Taitel Stability of Severe Slugging

8/13/2019 Taitel Stability of Severe Slugging

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Int. J. Mu/r/phase F/ow Vol. 12, No. 2, pp. 203-217, 1986 0301-9322/86 3.00+.(]0Printed in Grest Britain. 1986 Pall smOnl ~sev ier

S T A B I L I T Y O F S E V E R E S L U G G I N G

Y E H U D A T A I T E LF a c u l ty o f E n g in e e r in g , T e I-A v iv U n iv e r s i ty , R a m a t -A v iv 6 9 97 8 , I s r a e lReceived 25 September1984; in revised orm 30 May 1985

A b s t r a c t - - F o r a c o n s t a n t f l o w r a t e o f l iq u i d a n d g a s i n a p i p e o n e e x p e ct s t h e c o n d it i o n s a l o n gth e p ip e to b e o f a s t e a d y s t a t e n a tu re . H o w e v e r , fo r a p ip e in a h i l l y t e r r a in o r i n a n o f f s h o rep ipe l ine -r ise r sys tem, a s teady s ta te opera t ion is o f ten no t poss ib le , and condi t ions o f severe o rt e r r a in s lu g g in g d e v e lo p . T h i s c a u s e s th e s y s t e m to o p e ra t e in a n u n d e s i r e d c y c li c f a s h io n in w h ic ha l t e rn a te lo n g l iq u id s lu g s a re fo llo w e d b y th e p ro d u c t io n o f h ig h g a s f lo w ra t e . T h e p re s e n t w o rkd e a l s w i th th e c o n d i t io n u n d e r w h ic h s t e a d y s t a t e o p e ra t io n i s p o s s ib l e . I t s h o w s th e o re t i ca l ly th a ti t i s p o s s ib le to s t a b l i z e th e f lo w b y in c re a s in g th e b a c k p re s s u re o f t h e s e p a ra to r o r b y e mp lo y in ga c o n t ro l l e d c h o k in g a t t h e p ip e e x it .

I N T R O D U C T I O NSteady s ta te ope ra t ion o f two-phase f low in p ipes usua l ly means tha t the f low ra te o f l iqu idand gas a re cons tan t . As a re su l t , the cond i t ions a t any po in t in the p ipe remain cons tan t ;namely , the f low pa t te rn , ave rage vo id f rac t ion , ave rage p ressu re d rop and ave rage loca lf low ra te s do no t va ry wi th t ime .

The te rm average is u sed he re because two-phase f low i s se ldom a t ru ly s tead y s ta tef low and ave rag ing va lues a re used ov er a t ime pe r iod cha rac te r i st ic o f the f low pa t te rn . Atyp ica l example i s the s lug f low pa t te rn , fo r wh ich ave rage va lues a re taken dur ing one o ra few slug passages.

Ho wev er , under ce r tain s i tua t ions a s tead y s ta te ope ra t ion i s no t poss ible . For example ,when a subsea l ine wi th downwards inc l ina t ion ends wi th a ve r t ica l r i se r to a p la t fo rm, o rwhen a p ipe i s la id in a h i l ly te r ra in , under ce r ta in cond i t ions the low er sec t ion o f the p ipeaccumula tes l iqu id and b locks the gas passage . The gas ups t ream is compressed un t i l i tove rcom es the g rav i tat iona l head o f the liqu id , the reb y c rea t ing a long l iqu id s lug tha t i spushed in f ron t o f the expand ing gas ups t ream. Un der such cond i t ions a cyc l ic ope ra t ionis obta in ed, te rm ed severe or terra in s lugging. Severe s lugging is consid ered to b e an u nstab lef low reg ime in the sense tha t i t i s a ssoc ia ted wi th la rge and ab rup t f luc tua t ions in the p ipepressu re and in the gas and l iqu id f low ra te s a t the ou t le t .

The p rocess o f seve re s lugg ing fo rmat ion can be desc r ibed as tak ing p lace accord ingto the fo l lowing s teps.

Th e f irs t s tep is the s lug form ation (figure 1). In th is s tep l iquid enter ing the pipel ineaccumula tes a t the bo t tom of the r i se r , b lock ing the gas passage and caus ing the gas tocom press . W hen th e l iquid height in the r iser, z reaches the top of the r iser, z = h the seconds tep o f s lug movement in to the sepa ra to r s ta r t s ( f igu re 2 ) . Af te r the gas tha t i s b locked inthe p ipe l ine reaches the bo t tom of the r iser , the l iqu id s lug con t inues to f low in to theseparator with a ra ther fast veloci ty , termed blowout (f igure 3) . In the las t s tep , f igure 4 ,the remain ing l iqu id in the r i se r fa l l s back to the bo t tom of the r i se r and the p rocess o fs lug fo rmat ion s ta r t s aga in .

The severe s lugging pat tern is typical of re la t ively low l iquid and gas f low ra tes . I trequires that the f low pattern in the p ipel ine be s tra t i f ied . In addit ion, i t requires that thel iqu id reaches the top o f the r i se r p ipe be fo re the gas reaches the b o t tom of the r i se r du r ings lug fo rmat ion . The la t te r cond i t ion can be ca lcu la ted us ing the Schmid t e t a l mo del (1980).A s impl if ied ve rs ion o f the Schm id t m ode l i s u sed he re (append ix A ) to de te rmine the f lowrate of l iquid and gas a t which severe s lugging wil l not occur .

