Prediction of Liquid Holdup in Horizontal Stratified Two ...

Thammasat Int. J. Sc. Tech., Vol.3, No.2, July 1998

Prediction of Liquid Holdup inHorizontal Stratified Two-Phase Flow

S. Wongwises, W. Khankaewr'W. VetchsupakhunDepartment of Mechanical Engineering,

King Mongkut's University of Technology Thonburi

Bangmod, Bangkok 10140, Thailand

Abstract

This paper provides a combined theoretical and experimental investigation into the prediction

ofhold-up ioi a stratified two-phase concurrent flow in a horizontal circular pipe. The test section,

l0 m long, with an inside diameter 54 mm was made of transparent acrylic glass to permit visual

observation of the flow patterns. The experiments were carried out under various air and water flow

rates in the regime of smooth and wavy stratified flows. Stainless ring electrodes were mounted

flush in the tube wall for measuring the liquid hold-up which is defined as the ratio of the cross-

sectional area filled with liquid to the total crossectional area of the pipe. Calculation method for

predicting the liquid hold-up was developed by using the Taitel and Dukler momentum balance.The

iatio of interfaciat friction factor and superfrcial gas-wall friction factor,(f1lfs6) was assumed to be

constant. Hold-up curves calculated by this method are compared with present experimental data

and those of other researchers. A ratio of f;/f56 ,which corresponds with the flow conditions,

(laminar or turbulent) are presented.

Key Words: Two-Phase Flow, Co-Current Ftow, Stratified Flow, Liquid Hold-Up

1. Introduction

Stratified two-phase flow regime is

frequently encountered in various chemical and

industrial proaesses; e.g. the flows of steam and

water, or oil and natural gas in pipelines etc.

One of the main problems in two-phase flow is

the calculation to determine the liquid hold-upand pressure loss. Lockhart and Martinelli [l]have developed a procedure for calculating thefrictional pressure loss for adiabatic two-phaseflow using their data on the horizontal flow of

air and water and various other liquids at

atmospheric pressure. Their correlations have

been applied to all regions of two-phase flow

both by the originators and by several otherinvestigators. Chishotm [2] has developed the

Martinelli models in such a way that the

original Martinelli curves for the various flow

regimes can be fitted quite well by selecting a

fixed value of a parameter for each flow regime'

Johannessen [3] has developed a theoretical

solution of the original Lockhart and Martinelli

flow model for calculating two-phase pressure

drop and holdup in the stratified and wavy flow

region. He has shown that his theoretical

solutions of pressure drop and holdup agree

much better than those of Lockhart and

Martinelli in the separated flow region.

48

Thammasat Int. J. Sc. Tech., Vol.3, No.2, July l99E

Mixing section

2 m

Tcd S.dlon

I'ilixing S.clirn

&D(

IT

Gat vatv.

Glob.ValveFilt r

W.bt Ftolv

-r_I

r lv Dh L l

AI

Figure l. Testfacility

T*c-

Ti

UG----->

U

Figure 2. Stratified co-current two-phase flow

49

The semi-empirical methods forcalculating the two-phase flow pressure drophave been proposed by numerous investigators.Wallis [4] correlation which has been improvedfurther by Hewitt and Hall-Taylor [5] can beused in the annular flow region. Hughmark [6]developed a semi-empirical pressure dropcorrelation independently which is applicable inslug flow region. Kadambi [7] proposed ananalytical procedure to determine the pressuredrop and void fraction in two-phase stratifiedflow between parallel plates.

Most stratified flow models were basedon an iterative solution of the two phasemomentum balance, but differed in the model ofthe interfacial shear stress. To solve thisproblem, Taitel and Dukler [8] made theassumption that the interface was smooth andinterfacial friction factor equal to the gas-wallfriction factor and the gas interfacial shearstress was evaluated with the same equation asthe gas wall shear stress.

