# The Effect of the Angle of Pipe Inclination on the Average Size and Velocity of Gas Bubbles Injected from a Capillary into a Liquid

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## Abstract

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## 1. Introduction

^{3}and 300 × 150 × 500 mm

^{3}was carried out. Bubbles with a diameter of 3–5 mm were considered (depending on the capillary and fluid flow, the diameter of the bubbles changed slightly). The dependences of the effect of gas flow rate, height of the liquid column, and nozzle diameter on bubble ascent trajectory and deformation of the bubble surface are shown.

## 2. Experimental Setup and Technique

_{g}= 3.3 mL/min.

_{d}is the drag coefficient. Bubbles with a small diameter (up to 3–4 mm) do not experience pulsation of shape, and their movement can be described as the movement of a rigid sphere. When liquid flows around a rigid sphere, the drag coefficient is equal to [24]:

_{b}is calculated relative to the velocity V

_{b}and diameter D

_{b}of the gas bubble. Measuring values were compared with the calculated bubble rise velocity for the same bubble diameters (Figure 3).

## 3. Results and Discussion

#### 3.1. Bubble Chain Movement Mode

_{g}= 3.3 mL/min) and equal distances from the gas input point to the shooting point (L = 200 mm) for different angles of inclination of the pipe (θ = 30°–60°).

_{g}= 3.3 mL/min. For cluster modes of bubble movement, the average values of bubble size and velocity were characteristically close to the values of size and velocity for a single bubble in a chain (deviation of no more than 5%). For modes without cluster formation, there were several typical bubble diameters that moved with different velocities.

#### 3.2. Average Size of Gas Bubbles in the Inclined Tube

_{b}= 1.4–2.3 mm for the gas flow rate Q

_{g}= 3.3 mL/min and D

_{b}= 1.5–2.2 mm for the gas flow rate Q

_{g}= 5.5 mL/min.

_{g}= 3.3 mL/min (Figure 7a), the average diameter decreased with an increase in the angle of inclination of the pipe, since the departure diameter decreased due to the angle of inclination of the capillary. At a distance of 200 mm and pipe inclination angles up to 50°, the average diameter decreased due to the decrease in the departure diameter. However, at angles of inclination of more than 50°, bubbles were actively grouped when moving along the upper wall, and the average diameter increased due to coalescence. At distances of 400 and 600 mm, the effect of bubble coalescence along the upper wall of the inclined pipe compensated for the decrease in the departure diameter in the range of channel inclination angles from 30° to 50°. The average diameter of gas bubbles at the flow rate of 5 mL/min (Figure 7b) demonstrated similar behavior.

#### 3.3. Average Velocity of Gas Bubbles in the Inclined Tube

_{g}= 3.3 mL/min (Figure 8a), the velocity of the bubbles decreased throughout the selected range of angles. At a distance L = 200 mm for angles of 55° and 60°, an increase in the velocity of the bubbles was observed. This was due to the fact that the bubbles were actively coalescing when moving along the upper wall of the pipe, and the average velocity of the bubbles increased due to the Archimedes force. The bubble chain mode for angles greater than 55 was not obtained. For the gas flow rate Q

_{g}= 5.0 mL/min (Figure 8b), the dependences of the average velocity of the bubbles for different angles of inclination of the pipe demonstrated similar behavior. An increase in the average velocity for the same distance L was associated with the increase in the average diameter of the bubbles as the gas flow rate increased. The nonlinearity in the average velocity graphs for the distance L = 400 mm was associated with the coalescence of bubbles when moving along the upper wall of an inclined pipe. This was also typical for gas flow rate Q

_{g}= 3.3 mL/min.

_{b}= ρV

_{b}D

_{b}/μ, where ρ is the density of the fluid, μ is the dynamic viscosity of the fluid, V

_{b}is the mean velocity of bubbles, and D

_{b}is the mean diameter of the bubbles. Figure 9 shows the values of the Reynolds number for gas bubbles, depending on the distance between the gas input point and the point of measurement.

_{b}= 390–450 for a gas flow rate Q

_{g}= 3.3 mL/min (Figure 9a). The behavior of the Reynolds number at the channel angle of 60° differed significantly from its behavior at other angles. From this angle of inclination, the velocity of bubbles due to coalescence and the growth of the average diameter were significantly reduced, leading to a decrease in the Reynolds number.

_{g}= 5.0 mL/min (Figure 9b), the graphs tended toward the value of the Reynolds number Re

_{b}= 390–450 at a distance from the gas input point to the camera shooting point L = 400 mm. With an increase in the distance to L = 600 mm, due to coalescence, the differences in the diameters and velocity of the bubbles increased, leading to a discrepancy in the graphs of the Reynolds number.

