Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 431–438

Experimental study on boiling flow of liquid nitrogen in inclined tubes—Velocitiesand length distributions of Taylor bubbles

Hua Zhang, Shuhua Wang, Jing Wang *

Institute of Engineering Thermo-physics, School of Mechanical and Power Engineering, Shanghai Jiaotong University, Shanghai 200240, China

A R T I C L E I N F O

Article history:

Received 21 July 2008

Received in revised form 16 December 2008

Accepted 18 December 2008

Keywords:

Nitrogen Taylor bubble

Velocity and length distribution

Statistical method

Inclined tube

A B S T R A C T

An experimental study was carried out to understand the phenomena of the boiling flow of liquid

nitrogen in inclined tubes with closed bottom by using the high-speed motion analyzer. The tubes in the

experiment are 0.014 and 0.018 m in inner diameter separately and both 1.0 m long. The range of the

inclination angles is 0–458 from the vertical. The statistical method is employed to analyze the

experimental data. The experiment focused on the effect of the inclination angles on the velocities and

the length distributions of the nitrogen Taylor bubbles at different positions along the tubes. The trend of

velocity along the tubes is related with u. At the same position of the tubes, the velocity of the Taylor

bubbles increases first, and then decreases with the increase of u. And the mean Taylor bubble length

increases with the increasing x/D along the tubes at various u. At the same position of the tubes, the mean

Taylor bubble length increases first, and then decreases with increasing u, maximum at 308. In vertical

tubes, standard deviations of the nitrogen Taylor bubble lengths increase with the increasing x/D. For

inclined tubes, standard deviations of the nitrogen Taylor bubble lengths increase first, and then

decrease with the increasing x/D.

� 2008 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Contents lists available at ScienceDirect

Journal of the Taiwan Institute of Chemical Engineers

journa l homepage: www.e lsev ier .com/ locate / j t i ce

1. Introduction

Gas–liquid slug flow is highly complex with an inherentunsteady behavior. It is characterized by long bullet-shapedbubbles separated by liquid slugs that may be aerated by smalldispersed bubbles. Slug flow is found in many industrialapplications, such as heat exchangers, emergency core cooling,transport and handling of cryogenic fluids.

With the development of current aerospace technology, cryo-genic propellants are increasingly used in the rocket industry.Geysering is an important problem in long cryogenic propellantfeeding tube which is connecting the launch vehicle propellant tankand the rocket engines (Hands, 1988). It can be described as theunstable expulsion of a boiling liquid and its vapor from a tube.When the cryogenic fluid in the feeding tube is heated with heat leak,the bubbles are formed at the wall of the tube, and eventually thebubbles coalesce into a Taylor bubble, which fills the cross-section ofthe tube. As the Taylor bubble rises in the tube, it expels the liquidfrom the line into the tank ahead of the tube; cold liquid at thebottom of the tank then rushes into the empty line propelled notonly by gravity, but also by condensation of the vapor in the tube.This column of liquid impacts a closed valve or other obstructions at

* Corresponding author. Tel.: +86 21 34206089.

E-mail address: wangj@sjtu.edu.cn (J. Wang).

1876-1070/$ – see front matter � 2008 Taiwan Institute of Chemical Engineers. Publis

doi:10.1016/j.jtice.2008.12.004

the bottom of the tube with a sufficiently high velocity creating apotentially destructive water hammer with surge pressure, which insome cases will be large enough to damage equipment.

These bring new challenges on the application of the multi-phase flow theory in cryogenic engineering. The identification offlow patterns in vertical and inclined conveying tube is animportant issue in many research fields of cryogenic two-phaseflow. Many researches address the Taylor bubble lengths and theTaylor bubble’s translation velocity which used normal atmo-spheric temperature liquid such as air–kerosene and air–water,and most of that studies were carried out mainly for horizontal orslightly inclined slug flow and for vertical flow (Andreussi andBendiksen, 1993; Brill et al., 1981; Cook and Behnia, 2000; Griffithand Wallis, 1961; Hasanein et al., 1996; Hout et al., 1992, 2001,2003; Mao and Dukler, 1989; Nicholson and Aziz, 1978; Nydalet al., 1992). The Taylor bubble length distribution can be describedby positively skewed distributions, such as the log-normal, thegamma, or the inverse Gaussian (Brill et al., 1981; Hout et al., 2001,2003; Nydal et al., 1992).

