Motion of Taylor Bubbles and Taylor Drops in LiquidLiquid Systems

7048

Ind. Eng. Chem. Res. 2008, 47, 7048–7057

Motion of Taylor Bubbles and Taylor Drops in Liquid-Liquid Systems Tapas K. Mandal Department of Chemical Engineering, National Institute of Technology, Durgapur-713209, India

Gargi Das* Department of Chemical Engineering, Indian Institute of Technology, Kharagpur-721302, India

Prasanta K. Das Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur-721302, India

The present work reports an experimental study on the shape and stability of liquid Taylor bubbles and liquid Taylor drops in vertical and inclined tubes. Experiments have been performed with five liquid pairs, namely, water-kerosene, brine-kerosene, water-benzene, water-cyclohexane, and water-2-heptanone, in five different tube diameters ranging from 0.012 to 0.046 m and inclinations of 0°, 15°, 30°, 45°, 60°, and 75° with vertical. The effect of tube inclination, tube diameter, and pipe material on shape, stability, and velocity of a liquid Taylor bubbles and liquid Taylor drops has been explained qualitatively from basic physics. The existing correlations generally used for gas-liquid system have been modified to predict the rise velocity in vertical tubes. Introduction A volume of gas introduced at the bottom of a stationary liquid column rises up as a single, elongated bubble. These axisymmetric bullet-shaped bubbles occupying almost the entire cross section of the pipe are commonly known as Taylor bubbles. A photograph of such a bubble for air-water system is shown in Figure 1a. It is comprised of a spherical nose and a cylindrical tail. A wake region comprised of densely packed small bubbles follows the tail. A similar phenomenon is observed when a volume of lighter liquid is introduced in a stationary column of heavier liquid (Figure 1b). Following gas-liquid terminology, the rising drop of the lighter liquid is termed as liquid Taylor bubble. Another interesting phenomenon is observed in liquid-liquid systems. When a volume of a heavier liquid is introduced at the top of a stationary column of a lighter one, it propagates downward as a single elongated drop. Figure 1c shows that the drop resembles the shape of an inverted Taylor bubble. Accordingly, these have been referred to as liquid Taylor drops in the present study. A survey of the past literature shows that although several studies, both experimental1–11 and theoretical,12–33 have been carried out to understand the rise of Taylor bubbles in gas-liquid systems, not much is known about the hydrodynamics of liquid Taylor bubbles and drops. To the best of the authors’ knowledge, the only work is from Zukoski,1 who estimated the velocity of liquid Taylor bubbles in vertical tubes through experiments. He also discussed the influence of inclination angle on rise velocity of such bubbles. In the present study, an interest was felt to investigate the characteristics of liquid Taylor bubbles and liquid Taylor drops. Efforts have been directed to determine the influence of fluid properties, tube diameter, and tube inclination on the propagation * To whom correspondence should be addressed. Tel.: +91 3222 283952. Fax: +91 3222 255303. E-mail: [email protected].

rate, stability, and shape of a liquid Taylor bubble rising through a heavier fluid and a liquid Taylor drop falling through a lighter liquid. Experimental Details The experiments have been performed with different pairs of immiscible liquids, namely, (1) water and kerosene, (2) brine solution and kerosene, (3) water and benzene, (4) water and cyclohexane, and (5) water and 2-heptanone. The physical properties of the different liquids, namely, their viscosity, surface tension, and density, have been measured by Ostwald viscometer, stalagmometer, and specific gravity bottle, respectively. The interfacial tension between the two liquids has been measured by a ring tensiometer. The values of the properties have been listed in Table 1. For ease of reference, the liquid filling the tube is termed as the primary liquid and the liquid comprising the bubble or drop is designated as the secondary liquid for both liquid Taylor bubbles and liquid Taylor drops. The experiments have been conducted in borosilicate glass tubes of five different diameterss0.012, 0.0176, 0.0257, 0.0358, and 0.0461 m. A few experiments have been repeated in Perspex tubes of the same diameters to note the effect of pipe material on the rise velocity and shape of bubbles and drops. The average diameter of each tube has been checked from the volume of water required to fill a known length of the tube. The measurements have been repeated several times for an accurate prediction. It has been noted that the surface properties of the tube material play an important role in the propagation of bubbles and drops. Any trace of impurity can alter the hydrodynamics of flow. So special care has been taken to remove contaminants from the tube walls prior to each experimental run. For this, the tubes have been washed with soapy water and distilled water. The glass tubes are then rinsed with acetone and dried. A few experiments without this exercise have also been performed to note the effect of tube cleanliness on the dynamics of bubbles and drops. A schematic diagram of the test rig is shown in Figure 2. The figure shows a 1.5-m-long glass tube G centrally pivoted