## 4. Conclusions

_{b}= 1.4–2.3 mm for the gas flow rate Q

_{g}= 3.3 mL/min and D

_{b}= 1.5–2.2 mm for the gas flow rate Q

_{g}= 5.5 mL/min. With an increase in the angle of inclination of the channel, the average diameter of the gas bubbles decreased. The nonlinearity of the graphs of the average diameter was caused by the influence of the angle of inclination on the departure diameter, the coalescence of bubbles near the capillary, and the coalescence of bubbles when moving along the upper wall of the inclined pipe.

_{g}= 3.3 mL/min and to ~0.18 m/s for the gas flow rate Q

_{g}= 5.0 mL/min. The nonlinearity in the average velocity graphs for distances L = 400 mm and greater was associated with the coalescence of bubbles when moving along the upper wall of an inclined pipe.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Lobanov, P.D. Wall Shear Stress and Heat Transfer of Downward Bubbly Flow at Low Flow Rates of Liquid and Gas. J. Eng. Thermophys.
**2018**, 27, 232–244. [Google Scholar] [CrossRef] - Lobanov, P.; Pakhomov, M.; Terekhov, V. Experimental and Numerical Study of the Flow and Heat Transfer in a Bubbly Turbulent Flow in a Pipe with Sudden Expansion. Energies
**2019**, 12, 2735. [Google Scholar] [CrossRef][Green Version] - Feng, Z.C.; Leal, L.G. Nonlinear Bubble Dynamics. Annu. Rev. Fluid Mech.
**1997**, 29, 201–243. [Google Scholar] [CrossRef] - Liow, J.L. Quasi-equilibrium bubble formation during topsubmerged gas injection. Chem. Eng. Sci.
**2000**, 55, 4515. [Google Scholar] [CrossRef] - Jamialahmadi, M.; Zehtaban, M.R.; Müller-Steinhagen, H.; Sarrafi, A.; Smith, J.M. Study of Bubble Formation Under Constant Flow Conditions. Chem. Eng. Res. Des.
**2001**, 79, 523–532. [Google Scholar] [CrossRef] - Li, M.; Hu, L. Experimental investigation of the behaviors of highly deformed bubbles produced by coaxial coalescence. Exp. Therm. Fluid Sci.
**2020**, 117, 110114. [Google Scholar] [CrossRef] - Houston, S.D.; Cornwell, K. Heat transfer to sliding bubbles on a tube under evaporating and non-evaporating conditions. Int. J. Heat Mass Transf.
**1996**, 39, 211–214. [Google Scholar] [CrossRef] - Dabiri, S.; Tryggvason, G. Heat transfer in turbulent bubbly flow in vertical channels. Chem. Eng. Sci.
**2015**, 122, 106–113. [Google Scholar] [CrossRef] - Chakraborty, S.; Das, P.K. Characterisation and classification of gas-liquid two-phase flow using conductivity probe and multiple optical sensors. Int. J. Multiph. Flow
**2020**, 124, 103193. [Google Scholar] [CrossRef] - Zhou, Y.; Kang, P.; Huang, Z.; Yan, P.; Sun, J.; Wang, J.; Yang, Y. Experimental measurement and theoretical analysis on bubble dynamic behaviors in a gas-liquid bubble column. Chem. Eng. Sci.
**2020**, 211, 115295. [Google Scholar] [CrossRef] - Razzaque, M.M.; Afacan, A.; Liu, S.; Nandakumar, K.; Masliyah, J.H.; Sanders, R.S. Bubble size in coalescence dominant regime of turbulent air–water flow through horizontal pipes. Int. J. Multiph. Flow
**2003**, 29, 1451–1471. [Google Scholar] [CrossRef] - Kadri, U.; Henkes, R.A.W.M.; Mudde, R.F.; Oliemans, R.V.A. Effect of gas pulsation on long slugs in horizontal gas–liquid pipe flow. Int. J. Multiph. Flow
**2011**, 37, 1120–1128. [Google Scholar] [CrossRef] - Sarafraz, M.M.; Shadloo, M.S.; Tian, Z.; Tlili, I.; Alkanhal, T.A.; Safaei, M.R.; Goodarzi, M.; Arjomandi, M. Convective Bubbly Flow of Water in an Annular Pipe: Role of Total Dissolved Solids on Heat Transfer Characteristics and Bubble Formation. Water
**2019**, 11, 1566. [Google Scholar] [CrossRef][Green Version] - Pokusaev, B.G.; Kazenin, D.A.; Karlov, S.P.; Ermolaev, V.S. Motion of a gas slug in inclined tubes. Theor. Found. Chem. Eng.
**2011**, 45, 640–645. [Google Scholar] [CrossRef] - Tihon, J.; Pěnkavová, V.; Vejražka, J. Wall shear stress induced by a large bubble rising in an inclined rectangular channel. Int. J. Multiph. Flow
**2014**, 67, 76–87. [Google Scholar] [CrossRef] - Wang, H.; Zhang, C.; Xiong, H. Growth and Collapse Dynamics of a Vapor Bubble near or at a Wall. Water
**2021**, 13, 12. [Google Scholar] [CrossRef] - Donnelly, B.; Meehan, B. O’Reilly; Nolan, K.; Murray, D.B. The dynamics of sliding air bubbles and the effects on surface heat transfer. Int. J. Heat Mass Transf.
**2015**, 91, 532–542. [Google Scholar] [CrossRef] - Augustyniak, J.; Perkowski, D.M. Compound analysis of gas bubble trajectories with help of multifractal algorithm. Exp. Therm. Fluid Sci.
**2021**, 124, 110351. [Google Scholar] [CrossRef] - Kamp, A.M.; Chesters, A.K.; Colin, C.; Fabre, J. Bubble coalescence in turbulent flows: A mechanistic model for turbulence-induced coalescence applied to microgravity bubbly pipe flow. Int. J. Multiph. Flow
**2001**, 27, 1363–1396. [Google Scholar] [CrossRef] - Lahey, R.T.; Baglietto, E.; Bolotnov, I.A. Progress in multiphase computational fluid dynamics. Nucl. Eng. Des.
**2021**, 374, 111018. [Google Scholar] [CrossRef] - Chinak, A.V.; Gorelikova, A.E.; Kashinsky, O.N.; Pakhomov, M.A.; Randin, V.V.; Terekhov, V.I. Hydrodynamics and heat transfer in an inclined bubbly flow. Int. J. Heat Mass Transf.
**2018**, 118, 785–801. [Google Scholar] [CrossRef] - Haynes, W.M. CRC Handbook of Chemistry and Physics; CRC: Boca Raton, FL, USA, 2014. [Google Scholar]
- Fu, Y.; Liu, Y. Development of a robust image processing technique for bubbly flow measurement in a narrow rectangular channel. Int. J. Multiph. Flow
**2016**, 84, 217–228. [Google Scholar] [CrossRef] - Bhunia, A.; Pais, S.C.; Kamotani, Y.; Kim, I.-H. Bubble formation in a coflow configuration in normal and reduced gravity. AIChE J.
**1998**, 44, 1499–1509. [Google Scholar] [CrossRef]

