New Correlations for the Prediction of Terminal Velocity and Drag Coefficient of a Bubble Rising

Document Type : Research Paper


1 Corresponding Author, Assistant Professor, Chemical Engineering Department, Jundi-Shapur University of Technology, Dezful, Iran

2 Researcher, Chemical Engineering Department, Jundi-Shapur University of Technology, Dezful, Iran

3 Assistant Professor, Chemical Engineering Department, Jundi-Shapur University of Technology, Dezful, Iran

4 Researcher, Mechanical Engineering Department, Jundi-Shapur University of Technology, Dezful, Iran,


The present experimental study was done aimed to investigate dynamic of
a single bubble rising through wall-bounded flow at high Reynolds
number. Thus, Rhamnolipid biosurfactant was added to stagnant fluid and
bubble diameter was controlled between 2.5 and 3.5mm. The resulted
Reynolds number was in the range of 400 to 900 depends on biosurfactant
concentration. Rhamnolipid has a low toxicity, a high biodegradability
and good stability at a wide range of temperatures. The results showed
that terminal velocity linearly depends on Reynolds number. Furthermore,
drag coefficient is related to Eötvos number and is autonomous to
Reynolds number. Finally, to estimate terminal velocity and drag
coefficient, four empirical correlations were developed. Relative errors of
the proposed correlations were less than of 3.35% and 1.97% for velocity
and dimensionless velocity equations, respectively, and average errors of
two equations proposed for drag coefficient were 4.44% and 3.26%.