The mean Taylor bubble lengths have a minimum at about 308from the vertical and extend to larger values for the small tube forthe inclined tubes (Hout et al., 2003). For the vertical tubes, the meanTaylor bubble lengths vary between 3 and 48D at various flowconditions and positions (Hout et al., 1992; Mao and Dukler, 1989).

The bubble rise in stagnant fluids has been studied extensively,and numerous publications can be found in the technical literature.

hed by Elsevier B.V. All rights reserved.

Nomenclature

D the inner diameter of the tube (m)

f the frame rate (f/s)

g the gravitational constant (m2/s)

LB the length of Taylor bubble (m)

LBmean the mean length of Taylor bubble (m)

n the frame difference between two frames

n1, n2 the frame number of picture

S.D. the abbreviation of standard deviation

Dt the time interval between two consecutive frames

(s)

Ud the drift velocity of a single Taylor bubble (m/s)

UNTB the rising velocity of a single Taylor bubble (m/s)

x the position along the tube (m)

x the arithmetic mean (m)

x1 the bubble’s nose position of frame n1

x2 the bubble’s nose position of frame n2

Greek symbols

u the inclination angle (8)

l the mean logarithmic of x/D

j the logarithmic standard deviation of x/D

Dr the density difference between liquid and gas

phases (kg/m3)

s the surface tension (N/m)

Fig. 1. Experimental apparatus. 1: Power supply. 2: Nitrogen Dewar. 3: Electric

heating rod. 4: Power cord. 5: Liquid nitrogen delivery tube. 6: Ball valve. 7: Flexible

tube. 8: Upper tank. 9: Vacuum valve. 10: Test section. 11: Vacuum tube. 12:

Vacuum pump.

H. Zhang et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 431–438432

In an early study, Gibson (1913) found that bubble rise depends onthe size of the bubble and its size is relative to that of the tube. Barr(1926) investigated the effect of bubble size, tube size, andviscosity on bubble velocity. Dumitrescu (1943) proposed acorrelation for the calculation of the drift velocities Ud of a singleTaylor bubble in stagnant liquid, when the surface tensionparameter S4s/DrgD2 < 0.001,

Ud ¼ 0:35ffiffiffiffiffiffigD

p(1)

Davies and Taylor (1950) provided a theoretical foundation forthe study of bubbles in vertical tubes. By assuming potential flow,they concluded that the Froude number (Fr) reaches a constantvalue in a vertical tube. Other important studies of bubble rise invertical tubes have been conducted by Bretherton (1961), Gold-smith and Mason (1962), White and Beardmore (1962), and Brown(1965), among others.

Bubble motion in inclined tubes has been studied by severalauthors. White and Beardmore (1962) noted the influence of theangle of inclination on bubble rise velocity. Zukoski (1966) studiedthe influence of u as well as the effects of viscosity and surfacetension on the rise velocity. Bubble motion in inclined tubes(including vertical and horizontal) has also been studied by severalother researchers including Maneri and Zuber (1974), Bendiksen(1984), Weber et al. (1986), Crouet and Stumolo (1987), Alves et al.

(1993), and Hout et al. (2001, 2003). All of these authors found thatthe bubble velocity first increases and then decreases as the angleof inclination increasing.

Cryogenic vapor–liquid slug flow is seldom studied ininclined tube with closed bottom. Compared with normalatmospheric temperature liquid, cryogenic liquid has lowdensity difference between vapor and liquid and low latentheat of vaporization. There are large differences on bubblemotion in cryogenic two-phase flow and normal temperaturetwo-phase flow.

The purpose of the present study is to investigate experimen-tally the velocity and the length distributions of the nitrogen Taylorbubble in inclined tubes with closed bottom. The liquid nitrogen isused as working medium.

2. Description of the experiment

During the experiment, the range of inclination angles is 0–458from the vertical. The positions of 20D–60D from the bottom oftube with inner diameter 0.014 m and the positions of 20D–55D

from the bottom of tube with inner diameter 0.018 m areseparately measured with high-speed motion analyzer.

