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Experimental Study on the Initial Position Distribution of Taylor Bubbles in Cryogenic Upward Inclined Tubes
RESEARCH NOTES Chinese Journal of Chemical Engineering, 17(2) 340ü343 (2009)
Experimental Study on the Initial Position Distribution of Taylor Bubbles in Cryogenic Upward Inclined Tubes* ZHANG Hua (**)ܟ, WANG Shuhua (ฆೞ)ܟ, LIU Yiping (ঞྫ଼) and WANG Jing (ฆ࠼) Institute of Engineering Thermo-physics, Shanghai Jiao Tong University, Shanghai 200240, China Abstract 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 digital camera. The tubes in the experiment are 0.018 m and 0.014 m in inner diameter and 1.0 m in length. The range of the inclination angles is 045° from the vertical. The statistical method is employed to analyze the experimental data. The experiment was focused on the effect of the inclination angle on the initial position distribution of Taylor bubbles. The formation criterion of Taylor bubbles was confirmed by analyzing the images of Taylor bubbles. The experimental results show that the initial position of Taylor bubble increased first, and then decreased with the increasing inclination angle, with the maximum at 30°. The standard deviation of the initial position of Taylor bubble in tubes was different with different inner diameters. The lognormal shape was fitted to the measured the initial position distributions of Taylor bubbles in the cryogenic tubes. Keywords cryogenic, nitrogen, boiling, initial position distribution, Taylor bubble, inclined tube
1
INTRODUCTION
Gas-liquid slug flow is highly complex with an inherent unsteady behavior. It is characterized by long bullet-shaped bubbles separated by liquid slugs that may be aerated by small dispersed bubbles. In cryogenic engineering, superheating always exists in conveyor and storage system of cryogenic fluids. So the cryogenic two-phase flow is unavoidable. The propagation and storage of cryogenic fluids bring many problems, such as stratification, geysering and rollover [1]. And these problems can cause high transient pressures and vapor flow rates, in some cases large enough to damage equipment. The diagnosis of flow patterns in vertical and inclined conveying pipe is an important issue in many research fields of cryogenic two-phase flow. Most of researches are focused on the Taylor bubble length, void fraction, liquid slug lengths and the Taylor bubble translation velocity, and often the ambient fluids such as air-kerosene and air-water are used. Most of those researches are carried out mainly for horizontal or slightly inclined slug flow and for vertical flow in developed slug flow [29]. The slug length, void fraction and Taylor bubble length are studied extensively. The mean liquid slug length in inclined tubes (including vertical and horizontal) has been studied by several researchers including Gu and Guo [2], Wang et al. [3], Barnea & Taitel [4], Mao & Dukler [5, 6], Xia et al. [7] and van Hout et al. [8]. They found that the mean liquid slug length decreases with increasing inclination angle. The mean Taylor bubble length in inclined tubes (including vertical and horizontal) has been studied by
Mao & Dukler [5] and van Hout et al. [8, 9]. And the mean Taylor bubble length has a minimum at about 30º and extends to much larger values for the small tube for the inclined tubes [8]. Bubble motion in inclined tubes has been studied by several authors. White and Beardmore noted the influence of the angle of inclination on bubble rise velocity [10]. Zukoski studied the influence of ș as well as the effects of viscosity and surface tension on the rise velocity [11]. Bubble motion in inclined tubes (including vertical and horizontal) has also been studied by several other researchers including Maneri and Zuber [12], Bendiksen [13], Weber et al. [14], Couët and Strumolo [15], Alves et al. [16], and van Hout et al [8, 17]. All of these authors found that the bubble velocity first increases and then decreases as the angle of inclination increases. The normalized velocity would decrease to a more or less constant value at the exits of the vertical or inclined pipes. Visualization study of cryogenic vapor-liquid slug flow is seldom studied in inclined tubes with closed bottom. Compared with normal atmospheric temperature liquid, cryogenic liquid has high compressibility, low density difference between vapor and liquid and low latent heat of vaporization. There are large differences on bubble motion in cryogenic two-phase flow and normal atmospheric temperature two-phase flow. Few investigations are performed to understand Taylor bubble initial position distribution in cryogenic liquid in upward inclined pipes. The purpose of the present study is to investigate experimentally the initial position distributions of Taylor bubbles in inclined tubes with closed bottom. The liquid nitrogen is used as working medium.
