d f o r 0 < 0 .6 .

T h e v a l u e o f F ro nt, i.e . t h e F r w h e r e a n n u l a r f l o w st a r t s t o o c c u r , d e c r e a s e s s l ig h t ly w i t h

d e c r e a s i n g s u r f a c e t e n s i o n a . T h e r e i s h o w e v e r a p o o r c o r r e l a t i o n b e t w e e n F r ¢ , , a n d tr. T h e r e f o r e ,

i t s h a l l b e s t a t e d t h a t f o r F r < 2 a d e c r e a s e i n t h e s u r f a c e t e n s i o n h a s l i tt le e f f e c t o n t h e v a l u e o f

t h e w e t t e d w a l l f r a c t io n . F o r t h e r e g i o n F r < 2 , i t i s u n l i k e l y t h a t a n n u l a r f l o w o c c u r s in g a s - l i q u i d

f l o w t h r o u g h a h o r i z o n t a l , s t r a ig h t t u b e . F o r t h e r e g io n 2 < F r < 4 , t h e b o u n d a r y b e t w e e n w a v y

a n d a n n u l a r f lo w d e p e n d s o n t h e v a l u e o f t h e s u rf a c e t e ns io n . F o r F r > 4 , a n n u l a r f lo w m a y b e

ex p ec t ed , i . e . 0 = 1 .

F r o m e x p e r i m e n t s w i th t h e a i r - w a t e r a n d a i r - w a t e r + g l y c o l m i x t u r e s t h e v a lu e o f t h e c o n s t a n t

C 2 i n [1 3 ] t u r n e d o u t t o h a v e a b e s t - f it v a l u e o f C 2 = 0 . 2 6 . I n f ig u r e 5 , t h e 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 th e w e t t e d w a l l f r a c t io n h a v e b e e n p l o t t e d a s a f u n c t io n o f th e v a l u e s c a l c u l a t e d w i t h [1 3]

a n d [1 4] . A n a v e r a g e r e l a t i v e d e v i a t i o n ( R D ) o f a b o u t 1 9 % i s o b t a i n e d i f [ 13 ] a n d [ 14 ] a r e u s e d

( s ee a ls o s e c t i o n 4 .4 ) . T h i s i s c o n s i d e r a b l y b e t t e r t h a n t h e a v e r a g e R D o f a b o u t 8 0 % i f [ 11 ] i s

a p p l i e d . T h u s , t h e p r e s e n t m o d e l g i v e s a b e t t e r d e s c r i p t i o n o f t h e w e t t e d w a l l f r a c t i o n 0 th a n t h a t

p u b l i s h e d p r e v io u s l y ( H a m e r s m a & H a r t 1 98 7).

4 .2 . The l i qu id ho ldup q .

A c c o r d i n g t o t h e c o r r e l a t i o n o f se c t i o n 2 .3 , th e v a l u e o f t h e l iq u i d h o l d u p i n h o r i z o n t a l g a s - l i q u i d

p i p e f l o w c a n b e o b t a i n e d f r o m [ 16 ]:

E___L_L_ UL i fo r EL ~ 0 . 0 6 .

- U o

R e a r r a n g i n g [ 16 ], w e o b t a i n

fL UG E__ 1 . P._~C

f i = I L P L

[17]

O u r e x p e r i m e n t a l r e s u l t s s h o w e d t h a t t h e r e i s a p r o n o u n c e d c o r r e l a t i o n b e t w e e n t h e r a t io f L / f

c a l c u l a t e d f r o m [ 17 ], a n d t h e su p e r f ic i a l R e y n o l d s n u m b e r o f th e l iq u i d p h a s e R e s L ,

fL = 108 r e s °7 -'6, [18]

f

i n w h i c h 1 08 a n d - 0 . 7 2 6 a r e e m p i r ic a l c o n s t a n t s . T h i s e q u a t i o n i s m u c h s i m p l e r t h a n c o r r e l a t i o n s

o b t a i n e d b y u s i n g t h e s e p a r a t e f r i c t i o n f a c t o r s f L a n d f , , a n a p p r o a c h w h i c h i s g e n e r a l l y a c c e p t e d

i n t h e li t e r a t u r e ( L o c k h a r t & M a r t i n e l l i 1 9 49 ; T a i t e i & D u k l e r 1 9 76 ; C h e n & S p e d d i n g 1 9 83 ;

O l i e m a n s 1 9 8 7 ) .

S u b s t i t u t i o n o f [ 1 8 ] i n t o [ 16 ] g i v e s t h e f o l lo w i n g c o r r e l a t i o n c o n t a i n i n g t w o e m p i r i c a l c o n s t a n t s :

__'L = UL~ 1 + [1 0 .4 ResL0.S6s(p~_)tal}. [19]

1 - - CL UG l

F i g u r e 6 s h o w s t h a t f o r fi ve d i ff e r e n t g a s - l i q u i d s y s t e m s a g o o d a g r e e m e n t is o b t a i n e d b e t w e e n t h e

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 t h e li q u i d h o l d u p a n d t h e v a l u e s c a l c u l a t e d w i t h [1 9] .


8/18

95 J . HAR T

e t a l

1 00

0 . 7 5

O. 50

0ex

+ 2 0 / / /

/

-20

, / /

÷

÷ • 4

/ / / +

i÷ , , ,+ ~,.

.08

.06

.04

EL,ex

+10

/ / / / - I 0

o.25 t- . ~

0 0.25 0.50 0.75 1.00

F i g u r e 5 . 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 t h e w e t te d

wall fraction 0 for the a ir-water an d a ir-water + ethylene-

glycol systems as a function of the values calculated with

[13], where C : = 0.26.

