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Nuclear Engineering and Design 238 (2008) 3528–3541
Contents lists available at ScienceDirect
Nuclear Engineering and Design
journa l homepage: www.e lsev ier .com/ locate /nucengdes
roperties of disturbance waves in vertical annular two-phase flow
ravin Sawanta,∗, Mamoru Ishii a, Tatsuya Hazukub, Tomoji Takamasab, Michitsugu Moric
Purdue University, School of Nuclear Engineering, 400 Central Dr., West Lafayette, IN 47907-2017, USAFaculty of Marine Technology, Tokyo University of Marine Science and Technology, Etchujima, Koto-ku, Tokyo 135-8533, JapanTokyo Electric Power Co., Inc., 4-1 Egasaki-cho, Tsurumi-ku, Yokohama 230-8510, Japan
r t i c l e i n f o
rticle history:eceived 25 July 2007eceived in revised form 22 May 2008ccepted 17 June 2008
a b s t r a c t
Disturbance waves play an important role in interfacial transfer of mass, momentum and energy in annulartwo-phase flow. In spite of their importance, majority of the experimental data available in literature ondisturbance wave properties such as velocity, frequency, wavelength and amplitude are limited to nearatmospheric conditions (Azzopardi, B.J., 1997. Drops in annular two-phase flow. International Journal ofMultiphase Flow, 23, 1–53). In view of this, air–water annular flow experiments have been conductedat three pressure conditions (1.2, 4.0 and 5.8 bar) in a tubular test section having an inside diameter9.4 mm. At each pressure condition liquid and gas phase flow rates are varied over a large range so thatthe effects of density ratio, liquid flow rate and gas flow rate on disturbance wave properties can be studiedsystematically. A liquid film thickness is measured by two flush mounted ring shaped conductance probeslocated 38.1 mm apart. Disturbance wave velocity, frequency, amplitude and wavelength are estimated
from the liquid film thickness measurements by following the statistical analysis methods. Parametrictrends in variations of disturbance wave properties are analyzed using the non-dimensional numbers;liquid phase Reynolds number (Ref), gas phase Reynolds number (Reg), Weber number (We) and Strouhalnumber (Sr). Finally, the existing correlations available for the prediction of disturbance wave velocityand frequency are analyzed and a new, improved correlation is proposed for the prediction of disturbancewave frequency. The new correlation satisfactorily predicted the current data and the data available inee
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literature.
. Introduction
Annular two-phase flow regime is important for the operationf several industrial equipments such as nuclear reactors, steamoilers, refrigeration systems and many others. It is characterizedy the presence of a liquid film adjacent to the wall and a gas phaseowing through the center. The gas phase usually contains liquidroplets entrained from the liquid film surface. The waves travers-
ng the liquid film interface in annular flow have been classifiednto two main categories by several researchers; ripple and dis-urbance waves (Hewitt and Hall-Taylor, 1970; Azzopardi, 1997).ipple waves are relatively small wavelength and small ampli-ude dynamic waves traveling at much lower velocity comparedo the velocity of disturbance waves and gas phase. These waves
re also non-coherent and loose their identity after traveling ahort distance. On the other hand, disturbance waves are largemplitude kinematic waves and travels at velocity much higherhan the velocity of liquid film and ripple waves. They are coher-∗ Corresponding author. Tel.: +1 765 496 3571; fax: +1 765 494 9570.E-mail address: [email protected] (P. Sawant).
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029-5493/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.nucengdes.2008.06.013
© 2008 Elsevier B.V. All rights reserved.
nt and dominate the interfacial transfer of mass, momentum andnergy.
Disturbance waves have been identified as the major sourcef entrained droplets in annular flow (Ishii and Grolmes, 1975;ousins and Hewitt, 1968). The experimental evidence shows thathe droplets in annular flow are generated by shearing off the crestf disturbance waves (Ishii and Grolmes, 1975). An accurate pre-iction of amount of these entrained droplets (i.e. entrainmentraction) is essential for the estimation of dryout in nuclear reactorsnd boilers (Sawant et al., 2008). In addition to this, the accelerationf droplets significantly contributes to the momentum transfer andotal pressure drop in annular flow (Fore and Dukler, 1995). Since
ajority of the droplets are generated from disturbance waves,knowledge of disturbance wave properties is essential for the
evelopment of mechanistic models for the prediction dryout andressure drop in annular flow.