In another paper (Taitel and Dukler [9]),they demonstrated that the hold up and thedimensionless pressure drop for stratified floware unique functions of X under the assumptionthat fiy'fl = constant. Kawaji [l0] predictedholdup successfully by substituting the ratio ofthe gas-wall friction factor and the gasinterfacial shear stress into the Taitel andDukler momentum balance.

Inaccuracies in previous shatified flowmodels are found to be a result of theinterfacial shear stress used in the model. In thepresent study, the method for prediction ofliquid hold-up will be presented. The method isbased on that of Spedding et al. [11,12] andWongwises tl3l where the ratio of theinterfacial friction factor and gas-wall frictionfactor is assumed to be a constant. With thistechnique a mathematical model of interfacialfriction factor is not nec€ssary. The value of theconstant depends on whether the phases are inturbulent or laminar flow.

2. Experimental Apparatus and Method

The experimental facility used is shown

schematically in Fig LThe main components of

the system consisted of the lest section, airsupply, water supply, instrumentation, and dataacquisition system.The horizontal test section,with an inside diameter of 54 mm and length ofl0 m was made of transparent acrylic glass topermit visual observation of the flowpatterns.Water was pumped from the storagetank through the rotameter to the water inletsection at the bottom of the pipe. Aif wassupplied to the test section by a suction-typeblower. The air flow could be controlled by avalve at the outlet of the blower. Many smallrods were used as guide vanes at the air inletsection to maintain a uniform flow. Both the airand water streams were brought together in amixer and t}ten passed through the test sectionconcurrently.The inlet flow rate of. air wasmeasured by means of a round-type orifice andthat of water was measured by two sets ofrotameters.

The temperature of the air and water wasmeasured by thermocouples. Stainless ringelectrodes were mounted flush in the tube wallfor measuring the liquid hold up. They operateon the principle of the variation of electricalresistance following changes in the water levelbetween two parallel electrode rings. The samedescription of the calibration procedures forstratified flow can be found in Andreussi [4].Due to the variation of conductivity caused bytemperature change and coating of theelectrodes with impurities, the gauges werecalibrated before and after each run.

Experiments were conducted with variousflow rates of air and water at ambient condition.In the experiments the air flow rate wasincreased by small increments while the waterflow rate was kept constant at a preselectedvalue. After each change in inlet air flow rate,both the air and water flow rates were recorded.The liquid hold-up was registered through thetransducers. The flow phenomena was detectedbv visual observation.

50

3. Mathematical Model

Consider an equilibrium horizontal stratifiedflow as shown in Fig. 2. A momentum balanceon each phase yields:

. ( a p \-n r l i ) - rwtSt+r ,S ,= 0 ( l )

' ( a P \-n " l ; ) - rwcSc - r iS i= 0 (2 )

Equating pressure drop in the two phases andassuming that the hydraulic gradient in theliquid is negligible, the following result isobtained:

- 0 ( 3 )

The shear stresses are evaluated in aconventional manner


(10)

DL=!41 (e)" sL

The gas is visualized as flowing in a closed ductand thus

,"=#

'*oZ-"*,+,.r,t,(t.+)

Normally for equilibrium flow u6 2u 1 suchthat u 1 in eq.(6) can be neglected. A widelyused method for the correlation of the liquid andgas friction factors is in the form of Blasiusequation:

Furthermore, the coefficients C1, n, C6 and mused in Eq. (7) and Eq. (8) are those used byTaitel and Dukler [8] in their co-current studies,

in turbulent flows; C6: Cr_ = 0.046,m : n = 0 . 2 0

in laminar flows; C6 = C1 = 16,m = n = 1 . 0 .

Turbulent or laminar flow conditions in eachphase are identified by calculating the Reynoldsnumber for each phase using the superfrcialvelocity and diameter of the pipe, i.e.