.02

~L,th

0 .02 .04 .06 .08

Figure 6. Experimentally determined values of the liquid

holdup ~L for the air-water and four different air-water

ethyleneglycol systems as a function of values calculated

with [19]. The t ranspor t properties are listed in table 1.

A d d i t i o n a l m e a s u r e m e n t s s h o w e d t h a t t h e l iq u i d - h o l d u p c o r r e l a t i o n [1 9] a l so h o l d s f o r m e a s u r e -

m e n t s c o n c e r n i n g lo w l i q u i d - h o l d u p v a l u e s p e r f o r m e d a t o u r l a b o r a t o r y i n a 1 5 m m i .d . t u b e o n

a n a i r - w a t e r s y s te m . T h i s r e s u l t is im p o r t a n t b e c a u s e i t d e m o n s t r a t e s t h a t t h e e f fe c t o f t h e t u b e

d i a m e t e r o n t h e v a l u e o f t h e l i q u i d h o l d u p is a ls o i n c l u d e d i n [1 9 ].

B e s id e s t h e c o m p a r i s o n o f l i te r a t u r e c o r r e la t i o n s , p r e s e n t e d in a p r e v i o u s p a p e r ( H a m e r s m a &

H a r t 1 98 7) , w e h a v e c o m p a r e d E a t o n ' s c o r r e l at i o n ( E a t o n e t al 1967) a nd c or re l a t i on [19] i n t he

p r e s e n t p a p e r u s in g th e d a t a o f A n d r e w s ( 1 9 6 6 ; t h e E a t o n d a t a ) , M i n a m i ( 1 9 8 3) a n d o u r

e x p e r i m e n t a l r e s u l t s . I t w a s f o u n d t h a t :

( 1 ) T h e h o l d u p v a l u e s p r e d i c t e d w i t h [ 1 9 ] a g r e e w i t h

• o u r e x p e r i m e n t a l r e s u lt s ( 0 .0 0 1 2 < EL < 0 . 0 6 ) w i t h i n a v e r a g e R D l i m i t s, R D ( q )

( s ee a l s o s e c t i o n 4 .4 ) , o f a b o u t 1 0 % ,

• A n d r e w s ' s e x p e r i m e n t a l r e s u lt s ( n a t u r a l g a s - w a t e r s y s te m ; D = 0 . 0 52 5 m ;

1 5 < P < 4 0 b a r ; 0 . 0 0 6 < EL < 0 . 7 5 ) w i t h i n a v e r a g e R D l im i t s o f a b o u t 1 7 %

(see f igure 7) ,

• M i n a m i ' s e x p e r i m e n t a l r e s u lt s ( a i r - k e r o s e n e s y s te m ; D = 0 . 0 7 8 m ;

3 < P < 7 b a r ; 0 .0 1 < ¢ t < 0 .4 3 ) w i t h i n a v e r a g e R D l im i t s o f a b o u t 2 3 % ( s ee

f igure 7) ,

( 2 ) T h e h o l d u p v a l u e s p r e d i c t e d w i t h E a t o n ' s m u l t i - p a r a m e t e r c o r r e l a t i o n ( E a t o n

et al 1967)

• a g r e e w i t h A n d r e w s ' s h o l d u p d a t a ( 0 . 0 0 6 < EL < 0 . 7 5 ) w i t h i n a v e r a g e R D l im i t s

o f a b o u t 2 0 % ,

• d i f fe r c o n s i d e r a b l y f r o m o u r e x p e r i m e n t a l re s u lt s (0 .0 0 1 2 < E L < 0 . 0 6 ) ; a n

a v e r a g e R D o f a b o u t 4 0 0 % h a s b e e n o b ta i n e d .

F u r t h e r , o u r m e a s u r e m e n t s s h o w e d t h a t t h e v a lu e o f t h e li q u id h o l d u p is n o t a f f e c t e d b y t h e v a l u e

o f t h e i n t e r f a c i a l t e n s i o n 3 8 < a < 7 2 m P a m . T h i s h o l d s o n l y f o r c o n d i t i o n s w h e r e t h e l i q u i d

e n t r a i n m e n t is n e g li g i b le . W i ll e ts e t al ( 1 9 8 7 ) s h o w e d t h a t a b o v e a c r i t i c a l l i q u i d - f i l m f l o w r a t e t h e

l iq u i d e n t r a i n m e n t is s tr o n g l y d e p e n d e n t o n t h e v a l u e o f t h e i n t er f a c ia l t e n s io n .

4 3 T h e p r e s s u r e g r a d i e n t A P r p / L

T h e v a l u e o f t h e t w o - p h a s e p r e s s u r e g r a d i e n t A P T p / L c a n b e o b t a i n e d w i t h t h e A R S m o d e l

a c c o r d i n g t o t h e p r o c e d u r e g i v e n i n a p p e n d i x D .

F i g u r e 8 s h o w s 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 es o f t h e t w o - p h a s e f r i ct i o n a l p r e s s u r e g r a d i e n t

A P T p / L

f o r a i r - w a t e r a n d a i r - w a t e r + e t y l e n e g l y c o l s y s t em s a s a f u n c t i o n o f th e v a l u e s c a lc u l a t e d

a c c o r d i n g t o t h e a b o v e - m e n t i o n e d p r o c e d u r e a f t e r s u b s t it u t i o n o f th e c a l c u la t e d v a l u e s o f t h e li q u id


9/18

PRESSURE DROP AND LIQUID HOLDUP IN GAS LIQUID FLOW 955

/ /

+ 2 0 /

0 . 8 /

E L . e x / / / / / l /

, J , ~ ~ . .

