effect of liqud viscosity

8/10/2019 Effect of Liqud Viscosity

1/16

Int. J. Muhiphase FlowVo l. 15, No. 6. pp. 877-892, 1989 0301-9322/89 3.00 + 0.00

Printed in Great Britain. All rights reserved Copy right 1989 Pergamon Press/Elsevier

E F F E C T O F L I Q U I D V I S C O S IT Y O N T H E

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

H O R I Z O N T A L P IP E F L O W

N

ANDP.II'SOS

Chemica l Process Engineer ing Research Inst i tu te , P .O. Box 19517, 540 06 Thessa lonik i , Greece

L

WILLIAMS an d

T J

HANRATTY

D e p a r tm e n t o f Ch e m ic a l En g in e er in g , U n iv e r s i ty o f I ll in o is , U rb a n a , IL 6 1 8 0 1 , U .S .A .

(Received 21 M a r c h 1988; in revised orm 27 December 1988)

Alwtract The

effec t of l iqu id v iscosity on the in i t ia t ion of s lug f low was s tudied in ho r izonta l 2 .52

and

9.53 cm pipe l ines. The resu l ts show the s tab i l iz ing e ffect of v iscosi ty predic ted by Lin & H anra t ty , and

are a t var iance wi th ana lyses which use a long-wave length invisc id approxim at ion . Fo r very v iscous liquids

a s tab i l i ty ana lysis which recognizes tha t s lugs or ig ina te f rom a t ra in of smal l -wave leng th s inusoida i waves

seems consis ten t wi th the measurements .

Key Words: two-phase f low, hor izonta l , gas- l iquid , t ransi t ion , s lug f low, exper imenta l , l iqu id v iscosi ty

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

This pa per presents the resu l t s of exper imen ts on the e f fect of l iqu id v i scos ity on the t ran s i t ion f ro m

a s t ra t i f ied pa t te rn to a s lug pa t te rn for gas- l iqu id f low in a hor izonta l p ipe l ine . They sugges t a

new m echan i sm, whe reby t he p r ecu r so r t o s l ug fo rma t ion i s t he appea ran ce o f sma l l -wave leng th

K elv in -H elm hol tz (K H) waves . This me chan ism i s appl icable to l iqu ids wi th v iscos it ies > 20 cP,

and could be impor tan t in la rge d iameter p ipe l ines .

Th e t rans i t ion f ro m a s t ra t if ied pa t te rn to a s lug pa t te rn fo r a hor izonta l gas- l iqu id f low has been

the sub j ec t o f a num ber o f t heo re ti c a l s tud ie s . Ea r ly works by Ko rdyb an Ran ov (1970) and

Wal l i s Do bso n (1973) explored a s tab i li ty me chan ism, wh ereby the s lugs a r i se f rom the grow th

of inf in i tes imal d i s turban ces a t the in te r face . S ince the spac ing be twee n the s lugs i s la rge com pare d

to the p ipe d ia m eter it was na tu ra l to res t r ic t the ana lys i s to w aves which have a la rge wavelength

compared to the he ight of the gas space . I f v i scous e f fec t s a re neglec ted the fo l lowing condi t ion

for ins tab i l i ty i s ob ta ined for a hor izonta l rec tangular channel of he ight B:

k p L (U - - C ) 2

c o t h

k h L -t- k p G ( U - - C ) 2

c o t h

k hG

= g ( P L - - P G ) + t r k 2 , [ 1 ]

wh ere k i s the w aven um ber , U is the gas ve loc i ty , u is the l iqu id ve loc i ty , hG i s the he igh t o f the

gas space , h is the height of the l iquid space, PG is the gas densi ty, PL is the l iquid densi ty and

g i s the acce le ra t ion of gravi ty . Fo r la rge w a v e s

( k h L ~ .

l , khC ,~, l ) and for negl ig ib le sur face tens ion

ef fec t s the fo l lowing ins tab i l i ty condi t ion i s ob ta ined f rom [1] :

US PG -- 1 >

ghG P P

and the wa ve ve loc i ty , C, i s

C = Up 6hL + UpLhG

PLhG + hLPG

I f PGhL/pLhG i s smal l com par ed to uni ty [2] and [3] s impl i fy to

(U -- u ) > [ g( P L -- pGhG2~g . j

a n d

C = u + U ~ .

ghL

- - 1 ; [2]

[ 3

[ ]

[ s ]

877


2/16

878 N ANDRITSOS e t a l

The phys ica l i n t e rpre t a t ion of t he K H ins t ab i li t y represented by [4] is as fo l lows: t he presence

of waves a t t he in t e r face causes a ve loc i ty ma ximu m in the gas a t t he c res t and a ve loc i ty min im um

at the t rough . Acc ord ing to the Bernoul l i equ a t ion th is f low var i a t ion i s assoc ia t ed w i th a pressure

minim um a t t he li qu id surface a t t he c res t and a pressure max imu m a t t he t rough . I f t he

des t ab i li z ing inf luence o f t hese forces i s l a rge enou gh to o verc om e the s t ab i li z ing e f fec t o f g rav i ty ,

ins t ab i l it y occurs .

W al li s D ob so n (1973) perfo rm ed t rans i t ion s tud ies in rec t angu lar channel s and conc luded

tha t [4] overpre d ic t s t he c r it ica l gas ve loc i ty by a fac tor of abou t two . A nu m ber o f au th ors have

argued tha t t h i s d i screpancy can b e expla ined b y cons ider ing n on- l inear e f fec ts . The ana lyses of

Ko rdy ba n (1977), M ish ima Ish ii (1980) and W al li s D ob so n (1973) sugges t a c r i t e r ion of t he

sam e form as [4] wi th factors of 0.74, 0.49 and 0.50 on the r .h.s. T ai tel D uk ler (1976) suggest

tha t i ns t ab i l i t y occurs in a channel when

p ~

J

[61

For gas- l iqu id f low in a p ipe they g ive

[ -g 0 L - - 0 ) X T :

L ~ S j . [7]

Here S , i s t he l ength of t he in t e r face

h L _ i )2 3 ''2 ,

hL is the he ight mea sured f rom the bo t tom of the p ipe , D i s t he d i ameter and Ao is t he a rea occ upied

by the gas .

Recen t ly , L in H anr a t ty (1986), H an ra t ty (1987) and W u

et al.

(1987) have re -examined the

poss ib i l it y of expla in ing the t rans i t ion f rom s t ra ti f ied to in t e rm i t ten t f low by l inear s t ab i l it y theory .

T h ey r e t a i n ed t h e a s su m p t i o n t h a t t h e w av e l en g t h i s l o n g co m p ared t o

h o,

o r h L , b u t ab an d o n ed

the assu mp t ion of i nv i sc id f low by inc luding the e f fec t s o f t he drag of t he gas o n the l i qu id , T i, and

of the res is t ing s t ress o f t he w al l on the l iqu id , ~WL. Eq uat io n

[ 2

s t i l l ho lds provided the ve loc i ty

prof i les in the l iqu id and the gas can be app rox ima ted by p lug f lows. The pr inc ipa l d i f fe rences foun d

by Lin H an ra t ty i s t ha t t he wav e ve loc i ty i s no t g iven by [3] or [5] .

The f i rs t te rm on the l .h .s , o f [1] represent s t he des t ab i l iz ing e f fec t o f l i qu id iner ti a . Fo r a i r -w ater

flows,

PG/PL

= 1.2 X 10 -3 and [5] simpl if ies to C = u for the range o f con di t ion s o ver w hich the

t rans i t ion i s observ ed . Co nseq uent ly , t he inv isc id K H ana lys i s p red ic ts no inf luence of li qu id iner ti a

on neutral s tabi l i ty since i t predicts

C/u -

1) ~ 0. I t repre sents a stat ic instabi l i ty . In co ntras t , the

v i sco u s an a l y s is o f L i n H a n ra t t y g i ve s n o n -ze ro v a lu e s o f

C/u -

1) because of t he inc lus ion of

the effects of ~, and 3w. As a resul t , an important destabi l izing effect of l iquid inert ia i s predicted

by the long-wavelength v i scous K H ana lys i s for l ow v i scos ity l iqu ids . The inc lus ion of v i scous

s t resses , t herefore , g ives the surpr i s ing e f fec t o f caus ing the a i r -w ater sys t em to be mo re u ns t ab le ,

in tha t t he in i t ia t ion o f l ong-wa velength in t e r fac ia l d i s turba nces i s p red ic t ed to oc cur a t muc h low er

gas ve loc i t i es (cons i s t en t wi th exper imenta l observa t ion) tha t i s g iven by an inv i sc id KH ana lys i s .

The theore t i ca l s ign if i cance is tha t a mech ani sm for the in it i a tion of i n t e rmi t ten t f low by the grow th

of in f in it es imal d i s turbanc es can not be ru l ed out .

These theore t i ca l resu lt s o f L in H anr a t ty (1986) a re i l lus t ra t ed in f igure I . The sys t em be ing

cons idere d i s a conc urren t f low of gas and l iqu id in a hor i zon ta l rec t angular channel o f he ight B .

The ord ina te i s t he d imensionless l i qu id he ight ,

hL/B. The

dashed curve i s t he superf i c i a l gas

ve loc i ty , Usc , p red ic t ed b y an inv isc id ana lys i s for the grow th o f l ong-wav elength d i s turbance s . Th e

sol id curves a re the v i scous pred ic t ions for a i r -water f low. In th i s case the in t e r face i s covered by

smal l -wavelength waves when in t e rmi t t en t f low i s i n i t i a t ed . Therefore , t he ana lys i s envi s ions a

l o n g -w av e l en g th d i s t u rb an ce su p e r i m p o sed u p o n t h e ro u g h en ed i n te r f ace . T h e e f fec t o f t h e se w av es

is fe l t by an increase in the in te r fac i a l s t ress . The p ara m ete r f / f ~ i s t he ra t io o f t he f r i c tion fac tor

f PU2

[9]


3/16

L I Q U I D V I SC O S IT Y O N T H E S T R A T I F I E D S L U G T R A N S I T I O N

8 9

to the f r ic t ion fac tor for a sm oo th wal l , f , . A t rans i t ion a t lower gas ve loc i ties predic ted by the

viscous analysis is clearly indicated.