10.1021/ie8004429 CCC: $40.75  2008 American Chemical Society Published on Web 08/26/2008

Ind. Eng. Chem. Res., Vol. 47, No. 18, 2008 7049

Figure 1. Photograph of Taylor bubble and droplet in different systems. Table 1. Physical Properties of Test Liquids fluid

density (kg/m3)

viscosity (mPa · s)

interfacial tension with water (N/m)

water kerosene benzene cyclohexane 2-heptanone brine solution

1000 787 879 775 810 1200

1 1.2 0.73 0.96 0.71 1.75

0.0385 0.0356 0.0585 0.0208 0.0378 (with kerosene)

to the frame F to facilitate free rotation. The tube is closed at both ends. Near the ends, two small side tubes N1 and N2 are provided with valve connections. They are used for filling and emptying the tube with the test liquids. For the experiments, the tubes are initially filled with the primary liquid. A measured volume of the secondary liquid is then injected through the side tube N2 keeping G at a slightly inclined position from the vertical. During this process, an identical amount of primary liquid is discharged through tube N1. Both the valves are closed once a required amount of

secondary liquid is introduced in the tube. Extreme care is taken to ensure that no air bubble gets entrapped inside the tube during the process of filling. It has been observed that small air bubbles get attached to the tip of the liquid Taylor bubble and render the tip pointed. This increases the rise velocity substantially. The tube is positioned with N2 on the top when bubble rise has been studied and vice versa for falling of the drop. After allowing the secondary liquid to settle, the tube is inverted and the movement of the secondary liquid is observed through the transparent test section. The propagation of the secondary liquid is estimated from the time of translation of the bubbles/drops between two points P1 and P2 0.5 m apart from one another. Each velocity reading has been carefully repeated five times in clean tubes to check the reproducibility of data. Uncertainty in determination of rise velocity of the Taylor bubble or droplet has been calculated and it lies within (0.5%. The experiments have been performed for different inclinations ranging from 0° to 75° with respect to the vertical alignment. The inclinations are noted from the graduations in the frame F, and the tube is supported by clamp C at any desired inclination. The shape of the bubbles and drops are noted from visualization and photographic techniques. A digital camera (Sony, DSC F717) was used for photography. Results and Discussion

Figure 2. Schematic diagram of the experimental facility.

The results obtained from the experiments are reported in the following sections. Initially, the shape and stability of liquid Taylor bubbles and liquid Taylor drops as observed in vertical tubes have been discussed. Next, the influence of inclination has been detailed. The experimental results and a suitable empirical relationship to predict the velocity of propagation have been reported in the subsequent sections of the paper. Liquid Taylor Bubble in a Vertical Tube. The Taylor bubble consists of a nose and a tail as shown in Figure 3. The tail is followed by a wake region, which is often distinguishable from the other regions of the primary fluid due to the presence of satellite bubbles shed by the liquid Taylor bubble. Figure 3a indicates a sketch of a liquid Taylor bubble, and Figure 3b represents the photograph of a typical bubble recorded during experiment. The figure shows that the bubble is axisymmetric

7050 Ind. Eng. Chem. Res., Vol. 47, No. 18, 2008

Figure 3. Shape of a typical liquid Taylor bubble in vertical tube. (a) Sketch of a Taylor bubble. (b) Photograph of a Taylor bubble.

with a spherical nose, a cylindrical body, and a rather flat tail. The film thickness of the primary liquid between the wall and bubble surface gradually decreases in the nose region and attains a uniform value toward the tail. Moreover, there is an increase in interfacial waviness with increase in length of the bubble. Due to the waviness, ripples have been observed at the tail region of the bubble. Such ripples are absent in the nose region. Wakes generated by the bubble are present behind the tail as has been reported for Taylor bubbles in gas-liquid systems. This wake region contains densely packed small bubbles of the secondary liquid. They are generated by the shearing action of the interfacial waves at the lower portion of the liquid Taylor bubble. The dynamics of the formation of these bubbles from the edge of the liquid Taylor bubble and their merger with the central region of the tail have been described elsewhere.34–36 The length of this region increases with tube diameter as is evident from Figure 4. Almost no wake region is formed in a 0.012-m tube whereas intense wake formation leads to smaller Taylor bubbles in a 0.0461 m tube. Interestingly, stable liquid Taylor bubbles are not formed in glass tubes larger that 0.0461 m diameter. In this case, the long bubble breaks into a large number of smaller bubbles, which form a zone of dispersed phase. Properties like viscosity of the secondary liquid and interfacial tension also play an important role in the formation of this bubble-rich wake zone. The tendency of wake formation has been observed to decrease with increasing viscosity of the secondary liquid. This is evident from a comparison of the photographs presented in Figures 4 and 5. Figure 5 shows the negligible wake region behind a lubricating oil (density, 920 kg/m3; viscosity, 210 mPa · s) Taylor bubble rising through water while Figure 4b shows the intense wake behind a kerosene Taylor bubble rising through the same water column. Bubbles