Main Subjects

[1] Silva, M.,Campos, J., and Araújo, J., "General Correlations for Gas-liquid Mass Transfer in
Laminar Slug Flow", International Communications in Heat and Mass Transfer, Vol. 120,
pp. 104998, (2021).
[2] Yao, C., Ma, H., Zhao, Q., Liu, Y., Zhao, Y., and Chen, G., "Mass Transfer in Liquid-liquid
Taylor Flow in a Microchannel: Local Concentration Distribution, Mass Transfer
Regime and the Effect of Fluid Viscosity", Chemical Engineering Science, Vol. 223,
pp. 115734, (2020).
[3] Fayzi, P., Bastani, D., and Lotfi, M., "A Note on the Synergistic Effect of Surfactants and
Nanoparticles on Rising Bubble Hydrodynamics", Chemical Engineering and
Processing-Process Intensification, Vol. 155, pp. 108068, (2020).
[4] Fayzi, P., Bastani, D., Lotfi, M., and Miller, R., "Influence of Surface-Modified
Nanoparticles on the Hydrodynamics of Rising Bubbles", Chemical Engineering &
Technology, Vol. 44, pp. 513-520, (2021).
[5] Hayashi, K.,and Tomiyama, A., "Effects of Numerical Treatment of Viscous and Surface
Tension Forces on Predicted Motion of Interface", The Journal of Computational
Multiphase Flows, Vol. 6, pp. 111-126, (2014).
[6] Li, H., Liu, Z., Chen, J., Sun, B., Guo, Y., and He, H., "Correlation of Aspect Ratio and
Drag Coefficient for Hydrate-film-covered Methane Bubbles in Water", Experimental
Thermal and Fluid Science, Vol. 88, pp. 554-565, (2017).
[7] Bora, M., Behera, S.K., and Deb, P., "Dynamic Coalescence and Implosion of Internal
Microbubbles in Immobile Droplet", In: editor.^editors. AIP Conference Proceedings,
ed.: AIP Publishing LLC. pp. 030014, (2020).
[8] Kim, S., Oshima, N., Murai, Y., and Park, H.J., "Numerical Investigation of a Single
Intermediate-sized Bubble in Horizontal Turbulent Channel Flow", Journal of Fluid
Science and Technology, Vol. 15, pp. JFST0020-JFST0020, (2020).
[9] Vitasari, D., Cox, S., Grassia, P., and Rosario, R., "Effect of Surfactant Redistribution on
the Flow and Stability of Foam Films", Proceedings of the Royal Society A, Vol. 476,
pp. 20190637, (2020).
[10] Moore, D., "The Boundary Layer on a Spherical Gas Bubble", Journal of Fluid Mechanics,
Vol. 16, pp. 161-176, (1963).
[11] Lehrer, I.H., "A Rational Terminal Velocity Equation for Bubbles and Drops at
Intermediate and High Reynolds Numbers", Journal of Chemical Engineering of Japan,
Vol. 9, pp. 237-240, (1976).
[12] Almatroushi, E., and Borhan, A., "Surfactant Effect on the Buoyancy‐Driven Motion of
Bubbles and Drops in a Tube", Annals of the New York Academy of Sciences, Vol.
1027, pp. 330-341, (2004).
[13] Liu, L., Yan, H., Zhao, G., and Zhuang, J., "Experimental Studies on the Terminal Velocity
of Air Bubbles in Water and Glycerol Aqueous Solution", Experimental Thermal and
Fluid Science, Vol. 78, pp. 254-265, (2016).
[14] Zhang, C., Zhou, D., Sa, R., and Wu, Q., "Investigation of Single Bubble Rising Velocity
in LBE by Transparent Liquids Similarity Experiments", Progress in Nuclear Energy,
Vol. 108, pp. 204-213, (2018).
[15] Kurimoto, R., Hayashi, K., and Tomiyama, A., "Terminal Velocities of Clean and Fullycontaminated
Drops in Vertical Pipes", International Journal of Multiphase Flow, Vol.
49, pp. 8-23, (2013).
[16] Batchvarov, A., Kahouadji, L., Magnini, M., Constante-Amores, C.R., Craster, R.V., Shin,
S., Chergui, J., Juric, D., and Matar, O.K., "Effect of Surfactant on Elongated Bubbles
in Capillary Tubes at High Reynolds Number", Phys Rev Fluids, Vol. 5, pp. 093605,
[17] Barbosa, C., Legendre, D., and Zenit, R., "Sliding Motion of a Bubble Against an Inclined
Wall from Moderate to High Bubble Reynolds Number", Physical Review Fluids, Vol.
4, pp. 043602, (2019).
[18] Garnier, C., Lance, M., and Marié, J., "Measurement of Local Flow Characteristics in
Buoyancy-driven Bubbly Flow at High Void Fraction", Experimental Thermal and
Fluid Science, Vol. 26, pp. 811-815, (2002).
[19] Karimi, S., Shafiee, M., Abiri, A., and Ghadam, F., "The Drag Coefficient Prediction of a
Rising Bubble through a Non-Newtonian Fluid", Amirkabir Journal of Mechanical
Engineering, Vol. 52, No. 4, pp. 71-80, (2019).