-2o

.6 ,, / ;,,,

,,,g'~ - +,'

O

2

~ 1 I 5 t h

0 0.2 0.4 0.6 0.8

Figur e 7 . Expe r imenta l ly de te r mined l i t e r a tur e da ta of the

l iqu id ho ldu p ~ t f or the na tu r a l gas - w a te r + ; A ndr ew s

1966) and a i r - ke r ose ne O ; M inam i 1983) systems as a

f unc t ion of va lues ca lcu la ted w i th [ 19] of the A R S mode l.

+ 1 0 %

/

/

6 0 0 [

//÷÷+/~,/**~

+/ / -10

/ ÷ / +

4 0 0

.oo ]

0 100 200

3 0 0

400 500 600

Figur e 8 . Exper imenta l ly de te rmined va lues of the pr es sur e

gr ad ien t APtr /L for the air -water and four dif ferent

air -w ater + ethyleneglycol systems as a func tion of values

ca lcu la ted w i th the A R S mo de l s ee appendix D ) .

h ol d u p e L , [19 ] , an d th e w e t te d w a l l f r ac t i on 0 ( [13 ] an d [14 ] an d

C 2

= 0 .26 ) . I t c an b e s e e n th at

a g o o d a g r e e m e n t i s f o u n d b e t w e e n p r e d i c t e d a n d m e a s u r e d v a l u e s o f t h e p r e s s u r e g r a d i e n t .

A d d i n g s u r f a c e - a c t i v e a g e n t s t o t h e w a t e r h a s s o m e e f f e c t o n 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 t h e p r e s s u r e g r a d i e n t c o m p a r e d w i t h 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 w i t h t h e a i r - w a t e r s y s t e m .

A s m e n t i o n e d i n t h e i n t r o d u c t i o n , t h e e x i s t en c e o f w a v e s i n g a s - l i q u i d f l o w i s o f i m p o r t a n c e f o r

t h e p r e s su r e g r a d i e n t o f t h e f l o w i n g t w o - p h a s e s y s t e m . I t i s k n o w n ( F r i ed e l 1 9 77 ) t h a t a d e c r e a se

i n th e s u r fa c e t e n s io n o f th e fl o w i n g g a s - l i q u i d s y s t e m p r o m o t e s t h e f o r m a t i o n a n d m a g n i f i c a t io n

o f w a v e s o n a l i q u id s u rf a c e . T h e r e f o r e , i t c a n b e e x p e c t e d t h a t t h e r o u g h n e s s k ( k

= 6 M A X - - 6 M I N

of th e l i q u i d f i l m an d , c on se q u e n t l y , th e p r e s su r e gr ad i e n t ar e i n c r e ase d . Th i s b e h av i ou r i s ve r i f i e d

i n f i g u r e 9 . I n t h i s f i g u r e a c o m p a r i s o n h a s b e e n m a d e b e t w e e n m e a s u r e m e n t s o n t h e a i r - w a t e r

6 0 0

5 0 0

4 0 0

300

200

100

( a P T P / L) x

- I

P a m

+ 1 5 %

/

/

e

~ T ~ r.)

P a m I

o lo o 200 3o0 4o0 5o0 6o0

Figur e 9 . Exper imenta l ly de te r mined va lues of the pr es sur e

g r a d i e n t APrp/L s a f u n c t i o n o f t h e v a l u e s c a l c u l a t e d i t h

t h e A R S m o d e l ( s e e a p p e n d i x D ) : + , a i r - wa t e r s y s t em

( a = 7 2 m N m ); ( 9, a i r - wa t e r + 0 . I w t % T w c c n 8 0 sy s -

t e m (or = 3 8 m N m - ' ) .

6 0 0

+ e L = 0 . 0 4 0

, , ' - I -

P .

~ - 1 x

~ C o o o s

, g - + /

I ~ . J / - , '

4 0 0 , ~ + ~ ,

2 ~ ~

( m s I )

0 i 0 2 0 3 0 4 0

F i g u r e I 0. P r e s s u r e g r a d i e n t a l u e s

A P r p L

d u r i n g a i r - w a t e r

f l o w i n a h o r i z o n t a l t u b e ( d i a 5 1 r a m ) a s a f u n c t i o n o f t h e

s u p er f i ci a l a s v e l o c i t y U G f o r d i f fe r e n t a l u e s o f t h e l i q u i d

h o l d u p ~ L T h e m a r k e d p o i n t s r e f e r t o e x p e r im e n t al l y

d e t e r m i n e d v a l u e s o f t h e p r e s s u r e g r a d i e n t. T h e s o l i d l i n e

r e p r e s en t s t h e c a l c u l a t e d p r e s s u r e g r a d i e n t i n a h o r i z o n t a l

s t r ai g h t s m o o t h t u b e d u r i n g s i n g l e - p h a s e a i r f l o w . T h e

d a s h e d l i n es h a v e b e e n c a l c u l a t ed w i t h t h e A R S m o d e l

( s cc A p p e n d i x D ) u s i n g a v e r ag e v a l u e s o f t h e t r an s p o r t

p r o p e rt i e s f o r t h e a i r -w a t e r s y s t e m ( P o = 1 . 2 0 g m - 3 ; P L =

9 9 8 k g m - 3 ; r / o = 1 . 7 8 . 1 0- s P a s ; ~ L = 1 . 0 2 ' 1 0 - 3 P a s ) .