Figure 4. Wake region of a kerosene Taylor bubble in different diameter vertical tubes.

are also more stable for higher interfacial tension. Due to this, 2-heptanone bubbles rising in water are less stable as compared to kerosene or benzene bubble. In order to understand the influence of pipe material on the stability of liquid Taylor bubbles, some experiments have been performed in Perspex tubes of similar diameters. Perspex is known to be hydrophobic while glass is hydrophilic in nature. It has been observed that stable kerosene Taylor bubbles are not formed in Perspex tubes of diameter 0.012 m and lower. Some preliminary studies in tube diameters of 0.003 and 0.006 m have indicated that kerosene sticks to the wall and does not


Figure 5. Taylor bubble of a high viscous liquid (Lube oil ASE 40) in a water column of 0.0461-m i.d.

Figure 6. Photographs during the rise of a kerosene Taylor bubble through water in a 0.012-m-i.d. Perspex tube.

rise as elongated bubbles. An interesting phenomenon occurs in these cases. This has been shown by means of photographs (Figure 6) for kerosene-water pair in a 0.012-m tube. The photographs indicate that a bubble is formed initially. After a few seconds, kerosene wets the inner surface of the Perspex tube and the nose of the bubble becomes flat. As a result, the

kerosene mass appears to stick to the pipe wall and does not move at all. The effect decreases with increasing tube diameter and a well-defined Taylor bubble has been observed in a Perspex tube of 0.0257-m i.d. This can be attributed to the decreasing influence of wettability in larger pipe diameters. Liquid Taylor Drops in a Vertical Tube. Although the liquid Taylor drops are similar in shape to the corresponding bubbles, there are certain points of difference. The surface of a liquid Taylor drop has been observed to be smoother as compared to that of a liquid Taylor bubble. Moreover, although the liquid film around the tail of a liquid Taylor drop is usually symmetric, it does lose its symmetry at times. This can be attributed to the hydrophilic nature of the glass wall. The water drop in its attempt to climb along the inner wall of the glass tube shifts to one side and most of the primary liquid flows through the other side. This makes the liquid Taylor drop asymmetric with respect to the axis of the tube. A sketch and photograph of this phenomenon is shown in Figure 7a and b, respectively. The phenomenon of channeling is more pronounced for smaller diameter tubes, probably due to the increased effect of surface properties in these cases. As a result, this phenomenon has been noted in the 0.012 m but not in the 0.0257-m-i.d. glass tube. It may be noted that similar phenomena have also been observed for liquid Taylor bubbles in a dirty tube when the wall becomes hydrophobic. However, despite the smooth interface, the liquid Taylor drop is less stable as compared to a liquid Taylor bubble. It breaks into small drops in a 0.0257-m-diameter tube, and no welldefined liquid Taylor drop has been observed in larger tubes. This probably arises because a liquid Taylor drop moves faster than a liquid Taylor bubble, thus resulting in a higher interfacial shear, which tries to break the long water drop into smaller ones. Influence of Inclination. The shape of a Taylor bubble as well as a Taylor drop becomes distorted in an inclined tube. As is evident from Figure 8, the distortion arises because most of the heavier liquid tries to move down the lower side of the tube through the crescent-shaped passage between the bubble and the tube wall. As a result, the bubble is no longer axisymmetric. It becomes semicylindrical, with a straight upper portion and a wavy lower interface. The amplitude of the waves is much higher as compared to that in a vertical tube, and it is maximum at an inclination of 50°-60° from the vertical. Probably, this is due to the maximum velocity of the bubble observed at this range of inclination. The liquid film thickness increases gradually with tube diameter and inclination and is maximum at an inclination angle of 75° from the vertical. The nose also appears to be more pointed at higher tube inclinations. Propagation of Liquid Taylor Bubbles and Liquid Taylor Drops in Vertical Tubes. Experiments were next performed to note the velocity of propagation of liquid Taylor bubbles and liquid Taylor drops through stationary liquid columns. The velocities have been measured for different volumes of the secondary liquid as a function of liquid properties, tube diameter, and inclination. Figure 9 shows that in vertical tubes the velocity decreases asymptotically with volume for very small volumes of the secondary liquid and henceforth attains a constant value. The volume at which the velocity reaches a constant value is almost independent of the liquid pair but is higher in larger diameter pipes. A similar trend has also been obtained in Figure 9c for the fall of liquid Taylor drops through liquid filled columns. It may be noted that the forces acting on a rising bubble are buoyancy, surface tension, and viscosity. The past researchers have considered the effect of the aforementioned forces in terms