Lin H an ra t ty (1986) a l so present s tab i li ty ca lcu la t ions for l iqu ids o the r than water . The

predic ted e f fec t of a n increase in l iqu id v iscosi ty , when presented in a p lo t su ch as f igure 1 , i s to

t rans la te the so l id curves upwa rd . T he v iscous ana lys is thus p redic t s a s tab i liza t ion wi th increas ing

viscos i ty (when v iewed in th i s type of p lo t ) . However , the reason for th i s i s no t because of the

inc rea sed damping o f d i s tu rbances . S ince compar i sons a r e m ade a t f ixed hL/B increases in viscosi ty

are acco m pan ied by decreases in the l iqu id f low ra te for a g iven U sc . The iner t ia of the l iqu id and ,

there fore, i ts destabli l izing effect decre ases with in creasing l iquid viscosi ty. Fo r su ff icient ly high

l iquid v iscos i ty , des tab i l iz ing e f fec ts of l iqu id iner t ia beco me negligible and the L in Ha nra t ty

ana lys is is represented by the dash ed curve . I t , therefore , g ives the surpr i s ing resu l t tha t the inv isc id

theory becomes more accura te as the l iqu id v iscos i ty increases .

W hen the L in Ha nra t t y ana ly s is i s exh ib it ed i n M and han e coo rd ina t e s (UsL vs Us~), r a the r

than in the coord in a tes used in figure 1, the in te rpre ta t ion b ecom es m ore comp l ica ted . An increase

in l iquid viscosi ty has tw o op pos ing effects: for a f ixed Usc, an increase in the l iquid viscosi ty will

be assoc ia ted wi th the des tab i l iz ing e f fec t of an increase in hL for an y g iven USL. Ho wev er , an

increase in viscosi ty is acco m pa nie d by a sm aller USL for an y hL. This d ecreas es the de stabil izing

effect of l iquid inert ia .

L in H anra t t y (1986) t e st ed the i r ana ly si s by ca r ry ing ou t l abo ra to ry s t ud ie s o f a i r and w a te r

f lowing in a 2.52 cm i .d. pipe with a length of 600 pipe diameters and in a 9.53 cm i .d. pipe with

a length of 260 p ipe d iam eters . The y foun d tha t for UsG > 3 .3 m/s the s lugs were fo rm ed by a

non l inea r m echan i sm invo lv ing t he coa l e scence o f l a rge -ampl i tude i r r egu l a r waves . C onsequen t ly ,

i t could be expec ted tha t a t rans i t ion caused by the growth of inf in i tes imal d is turbances could be

appl icable only for UsG < 3 .3 m/s . A co mp ar ison o f the t rans i t ion fo r a i r -wa ter observed un der th i s

res t r ic t ion agrees qui te wel l wi th predic t ions based on the v iscous , long-wavelength KH ana lys is

i f f / f~ i s t aken equa l t o 2 , a r e a sonab l e a s sumpt ion fo r a i r -wa te r f l ow in t h i s r ange o f hL/D

(And r i t sos H an ra t ty 1987b).

How eve r , t he se t e st s w i th a i r and wa te r a r e a l so i n app rox im a te ag reem en t w i th t he non - l i nea r

ana ly se s d is cussed above . Th i s po in t ed ou t t he need fo r add i t i ona l expe r imen t s and p rom pted t he

in i t ia tion of a labo ra tor y s tud y o f the e f fec ts of change s of the l iqu id v iscosi ty . The resu l ts of th i s

ef for t a re presented in th is paper . Th e s t rong effec t of l iqu id v iscos i ty predic ted by Lin Ha nra t ty ,

when t he t r ans i t i on da t a a r e p lo t t ed a s

hL/D

vs Use , i s observed . However , a c loser compar ison

shows d i ff e rences be tween l abo ra to ry obse rva t ions and t he L in Ha nra t t y ana ly s is . Th i s p rompted

addi t iona l de ta i led s tudies of the mec hanism s for the in i tia t ion o f slugs . On the bas is of these tes t s

1.0

0.9

0,8

l j

m

0.7

0.6

. a 0 5

0 . 4

0 . ~

0 . 2

0 . I -

~ f t/ft 1.0

- f i / f 2 . 0

i l l I t t i l l I l

OOl 0~3~ QO3 005 0.07 0.10 0.20 0.30 3,50

U S G

F i g u r e I . S t a b i l i t y a n a l y s i s f o r s t r a ti f i e d fl o w in a c h a n n e l o f h e i g h t B : - - - i n v i s c id K H a n a l y s i s ; - -

v i s c o u s K H a n a l y s is .

Po PL =

1.12 x 10 -3

VG V L =

16.1.


4/16

8 8 0 N ANDR I TSOS e t a l

it is argued that a KH stability analysis can be used for high viscosity liquids only if the assumption

of long-wavelength disturbances is abandoned.

An important ingredient in developing these arguments is the recent study by Andritsos

Hanrat ty (1987a), over a range of liquid viscosities of 1-80 cP, of the initiation of waves in stratified

gas-liquid flows. For air and water the first waves observed with increasing gas velocity are

generated by a sheltering mechanism involving pressure variations in phase with the wave slope.

At larger gas velocities irregular, large-amplitude waves appear. These are initiated by gas-phase

pressure variations 180 out of phase with the wave height (a KH mechanism). The influence o f

pressure variations in-phase with the wave slope becomes less important with increasing liquid

viscosity. In fact, at sufficiently high liquid viscosities the first waves that appear with increasing

liquid viscosity are the KH waves.

This influence of liquid viscosity has been pointed out some years earlier by Francis (1954, 1956)

and Miles (1959). Both Francis and Andritsos Hanratty observed the same mechanism for the

initiation of waves on a very viscous liquid. The first disturbances observed with increasing gas

velocity are small-amplitude, small-wavelength, rather regular 2-D waves. With a slight increase

in gas velocity, these give way to a few large-amplitude waves with steep fronts and smooth troughs,

and with spacings that can vary from a few centimeters to a meter. Ocassionally, several small 2-D

waves can be seen in front of the large waves.

In this paper evidence will be presented that events leading to the appearance of intermittent flow

at low gas velocities or to the appearance of large-amplitude irregular waves at large gas velocities

are the same. Consequently, an attempt will be made to interpret both phenomena by the same

stability analysis. Similar approaches have been adopted previously by Wallis Dobson (1973)

and by Andreussi Persen (1987). The large-amplitude waves seem to have been given a number

of names: slugs by Wallis Dobson (1973); roll waves, by Lin (1985); large disturbance waves,

by Andreussi Persen (1984); and irregular large-ampli tude waves, by Andri tsos (1986). Since the

initiation of these waves has been shown by Miles (1959) and by Andri tsos Hanrat ty (1987a)

to be due to a Kelvin-Helmholtz instability it seems appropriate to call them Keivin-Helmhol tz

(KH) waves.

DESCRIPTION OF THE EXPERIMENTS

The experimental results, discussed in this paper, for the 10 m horizonta l 2.52 cm pipe are from

the studies of Andritsos Hanrat ty (1987a) for the flow of air and liquids with viscosities of 1,

4.5, 16 and 70 cP. The results for 25 m horizontal 9.53 cm pipeline were obtained recently in a new

facility constructed by L. Williams. Transitions were observed with glycerine-water solutions

having viscosities of 1, 20 and 100 cP. Simple T-junction entries were used for mixing the two

phases at the inlet in both studies. In the 9.53 cm pipe the gas and liquid were introduced at two

different locations.

The thickness of the liquid layer flowing along the bottom of the pipe, hL, was measured in one

test section by two parallel-wire conductance probes which extended vertically across the whole

pipe cross section. For experiments in which small-wavelength waves were present the liquid height

was represented as the time-average of the signal from the conductance probes. A second test

section used pairs of short parallel 0.51 mm chromel wires at 45 c intervals to measure the variation

of the liquid height around the pipe circumference. In the 2.52 cm pipe the two sections were

separated by 10.2 cm, and in the 9.23 cm pipe, by 26.7 cm.

EXPERIMENTAL RESULTS

a) Mandhane p lo t s

A map of the different flow regimes observed in the 2.52 cm pipe is given in figure 2. The solid

line is for a liquid of 4.5 cP; the dashed line for 1.0 cP. Both shown a large region where the

pseudo-slugs, described by Lin Hanrat ty (1987a), occur. These are large-amplitude waves which

touch the top wall. They differ from slugs in that they are not accompanied by large pressure

fluctuations and they can lose their coherency as they move down the pipe.


5/16

L I Q U I D

V I S CO S I T Y

O N T H E S T R A T I F I E D - - S L U G T R A N S I T I O N

881

_1

I ,O t

j . ~ .

