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Available online at www.sciencedirect.com

Nuclear Engineering and Design 238 (2008) 1342–1352

Droplet entrainment correlation in vertical upwardco-current annular two-phase flow

Pravin Sawant a,∗, Mamoru Ishii a, Michitsugu Mori b,1

a Purdue University, School of Nuclear Engineering, 400 Central Dr., West Lafayette, IN 47907-2017, USAb Tokyo Electric Power Co., Inc., R & D Center, 4-1 Egasaki-cho, Tsurumi-ku, Yokohama 230-8510, Japan

Received 31 May 2007; received in revised form 5 October 2007; accepted 9 October 2007

bstract

Upward annular two-phase flow in a vertical tube is characterized by the presence of liquid film on the tube wall and entrained droplet ladenas phase flowing through the tube core. Entrainment fraction in annular flow is defined as a fraction of the total liquid flow flowing in the formf droplets through the central gas core. Its prediction is important for the estimation of pressure drop and dryout in annular flow. In the followingtudy, measurements of entrainment fraction have been obtained in vertical upward co-current air–water annular flow covering wide ranges ofressure and flow conditions. Comparison of the experimental data with the existing entrainment fraction prediction correlations revealed theirnadequacies in simulating the trends observed under high flow and high pressure conditions. Furthermore, several correlations available in theiterature are implicit and require iterative calculations.

Analysis of the experimental data showed that the non-dimensional numbers, Weber number (We = ρg〈jg〉2D/σ(�ρ/ρg)1/4) and liquid phaseeynolds number (Ref = ρf〈jf〉D/μf), successfully collapse the data. In view of this, simple, explicit correlation was developed based on theseon-dimensional numbers for the prediction of entrainment fraction. The new correlation successfully predicted the trends under the high flow

nd high pressure conditions observed in the current experimental data and the data available in open literature. However, in order to use theroposed correlation it is necessary to predict the maximum possible entrainment fraction (or limiting entrainment fraction). In the currentnalysis, an experimental data based correlation was used for this purpose. However, a better model or correlation is necessary for the maximumossible entrainment fraction. A theoretical discussion on the mechanism and modeling of the maximum possible entrainment fraction conditions presented.

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2007 Elsevier B.V. All rights reserved.

. Introduction

A maximum heat flux just before the boiling transition, i.e.,ritical heat flux (CHF) is one of the thermal limits for the nucleareactor design. It is essential for the nuclear reactor vendors andtilities to fully understand the mechanism of boiling transi-ion and to be able to predict it with reasonable accuracy overide ranges of conditions to ensure the safe operation of nuclear

eactors. Recently, this requirement has become more importantince there is a significant desire in nuclear utilities to up-rate theeactor power. The critical heat flux in high quality annular flow

∗ Corresponding author. Tel.: +1 765 496 3571; fax: +1 765 494 9570.E-mail addresses: psawant@purdue.edu (P. Sawant),

ichitsugu.mori@tepco.co.jp (M. Mori).1 Tel.: +81 45 613 6704; fax: +81 45 613 7899.

wedrarmvm

029-5493/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.nucengdes.2007.10.005

s called dryout and it is important to the operation of boilingater reactors (BWRs).In annular flow, part of the liquid flows in the form of a

iquid film adjacent to the wall and remaining liquid flowshrough the gas core as entrained droplets. Continuous thin-ing of the liquid film take place due to the combined effectf droplet entrainment, droplet deposition and evaporation pro-esses. In view of the fact that the evaporation process is quiteell understood, the accurate correlations are necessary for the

ntrainment and deposition rates of droplets in order to predictryout. However, compared to the entrainment and depositionate, entrainment fraction is more stable parameter to measurend correlate (Kataoka and Ishii, 1982). Entrainment fraction

epresents the integral effects of the rate processes of entrain-ent and deposition. The knowledge of entrainment fraction is

ery useful for the development of correlations for the entrain-ent and deposition rate. Furthermore, the usefulness of the

P. Sawant et al. / Nuclear Engineering a

Nomenclature

D diameter (m)E entrainment fractionEm maximum entrainment fractiong gravitational acceleration (m s−2)<jf> superficial liquid velocity (m/s)<jg> superficial gas velocity (m/s)<jgc> superficial critical gas velocity at the onset of

entrainment (m/s)ktp two-phase Von Karman constantp pressure (N m−2)Ref liquid phase Reynolds numberReff liquid film Reynolds numberRefflim limiting liquid film Reynolds numberT temperature (K)Wlf liquid film flow rate (kg/s)Wlfc limiting liquid film flow rate (kg/s)Wf liquid phase flow rate (kg/s)We Weber number