Resx =

where K: G, L

U t*D

Laminar flow is also assumed for superficialReynold number < 2000.Substituting rryq ry6.ri from Eq.(4), Eq.(5) andEq.(6) into Eq.(3),the following equation isobtained;

fcpc4so -,frpru?.s, *2Ao 2A,

f ,polr ts, l r* ! l =o (u)2 l_t, Ao l

In the case of the single phase flow, the pressuregradient is determined from;

- - , PLuZLL W L - J L - n -

- - , P c u L. w G - J G 2

- - t Pc ( ' o -u r ) '. i - J i -

2 -

(4)

UK(s)

(6)

, , =r , (o : : ' ) ' (7)

fo =co(4s!s-)-^\ u 6 )

(8)

where Dp and D6 are the hydraulic diameterevaluated in the manner as suggested byAgrawal et al.[5]. The liquid is visualized as ifit was flowing in an open channel .

( a p \\d r ) ro

5 l

(r2)

Thammasat Int. J. Sc. Tech., Vol.3, No,2, July 1998

where ,*=""(T)

Equation (l l) is non-dimensionalized

dividingrr f+l\ ax ) sc

Finally the following equation is obtained;

f ouZsoo _ f rp ru?s rD .47;A"E- +Jropot*;-

f , p o r Z s , ? [ t * t l = o ( r 3 )4fropoulolA, Ao )

or in dimensionless form

(ro)'(Dotol^ X

It )'{r,r,)-" Z)*' .

*r,"rl*-*] = o (,4)

ac1(us.D\-i or("rr) 'D \ ; ) 2

diameter.Liquid hold up can be calculated fromh/D which is in the form of Ac, Al.

All dimensionless variables with the superscriptcan be seen from

Z = n / 4 ,

7 r = A t / D 2 'xS r = S r l D ,

7 o = A c / D 2 'v _ f i 3 )S c = S c / D " '

S i t =S , /D ,

Dt ='DL I D'

D 6 = D o / D '7 ,h L = h L l D

Sr= r-cos-'(z6l- l),

Sc = coS-'(zhl-l),r *

S, = r / l - (zh l - l ) ' ,

0.=2.- ( ' io '

A, =Z' A L

where y2 = 1dn I dr)sL / @P / dx)sc is the

ratio of the frictional pressure gradient of theliquid to that of the gas when each phase flowsalong in the pipe. cos-t(2EL- tl ]*

)rr-6;fl

Ac = o.zslcos-, (zEL- t)]-

0.25|(2h, - r)Jl -(2h, - l)' IL - J

In order to solve Eq.(la) for liquid hold up, gashold up and pressure drop, an iterative computerprogram is required. A flow chart of thisprogram is shown in Fig 3.

7r=o.25ln-

o.zsl<zL -r

X 2 =D \ u r

T ( ls)ac6 ( us6D\-^ po(uro)'

D \ % ) zD \ u c

X is recognized as the parameter introduced byLockhart and Martinelli tll and can becalculated unambiguously with the knowledgeof the flow rate, fluid properties and tube

52


Figure 3. Flow chart for calculation of liquid hold-up and pressure drop

P. ' P,t. F... lr,, ' Q.' Qt '

L:G . T:T

f/-.=Const.l

53

4. Results and Discussion

To handle practical problems, it is necessary

to gain a better understanding of flow

characteristics. Visual observation shows that

different flow patterns may occur with gas-

liquid cocurrent flow in horizontal pipes. In

accordance. with results obtained from this

experiment, the following flow patterns were

obtained:

a) Stratified flow: The water flows in the

lower part of the pipe and the air over it with a

smooth interface between the two phases.

b) Two-dimensional wavy flow: Similar to

stratified flow except for a wavy interface, due

to a velocity difference between the two phases

and two-dimensional steady waves travel with a

relatively regular pitch.

c) Three-dimensional wavY flow: At a

higher air flow rate, the water surface is.

disturbed and three-dimensional waves occur'

which have small irregular ripples on the

fundamental waves.d) Violent wavy flow: The interface is

violently disturbed by the air stream. This flowpattern occurs at a relatively high air flow rate.