Figure 7. Channeling during motion of water Taylor drop in kerosene.

of different dimensionless groups. The most commonly used groups are Froude number (Fr), Eotvos number (Eo), and Morton number (Mo). Wallis31 has shown that the influence of Mo is negligible for Mo < 10-8. Since, in the present case, Mo has been obtained in the range of 2.07 × 10-10-3.66 × 10-11 for all the liquid pairs, attempts have been made to correlate Fr with Eo (Figure 10). The figure shows that, for all the liquid pairs, Fr increases sharply with Eo and gradually reaches an asymptotic value as expected. However, separate curves for the different liquid pairs in the figure indicate the importance of fluid properties other than those included in Eo. A close observation of the curves indicates the existence of three distinct regions, namely, (a) a surface tension dominated flow with Eo < 20, where Fr increases steeply with Eo, (b) an intermediate range, 20 e Eo e 70, where the rate of increase of Fr with Eo becomes gradual, and (c) an inertia-dominated range for Eo > 70, where Fr is almost independent of Eo. A comparison of the Froude number for upflow and downflow of the secondary liquid in Figure 11 shows that the difference in velocity is not significant for small-diameter tubes but increases gradually with tube diameter.

past researchers to predict the rise velocity of gas Taylor bubbles. The mathematical expression of the correlations and their range of applicability are as follows, Viana et al.30

Empirical Correlations To Predict the Rise Velocity of Liquid Taylor Bubbles. Considering the similarity between the rise velocity characteristics of gas and liquid Taylor bubbles, attempts have been made to extend the correlations available for gas-liquid systems to predict the rise velocity for liquid Taylor bubbles. A thorough survey of the past literature has revealed that several correlations have been proposed by the

for intermediate range of Reb (10 < Reb 200

(1)

for Eo > 70

(2)

28

Nickens and Yannitell

Fr ) 0.361(1 - 14.68 ⁄ Eo)0.5 Wallis31

Fr ) 0.345[1 - exp(-0.01Reb ⁄ 0.345)] × [1 - exp{(3.37 - Eo) ⁄ m}] for Reb > 250, m ) 10

(3)

30

Viana et al. Fr )

( (

0.34 ⁄ (1 + 3805 ⁄ Eo3.06)0.58

(

Reb 778.76 1+ 1+ 31.08 Eo1.96

) )

(

-0.49 -1.45 1 +

7.22 × 1013 Eo9.33

)

0.094

)

(

0.71 1 +

7.22 × 1013 Eo9.33

)

-0.094

(4)

Frz(Rz,Σ) ) Frz(R, Σ)f(Rz)

(5)


Figure 8. Shape of a kerosene Taylor bubble in water for different inclinations (0.0461-m-i.d. tube).

where Frz(R, Σ) ) 0.4664 + 0.3473Σ - 5.3928Σ2 + 10.532Σ3 6.7095Σ4

for Σ < 0.6

1 f(Rz) ) (1 + 44.72 ⁄ Rz1.8)0.279 Σ)

4σ FpgD2

Rz )

FpUD 2µp

In the above equations, Reb, Fr, and Eo represent buoyancy Reynolds number, Froude number, and Eotvos number, respectively. In the present study, the buoyancy Reynolds number for all the cases is greater than 200 and Figure 10 indicates that the majority of the experimental data is confined to 20 e Eo e 70. Further, a close observation of eq 5 reveals a negative Fr for large Σ (Σ > 6). Therefore, only eqs 1 and 3 are used to calculate rise velocity of liquid Taylor bubbles in vertical tubes from Froude number expressed as Fr )

U√Fp

√∆FgD

(6)

A comparison of the calculated and experimental data as listed in Table 2 reveals that both the correlations can predict rise velocity with sufficient accuracy and the predictions improve at higher Eo for a particular liquid pair. Moreover, a close observation of eqs 1 and 3 denotes that eq 1 does not have a viscosity correction factor while the first term in eq 3 does not provide any substantial improvement in prediction. Accordingly, it is felt that a further improvement can be obtained by incorporating a suitable viscosity correction factor in the equations. The viscous correction term has been obtained form the analysis proposed by Brown.33 He has considered viscous effect at the film region and potential flow at the nose. The expression of rise velocity as obtained from the analysis is given by

U ) 0.496

Fp - Fs gR Fp

1-

-1 + √1 + 4NR 2NR

(7)

where N ) [1.81(Fp-Fs)Fp/µp2g]1/3 is a dimensional property parameter and µp is the viscosity of primary fluid. It is evident from eq 7 that the term

-1 + √1 + 4NR 2NR accounts for viscous effects. This term is incorporated in eqs 1 and 3 as the viscous correction term in place of the original 1-


Figure 10. Froude number as a function of Eotvos number for different liquid pairs.