2.1. Experimental apparatus

Fig. 1 shows the schematic diagram of the experimentalapparatus. The apparatus consists of a liquid nitrogen Dewar, testpart and the vacuum pump. The test part is made of double layerPyrex glass. And it includes an upper tank and the cryogenicvertical conveying tube (testing section). The upper tank is 0.4 mlong with inner diameter of 0.1 m. The experimental tubes are1.0 m long with inner diameters of 0.014 and 0.018 m. The test partcan be rotated around its axis and fixed at 0–458 inclination anglesfrom the vertical. The vacuum interlayer is 0.021 m, thick, which isvacuumized by vacuum pump to serve as the thermal insulation todecrease the convection heat transfer. The degree of vacuum invacuum interlayer is 6 � 10�2 Pa. The test part as a whole, only hasthe upper end pipeline connected, so the heat leakage caused bythe heat conduction is very small. The main heat leakage on thepipeline is the radiation heat transfer. Through the experiment, theroom temperature is 278 K, so the heat flux of leak heat inexperiment is about 300 W/m2. And vapor generation rate is about0.089 � 10�3 kg/s.

The liquid nitrogen is stored in a Dewar. In order to pump liquidnitrogen to the test section by pressurizing the Dewar withnitrogen vapor, the liquid nitrogen is heated by the electric heatingrod. The liquid level of the upper tank is controlled between 1.16and 1.18 m from the bottom of tubes. The heating is controlled by apower switch. The power switch is off when the liquid level of theupper tank is about 1.18 m high, and is on when the liquid level ofthe upper tank is about 1.16 m.

Fig. 2. The schematic sketch of image processing system. 1: Computer. 2: High-

speed motion analyzer. 3: Light. 4: Screen. 5: Taylor bubble. 6: Vacuum interlayer.

Fig. 3. Determination of the bubble velocity with processed images. Example for

D = 0.018 m and u = 458, frame numbers 10 and 40. Frame rate = 1000 fps.

H. Zhang et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 431–438 433

2.2. Image processing system

Fig. 2 is the schematic design of the image processing system.The digital camera photography system is composed of the high-speed digital camera, the monitor, the light source and thecomputer. The high-speed digital camera (REDLAKE Motion-Pro1X3, 1280 � 1024 pixels resolution, 1000 frames/s with thefull resolution) is employed in the experiment, together with a lens(AI NIKKOR 50/F1.2S). In the experiment, 512 � 512 pixelsresolution was used with 1000 frames/s. The recorded imagesare transmitted to the computer for further analysis. Twophotoflood lamps directly irradiated the screen, and the reflectedlight is irradiated the test tube as light source. And the power ofphotoflood lamp is 1000 w.

2.3. Image processing

In the experiments, more than 90,000 images are obtained at aposition along experimental tubes, and the size of ensembles ofTaylor bubbles that served for extracting all quantitativeinformation presented is about 150–300 at each measuringposition along the tube.

The local propagation bubble velocity of the bubble interface iscalculated as the shift of the corresponding interface, divided bythe time elapsed between the frames:

UNTB ¼x2 � x1

nDt(2)

where n = n2 � n1 (in Fig. 3), Dt = 1/f.Before the calculating method of Taylor bubbles velocity is

determined, the velocities of the nose of Taylor bubble and thebottom of the Taylor bubble are calculated, and the arithmeticmean of velocities of the nose of Taylor bubble and the bottom ofthe Taylor bubble is also calculated. These three velocities arealmost same, and the difference among them is less than 3%, so inthis paper the velocity of the nose of Taylor bubble is used as thevelocity of Taylor bubble.

The length of Taylor bubbles is obtained through two ways,when the images are processed. First, when the nose of Taylorbubble and the bottom of Taylor bubble reach the same position,the length of Taylor bubble could be obtained through calculatingthe number of pixels. And the arithmetic mean of two lengths iscalculated as the length of Taylor bubble at the position in the tube.

Second, the length of Taylor bubble is determined by multiplyingthe residence time of the vapor bubble over the measured positionby the bubble nose velocity. And the kind of length is a littlesmaller than the first kind of length, but the difference betweenthem is less than 5%, so the second method is taken to calculate thelength of Taylor bubbles.

3. Results

3.1. Nitrogen Taylor bubble video images along the tube at various

inclination angles

Fig. 4 shows the evolution of the nitrogen Taylor bubble alongthe tube at various u. Example for D = 0.018 m. The Taylor bubblesin Fig. 4 were not the same one, but the nitrogen Taylor bubbles inFig. 4 could show a greater difference with the change of u along thetube, that is to say, the whole nitrogen Taylor bubbles are near thecenter of tube at 08, 108 and 208, while near the tube upper wall at308 and 458. It’s apparently in Fig. 4 that Taylor bubbles lengthincreases along the tube. The possible reason is that after Taylorbubble formation, interaction effects among the Taylor bubbles arestrong due to the separation distance is not large enough betweenthe Taylor bubbles. The Taylor bubbles will coalesce and eventuallyform much longer slug bubble. Therefore, the length of the Taylorbubble increases along the tube.