Disturbance waves also act as a roughness to the central gas
ow and contribute to the frictional pressure drop in annular flow.t has been also found that the motion of disturbance waves affectshe interfacial momentum transfer significantly (Hori et al., 1979;umar et al., 2002; Wang et al., 2004; Zabaras et al., 1986). Zabarast al. (1986) notes that merely treating the interfacial waves as
P. Sawant et al. / Nuclear Engineering an
Nomenclature
a constant in Eq. (6)A constant in Eq. (15)B constant in Eq. (15)C constant in Eqs. (5) and (6)Cf interfacial friction factorDt test section diameter [m]Eo Eötvös number (Eq. (12))f average disturbance wave frequency [Hz]Frg gas phase Froude number (Eq. (14))g gravitational acceleration [m/s2]Glf liquid film mass flux [kg/m2 s]h average disturbance wave amplitude [�m]〈jf〉 area averaged superficial liquid velocity [m/s]〈jg〉 area averaged superficial gas velocity [m/s]K constant in Eq. (7)p pressure [N m−2]Re Reynolds numberRxy cross-correlation function [�m2]Sr Strouhal number (Eq. (9))Sxx spectrum of autocorrelation function [�m/Hz]t time [ms]ulf average liquid film velocity [m/s]Vw wave velocity [m/s]We Weber number (Eq. (3))z∞ developing length [m]
Greek symbols� liquid film thickness [m]� average disturbance wave wavelength [mm]� dynamic viscosity [kg m−1 s−1]� mass density [kg m−3]� surface tension [N m−1]
Subscriptsavg averageg gas phase
atfcswvcfeaestmtuiFtmo
dfl
ampVbe(aasta
v1rlbtetwtrTlhwra(
2
i(tasfpf
2
sstoef
z
f f t
f liquid phaself liquid film
rough surface and not considering their dynamic characteris-ics requires unrealistically large equivalent roughness to accountor the observed interfacial shear. Hori et al. (1979) derived aorrelation for the interfacial friction factor in annular flow andhowed that the interfacial shear depends on wave amplitude (i.e.ave roughness) as well as wave frequency and wave propagation
elocity. Moalem Maron and Brauner (1987) correlated interfa-ial friction factor to disturbance wave amplitude, velocity andrequency. Recently, Wang et al. (2004) performed annular flowxperiments in 9.5 mm diameter tube and measured pressure dropnd film thickness. They estimated the wave roughness from thexperimental measurements and calculated friction factor usingingle-phase friction factor correlation. Their analysis showed thathe calculated friction factor is always less than the experimentally
easured friction factor by 18–30%. Since the entrainment frac-ion was negligible in their experiment, it was concluded that thender prediction was mainly due to neglecting the wave mobil-
ty effects, i.e. effects of disturbance wave frequency and velocity.urthermore, dynamic properties of disturbance waves also affecthe mixing characteristics of the liquid film and hence heat and
ass transfer in annular flow. Therefore detailed understandingf disturbance wave properties can significantly contribute to the
R
W
d Design 238 (2008) 3528–3541 3529
evelopment of various mechanistic models in annular two-phaseow.
A comprehensive literature review by Azzopardi (1997) onnnular two-phase flow data and modeling revealed that theajority of the available experimental data on disturbance wave
roperties are obtained at near atmospheric pressure conditions.ery few of them are obtained for the superficial liquid velocityeyond 0.1 m/s. Also several of the previous experimental studiesmployed subjective methods for the estimation of these propertiesAzzopardi, 1986). In view of this, in the current study, experimentsre performed over a wide range of pressure and flow conditionsnd disturbance wave properties are measured using the objectivetatistical analysis methods. A comprehensive database on dis-urbance wave properties developed in this study is a significantddition to the existing database on disturbance wave properties.
The recent studies by Hawkes et al. (2000) and some earlier flowisualization studies (Bennett et al., 1965; Sekoguchi and Takeishi,989) have shown that in annular flow regime as the liquid flowate is increased at a given gas velocity, a different form of annu-ar flow known as wispy annular flow is observed. A transitionoundary between the regular annular flow (observed at rela-ively low liquid flow rate) and wispy annular flow is not wellstablished. The experimental observations have shown the exis-ence of wisps or agglomerated liquid structure in the gas core ofispy annular flow. However, surprisingly very limited informa-
ion is available on the nature of interfacial waves in the transitionegion as well as in wispy annular flow regime. Sekoguchi andakeishi (1989) reported the existence of huge waves which arearger compared to the regular disturbance waves and travel atigher velocity. While the studies of transition from annular toisp annular flow and characteristics of interfacial waves in this
egion are the part of future work, the current study is limited tonnular flow observed at relatively low liquid flow rate conditions500 ≤ Ref ≤ 5700).
. Experimental facility and instrumentation
The experiments have been performed on a air–water test facil-ty available in Thermal-hydraulics and Reactor Safety LaboratoryTRSL), Purdue University. Originally the facility was designed forhe measurement of entrainment fraction and droplet entrainmentnd deposition rates in annular flow. An acrylic film thickness mea-urement port is newly designed and installed on the test facilityor the measurement liquid film thickness and disturbance waveroperties. The film thickness measurement port can be also usedor the flow visualization and dry-patch study.