M F I ~ '6 - -G


10/18

956 J HARTet al

sy s tem an d t h e a i r -w a t e r + 0 . 1 1 w t % T w een ( a = 3 8 m N m -~) sy s tem . A n i n c rea se o f ab o u t 1 5 %

in the exper imenta l ly de t e rmined pressure grad ien t i s obse rved i f t he surface t ens ion i s decrease d

b y ab o u t 5 0 % . S i nce th e u se o f su r f ace -ac ti v e ag en ts d o es n o t p e rm i t q u an t i t a ti v e en u n c i a ti o n s fo r

t h e sy s t em co n s i d e red , w e w i l l o n l y r em ark o n t h e p h en o m en o n .

The l a rge increase in the pressure grad ien t due to the presence o f smal l quant i t i es o f li qu id , as

com par ed wi th s ing le-phase gas f low, i s c l ear ly dem ons t ra t ed in f igure 10 for t he a i r -w ater sys t em.

Fro m f igure 10 i t is c l ear t ha t t he presence of smal l quant i t i es o f li qu id has s ign i f ican t conseq uence s

for the ope ra t ion of na tura l gas p ipe lines . There fore , an ac cura t e pred ic t ion of the l iqu id ho ld up

and the pressure grad ien t is imp or t an t i n p ipe l ine des ign , espec ia lly when we t -gas sys t ems m ay

be expec ted .

The resu l t s o f p ressure-d rop m easu rem ent s pu bl i shed in the l i te ra ture genera l ly re fe r t o

cond i t ions which d o n ot cov er our a rea of i n t e res t (0 < EL < 0 .06 ; s t ra t if i ed , wav y and a nnular f low

regimes). Fo r the Ea ton da ta (And rews 1966) , fo r examp le , on ly

two

of the 130 pressure-drop

mea surem ent s re fe r to s t ra ti f ied , w avy or annu lar f low wi th ho ldup va lues EL < 0 .06 . The two

pressu re-dro p va lues re fe r r ing to the na tura l g as- wa ter sys t em agree well wi th the va lues ca l cu la ted

w i t h t h e A R S m o d e l . O t h e r ex p e r i m en t a l p re s su re -d ro p v a l u e s i n t h e E a t o n d a t a r e f e r t o l a rg e r

holdup va lues and/or s lug f low and mis t f l ow reg imes and , consequent ly , t hese va lues of t he

physica l quant i t i es cannot be pred ic t ed accura t e ly by the procedure g iven in appendix D.

4.4. Accuracy o f the proposed models

F o r g a s - l i q u i d f l o w i n h o r i zo n t a l s t r a ig h t t u b e s t h e R D s b e t w een ex p e r i m en t a l d a t a an d r e su lt s

obta in ed wi th the c orre l a t ions for t he w et t ed wal l f rac t ion 0 , [13] , t he l i qu id ho ld up EL, [20] ,

an d t h e t w o -p h ase p re s su re g rad i en t

APTp/L

[4] , a re g iven in t ab le 2 . The respec t ive RD s a re

def ined by

R D (0 ) = / ~ 10ex- 0thl

100%, [201

a n d

l i o o [ 2

RD (E L) = ~ eL.~h

[ ]

in which N i s t he num ber of exper iment s .

5 . CONCLUSIONS

1. The AR S mo del deve loped in the presen t pape r for p red ic t ing low va lues of t he l iqu id ho ld up

EL and va lues of t he f r i c t iona l p ressure grad ien t

APTp/L

d u r i n g g a s - l i q u i d p i p e f l o w h as b een

Table 2. The RD between experimental data and results calculated with correla-

tions for 0, ~L and

A P T p / L

of the ARS model (the number of observations is given

in parentheses)

RD(0) RD(~ L RD(APn~/L)

Gas-l iquid systems used ( ) ( ) ( )

Air -wa ter (638) 15.6 8.4 9.2

Air-wate r + glycol 11.5 wt (135) 19.7 5.5 11.7

Air-wate r + glycol 25 wt (123) 14.0 6.0 8.6



Air -water + Tween, al I (303) 26.6 8.4 11.5

All dat a (1521) 20.7 8.1 9.5


11/18

PRESSURE DROP AND LIQUID HOLDUP IN GAS LIQUID FLOW 95 7

ver i f i ed wi th exper imenta l da t a . These da ta re fe r t o s t eady-s t a t e gas- l iqu id f low through

h o r i zo n t a l s t r a i g h t sm o o t h t u b es an d h av e b een o b t a i n ed b o t h f ro m t h e l i t e r a t u re (A n d rew s

1966; Minam i 1983) and f rom ou r exp er iment s cover ing the dom ain 0 < EL ~< 0 .06 and d i f fe ren t

range s of f low rates (5


12/18

9 5 8 J H RT e t a l

HAMERSMAP. J . 19 83 Ga s- l iqu id p ipe f low wi th drag reducers . Ph .D. Thes i s , Univ . o f Am sterdam .

HAMERSMA, P. J . HART, J . 198 7 A pres sure d ro p cor relat io n f or gas/ l iquid p ipe f low wi th a smal l

l i qu id ho ldup .

Chem. Engng Sci.

42, 1187-1196.

HART, J . 198 8 S ingle-phase and two -pha se p ipe f low. Ph .D. Thes i s , Univ . o f Am sterdam .

HASHIZUME, K. 1985 F low p a t t e rn , vo id f rac t ion and pressure d rop of re f r igeran t two -pha se f low

in a hor i zonta l p ipe I I . Analys i s o f p ressure d rop .


11, 643--658.

HOOGENDOORNC. J. 1965 The charac te r i s t i cs of annu lar -m is t f l ow in hor i zonta l p ipes . Presen ted

at a

Syrup. on Two-phase Flow,

Exeter , Devon, Paper C3.