S l u / ~

O . I - - / / P l e u do - S lu g

~ t A nnu la r

mlzal ion

o . o o l I I I

1.0 IO.0 IO0,0

OsG m/s )

F i g u r e 2 . F l o w r e g i m e s f o r t h e 2 . 52 c r n h o r i z o n t a l p i p e :

- - , 4 c P l i q ui d ; - - - , I c P l i qu i d . T h e t ra n s i t i o n t o

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

1 .0

i

Slug

o o i i i

" I ~lon

o . o o l I ' - I

1.0 I0.0 IOOD

USG m /s l

F i g u r e 3 . F l o w r e g i m e s f o r a 2 0 c P ) a n d a I c P - - - )

l i q u id i n t h e 9 .5 3 c m h o r i z o n t a l p i p e . T h e t r a n s i t i o n t o

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

As d i scussed by An dr i t sos Ha nra t ty (1987a) , t he f i rs t wav es tha t appe ar on a s t ra ti f ied f low

wi th increas ing gas ve loc i ty a re smal l -ampl i tude , regular 2 -D waves assoc ia t ed wi th gas-phase

pressure var i a t ions in phase wi th the wave s lope . Thi s t rans i t i on i s no t shown. At l a rger gas

ve loc i ti es l a rge-am pl i tude i r regular waves appea r tha t a re assoc ia t ed wi th pressure var i a t ions in

p h ase w i th t h e w av e h e i g h t (K H w av es ) .

An dr i t sos H an ra t ty (1987a) g ive resu l ts s imi la r t o those show n in f igure 2 for 16 and 70 cP

l iqu ids . These show a decrease in the l i qu id ve loc i ty requi red to in i t i a t e s lug f low a t l ow gas

ve loc it ies . For the 16 cP l iqu id the regular w aves appea r ov er a very narro w range o f gas ve loc it i es

an d fo r t h e 7 0 cP l i q u id t h ey d o n o t ap p ea r a t a ll.

Ob serve d t rans i t i ons in the 9 .53 cm p ipe a re show n in f igures 3 and 4 for 20 and 100 cP l iqu ids .

I t is no te d th a t t he p seud o-s lug reg ion decrease s in size wi th increas ing p ipe d i ameter and increas ing

l iqu id v i scos i ty . In fac t , i t is no t obse rved for t he 100 cP l iqu id in the 9 .53 cm p ipe .

T h e b eh av i o r a t t h e t r an s i ti o n t o s l u g fl o w can b e s ep a ra t ed i n t o t h ree r eg io n s (w h i ch a re m o s t

ev ident for t he h igh v i scos i ty l i qu ids) . The f i r s t i s for gas ve loc i t i es which a re too low for t he

app eara nce of K H wav es in a s t ra ti f ied f low (UsG < 4 m/s) . The second ex tends to the gas ve loc i ty

a t w h i ch p seu d o - s l u g s f i rs t ap p ea r i n t h e 9 .5 3 cm p i p e (U sG = 4 - 1 2 m / s ) . T h e t h ird is fo r

UsG > 12 m/s.

Fo r m ost o f t he f i r s t reg ime the f low i s no t fu l ly -deve loped and the l i qu id f low i s g rea t ly

inf luenced by the hy draul i c gra d ien t t ha t ex i s ts a long the en t i re p ipe , even thoug h i ts length i s 400

pipe d i am eters fo r t he 2 .52 cm p ipe and 260 p ipe d i ameters fo r t he 9 .53 cm p ipe . W hen such

hydraul i c grad ien t s ex i s t t he l i qu id f lowing for a g iven hL/D s l a rger s ince i t i s be ing moved both

by the gas drag a t t he in t e r face and the hydrau l i c grad ien t . B ecause the in i t ia t ion o f s lugs i s

p r i m ar i l y d ep en d en t o n

hL/D

t he USL va lues corre spo nding to s lug f low t rans i t i ons a t l ow UsG in

f igures 2-4 a re h igher tha n wo uld be ob serve d for a fu l ly -dev e loped f low. Thi s e f fec t i ncreased wi th

increas ing l i qu id v i scos i ty , increas ing p ipe d i am eter an d decreas ing l i qu id ve loc i ty . Thi s no n-devel -

ope d co ndi t ion was obse rved for l ow l iqu id v i scos i ti es a t UsG < 2 m/s in the 2 .52 cm p ipe an d a t

Us~ < 3.5 m/s in the 9 .53 cm p ipe . I t was obse rved for a l iqu id wi th a v i scos i ty of 100 cP a t

UsG < 4.5 m/s in the 9.53 cm pipe. Figu re 5 i l lust rates this effect for ai r-w ate r f lo w in the 9.53 cm

pipe . Here i t i s no ted tha t a t l ow gas ve loc i t i es t he he ight o f t he l i qu id i s i nsens i t i ve to changes

in UGs and dep end s on ly on the loc a t ion in the p ipe .

T h e ap p ea ran ce o f K H w av es i n a s t r a t i f i ed f l o w i s a cco m p an i ed b y a l a rg e i n c rea se i n t h e

in t e r fac ia l s t ress and a dras t i c t h inn ing of t he l i qu id l ayer , as i ll us t ra t ed in f igure 6 . Because of t h is

th inn ing of t he l i qu id mu ch l a rger USL are requi red to in i t ia t e s lugs . Th i s i s re f l ec ted by the sharp

rise in the t ran si t ion cur ve f or UsG = 4.5 -- 12 m/s in f igure 4 for a 100 cP l iquid. Th e t ransi t ion

to s lug f low observed wi th increas ing l i qu id f low in th i s reg ion in the 9 .53 cm p ipe for a l l l i qu ids


6/16

882

1.0

N A N D R I T S O S

e l a l

i

E

Slug

_ ~

~ . ~ ~ . / . ~ ~ \ / P s e ud o - Sug. . . . .

o . . .. r . . . .

.

\ ~ Annulor

llti /

~ K - H I ~

o . o o ,

i - - I

1,0 I0.0 I00.0

USG (m/s)

F i g u r e 4 . F l o w r e g i m e s f o r a 1 0 0 c P ) a n d a l c P

-- - ) l i q u i d i n t h e 9 .5 3 c m h o r i z o n t a l p i p e .

-J

.c

0. 7 ----~

0.6

H

0 . 5 - -

o

0 . 4 - -

0 . 3 - -

0.2 ~

0.0

D D

0 0

1380 Dio

160 Dia

o 2; 0 Dia

D =9.53cP

p,L = I CP

g

0 0

g

1.0 I .0

UsG (m/s)

F i g u r e 5 . P l o t s h o w i n g t h a t a t l o w g a s v e l o c i t i e s t h e h e i g h t

o f t h e l i q u i d i s i n s e n s i t i v e t o c h a n g e s i n g a s v e l o c i t y a n d i s

m a i n l y d e p e n d e n t o n t h e l o c a t i o n i n t h e p i p e .

an d i n th e 2 .5 2 cm p i p e f o r 1 6 an d 7 0 cP l i q u i d s is v e r y s h a r p an d acco m p an i ed b y l a r g e p r e s s u r e

p u l s a t i o n s . F o r t h e l o w er v i s co s i ty l i q u i d s in t h e 2 . 5 2 cm p i p e t h e ex i s t en ce o f a l a rg e p s eu d o - s l u g

r eg i o n m ak es t h e d e f i n i t i o n o f t h e t r an s i t i o n m o r e d i f f i cu l t .

T h e t r an s i t i o n t o s l u g f l o w w i t h i n c r ea s i n g l i q u i d v e l o c i t y a t USG > 1 2 m / s is h a r d e r t o d e f i n e

b ecau s e o f t h e p r e s en ce o f p s eu d o - s l u g s . S l u g s m o v e ap p r o x i m a t e l y w i t h t h e g a s v e l o c i t y . A t

t r an s i t io n o n ly one s lug w i l l ex i s t in the p ipe l ine . Af te r i t s pas sag e the l iqu id l eve l in the p ipe i s

g r ea t l y r ed u ced . P s eu d o - s l u g s , t h a t a r e p r e s en t b e t w een t h e ap p ea r an c e o f s lu g s , t r av e l a t v e l o c i ti e s

u p t o 3 0 o f th e g a s v e l o c it y a n d c a r r y l i q u id w i th t h e m . C o n s e q u e n t l y , i t m a y t a k e a s m u c h a s

5 m i n a f t e r th e p a s s a g e o f a s l u g f o r th e l i q u i d to t h i c k e n e n o u g h f o r a n o t h e r s lu g t o a p p e a r . T h e

p r e se n c e o f a s lu g w a s d e t e c t e d f r o m m e a s u r e m e n t s o f p r es s u re f l u c tu a t i o n s , a s d is c u s se d b y L i n

& H an r a t t y ( 1 9 87 b ) . I t is n o t e d f r o m f i g u re s 2 - 5 an d f r o m f i g u r e s 1 3 an d 1 4 i n t h e p ap e r b y

A n d r i t s o s & H an r a t t y ( 1 9 8 7 a) t h a t t h e t r an s i t i o n cu r v e i n a M an d h an e p l o t a t U sG > 1 2 m / s i s

a p p r o x i m a t e l y i n d e p e n d e n t o f li q u i d v i c os i ty a n d o f p i p e d i a m e t e r f o r p i p e si ze s f r o m 2 . 5 4 t o

9.53 cm.

b) Effect of liquid height on transitions

I n o r d e r t o o b t a i n a m e ch an i s t i c u n d e r s t an d i n g o f t h e t r an s i t i o n t o s l u g f l ow , it i s u s e f u l t o p l o t

t h e r a t i o o f t h e l i q u id h e i g h t a n d t h e p i p e d i a m e t e r , hL/D, vs super f i c i a l gas ve loc i ty , Usc. Thi s typ e

o f p l o t m ay a l s o h av e t h e ad v an t a g e t h a t i t i s le ss s en s it i v e t o w h e t h e r t h e f l o w i s u n d e r d ev e l o p ed .