Received 2008-08-27, accepted 2008-12-14. * Supported by the National Natural Science Foundation of China (50476015) and National High-Tech Research and Development Program of China (2006AA09Z333). ** To whom correspondence should be addressed. E-mail: [email protected]
Chin. J. Chem. Eng., Vol. 17, No. 2, April 2009
2
EXPERIMENTAL
3
2.1 Experimental apparatus and image processing system The experimental apparatus consists of a liquid nitrogen Dewar, a test jacketed Pyrex glass tube and a vacuum pump. The experimental tubes are 1.0 m long with inner diameters 0.014 m and 0.018 m. The test section can be inclined at 045º inclination angles from the vertical. More detailed information about the facility and the measurement method can be found in ˉ Ref. [18]. The vacuum in the jacket is kept at 6×10 2 Pa to serve as the thermal insulation to decrease the convection heat transfer. 2.2
341
RESULTS AND DISCUSSION
3.1 Detection of initial Taylor bubbles
It is necessary to set a criterion for the occurrence of an incipient Taylor bubble in a horizontal or inclined tube. In this work, if a bubble is characterized with bullet-shaped nose, its body is columned, its bottom outline is clear, there is a dispersion of smaller bubbles after the bubble, the diameter of bubble is greater than the tube radius, and it is longer than the tube inner diameter D, the bubble is identified as a Taylor bubble. Fig. 1 shows the initial Taylor bubbles in the 0.018 m tube at various inclination angles.
Experimental condition
In the experiment, leak heat causes the boiling flow of liquid nitrogen. Due to the vacuum jacket, the convective heat transfer could be neglected. There is only a connection at the top of the upper tank, so the conducted heat transfer could be also neglected. Radiant heat transfer is the main mode of leak heat in the experiment. The heat flux of radiant heat transfer is calculated with
q
V u T24 T14
(1)
§ 2 · ¨ H 1¸ © 0 ¹ where ı is Stefan-Boltzmann’s constant, T2 is the outer wall surface temperature of the tube, T1 is the inner wall surface temperature of the tube, İ0 is emissivity of the Pyrex glass. In the experiment, heat flux leaked into the test ˉ tube is estimated to be about 300 W·m 2 for two tubes, which then generates about 130000 and 165000 bubbles (initial size below 0.6 mm) per second respectively as estimated from image analysis. During the experiment, the range of inclination angle is 045º from the vertical. The test tube in the range of 5D to 15D from the bottom was measured with a high speed digital camera (REDLAKE Motion-Pro® ˉ X3, 1280 ×1024 pixels resolution, 1000 frames·s 1).
Figure 2 0.018 m
Figure 1 Taylor bubble images of the initial positions at various inclination angles (Example for D 0.018 m)
3.2 The initial position distribution of Taylor bubbles
The histograms showing the initial position distribution of Taylor bubbles in tubes with inner diameters 0.018 m and 0.014 m at different inclination angles are respectively given in Figs. 2 and 3. The mean and the most mode column section move rightward first, and then leftward with increasing ș where ș is the inclination angle of the tube. Figure 2 shows that with the increase of ș, the initial position distribution of Taylor bubbles agrees well with lognormal distribution. Fig. 3 shows that in the tube with the diameter 0.014 m, the initial position distribution of Taylor bubbles is in line with lognormal
The initial position distribution of Taylor bubbles at various inclination angels in the tube with inner diameter
342
Figure 3 0.014 m
Chin. J. Chem. Eng., Vol. 17, No. 2, April 2009
The initial position distribution of Taylor bubbles at various inclination angels in the tube with inner diameter
distribution at all inclination angles. 3.3 Mean initial position of Taylor bubbles and standard deviation
the bottom of the tube are gradually gathering at the center of the tube through ascending along the tube. At this condition, the disturbance in tube is weak, so that the initial Taylor bubbles are formed at a lower position, as shown in Fig. 5.