HUGHMARK

G. A.

PRESSBURG

B. S . 1961 H old up and pressure dro p w i th gas- l iqu id f low in a

vert ical pipe. AIChE Jl 7, 677-682.

KADAMBIV. 1981 Void f rac t ion and pressure d rop in two -pha se s t ra ti f ied f low.

Can. J. Chem.

Engng

59, 584-589.

KORDBYAN, E . 1985 Som e de ta i l s o f deve loping s lugs in hor i zonta l two- phas e f low.

AIChE Jl

31,

802-806 .

LOCKHART, R. W . MARTINELLI, R. C. 194 9 Pr op os ed correla t ions f or i sotherm al tw o-p ha se f low

in pipes.

Chem. Engng Prog.

45, 39--48.

MANDHANE, J . M. , GREGORY, C. A. AZlZ, K. 19 74 A flow pat ter n m ap for g as-l iq uid f low in

hor i zonta l p ipes .

Chem. Engng Prog.

45, 39-40 .

MINAM1, K. M . 19 83 Liqu id h old up in wet gas pipel ines. Th esis , Un iv. o f Tulsa , O kla.

Mf3LLER-STEI~HAGE~, H. HEC K, K. 198 6 A simp le fr ict ion pressu re dr op co rrela t ion fo r

two-phase f low in p ipes .

Chem. Engng Process.

20, 297-308 .

OLmMANS, R. V. A. 1987 M ode l l ing o f gas--cond ensate f low in ho rizon tal and incl ined pipes.

Paper presen ted a t t he 1987 ETCE Conf. (6th A. Pipeline Engineering Syrup.), Dallas, Tex.

OLUJI¢ , Z . 1985 Pred ic t ing two -phase-f low f r i c tion losses in hor i zonta l p ipes .

Chem. Engng 24,

45-50 .

PERRY, R. H. CHILTON, C. H. 19 73

Chemical Engineers Handbook,

5 t h ed n . M cG raw -H i l l ,

N e w Y o r k .

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Boundary Layer Theory,

7th edn , p . 625 . McGraw-Hi l l , New York .

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Conf. on Muhiphase Flow,

T h e H ag u e , T h e N e t h e r l an d s , P ap e r A 3 , p p . 2 3 -3 6 .

SPEDDING, P. L. CHEN, J . J . J . 1 98 4 H ol du p in two -ph ase f low.


10,

307-339 .

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hor i zonta l en and ver t ika l en Rohren .

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599, 1-36.

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Encyclopaedia

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G u l l H o u s t o n , T e x.

TAITEL, Y. DUKLER, A. E. 19 76 A mo del for predict ing flow regime t ransi t ions in horiz onta l an d

near hor i zonta l gas- l iqu id f low. AIChE Jl 22, 47-55.

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Trans. ASME Jl Basic Engng

59-72 .

WlLLETS, I. P. , AZZOPARDL B. J. WrtALLEY, P. B. 1 98 7 Th e effect o f gas an d l iquid p ro pe rtie s

on annular two-phase f low. Presented a t t he 3rd Int. Conf. on Multiphase Flow, T h e H a g u e , T h e

Nether l ands , Paper C2, pp . 85-95 .

Wu, H. L., PoTs, B. F. M., HOLLENBERG,J. F. MEERHOFF, R. 19 87 Flo w pat te rn t rans i t ions in

two- phas e gas / cond ensa te f low a t h igh pressure in an 8- inch hor i zonta l p ipe . Presen ted a t t he

3rd Int. Conf. on Muhiphase Flow, The Hague , The Nether l ands , Paper A2, pp . 13-21 .

A P P E N D I X A

Frictional Pressure Drop APre During Co-current Steady-state Gas-Liquid Pipe Flow

with Small Liquid-holdup Values (0 < ~L < 0.06)

In the s t eady-s t a t e , t he fo l lowing ba lance of forces ac ting on the gas phase h o lds (see a l so


13/18

P R E SS U R E D R O P A N D L IQ U ID H O L D U P I N G A S L I Q U I D F L O W 9 5 9

f igure 1 ) :

/ d P ~ \

- A c ~ - - ~ E - ) G - * w c S c - T i S i - p G A c g s in ( f l ) = O,

[ A . i ]

w h e r e A r e f e r s t o t h e c r o s s - s e c t i o n a l a r e a , z t o t h e s h e a r s t r e ss , S t o t h e p e r i m e t e r a n d f l t o t h e

u p w a r d l y i n c li n e d a n g l e . T h e i n d i c e s G a n d i r e f e r to g a s p h a s e a n d i n te r f a c e , r e s p e c t iv e l y .

S u b s t i t u t in g t h e r e l a t io n s o f [C . 7] , g i v e n in a p p e n d i x C , a n d a s s u m i n g

6 / D

~ l , [ A . I ] r e s u l t s

in

d PT v I ~ 2 4 ( 1 - - 0 ) 4 0

= 2 J G P G V G - ~G -D - + ~ f i P c ( V G - - V ) : - T - - ~ + P G g s in ( f l ) .

d L

~GL,

[A . 2 ]

I f t h e f r i c t i o n a l p r e s s u r e g r a d i e n t i s c o n s t a n t , [ A . 2 ] r e s u l t s in

L i 2 L 1

wh r

+ EGLpGg

s i n (B) , [A . 3 ]

A P = 1 - 0 ) f o + o f , .