Severe s lugging is an undesired phenomenon. One of the methods of a l levia t ing severes lugging i s by inc reas ing the sepa ra to r back p ressu re (Yoc um 1973) . Chok ing the f low(Schmid t e t a l . 1979b, 1980) was a lso found to a l levia te severe s lugging w ith minimalincrease in the p ipel ine pressure (for the same f low ra tes of l iquid and gas) . Once severePlF 12:2-D

203


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204 ~NI IDA TAITI~PO[ P

Figure 1. Slug formation.

slugging was eliminated, a steady state operation was achieved as shown in figure 5. In thissteady state operation the pipeline is in stratified flow while the riser is in bubble or slugflow. The pressure of the pipeline remains constant and the liquid does not penetrateupstream into the pipeline to form the long liquid slug.

In spite of the progress achieved in eliminating severe slugging, it seems that thisprocess is not well understood and the conditions under which severe slugging can betransformed into steady state flow are still not clear. The statment that the process inwhich severe slugging has been eliminated successfully has been repeated often enough toprove the value of choking as probably the most practical method of eliminating slugging(Schmidt e t o l 1980), reveals the need for a better understanding of this process.

In this work we examine the conditions under which severe slugging will take placeand find under what conditions and how severe slugging could be eliminated and transformedinto steady state operation. Furthermore, the stability of steady state operation is analysedand the conditions under which steady state operation will take place are established.

po

: T I ,~ Lh it , , , i

i

G A S J ~ il

L I Q U I D P ;

Figure 2. Slug movement nto the separator.


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STABILITYOF SEVERE SLUGGING

G A S

I ~ .I , :iI

L I Q U I D / . /~ p p i? .:

h

Figure 3. Blowout.

205

ANALYSISSevere s lugg ing occurs due to the compress ib i l i ty o f the gas . The gas compress ib i l i ty

man i fes t s i t sel f in the b low out s tep o f the seve re s lugging cyc le f igu re 3 ). In th i s s tep thel iqu id co lumn he igh t is r educed and an uns tab le s i tua tion can b e reached where the p ressu rein the p ipel ine , pp, wil l exceed the back pressure provided by the separator and the l iquidc o l u m n h - - y ) . I f t h e s y s te m i s n o t s t a b l e th e l iq u i d w ill b e b l o w n - o u t b y th e g a s, t h e r eb ycausing the severe s lugging cycle to take place .

Th is s i tua t ion can be ana lysed as fo l lows : Assume tha t the cyc le o f seve re s lugg ingreaches the po in t a t wh ich the s lug ta i l has ju s t en te red the r i se r and the r i se r i s now l iqu idful l. As sum e a sma l l d i s tu rbance y tha t m ay ca r ry the l iqu id some wh a t h igher see f igure3 , where y can a l so be cons ide red the d i s tu rbed leve l) and tha t the d i s tu rbance i s fa s t enoughso tha t the s low f low ra te o f l iqu id and gas i s ignored whi le y changes.

The n e t fo rce pe r un i t a rea ) ac ting on th e l iqu id in the r ise r i sl- - [ P , + p ~ e h ) ~ ] - [P , + p ~ g h - y ) ]a L - r c t y x ]

Po

F A L L IN G F I L M ' - ~G A S

Figure 4. Liquid bl ll ~k .


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206 YEHUD T ITEL

BUBBLE RSLUG FLOW

GASL I Q I U ~Figure 5 Steady s tate operat ion

- -. Po

The f i r s t t e rm on the rhs in the square pa ren thes i s i s the p ipe l ine p ressu re d r iv ingfo rce. The p ressu re va r ie s wi th y a s a re su l t o f the expans ion o f the gas in the p ipe line . Thes e c o n d t e r m c o r r e s p o n d s t o t h e b a c k p r e s s u r e f o r c e a p p l i e d b y t h e s e p a r a to r p r e s s u r e a n dthe l iqu id co lum n o f dens i ty PL and he igh t h - y ) . N o t e t h a t f o r y = O t h e sys tem is inequ i l ib r ium a nd A F-- 0 . I and h a re the p ipe l ine and r i ser lenBlhs , re spec t ive ly . P , i s thep ressu re in the sepa ra to r , a i s the gas ho ldup in the l ine which i s in s t ra t i f ied f low. a ' i sthe gas ho ldup in the gas cap pene t ra t ing the l iqu id co lumn, a can be ca lcu la ted on thebas is o f a s t ra t i f ied f low mode l desc r ibed in append ix B. a ' i s ca lcu la ted on the bas i s o ft h e s l u g f lo w m o d e l d e s c r i b e d i n a p p e n d ix C . a a n d a ' h a v e v a lu e s t y p i c a ll y r a n g in g f r o m0.8 to 1 .0 . The i r exac t va lues on ly s l igh t ly e f fec t the re su lt s . In th i s ana lys is , shea r s t re ssesa re neg lec ted due to the low ra te s typ ica l o f seve re s lugg ing opera t ion . Also , the gas i s

ssumed to expan d i so the rmal ly fo l lowing the idea l gas law.The l iqu id co lumn wi ll be b low n ou t o f the p ipe i f A F inc reases wi th y , wh ich i s anecessa ry cond i t ion fo r seve re s lugg ing f low. Thus the cond i t ion fo r s tab i l i ty , namely thecond i t ion under wh ich seve re s lugg ing i s no t poss ib le i s

a ( A F ) < a t y = O [ 2]ayThis leads to the c r i te r ion fo r s tab i l i ty

P ~ > ( a / a ) 1 - h , [ 3 ]Po Po PLg

where P0 i s the a tm ospher ic p ressu re .Th is i s a ve ry s imple re su l t s ta t ing tha t when the sepa ra to r p ressu re inc reases to thelevel that sa t isf ies [3], severe s lugging w ill be e l ihainated and s tea dy s ta te cond it ion wil l bereached . I t i s a l so in te res t ing to obse rve tha t the sys tem becom es le ss s tab le fo r inc reas ingpipel ine length and more s table for increasing r iser length .Stability o f steady state operationThe s tab i l i ty o f the s teady s ta te ope ra t ion shown in f igu re 5 cou ld be ana lysed thesame way , excep t wi th the l iqu id dens i ty rep laced by the ave rage co lum n dens i ty . Des igna t ingthe l iqu id ho ldu p in the r iser as ~b, the av erage den si ty (neglect ing the g as dens i ty) is ~bpLand [3 ] takes the fo rm


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ST BILITY O F S V R S L U G G I N G 2 7

> ( ( a / a, ) , h ) [ 4 1Po Po PLgSince qb is less than uni ty , th is resul t indicates that s tead y s ta te ope rat ion is mo re s tablethan s tep 3 in the severe s lugging cycle . This a lso suggests that lower separator pressurecan be used on ce s teady s ta te ope ra t ion i s r eached .