Greek lettersδ liquid film thickness (m)μ dynamic viscosity (kg m−1 s−1)ν kinematical viscosity (m2 s−1)ρ mass density (kg m−3)σ surface tension (N m−1)τi interfacial shear stress (kg m−1 s−2)

Subscriptsg gas phase

pmsi

olrotmmuucafse

fT

meaorl

2

dmIbtmce

E

wlLa

R

W

wdslrerTiahcwieetith

t

E/Em kAD < jg> S(ρgρf)

f liquid phase

rediction of entrainment fraction is not only limited to the esti-ation of dryout but many general thermal hydraulic predictions

uch as pressure drop in annular flow can also be significantlymproved (Kataoka and Ishii, 1982).

Most of the models or correlations available for the predictionf entrainment fraction are developed based on the low pressure,ow flow rate air–water data. There is a significant uncertaintyegarding the applicability of these models over wide rangesf conditions encountered in BWR operational and accidentalransients. Furthermore, power up-rate of the existing reactors

ay lead to the increased mass flux of both the phases whichay push the available entrainment fraction correlations to the

pper limit of the experimental database leading to the largencertainty in prediction of dryout. Therefore there is a need toheck the applicability of these correlations under high pressurend high flow conditions. Also a correlation for the entrainmentraction should be simple to use, explicit and non-dimensionalo that it can be incorporated into the system analysis codes very

asily.

In what follows, non-dimensional and explicit entrainmentraction correlations available in literature are summarized.he air–water test facility and entrainment fraction measure-

we

nd Design 238 (2008) 1342–1352 1343

ent technique used in this study are explained. The measuredntrainment fraction is compared with the existing correlationsnd finally, a simple and explicit correlation for the predictionf entrainment fraction is developed and compared with the cur-ent experimental data and the experimental data available initerature.

. Previous work

Experimental observations of the annular flow shows that theisturbance waves traversing the liquid film interface are theain source of droplet entrainment (Ishii and Grolmes, 1975).

shii and Grolmes (1975) proposed shearing off of the distur-ance wave crest as the primary mechanism of entrainment forhe low viscosity liquid such as water. Based on this entrain-

ent mechanism, Ishii and Mishima (1989) developed followingorrelation for the prediction of the entrainment fraction in quasi-quilibrium annular flow region.

= tanh(7.25 × 10−7We1.25Re0.25f ) (1)

here E, Ref and We are the equilibrium entrainment fraction,iquid phase Reynolds number and Weber number, respectively.iquid phase Reynolds number and Weber number are defineds

ef = ρf < jf > D

μf(2)

e = ρg < jg>2D

σ

(�ρ

ρg

)1/3

(3)

here D, <jf>, <jg>, ρf, ρg, μf, σ and �ρ are test sectioniameter, area average superficial liquid velocity, area averageuperficial gas velocity, liquid phase density, gas phase density,iquid phase viscosity, surface tension and density difference,espectively. Equilibrium entrainment fraction is defined as thentrainment fraction in an equilibrium annular flow where theate of droplet entrainment is equal to the rate droplet deposition.he above correlation is non-dimensional and explicit. However,

t is developed based on the low pressure air–water data. Assad etl. (1998) analyzed this correlation using the high pressure andigh flow rate air–water and Freon-113 data. They found that theorrelation failed to predict the trends for We1.25Re0.25

f > 106. Itas concluded that the correlation do not account for the fact that

t is not possible to entrain all of the liquid from the liquid filmven at very high gas velocity. Ishii and Mishima (1989) used thentrainment fraction of 1.0 as the upper boundary condition forhe development of the correlation. However, the current exper-mental data showed that the entrainment fraction approacheshe limiting entrainment fraction value much lower than 1.0 atigh gas velocity.