[A.4]

I t i s o f t e n a s s u m e d t h a t v i ~ V L i f t h e l i q u i d f l o w i s t u r b u l e n t , a n d v~ ~ 2V L i f th e l i q u i d f l o w i s

l a m i n a r ( e .g . O l i e m a n s 1 9 8 7 ) . S i n c e it i s r a t h e r d i f f i c u l t t o e s t a b l i s h w h e t h e r t h e l i q u i d f l o w

i s l a m i n a r o r t u r b u l e n t w e h a v e a s s u m e d t h a t vi ~ CL F o r h o r i z o n t a l p i p e f l o w ( fl = 0 ) , [A . 3 ]

b e c o m e s

L 1 2 L I

I f V G >> VL an d CL ~< 0 . 0 6 , w e o b t a i n fo r h o r i zo n t a l p i p e f l o w :

L i 2

A P ~ = 4 f n , - ~ )p G V G .

[A.5]

[A.6]

A P P E N D I X B

Wetted W all Fraction 0 During Co-current Steady-state Gas-Liquid Flo w in a Ho rizontal Pipe

with Sm all Liquid-holdup Values 0 < EL < 0.06)

C o n s i d e r t h e s it u a t i o n s k e t c h e d i n f i g u re B 1 . I n c y li n d r i c a l c o o r d i n a t e s t h e d is t a n c e M Z f r o m

t h e c e n t r e M o f th e t u b e t o t h e c e n tr e o f g r a v i ty Z o f th e l iq u i d p h a s e c a n b e o b t a i n e d f r o m

Fi gu r e B I . Sc he m a t i c r e p r e s e n t a t i on o f t he c r os a a e c t i on o f t he ge om e t r y c ons i de r e d f o r ga s l i qu i d p i pe

f l ow w i t h s m a l l l i qu i d ho l dup va l ue s . M i s t he c e n t r e o f t he t ube .


14/18

96 J . H A R T e t a l

f f 2

r [ r

cos(tp)] dtp dr

M Z

= R - 5 2

L f 2

r dip dr

R - ~ , 2

w h er e r is t h e r ad i a l d i s t an ce i n cy l i n d r i ca l co o r d i n a t e s . T h e r e s u l t o f [B . I] i s

[B. l ]

2 r t / 9 ~ [ l - - ( l - ¢ L'1 3 2 1

t an(n00) =

OoJ _J

[B.51

( L ~ I /2

3EL( 1

- ~ }

Solv in g [B.5] it e ra t ive ly for 00 an d sub s t i tu t i ng th i s v a lue in to [B .3] g ives the va lu e of 40 for a f ixed

v a l u e o f EL . T h u s , t h e v a r i ab l e 0 o r ep r e s en t s v a l u e s o f 0 w h e r e t h e cen t r e o f g r av i t y Z o f t h e l i q u i d

is lo c a t e d a t t h e m a x i m u m d i s t a n c e f r o m t h e c e n tr e M o f t h e t u b e , o r a t t h e s m a l l e st d is t a n c e

f r o m t h e b o t t o m o f t h e t u b e , p r o v i d e d t h a t t h e l i qu i d h a s a g e o m e t r y a s i n d i c a te d i n f i gu r e B I .

T h e r e f o r e , 00 c a n b e i n t e r p r e te d a s a m i n i m u m v a l u e o f 0 f o r a c e r ta i n v a l u e o f t h e l iq u i d h o l d u p

EL- I n o t h e r w o r d s , i f t h e l i q u i d o ccu p i e s a f r a c t i o n 00 o f t h e t u b e w a l l , th e l i q u i d h a s m i n i m a l

p o t en t i a l en e r g y .

T h e i n c r ea s e in p o t en t i a l en e r g y o f t h e l i q u i d p h as e p e r u n i t o f l iq u i d v o l u m e A E p a s a r e s u l t

o f a s h i f t o f t h e c en t r e o f g r av i t y o f t h e l i q u i d f r o m 7-0 t o Z can b e ca l cu l a t ed f r o m

aE p = p g l M Z - M Z o [

= V L A p g R ( C o - C ) .

[B.6]

A s s u m i n g t h a t a f r a c t i o n C K o f t h e k i n e t i c en e r g y p e r u n i t o f l i q u id v o l u m e E K.L i s u s ed t o a cco u n t

f o r t h e i n c r ea s e i n A E p , w e o b t a i n

C K E K ,L -~ A p g R ( 4 o - - 4 ) . [B.7]

S u b s t i t u t i n g t h e k i n e t i c en e r g y p e r u n i t o f l i q u i d v o l u m e i n t o [ B . 7 ] r e s u l t s i n

T h i s eq u a t i o n can b e r ew r i t t en a s

i n w h i ch

1 2 = A p g R ( C o 4 ) .

CK 2PL /)

CK F r = Co - C, [B.9]

P L ~ [B 10I

F r = \ ~ p ] ( g D )

[B.8I

M__ZZ= 4(1 - x )sin ~ [B.21

R 3(I - - K 2 0 [

W ith the de f in i t ion 0 = ~ / (2n ) an d the a s su m pt io n CL = g ( i - - ~C2), [B .2] can be w r i t t en as

-= C = 1 - 1 - sin (n0 ). [B.3]

F o r t h e s i t u a t i o n w h e r e an n u l a r f l o w o ccu r s i n t h e p i p e ( 0 = i ) , it f o l lo w s f r o m [B .3 ] t h a t C = 0

o r , i n o t h e r w o r d s , t h e c en t r e Z o f g r av i t y o f t h e l i q u i d co i n c i d e s w i t h t h e cen t r e M o f t h e t u b e .

T h i s i s i n ag r eem en t w i t h w h a t i s t o b e ex p ec t ed .