.1. Air–water test facility
The test facility is comprised of a stainless steel tubular testection with 9.4 mm inside diameter. A height of the original testection is increased from 3.2 to 3.7 m in order to install the filmhickness measurement port. Fig. 1 shows the schematic drawingf air–water test facility. A developing length (z∞) necessary for thestablishment of an equilibrium annular flow is calculated using theollowing correlation given by Kataoka and Ishii (1982),
∞ ∼= 440.0DtWe0.25
Re0.5f
(1)
� 〈j 〉D
ef =�fReynolds number (2)
e = �g〈jg〉2Dt
�
(�f − �g
�g
)1/3
Weber number (3)
3530 P. Sawant et al. / Nuclear Engineering and Design 238 (2008) 3528–3541
ic of t
wggvptamrmficeiaceetbTfifoedati
flsmp
2
f2cocpsupacmamom
Fig. 1. Schemat
here Dt, �g, �f, 〈jg〉, 〈jf〉, �f and � are test section diameter,as phase density, liquid phase density, area average superficialas velocity, area average superficial liquid velocity, liquid phaseiscosity and surface tension, respectively. For the experimentserformed in the current study, the ratio z∞/Dt calculated is lesshan 210. As shown in Fig. 1, all the measurements are performedfter the developing length of L/Dt = 270. This insured that theeasurements are corresponding to the fully developed equilib-
ium annular flow. Major components of the test section are aixer/injector unit, a liquid film thickness measurement port and
rst and second extraction units. The mixer/injector consists of aentral porous tube having 100 �m porosity and an inside diam-ter similar to the test section inside diameter. Water is injectednto the test section through the wall of this porous tube whileir is injected from the bottom of the mixer unit directly into theore of the test section. This method of injection is helpful in gen-ration of annular flow with minimum initial entrainment andntrance effect. The liquid film thickness measurement port is usedo measure instantaneous liquid film thickness and various distur-ance wave properties using two ring shaped conductance probes.he same port is going to be used for the observation of liquidlm breakdown and generation and observation of dry-patch in
uture studies. The extraction units are used for the measurementf liquid film flow rate and other entrainment parameters such as
ntrainment fraction and entrainment and deposition rates. Theetails of measurement of these entrainment parameters are avail-ble in Sawant et al. (2007, 2008). Other instrumentation on theest facility includes high speed CCD camera for the flow visual-zation, magnetic flow meter and rotameters for the inlet watertablt
he test facility.
ow measurement, rotameters for the air flow measurement, pres-ure transducers for the gage pressure and differential pressureeasurement and thermocouples for the inlet air and water tem-
erature measurement.
.2. Film thickness measurement
Several techniques have been devised by previous researchersor the measurement of liquid film thickness in annular flow (Clark,002). Examples are conductance probe, capacitance probe, opti-al probe, needle probe and laser focus displacement meter. Eachf the probes has some advantage over the others under certainonditions. If the liquid phase is conducting, use of conductancerobe is more suitable. Due to their simple design and good mea-urement reliability, conductance probes have been extensivelysed by several researchers (Clark, 2002). Working of conductancerobe depends on a linear relation between the film thicknessnd total conductance. Two kinds of conductance probes areommonly used; a parallel wire conductance probe and a flushounted conductance probe. The parallel wire conductance probes
re employed for the measurement of thick films while the flushounted probes are capable of measuring very thin film with-
ut disturbing the film flow. This non-intrusive nature of the flushounted probes make them suitable for the measurement of very
hin films (of the order of few microns) observed in the currentnnular flow experiments. Several types of electrode designs haveeen used for the flush mounted probes such as probes with paral-
el strip electrodes, concentric circle type electrodes, circular rodype electrodes, ring shaped electrodes, etc. Depending on the
P. Sawant et al. / Nuclear Engineering and Design 238 (2008) 3528–3541 3531
ttchtrwctiap
T3d
abc
IbadTatmtsio
ivtTomtpham(vfmsa
a
Fig. 2. Schematic of the ring shaped conductance probe.
ype of measurement, i.e. average or localized and the test sec-ion geometry, a particular probe design can be selected. In theurrent experiments, flush mounted ring shaped electrode designas been selected. Fig. 2 shows the schematic of ring type conduc-ance probes used in the current experiments. Two flush mounteding electrodes separated by 3.048 mm are installed concentricallyith the test section. As the liquid film thickness changes, the
onductance between the two electrodes varies. This variation inhe conductance is measured and converted into the correspond-ng change in output voltage using an electrical circuit. The designnd working of the electrical circuit are very similar to the designresented by Coney (1973).
Fig. 3 shows a schematic of the liquid film measurement port.he port consists of two ring-shaped conductance probes located8.1 mm apart. Following parameters have been considered for theesign of conductance probes;
. A separation between the two electrodes of the probe.
. A width of each electrode.