Pan and Hanratty (2002) proposed following correlation forhe prediction of equilibrium entrainment fraction,

n+2 0.5

1 − (E/Em)=

4kd(4)

here Em = (1 − (Wlfc/Wf)) is the maximum (or limiting)ntrainment fraction, Wlfc the limiting liquid film flow rate below

1 ring a

wdttcaw29dCevatb

woA

<

cc

rwuttufin

3

teawWt

344 P. Sawant et al. / Nuclear Enginee

hich there is no entrainment, kA and kd are atomization andeposition coefficients, respectively and S is the slip betweenhe droplets and gas phase. It can be observed that the correla-ion accounts for the existence of limiting entrainment fractionondition. However, the dependence of empirical constants kAnd kd on diameter, flow conditions and fluid properties is notell understood (Dykhno and Hanratty, 1996; Pan and Hanratty,002). Dallman et al. (1979) correlated the air–water data in.5 mm diameter test section using kA/4kd = 6.7 × 10−6 and theata in 31.8 mm diameter test section using kA/4kd = 6.5 × 10−5.orrelation was not proposed for the prediction of limitingntrainment fraction (Em). They used experimentally measuredalues of limiting liquid film flow rate (Wlfc) to obtain Em. Pannd Hanratty (2002) also presented another simplified correla-ion in order to account for the existence of critical gas velocityelow which no entrainment and the effect of surface tension.

E/Em

1 − (E/Em)= A1

D(< jg > − < jgc >)2 < jg > S(ρgρf)0.5

σ(5)

here <jgc> is the superficial critical gas velocity at the onsetf entrainment. Based on the experimental data they obtained

−5

1 = 6 × 10 and proposed the following correlation for <jgc>:

jgc >∼= 40σ0.5

D0.5(ρfρg)0.25 (6)

(pca

Fig. 1. Schematic of the ai

nd Design 238 (2008) 1342–1352

This correlation is empirical and dimensional. It is necessaryheck its applicability under the high pressure and high flowonditions.

Owen et al. (1985) developed correlations for the entrainmentate and deposition rate and equilibrium entrainment fractionas calculated by equating these two rates. Their correlationses a non-dimensional parameter τiδ/σ. In many experiments,he parameters like interfacial shear stress (τi) and liquid filmhickness δ are not measured directly. Also there is a largencertainty in the measurement and calculation of mass trans-er coefficient used in the model. Moreover their correlation ismplicit and requires iterative calculations. This correlation isot considered in the current analysis.

. Experimental facility

Fig. 1 shows schematic of the air–water test facility used forhe current experiments. A stainless steel tube with inside diam-ter 9.4 mm was used as a test section. Water was circulated byhigh pressure, multistage, 5 horse power pump while air flowas derived from the compressor with 150 psi supply pressure.ater flow rate was measured using a high accuracy rotame-

ers and a magnetic flow meter. Air flow measurement system

not shown in the figure) consisted of a set of rotameters and aressure gage for the measurement of back pressure. The majoromponents of the test section were a mixture unit, a flow visu-lization port, a first extraction unit and a second extraction unit.

r–water test facility.

ing and Design 238 (2008) 1342–1352 1345

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P. Sawant et al. / Nuclear Engineer

he mixture unit and both the extraction units consisted of a cen-ral porous tube having the porosity of 100 �m. In the mixturenit, water was injected into the test section through the wall ofhe porous tube and air was injected directly into the test sectionore from the bottom of the mixture unit. This injection methodelps in early establishment of the equilibrium annular flow. Inhe first extraction units, the liquid film on the inner wall of theest section was extracted completely through the porous tube byreating a small pressure difference across the porous tube wall.his pressure difference was created by venting a very smallmount air using the needle valve. As the air venting flow ratencreased, the pressure drop across the porous tube increasedeading to the increased extracted liquid flow rate. Eventuallyhe point was reached at which the further increase in the airenting flow rate did not affected the extracted liquid flow rate.t this condition the complete extraction of film was assumed.he extracted water flow rate was measured in the measuringylinders. After the extraction of the liquid film in the first extrac-ion unit the droplets remaining in the gas core deposited on theall forming a new liquid film which was again extracted using

he second extraction unit and measured. Further details of theesign, operation and working of the extraction units are avail-ble in Ishii and Sawant (2006). The two-phase mixture fromxit of the test section was separated in a cyclone separator. Their was vented into the atmosphere while the separated waterowed back to the storage tank. Temperature was measured at

he second extraction unit. The locations of the gage pressurend the differential pressure measurements are shown in Fig. 1.

developing length of 57 cm (60D) was provided for the airow before injecting it into the mixture unit. After the mixturenit, a developing length of 210D was provided before perform-ng any measurements. This insured that all the measurementsere obtained in the fully developed annular flow.The liquid film extraction technique utilized in the current

xperiments is widely used for the measurement of entrain-ent fraction and entrainment rate in annular flow (Cousins

nd Hewitt (1968), Assad et al. (1998)). The liquid film flowate measurement from the first extraction unit was used for thestimation of entrainment fraction while the liquid film flowate measured from the second extraction unit was used forhe estimation droplet deposition rate (the deposition rate dataot presented here). The detailed explanation of the liquid filmxtraction technique is available in Assad et al. (1998).