e) Plug flow: Air moves along the uppersideof the pipe. This flow pattern occurs at a

relatively low air flow rate. The interface is

smooth and no bubbles are contained in a water

plug.

f) Slug flow: Splashes or slugs of water

occasionally pass through the pipe with a higher

velocity than the bulk of the water. The tail of

water slug is relatively smooth and sometimes

contains some small bubbles. The upstream

portion of the water slug is similar to the wavy

flow, and the downstream portion to the

stratified flow or wavY flow.

g) Pseudo slug flow: The semi-slug is

defined as a highly agitated long wave which

contains many bubbles. Its upstream and

downstream portions are similar to the wavy

flow.


The typical photographs of flow Patterns are

shown in Figure 4. The focus of the study was

on the stratified and small wavy flow. Figures 5'and 6 show the relation between the liquid

holdup,es against the Lockhart-Martinelliparameter, X for a laminar liquid-turbulent gas

flow in the 0.054 m. diameter pipe and Q1 :

1.67 x l0-5, 6.67x10'5 .m3 ls respectively. The

values C6:Cy:0.046, n:m:0.2 for turbulent

flow and C6=C1:16,n:m:l'0 for laminar flow

are used. The figures show a comparison of the

experimental data with the present model

where the ratio, f1lfs6 is assumed. It is found

that an agreement of the present model with the

experimental data is obtained by using filfsa =

0.30-1.0. The data obtained by Spedding et al'

[11] who tested the model against wavy and

stratified flow data from 93.5 and 45.5 mm

diameter pipes are compared with the

predictions from the present model. Their data

points were taken from log scale, thus were a

cause of some uncertainties. Their data can be

accurately predicted with f1lfs6 = 0.6 for

laminar liquid-turbulent gas flow' Theirpredicted {/fs6 are in the recommended range

in this work. The scatter of Spedding et al. data

for the smaller diameter pipe is much greater

than the large diameter.

Figures 7 and 8 show also the relation

between sL against X for a turbulent liquid-

turbulent gas flow for Q; = 8.3x10-' and l.67xl

0-4.m3/s respectively. They show that the liquidholdup can be accurately predicted by assuming

4/fsc = 2.0-4.0. The data shows that the

assumption of filfs6 = 1.0 overpredicted liquid

holdup for the stratified flows.The results

correspond to those from Kawaji [10] who

predicted holdup successfully by substituting

filfsc : 3.0 and also from Spedding et.al.[l1] by

substituting f;/fsc = 4 for turbulent liquid-

turbulent gas flow into the Taitel and Dukler [8]momemtum balance. Their predicted f;/fs6 are

also in the recommended range in this work.

However, for Spedding et al. results, a

discrepancy is found between the present

recommended ratio of f;/fsc and the

experimental data at greater Lockhart Martinelli

Parameter. This is because of a change of

interfacial phenomena. The amplitude of the

water layer fluctuation increases slightly with

54


Test Section<-<-

Ai rWater

a. Stratified flowb. Two-dimensional wavy flowc. Three-dimensional wavy flowd. Violent waw flowe. Plug flowf. Slug Flowg. Pseudo slug flow

Figure 4. Photographs of flow Patterns

Qr- = 1.67x 10's m3/sLiquid : LaminarGas: Turbulent

Present Data

fy'fsc = 0.3

fy'fs6 = 9.6

fy'fs6 = 1.9

Thammasat Int. J. Sc. Tech., Vol.3, No.2, July l99E

0.40

0.35

0.30

o,3 o.2s

G'

6* 0.20p5.g 0.15J

0 .10

0.05

0.000 .10

Lockhart Martinelli Parameter

Figure5. esagainstlog(X)forq1=1.67x10-5m37';Liquid-LaminarandGas-Turbulent

0.40

0.35

0.30

0.25

0.20

0 .15

0.'10

0.05

0.000.10


Figure 6. e1 against 16g (X) for Q= 6.67x10'5 m3A ; Liquid-L^aminar and Gas-Turbulent