Figure 4 shows the mean initial position of Taylor bubbles in different tube diameters at various ș. With the increase of ș, the mean initial position of Taylor bubbles in tubes increases first, and then decreases, with the maximum at T 30q . It also indicates that when the diameter is large, with the inclination angle increasing, the mean initial position of Taylor bubble changes more sharply than that in small tube. Figure 5 The images of bubbles at ș D 0.018 m)
Figure 4 The mean initial position of Taylor bubbles at various inclination angels ƶ D 0.014 m; ƻ D 0.018 m
0° (Example for
When the tube began to incline (the inclination angles were 10°, 20° and 30° from the vertical), the disturbance in pipe is gradually strong, vortices occurred in the pipe. The small bubbles are affected by the vortices, so that the initial Taylor bubbles are formed at a higher position, and with the inclination angle increasing, the position would be higher and higher. And the vortices could be found in the area marked in Figs. 6 and 7.
Figure 4 also shows that with the increase of ș, the standard deviation has the trend that it increases first, and then decreases in the tube with inner diameter 0.018 m; and the mean standard deviation is almost the same when the inclined pipe with inner diameter 0.014 m changes from 0° to 45°. 3.4 Variation of initial position of Taylor bubbles with inclination
The cause of variation of the initial position of Taylor bubbles is explored by analyzing the images shot in the experiment with pipe of 0.018 m diameter as an example. When the tube is vertical, the small bubbles at
Figure 6 The images of bubbles at ș D 0.018 m)
10° (Example for
When the inclination angle was 45°, the small bubbles rise rapidly to the pipe upper wall after formation. For so many small bubbles gather at the tube
343
Chin. J. Chem. Eng., Vol. 17, No. 2, April 2009
Table 1 for all cases. Table 1
ȟ
Ȝ
ȟ
0°
0.24
1.97
0.07
2.26
10°
0.13
2.40
0.08
2.4
20°
0.11
2.50
0.05
2.52
30°
0.10
2.63
0.06
2.54
45°
0.11
2.31
0.07
2.26
REFERENCES 1 2
4 5 6 7
45° (Example for
8 9
Figures 58 are the consecutive images shot in the experiment when the inclination angles are 0°, 10°, 30° and 45° respectively (time interval between images is 1 ms).
10
3.5 Lognormal distribution of the Taylor bubble initial position
12
11
13
From Figs. 2 and 3, the initial position distributions of Taylor bubbles are right-skewed. The lognormal shape is fitted to the measured distributions and is depicted in Figs. 2 and 3 as a solid line. The probability density function of the lognormal distribution is 2º ª 1 (2) x 1 exp « ¨§ ln x [ ¸· » 2Sb ¬ 2© O ¹ ¼ where x l / D , l is the length of the liquid slug or the Taylor bubble, D is the inner diameter, Ȝ is the standard deviation of lnx, ȟ is the mean value of lnx, y x . The parameters Ȝ and ȟ in Eq. (2) are given in
f ( y)
1
D=0.014 m
Ȝ
3
Figure 8 The images of bubbles at ș D 0.018 m)
D=0.018 m
Inclination angle
Figure 7 The images of bubbles at ș 30° (Example for D 0.018 m)
wall, the bubbles movement along the pipe is hindered, and the small bubbles coalesce rapidly. So the initial position of Taylor bubble is lower than those in the conditions with inclination angles 10°, 20°and 30°. In the areas in the loop in Fig. 8, the small bubbles mass near the upper wall of the pipe.