. A separation between two probes.
n order to obtain the localized film thickness measurement (i.e.etter spatial resolution) two electrodes should be located as closes possible. However, as the separation between the electrodesecreases, a linear measurement range of the probe decreases.hus there is a tradeoff between the spatial resolution of the probend liquid film thickness measurement range. The second impor-ant design parameter is the width of electrodes. Wider electrodes
ake the probe more sensitive to the small changes in the film
hickness. However, the increased electrode width also reduces thepatial resolution of the probe. The width of electrode is also lim-ted by the machining capability. In the current design, the widthf electrode is 2.54 mm. The separation between the two probesFig. 3. Liquid film thickness measurement port.
pitaf
3
teflctreoo
tt
Fig. 4. Calibration of conductance probe.
s important parameter for the measurement of disturbance waveelocity. Two probes should be located as close as possible sohat the same disturbance wave is measured by both the probes.his insures good cross-correlation between the measurementsf two probes and improves the accuracy of velocity measure-ent. However, as the distance between two probes decreases,
he possibility of interference or cross-communication between therobes increases. Optimum values for the above design parametersave been obtained from the numerical analysis performed overn annular control volume representing the liquid film and flushounted conductance probes. Two-dimensional Laplace equations
cylindrical co-ordinates) were discretized over an annular contrololume with the constant potential boundary at the electrode sur-aces. The discretization was carried out using the control volume
ethod. Tri-Diagonal Matrix Algorithm (TDMA) was employed forolving the system of simultaneous equations. The results of thenalysis are available in Ishii and Sawant (2006).
A calibration set-up for the conductance probes included sep-rately machined acrylic port with one ring shaped conductancerobe. The design of the calibration probe was similar to the probes
nstalled in the film thickness measurement port. A film of knownhickness was created using the acrylic plug as shown in Fig. 4. Eightcrylic plugs were designed in order to create liquid film thicknessrom 25 to 1270 �m. Fig. 5 shows the typical calibration curve.
. Experimental data and analysis
In the overall total 157 experiments have been carried out athree pressure conditions, 1.2, 4.0 and 5.8 bar. At each pressure,xperiments have been performed at five different liquid phaseow rates; 0.21, 0.40, 0.60, 1.22 and 2.25 m3/min. The experimentalonditions correspond to pressures 1.2, 4.0 and 5.8 bar are plot-ed on Mishima and Ishii (1984) flow pattern map in Fig. 6(a–c),espectively. A dashed line in the figures indicates the onset ofntrainment criteria given by Ishii and Grolmes (1979). It can be
bserved that most of the experiments are performed after thenset of entrainment.The lowest gas velocity experiment conditions are very closeo the churn-annular flow regime transition boundary. The filmhickness data and flow visualization results showed that the liq-
3532 P. Sawant et al. / Nuclear Engineering an
uMbflwg
rmflRrit
3
comitEbpaflmature.
Fig. 5. Calibration curve for conductance probe.
id film in these experiments is very unstable. A theoretical study ofoalem Maron and Dukler (1984) and experimental data obtained
y Zabaras et al. (1986) showed that in this region wall shear stressuctuates around zero and takes both negative and positive signs,hich results in an internal recycle in the liquid film. In the higher
as velocity experiments, a liquid film interface is dominated by the
1vt
Fig. 6. Range of experimental parameters (a)
d Design 238 (2008) 3528–3541
egular and uniform disturbance waves. Furthermore, the experi-ental data available in literature shows that in air–water annular
ow, onset of disturbance waves takes place when the liquid phaseeynolds number (Ref) exceeds 200 (Azzopardi, 1997). Since theange of liquid phase Reynolds number in the current experimentss 500 ≤ Ref ≤ 5700, the disturbance waves were always present onhe liquid film interface.
.1. Data analysis procedure
In each experiment, the measurement of liquid film thickness isarried out by sampling the data at 2000 Hz over a time periodf 5 s. The raw voltage data is converted to the film thicknesseasurements by applying the calibration curve. In the follow-
ng section, the data analysis procedure employed in this study forhe estimation of various disturbance wave properties is explained.xperiments performed at 1.2 bar and liquid phase Reynolds num-er 950 are considered in the following explanation. The liquidhase Reynolds number values reported in the following discussionre representative values for the particular liquid phase volumetricow rate. Actual liquid phase Reynolds number in each experimentight be slightly different depending on the liquid phase temper-
Fig. 7 shows the liquid film thickness time trace measured at.2 bar and liquid phase Reynolds number 950. A superficial gaselocity is increased from 6 to 88 m/s. The data corresponding tohe superficial gas velocities 6, 12, 25, 40, 55 and 75 m/s are shown
P 1.2 bar, (b) P 4.0 bar and (c) P 5.8 bar.
P. Sawant et al. / Nuclear Engineering and Design 238 (2008) 3528–3541 3533
itetMfloitsv
dwdsffosatso
fsrltsasv
cncrcdSaveral previous researchers (Azzopardi, 1986; Webb, 1970). At lowgas velocities where the waves are less regular, the peak in cross-correlation function is shorter and wider. However, as the gasvelocity increases more regular waves appear on the interface and
Fig. 7. Liquid film thickness time trace (Ref 950, P 1.2 bar).
n the figure. The figure also shows the time averaged liquid filmhickness (ıavg) measured at each flow condition. As mentionedarlier, the lowest gas velocity data near the churn-annular transi-ion boundary shows that the liquid film is very thick and unstable.
ore regular disturbance waves appear on the liquid film inter-ace at higher gas velocities. As noted by previous researchers, theiquid film can be divided into two parts, a wavy part and a baser substrate. As the gas velocity increases, frequency of the wavesncreases while their amplitude decreases. It can be also observedhat the average liquid film thickness (ıavg) and thickness of the sub-trate decreases with the increase in gas velocity. These trends areery similar to the observations summarized by Azzopardi (1997).