F r o m [ B.3 ] i t c an b e d e r i v ed t h a t f o r a f i x ed v a l u e o f EL a c e r t a i n m ax i m u m v a l u e o f 4 ex is ts ,

i.e. C0 =

M Z o / R .

A t C = C0 t h e p o t e n t i a l en e r g y o f t h e l i q u id h a s a m i n i m u m v a l u e . T h e v a l u e o f

40 a t a c o n s t an t v a l u e o f CL can b e o b t a i n ed f r o m

d

= 0 . I S . 4 ]

d 0

C o m b i n a t i o n o f [B .3 ] an d [ B.4 ] r e s u l ts i n t h e f o l l o w i n g eq u a t i o n , w h i ch d e s c r i b e s t h e r e l a -

t i o n b e t w een v a l u e s o f t h e w e t t ed w a l l f r a c t i o n w h e r e C = C0 ( d en o t e d a s 0 0 ) an d t h e l i q u i d

holdup EL:


15/18

PRESSURE DROPAND LIQUIDHOLDUP N GAS-LIQUIDFLOW 961

0 . 4 0

0 . 3 0

0.20 ~

O. I0

t

0 0.05 0.10 0.15 0.20 0.25

Figure B2. The m inimumvalue of the wetted wall fraction

8o as a functionof e,: the dashe d line refers o [B.5]; the solid

line represents the approx imation given by [B.I 1].

1 . 0 0

0 . 7 5 t

0.50 , /

¢L 0 2 =0.01

0 . 2 5

I 0 0° I

o 0.2s o.5o 0.75 t .oo

Figure B3. The value of ~ o- ~ as a function of 0 -Oo : the

dashed lines have been calcu lated with [B.3I and [B.5]; the

solid line represents the approx imation given by [B.12].

T h e v a l u e o f C K h a s t o b e o b t a i n e d f r o m e x p e r i m e n t s . I f th e v a l u e o f t h e l iq u i d h o l d u p EL i s k n o w n ,

the va lues o f 00 [B .5], G0 ( lB .3] a nd 0 = 00) an d Fr [B .10] can b e ca lcu la ted . T he n the v a lue o f

can b e o b t a i n ed f r o m [B .9 ] an d , f i n a l ly , t h e v a l u e o f t h e w e t t ed w a l l f ra c t i o n 0 f r o m [ B.3 ]. H o w ev e r ,

i t m u s t b e c l e a r t h a t [ B . 5 ] , f o r t h e d e t e r m i n a t i o n o f 0 0 , an d l B . 3 ] , f o r t h e d e t e r m i n a t i o n o f 0 , a r e

d i ff ic u l t t o h a n d l e . T h e r e f o r e a p p r o x i m a t i o n s w il l b e m a d e f o r e a c h o f t h e se q u a n t i t i e s.

Fo r 0 ~< EL ~< 0 .20 , lB .5] m ay be a pp rox im ate d by th e fo l low ing equa t ion , as sho wn in f igure B2:

00 = 0.52 E~ 374. [B.

11]

I t c an b e v e r i fi ed t h a t t h e q u an t i t y G o - ~ i s a f u n c t i o n o f 0 - 0 o, w h i ch i s a l m o s t i n d e p en d en t o f

EL for 0 < EL ~< 0 .20 . F ur the rm ore , f r om the exp er im ents i t app ear ed th a t 0 ~< 0 - 00 < 0 .6 . Fo r these

i n t er v a ls , G o - ~ m a y b e a p p r o x i m a t e d b y

G o - ~ -'~ 1.6(0 - 0o) 17'. [B.12 ]

T h i s r e l a t i o n i s p l o t t ed i n f i g u r e B 3 , t o g e t h e r w i t h t h e r e l a t i o n b e t w een Go - ~ an d 0 - 0 o, o b t a i n ed

f ro m [B .3] an d [B .5]. C om bin a t ion of [B .9], [B .12] a nd C2 - - {Cz/ l .6 } °'58, r esu l t s in

0 = 0 o + C 2 F r ° 5 8 , [ B . 1 3 ]

i n w h i ch t h e v a l u e C ~ = 0 . 2 6 h a s b een o b t a i n ed ex p e r i m en t a l l y (s ee s ec t i o n 4 . 1 ).

A P P E N D I X C

Liquid Holdup During Co-current Steady-state Gas-Liquid Pipe Flow with

Sm all Liqu id-Holdup Values 0 < EL < 0.06)

A g e n e r a ll y a c c e p te d m e t h o d f o r t h e d e r i v a t io n o f a l i q u i d - h o l d u p c o r r e l a t io n f o r s t e a d y - s ta t e

g as - l i q u i d f l o w i s s o l v i n g t h e f o r ce b a l an ce o v e r t h e l i q u i d p h as e an d t h e g a s p h as e ( e . g . C h en &

S p ed d i n g 1 9 8 3 ; W u

et ai.

1987) (see a l so f igure l ) . P res sure los ses du e to acce le ra t ion a re neg lec ted .

R e f e r r i n g to f ig u r e l , w e m a y s t a t e t h a t f o r a s t e a d y - s t a te g a s - l iq u i d f l o w t h r o u g h a h o r i z o n t a l o r

u p w a r d l y i n c l in e d s t r a i g h t s m o o t h t u b e t h e f o l lo w i n g f o rc e b a l a n c e s h o l d :

a n d

- ~ - ~ - , / L - - Z w L S L + z ~ S ~ - - p L A L g

s i n ( B ) - - 0

A o ( d P ~

- \ - ~ - } o - z wo SG - z~S~- poA og sin(B) = O.