The ana lysi s o f the s t ab i l ity o f a s t eady s ta te ope ra t ion requ i res, however , the kn owledgeof the av erage l iquid ho ldup in th e pipe , ~b. This can be calcula t.ed on the basis o f s teadys ta te mod e ls o r co r re la t ions which y ie ld the liqu id h o ldup as a func t ion o f the opera t ingcondi t ions .S t e a d y s t a t e o p e r a t i o nIn s t eady s ta te ope ra t ion , the f low in the r i se r p ipe wi ll t ake the fo rm o f e i the r bubb leof s lug f low. T he average l iquid holdu p, ~b, depen ds o n the f iquid superf ic ia l veloci ty , gasmass f low ra te o r superf ic ia l ve loc ity und er a tmosph er ic cond i t ions ) and the sepa ra to rpressure . I t i s convenient , however , to express th is funct ional re la t ionship in the form ofas in [4])

P~oo =f qb , Uza, UGso , [5]wh ere ULs is the superf ic ia l veloci ty of the 5qu id assum ed incompressible) an d Uoso is thesuperf ic ia l veloci ty of the gas under a tmospheric condi t ion Po.Fo r th i s w e need mo de ls fo r bubb le f low and s lug f low tha t w i l l r e su lt in the re la t iongiven in [5] . This re la t ions hip is descr ibed below for bubble f low. The m ore com plex re la t ionfor s lug f low is g iven in appendix C.For bubble f low the l iquid and gas veloci t ies are

[ ]u L [ 7 ]

wh ere qb is the l iquid ho ldup. Th e superf ic ia l gas veloci ty depen ds on the se parato r pressure ,U G s = U 6 s o P o / P , . Assuming tha t the s l ip ve loc i ty Uo= U G - U L ~ cons tan t , th e fo l lowingrela t ion is obta ined,

- P s U a s oP o ( U o - F - ~ ) ( I - - ~ ) [8]

whe re Uo can be ca lcu la ted by Har m athy 1960)U o 1 . 5 3 g ( P L ; ~ ) ]/4 [ 9 ]

No te tha t fo r s impl ic ity the gas de ns i ty in the r i se r is approx im ated by the gas dens i ty a tthe se pa ra to r p ressu re .The ana lys is o f the poss ib le s t eady s ta te s and the i r st ab i li ty can be de mo ns t ra ted wi than exam ple for an a i r - w at er system with superf ic ia l l iquid veloci ty of ULs----0 .1 m /s andsuperficial gas veloci ty a t a tm osph eric condi t ions) Uas0 in the range of 0 .05 to 0 .2 m /s .P , / P o for s tead y s ta te ope rat ion as a funct ion of qb [8]) is p lot ted in f igure 6 forU~so-----0.05, 0.1 an d 0.2 m /s . Th e bub ble flow p atter n cha nge s to slug flow at abo ut l iquidh ol d up qb __~ 0.7 Ta itei e t a L 1980). Therefore the curves in f igure 6 repre sent th e resul t of


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2 0 8 Y E H U D A T A I T E L

PsPo

S L U G F L O W

Ls = 0 1 mls

2

1 -

o 0 0 5

I~ 1 B U B B L E~ F L O W v

I

/ o lc y IFB

L I

F i g u r e 6 . S t a b il i ty a n a l y s is w a t e r - a i r D = 5 c m d ia m . I = 3 0 m h 1 5 m / 3 = 2 .

the bubb le f low pa t te rn f rom ~b= 1 to 0 .7 w h i le fo r ~b < 0 .7 the mo de l o f s lug f low i s used(append ix C) .

I f we now p lo t the s t ra igh t l ine o f P,/Po vs ~b as g iven in [4], which represen ts th es tab i l i ty c r ite r ia fo r s teady s ta te ope ra t ion , w e can lea rn when the f low i s stab le . The s t ra igh tl ine in f igure 6 , the s tabi l i ty l ine , represents the resul t of [4] for a r iser length of 15 m andp ipe line leng th o f 30 m wi th 20 dow nw ards inc l ina t ion (a sys tem s imi la r to the one usedb y S c h m id t et aL 1980). The vo id f rac t ion in the p ipe l ine a , ca lcu la ted us ing Ta i te l Du k le r(1976) ( see append ix B) , was found to y ie ld a= 0 . 8 7 , a ' can be ca lcu la ted on the bas i s o ft h e s l u g f l o w m o d e l s i n c e t h e Ta y lo r b u b b l e t h a t p e n e t r a t e s t h e c o lu m n i s t h e s a m e a sTay lo r bubb les in no rmal s lug f low. To use the s lug f low mode l (append ix C) fo r th i spurpose we need as inpu t the f low ra te s o f the l iqu id and gas , wh ich a re unknown fo r thed is tu rbed va r iab le . Ho we ver , u s ing the s lug f low mode l fo r Uzs and Uos, bo th rang ing f rom0.01 to 10 m /s , show s tha t the a ' ob ta ined i s no t sens i t ive to the f low ra te s and the re su l tfo r th i s pa r t icu la r sy s tem ( fo r a ll f low ra te s ) was tha t p rac t ica l ly = c o n s t a n t = 0 .89 . No tea l s o t h a t t h e e x a c t v a lu e o f a ' i s n o t im p o r t a n t a n y w a y .S teady s ta te ope ra t ion abov e the s tab i l i ty l ine i s s tab le w hereas be low i t , i t i s uns tab le .