1.000.01

o.5Io

tt:'6

J

Liquid: LaminarGas : TurbulentPresentData :D=54mmData of Spedding et.al. (1990) : D = 93.5 mmData of Spedding et.al. (1990) : D = 45'5 mm

fy'fsa = 0.3

fy'fs6 = 9.6

fy'fe6 = 1.9

56

o.ItE 0.4

t,'5 0.3

.gJ

0.2


1 . 0


Figure 7. €L against log (X) for Q1= 8.33x10-5 m3/s ; Liquid-Turbulent and Gas-Turbulent

o.!,o

E

ctJ

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Liquid : TurbulentGas :Turbulent

o Present Data : D = 54 mm

r Data of Spedding et.al. (1990)D = 93.5 mm

,4;/4-1:I / ". . ;t t

I

r Data of Spedding et.al. ('1990) / 7D = 45 .5 mm . / t ' t/ t : .

, / i : . ',/. ..r ' ̂

,/(r*'/ ' z : ' ^ '

{", ' .; i t4r: ' ; 'fy'fs6 = 1.9

fy'fsc = 2.0

fy'fe6 = 4.9

l

0 .1' " 1

1 . 0


' l

10 .0

Figure 8. eg against log (X) for Qr= l.67xl0a m3a ; Liquid- Turbulent and Gas-Turbulent

Qr- = 8.33x 10-s m3/sLiquid : TurbulentGas : Turbulent

Present Data

fy'fsc - 1.0

fy'fsc - 2.0

f,/fs6 = 4.9

57

air flow. Two-phase pressure drop can be

determined further by substituting hr,/D into

Eq. (1) or (2). In this work, the situation when

gas flow was laminar, was not considered.

5. Conclusion

This paper presents new data to predict the

liquid holdup in horizontal concurrent stratifiedflow in a circular pipe. It has been

demonstrated that the liquid holdup can bepredicted by using Taitel and Duklermomemtum balance between both phases. Theratio of the friction factor of the gas at theinterface and the gas at the pipe wall, fi /fsc isassumed to be constant. The constant dependson the phase being either turbulent or laminar.With this method a model of interfacial frictionfactor is not necessary. For turbulent liquid-turbulent gas flows, the former assumption thatfi = fsc is shown to give a result which doesnot agree with the experimental data.Futurework should examine the effect of pipediameter. It may be also worthwhile to study incountercurrent flow for comparison withconcurrent flow data.

Nomenclature

A Crossectional area of pipe, m2Ac, Al Crossectional area of gas and

liquid phase, m'Cc, Cl Constant in Eq.(7) and (8)

D Pipe diameter,mDc, Dr Hydraulic diameter of gas and

liquid phase, mfo fr, Gas-wall and liquid-wall

friction factorfi Interfacial friction factorfsc Superficial gas-wall friction factorg Gravitational acceleration, m/s'h Liquid height, mtr,D Constant in Eq.(7) and (8)

P Pressure, N/m'dP/dx Two phase pressure gradient, N/m'(dP/dx)sc Pressuregradlentofsingle

gas phase, N/m'(dP/dx)s1 Pressure gradientofsingle

liquid phase, N/m'

Qc Volume flow rate of gas,m'/s


R"st

Volume flow rate of liquid,m3/sGas phase Reynolds numberLiquid phase Reynolds numberSuperficial gas phase

Reynolds numberSuperficial liquid phase

Reynolds numberGas phase perimeter,mLiquid phase perimeter,mInterfacial width,mAverage velocity of gas, m/sAverage velocity of liquid, misSuperficial velocity of gas, m/sSuperficial velocity of liquid, m/sLockhart-Martinelli parameter

Greek Svmbols

QrRecRel

Resc

s^S1S1

U6

U1

UscUslX

put

t

GLiWLWGSGSL

Density, kg/m3Kinematic viscosity, m'lsShear stress, N/m'Liquid hold up

Subscripts

Gas phaseLiquid phaseInterfaceLiquid-wallGas-wallSuperficial gasSuperficial liquid

Superscripts

- dimensionless term

Acknowledgement

This work was given financial support bythe Thailand Research Fund (TM), whoseguidance and assistance are gratefully

acknowledged. The authors also wish to thank

students and staff of the Department of

Mechanical Engineering, King Mongkut's

University of Technology Thonburi for

tremendous assistance given during their work.