F i g u r e 7 p r e s en t s r e s u l ts o b t a i n ed i n t h e 9 .5 3 cm p i p e . T h e h L w e r e m e as u r e d j u s t p r i o r t o

t r an s i t i o n a t l o ca t i o n s i n t h e p i p e w h e r e s l ug s w e r e fi r st o b s e r v ed . T r an s i t i o n s w e r e o b s e r v ed a t t h e

m i d d l e o f t h e p i p e f o r l i q u i d s w i t h v i s co s it i es < 2 0 cP an d u p s t r e am o f t h e m i d d l e f o r l a r g e l iq u i d

v i sco s it ie s . I n s o m e ca se s t h e g a s v e l o c i t y a t w h i ch t h e l i q u i d h e i g h t w as m e as u r e d w as 2 0 s m a l l e r

t h an t h a t r eq u i r ed t o i n i t i a t e s lu g s . T h e r e f o r e , t h e v a l u e s o f h L m ay b e s o m ew h a t h i g h e r t h an t h e

t rue va lues .

T h e o p en s y m b o l s i n fi g u r e 7 r ep r e s en t co n d i t i o n s w h e r e a t r an s i t i o n t o s l u g f lo w w as o b s e r v ed .

T h e s o l i d s y m b o l s r e p r e s e n t c o n d i t i o n s w h e r e a t r a n s i t i o n t o K H w a v e s w a s o b s e r v e d . C o n s i d e r

the r esu l t s fo r a 100 cP l iqu id . I t is s een tha t fo r UsG < ca . 5 m /s an d h , / D > ca . 0 . 4 t h a t t h e h , / D

r eq u i r ed f o r t h e ap p ea r a n ce o f r o ll w av es v a ri e s s t r o n g l y w i t h U se . F o r t h e s e co n d i t i o n s t h e

i n t e r f ace is s m o o t h a t t h e t r an s i t i o n t o s l u g g i n g . H o w e v e r , f o r UsG > ca . 5 m / s t h e i n t e r f ace o f t h e

1 00 c P l i q u id i s r o u g h e n e d w i t h l a r g e - a m p l i t u d e w a v e s w h e n s lu g s a p p e a r . F o r t h e r a n g e o f g a s

v e l o c it i es o f 5 - 9 m / s t h e h L / D a t t r a n s i t i o n i s a p p r o x i m a t e l y c o n s t a n t a n d e q u a l t o 0 . 4 . A t

UsG > ca. 9 m /s the hL/D ag a i n d ec r ea s e s w i t h i n c r ea s i n g U SG - F o r h , / D < ca . 0 .4 the f ir s t t r an s i t ion

w i t h a 1 00 cP l i q u id t h a t i s o b s e r v ed i s t h e i n i t i a t i o n o f l a r g e - am p l i t u d e w av es . F o r ex a m p l e , f o r


7/16

LIQUID

V I S C O S I T Y O N

TH E

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

'-, . I0

5

0 . 4

0 . 3

0 .2

0.1

0 , 0

2 0

1 5 - -

1

0 9 . 5 3 c m

F L ( c P ) U ~ ; L ( r o l l )

I 0 , 0 2 5

8 0 . 0 2

LI A A

e l l

2

,

2 0

15

I

~_=

- I 0 -

A A /

2 . -

o

t , o

d

K - H I

A

Wove

i t

5 t O 2 0 3 0

U s G ( m / s )

0 . 9

0 . 7

0 . 5

0 . 3

0.1

0 , 2

i i i ] I I

- ~ . . ~ ~ 2 o

~ , ~ ~ ~ O A 1 0 0

_

.

4

I I I i k l N

0 . 5 1 ,0 2 ~ ) 5 . 0 I0 2 0

U SG ( m / s )

F i g u r e 6 . E f f e c t o f t h e a p p e a r a n c e o f K H

w a v e s o n t h e h e i g h t o f th e l i q u id a n d t h e

i n t e r f a c i a l d r a g .

8 8 3

F i g u r e 7 . I n i t i a t i o n o f s l u g f l o w o r K H w a v e s in t h e 9 . 5 3 c m

p i pe : - - - , T a i te l D u k l e r t r a n s i t io n s ; - - , a p p r o x i m a t e

c a l c u l a t io n s o f t r a n s i t i o n s t o s l u g s ( o p e n s y m b o l s ) o r K H

w a v e s ( s o l i d s y m b o l s ) .

5 0

hL D

= 0 . 3 l a r g e - am p l i t u d e w av es ap p ea r o n a 1 00 cP l i q u i d a t U s e = ca . 6 .5 m / s an d s l u gs d o n o t

a p p e a r u n t i l U ~ = c a . i0 m / s.

F o r s m a l l hL D h e r e i s a t r an s i t i o n f r o m s t r a t if i ed - w av y f l o w t o an n u l a r f lo w a t l a r g e U s c r a t h e r

t h an t o s l u g f l o w . T h i s i s n o t s h o w n i n f i g u r e 7 .

F i g u r e 8 p r e s en t s t r an s i t i o n r e s u l t s f o r t h e 2 . 5 2 cm p i p e . I n t h i s f i g u re t h e i n i t i a t i o n o f s l u g g i n g

o r o f la r g e - a m p l i t u d e w a v e s a r e t r e a t e d t h e s a m e . N o o b s e r v a t i o n s a r e p r e s e n t e d o f t h e i n it i a t io n

o f s lu g s o n a l i q u i d i n t e r f ace t h a t h ad l a r g e - am p l i t u d e w av es p r e s en t . T h e r e s u l t s s h o w n i n fi g u r e

8 t h u s r ep r e s en t e i t h e r a t r an s i t i o n t o sl u g s o r f o r U ~ > 4 m / s t o a s t r a t if i ed f l o w w i t h K H w av es .

T h e i m p o r t an t r e s u l t t o b e n o t ed i n b o t h f i g u re s 7 an d 8 is t h a t l i q u i d v i sco s i t y h a s a s t ab i l i z in g

e f f ec t i n t h a t l a r g e r v a l u e s o f

hL D

a r e r eq u i r ed f o r l a r g e v i s co s i t i e s . I t i s a l s o n o t ed t h a t an

a s y m p t o t i c b eh a v i o r i s o b s e r v ed f o r v i s co si ti e s > ca . 20 cP w h e r eb y t h e t r an s i t i o n i s i n s en s it i v e t o

ch an g es i n v i s co s i t y .

T h e r e s u l t s s h o w n i n f i g u re s 7 an d 8 co u l d h e l p ex p l a i n t h e o b s e r v ed e f f ec t o f l i q u i d v i s co s it y

o n t h e i n i t i a t i o n o f s l u g f lo w d i s cu s s ed in t h e p r ev i o u s s ec t i o n . A n i n c r ea s e i n li q u i d v i s co s i t y

0 , 9

i i 1 i i

O 2 . 5 2 c m F ' L ( c P )

0 .7 ~ ; , . , ~ . ~ . . . . . .

-

0 . 5 - -

0 . 3 - ~ ~ ~ l k ) ~

o I i

0.2 0 5 1.0 2 0 5.0 I 2 0 5 0

U S G m / s )

F i g u r e 8 . I n i t i a t i o n o f s l u g f l o w o r K H w a v e s in t h e 2 . 5 c m p i p e: - - - , T a i t e l D u k l e r t r a n s i t i o n s ; - -

a p p r o x i m a t e c a l c u l a t i o n s o f t r a n s i t io n s t o s l u g s ( o p e n s y m b o l s ) o r K H w a v e s ( so l id s y m b o l s ) .


8/16

884 N ANDRITSOS t a l

i ncreases the hL/D co r r e s p o n d i n g t o a g i v en U SL an d , b ecau s e o f t h is , t en d s t o c au s e t r an s i t i o n a t

l o w er U SE . H o w ev e r , f o r v i s co s i ty ch an g es f r o m 1 t o 2 0 cP t h i s i s co u n t e r b a l an ced b y t h e s t ab i li z i n g

ef fec t sho wn in f igures 7 and 8 . Th i s exp la ins the sma l l e ff ec t on the USL requ i r ed for t r an s i t ion

cau s ed b y v i s co s i ty ch an g es f r o m I t o 4 .5 cP , s h o w n i n fi g u r e 2 , an d f r o m 1 t o 2 0 cP , s h o w n i n

f ig u r e 3 . H o w ev e r , f i g u r e 4 s h o w s a l a r g e e ff ec t o n t h e U sE a t t r an s i t i o n f o r a v i s co s i t y ch an g e f r o m

1 t o 1 00 cP , b ecau s e t h e d e s t ab i l i z in g e f f ec t o f i n c r ea s i n g h L D o u t b a l an ce s t h e s t ab i li z i n g e ff ec t f r o m

an increase in l iqu id v i scos i ty , shown in f igures 7 and 8 .

c) Visual observations

T h e f ir s t s l ug s f o r l i q u i d v i s co s i ti e s < 2 0 cP w e r e a l w ay s o b s e r v ed i n t h e d o w n s t r ea m h a l f o f

t h e p i p e l in e , d e s p i t e t h e f a c t t h a t h e w as l a r g e r i n t h e u p s t r ea m s ec t io n . F o r t h e 1 00 cP l i q u i d

i n t h e 9 .5 3 cm p i p e t h e f i r st s lu g u s u a l l y w as o b s e r v ed s l ig h t l y d o w n s t r e am o f t h e en t r y ,

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

p ipe l ine .

T h e m ech a n i s m s f o r t h e t r an s i t i o n f r o m a s t r a t if i ed fl o w t o a s lu g f l o w f o r U s~ < 4 m / s i s m o s t

c l ea r l y seen f o r r u n s w i t h h i g h v i s co s it y li q u i d s b ecau s e t h e i n t e r f ace o f t h e s t r a ti f ied f l o w is s m o o t h .