. Results and discussion

Air–water experiments were performed at three pressure con-itions, 1.2, 4.0 and 6.0 bar. Table 1 shows the ranges of liquid

able 1anges of experimental parameters

ressure P (bar) Superficial liquidvelocity <jf> (m/s)

Superficial gasvelocity <jg> (m/s)

.2 0.05–0.50 15–100

.0 0.05–0.75 15–100

.0 0.05–0.50 15–65

tpPrc

4

oIo

ig. 2. Experimental conditions on Mishima and Ishii (1984) flow pattern map:a) pressure 1.2 bar and (b) pressure 6 bar.

nd gas phase superficial velocities covered at each pressureondition. The experimental conditions corresponding to 1.2nd 6.0 bar are shown in Mishima and Ishii (1984) flow pat-ern map in Fig. 2. The dotted line in the figure shows the onsetf entrainment criteria given by Ishii and Grolmes (1975). It cane observed that the most of the experiments were performedfter the onset of entrainment in annular mist flow regime. Com-ared to the experimental data used for the development of Ishiind Mishima (1989) correlation (Eq. (1)), the current data coversigher pressure condition. Ishii and Mishima correlation utilizedir–water data at pressures up to 4 atm. In the following section,he trends observed in the experimental data are explained. Com-arison of the data with Ishii and Mishima (1989) correlation andan and Hanratty (2002) correlations is presented. The new cor-elation is developed based on the current experimental data andompared with the data available in literature.

.1. Trends observed in the experimental data

Detailed analysis of the experimental data and a methodol-gy for the modeling of entrainment fraction are presented inshii and Sawant (2006) and Sawant et al. (in press). The effectsf pressure, superficial gas velocity and liquid phase Reynolds

1346 P. Sawant et al. / Nuclear Engineering and Design 238 (2008) 1342–1352

Fl

nafbdcc

W

dmWa(yienWtnotrRfRt(a

4

sbi

Fc

sfidrimtRpIttdtlTnr

ig. 3. The experimental data correlated using modified Weber number andiquid phase Reynolds number.

umber on entrainment fraction were analyzed in Sawant etl. (in press). It was found that the Weber number in Eq. (3)ailed to predict the pressure effect. Therefore the Weber num-er definition was modified based on the current experimentalata. The modified Weber number shown in Eq. (7) successfullyollapsed the experimental data corresponding to all pressureonditions.

e = ρgj2gD/σ(�ρ/ρg)1/4 (7)

In Fig. 3, the experimental entrainment fraction measured atifferent liquid phase Reynolds numbers is plotted against theodified Weber number (Eq. (7)) raised to 1.25. The power of theeber number 1.25 is selected same as the power used in Ishii

nd Mishima (1989) correlation (Eq. (1)). Ishii and Mishima1989) proposed the above power based on the theoretical anal-sis. The data corresponding to all pressure conditions is plottedn Fig. 3. It can be observed that, at low Weber number thentrainment fraction is independent of liquid phase Reynoldsumber and it increases linearly with We1.25. However, at higheber number, corresponding to the limiting entrainment frac-

ion condition, the entrainment fraction do not depends on Weberumber. In this region, the entrainment fraction is functionf only liquid phase Reynolds number and it increases withhe increase liquid phase Reynolds number. In the transitionegion, between the Weber number dependant and liquid phaseeynolds number dependant limiting regions, the entrainment

raction is function of Weber number as well as liquid phaseeynolds number. From the figure it can be concluded that the

wo non-dimensional numbers, liquid phase Reynolds numberRef) and modified Weber number (We) adequately describe their–water entrainment fraction data.