3.2. The Taylor bubble velocities along the tubes at various u

The measurement rising velocities of the nitrogen Taylorbubbles as a function of position along the tubes with innerdiameter of 0.014 and 0.018 m at various u are presented in Figs. 5and 6 separately. To facilitate comparison of data obtained indifferent experiments, the velocities were normalized by thevelocities calculated by Eq. (1). Figs. 5 and 6 indicate that near thetubes exit, the Taylor bubble rise velocity is almost constant. InFigs. 5 and 6, when u is less than 458, the Taylor bubble risevelocities near the tubes exit are almost the same in each tube andthe normalized values are about 3.15 and 2.80 separately. While uis equal to 458, the value is about 2.37 in two tubes. Figs. 5 and 6also show that when u is 08, 108 or 208, the velocity increases alongthe tubes. When u is 308, the velocity increases first, and thendecreases along the tubes, and the velocity could reach a maximumat the position 40D along the tubes. While u is 458, the velocitydecreases along the tubes. In Figs. 5 and 6, when u is 08, 108, 208 or308, the results are different with the results of undeveloped air–water slug flow (Aladjem et al., 2000; Hout et al., 2001, 2003). Thepossible reasons for this could be attributed to the undevelopedslug flow and the presence of dispersed bubbles in the liquid slugregion as well as liquid from upper tank refilling the tube when theliquid slug above the region of the nitrogen Taylor bubble is forced

Fig. 4. Nitrogen Taylor bubble images along the tube at various inclination angles. Example for D = 0.018 m.

Table 1Standard deviation of the nitrogen Taylor bubbles rising velocity in inclined tubes.

x/D S.D. (m/s) (D = 0.014 m) S.D. (m/s) (D = 0.018 m)

08 108 208 308 458 508 08 108 208 308 458 508

20 0.06 0.07 0.1 0.1 0.12 0.08 0.07 0.08 0.09 0.12 0.11 0.08

30 0.08 0.1 0.11 0.11 0.12 0.12 0.06 0.09 0.06 0.12 0.09 0.12

40 0.11 0.11 0.09 0.12 0.11 0.13 0.08 0.09 0.07 0.09 0.08 0.1

50 0.06 0.11 0.11 0.13 0.13 0.11 0.08 0.07 0.06 0.07 0.08 0.09

55 0.09 0.08 0.08 0.1 0.1 0.11

60 0.1 0.09 0.11 0.1 0.08 0.1

H. Zhang et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 431–438434

out of the tube at a speed. While when u is 458, the results aresimilar to the results of undeveloped air–water slug flow.

Fig. 5 also shows that the velocity increases first, and thendecreases with the increase of u at different position along the tubewith inner diameter of 0.014 m. Fig. 6 shows that for the tube withinner diameter of 0.018 m, except the exit of the tube, the velocityincreases first, and then decreases with the increase of u atdifferent position along the tube. Both the results are similar to theresults of inclined tubes (Alves et al., 1993; Bendiksen, 1984;Crouet and Stumolo, 1987; Maneri and Zuber, 1974; Weber et al.,1986). And at the exit of the tube with inner diameter of 0.018 m,the velocity decreases with the increase of u. And when u is 308, thevelocity reaches maximum at various positions along tubes. Theresult at exit of tube with inner diameter of 0.018 m is differentwith the results of inclined tubes. The possible reason for this could

Fig. 5. The Taylor bubble velocities along the tube with inner diameter 0.014 m at

various inclination angles.

attribute to the liquid nitrogen refilling the tube from the uppertank.

Standard deviation (S.D.) of the nitrogen Taylor bubble meanrise velocity in the inclined tubes is listed in Table 1. The standarddeviation of UNTB is less than 30%. When u is less than 308, thestandard deviation is relatively less than that of the largeinclination angles in two tubes. It indicates that the flow conditionis more stable at small inclination angles. Table 1 shows small risevelocity variability between Taylor bubbles.

3.3. Nitrogen Taylor bubbles length distributions

The histograms showing the distributions of nitrogen Taylorbubbles lengths in tubes with inner diameter of 0.014 and 0.018 mare given in Figs. 7 and 8 separately. In general, the mean and the

Fig. 6. The Taylor bubble velocities along the tube with inner diameter 0.018 m at

various inclination angles.