[20] Ahmadpour, A., Amani, E., and Esmaili, M., "Numerical Simulation of Shear Thinning
Slug Flows: the Effect of Viscosity Variation on the Shape of Taylor Bubbles and Wall
Shear Stress", Journal of the Brazilian Society of Mechanical Sciences and Engineering,
Vol. 41, pp. 48, (2019).
[21] Bozzano, G.,and Dente, M., "Shape and Terminal Velocity of Single Bubble Motion: a
Novel Approach", Computers & Chemical Engineering, Vol. 25, pp. 571-576, (2001).
[22] Riboux, G., Risso, F., and Legendre, D., "Experimental Characterization of the Agitation
Generated by Bubbles Rising at High Reynolds Number", Journal of Fluid Mechanics,
Vol. 643, pp. 509, (2010).
[23] Yan, X., Jia, Y., Wang, L., and Cao, Y., "Drag Coefficient Fluctuation Prediction of a
Single Bubble Rising in Water", Chemical Engineering Journal, Vol. 316, pp. 553-562,
[24] Tihon, J., and Ezeji, K., "Velocity of a Large Bubble Rising in a Stagnant Liquid Inside an
Inclined Rectangular Channel", Physics of Fluids, Vol. 31, pp. 113301, (2019).
[25] Li, M., and Hu, L., "Experimental Investigation of the Behaviors of Highly Deformed
Bubbles Produced by Coaxial Coalescence", Experimental Thermal and Fluid Science,
Vol. 117, pp. 110114, (2020).
[26] Pietrasanta, L., Mameli, M., Mangini, D., Georgoulas, A., Michè, N., Filippeschi, S., and
Marengo, M., "Developing Flow Pattern Maps for Accelerated Two-phase Capillary
Flows", Experimental Thermal and Fluid Science, Vol. 112, pp. 109981, (2020).
[27] Kelbaliyev, G., and Ceylan, K., "Development of New Empirical Equations for Estimation
of Drag Coefficient, Shape Deformation, and Rising Velocity of Gas Bubbles or Liquid
Drops", Chemical Engineering Communications, Vol. 194, pp. 1623-1637, (2007).
[28] Baltussen, M., Kuipers, J., and Deen, N., "Direct Numerical Simulation of Effective Drag
in Dense gas–liquid–solid Three-phase Flows", Chemical Engineering Science, Vol.
158, pp. 561-568, (2017).
[29] Chung, B., Cohrs, M., Ernst, W., Galdi, G.,and Vaidya, A., "Wake–cylinder Interactions
of a Hinged Cylinder at Low and Intermediate Reynolds Numbers", Archive of Applied
Mechanics, Vol. 86, pp. 627-641, (2016).
[30] Wolf, A., Rauh, C., and Delgado, A., "Dynamics and Long-time Behavior of a Small
Bubble in Viscous Liquids with Applications to Food Rheology", Archive of Applied
Mechanics, Vol. 86, pp. 979-1002, (2016).
[31] Chen, Q., Liu, Y., Wu, Q., Wang, Y., Liu, T., and Wang, G., "Global Cavitation Patterns
and Corresponding Hydrodynamics of the Hydrofoil with Leading Edge Roughness",
Acta Mechanica Sinica, Vol. 36, pp. 1202-1214, (2020).
[32] Deng, C., Huang, W., Wang, H., Cheng, S., He, X., and Xu, B., "Preparation of Micron-
Sized Droplets and their Hydrodynamic Behavior in Quiescent Water", Brazilian
Journal of Chemical Engineering, Vol. 35, pp. 709-720, (2018).
[33] Hayashi, K.,and Tomiyama, A., "Effects of Surfactant on Lift Coefficients of Bubbles in
Linear Shear Flows", International Journal of Multiphase Flow, Vol. 99, pp. 86-93,
[34] Tagawa, Y., Takagi, S., and Matsumoto, Y., "Surfactant Effect on Path Instability of a
Rising Bubble", Journal of Fluid Mechanics, Vol. 738, pp. 124-142, (2014).
[35] Arkhipov, V., Vasenin, I., and Usanina, A., "Dynamics of Bubble Rising in the Presence
of Surfactants", Fluid Dynamics, Vol. 51, pp. 266-274, (2016).
[36] Li, X., Jiang, M., Huang, Z., and Zhou, Q., "Effect of Particle Orientation on the Drag
Force in Random Arrays of Oblate Ellipsoids in Low‐Reynolds‐number Flows",
AICHE Journal, Vol. 67, pp. e17040, (2021).
[37] Chen, L., and Dong, Y., "Numerical Investigation on Fluid Forces of Piggyback Circular
Cylinders in Tandem Arrangement at Low Reynolds Numbers", Acta Mechanica Sinica,
Vol. pp. 1-14, (2021).
[38] Ganguli, S., and Lele, S.K., "Drag of a Heated Sphere at Low Reynolds Numbers in the
Absence of Buoyancy", Journal of Fluid Mechanics, Vol. 869, pp. 264-291, (2019).
[39] Tomiyama, A., Celata, G., Hosokawa, S., and Yoshida, S., "Terminal Velocity of Single
Bubbles in Surface Tension Force Dominant Regime", International Journal of