.2. Comparison with the existing correlations

The measured entrainment fraction is plotted against theuperficial gas velocity at different liquid phase Reynolds num-er in Fig. 4. The data corresponding to 1.2 and 4 bar are plottedn Fig. 4a and b, respectively. As explained in the previous

IM1t

ig. 4. Comparison of the experimental data with Ishii and Mishima (1989)orrelation: (a) pressure 1.2 bar and (b) pressure 4.0 bar.

ection, at the lower superficial gas velocity, the entrainmentraction increases with the increase in gas velocity and it isndependent of liquid phase Reynolds number. A strong depen-ence on liquid phase Reynolds number is clearly visible atelatively high gas velocities. Eventually at very high gas veloc-ty, the entrainment fraction reaches a limiting condition of

aximum possible entrainment fraction. It can be also observedhat the limiting entrainment fraction is function of liquid phaseeynolds number and it increases with the increase in liquidhase Reynolds number. The figure also shows the prediction ofshii and Mishima (1989) correlation (Eq. (1)). The solid lines inhe figure indicate the predictions of the correlation. The correla-ion satisfactorily predicted the overall trends in the experimentalata at low gas velocity. However, at high gas velocity it predictshe continuously increasing entrainment fraction contrary to theimiting entrainment fraction observed in the experimental data.hus the correlation failed to predict the liquid phase Reynoldsumber dependence under high gas velocity condition. Similaresults were obtained by Assad et al. (1998) when they comparedshii and Mishima (1989) correlation with their data. Ishii and

ishima (1989) assumed the upper limit of entrainment fraction. However the experimental data shows that it is not possibleo entrain all of the liquid even at very high gas velocity.

P. Sawant et al. / Nuclear Engineering and Design 238 (2008) 1342–1352 1347

Fc

pii9rfsHispsttRtfp(ilr

Fc

rnffeiREtHcetl

ooI

ig. 5. Comparison of the experimental data with Pan and Hanratty (2002)orrelation Eq. (4): (a) pressure 1.2 bar and (b) pressure 4.0 bar.

Comparison of the measured entrainment fraction with theredictions of Pan and Hanratty (2002) correlation (Eq. (4))s shown in Fig. 5. kA/4kd = 6.7 × 10−6 and n = 1 were usedn the calculations as proposed by Dallman et al. (1979) for.5 mm diameter test section. Since there is no established cor-elation available for the prediction of limiting entrainmentraction (Em) or limiting liquid film flow rate (Wlfc), a con-tant value, Wlfc/πD = 0.046 kg/ms, proposed by Dykhno andanratty (1996) was used in the current calculations. As shown

n Fig. 5a, the low pressure experimental data is predictedatisfactorily by the correlation but it under predicts the highressure data corresponding to 4.0 bar (Fig. 5b) and 6.0 bar (nothown in figure) pressures. Since this correlation accounts forhe existence limiting entrainment fraction, it performs betterhan Ishii and Mishima (1989) correlation in the liquid phaseeynolds number dependent region or limiting entrainment frac-

ion region. However, it failed to predict the limiting entrainmentraction. The same conclusion can be drawn from the com-arison of another Pan and Hanratty (2002) correlation (Eq.

5)) shown in Fig. 6. The limiting entrainment fraction (Em)s again calculated assuming the constant value for the limitingiquid film flow rate Wlfc/πD = 0.046 kg/ms. However, the cur-ent experimental results show that the limiting liquid film flow

cFwl

ig. 6. Comparison of the experimental data with Pan and Hanratty (2002)orrelation Eq. (5): (a) pressure 1.2 bar and (b) pressure 4.0 bar.

ate (Wlfc) increases with the increase in liquid phase Reynoldsumber (see Fig. 14). It was found that the limiting entrainmentraction calculated assuming Wlfc/πD = 0.046 is very differentrom the experimentally measured values. Based on the currentxperimental data, an empirical correlation relating the limit-ng liquid film Reynolds number (Refflim) to the liquid phaseeynolds number (Ref) is developed in the next section (Seeq. (18)). This correlation along with Eq. (15) are used for

he prediction of limiting entrainment fraction. The Pan andanratty (2002) correlation Eq. (4) is again compared with the

urrent experimental data in Fig. 7. It can be seen that the use ofxperimental limiting entrainment fraction worsen the predic-ion of Pan and Hanratty (2002) correlations. Even the data atow pressure condition is not predicted well.