The film thickness measurement from any one the two con-uctance probes can be utilized for the estimation of disturbanceave frequency and amplitude. However, in order to estimateisturbance wave velocity and wavelength, the film thickness mea-urements from both the probes are necessary. Disturbance waverequency can be calculated either by plotting the autocorrelationunction or by plotting the spectrum of autocorrelation functionf film thickness measurement. A time lag corresponding to theecond peak in the autocorrelation function represents the aver-
ge time period between the two successive disturbance waves. Inhe plot of spectrum of autocorrelation function, a frequency corre-ponding to the tallest peak in the plot gives the average frequencyf disturbance waves. Fig. 8 shows the spectrum of autocorrelationFig. 8. Spectrum of autocorrelation function (Ref 950, P 1.2 bar).
unction (Sxx) calculated from the film thickness measurementshown Fig. 7. This approach has been used by several previousesearchers (Azzopardi, 1986; Chu, 1973; Webb, 1970) for calcu-ation of disturbance wave frequency. Azzopardi (1986) noted thathis approach is the most objective approach available for the mea-urement of disturbance wave frequency. Since disturbance wavesre very regular under high gas velocity conditions, the peak in thepectrum becomes more sharp and thin with the increase in gaselocity (see Fig. 8).
Disturbance wave velocity is calculated using the plot ofross-correlation function (Rxy) obtained from the film thick-ess measurements of two conductance probes. Fig. 9 shows theross-correlation functions plotted against a time lag (1 time lagepresents 1/2000 s). A time lag corresponding to the peak in theross-correlation function represents the average time required foristurbance wave to travel from the first probe to the second probe.ince the distance between the two probes is known, the aver-ge disturbance wave velocity can be calculated. This method ofelocity measurement is objective and it has been used by sev-
Fig. 9. Cross-correlation function (Ref 950, P 1.2 bar).
3 ing an
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mel
4
4
i
odwloitlisomfiT1tIWnsce
534 P. Sawant et al. / Nuclear Engineer
he peak becomes narrower and taller indicating more uniformave velocities.
The accurate measurement of disturbance wave amplitude isot possible using the current conductance probe design. Sincehe probe has finite length in axial direction, the measured filmhickness is an average over the probe width and the measuredave heights are lower than the actual wave heights (Hall Taylor,
966; Martin, 1983). However, this limitation should not affect thetatistical characteristics of disturbance waves such as frequency,elocity and wavelength. Thus the amplitude measurements pre-ented in this study do not represent the absolute amplitude andhould be used only to study the trends in variation of disturbanceave amplitude with the change in boundary conditions. Distur-ance wave wavelength is defined as the ratio of wave velocity andrequency. It is calculated from the above estimated disturbanceave frequency and velocity.
All the above calculations have been performed using com-ercially available software, MATLAB. In the following section,
xperimental data on disturbance wave velocity, amplitude, wave-ength and frequency are analyzed in detail.
. Results and discussion
.1. Disturbance wave velocity
Analysis of the experimental data on disturbance wave veloc-ty (Vw) is shown in Fig. 10. In order to investigate the effects
nsant
Fig. 10. Disturbance wave velocity: effects of pr
d Design 238 (2008) 3528–3541
f gas velocity (〈jg〉) and liquid phase Reynolds number (Ref) onisturbance wave velocity, in Fig. 10(a) the measured disturbanceave velocity is plotted against the superficial gas velocity with
iquid phase Reynolds number used as a parameter. It can bebserved from the figure that the wave velocity increases with thencrease in gas velocity and liquid phase Reynolds number. Also,he wave velocity shows linear dependence on gas velocity underow gas velocity conditions. However, as the gas velocity increasest appears that the wave velocity is asymptotically approachingome constant value at high gas velocity. Furthermore, the effectf pressure on disturbance wave velocity is shown in Fig. 10(b). Theeasured disturbance wave velocity is plotted against the super-
cial gas velocity with test section pressure used as a parameter.he data corresponding to liquid phase Reynolds number 3100 and.2, 4.0 and 5.8 bar are shown in the figure. The figure shows thathe wave velocity increases with the increase in system pressure.n Fig. 10(c), the same data shown in Fig. 10(b) are plotted against
eber number (Eq. (3)). Ishii and Mishima (1989) used this Weberumber to correlate air–water entrainment fraction data. A den-ity ratio term in the Weber number definition accounts for thehange in inertia of gas phase due to the increase of pressure andntrained droplet concentration. As observed in Fig. 10(c), Weber
umber successfully collapses the data corresponding to all pres-ure conditions. Also it can be observed that the wave velocitysymptotically approaches some constant value at higher Weberumber conditions. Finally, all the wave velocity data measured inhis study are plotted against Weber number in Fig. 10(d) with liq-essure, liquid flow rate and gas flow rate.
ing an
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dtardddcib
V
w
C
we
m
C
FecPol
V
wev
u
w
P. Sawant et al. / Nuclear Engineer
id phase Reynolds number used as a parameter. It can be seenhat just two non-dimensional numbers, Weber number and liq-id phase Reynolds number completely describe the dependencef disturbance wave velocity on pressure, gas flow rate and liquidow rate boundary conditions.