S ince the sepa ra to r p ressu re wi l l a lways exceed the a tmospher ic p ressu re (PJPo > 1),the sys tem wi ll be s tab le fo r Uo s0=0 .2 and 0 .1 . At a tmo spher ic p ressu re fo r Uoso=0.2 ther i ser wil l be under s lug f low (~b


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ST BILITY OF SEVERE SLUGGING 209Steady unstable operation

An uns tab le sys tem can s t i l l ope ra te a round the equ i l ib r ium s teady s ta te p rov ided afeedback con t ro l sys tem i s used to s tab i l i ze the sys tem. Schmid t et aL (1980) found exper -imen ta l ly tha t they cou ld s tab il i ze the f low by chok ing the f low a t the to p o f the r i se r be fo reenter ing the separator (see f igure 7) .Fo l lowing th i s ana lys i s i t i s c lea r tha t i f chok ing can be used to inc rease the backpressu re Ps propor t iona l ly to the d i s tu rbance movement y , a con t ro l led s tab le sys tem canresu l t . Us ing a con t ro l sys tem to p rov ide

P ~ - P = g y [ l O ]the ne t fo rce tha t ac t s on the co lumn i s now (see [1 ] )

A F = [(P, + ~ p L g h ) ~ ] - - [ P,-Ky)+~ppLg h-y)] [ 1 1 ]The condit ion for s tabi l i ty g iven by [2] y ie lds

_ _ _ K __ . a t ( l - ~ , ha PLg~ > [12]Po PolpLgEqua t ion [12] can be used to de te rmine the des i red combina t ion o f sepa ra to r p ressu re andthe s tab i l i ty coe f f icien t K to ensu re s teady s ta te ope ra t ion .

The co n t ro l sys tem sho u ld be des igned in such a way tha t the chok ing va lve wi ll bead jus ted to p rov ide an inc rease o f the back p ressu re accord ing to [10 ] . Such a con t ro lsys tem needs an inpu t o f the va lue o f the d i s tu rbance y . Th is cou ld be done in va r iousways , e i the r measur ing y d i rec t ly (us ing v o id f rac t ion de tec to rs ) o r b y co r re la ting the p ressu red i f fe rence be tween the bo t tom of the r i se r and som e loca t ion h igher than the y pos i tion .

I t i s in te res t ing to obse rve tha t , to a g ood approx imat ion , l i tt l e movem ent o f thechok ing va lve i s needed fo r such a con t ro l sys tem. Th is mak es i t poss ib le to se t the chok ingvalve in a precalcula ted co nsta nt value .The con t ro l va lve func t ion approx im ate ly fo l lows the re la t ion

P ~ - P = c t r t [ 1 3 ]

CHOKINGV A L V EB A C K R E S S U R EP b ~ Y Pb~

G A Sp p ~

L I O U I D ~ ~ _- - - - - -4 -

Fi g u r e 7 S t a b i l i z i n g u n s t a b l e s te a d y s t a t e o p e r a t i o n


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1 1 0

STABILITY OF SEVERE SLUOGING

TURBULENT SMOOTHLAMINAR BMOOTH~ m F UL LY T UR BU LE NT

1 0 0 1 0 0 0 1 0 , 0 0 0 1 0 0 , 0 0 0PL.O sin( d P / d X ) L $

F i g u r e 8 . V o i d f r a c t io n i n d o w n w a r d s i n c l in e d p i p e ,

2 1 1

A s seen the results o f [A. 13] i s fa ir ly c lose to the experimenta l bou nda ry to severe s luggin gI a s g iv en by Schmidt e t a l . (197 ) ( the broken curve) . Bou ndar y b is the transi tionbo und a ry f ro m s t ra t if ied f lo w a s ca l cu la t ed us ing T a it el a n d D ukler ( 1 9 7 6 ) m o de l . T h e f lo win the incl ined pipe must be strat i f ied in order for severe s lugging to be possible . The reg ionenc lo sed by a a nd be lo w b i s t he reg io n where sev ere s lug g ing i s po ss ib l e .

Although severe s lugging is possible , in this reg ion, i t wi l l not take place i f the stabi l i tycr i ter ion [3] i s sat isf ied. Fo r a g iven l iquid f lo w rate the pipel ine vo id fract ion a can beca lcu la t ed us ing f ig ure 8 ( a ppendix B ) , whi l e a ' , t he v o id f ra c t io n o f a T a y lo r bubble , i sf a i r ly co nst a nt ( see a ppendix C) a nd wa s t a ken a s a - - 0 .8 9 . E qua t io n [ 3 ] ca n no w be usedto ca lculated the separator pressure rat io t , / t o a bo v e w hich sev ere s lug g ing i s no t po ss ib le .In f igure 9 three such pressure rat ios are plot ted: 0 .1 , 0 .5 and 1 .0 . This indicates that , forex a m ple , wh en t he sepa rat o r pressure ra t io i s 1 o n ly in t he reg io n be lo w t he P , / P o = 1 l inesevere s lugging is possible . Interesting ly enou gh, S chm idt data fa l l r ight in the range of

s

o SEVERE SEVERE

O ~ TRANS

O

] JSLUGGING Z x SLUG FLOWSLUGGING X ANNULAR FLOW

TO SLUGGING O BUBBLE FLOW

0 0 O 0 O O O

S T ~ X x x x x x

o Oo..o/,,X.x ' * * IIS EvEREi ~_~SLUGGING b

e m

O I I IUGS Ira/S]

I 0 I

F i g u r e 9 . E x p e r i m e n t a l r e s u lt s , k e r o s i n e - a i r s y s t m n , 3 1 - m m - d i a m . p i p e , 5 p i p e l i n e i n c l in c a t i o n ,r i se r 1 5 m l o n g , p i p e l i n e 3 0 m l o n g ( S c h m i d t 1 9 7 7 ; S c h m i d t e t a L 1 9 7 9 a ) .