[C . l ]

[C.2]


16/18

962

1. ART et

It is assumed, and also measured (see section 3), that

gL= gG.

Therefore, combination of [C.l] and [C.2], results in

AL CL ~wLSL-~~&+AL(PL-PG)~ sin(8)

-_=-_=

AG cG

7WGb ?G+7iSi

1C.31

r 41

Adding 1 to the 1.h.s of [C.4] and (rwoSo + riSi)/(

rwoSo + rigi) to the r.h.s. results in

1 7WLSL+7WGSG+AL(PL-PGksin(8)

7WGsG+7isi

K.4

Rewriting [C.5] gives

-~+AL(pL_p,)gsin(8)=0.

W

For the situation of small liquid-holdup values (0 < cL< 0.06) and flow regimes where the liquid

flows along the tube wall (see figure l), the terms of [C.6] are given by

7WL

=SfLPL~~, S~=elrO, AL=~L~D’, 7i=t~pG(Vg-ui)2,

si=h(D -26)~

7WG’i GPGV~,

f

SG= (1 - O)ICD,

AG = tG %D 2.

LC.71

For the situation 6 4 D, and taking into account [C.7], [C.6] can be rewritten into

i f Lmt D

- zffGPG itl

- e)7cD

- iXpG(uG - Ui)281CD

+ cL a D2(pL - p& sin(b) = 0.

[C.8]

Dividing [C.8] by feaDfLpLu& and assuming Uik: uL, leads to

For small

liquid-holdup values and/or annular flow (8 =

1) the first term on the r.h.s. can be

neglected. Substitution of trL= uL/cLand no = u&, results in

u:


17/18

PRESSURE D R O P AN D L I Q U I D H O L D U P I N G A S L I Q U I D F L O W 9 6

Su bsti tutin g VL = UL/eL an d VG = UG/eG in [C.12] and tak ing the squa re ro ot g ives the fo l lowing

equat ion for the l iqu id holdup dur ing gas- l iqu id f low in s t ra ight p ipes :

{ + ( e o f L P L ] T i

L UL 1 -~ [C.13 ]

eo uo \ f - - ~ G / k f t F r , J J

This eq ua t ion can only be so lved by me ans o f an i te ra t ive procedure . A s a f i r s t guess [C. 13] m ay

be solved for /7 = 0. F or ho rizon tal pipe f low (8 = 0) and eL ~< 0.06, [C .13] beco me s

- - ' F ,

l U G L

~ , f P c l A

i t 1 4 ]

The r a t i o f L / f has to be obta ine d e xper im enta l ly ( see [18]) .

A P P E N D I X D

A R S M o l

The p roced u re fo r t he ca l cu l a t ion o f t he l i qu id ho ldup EL and t he p r e s su re d rop APTp

dur ing hor izonta l co-cur ren t s teady-s ta te gas- l iqu id p ipe f low wi th smal l l iqu id-holdup va lues

(0 < EL < 0.06) is as follo ws .

. Calcula te the va lue of the l iqu id holdu p wi th

'L UL{I+[IO.4Re~L O.~JP \ '/2q~

l C L u o

2. D eter m ine the values of the avera ge local gas and l iquid velocit ies , respect ively:

U UL

V G = - - ; V L = - - [D .2]

1 - ~L ~L

3. Calcula te the m odi f ied Fr and the R e of the gas phase :

F r = v~ .. P__A. ReG = DVGPG [D.3]

gD Ap ' rio

4. Calcula te the va lue of the wet ted wal l f rac t ion 0 :

0 = 0o + 0.26 Fr °sS, in w hi ch 00 = 0.52E~

TM

[D.4]

I f

0 > 1 t ak e 0 = 1 .

5 . D e t e r m i n e t h e v a l u e o f t h e a p p a r e n t r e l a ti v e r o u g h n e s s o f t h e li q u i d fi lm k D f r o m

( : 0 )

, o , ,

D

6. Calcula te the va lue of the in te r fac ia l f r ic t ion fac tor f rom

0.0625

f = F / 1 5 + k 2"

Ll°g l° t~-~c 3 .7 iS D )} [O.6 ,

7 . C alcula te the va lue of the s ingle-phase f ric t ion fac tor f Fo r smooth tubes:

0.07725

' ° - - [ , o , 0 < o

o 7 ,

wh ich is val id for 2100 < ReG < 10s. Fo r rough tubes [ I ).6] m ust be appl ied w i th the prope r

va lues of the re la t ive p ipe roughness , ob ta ine d f ro m, fo r example , the Chem ical Engineers'

Handbook (Per ry & Ch i l ton 1973).

8 . Calcula te th e tw o-phas e f r ic t ion fac tor fTP f rom

fTP = (1 - 0 ) . f + 0 . f . [D.S]


18/18

96

J . H A R T

e t a l

9 . C a l c u l a t e t h e t w o - p h a s e p r e s s u r e d r o p

APTp:

1 I 2 L _ v L ) ]

A P T P = I - - ~ e L ) . I 4 f T p L ) I P G V G - - 4 O f - - D ) ~ p G 2 V G V

[D.9]

I

10. I f t o >> VL a n d 0 < EL < 0 .06 , we ob t a i n fo r ho r i z o n t a l p i pe f l ow:

L i ,

A P T p = 4 f T p ( ' - ~ ) ] p G V ~ • [D. 10]

A p p l i c a t io n o f t h is p r o c e d u r e r e s u l ts i n t h e v a l u e s o f

A P v p / L

p r e s e n t e d i n f i g u r e s 8 a n d 9 .

Documents

Correlations Predicting Frictional Pressure