Several correlations have been developed for the prediction ofisturbance wave velocity. Azzopardi (1997) reviewed the correla-ions proposed by Pearce (1979), Sawai et al. (1989) and Paras etl. (1994). He found that Pearce (1979) and Sawai et al. (1989) cor-elations failed to correlate the existing disturbance wave velocityatabase. Also he noted that Paras (1979) correlation, which waseveloped for the horizontal annular flow, could not predict theisturbance wave velocity in vertical annular flow data. Anotherorrelation developed for the prediction of disturbance wave veloc-ty by Kumar et al. (2002) using their experimental data is shownelow,
w = Cjg + jf1 + C
(4)
here C is correlated by the following equation;
√
= �g�f
Cfi |gCfi |f
(5)
here Cfi is interfacial friction factor. They obtained followingmpirical correlation for the parameter C based on their experi-
nKidt
Fig. 11. Comparison of disturbance w
d Design 238 (2008) 3528–3541 3535
ental data obtained in rectangular test section.
= a(
�g
�f
)1/2(
Ref
Reg
)1/4
(6)
or the constant a they assigned value of 5.5 based on thexperimental data. Recently, Omebere-Iyari and Azzopardi (2007)orrelated the existing database on disturbance wave velocity usingearce (1979) correlation and evaluated the diameter dependencef an adjustable parameter in the correlation. Pearce (1979) corre-ation is shown in Eq. (7).
w = Kulf + 〈jg〉√
�g/�f
K +√
�g/�f
(7)
here ulf and K are time averaged liquid film velocity and diam-ter dependent parameter, respectively. Time averaged liquid filmelocity is obtained form the following correlation;
¯ lf = Dt × Glf
4 × �f × ı(8)
here Glf and ı are mass flux in liquid film and liquid film thick-
ess, respectively. Omebere-Iyari and Azzopardi (2007) found thatincreases from 0.51 to 0.9 when the test section diameter isncreased from 5 to 25 mm and it remains constant for the largeriameters. They proposed that this dependence of K on pipe diame-er is due to the effect of pipe diameter on coherence of disturbance
ave velocity with correlations.
3 ing an
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Ktava(ct(dAtclFplUdwv(ml
paeaPK(ddbopu
4
wwsoi
536 P. Sawant et al. / Nuclear Engineer
aves. Disturbance waves become less coherent as the pipe diam-ter increases.
The current disturbance wave velocity data is compared withumar et al. and Pearce’s correlations in Fig. 11. The predicted dis-urbance wave velocities by Kumar et al. and Pearce’s correlationsre plotted against the experimentally measured disturbance waveelocities in Fig. 11(a and b), respectively. For the calculation of timeveraged liquid film velocity (ulf) in Eq. (8), the liquid film mass fluxGlf) is estimated using Sawant et al. (2008) entrainment fractionorrelation and the experimentally measured values are used forhe liquid film thickness (ı). Since Omebere-Iyari and Azzopardi2007) correlated the disturbance wave velocity data in 10 mmiameter test section using K = 0.61, this value is used in Eq. (7).s shown in Fig. 11(a), although Kumar et al. correlation satisfac-
orily collapses the experimental data correspond to all pressureonditions, the correlation under-predicts the data. Pearce corre-ation also collapses the data at different pressure conditions (seeig. 11b); however, there is more scatter in the data compared to theredictions of Kumar et al. correlation. Furthermore, Pearce corre-
ation also failed to predict the wave velocities higher than 6 m/s.nder high flow rate conditions, the experimental data shows thatisturbance wave velocity gradually approaches a constant value
hile the correlation predicts the continuously increasing waveelocities. Omebere-Iyari and Azzopardi (2007) compared Pearce1979) correlation with several experimental data. However, the
easured disturbance velocities in their experimental data wereess than 6 m/s and the data were obtained at near atmospheric
ggsal
Fig. 12. Disturbance wave amplitude: effects of p
d Design 238 (2008) 3528–3541
ressure conditions. One such a data compared by Omebere-Iyarind Azzopardi (2007) is Schadel’s (1988) data obtained in air–waterxperiments carried out in 25.4 mm diameter test section at neartmospheric pressure condition. Schadel’s data is compared withearce correlation along with the current data in Fig. 11(c) using= 0.94, a similar value proposed by Omebere-Iyari and Azzopardi
2007). It can be observed that Schadel’s data is limited to theisturbance wave velocities less than 6 m/s and the correlation pre-icts it satisfactorily. The predictions of Kumar et al. correlation cane improved significantly by adjusting the parameter a. It can bebserved from Fig. 11(d), Kumar et al. correlation gives very goodredictions of the current experimental data and Schadels’s datasing a = 9 and 4.8, respectively.