Fig. 7. Nitrogen Taylor bubble length distribution along the tube with inner diameter 0.014 m at various u.

H. Zhang et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 431–438 435

mode of the length distributions increase along the tubes and areright-skewed in all cases.

The effect of the inclination angle on the measured nitrogenTaylor bubbles length distributions is shown in Figs. 7 and 8 atdifferent locations along the tubes. The mean and the most modeTaylor bubbles lengths increase first, and then decrease withincreasing u, maximum at 308, and the distributions become moreinhomogeneous with increasing u.

At high x/D positions, the histograms of distribution of nitrogenTaylor bubbles lengths are somewhat flat for two pipes. This showsthe distribution range of nitrogen Taylor bubbles lengths aregreater for smaller pipe.

3.4. Nitrogen Taylor bubbles mean length and standard deviation

Fig. 9 presents the evolution of the dimensionless nitrogenTaylor bubbles mean lengths, LBmean, as a function of inclinationangle. For the tube with inner diameter of 0.014 m, the value ofLBmean is 1.5–2.2D at x/D = 20 and increases to 4.8–7.7D atx/D = 60, where x is the axial pipe distance measured from thebottom of the tube in all cases. And in the tube with diameter0.018 m, the values of LBmean are 1.4–1.8D at x/D = 20 andincrease to 3.4–4.3D at x/D = 50. In both tubes, the Taylorbubbles mean length increases first, and then decreases withdecreasing u, maximum at 308, which shows Taylor bubble is

Fig. 8. Nitrogen Taylor bubble length distribution along the tube with inner diameter 0.018 m at various u.

H. Zhang et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 431–438436

easier to coalescence with inclination angle decreasing, butcoalescence lessening at 458. Fig. 9 also shows that the meanlength of Taylor bubbles in pipe with small diameter is greaterthan those in pipe with large diameter.

Fig. 10 presents the evolution of the dimensionless nitrogenTaylor bubbles lengths standard deviations as a function ofinclination angle. S.D. is the abbreviation of standard deviation,and it is used to measure the deviation of data value from thearithmetic mean.

S:D: ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1 ðxi � xÞ2

n� 1

s(3)

where x is arithmetic mean.In vertical tubes, the nitrogen Taylor bubbles lengths standard

deviation increases with increasing x/D. For inclined tube, thenitrogen Taylor bubbles lengths standard deviation increases first,and then decreases with increasing x/D, which shows Taylor

bubbles length distributions are more homogeneous when x/D isgreater than 40D. The possible reason is that at different inclinationangle, the effect of liquid nitrogen from upper tank is different.When the tube is vertical, the refilling of liquid nitrogen wouldmake Taylor bubbles inhomogeneous, while at inclined tubes, therefilling of liquid nitrogen would help Taylor bubbles homo-geneous.

Fig. 10 also shows that the standard deviation of mean lengthsof Taylor bubbles in pipe with small diameter is greater than that inpipe with large diameter. And the standard deviation of LBmean isless than 50% for two pipes.

3.5. The log-normal distributions of nitrogen Taylor bubble lengths

Figs. 7 and 8 show the nitrogen Taylor bubble lengthdistributions in different tubes are right-skewed in all cases. Thelog-normal shape is fitted to the measured distributions and isdepicted in Figs. 7 and 8 as a solid line. The probability density

Fig. 9. Nitrogen Taylor bubble mean length along the tube at various u. (a) Nitrogen

Taylor bubble mean length for D = 0.014 m. (b) Nitrogen Taylor bubble mean length

for D = 0.018 m.

Fig. 10. Nitrogen Taylor bubble length standard deviations along the tube at various

u. (a) Nitrogen Taylor bubble length standard deviation for D = 0.014 m. (b) Nitrogen

Taylor bubble length standard deviation for D = 0.018 m.

Table 2Parameters l and j of log-normal fit.