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212 YEHUDATAITELthese pressure ratios, and although the pressure at the top of the riser was not controlled,nor reported by Schmidt (1977), it was in the range of 0.1 to 1. This may explain why inthe upper region of the severe slugging zone severe slugging was not observed. Obviouslymore carefully planned experiments are needed to fully verify this proposed theory.

SUMMARY AND CONCLUSIONSSevere slugging may occur whenever a pipeline is laid with a downward inclination

followed by an upward slope that allows liquid to accumulate at the lower section. Thisphenomenon is limited to low liquid and gas flow rates at which the flow pattern in thedownhill section is stratified. The system analysed here is that of a downsloping pipelinethat ends with a vertical riser. For this system severe slugging is possible when the flowpattern in the pipeline is stratified and the liquid flow rate is above the value given by[A.13].

It has been shown that a steady state operation is stable if the separator pressure ishigh enough to satisfy [4]. If the steady state operation is unstable, a proportional control ledchoking valve that follows [10] can stabilize the flow if the proport ionality constant ischosen to satisfy [12].There are situations, however, when the system can operate both under steady stateand under unstable severe slugging conditions. Transition from severe slugging to steadystate operation can be achieved by temporarily increasing the separator pressure or theproportionality constant K when a choking valve is used.Acknowledgement This work was carried out while the author stayed at the University of Houston,Department of Chemical Engineering. The support of Dr. A. E. Dukler is greatly acknowledged.

REFERENCESALLEN, T., JR. DITSWORTH, R.L. 1972 Fluid Mechanics. McGraw-Hill, New York.BARNEA, D. BRAUNER, N. 1985 Hold-up of the liquid slug in two phase intermittent

flow. In t . J . Mult iphase Flow 11, 43-49.BENDIKSEN, K., MALNES, D. NULAND, S. 1982 Severe slugging in two-phase flow

systems. Report prepared for STATOIL--Den norske stats oljeselskap a.s., ISSN 0333-2039, ISBN 82-7017-018-6.

BELKIN, H.H., MACLEOD, A.A., MONARAD, C.C. ROTHFUS, R.R. 1959 Turbulentliquid flow down vertical walls. A L C h E J . 5, 245-248.

BROTZ, W. 1954 Uber die vorausberechnung der absorptionsgeschwindigkeit yon gasen instromnden flussigkeitsschichten. Chem. Ing. Tech. 26, 470.FERNANDeS, R.C., SEMIAT, R. DUKLER, A.E. 1983 Hydrodynamic model for ga s-liquid slug flow in vertical tubes. A I C h E J. 29, 981-989.NICKLIN, D.J., WILKES, J.O. DAVIDSON, J.F. 1962 Two-phase flow in vertical tubes.Tran~ Inst . Chem. Engng 40, 61-68.SCHMIDT, Z. 1977 Experimental study of two-phase slug flow in a pipeline-riser pipe system.Ph.D. dissertation, The University of Tulsa.SCHMIDT, Z., BRILL, J.P. BEGGS, H.D. 1979a Experimental study of severe slugging ina two-phase flow pipeline-riser pipe system. SPE 8306.SCHMIDT, Z., BRILL, J.P. BEGOS, H.D. 1979b Choking can eliminate severe pipelineslugging. Oil Gas J. 12, 230-238.SCHMIDT, Z., BRILL, J.P. BEOOS, H.D. 1980 Experimental study of severe slugging ina two-phase flow pipeline-riser pipe system, Soc. Petrol . Engng J . 407-414.

TAITEL, Y. DUKLER, A.E. 1976 A model for predicting flow regime transitions inhorizontal and near horizontal gas liquid flow, A I C h E J . 22, 47-55.TAITEL, Y., BARNEA, D. DUKLER, A.E. 1980 Modelling flow pattern t ransit ions forsteady upward gas-liquid flow in vertical tubes, A I C h E J. 26, 345-354.TAITEL, Y. BARNEA, D. 1983 Counter current gas liquid vertical flow-model for flowpattern and pressure drop. In t . J . Mult iphase Flow 9, 634-648.WALLIS, G.B. 1969 One Dimen s iona l Two Phase F low. McGraw-Hill, New York.


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STABILITYOF SEVERESLUGGING 213YOCUM, B.T. 1973 Offshore r ise r s lug f low avo idance , M athemat ica l mo de l fo r des ign and

op t imiza t ion , Paper SPE 4312 p resen ted a t the SPE Europ e Mee t ing , London , Apr i l 2 -3 .

APPENDIX AS i m p l i f i e d s e v er e s l u g g i n g m o d e l

A mode l fo r seve re s lugg ing was p rsen ted by Schmid t e t a L (1980) . The purpose o fsuch a mod e l i s to p red ic t the s lug leng th , s lug cyc le t ime , p ressu re f luc tua t ions , e tc .A somewha t s imple r rep resen ta t ion o f th i s mode l i s desc r ibed be low wi th the p r imeob jec t ive to p red ic t the f low ra te o f l iqu id above which the ta i l o f the s lug reaches the

bo t tom of the r ise r be fo re the f ron t o f the s lug reaches the top o f the ri se r. U nde r suchcondit ions , severe s lugging is not possible .