.2. Disturbance wave amplitude
Fig. 12 shows the analysis of experimental data on disturbanceave amplitude. Variations of the disturbance wave amplitudeith gas velocity, liquid phase Reynolds number and pressure are
hown in the figure. As observed from the Fig. 12(a), the amplitudef disturbance wave decreases with the increase in gas velocity andncreases with the increase in liquid phase Reynolds number. As the
as velocity increases, more and more liquid from the liquid filmets entrained into the gas core due to the increased interfacialhear stress. This results in the reduction of liquid film thicknesss well as disturbance wave amplitude. On the other hand as theiquid phase Reynolds number increases for a given gas velocity,ressure, liquid flow rate and gas flow rate.
ing an
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4
dwssowtwtvotv
e1giIilcdFdvwFttc
4
P. Sawant et al. / Nuclear Engineer
he liquid film thickness and wave amplitude increases. Similar toisturbance wave velocity, disturbance wave amplitude also showshe effect of pressure and the same Weber number successfullyollapses the wave amplitude data correspond to all pressure con-itions (see Fig. 12b and c). Finally, in Fig. 12(d) all the experimentalata on disturbance wave amplitude are plotted using Weber num-er and liquid phase Reynolds number. It can be observed that thewo non-dimensional numbers fully describes the wave amplitudexperimental data.
.3. Disturbance wave wavelength
The parametric trends observed in the experimental data onisturbance wave wavelength are shown in Fig. 13. Disturbanceave wavelength represents the average distance between the two
uccessive waves. In the current analysis it is derived from the mea-urement of disturbance wave velocity and frequency. The effectsf gas velocity and liquid phase Reynolds number on disturbanceave wavelength are presented in Fig. 13(a). It can be observed from
he figure that disturbance wave wavelength decreases sharplyith the increase in gas velocity at lower gas velocities and even-
ually it asymptotically approaches a limiting value under high gaselocity conditions. The wavelength shows very weak dependencen liquid phase Reynolds number. As the gas velocity increases, dis-urbance waves become more regular and achieve more uniformelocity. Due to uniform velocity, the waves can come very close to
obtid
Fig. 13. Disturbance wave wavelength: effects of
d Design 238 (2008) 3528–3541 3537
ach other without coalescence of the successive waves (Azzopardi,986). This results in reduction of wavelength with the increase inas velocity. A weak dependence on liquid phase Reynolds numbers visible under low liquid phase Reynolds number conditions only.t appears that the wavelength initially decreases with the increasen liquid phase Reynolds number, however eventually it approachesimiting value at higher liquid Reynolds number due to the coales-ence of successive waves (Azzopardi, 1986). Fig. 13(b) indicates theependence of disturbance wave wavelength on system pressure.or a given gas velocity, increase in pressure results in decrease ofisturbance wave wavelength. Contrary to the disturbance waveelocity and amplitude data, the effect of pressure on disturbanceave wavelength cannot be predicted using Weber number (see
ig. 13c). Finally, in Fig. 13(d) disturbance wave wavelength is plot-ed against gas phase Reynolds number (Reg). It can be observedhat, gas phase Reynolds number successfully collapses the dataorrespond to different pressure conditions.
.4. Disturbance wave frequency
Fig. 14 shows the dependence of disturbance wave frequency
n gas velocity, liquid phase Reynolds number and pressure. It cane observed from Fig. 14(a) that the wave frequency increases withhe increase in gas velocity and liquid phase Reynolds number. Alsot appears that, at the lowest liquid phase Reynolds number con-ition, the wave frequency asymptotically approaches a constantpressure, liquid flow rate and gas flow rate.
3538 P. Sawant et al. / Nuclear Engineering and Design 238 (2008) 3528–3541
ts of p
vwwoldWRfdgd
bn
S
w
R
wpasFr
observed that the non-dimensional groups proposed by Azzopardi(1997) failed to predict the dependence of disturbance wave fre-quency on pressure.
Hazuku et al. (2003) measured the frequency of disturbancewaves in air–water vertical annular flow experiments performed
Fig. 14. Disturbance wave frequency: effec
alue at higher gas velocities. The effect of pressure on disturbanceave frequency is shown in Fig. 14(b). For a given gas velocity, theave frequency increases with the increase in system pressure. Inrder to check whether Weber number defined in Eq. (3) can col-apse the disturbance wave frequency measurements correspond toifferent pressure conditions, the wave frequency is plotted againsteber number in Fig. 14(c). The data corresponding to liquid phase
eynolds number 1480 are shown in the figure. It can be observedrom the figure that Weber number failed to predict the pressureependence. Similarly it can be observed from Fig. 14(d) that evenas phase Reynolds number failed to predict the pressure depen-ence.
Azzopardi (1997) correlated disturbance wave frequency datay plotting Strouhal number (Sr) against excess liquid Reynoldsumber. Strouhal number is defined as follows;
r = fDt
〈jg〉 (9)
here f is disturbance wave frequency.Excess liquid Reynolds number is defined as;
eEXf = Ref − Relim
f (10)
here ReEXf , Ref and Relim
f are excess liquid Reynolds number, liquid
hase Reynolds number and critical liquid film Reynolds numbert the inception of disturbance waves, respectively. This approachuccessfully collapsed the low-pressure data available in literature.ig. 15 shows the comparison of this correlation with the cur-ent data obtained at three different pressure conditions. It can beFt
ressure, liquid flow rate and gas flow rate.
ig. 15. Comparison of disturbance wave frequency with Azzopardi (1997) correla-ion.