C o n s e q u e n t l y , sp e c ia l c a r e w a s t a k e n t o a v o i d t h e f o r m a t i o n o f s l ug s a t t h e e n t ry w h e r e

o b s e r v a t i o n s c o u l d n o t b e m a d e . T h i s w a s d o n e b y e l e v a ti n g th e e n t r y f r o m t h e r e st o f t h e p ip e

b y a b o u t o n e - t e n t h o f a p i p e d i a m e t e r . T h i s c a u se d t h e l i q u i d h e i g h t a t t he e n t r y t o b e s m a l le r t h a n

i n t h e r e st o f t h e p ip e . T h e f i r st in t e r f ac i a l d i s t u r b an ces w e r e t h en o b s e r v ed f a r d o w n s t r e am o f t h e

en t r y w h e r e h L h ad a f a i r l y co n s t a n t v a l u e .

F o r r u n s w i t h t h e h i g h es t v i s co s i ty l iq u i d s i n b o t h t h e 2 . 5 2 an d 9 . 52 cm p i p e s, t h e f i rs t

d i s t u r b an ces p r i o r t o t h e ap p ea r an ce o f s l u g s w e r e v e r y s m a l l s i n u s o i d a l d i s t u r b an ces w i t h a

w a v e l e n g t h o f 2 - 3 c m . T h e s e w e r e p r e se n t f o r a f r a c t i o n o f a s e c o n d a n d q u i c k l y g a v e w a y t o a

l a r g e - am p l i t u d e w av e w h i ch g r ew v e r y r ap i d l y t o b r i d g e t h e w h o l e p i p e . W i t h i n c r ea s i n g U s ~ t h e

l i q u id l ay e r i s t h i n n e r a t t h e t r an s i t i o n . E v e n t u a l l y , a t s u f f ic i en t ly l a r g e U s c th e l i q u i d l ay e r is s o

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

a r e n o t ab l e t o g r o w t o b r i d g e t h e p i p e c r o s s s ec t i o n . I n s t ead t h ey a r e t h e i r r eg u l a r w av es , i d en t if i ed

b y F r an c i s an d b y A n d r i t s o s & H an r a t t y , w h i ch cau s e l a r g e r i n c r ea s e s i n t h e i n t e r f ac i a l s t r e s s an d

a t h i n n i n g o f t h e l i q u i d l ay e r .

F o r i n t e r m ed i a t e l i q u i d v i s co s i ti e s an d f o r g a s v e l o c i t ie s < 3 m / s , t h e f i rs t s l u g s ap p ea r ed c l o se

t o t h e m i d s e c t i o n o f t h e p ip e b y t h e f o l l o w i n g m e c h a n i s m . W h e n t h e l iq u i d w a s t h ic k e n o u g h f o r

t h e ap p e a r an c e o f s l u gs , a s h o r t l en g t h o f t h e l iq u i d w as co v e r ed w i t h sm a l l s i n u s o i d a l d i s t u r b an ces .

T h es e p e r s i s t ed f o r a f ew s eco n d s b e f o r e p r o d u c i n g a l a r g e - am p l i t u d e w av e w h i ch ev o l v ed i n t o a

s lu g a f te r t ra v e l in g a d i st a n c e o f a b o u t 1 m . A t U s c = 3 - 4 .5 m / s th e s m a l l - a m p l i t u d e w a v es , w h ic h

f ir s t ap p ea r ed o n t h e s t r a t i f ied l i q u i d w i t h i n c r ea s i n g l i q u i d f lo w , co v e r ed t h e w h o l e l en g t h o f t h e

p i p e li n e . W i t h a s l i g h t i n c r ea s e in U s e t h e s m a l l - am p l i t u d e w av es ch an g e d t o l a r g e - am p l i t u d e K H

w av es . O n e o f t h e s e q u i ck l y ev o l v ed i n t o a s lu g ; o ca s s i o n a l l y , a s l u g w as p r o d u c ed b y t h e

c o a l e sc e n c e o f tw o w a v e s . A s l ug , o n c e f o r m e d , m o v e d q u i c k l y d o w n t h e p i p e a n d t h i n n e d o u t t h e

l i q u i d l a y e r b e f o r e a n o t h e r c o u l d a p p e a r .

F i g u r e 9 a i s a p h o t o g r a p h o f r e g u l a r 2 -D w a v e s w h i c h c o v e r e d t h e w h o l e l e n g th o f th e p i p e, w h e r e

t h e d i r ec t i o n o f fl o w is f r o m f i g h t t o l e f t. I f th e l i q u i d is t h en en o u g h t h e s e r eg u l a r w av es can ev o l v e

i n t o i r r eg u l a r l a r g e - am p l i t u d e w av es s u ch a s s h o w n i n f i g u r e 9 ( f ) . F i g u r e s 9 ( d , e ) s h o w t h e f r o n t

an d t a il o f a l a r g e - am p l i t u d e w av e w h i ch h a s j u s t g r o w n t o f o r m a s l u g o n a t h i ck l i q u i d l ay e r .

F i g u r e s 9 (b , c ) s h o w t h e f r o n t a n d b a c k o f a s lu g t h a t w a s f o r m e d 8 0 p i p e d i a m e t e r s u p s t r e a m f r o m

w h e r e t h e p h o t o g r a p h w a s t a k e n .

T h e f i r st s lu g s t h a t a r e f o r m ed ac t a s a s w eep b y t ak i n g f l u id i n a t t h e f r o n t to i n c rea s e in

s ize and l eav ing a th in l iqu id l ayer beh ind . As shown in f igures 9 (d , e ) there i s a d ip in the f i lm

t h i ck n es s b eh i n d t h e s l u g . A s s h o w n i n f ig u r e 1 0, th e s l u g v e l o c i t y a t l a r g e g a s v e l o c i t y , m e as u r ed

as t h e t i m e f o r t h e s l u g t o m o v e t h e d i s t an ce b e t w een t h e t w o p a r a l l e l - w i r e co n d u c t an ce p r o b es ,

is ap p r o x i m a t e l y eq u a l t o t h e ac t u a l g a s v e l o c i t y j u s t b eh i n d t h e s l u g . A t l o w g as v e l o c it ie s , f o r

w h i ch t h e r e a r e l a r g e h y d r au l i c g r ad i en t s i n t h e p i p e , t h e s l u g v e l o c i t y ap p e a r s t o b e g r ea t e r

t h a n t h e g a s v e lo c i ty a t t h e d o w n s t r e a m l o c a t i o n a t w h i c h w a v e v e l o c i ty m e a s u r e m e n t s w e r e

m a d e .


9/16

LIQUID VISCOSITY ON T H STRATIFIED SLUG TRANSITION 885

For conditions under which at most one slug existed in the pipe, the exit of a slug from the pipe

resulted in a pressure release, even at low gas velocities. The gas acceleration that resulted from

this pressure release instantaneously triggered the formation of regular sinusoidal surface distur-

bances. If the film thicknesses was high enough, a s econd and, sometimes, a third slug were formed.

These "satellites" moved at lower velocities than the original slug because o f the lower liquid levels

in front and behind the "satellite".

For UsG > 4.5 m/s K H waves are present on the liquid layer prio r to the init iation of slug flow.

Lin & Hanr att y (1987a) poin ted o ut that, for these values of Us~, slugs are formed by the

coalescence of the large-ampli tude waves. This process is very clearly seen in experiments with high

viscosity liquids. At low liquid velocities the K H waves are observed to coalesce and accelerate and

then break up and decelerate. However, at sufficiently large liquid flows (or sufficiently high liquid

viscosities) these waves come together in a group and form a disturbance which grows to a

sufficiently large height to cover the pipe cross section. Slugs formed in this way have a short length

and are highly aerated. The film thickness in front and behind them are seen to be approximately

he same, so they do not increase in size as they move down the pipe.

-.

.... - . . . . . . ,

Figures 9(a-c). (a) Regular waves observed 180 pipe diameters from the entry of the 9.53 cm pipe for

ULS = 0.06 m/s, UGs = 4.3 m/s and /~L = 20 cP. (b) Front of the slug that was formed 100 pipe diameters

from the entry of the 9.53 cm pipe, observed at 180 pipe diameters for Uts '-0.08 m/s, UGs = 4.3 m/s and

gL = 20 cP. (c) Tail of the same slug as in (b).


10/16

g8 6 t~. ANDRIT SOS e t a l

,~

: . . : ,

. : . , : q . - . ) .t :- '. ;: ~ =. O .: . ' ~ . - - - . . ' , ~ ~ . i ~ , . : , : . . " " : ' . . . ? : ' . , ~ ' : , ~ .

. . . . . . . . . . . ~ . . j ~ . ~_ ~ . ~ . . . T ~ . ~ . . ~ . q ~ . . ~ . . z ~ . ~ : -- . .. . . . .

.... : .~.. ~.~ ~.~.;;:~ V.~M

F i g u r e s 9 ( d - f ). ( d ) S l u g t h a t h a s j u s t b e e n f o r m e d 3 0 p i p e d i a m e t e r s f r o m t h e e n t r y o f t h e 9 .5 3 c m p i p e

for L ' t .s = 0 . 0 3 3 m ,.'s , L 'o s = 4 m : s a n d ,u t. = 1 [.1 0 c P . ( e ) T a i l o f t h e s a m e s l u g a s i n ( d ) . ( f ) L a r g e - a m p l i t u d e

w a v e o b s e r v e d 3 0 p i p e d i a m e t e r s f r o m t h e e n t r y o f t h e 9 .5 3 c m p i p e f o r U ~ s = 0 . 0 3 3 m ' s U ~is = 7 m / s a n d

/a = 100 c P .