Figure 11. Froude number versus Eotvos number for a liquid Taylor bubble and a liquid Taylor drop in water-kerosene system.

Figure 9. Variation of velocity with volume of a liquid Taylor bubble in a (a) 0.012- and (b) 0.0461-m-i.d. pipe. (c) Velocity-volume curve of a liquid Taylor drop in water-kerosene system (0.012-m-i.d. tube).

correction term as proposed by Wallis.31 The final form of the correlations (after introducing the necessary correction terms) are Fr ) 0.34 ×

1-

-1 + √1 + 4NR ( ⁄ 1 + 3805 ⁄ Eo3.06)0.58 2NR (1a)

-1 + √1 + 4NR × 2NR [1 - exp{(3.37 - Eo) ⁄ 10}] (3a) The improvement in the results on incorporating each of the correction terms is shown in Figure 12. In the figure, percent error is plotted against predicted Froude number where the percent error is calculated based on the experimental value. The figures show that the modified form of eq 3 (i.e., eq 3a) gives better prediction for some of the liquid pairs, namely, water-kerosene and brine-kerosene. On the other hand, eq 3 is satisfactory for other liquid pairs. The diameter effect on the predicted results is evident from Table 2. The table shows that the modified correlations can predict rise velocity within (10% for most of the cases and the percent deviation is higher for tube diameters of 0.012 and 0.0176 m. This indicates the need for further modifications of the correlations for a correct representation of the hydrodynamics in small tubes. Fr ) 0.345 ×

1-

Influence of Inclination on the Velocity of Propagation. The rise velocity of liquid Taylor bubbles as a function of tube inclination is shown in Figure 13. The figure shows that the curves are parabolic in nature and the rise velocity attains a maximum value at an inclination angle of 50°-60° from the vertical. A similar observation has also been reported in the literature (White and Beardmore,3 Zukoski,1 Maneri and Zuber4) for gas Taylor bubbles in inclined pipes and has also been obtained for liquid Taylor drops in the present work. The reason behind this can be attributed to the opposing effects of gravity and shape factor. The inclined orientation reduces the effect of gravity as compared to the vertical orientation but renders the nose of a liquid Taylor bubble and a liquid Taylor drop more pointed. A qualitative presentation of the two effects is depicted in Figure 14. The figure shows the evolution of a parabolic profile with inclination due to the opposing effects. In order to predict the rise velocity of liquid Taylor bubbles in inclined geometries, the change in nose shape with inclination needs to be considered. Since there is no straightforward way to predict this factor, attempts have been made to propose graphical correlations to obtain velocity in an inclined pipe with respect to the vertical orientation. Considering the fact that the rise velocity in an inclined tube Uθ is a function of buoyancy Reynolds number (Reb), Eotvos number (Eo), and angle of inclination (θ) while in a vertical tube U depends on Reb and Eo, the ratio (Uθ/U) can be expressed as a function of Reb, Eo, and θ, viz Uθ f1(Reb, Eo, θ) ) U f(Reb, Eo, 0)

(8)


Figure 12. Percent error in prediction of rise velocity of a liquid Taylor bubble. (a) Percent error observed in eq 1 and eq 1a. (b) Percent error observed in eq 3 and eq 3a. Table 2. Prediction of Rise Velocity from Eqs 1, 3, 1a, and 3a predicted Fr from liq pair kerosene bubble in water

kerosene bubble in brine

benzene bubble in water

cyclohex-ane bubble in water

2-heptan-one bubble in water

% error

predicted Fr from

% error

D × 10 m

expt Fr

Eo

eq 1

eq 3

eq 1

eq 3

eq 1a

eq 3a

eq 1a

eq 3a

1.2 1.76 2.57 3.58 4.61 1.2 1.76 2.57 3.58 4.61 1.2 1.76 2.57 3.58 4.61 1.2 1.76 2.57 3.58 4.61 1.2 1.76 2.57 3.58 4.61

0.111 0.219 0.302 0.307 0.332 0.180 0.240 0.289 0.291 0.290 0.055 0.234 0.320 0.339 0.345 0.082 0.214 0.311 0.344 0.338 0.254 0.341 0.342 0.349 0.337