Parameters Ȝ and ȟ of lognormal fit
14 15 16 17 18
Hands, B.A., “Problems due to superheating of cryogenic liquids”, Cryogenics, 28 (12), 823829 (1988). Gu, H.Y., Guo, L.J., “Experimental investigation of slug development on horizontal two-phase flow”, Chin. J. Chem. Eng., 16 (2), 171177 (2008). Wang, X., Guo, L.J., Zhang, X.M., “Development of liquid slug length in gas-liquid slug flow along horizontal pipeline: Experiment and simulation”, Chin. J. Chem. Eng., 14 (5), 626633 (2006). Barnea, D.A., Taitel, Y., “A model for slug length distribution in gas-liquid slug flow”, Int. J. Multiphase Flow, 19 (5), 829838 (1993). Mao, Z.S., Dukler, A.E., “An experimental study of gas-liquid slug flow”, Exp. Fluids, 8 (2), 169182 (1989). Mao, Z.S., Dukler, A.E., “Improved hydrodynamic model of two-phase slug flow in vertical tubes”, Chin. J. Chem. Eng., 1 (1), 1829 (1993). Xia, G.D., Zhou, F.D., Hu, M.S., “An investigation on the void fraction for upward gas-liquid slug flow in vertical pipe”, Chin. J. Chem. Eng., 9 (4), 436440 (2001). van Hout, R., Shemer, L., Barnea, D., “Evolution of hydrodynamic and statistical parameters of gas–liquid slug flow along inclined pipes”, Chem. Eng. Sci., 58 (1), 115133 (2003). van Hout, R., Shemer, L., Barnea, D., “Spatial distribution of void fraction within the liquid slug and some other related slug parameters”, Int. J. Multiphase Flow, 18 (6), 831845 (1992). White, E.T., Beardmore, R.H., “The velocity of rise of single cylindrical air bubbles through liquids contained in vertical tubes”, Chem. Eng. Sci., 17 (5), 351361 (1962). Zukoski, E.E., “Influence of viscosity, surface tension, and inclination angle on motion on long bubbles in closed tubes”, J. Fluid Mech., 25, 821837 (1966). Maneri, C.C., Zuber, N., “An experimental study of plane bubbles rising at inclination”, Int. J. Multiphase Flow, 1 (5), 623 645 (1974). Bendiksen, K.H., “An experimental investigation of the motion of the long bubble in inclined tubes”, Int. J. Multiphase Flow, 10 (4), 467483 (1984). Weber, M.E., Alarie, A., Ryan, M.E., “Velocities of extended bubbles in inclined tubes”, Chem. Eng. Sci., 41 (9), 22352240 (1986). Couët, B., Strumolo, G.S., “The effects of surface tension and tube inclination on a two-dimensional rising bubble”, J. Fluid Mech., 184, 114 (1987). Alves, I.N., Shoham, O., Taitel, Y., “Drift velocity of elongated bubbles in inclined pipesüexperimental and modeling”, Chem. Eng. Sci., 48 (17), 3063-3070 (1993). van Hout, R., Barnea, D., Shemer, L., “Evolution of statistical parameters of gas-liquid slug flow along vertical pipes”, Int. J. Multiphase Flow, 27 (9), 15791602 (2001). Wang, S.H., Zhang, H., Wang, J., “Experimental study of boiling flow of liquid nitrogen in inclined tube-liquid slug and Taylor bubble length distributions”, Chin. J. Chem. Eng., 17 (1), 2026 (2009).
Experimental Study on the Initial Position Distribution of Taylor Bubbles in Cryogenic Upward Inclined Tubes* ZHANG Hua (**)ܟ, WANG Shuhua (ฆೞ)ܟ, LIU Yiping (ঞྫ଼) and WANG Jing (ฆ࠼) Institute of Engineering Thermo-physics, Shanghai Jiao Tong University, Shanghai 200240, China Abstract 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 digital camera. The tubes in the experiment are 0.018 m and 0.014 m in inner diameter and 1.0 m in length. The range of the inclination angles is 045° from the vertical. The statistical method is employed to analyze the experimental data. The experiment was focused on the effect of the inclination angle on the initial position distribution of Taylor bubbles. The formation criterion of Taylor bubbles was confirmed by analyzing the images of Taylor bubbles. The experimental results show that the initial position of Taylor bubble increased first, and then decreased with the increasing inclination angle, with the maximum at 30°. The standard deviation of the initial position of Taylor bubble in tubes was different with different inner diameters. The lognormal shape was fitted to the measured the initial position distributions of Taylor bubbles in the cryogenic tubes. Keywords cryogenic, nitrogen, boiling, initial position distribution, Taylor bubble, inclined tube
1
INTRODUCTION
Gas-liquid slug flow is highly complex with an inherent unsteady behavior. It is characterized by long bullet-shaped bubbles separated by liquid slugs that may be aerated by small dispersed bubbles. In cryogenic engineering, superheating always exists in conveyor and storage system of cryogenic fluids. So the cryogenic two-phase flow is unavoidable. The propagation and storage of cryogenic fluids bring many problems, such as stratification, geysering and rollover [1]. And these problems can cause high transient pressures and vapor flow rates, in some cases large enough to damage equipment. The diagnosis of flow patterns in vertical and inclined conveying pipe is an important issue in many research fields of cryogenic two-phase flow. Most of researches are focused on the Taylor bubble length, void fraction, liquid slug lengths and the Taylor bubble translation velocity, and often the ambient fluids such as air-kerosene and air-water are used. Most of those researches are carried out mainly for horizontal or slightly inclined slug flow and for vertical flow in developed slug flow [29]. The slug length, void fraction and Taylor bubble length are studied extensively. The mean liquid slug length in inclined tubes (including vertical and horizontal) has been studied by several researchers including Gu and Guo [2], Wang et al. [3], Barnea & Taitel [4], Mao & Dukler [5, 6], Xia et al. [7] and van Hout et al. [8]. They found that the mean liquid slug length decreases with increasing inclination angle. The mean Taylor bubble length in inclined tubes (including vertical and horizontal) has been studied by