DISCUSSION

a) Comparison with previous analyses

The non-linear analyses that have been presented in the literature predict that a plot of

hL/D

at the stratified-slug flow transition vs Us~, for a given pipe diameter, should be independent of

liquid viscosity. Figures 7 and 8 clearly are contradic tory to this.

For example, the Taitel Dukler (1976) analysis predicts the appearance of intermit tent flow

to the right of the dashed curve and above the horizontal dashed line. It predicts the appearance

of annular flow beneath the horizontal dashed line. It is noted that the results for water at low

Use, are in good agreement with the analysis. However, the analysis underpredicts the

hL/D

required

for the initiation of slugs in high viscosity liquids.

The results in figures 7 and 8 would seem to support the large-wavelength viscous analysis of

Lin Hanrat ty (1986). See, for example, the calculations for a rectangular channel in figure I.


11/16

LIQUID VISCOSITY ON THE STRATIFIED--SLUG TRANSITION 887

E

t . )

2 0 i i i

/

I 0 - - D = 9 , 5 3 c m /

J

5 - ,

~:

& 7 e

&

(CP)

2 - - I

k / 10(3

i I I I

2 5 I 0 2 0

UG ( m / s )

Figure 10. Slug ve loc i ty as a funct ion of the gas

veloci ty behind the s lug.

0 . 9

0 . 7

(3

~ o , 5

0 . 3

i i i , i

- - D,9.53 cm

O& ' ~ - . .

, , o D ,2 ,5 2 c m

Kelv in- Helmholtz

In l tabi l ty ~ ~

. K ~ l , i , : . , ~ . ~ h , . L n J ~ o , ' , ' ~

I n l . j o o i | i l y

I o l " ' 1

0.5 I.O 2 .0 5 ,0 I0 20 50

US ( m / s )

F i g u r e 1 1. C o m p a r i s o n o f K H s t a b il i ty c a l c u l a ti o n s w i t h

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

a 7 0 c P D = 2 . 5 2 e r a ) a n d a 1 0 0 c P D = 9 5 3 c m ) l i q u i d .

0.1

0.2

H o w ev e r , a c l o s e r ex am i n a t i o n r ev ea l s d i s c r ep an c i e s . T h i s i s p a r t i cu l a r l y t r u e f o r ex p e r i m en t s i n

the 9 .53 cm p ipe w i th USG c lose to 3 m /s an d e xpe r ime nts w i th h igh l iqu id v i scos i ti es .

(b) Expe riments w ith high liquid viscosities

F i g u r e s 7 a n d 8 s h o w t h a t c u r v e s r e p r e s e n t i n g t h e t r a n s i t i o n t o K H w a v e s a r e c o n t i n u o u s w i t h

t h e cu r v es r ep r e s en t i n g t h e t r an s i t i o n t o a s l u g f l o w . T h i s r e s u l t w o u l d s eem t o s u g g es t t h a t t h e

s am e i n s t ab i l i t y m ech an i s m i s r e s p o n s i b l e f o r b o t h t r an s i t i o n s .

F i g u r e 11 is a p l o t o f t h e h i g h v i s co s i t y r u n s s h o w n i n f ig u r e s 7 an d 8 . T h e s t r i k i n g f ea t u r e o f

t h i s p l o t i s t h a t t h e d a t a f o r t h e t w o p i p e d i am e t e r s f a l l o n t h e s am e cu r v e . T h i s i s i n co n t r ad i c t i o n

t o t h e a n a l y s is o f L i n & H a n r a t t y a n d T a i t e l & D u k l e r , w h i c h w o u l d p r e d ic t

hL/D

t o b e a f u n c t i o n

o f t h e d i m e n s i o n l e s s g r o u p [(UsG/(gD) 2] [(PG/PL)I'Z]

T h e v i s u a l o b s e r v a t i o n s p r e s en t ed i n t h e p r ev i o u s s ec t i o n s h o w n t h a t a p r ecu r s o r t o t h e

ap p ea r an ce o f s l u g s o r o f K H w av es o n h i g h v i s co s i t y l i q u i d s i s an i n s t ab i l i t y w h i ch l e ad s t o

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

b y L i n & H an r a t t y t h a t s l u g s a t UsG < 4 m / s a r e t h e r e s u l t o f t h e g r o w t h o f an i n f i n i te s i m a l

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

T h e r e f o r e , s t ab i l i t y c a l cu l a t i o n s , w h i ch a r e n o t r e s t r i c t ed t o t h e a s s u m p t i o n o f la r g e - w av e l en g t h

w av es , w e r e ca r r i ed o u t . N eu t r a l s t ab i l i t y cu r v es o b t a i n ed b y u s i n g a co m p l e t e l y i n v i s c i d an a l y s i s

( [1] w i th ho ap pro xim ate d as in [10] an d [ I 1 ]) a re g iven in f igures 12-14 . Jus t i f i ca t io ns for us in g

an i n v i s ci d an a l y s i s a r e t h a t A n d r i t s o s & H an r a t t y ( 1 9 87 a ) h av e s h o w n t h a t t h i s a s s u m p t i o n g i v e s

r e s u l t s i n ap p r o x i m a t e ag r eem en t w i t h t h e o b s e r v ed c r i t i c a l g a s v e l o c i t y f o r t h e i n i t i a t i o n o f

l a r g e - am p l i t u d e w av es o n r e l a ti v e l y t h i n l iq u i d l ay e rs an d t h a t L i n & H an r a t t y ( 19 86 ) h av e f o u n d

t h a t t h e s t ab i l i ty o f s t r a t if i ed f lo w s t o l o n g - w a v e l en g t h d i s t u r b an ce s i s i n d ep en d en t o f v i s co s it y f o r

h igh v i scos i t i es .

A l l o f t h e s e n eu t r a l s t ab i l i t y cu r v es s h o w a c r i t i c a l U s w h i ch i s ap p r o x i m a t e l y i n d ep en d en t o f

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

. 5 - 3 . 0 cm . I t is a l s o n o t ed t h a t t h e v a l u e o f U s n eed ed f o r th e g en e r a t i o n o f la r g e - w av e l en g t h

w av es ( g i v en b y [ 4 ]) i s l a r g e r t h an t h e c r it i c al U sG an d is s t r o n g l y d ep e n d e n t o n p i p e d i am e t e r . F o r

hL/D = 0 .7 , t h e d i f f e ren ce b e t w een t h e c r it i ca l U so an d t h e s t ab i l i ty co n d i t i o n f o r l a r g e w av e l en g t h s

i s q u i t e s m a l l . H o w e v e r , a s hL/D d ec r ea s e s t h e d i f f e r en ce b eco m es s i g n i f ic an t an d can b e q u i t e l a r g e

f o r l a r g e d i am e t e r p i p e s .

Th e gas v e loc i t i es a t w hich s lugs were f i rs t obse rved in the 2 .52 an d 9 .53 cm p ipes for h ig h

v i s co s it y l i q u i d s a r e i n d i ca t ed a t t h e b o t t o m o f g r ap h s . I t i s n o t e d t h a t t h e i n i t i a t i o n o f s l u g s

co r r e s p o n d s t o t h e c r it i c a l U s an d n o t t o t h e v a l u e p r ed i c t ed b y [ 4 ] . F o r t h i n l i q u i d l ay e r s

(h/D = 0 .3 ) , it i s s een t h a t t h e c r i ti c a l U so co r r e s p o n d s t o t h e i n i t i a t i o n o f K H w av es . T h e

a g r e e m e n t o f t h is s t a b il i ty a n a l y s i s w i t h t h e o b s e r v e d a p p e a r a n c e o f s lu g s o r K H w a v e s i s a g a i n

s h o w n i n f i g u r e I I w h e r e t h e cu r v e r ep r e s en t s t h e c a l cu l a t ed c r i t ic a l U sG . T h e d a s h ed cu r v es i n d i ca t e


12/16

N A N D R I T S O S t

a l

l O 0 . O

J

0 . 7 0 h h 3 0

I 0 , 0

I , O - -

K - H

S l u o s S l u o s

Waves

o . t l t t

0 . 0 2 . 0 4 0 6 . 0 8 . 0 I 0 0

U S G m / s )

Figure 12. Neutral stability curves for the flow of air and a very viscous iquid (PL = 1.22 g;cm3,/aL= :c.)

through a 2.52 cm pipe.

c a l c u l a t i o n s b a s e d o n [ 4 ] . T h e s e i n d i c a t e t h a t l o n g - w a v e l e n g t h s t a b i l i t y t h e o r y d i s a g r e e s w i t h

e x p e r i m e n t s a n d p r e d i c ts a s t r o n g e f f ec t o f p i p e d i a m e t e r .

(c) Experiments with low l iquid v iscosi t ies

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

h i g h v i s c o s i ty l iq u i d s . A s a l r e a d y s h o w n b y L i n H a n r a t t y , th e U sG r e q u i r e d f o r t h e i n i t i a ti o n o f

s lu g s is d e p e n d e n t o n p i p e d i a m e t e r . W a t e r t r a n s i t i o n r e s u l ts a t U SG < 3 m / s f o r b o t h t h e 9 .5 3 a n d

2 .5 2 c m p i p e s c o m e t o g e t h e r o n a s i ng l e c u r v e o f

h L / D

v s

[UsG/(gD)~2][pG/p,)~ -].