P. Sawant et al. / Nuclear Engineering and Design 238 (2008) 3528–3541 3539
Ft
aTmft
S
wg
E
�
F
wdod
Fig. 17. Dependence of Strouhal number on gas phase Reynolds number and pres-sure.
ig. 16. Comparison of disturbance wave frequency with Sekoguchi (1985) correla-ion.
t atmospheric pressure condition in 11 mm diameter test section.hey used Laser Focus Displacement (LFD) meter for the measure-ent of local instantaneous film thickness. They found that the
ollowing correlation proposed by Sekoguchi et al. (1985) predictedheir disturbance wave frequency data within ±25% of deviation;
r = fDt
〈jg〉 = f1(Eo)g1(�) (11)
here f1(Eo) = Eo−0.5(0.5 ln(Eo) − 0.47) and1(�) = 0.0076 ln � − 0.051
o = gD2t (�f − �g)
�(12)
= Re2.5f
Frg(13)
rg = 〈jg〉√gDt
(14)
here Eo, Frg and � are Eötvös number, Froude number and non-imensional parameter in Eq. (13), respectively. The comparisonf this correlation with the current disturbance wave frequencyata obtained at three pressure conditions is shown in Fig. 16.
Ad
o
Fig. 19. Comparison of disturbance wave frequ
Fig. 18. Dependence of Strouhal number on liquid phase Reynolds number.
lthough the correlation successfully predicted the low-pressureata, it failed to predict the high-pressure data.
Since the existing correlations failed to predict the dependencef disturbance wave frequency on pressure, a new correlation is
ency with the new correlation Eq. (16).
3 ing an
pt(pdtditwaRi
S
Ee
S
Ttt±iWadfm
5
tatwcgdnwaipraewtfdFbdt
A
Ca
R
A
A
B
C
C
C
C
F
H
H
H
H
H
I
I
I
K
K
MM
M
M
O
P
P
S
S
S
S
S
540 P. Sawant et al. / Nuclear Engineer
roposed in this study based on the current data. The experimen-al Strouhal number is plotted against gas phase Reynolds numberReg) in Fig. 17. The figure shows that Strouhal number is inde-endent of gas phase Reynolds number. The figure also shows theependence of Strouhal number on pressure. It can be observedhat Strouhal number increases with the increase in pressure. Theependence of Strouhal number on liquid phase Reynolds number
s shown in the Fig. 18 where the data correspond 1.2 bar are plot-ed against gas phase Reynolds number. Strouhal number increasesith the increase in liquid phase Reynolds number. Since the above
nalysis revealed that Strouhal number is function of liquid phaseeynolds number and pressure, the following form of correlation
s proposed,
r = A(Ref)B ×
(�f
�g
)n
(15)
mpirical constants A, B and n are obtained based on the currentxperimental data using regression analysis,
r = 0.086(Ref)0.27 ×
(�f
�g
)−0.64
(16)
he above correlation is compared with the current experimen-al data in Fig. 19(a). The newly developed correlation predictedhe experimental data correspond to all pressure conditions within25% deviation. The correlation is also compared with the exper-
mental data of Schadel (1988) and Willetts (1987) in Fig. 19(b).illetts carried out air–water and helium–water experiments
t 1.5 bar in 10 mm diameter test section. Their helium–waterata represents the effect of density ratio on disturbance waverequency. The new correlation satisfactorily predicted the experi-
ental data.
. Conclusion
In depth understanding of various disturbance wave proper-ies is essential for the development of advanced correlations innnular two-phase flow. In view of this, comprehensive experimen-al data on measurements disturbance wave velocity, amplitude,avelength and frequency have been obtained. The experiments
overed three pressure conditions and wide range of liquid andas velocities. The analysis of the data revealed that only two non-imensional numbers, Weber number and liquid phase Reynoldsumber, can satisfactorily predict the dependence of disturbanceave velocity and amplitude on pressure, gas phase flow rate
nd liquid phase flow rate. Disturbance wave wavelength is sat-sfactorily described by liquid phase Reynolds number and gashase Reynolds number and disturbance wave frequency is cor-elated using Strouhal number, liquid phase Reynolds numbernd density ratio which accounts the effect of pressure. Thexisting correlations available for the prediction of disturbanceave velocity only partially predicted the current experimen-
al data. Furthermore, the correlations available in literatureor the prediction of disturbance wave frequency failed to pre-ict the effect of pressure observed in the experimental data.inally, a new correlation is developed for the estimation distur-ance wave frequency. The new correlation predicted the currentata and the data available in literature within ±25% of devia-ion.
cknowledgement
The research project was supported by Tokyo Electrical Powerompany (TEPCO). The authors would like to express their sincereppreciation for support from the TEPCO.
S
d Design 238 (2008) 3528–3541
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