In their analysis, Pan and Hanratty (2002) studied the effectf gas density on entrainment fraction using the air–water dataf Assad et al. (1998) obtained in 9.5 mm diameter test section.n order to check whether their correlations (Eqs. (4) and (5))

an predict the high pressure data of Assad et al. (1998), inig. 8 the entrainment fraction measured at 3.7 bar is comparedith Pan and Hanratty (2002) correlations Eqs. (4) and (5). The

imiting entrainment fraction is calculated assuming the constant

1348 P. Sawant et al. / Nuclear Engineering and Design 238 (2008) 1342–1352

Fc

lpttA

tcp

4m

pitRiadto

F3

ampneed to consider the effect Weber number and liquid phaseReynolds number on different disturbance characteristics. Acomprehensive database on variations of disturbance wave prop-erties with Weber number and liquid phase Reynolds number is

ig. 7. Comparison of the experimental data with Pan and Hanratty (2002)orrelation Eq. (4): (a) pressure 1.2 bar and (b) pressure 4.0 bar.

imiting liquid film flow rate Wlfc/πD = 0.046 kg/ms. Since therediction of the correlation Eq. (4) and (5) are very similar, onlyhe comparison results of Eq. (4) are shown. It can be observedhat the correlation failed to predict the high pressure data ofssad et al. (1998).From the above comparisons of the experimental data with

he existing correlations it can be concluded that the existingorrelations failed to predict the trends observed in the highressure and high flow rate entrainment fraction data.

.3. Explanation of the mechanism and modelingethodology

The methodology for the modeling of entrainment fractionroposed by Sawant et al. (in press) is shown in Fig. 9. A curven the figure is schematic representation of entrainment frac-ion variation with Weber number at a constant liquid phaseeynolds number. The entrainment curve O-A-B-C is divided

nto three regions; a Weber number dependent region O-A,

transition region A-B and a liquid phase Reynolds number

ependant region B-C. Since the disturbance waves traversinghe liquid film interface are the main source of entrainment, vari-us disturbance wave characteristics such as frequency, velocity,

ig. 8. Comparison of the Assad et al. (1998) experimental data at pressure.7 bar with Pan and Hanratty (2002) correlation: (a) Eq. (4) and (b) Eq. (5).

mplitude and wave-length significantly affects the entrainmentechanism. In order to understand and model the entrainment

henomenon in various regions of the entrainment curve, we

Fig. 9. Correlation methodology (Sawant et al., in press).

ing a

nc

tbrbimttlmRieflmeniptiomagntcmlitRrfRi

i

E

wefR

E

wg

E

fl

W

W

ic

E

p

W

lf

E

wi

otepsi

4

le

E

w(stf

a

opt

R

(

P. Sawant et al. / Nuclear Engineer

ecessary. Currently such database covering wide ranges of flowonditions in not available.

Based on the limited experimental data available in litera-ure a qualitative explanation of the phenomenon was proposedy Sawant et al. (in press). In the Weber number dependantegion O-A, a liquid film is relatively thick. As the Weber num-er or superficial gas velocity increases, entrainment fractionncreases. At the same time, liquid film flow rate decreases asore and more liquid gets entrained in to the gas core. However,

his decrease in the liquid film flow rate do not affects the dis-urbance wave characteristics and hence entrainment fraction, asong as the liquid film is sufficiently thick. Therefore the entrain-

ent fraction in this region is independent of the liquid phaseeynolds number. Further increase of the superficial gas veloc-

ty or Weber number results in higher entrainment fraction andventually at point A, the first transition point, the liquid filmow rate decreases sufficiently and starts affecting the interfacialomentum transfer. Thus, in this part of the curve (A-B), the

ntrainment fraction depends on both the liquid phase Reynoldsumber and the Weber number. As the superficial gas veloc-ty increases, the liquid film flow rate decreases further and atoint B, the second transition point, there is no more interac-ion between the gas core and the liquid film. The liquid filmn this region (B-C) gets submerged in to the viscous sub-layerf the core gas flow. This leads to the suppression of entrain-ent. In this region the entrainment fraction remains constant

nd do not increase with the further increase in the superficialas velocity. It was also found that as the liquid phase Reynoldsumber increases, the first and second transition points shiftso the higher superficial gas velocity and entrainment fractiononditions. With the increase in liquid phase Reynolds number,ore amount of liquid needs to be entrained in order to reduce the

iquid film flow rate below the critical value. Similarly the limit-ng entrainment fraction at which the liquid film submerges intohe gas flow viscous sub-layer increases with the liquid phaseeynolds number. It was also observed that the liquid film flow

ate at both the transition points and at the limiting entrainmentraction region increases with the increase in the liquid phaseeynolds number. This increase is may be due to the effect of

ncreased droplet concentration on the gas phase turbulence.In the Weber number dependent region, entrainment fraction

s correlated by the following relation:

= aWe1.25 (8)

here a = 9.0 × 10−6 is obtained from the current air–waterxperimental data. In the transition region, since entrainmentraction is function of both Weber number and liquid phaseeynolds number, it is correlated by

= Ftr(We1.25, Ref) (10)

here Ftr is a transition function. The first transition criterion isiven by the following correlation

1 = aWe1.251 = Ftr(Ref, We1.25

1 ) (11)

This is the transition point at which decrease in the liquid filmow rate starts affecting the entrainment fraction. The transition

brta

nd Design 238 (2008) 1342–1352 1349

eber number, We1 is function of liquid phase Reynolds number

e1.251 = f (Ref) (12)

The second transition criterion is the point at which dynamicnteraction between the liquid film and the central gas coreeases. It is given by the following equation

2 = Elim = Ftr(Ref, We1.252 ) (13)

The transition Weber number (We2) is function of the liquidhase Reynolds number

e1.252 = f (Ref) (14)

Finally, the limiting entrainment fraction (Em) is function ofiquid phase Reynolds number and is given by the followingorm of correlation

m = 1 − Reff lim

Ref= f (Ref) (15)

here Refflim is the limiting liquid film Reynolds number whichs also the function of the liquid phase Reynolds number, Ref.

In order to develop a model based on the above methodol-gy, correlations for the first and second transition criteria andhe limiting entrainment fraction are necessary. A very extensivexperimental data on variations of different disturbance waveroperties is necessary for this purpose. Since such a comprehen-ive database is not available currently, a simplified correlations proposed in this study.

.4. New correlation for entrainment fraction

Based on the trends observed in the experimental data fol-owing simple correlation is proposed for the prediction ofntrainment fraction:

= Em tanh(aWe1.25) (16)

here Em is the function of liquid phase Reynolds number (Eq.15)) and coefficient ‘a’ accounts for the dependence of the tran-ition points A and B on liquid phase Reynolds number. Based onhe current experimental data following correlation is obtainedor the coefficient ‘a’;

= 2.31 × 10−4Re−0.35f (17)

Since no satisfactory correlation is available for the predictionf limiting entrainment fraction, the following correlation for therediction limiting liquid film Reynolds number obtained fromhe experimental data is proposed.

efflim = 250 ln(Ref) − 1265 (18)

Limiting entrainment fraction is derived from Eqs. (15) and18).

Eqs. (17) and (18) are purely empirical correlations derived

ased on the current experimental data. Even though the cor-elations use non-dimensional numbers it is necessary to checkheir applicability for the fluids having different surface tensionnd viscosity. Effects of diameter and fluid properties on the

1350 P. Sawant et al. / Nuclear Engineering and Design 238 (2008) 1342–1352

Fd

tml

tcdfceap(tmpftts2l(3umausTd

5

Ro

F

fiplal. (1985) in 3.2 cm diameter test section and Schadel (1988) in4.2 cm diameter test section are plotted against the liquid phaseReynolds number along with the current experimental data. Itcan be observed that the Refflim initially increases with the liquid

ig. 10. Comparison of the proposed correlation with the current experimentalata.

ransition criteria (i.e. coefficient ‘a’) and the limiting entrain-ent fraction are not known. Further discussion on the modeling

imiting entrainment fraction is presented in the next section.Fig. 10 shows the comparison of the new correlation with

he current experimental data. It can be observed that theorrelation predicts the liquid phase Reynolds number depen-ency in the transition as well as in the limiting entrainmentraction region satisfactorily. The proposed correlation is alsoompared with the entrainment fraction data available in lit-rature. Experimental data of Assad et al. (1998), Owen etl. (1985) and Schadel (1988) are compared with the pro-osed correlation in Figs. 11–13, respectively. Assad et al.1998) performed air–water experiments in 9.5 mm diameterest section. They used the film extraction technique for the