**Figure 1.**Experimental setup.: 1—flow rate meter; 2—capillary; 3—test section; 4—camera; 5—LED matrix.

**Figure 3.**Comparisons of bubble rise velocity in vertical pipe (θ = 0°) in experiments, and calculated bubble rise velocity.

**Figure 4.**Characteristic images of bubbles. Gas flow rates Q

_{g}= 3.3 mL/min; distances from the gas input point to the shooting point L = 200 mm; angle of inclination of the pipe θ = 30°–60°.

**Figure 5.**Bubble movement in modes without cluster formation (θ = 35°) and with cluster formation (θ = 50°); gas flow rate Q

_{g}= 3.3 mL/min.

**Figure 6.**Map of the modes of movement of bubbles. The points correspond to the parameters for which the bubble chain mode appeared.

**Figure 7.**Dependence of the average size of gas bubbles as a function of the pipe inclination angle. Gas flow rate (

**a**) Q

_{g}= 3.3 mL/min; (

**b**) Q

_{g}= 5.0 mL/min.

**Figure 8.**Dependence of the average velocity of gas bubbles as a function of the pipe inclination angle. Gas flow rate (

**a**) Q

_{g}= 3.3 mL/min; (

**b**) Q

_{g}= 5.0 mL/min.

**Figure 9.**The dependence of the Reynolds number Re

_{b}on the distance between the gas input point and the camera shooting point. Angle of inclination of the pipe θ = 30°–60°, gas flow rate (

**a**) Q

_{g}= 3.3 mL/min; (

**b**) Q

_{g}= 5.0 mL/min.

Fluid | Density, ρ (kg/m^{3}) | Viscosity, μ (mPa·s) | Surface Tension, σ (N/m) |
---|---|---|---|

Distilled water | 998 | 1 | 0.072 |

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**MDPI and ACS Style**

Gorelikova, A.E.; Randin, V.V.; Chinak, A.V.; Kashinsky, O.N. The Effect of the Angle of Pipe Inclination on the Average Size and Velocity of Gas Bubbles Injected from a Capillary into a Liquid. *Water* **2023**, *15*, 560.
https://doi.org/10.3390/w15030560

**AMA Style**

Gorelikova AE, Randin VV, Chinak AV, Kashinsky ON. The Effect of the Angle of Pipe Inclination on the Average Size and Velocity of Gas Bubbles Injected from a Capillary into a Liquid. *Water*. 2023; 15(3):560.
https://doi.org/10.3390/w15030560

**Chicago/Turabian Style**

Gorelikova, Anastasia E., Vyacheslav V. Randin, Alexander V. Chinak, and Oleg N. Kashinsky. 2023. "The Effect of the Angle of Pipe Inclination on the Average Size and Velocity of Gas Bubbles Injected from a Capillary into a Liquid" *Water* 15, no. 3: 560.
https://doi.org/10.3390/w15030560