With reference to f igure 1 , x t ) a n d z t ) can be ca lcu la ted us ing the fo l lowing .H y d r o s t a t i c p r e ss u r e

Pp = pm z - x si n a ) + P ,V o l u m e o f t h e g a s i n t h e p i p e l i n e

[ A I ]

V e = 1 - x ) a A , [A.2]where A is the cross-sect ional area of the p ipel ine .

S t a t e e q u a t i o n (assuming ideal gas)m GP P = -~ a R T , [A.31

where me i s the mass o f the gas and R i s the idea l gas cons tan t .Cons erva tion o f l iqu id

m L = m ~ + U L sp L d tCons erva tion o f gas

[A.4]

~m a = m c ~ + U a s o P eo d t , [A.5]where i refers to the in i t ia l condit ion.

The m asses o f the liqu id and gas a t any t ime can be g iven in te rms o f x and z a sfo l lows :

m L = p L A X + Z ) + (1 - a ) p f f l ( / - x ) ,P , + p L g z - x s i n f l ) 1 - x A am e = p a v e = R T

[ A 6 ][ A 7 ]

N ote th at the in i tia l v alues of the l iquid and gas masses , r ea l and m u , can a l so be ca lcu la tedby [A.6] and [A.7] with x = x~ and z- - zt . Th e determ inat ion of these in it ia l values will bed iscussed la te r . No te a l so tha t the vo id f rac t ion in the p ipe , a , i s cons ide red to be knownand i ts ca lcula t ion is g iven in appen dix B ( i t is, in principle , the same as suggested b ySchmid t e t a l . 1980).

Sub st i tu t ing moi f rom [A.7] in to [A.5] and then subst i tu t ing [A.1] , [A.2] and [A.5]into [A.3] givesP[PLg + z - x s i n B ) ] 1 - x ) a

= [ PPLg + (zl - xl sin/3)](1 - x l ) + - ~ - o U a s oP o o d t [ A . 8 1


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214 ~ d / U D A T A I T t LS u b s t it u t i n g [ A 6 ] f o r m L a n d m ~ i n t o [ A 4 ] y i e l d s t h e r e l at i o n o r t h e l i q u id c o n s e r v a t i o n

z = z i - c t (x - x~) + ULS d t [A.9]S u b s t i t u t i n g [ A . 9 ] i n [ A . 8 ] y i e l d s a s i m p l e q u a d r a t i c e q u a t i o n f o r x ( t ) as we l l a s z ( t ) .

T h e p r e d i c t i o n o f x ( t ) a n d z ( t ) g i v e n b y [ A . 8 ] a n d [ A . 9 ] c o r r e s p o n d s t o t h e s l u gf o r m a t i o n s t e p ( f i g u r e 1 ) . O n c e t h e s l u g r e a c h e s t h e t o p o f t h e r i s e r ( z = h ) t h e p r o c e s s i sc o n t i n u e d a s s h o w n i n s t e p 2 ( f ig u r e 2). T h u s a f t e r z = h t h e s o l u t i o n f o r x ( t ) i s o b t a i n e dd i r e c t l y f r o m [ A .8 ] w i t h z = h .

T h e i n i ti a l c o n d i t i o n f o r x~ a n d z~ d e p e n d s o n h o w m u c h l i q u i d f a ll s b a c k i n s t e p 4( f ig u r e 4 ) w h i c h , in t u r n , d e p e n d s o n t h e a m o u n t o f l i q u i d t h a t s t a y s a s a f i lm i n t h e b l o w o u ts t e p ( f i gu r e 3 ). S in c e th e b l o w o u t s t e p i s s i m i la r t o a T a y l o r b u b b l e m o t i o n i n n o r m a l s l u gf l o w , th e a m o u n t o f l iq u i d l e f t c a n b e c a l c u l a t e d u s i n g a s l u g f lo w m o d e l ( se e a p p e n d i x C ) .A s n o t e d p r e v i o u s l y t h i s is n o t a s t r a i g h t f o r w a r d s o l u t i o n s i n ce t h e e f f e ct i ve g a s a n d l i q u i df l o w r a t e s a r e n o t k n o w n . H o w e v e r , t h e v o i d f r a c t i o n i n a T a y l o r b u b b l e i s i n s e n s i t i v e t ot h e f l o w r a t es . F o r e x a m p l e , f o r w a t e r a n d a i r f lo w i n g i n 5 c m p i pe , a ' = 0 . 8 9 t o a v e r yg o o d a p p r o x i m a t i o n f o r a l l p r a c t i c a l f l o w r a t e s . T h i s m e a n s t h a t f o r a w a t e r - a i r s y s t e m ,a b o u t 1 0 o f th e l i q u i d f a l ls b a c k . C o n s i d e r t h a t t h e f a l l b a c k is fa s t . I f n o t , t h e r e w i ll b es o m e d i f fe r e n c e i n t h e s o l u t i o n f o r a s h o r t t i m e p e r i o d b u t n o d i f f e r en c e f o r l o n g e r t i m e so n c e t h e l i q u i d f a l l b a c k i s c o m p l e t e d . T h e n :H y d r o s t a t i c p r e s s u r e y i e l d s

Pp = pLg (Z~ - X; s in ~ ) + P~ [A .10]L i q u i d m a s s b a l a n c e r e q u i r e s

a x i + z ~ = ( 1 - - c t ' ) h [ A l q

w h i l e t h e c o m p r e s s i o n o f t h e g a s i n t h e p i p e li n e f o l lo w s t h e r e l a t i o nl [A 121P P = P ~ I - x i

Sub s t i t u t ing [A .12] an d [A .11] i n [A .10] y i e lds a s ing le equa t ion fo r x ; a s wa l l a s z~ (us ing[A.11] again) .Us in g the ca l cu la t ed va lues fo r x~ and z~ i t i s poss ib l e now to so lve fo r x an d z a s af u n c t i o n o f t i m e . I f z = h b e f o r e x = 0 , t h e n s e v e re s l u g g i n g w il l n o t t a k e p l a c e .