7.815 16.812 35.847 69.559 115.34 15.434 33.201 70.794 137.37 227.79 4.801 10.328 22.023 42.734 70.861 5.406 11.628 24.794 48.111 79.777 13.157 28.302 60.347 117.10 194.18

0.101 0.252 0.328 0.338 0.340 0.236 0.324 0.338 0.340 0.340 0.045 0.152 0.293 0.333 0.338 0.055 0.177 0.305 0.335 0.339 0.203 0.316 0.337 0.340 0.340

0.124 0.255 0.332 0.345 0.345 0.242 0.328 0.345 0.345 0.345 0.046 0.173 0.292 0.338 0.345 0.064 0.194 0.305 0.341 0.345 0.215 0.316 0.344 0.345 0.345

8.203 -14.885 -8.559 -10.024 -2.291 -31.352 -35.148 -17.152 -16.582 -17.100 16.964 35.075 8.485 1.799 1.999 32.608 17.242 1.849 2.602 -0.271 20.175 7.557 1.307 2.612 -0.779

-12.033 -16.273 -9.912 -12.059 -3.907 -34.616 -36.446 -19.306 -18.371 -18.838 15.687 26.174 8.786 0.113 0.199 22.829 9.2053 2.013 0.770 -2.036 15.357 7.281 -0.590 1.079 -2.284

0.096 0.241 0.316 0.328 0.331 0.223 0.310 0.326 0.329 0.331 0.043 0.145 0.281 0.322 0.329 0.053 0.169 0.295 0.325 0.330 0.193 0.302 0.325 0.329 0.331

0.118 0.244 0.320 0.334 0.336 0.229 0.313 0.332 0.334 0.336 0.043 0.165 0.280 0.327 0.335 0.060 0.186 0.294 0.331 0.336 0.204 0.303 0.332 0.335 0.336

12.828 -10.08 -4.791 -6.779 0.371 -24.43 -29.24 -12.89 -12.98 -13.91 21.550 38.051 11.970 4.978 4.801 35.973 20.671 5.227 5.449 2.317 24.273 11.494 4.798 5.539 1.895

-6.389 -11.413 -6.096 -8.754 -1.201 -27.526 -30.483 -14.974 -14.719 -15.601 20.344 29.558 12.260 3.345 3.051 26.683 12.967 5.385 3.671 0.597 19.702 11.229 2.968 4.052 0.429

2


Figure 13. Velocity variation with tube inclination in different diameter tubes: (a) 0.012-m-i.d. tube; (b) 0.046-m-i.d. tube.

Figure 15. Rise velocity as a function of inclination for liquid Taylor bubble at different Eo: (a) Eo < 20, (b) 20 e Eo e 70, and (c) Eo > 70.

Figure 14. Qualitative variation of Froude number with inclination.

Equation 8 represents a four-dimensional surface, which can be approximated over selected ranges of the Eotvos number by two-dimensional curves. The curves with three different regions as discussed above are shown in Figure 15a-c. The curves show that for all the three regimes of Eo [(a) Eo < 20, (b) 20 e Eo e 70, and (c) Eo > 70] the variation of bubble velocity with inclination has a similar trend. This probably indicates a more pronounced influence of inclination rather than Eotvos number on bubble velocity. Conclusion To the best of the authors’ knowledge, a comprehensive experimental study on the rise of liquid Taylor bubble and the fall of liquid Taylor drop through liquid filled tubes has been reported for the first time. The shape and velocity of these bubbles and drops have been noted in five tube diameters and

different inclinations. The observations bring out the following interesting features. (1) The shape of the nose in vertical tubes is spherical. Ripples are observed at the tail region of a Taylor bubble and drop but the interface of liquid Taylor drop is less wavy as compared to that of a liquid Taylor bubble. The tail is cylindrical in vertical tube and it becomes semicylindrical in an inclined tube. (2) The length of the wake region is smaller in inclined tubes. The opposite trend is observed for a liquid Taylor drop. Stability of a long water drop decreases with tube inclination. Channeling is observed during downward movement of a liquid Taylor drop. It does not occur during the upward motion of a liquid Taylor bubble in a clean tube. (3) The velocity increases with tube diameter, and it is higher for a liquid Taylor drop as compared to a liquid Taylor bubble. (4) Graphical relationships have also been proposed to obtain rise velocity in inclined tubes. Additional attempts have been made to modify the existing correlations for gas-liquid system in order to predict the rise of liquid Taylor bubbles in vertical tubes. Though the modified correlations can predict rise velocity reasonably well for most of the tube diameters and liquid pairs, the accuracy of the