Angles Parameters D = 0.014 m D = 0.018 m

20D 30D 40D 50D 60D 20D 30D 40D 50D

08 l 0.72 0.36 0.42 0.51 0.55 1.16 0.71 0.76 0.5

j 0.85 1.01 1.4 1.54 1.65 1.55 1.12 1.36 1.17

108 l 0.72 0.51 0.57 0.6 0.76 0.78 0.96 0.72 0.36

j 1.05 1.01 1.38 1.8 2.03 0.91 1.14 1.57 1.39

208 l 0.28 0.42 0.32 0.5 1.09 0.66 0.79 0.32 0.24

j 0.72 1.13 1.56 1.89 1.91 0.83 1.25 1.17 1.51

308 l 0.49 0.36 0.62 0.91 0.74 0.7 0.35 0.43 0.64

j 0.57 1.24 1.46 1.83 1.98 0.87 1.06 1.43 1.85

458 l 0.47 0.51 0.37 0.35 0.2 0.81 0.33 0.7 0.49

j 0.74 1.27 1.61 1.8 2.15 0.89 0.98 1.37 1.26

H. Zhang et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 431–438 437

function of the log-normal distribution is

f ðzÞ ¼ 1ffiffiffiffiffiffiffi2pp

l

x

D

� ��1

exp �1

2

lnðx=DÞ � jl

� �2" #

(4)

where x/D > 0, l > 0. The parameters l and j in Eq. (4) are given inTable 2 for all cases.

4. Conclusions

An experimental study of the evolution of continuous liquidnitrogen boiling flow in inclined tubes with internal diameters of0.014 and 0.018 m is presented. The hydrodynamic and statisticalparameters include nitrogen Taylor bubbles velocities and lengthdistributions. The measurements are carried out with high-speedmotion analyzer along the tube at various inclined angles.

H. Zhang et al. / Journal of the Taiwan Institute of Chemical Engineers 40 (2009) 431–438438

Through images analyzing, the whole nitrogen Taylor bubble isnear the center of tube at 08, 108 and 208, while near the tube upperat 308 and 458.

The effect of x/D on the velocity of Taylor bubbles is as follows.The mean velocity of nitrogen Taylor bubbles increases along thetubes when the inclination angle is 08, 108 and 208. While theinclination angle is 308, the velocity increases first, and thendecreases along the tubes; when the inclination angle is 458, thevelocity decreases along the tubes.

The effect of the inclination angles on the velocity of Taylorbubbles is as follows. At the same x/D position, the velocity of theTaylor bubbles increases first, and then decreases with uincreasing. When u is 308, the mean velocities of the Taylorbubbles reach the maximums at the positions along the tubes. Atthe exit of the tubes, the nitrogen Taylor bubble velocity reaches astable value separately at different inclination angles.

Measured length distributions are well described by the log-normal shape for all cases. The mean and the most modes Taylorbubble length increase first, and then decrease with increasing u,maximum at 308, and the distribution becomes more inhomoge-neous with increasing u.

In all cases, the mean Taylor bubble length increases first, andthen decreases with increasing u, maximum at 308, which showsTaylor bubble is easier to coalescence from vertical to inclined, butcoalescence lessening at 458.

In vertical tubes, the nitrogen Taylor bubbles length standarddeviation increases with increasing x/D. For inclined tubes, thenitrogen Taylor bubbles length standard deviation increases first,and then decreases with increasing x/D which shows the Taylorbubbles length distribution is more homogeneous when x/D isgreater than 40D.

Acknowledgement

This work was supported by China National Natural ScienceFoundation (grant no. 50476015).

References

Aladjem, T. C., L. Shemer, and D. Barnea, ‘‘On the Interaction between Two ConsecutiveElongated Bubbles in a Vertical Pipe,’’ Int. J. Multiphase Flow, 26 (12), 1905 (2000).

Alves, I. N., O. Shoham, and Y. Taitel, ‘‘Drift Velocity of Elongated Bubbles in InclinedPipes—Experimental and Modeling,’’ Chem. Eng. Sci., 48, 3063 (1993).

Andreussi, P. and K. H. Bendiksen, ‘‘Void Distribution in Slug Flow,’’ Int. J. MultiphaseFlow, 19 (5), 817 (1993).

Barr, G., ‘‘Air-bubble Viscometer,’’ Philos. Mag., 1, 395 (1926).Bendiksen, K. H., ‘‘An Experimental Investigation of the Motion of the Long Bubble in

Inclined Tubes,’’ Int. J. Multiphase Flow, 10 (4), 467 (1984).Bretherton, F. P., ‘‘The Motion of Long Bubbles in Tubes,’’ J. Fluid Mech., 10, 166 (1961).Brill, J. P., Z. Schmidt, W. A. Coberly, J. D. Herring, and D. W. Moore, ‘‘Analysis of Two-

Phase Tests in Large-Diameter Flow Lines in Prudhoe Bay Field,’’ Soc. Petrol. Eng. J.,21 (3), 363 (1981).