W e c a n s i m p l i f y t h e p r o c e d u r e b y a s s u m i n g x~, z~ = 0 . F o r a c o n s t a n t m a s s f l o w r a t eo f l i q u i d a n d g a s , [ A .8 ] a n d [ A .9 ] s h o w t h a t o n e c a n f i n d c o n d i t i o n s f o r w h i c h x i s i d e n t i c a l lyz e r o f o r a l l t i m e s . S u b s t i t u t i n g x = 0 i n [ A . 8 ] a n d [ A . 5 ] y i e l d s

ULs = PGR T U~so , [A . 13]p L g a lw h i c h s h o w s t h e l i q u i d f l o w r a t e b e l o w w h i c h s e v e r e s l u g g i n g w i ll n o t o c c u r .

T h i s i s r a t h e r a r e m a r k a b l e s i m p l e r e s u l t . I n d e e d , i n c l u s i o n o f t h e e f f e c t o f f a l l b a c kh a s o n l y a m i n o r e f f e c t o n t h i s r e s u lt .I t i s i n t e r e s t i n g t o s e e t h a t t h i s c o n d i t i o n d e p e n d s o n t h e p i p e l i n e l e n g t h ( t h o u g h n o to n t h e r i s e r l e n g t h ) . F r o m t h i s p o i n t i t i s s o m e w h a t m i s l e a d i n g t o c o r r e l a t e t h e b o u n d a r yo f s e v e r e s l u g g i n g o n a m a p w i t h U L S , U c s a s c o o r d i n a t e s w i t h o u t s p e c i f y in g t h e p i p e li n el eng th .

APPENDIX BS t r a ti f ie d f l o w m o d e l i n d o w n w a r d i n c li n at io n f l o wT h e l i q u i d d o w n f l o w i n t h e p i p e li n e t o w a r d s t h e r i s e r i s v e r y c l o se l y a p p r o x i m a t e d b ya f u l l y d e v e l o p e d o p e n c h a n n e l f l o w . S c h m i d t et al . ( 1 98 0 ) s u g g e s t e d t h e u s e o f th e M a n n i n g


13/15

ST BILITY OF SEVERE SLU OOIN G 2 5derivat ion (Allen 1972). Eq ual ly valid is the approa ch suggested by Tai te l Du kler (1976)for calcula t ing the equi l ibr ium level in s t ra t i f ied f low for the specia l case where the gasve loc ity i s neg ligible. In th i s case a m om entu m ba lance o f shea r s t r e ss and g rav i ty on thel iquid phase yie lds

T L S L p L g A L s i nwh ere (as in Tai te l Du kler 1976)

[ B 1 ]

P ~ [ B . 2 ]L=fL 2The f r ic t ion fac to r fL can be ca lcu la ted f rom the M ood y d iagram w i th the appropr ia tehydrau l ic d iamete r . For sm ooth p ipe , fo r example , the f r ic t ion fac to r can be ca lcu la ted by

A = c ~ ( ~ ) - , [ B . 3 ]w he re CL ---- 0.046, m=0.2 for turbulen t f low and CL = 16, m = 1 for laminar f low. AL,the cross-sect ional area of the l iquid and SL, the wet t ing per iphery are given in terms ofthe equilibrium liquid level hL:

AL ---- 0.25D2[~r -- co s- '(2 -- 1) + (2 -~ -- 1) 1- -(2 -- 1)2 , [B.41

= o [ , r - c o s - , ( 2 ~ - 1 ) ] [ B .5 1

Eq uat ion [B. 1] can now be solved by t r ia l and error fo r the equi l ibr ium level hL. On ce hLis given the void f ract ion a can be calcula ted bya = I AL/ A [ B . 6 ]

A general solut ion can be presented in a d imensionless forma , , . r p r p c ) g sin 8 ] [ B 7 ]

= ( d P / d x ) ~ '

where (dP /dx )zs is the p ressu re d rop when th e l iqu id f lows a lone in the p ipe, nam elyd P f z s P L U s( ~ ) z s = lB .8 ]

f ~ is the f r ic tion factor whe n the l iquid f lows a lone in the pipe.Th e resul t of [B.7] is p lot ted in f igure 8 for convenience. I t includes the sm ooth pipecase w here the f r ic t ion factor is g iven by [B.3] an d the ful ly turbulen t case w here fL iscons tan t OcL/f~= 1).APPENDIX C

Slug flow modelVert ical s lug f low consis ts of long Taylor bubbles separated by s lugs of l iquid . Thel iquid s lugs us ual ly conta in sm al l bubbles .M ode ls fo r s lug f low were p resen ted wi th va r ious degrees o f accuracy by Ta i te l , BarneaDu kle r (1980) , Ta i t el Barnea (1983) and recen t ly by Fem and es et al. (1983) , wh oproposed a de ta i led hyd rody nam ic m ode l fo r ve r ti cal upwa rd s lug f low.


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216 Ym4UDAT A F r ~In th i s append ix a s impl i f ied mode l , based p r imar i ly on the work o f Fe rnandes et al .(1983), is used. The s implif ica t ion a l lows a re la t ively s imple solut ion with l i t t le sacr if ice of

accuracy .The t rans la t iona l ve loc i ty o f a Tay lo r b ubb le i s a ssum ed to be g iven by (Nick l in e t a [ .

1962).