T h i s d i f f e r e n t e f f ec t o f p i p e d i a m e t e r f o r l o w a n d h i g h v i s c o s it y li q u id s c a n b e u n d e r s t o o d b y

c o n s i d e r i n g t h e f i g u r e 1 2 . N e u t r a l s t a b i l i t y d i a g r a m s f o r l o w v i s c o s i t y l i q u i d s , t h a t c o n s i d e r o n l y

t h e e ff e c t o f p r e s s u r e v a r i a t i o n s i n p h a s e w i t h t h e w a v e h e i g h t h a v e a f o r m , f o r

h t D >1

0 .4 , s i mi l a r

t o t h a t s h o w n i n f i g u r e 1 2 f o r h i g h v i s c o s i t y l i q u i d s f o r

hL/D >1

0 . 7 . T h e l o n g - w a v e l e n g t h v i s c o u s

s o l u t i o n , w h i c h i s d e p e n d e n t o n p i p e d i a m e t e r , w o u l d i n f l u e n c e t h e c a l c u l a t e d c r i t i c a l U ~

1 0 0 . 0

I I I I

I 0 . 0 - -

h h h

1 .0 - -

K H

Stuos Slugs Waves

o .

t l t t t l

0 . 0 2 , 0 4 , 0 6 , 0 8 , 0 I 0 . 0

U s G m / s )

F i g u r e 1 3 . N e u t r a l s t a bi l i ty c u r v e s f o r t h e f l o w o f ai r a n d a v e r y v i s c o u s l i q u i d P L = 1 . 2 2 g / c m ~, ~ L

= O O )

t h r o u g h a 9 . 5 3 c m p i p e .


13/16

LIQUID VISCOSITY ON THE STRATIFIED SLUG TRANSITION 889

I O O . 0

I 0 . 0 -- ~ ~.I

= i . = I - - I .

1.0 - -

0 . = 1 I I 1

a O 2 . O 4 . 0 6 , O 8 . 0 I 0 . 0

U S G m / s )

F i g u r e 1 4 . N e u t r a l s t a b i l i t y c u r v e s f o r t h e f l o w o f a i r a n d a v e r y v is c o u s l i q u i d P L = 1 .2 2 g / c m 3 , ,uL = oc )

t h r o u g h a 5 0 .8 c m p i p e .

f o r a i r - w a t e r f l o w s a t U s G

1

0.5 [10]

ri g = (0 .5D - HL ) + riG[hL O=0.5

h L / D

< 0.5 [I 1]

I n a d d i t i o n , t h e ~ /' t e r m d e f in e d b y L i n H a n r a t t y t o t a k e a c c o u n t o f t h e i n f lu e n c e o f l iq u i d

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

r e s u l t s o f A n d r i t s o s H a n r a t t y ( 1 9 8 7 b ) . T h e li q u i d f l o w s f o r I a n d 4 . 5 c P l iq u i d s i n t h e 2 . 5 2 c m

p i p e a n d f o r 1 a n d 2 0 c P l i q u id s i n t h e 9 . 53 c m p i p e w e r e c o n s i d e r e d t u r b u l e n t . A v a l u e o f t h e

f r i c t i o n f a c t o r r a t i o , f / f s e q u a l t o t w o w a s s e l ec t ed a s a r e a s o n a b l e a v e r a g e r e p r e s e n t a t i o n o f t h e

i n t e r f a c i a l s t r e s s f o r w a t e r .

C a l c u l a t e d t r a n s i t i o n c u r v e s f o r d i f f e r e n t l i q u i d v i s c o s it ie s a r e p l o t t e d a s t h e s o l i d c u r v e s in f i g u r e s

7 a n d 8 . A s h a s a l r e a d y b e e n i n d i c a t e d i n f ig u r e I I , th e 7 0 c P c u r v e f o r D = 2 . 5 2 c m a n d t h e 1 00 c P

c u r v e f o r D = 9 . 5 3 c m a g r e e . T h e c a l c u l a t e d c u r v e s f o r l o w v i sc o s i ti e s s h o w t h e s t r o n g e f f e c t o f p i p e

d i a m e t e r a l r e a d y d i s c u s s e d . T h i s i s i l lu s t r a t e d i n t h e c a l c u l a t i o n s f o r a l i q u i d v i s c o s i ty o f I c P s h o w n

i n f i g u r e 1 5 . A p p r o x i m a t e a g r e e m e n t b e t w e e n t h e c a l c u l a t i o n s a n d e x p e r i m e n t a l r e s u l t s i s s h o w n

i n th e f i g u r e s , e x c e p t t h a t t h e e x p e r i m e n t s s h o w n o d i f f e r e n c e b e t w e e n t h e 1 6 a n d 7 0 c P l i q u i d s in

t h e 2 . 5 2 c m p i p e a n d b e t w e e n t h e 2 0 a n d 1 0 0 c P l i q u i d s i n t h e 9 . 5 3 c m p i p e .


14/16

890 N. A N D R I T S O S t a l

0 . 9 , , , I ,

0 . 5

0 . 3

o i - I I 1

I ~ I

0.2 0 .5 1 .0 2 .0 5 .0 I0 20 50

U s o m / s )

F i g u r e 1 5. C a l c u l a t e d t r a n s i t i o n s t o s l u g f l o w o r K H w a v e s f o r a I c P l i q u id .

T h e ca l cu l a t ed r e s u l t s f o r a f u l l y - d ev e l o p ed fl o w a r e p r e s en t ed i n a M an d h an e p l o t i n fi g u re 1 6 .

T h e r e i s q u a l i t a t i v e ag r eem en t w i t h t h e r e s u l t s p r e s en t ed i n fi g u r e s 2 - 4 an d i n fi g u re s 1 2 - 1 4 o f th e

p ap e r b y A n d r i t s o s H a n r a t t y (1 9 8 7 a ). A n i n c r ea s e o f v i s co s it y f r o m I t o 4 .5 cP f o r D = 2 .5 2 cm

an d f r o m 1 t o 2 0 cP f o r D = 9 .5 3 cm h as a s m a l l e ff ec t b ecau s e o f th e co u n t e r b a l a n c i n g o f t h e

d es t ab i l i z a t i o n a s s o c i a t ed w i t h t h e i n c r ea s e i n h L / D an d t h e v i s co s it y s t ab i l i z a t i o n s h o w n i n f i g u re s

7 an d 8 . F u r t h e r i n c r ea se s i n v i s co s it y a r e s h o w n t o c au s e a s h a r p d ec r ea s e i n t h e USL r eq u i r ed f o r

t r a n s i ti o n t o a n i n t e r m i t t e n t f lo w s u c h a s s h o w n i n fi g u re 4. T h e d e t a i le d q u a n t i t a t i v e a g r e e m e n t

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

t h e f l o w is n o t w e l l -d ev e l o p ed a n d b ecau s e t h e m o d e l u s ed t o c a l cu l a t ed USL f o r a f i x ed h L / D an d

UsG is not exact .

C O M P A R I S O N W I T H O T H E R M E A S U R E M E N T S

W u et al (1987) r ecen t ly s tud ied g as - -cond ensa te f low in a hor iz on ta l 20 .3 cm p ipe a t 75 b

p r e s s u r e . T h e y r ep o r t ed t r an s i t i o n s t o s l u g f l o w a t s u p e r f i c i a l l i q u i d v e lo c i ti e s i n t h e r an g e o f I m / s

f o r su p e r f ic i a l g a s v e l o c it i es o f 0 . 7 -8 m / s . C o m p a r i s o n s s h o u l d b e m ad e a t t h e s am e v a l u e o f

1 ~

P G - U SG . B ecau s e o f t h e h i g h d e n s i t y o f t h e g a s , t h e i r t e s t s s h o u l d c o r r e s p o n d t o r e s u lt s i n t h i s p ap e r

1 , 0 I

I [

E

v

o . i

0 , 0 1

0 , 0 0 1

D

= 2 . 5 2 c m

/ . ~ L C P )

O = 9 . 5 3 c m

~L C P)

o.0 I ,o Io .o 0 .o I .O Io ,o

UsG m/s) USG m/s )

F i g u r e 1 6. C a l c u l a t e d t r a n s i t i o n s to s l u g f l ow o r K H w a v e s p r e s e n t e d i n a M a n d h a n e p l o t .


15/16

L I Q U I D V I S C O S I T Y O N T H E S T R A T IF I E D S L U G T R A N S I T I O N 891

1 0

,~ 0.1

: : : ) 0 .01

0.001

l i |

0 0 0 0 0 0 0 0

O 0 0 0

0 A

O 6

A

I

0 , 1 1 , 0 I 0 IO0

O s moot h s t r t i f i e d o e l o n g a t e d b u b b le w a v y a n n u l a r

w v y s t r t i f i e d e e l u g ~ a n n u l a r

Figure 17. Flow patterns for a 90 cP glycerine/waterand air m ixture in a horizon tal 3.8 era p ipe (Taitel

Dukler 1987).

i n t h e r eg i o n o f U s c > 5 m / s . I t is q u i t e p r o b ab l e t h a t t h ey w e r e o b s e r v i n g f l o w s f o r w h i ch t h e

m e ch a n i s m f o r t h e g en e r a t i o n o f sl u g s is th e co a l e s cen ce o f K H w av es . I f t h i s is t h e c a se , t h en l i n ea r

s t ab i l i ty p r ed i c t s t h e i n i t i a t i o n o f l a r g e - am p l i t u d e w av es , b u t n o t s l u g s .

F i g u r e 1 7 g i v e s a co m p a r i s o n o f t h e ca l cu l a t i o n s o u t l i n ed i n t h e p r ev i o u s s ec t i o n w i t h th e

m e as u r e m e n t s o f T a i t e l D u k l e r ( 19 87 ), d o n e i n a 3 .8 cm p i p e w i t h a l iq u i d v is co s i ty o f 9 0 cP .

G o o d a g r e e m e n t is n o t e d b o t h w i t h t h e o b se r v e d t r a n s i ti o n s t o K H w a v e s a n d t o i n t e r m i t t e n t f lo w .