Mao & Dukler [5] and van Hout et al. [8, 9]. And the mean Taylor bubble length has a minimum at about 30º and extends to much larger values for the small tube for the inclined tubes [8]. Bubble motion in inclined tubes has been studied by several authors. White and Beardmore noted the influence of the angle of inclination on bubble rise velocity [10]. Zukoski studied the influence of ș as well as the effects of viscosity and surface tension on the rise velocity [11]. Bubble motion in inclined tubes (including vertical and horizontal) has also been studied by several other researchers including Maneri and Zuber [12], Bendiksen [13], Weber et al. [14], Couët and Strumolo [15], Alves et al. [16], and van Hout et al [8, 17]. All of these authors found that the bubble velocity first increases and then decreases as the angle of inclination increases. The normalized velocity would decrease to a more or less constant value at the exits of the vertical or inclined pipes. Visualization study of cryogenic vapor-liquid slug flow is seldom studied in inclined tubes with closed bottom. Compared with normal atmospheric temperature liquid, cryogenic liquid has high compressibility, low density difference between vapor and liquid and low latent heat of vaporization. There are large differences on bubble motion in cryogenic two-phase flow and normal atmospheric temperature two-phase flow. Few investigations are performed to understand Taylor bubble initial position distribution in cryogenic liquid in upward inclined pipes. The purpose of the present study is to investigate experimentally the initial position distributions of Taylor bubbles in inclined tubes with closed bottom. The liquid nitrogen is used as working medium.
Received 2008-08-27, accepted 2008-12-14. * Supported by the National Natural Science Foundation of China (50476015) and National High-Tech Research and Development Program of China (2006AA09Z333). ** To whom correspondence should be addressed. E-mail: [email protected]
Chin. J. Chem. Eng., Vol. 17, No. 2, April 2009
2
EXPERIMENTAL
3
2.1 Experimental apparatus and image processing system The experimental apparatus consists of a liquid nitrogen Dewar, a test jacketed Pyrex glass tube and a vacuum pump. The experimental tubes are 1.0 m long with inner diameters 0.014 m and 0.018 m. The test section can be inclined at 045º inclination angles from the vertical. More detailed information about the facility and the measurement method can be found in ˉ Ref. [18]. The vacuum in the jacket is kept at 6×10 2 Pa to serve as the thermal insulation to decrease the convection heat transfer. 2.2
341
RESULTS AND DISCUSSION
3.1 Detection of initial Taylor bubbles
It is necessary to set a criterion for the occurrence of an incipient Taylor bubble in a horizontal or inclined tube. In this work, if a bubble is characterized with bullet-shaped nose, its body is columned, its bottom outline is clear, there is a dispersion of smaller bubbles after the bubble, the diameter of bubble is greater than the tube radius, and it is longer than the tube inner diameter D, the bubble is identified as a Taylor bubble. Fig. 1 shows the initial Taylor bubbles in the 0.018 m tube at various inclination angles.
Experimental condition
In the experiment, leak heat causes the boiling flow of liquid nitrogen. Due to the vacuum jacket, the convective heat transfer could be neglected. There is only a connection at the top of the upper tank, so the conducted heat transfer could be also neglected. Radiant heat transfer is the main mode of leak heat in the experiment. The heat flux of radiant heat transfer is calculated with
q
V u T24 T14
(1)
§ 2 · ¨ H 1¸ © 0 ¹ where ı is Stefan-Boltzmann’s constant, T2 is the outer wall surface temperature of the tube, T1 is the inner wall surface temperature of the tube, İ0 is emissivity of the Pyrex glass. In the experiment, heat flux leaked into the test ˉ tube is estimated to be about 300 W·m 2 for two tubes, which then generates about 130000 and 165000 bubbles (initial size below 0.6 mm) per second respectively as estimated from image analysis. During the experiment, the range of inclination angle is 045º from the vertical. The test tube in the range of 5D to 15D from the bottom was measured with a high speed digital camera (REDLAKE Motion-Pro® ˉ X3, 1280 ×1024 pixels resolution, 1000 frames·s 1).