Brown, R. A. S., ‘‘The Mechanics of Large Gas Bubbles in Tubes—I. Bubble Velocities inStagnant Liquids,’’ Can. J. Chem. Eng., 43 (5), 217 (1965).

Cook, M. and M. Behnia, ‘‘Slug Length Prediction in Near Horizontal Gas–LiquidIntermittent Flow,’’ Chem. Eng. Sci., 55 (11), 2009 (2000).

Crouet, B. and G. S. Stumolo, ‘‘The Effects of Surface Tension and Tube Inclination on aTwo-dimensional Rising Bubble,’’ J. Fluid Mech., 184, 1 (1987).

Davies, R. M. and G. Taylor, ‘‘The Mechanics of Large Bubble Rising through ExtendedLiquids and through Liquids in Tubes,’’ Proc. R. Soc. Lond., Ser. A, 200, 375 (1950).

Dumitrescu, D. T., ‘‘Stromung an Einer Luftblase im Senkrechten Rohr,’’ Zeitschrift furAngewandte Mathematik and Mechanik, 23, 139 (1943).

Gibson, A. H., ‘‘On the Motion of Long Air Bubbles in a Vertical Tube,’’ Philos. Mag., 26–156, 952 (1913).

Goldsmith, H. L. and S. G. Mason, ‘‘The Movement of Single Large Bubbles in ClosedVertical Tubes,’’ J. Fluid Mech., 14, 42 (1962).

Griffith, P. and G. B. Wallis, ‘‘Two-Phase Flow,’’ J. Heat Transf., 83 (3), 307 (1961).Hands, B. A., ‘‘Problems Due to Superheating of Cryogenic Liquids,’’ Cryogenics, 28 (12),

823 (1988).Hasanein, H. A., G. T. Tudose, S. Wong, and M. Malik, ‘‘Slug Flow Experiments and

Computer Simulation of Slug Length Distribution in Vertical Pipes,’’ AIChE Symp.Ser., 92 (310), 211 (1996).

Maneri, C. C. and N. Zuber, ‘‘An Experimental Study of Plane Bubbles Rising atInclination,’’ Int. J. Multiphase Flow, 1 (5), 623 (1974).

Mao, Z.-S, and A. E. Dukler, ‘‘An Experimental Study of Gas–Liquid Slug Flow,’’ Exp.Fluids, 8 (2), 169 (1989).

Nicholson, M. K. and K. Aziz, ‘‘Intermittent Two Phase Flow in Horizontal Pipes:Predictive Models,’’ Can. J. Chem. Eng., 56 (12), 653 (1978).

Nydal, O. J., S. Pintus, and P. Andreussi, ‘‘Statistical Characterization of Slug Flow inHorizontal Pipes,’’ Int. J. Multiphase Flow, 18 (3), 439 (1992).

van Hout, R., L. Shemer, and D. Barnea, ‘‘Spatial Distribution of Void Fraction within theLiquid Slug and Some Other Related Slug Parameters,’’ Int. J. Multiphase Flow, 18(6), 831 (1992).

van Hout, R., D. Barnea, and L. Shemer, ‘‘Evolution of Statistical Parameters of Gas–Liquid Slug Flow along Vertical Pipes,’’ Int. J. Multiphase Flow, 27 (9), 1579 (2001).

van Hout, R., L. Shemer, and D. Barnea, ‘‘Evolution of Hydrodynamic and StatisticalParameters of Gas–Liquid Slug Flow along Inclined Pipes,’’ Chem. Eng. Sci., 58 (1),115 (2003).

Weber, M. E., A. Alarie, and M. E. Ryan, ‘‘Velocities of Extended Bubbles in InclinedTubes,’’ Chem. Eng. Sci., 41, 2235 (1986).

White, E. T. and R. H. Beardmore, ‘‘The Velocity of Rise of Single Cylindrical Air Bubblesthrough Liquids Contained in Vertical Tubes,’’ Chem. Eng. Sci., 17, 351 (1962).

Zukoski, E. E., ‘‘Influence of Viscosity, Surface Tension, and Inclination Angle on Motionon Long Bubbles in Closed Tubes,’’ J. Fluid Mech., 25, 821 (1966).