C O N C L U D I N G R E M A R K S

A c o m p ar i s o n o f ex p e r i m e n t a l r e s u l t s o n t h e s t ab i l i t y o f a s t r a ti f i ed f l o w i s b e s t d o n e o n a p l o t

OfhL D

v s U s(:; r a t h e r t h a n w i t h t h e M an d h an e co o r d i n a t e s , USL VS U s ~ . T h i s i s so , n o t o n l y b ecau s e

an e v a l u a t i o n o f t h eo r i e s is m o r e ea s i l y d o n e , b u t a l s o b ecau s e e f fec t s a s s o c i a t ed w i t h a

n o n - d e v e l o p e d f l o w m a y b e l e s s i m p o r t a n t .

E x p e r i m en t a l r e s u l ts o n t h e e f f ect o f l i q u i d v i s co si t y o n t h e i n i t i a t i o n o f sl u g s s h o w a s t ab i l i z a t i o n

w i t h in c r ea s i n g l i q u i d v i s co s i ty i n t h i s t y p e o f p l o t , a s p r ed i c t ed b y L i n H a n r a t t y . T h i s w o u l d

s eem t o co n t r ad i c t i n t e r p r e t a t i o n s w h i ch u s e a l o n g - w av e l en g t h i n v i s c i d an a l y s i s .

L i n H a n r a t t y a r g u e d t h a t a t s m a l l USG t h e in i t i a ti o n o f sl ug s o c c u r s t h r o u g h t h e g r o w t h o f

l o n g - w av e l en g t h i n f in i t e s i m a l d i s t u r b an ces . T h e y s h o w ed t h a t l i q u i d i n e r t i a is d e s t ab i l i z in g i f

v i s co u s e f fec t s a r e i n c l u d ed an d t h a t t h i s e ff ec t d ec r ea s e s f o r f i x ed v a l u e s o f h E D a n d ( , ~ . F o r v e ry

l a r g e l i q u i d v i s co si t ie s w h e r e i n e r t i a is n e i t h e r s t ab l il i z in g n o r d e s t ab i l iz i n g t h e L i n H a n r a t t y

an a l y s i s g i v e s t h e s am e r e s u l t s a s a l o n g - w av e l en g t h K H an a l y s i s .

A t f i rs t g l an ce t h e o b s e r v ed e f f ec t o f l i q u i d v i sco s i t y an d t h e g o o d ag r eem en t o f ex p e r i m en t a l

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

H an r a t t y . H o w e v e r , a c l o s e r ex a m i n a t i o n o f t h e r es u l ts w i t h v e r y v i s co u s l i q u i d s s h o w s a d i f f e r en t

e f f ec t o f p i p e d i am e t e r t h an i s p r ed i c t ed .

V i s u a l o b s e r v a t i o n s o f t h e s t ab i l i t y o f a v e r y v i sco u s l iq u i d s h o w t h a t t h e f i rs t w av es t h a t a p p e a r

a r e s i n u s o i d a l an d h av e a s u f f i c i en tl y s h o r t w av e l en g t h t h a t t h e y a r e a f fec t ed b o t h b y g r av i t y an d

s u r f ace t en s i o n . W i t h a s l i g h t ch an g e i n t h e f lo w co n d i t i o n s l a r g e - am p l i t u d e i s o l a t ed d i s t u r b an ces

g r o w o u t o f th e s e r eg u l a r s m a l l - w av e l en g t h w av es . I f t h e l i q u i d l ay e r i s t h i ck en o u g h t h e

d i s t u r b an ces t o u ch t h e t o p w a l l an d f o r m s lu g s . O n t h e t h i n l ay e r s t h a t o ccu r f o r USG > 4 m / s t h e s e

d i s t u r b an ces ev o l v e i n t o l a r g e - am p l i t u d e i r r eg u l a r w av es t h a t c au s e a l a r g e i n c r ea s e i n t h e

i n t e r f ac i a l st re s s an d a l a r g e d ec r ea s e i n t h e t h i ck n es s o f t h e l i q u i d . A t s m a l l l i q u i d h e i g h t s ( o r l i q u i d

f l o w s ) t h e r e i s n o t en d en cy f o r t h e s e w av es t o co a l e s ce . H o w ev e r , a t s o m e c r i t i c a l co n d i t i o n t h ey

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

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

n o n - l i n e a r p r o c e ss e s w h i ch a r e n o t y e t u n d e r s t o o d . H o w e v e r , i f t h e a p p e a r a n c e o f t h e s m a l l- w a v e -

l en g t h s i n u s o i d a l w av e t r a i n i s a n ece s s a r y co n d i t i o n t h e n l i n ea r t h eo r y s ti ll c an b e u s ed t o p r ed i c t

s t ab i li t y . S ta b i l it y c a lc u l a t i o n s t h a t a b a n d o n t h e l a r g e- w a v e l e n g th a p p r o x i m a t i o n s u p p o r t t h i s

s u g g es t i o n i n t h a t t h e c r i ti c a l USG f o r t h e i n i t i a t i o n o f t h e s m a l l - w av e l en g t h w av es d u e t o g a s - p h as e

p r e s s u r e v a r i a t i o n s i n p h as e w i t h t h e w a v e h e i g h t i s t h e s am e a s t h e o b s e r v ed USG f o r t h e i n i t i a t i o n

o f s l u g s o r o f l a r g e am p l i t u d e i r r eg u l a r w av es .

MF. i ,'t~--C


16/16

89 N ANDRITSOS

e l a l

Observations of the mechanism for the initiation of slugs for liquids with viscosities close to that

of water is complicated because over a large range of flow conditions the interface of the stratified

flow is covered with waves (generated by gas-phase pressure variations in phase with the wave

slope) prior to the appearance of slugs. However, at very low gas velocities, where the interface

is smooth, Lin (1985) has reported the same phenomena that were observed on very viscous liquids.

This result, plus the agreement of experiments with the approximate calculations presented in this

paper, would tentatively suggest that the type of mechanism observed for very viscous liquids could

be operative for all viscosities.

Calculations with a more accurate representation of viscous effects in the liquid, than is used

in the section on approximate calculations, are needed to clarify this question.

Acknowledgements--This

work has been supported by the Shell Companies Foundation and by the

Department of Energy under DOE DEFG02-86E R 13556.

REFERENCES

ANDREUSSI

P.

PERSEN

L. N. 1987 Stratified gas-liquid flow in downwardly inclined pipes. Private

communication.

ANt)RI'rSOS, N. 1986 Effect of pipe diameter and liquid viscosity on horizontal stratified flow. Ph.D.

Thesis, Univ. of Illinois, Urbana.

ANDRITSOS, N. HANRATTY, T. J. 1987a Interracial instabilities for horizontal gas-liquid flows in

pipelines.

In t . J . Mul t iphase Flow

13, 583--603.

ANDRn'Sos, N. HANRATTY, T. J. 1987b Influence of interfacial waves on hold-up and frictional

pressure drop in stratified gas-liquid flows. A I C h E J l 33, ~4 A, 454.

CHOW, V. T. 1959 Open-channel Hydraul i cs . McGraw-Hill, New York.

FRANCIS, J. R. D. 1954 Wave motions and the aerodynamic drag on a free oil surface. Phi l . Mag.

45, 695-702.

FRANCIS, J. R. D. 1956 Wave mot ions on a free oil surface. Phi l Mag. l, 685~588.

HANRATTY, T. J. 1987 Gas-liquid flow in pipelines.

PhysicoChera. Hydrodynam.

9, 101-114.

KORDYBAN, E. S. 1977 Some characteristics o f high waves in closed channels approaching

Kelvin-Helmholtz instability. A S M E J I F lu id s E n g n g 9 9 339-346.

KORDVBAN, E. S. RANOV, T. 1970 Mechanism of slug formation in horizontal two-phase flow.

J. basic Engng 92, 857-864.

LIN, P. Y. 1985 Flow regime transitions in horizontal gas-liquid flow. Ph.D. Thesis, Univ. of

Illinois, Urbana.

LIN, P. Y. HANRATTY, T. J. 1986 Prediction of the initiation of slugs with linear stability theory.

In t . J . Mul t iphase Flow 12, 79-98.

LIN

P. Y. HANRATTY, Z. J. 1987a Effect of pipe diameter on the interracial configurations for

air-water flow in horizontal pipes.

Int . J . Mul t iphase Flow

13, 549-563.

LIN

P. Y.

HANRATTY

T. J. 1987b Detection of slug flow from pressure measurements.

Int . J.

Mul t iphase Flow 13, 13-21.

MILES, J. W. 1959 On the generation of surface waves by shear flows. Part 3, Kelvin-Helmholtz

instability.

J. Fluid Mech.

6, 583-598.

MlSrlIMA. K. ISHII, M. 1980 Theoretical prediction of onset of horizontal slug flow. T ra ns . A S M E

Jl Fluids Engng 102, 441-445.

TAITEL, Y. DUKLER, A. E. 1976 A model for predicting-flow regime transitions in horizontal and

near horizontal gas-liquid flow. A I C h E J l 22, 47-55.

TAITEL, Y. DUKLER, A. E. 1987 Effect of pipe length on the transition boundaries for

high-viscosity liquids. Int . J . Mul t iphase Flow 13, 577-581.

WALLIS, G. B. DOaSON, J. E. 1973 The onset of slugging in horizontal stratified air-water flow.

Int . J . Mul t iphase Flow

1, 173-193.

Wu, H. L., POTS, B. F. M., HOLLENBERG,J. F.

MEERHOF

R. 1987 Flow pattern transitions in

two-phase gas/condensation flow at high pressures in an 8-inch horizontal pipe. Presented at the

3rd Int . Conf . on Mu l t iphase Flow The Hague, The Netherlands.

effect of liqud viscosity

Documents