Figure 2 0.018 m
Figure 1 Taylor bubble images of the initial positions at various inclination angles (Example for D 0.018 m)
3.2 The initial position distribution of Taylor bubbles
The histograms showing the initial position distribution of Taylor bubbles in tubes with inner diameters 0.018 m and 0.014 m at different inclination angles are respectively given in Figs. 2 and 3. The mean and the most mode column section move rightward first, and then leftward with increasing ș where ș is the inclination angle of the tube. Figure 2 shows that with the increase of ș, the initial position distribution of Taylor bubbles agrees well with lognormal distribution. Fig. 3 shows that in the tube with the diameter 0.014 m, the initial position distribution of Taylor bubbles is in line with lognormal
The initial position distribution of Taylor bubbles at various inclination angels in the tube with inner diameter
342
Figure 3 0.014 m
Chin. J. Chem. Eng., Vol. 17, No. 2, April 2009
The initial position distribution of Taylor bubbles at various inclination angels in the tube with inner diameter
distribution at all inclination angles. 3.3 Mean initial position of Taylor bubbles and standard deviation
the bottom of the tube are gradually gathering at the center of the tube through ascending along the tube. At this condition, the disturbance in tube is weak, so that the initial Taylor bubbles are formed at a lower position, as shown in Fig. 5.
Figure 4 shows the mean initial position of Taylor bubbles in different tube diameters at various ș. With the increase of ș, the mean initial position of Taylor bubbles in tubes increases first, and then decreases, with the maximum at T 30q . It also indicates that when the diameter is large, with the inclination angle increasing, the mean initial position of Taylor bubble changes more sharply than that in small tube. Figure 5 The images of bubbles at ș D 0.018 m)
Figure 4 The mean initial position of Taylor bubbles at various inclination angels ƶ D 0.014 m; ƻ D 0.018 m
0° (Example for
When the tube began to incline (the inclination angles were 10°, 20° and 30° from the vertical), the disturbance in pipe is gradually strong, vortices occurred in the pipe. The small bubbles are affected by the vortices, so that the initial Taylor bubbles are formed at a higher position, and with the inclination angle increasing, the position would be higher and higher. And the vortices could be found in the area marked in Figs. 6 and 7.
Figure 4 also shows that with the increase of ș, the standard deviation has the trend that it increases first, and then decreases in the tube with inner diameter 0.018 m; and the mean standard deviation is almost the same when the inclined pipe with inner diameter 0.014 m changes from 0° to 45°. 3.4 Variation of initial position of Taylor bubbles with inclination
The cause of variation of the initial position of Taylor bubbles is explored by analyzing the images shot in the experiment with pipe of 0.018 m diameter as an example. When the tube is vertical, the small bubbles at
Figure 6 The images of bubbles at ș D 0.018 m)
10° (Example for
When the inclination angle was 45°, the small bubbles rise rapidly to the pipe upper wall after formation. For so many small bubbles gather at the tube
343
Chin. J. Chem. Eng., Vol. 17, No. 2, April 2009
Table 1 for all cases. Table 1
ȟ
Ȝ
ȟ
0°
0.24
1.97
0.07
2.26
10°
0.13
2.40
0.08
2.4
20°
0.11
2.50
0.05
2.52
30°
0.10
2.63
0.06
2.54
45°
0.11
2.31
0.07
2.26
REFERENCES 1 2
4 5 6 7
45° (Example for
8 9
Figures 58 are the consecutive images shot in the experiment when the inclination angles are 0°, 10°, 30° and 45° respectively (time interval between images is 1 ms).