58

6. References

[] Lockhart,R.W.and Martinelli, R.C. (1949),Proposed Correlation of Data for IsothermalTwo Phase,Two Component Flow in Pipes,Chem. Engg. Prog., Vol.45,pp.39-48.

l2l Chisholm,D.(1978), Influence of PipeSurface Roughness on Friction PressureGradient During Two Phase Flow,Mech.Eng.Sci.,Vol. 20, pp.353 -354.

[3] Johannessen, T. (1972), A TheoreticalSolution of the Lockhart-Martinelli Flow Modelto Calculate Two Phase Flow Pressure Drop andHoldup, Int. J. Heat Mass Transfer,Vol. 15, pp.1443-r449.

[4] Wallis,G.B.,(1969),One-Dimensional Two-Phase Flow,McGraw-Hill Book Co..New York.

[5] Hewitt, G.F. and Hall-Taylor,N.W. (1970),Annular Two-Phase Flow, Pergamon Press.

[6] Hughmark, G.A. (1965), Holdup and HeatTransfer in Horizontal Slug Gas-LiquidFlow,Chem. Eng.Sci., Vol.20, pp. 1007-1010.

[7] Kadambi, V. (1980), Prediction of VoidFraction and Pressure Drop in Two-PhaseAnnular Flow, GE Rep. No. 8OCRDI56.

[8] Taitel,Y., and Dukler,A.E.,1976,4 Modelfor Predicting Flow Regime Transitions inHorizontal and Near Horizontal Gas-LiquidFlow, AIChE., Y o1.22, pp.47 -55.

[9]Taitel,Y., and Dukler,A.E.,(1976),ATheoretical Approach to the Lockhart-Martinelli Correlation for Stratified Flow, Int. J.Multiphase Flow, Vol. 2, pp. 591 -595.

Thammasat Int. J. Sc. Tech., Vol.3, No.21 July 1998

[l0] Kawaji, M., Anoda, Y., Nakamura, H. andTasaka,T.(1987), Phase and VelocityDistributions and Holdup in High-PressureSteamAilater Stratified Flow in a LargeDiameter Horizontal Pipe, Int.J.MultiphaseFlow,Vol. 13, pp. 145- I 59.

tl ll Spedding,P.L. and Hand,N.P.,(l990),Prediction of Holdup and Pressure Loss fromthe Two Phase Momentum Balance forStratified Type Flows, in Advances in Gas-Liquid Flow,FED-Vol.99,HTF-Vol .l 5 5,pp.221 -228.

ll2) Spedding,P.L. and Hand,N.P.,(1997),Prediction in Stratified Gas-Liquid Co-currentFlow in Horizontal Pipelines, Int. J. Heat MassTransfer, Vol. 40, pp.1923-1935.

[3] Wongwises, 5.,(1997), Method forPrediction of Pressure Drop and Liquid Hold-Up in Horizontal Stratified Two-Phase Flow inPipes, Presented at the (1997) ASMESymposium on Gas Liquid Two-Phase FlowgJune 22-26, I 997, Vancouver, Canada.

[4] Andreussi, P., Donfrancesco,A.andMessia,M. (1988), An Impedance Method forthe Measurement of Liquid Hold-Up in Two-Phase Flow, Int.J.Multiphase Flow, Vol.l4, pp.777-78s.

[5] Agrawal, S.S., Gregory G.A., and GovierG.W. (1973), An Analysis of HorizontalStratified Two Phase Flow in Pipes, Can J.Chem. Eng.,Vol. 5 l, pp.280-286.

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