prediction falls as the tube diameter becomes narrower. This trend has also been observed by Das et al.37 while predicting the rise of Taylor bubbles through narrow annuli in an air-water system. It is likely that the wall effect of the narrow channel is not adequately captured in the conventional analysis. Finally, another interesting observation can be made from the present investigation. Though the root cause of the rise of liquid Taylor bubbles and fall of liquid Taylor drops is buoyancy, the hydrodynamics of these two phenomena are not identical. While for the Taylor bubbles the bulk fluid is water or brine, the Taylor droplet falls through organic fluids. This may be one of the causes of this difference. Further investigations, both through experimentation and theoretical analysis, are needed to explain the underlying physics satisfactorily. Notation Fr ) Froude number ) UFp1/2/(∆FgD)1/2, dimensionless Eo ) Eotvos number ) (Fp - Fs)gD2/σ, dimensionless ReG ) gravity Reynolds number ) (gD3/ν)1/2, dimensionless Reb ) buoyancy Reynolds number ) [D3g(Fp - Fs)Fp]1/2/µp, dimensionless Mo ) Morton number ) g(Fp – Fs)µp4/Fp2pσ3, dimensionless N ) dimensional property parameter ) [1.81(Fp - Fs)Fp/ µp2g]1/3, 1/m U ) rise velocity of Taylor bubble, m/s R ) tube radius, m Rc ) equilibrium bubble radius, m D ) pipe diameter, m g ) gravitational acceleration, m/s2 Greek letters Fp ) density of primary liquid, kg/m3 Fs ) density of secondary liquid, kg/m3 ∆F ) density difference of primary and secondary liquid, kg/m3 σ ) interfacial tension, N/m ν ) kinematic viscosity, m2/s µp ) viscosity of primary liquid, Pa · s Literature Cited (1) Zukoski, E. E. Influence of Viscosity, Surface Tension, and Inclination Angle on Motion of Long Bubbles in Closed Tubes. J. Fluid. Mech. 1966, 25, 821. (2) Davies, R. M.; Taylor, Sir. G. The Mechanics of Large Bubbles Rising Through Extended Liquids and Through Liquids in Tube. Proc. R. Soc. London, A 1950, 200, 375. (3) White, E. T.; Beardmore, R. H. The velocity of rise of single cylindrical air bubbles through liquids contained in vertical tubes. Chem. Eng. Sci. 1962, 17, 351. (4) Maneri, C. C.; Zuber, N. An Experimental Study of Plane Bubbles Rising Inclination. Int. J. Multiphase Flow 1974, 1, 623–645. (5) Bhaga, T.; Weber, M. Bubbles in Viscous Liquids: Shapes, Wakes and Velocities. J. Fluid. Mech. 1981, 105, 61. (6) Nigmatulin, T. R.; Bonetto, F. J. Shape of Taylor Bubbles in Vertical Tubes. Int. Commun. Heat Mass Transfer 1997, 24, 1177. (7) Salman, W.; Gavriilidis, A. Angeli, P. On The Formation of Taylor Bubbles in Small Tubes. Chem. Eng. Sci. 2006, 61, 6653. (8) Nogueira, S.; Riethmuler, M. L.; Campos, J. B. L. M.; Pinto, A. M. F. R. Flow in the Nose Region and Annular Film around a Taylor Bubble Rising through Vertical Columns of Stagnant and Flowing Newtonion Liquids. Chem. Eng. Sci. 2006, 61, 845. (9) Uno, S.; Kintner, R. C. Effect of Wall Proximity on the Rate of Rise of Single Air Bubbles in a Quiescent Liquid. AIChE J. 1956, 2, 420.