easurement of entrainment fraction. Since for the large liquidhase Reynolds number conditions this technique is not satis-actory due to the incomplete removal of the liquid film, onlyhe data points measured at liquid phase Reynolds number lesshan 6000 are considered for the comparison. Fig. 11a and bhows the comparison with Assad et al. (1998) data measured at.4 and 3.7 bar, respectively. It can be observed that the corre-ation satisfactorily predicts the experimental data. Owen et al.1985) obtained air–water entrainment fraction measurements in.18 cm diameter test section at atmospheric pressure. They alsosed film extraction technique for the measurement of entrain-ent fraction. Schadel (1988) carried out air–water experiments

t atmospheric pressure in 2.54 cm diameter test section andsed tracer mixing method for the entrainment fraction mea-urement. The correlation predicted all the data satisfactorily.he data covered diameter range from 0.95 to 3.18 cm and twoifferent entrainment fraction measurement techniques.

. Modeling of limiting entrainment fraction

A correlation for the limiting liquid film flow rate or limitingeynolds number (Refflim), is necessary for the determinationf limiting entrainment fraction (Eq. (15)). The limiting liquid F

ig. 11. Comparison of the proposed correlation with Assad et al. (1998) data.

lm flow rate occurs at high gas velocity at which the sup-ression of entrainment takes place. In Fig. 14, the limitingiquid film Reynolds number (Refflim) measured by Owen et

ig. 12. Comparison of the proposed correlation with Owen et al. (1985) data.

P. Sawant et al. / Nuclear Engineering a

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ig. 13. Comparison of the proposed correlation with Schadel (1988) data.

hase Reynolds number and asymptotically approaches a con-tant value at higher liquid phase Reynolds number. Very limitedxperimental data is available in literature on the Refflim. Manyesearchers assumed that the limiting liquid film Reynolds num-er at which the suppression of entrainment takes place is sames the liquid film Reynolds number at the onset of entrainment atery high gas velocity (Asali et al., 1985; Owen et al., 1985 andchadel, 1988). However, just before the onset of entrainment,

here are no entrained droplets in the gas phase. Therefore theorrelations developed for the prediction of onset of entrainmentre applicable when the gas phase is pure. While the mechanismf suppression of entrainment might be very similar to the onsetf entrainment under high gas velocity, we need to consider theffect of entrained droplets on the gas phase turbulence whileodeling the suppression of entrainment.There is an sample of evidence available in the literature

howing that the entrained droplets affect the velocity distri-ution and turbulence level in the gas core. Gill et al. (1964)

easured the local gas phase velocities in annular flow with

he entrained droplets using an isokinetic probe. He found thathat as the total liquid flow rate increases, the concentration of

Fig. 14. Variation of limiting liquid film Reynolds number.

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roplets in the gas core increases and the turbulent gas phaseelocity profile becomes more and more parabolic. Owen andewitt (1987) called this phenomenon as a laminerization of the

ore flow. They calculated two-phase Von Karman constant, ktpsing the experimentally measured turbulent velocity distribu-ion and found that as the droplet concentration increases, ktpecreases. It was concluded that the effect of droplets in theas core is to reduce the turbulent mixing length. For the pureas turbulent flow, Von Karman constant is 0.4 while with thentrained droplets it reduced to the value as low as 0.1. Thereforehere is a need to consider this effect of droplet concentrationhile modeling for the limiting liquid film Reynolds number.

. Conclusion

Air–water annular flow experiments were carried out in.4 mm diameter test section. The experiments covered threeressure conditions and wide ranges of air and water flow rates.ntrainment fraction was measured using the liquid film extrac-

ion technique. The analysis of the experimental data revealedhat the correlations available in literature failed to predict thentrainment fraction under the high flow and high pressureonditions. The trends in the data were explained qualitativelyonsidering the variations in liquid film thickness and distur-ance wave characteristics. Further analysis of the data showedhat the entrainment fraction can be completely described usinghe two non-dimensional numbers; a modified Weber numbernd a liquid phase Reynolds number. The new empirical corre-ation was developed using these two non-dimensional numbersnd the experimental data. The proposed correlation is simple,on-dimensional and explicit. It compared favorably with theurrent experimental data and the data available in open lit-rature. However, the correlation was developed based on their–water data. There is a need to improve the correlations forhe parameter ‘a’ and the limiting entrainment fraction by con-idering the effect of fluid properties.

cknowledgement

The research project was supported by Tokyo Electricalower Company (TEPCO). The authors would like to express

heir sincere appreciation for support from the TEPCO.

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