U , = 1.2 U s 0 . 3 5 ~ , [ C . I ]w h e r e U s i s t h e s u pe r f ic i al i x t u r e v e l o c i t y g i v e n b y

u s = u ~ u ~ s [ c. 2]A l i q u i d m a s s b a l a n c e r e l at i ve t o a c o o r d i n a t e s y s t e m t h a t m o v e s w i t h t h e t r a n s la t i o na lveloci ty U, y i e l d s

R f ( U , F U / ) = R , ( U , - - U L ) , [ C 3 ]

w h e r e U L i s t h e l i q u i d v e l o c i t y i n t h e s l u g , U / t h e i l m v e lo c i t y a r o u n d t h e T a y l o r b u b b l ep os i t i v e f o r d o w n w a r d f l o w ) , R , t h e l i q u i d h o l d u p i n t h e s l u g, a n d R f t h e l i q u i d h o l d u pi n a c r o s s - se c t i on a l a r e a o f t h e T a y l o r b u b b l e a n d t h e l i q u i d f i lm .

Th e v oid fract ion in the l iq uid is ver y c lose to 307o( Ba m e a & Br a u n e r 1 9 84 ; F e m a n d e se t aL 1 9 8 3 ) , n a m e ly Rs = 0 .7 .In the l iqu id s lug the re la t ive bub ble r ise veloci ty is U0 (given by [91) . The refore

u L = u s - u 0 1 - R ,) [c.4]T h e l i q u i d f i l m a r o u n d t h e T a y l o r b u b b l e i s c o n s i d e r e d t o b e a f r e e f a il i ng i l m f o r w h i c ht h e f i l m t h i c k n e s s i s g i v e n b y W a l l i s 1 9 6 9 )

8 k [ ~H ~.. 113 [ ~I ~] , [C.5]= D3g (P L-- Po)PL ~L 'where F i s the mass f low ra te pe r un i t pe r iphe r ia l l eng th , F = p L U / 8 . k and m fo r laminarf low equa l 0 .909 and 1 /3 . Fo r tu rbu len t f low R e = 4 F / ~ L > 1000) va r ious cons tan ts a res u g g e st e d . W a l l is s u g g e st e d k = 0 .1 1 5 , m = 0 .6 . A n a l te r n a ti v e r e la t io n p r o p o s e d b y Be lkin( 19 5 9) s u g ge s ts k = 0 . 0 6 3 , m = 2 / 3 . F e m a n d e s et al . (1983) , on the o the r hand , s t rong lyr e c o m m e n d s t h e u s e o f t h e Br o t z ( 19 5 4 ) r e la t io n w h ic h s u g g e s ts k = 0 .0 6 8 2 , m = 2 /3 w h ic hwere the cons tan ts used he re .Equ a t ion [ (2 .5 ] can b e rea r ranged in the fo rm

8 / D ) l- mU / = { k [ ~ / D 3 g ( p L - p o ) p L ] 1 : 3 [ 4 p L D / b ~ t .] m ] / m [ C . 6 ]

The l iqu id ho ldup in the f i lm, R: i s d i rec t ly re la ted to the f i lm th ickness

R / = 4 ~ - 4 ~)2 [C.7]E q u a t i o n [ C . 3 ] w i t h [ C . 7 ] , [ C . 6 ] a n d [ C . 4 ] c a n n o w b e s o l v e d b y t ri al a n d e r r o r u s i n gs t a n d a r d i t e r at i o n t e c h n i q u e s ) t o y i e l d t h e s o l u t i o n f o r t h e f i l m v e l o c i t y U / a n d t h e f i l ml i q u i d h o l d u p R/.

A con t inu i ty ba lance on the l iqu id f low ra te y ie lds

I, u:Rf1~t [ c . s ]t . s = U L R ~ u -


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S TA B ILITY O F S EV ER E S LU G G IN G 2 7

wh re 1 , is the l iquid s lug length and l~ the s lug un it length wh ich resul ts

= - [ u s - U o l - R , ) ] R , + U / R / [ c . 9 1

Fina l ly , the ave rage l iqu id ho ldup in s lug f low i s

[C lO]The s lug mode l p resen ted he re was used in the tex t fo r th ree d i f fe ren t pu rposes :(1 ) To ca lcu la te the opera t ing l ine P s / P o as a function of ~b for g iven l iq uid f low ra teULs and gas f low ra te , in term s of the a tm osph eric superf ic ia l veloci ty , U ~s0. (N ote th atU a s = U o s o P , / P o . T h e s e operat in g curves are val id for ~b--O to ~b=0.7 . Fo r ~b > 0 .7 thef low pa t te rn i s tha t o f bubb le f low and [8] was used in th i s region .

(2 ) To es t ima te the vo id f rac t ion a ' o f the gas bubb le tha t pene t ra te s the r i se r a s aresu l t o f the uns tab le s i tua t ion . As men t ioned fo r a wa t e r - a i r sys tem, 5 cm p ipe , R / i s 0.11f o r U ~ an d Uas var ia tion in the comple te range from 0 .01 to 10 m /s . Th is show s tha t theTay lo r bu bb le vo id f rac t ion i s p rac t ica lly independen t o f the l iqu id and gas f low ra te s andthus a cons tan t va lue fo r a ' cou ld be used .

(3 ) To p red ic t the am oun t o f fa l lback o f the l iqu id a f te r the b low out in seve re s lugging .A s m e n t io n e d fo r a w a t e r - a i r s y s t e m in 5 c m p ipe , R I ~ , 0 .1 w h ic h y i e ld s a b o u t 1 0 o fthe r i se r vo lum e as liqu id fa l lback . Schm id t e t aL (1980) co r re la ted the f iqu id fa l lhack w i thsuperf ic ia l gas veloci ty . A n averag e value of 10 of the r iser length is quite c lose to theirexperimental resul ts .

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Taitel Stability of Severe Slugging

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