(10) Maneri, C. C.; Mendelson, H. D. The Rise Velocity of Bubbles in Tubes and Rectangular Channels as Predicted by Wave Theory. AIChE J. 1968, 14, 294. (11) Clanet, C.; Heraud, P.; Searby, G. On the motion of bubbles in vertical tubes of arbitrary cross-sections: Some complements to the Dumitrescu-Taylor problem. J. Fluid. Mech. 2004, 519, 359. (12) Couet, B.; Strumolo, G. S. The Effects of Surface Tension and Tube Inclination on a Two-Dimensional Rising Bubble. J. Fluid. Mech. 1987, 184, 1. (13) Netto, J. R. F.; Fabre, J.; Peresson, L. Shape of long bubbles in horizontal slug flow. Int. J. Multiphase Flow 1999, 25, 1129. (14) Bi, C. Q.; Zhao, T. S. Taylor Bubbles in Miniaturized Circular and Noncircular Channels. Int. J. Multiphase Flow 2001, 27, 561. (15) Collins, R.; Moraes, F. F.; De.; Davidson, J. F.; Harrison, D. The Motion of the Large Gas Bubble Rising through Liquid Flowing in a Tube. J. Fluid. Mech. 1978, 89, 497. (16) Bendiksen, K. H. On the Motion of Long Bubbles in Vertical Tubes. Int. J. Multiphase Flow 1985, 11, 797–812. (17) Batchelor, G. K. The Stability of a Large Gas Bubble Rising through Liquid. J. Fluid. Mech. 1987, 184, 399. (18) Reinelt, D. A. The Rate at which a Long Bubble Rises in Vertical Tube. J. Fluid. Mech. 1987, 175, 557. (19) Funada, T.; Joseph, D. D.; Maehara, T.; Yamashita, S. Ellipsoidal Model of the Rise of a Taylor Bubble in a Round Tube. Int. J. Multiphase Flow 2005, 31, 473. (20) Harmathy, T. Z. Velocity of Large Drops and Bubbles in Media of Infinite or Restricted Extent. AIChE J. 1960, 6, 281. (21) Birkhoff, G.; Carter, D. Rising Plane Bubbles. J. Math. Mech. 1957, 6, 769. (22) Miksis, M. J.; Vanden-Broeck, J.-M.; Keller, J. B. Rising Bubbles. J. Fluid Mech. 1982, 123, 31. (23) Mao, Z.; Dukler, A. E. The Motion of Taylor Bubbles in Vertical Tubes. I. A Numerical Simulation for the Shape and Rise Velocity of Taylor Bubbles in Stagnant and Flowing Liquids. J. Comput. Phys. 1990, 91, 132. (24) Bugg, J. D.; Mack, K.; Rezkallah, K. S. A Numerical Model of Taylor Bubbles Rising through Stagnant Liquids in Vertical Tubes. Int. J. Multiphase Flow 1998, 24, 271. (25) Daripa, P. A Computational Study of Rising Plane Taylor Bubbles. J. Comput. Phys. 2000, 157, 120. (26) Tung, K. W.; Parlange, J. Y. Note on the motion of long bubbles in closed tubes-influence of surface tension. Acta Mech. 1976, 24, 313. (27) Vanden-Broeck, J.-M. Rising Bubbles in a Two-Dimensional Tube with Surface Tension. Phys. Fluids 1984, 27, 2604. (28) Nickens, H. V.; Yannitell, D. W. The Effects of Surface Tension and Viscosity on the Rise Velocity of a Large Gas Bubble in a Closed, Vertical Liquid-Liquid Tube. Int. J. Multiphase Flow 1987, 13, 57. (29) Garabedian, P. R. On steady-state bubbles generated by Taylor instability. Proc. R. Soc., London, A 1957, 241, 423–431. (30) Viana, F.; Pardo, R.; Yanez, R.; Trallero, J. L; Joseph D. D. Universal Correlation for the Rise Velocity of Long Bubbles in Round Pipes. J. Fluid Mech. 2003, 494, 379. (31) Wallis, G. B. One-Dimensional Two-Phase Flow; McGraw-Hill: New York, 1969. (32) Goldsmith, H. L.; Mason, S. G. The Motion of Single Large Bubbles in Closed Vertical Tubes. J. Fluid. Mech. 1962, 14, 42. (33) Brown, R. A. S. The Mechanics of Large Gas Bubbles in Tubes. I. Bubble Velocities in Stagnant Liquids. Can. J. Chem. Eng. 1965, Q2, 217. (34) Delfos, R.; Wisse, C. J.; Oliemans, R. V. A. Measurement of airentrainment from a stationary Taylor bubble in a vertical tube. Int. J. Multiphase Flow 2001, 27, 1769. (35) van Hout, R.; Gulitski, A.; Barnea, D.; Shemer, L. Experimental Investigation of the Velocity Field Induced by a Taylor Bubble Rising in Stagnant Water. Int. J. Multiphase Flow 2002, 28, 579. (36) Kockx, J. P.; Nieuwstadt, F. T. M.; Oliemans, R. V. A.; Delfos, R. Gas Entrainment by a Liquid Film Falling around a Stationary Taylor Bubble in a Vertical Tube. Int. J. Multiphase Flow 2005, 31, 1. (37) Das, G.; Purohit, N. K.; Mitra, A. K. Rise velocity of a Taylor bubble through concentric annulus. Chem. Eng. Sci. 1998, 53, 977.

ReceiVed for reView March 18, 2008 ReVised manuscript receiVed May 29, 2008 Accepted June 4, 2008 IE8004429

Motion of Taylor Bubbles and Taylor Drops in LiquidLiquid Systems

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