10
3.5 Lognormal distribution of the Taylor bubble initial position
12
11
13
From Figs. 2 and 3, the initial position distributions of Taylor bubbles are right-skewed. The lognormal shape is fitted to the measured distributions and is depicted in Figs. 2 and 3 as a solid line. The probability density function of the lognormal distribution is 2º ª 1 (2) x 1 exp « ¨§ ln x [ ¸· » 2Sb ¬ 2© O ¹ ¼ where x l / D , l is the length of the liquid slug or the Taylor bubble, D is the inner diameter, Ȝ is the standard deviation of lnx, ȟ is the mean value of lnx, y x . The parameters Ȝ and ȟ in Eq. (2) are given in
f ( y)
1
D=0.014 m
Ȝ
3
Figure 8 The images of bubbles at ș D 0.018 m)
D=0.018 m
Inclination angle
Figure 7 The images of bubbles at ș 30° (Example for D 0.018 m)
wall, the bubbles movement along the pipe is hindered, and the small bubbles coalesce rapidly. So the initial position of Taylor bubble is lower than those in the conditions with inclination angles 10°, 20°and 30°. In the areas in the loop in Fig. 8, the small bubbles mass near the upper wall of the pipe.
Parameters Ȝ and ȟ of lognormal fit
14 15 16 17 18
Hands, B.A., “Problems due to superheating of cryogenic liquids”, Cryogenics, 28 (12), 823829 (1988). Gu, H.Y., Guo, L.J., “Experimental investigation of slug development on horizontal two-phase flow”, Chin. J. Chem. Eng., 16 (2), 171177 (2008). Wang, X., Guo, L.J., Zhang, X.M., “Development of liquid slug length in gas-liquid slug flow along horizontal pipeline: Experiment and simulation”, Chin. J. Chem. Eng., 14 (5), 626633 (2006). Barnea, D.A., Taitel, Y., “A model for slug length distribution in gas-liquid slug flow”, Int. J. Multiphase Flow, 19 (5), 829838 (1993). Mao, Z.S., Dukler, A.E., “An experimental study of gas-liquid slug flow”, Exp. Fluids, 8 (2), 169182 (1989). Mao, Z.S., Dukler, A.E., “Improved hydrodynamic model of two-phase slug flow in vertical tubes”, Chin. J. Chem. Eng., 1 (1), 1829 (1993). Xia, G.D., Zhou, F.D., Hu, M.S., “An investigation on the void fraction for upward gas-liquid slug flow in vertical pipe”, Chin. J. Chem. Eng., 9 (4), 436440 (2001). van Hout, R., Shemer, L., Barnea, D., “Evolution of hydrodynamic and statistical parameters of gas–liquid slug flow along inclined pipes”, Chem. Eng. Sci., 58 (1), 115133 (2003). van Hout, R., Shemer, L., Barnea, D., “Spatial distribution of void fraction within the liquid slug and some other related slug parameters”, Int. J. Multiphase Flow, 18 (6), 831845 (1992). White, E.T., Beardmore, R.H., “The velocity of rise of single cylindrical air bubbles through liquids contained in vertical tubes”, Chem. Eng. Sci., 17 (5), 351361 (1962). Zukoski, E.E., “Influence of viscosity, surface tension, and inclination angle on motion on long bubbles in closed tubes”, J. Fluid Mech., 25, 821837 (1966). Maneri, C.C., Zuber, N., “An experimental study of plane bubbles rising at inclination”, Int. J. Multiphase Flow, 1 (5), 623 645 (1974). Bendiksen, K.H., “An experimental investigation of the motion of the long bubble in inclined tubes”, Int. J. Multiphase Flow, 10 (4), 467483 (1984). Weber, M.E., Alarie, A., Ryan, M.E., “Velocities of extended bubbles in inclined tubes”, Chem. Eng. Sci., 41 (9), 22352240 (1986). Couët, B., Strumolo, G.S., “The effects of surface tension and tube inclination on a two-dimensional rising bubble”, J. Fluid Mech., 184, 114 (1987). Alves, I.N., Shoham, O., Taitel, Y., “Drift velocity of elongated bubbles in inclined pipesüexperimental and modeling”, Chem. Eng. Sci., 48 (17), 3063-3070 (1993). van Hout, R., Barnea, D., Shemer, L., “Evolution of statistical parameters of gas-liquid slug flow along vertical pipes”, Int. J. Multiphase Flow, 27 (9), 15791602 (2001). Wang, S.H., Zhang, H., Wang, J., “Experimental study of boiling flow of liquid nitrogen in inclined tube-liquid slug and Taylor bubble length distributions”, Chin. J. Chem. Eng., 17 (1), 2026 (2009).