Multiphase Flow, Vol. 28, pp. 1497-1519, (2002).
[40] Hasadi, Y.E., and Padding, J., "On the Existence of Logarithmic Terms in the Drag
Coefficient and Nusselt Number of a Single Sphere at High Reynolds Numbers", arXiv
preprint arXiv:200710214, Vol. pp. (2020).
[41] Feng, J., Li, X., Bao, Y., Cai, Z., Gao, Z., "Coalescence and Conjunction of Two In-line
Bubbles at Low Reynolds Numbers", Chemical Engineering Science, Vol. 141, pp. 261-
270, (2016).
[42] Genç, M.S., Karasu, İ., and Açıkel, H.H., "An Experimental Study on Aerodynamics of
NACA2415 Aerofoil at Low Re Numbers", Experimental Thermal and Fluid Science,
Vol. 39, pp. 252-264, (2012).
[43] Guet, S., Ooms, G., and Oliemans, R., "Influence of Bubble Size on the Transition from
Low-Re Bubbly Flow to Slug Flow in a Vertical Pipe", Experimental Thermal and Fluid
Science, Vol. 26, pp. 635-641, (2002).
[44] Li, S. b., Fan, J. g., Li, R. d., and Wang, L., "Effect of Surfactants on Hydrodynamics
Characteristics of Bubble in Shear Thinning Fluids at Low Reynolds Number", Journal
of Central South University, Vol. 25, pp. 805-811, (2018).
[45] Massoud, E., Xiao, Q., El-Gamal, H., and Teamah, M., "Numerical Study of an Individual
Taylor Bubble Rising through Stagnant Liquids under Laminar Flow Regime", Ocean
Engineering, Vol. 162, pp. 117-137, (2018).
[46] Zhou, Y., Zhao, C., and Bo, H., "Analyses and Modified Models for Bubble Shape and
Drag Coefficient Covering a Wide Range of Working Conditions", International Journal
of Multiphase Flow, Vol. 127, pp. 103265, (2020).
[47] Karimi, S., Shafiee, M., Ghadam, F., Abiri, A., and Abbasi, H., "Experimental Study on
Drag Coefficient of a Rising Bubble in the Presence of Rhamnolipid as a Biosurfactant",
Journal of Dispersion Science and Technology, Vol. pp. (2020). DOI
[48] Shi, P., Rzehak, R., Lucas, D., and Magnaudet, J., "Hydrodynamic Forces on a Clean
Spherical Bubble Translating in a Wall-bounded Linear Shear Flow", Physical Review
Fluids, Vol. 5, pp. 073601, (2020).
[49] Ekanayake, N.I., Berry, J.D., Stickland, A.D., Dunstan, D.E., Muir, I.L., Dower, S.K., and
Harvie, D.J., "Lift and Drag Forces Acting on a Particle Moving with Zero Slip in a
Linear Shear Flow Near a Wall", Journal of Fluid Mechanics, Vol. 904, pp. (2020).
[50] Chhabra, R.P., "Bubbles, Drops, and Particles in Non-Newtonian Fluids", ed.: CRC
[51] Clift, R., Grace, J.R., and Weber, M.E., "Bubbles, Drops, and Particles", ed.: Courier
[52] Unnikrishnan, A., and Chhabra, R., "An Experimental Study of Motion of Cylinders in
Newtonian Fluids: Wall Effects and Drag Coefficient", The Canadian Journal of
Chemical Engineering, Vol. 69, pp. 729-735, (1991).
[53] Fayzi, P., Bastani, D., Lotfi, M., and Ghamangiz Khararoodi, M., "The Effects of Bubble
Detachment Shape on Rising Bubble Hydrodynamics", Scientia Iranica, Vol. 26, pp.
1546-1554, (2019).
[54] Zheng, K., Li, C., Yan, X., Zhang, H., and Wang, L., "Prediction of Bubble Terminal
Velocity in Surfactant Aqueous Solutions", The Canadian Journal of Chemical
Engineering, Vol. 98, pp. 607-615, (2019).
[55] Bird, R.B., Stewart, W.E., and Lightfoot, E.N., "Transport Phenomena", ed.: John Wiley
& Sons, (2007).
[56] Abbasi, H., Hamedi, M.M., Lotfabad, T.B., Zahiri, H.S., Sharafi, H., Masoomi, F.,
Moosavi-Movahedi, A.A., Ortiz, A., Amanlou, M., and Noghabi, K.A., "Biosurfactantproducing
Bacterium, Pseudomonas Aeruginosa MA01 Isolated from Spoiled Apples:
Physicochemical and Structural Characteristics of Isolated Biosurfactant", Journal of
Bioscience and Bioengineering, Vol. 113, pp. 211-219, (2012).
[57] Mahmoudi, S., Hashemi Shahraki, B., and Aghajani, M., "Experimental and Theoretical
Investigation of CO2 and Air Bubble Rising Velocity through Kerosene and Distilled
Water in Bubble Column", Journal of Dispersion Science and Technology, Vol. 40, pp.
33-42, (2019).
[58] Ziqi, C., Yuyun, B., and Zhengming, G., "Hydrodynamic Behavior of a Single Bubble
Rising in Viscous Liquids", Chinese Journal of Chemical Engineering, Vol. 18, pp. 923-
930, (2010).
[59] Harmathy, T.Z., "Velocity of Large Drops and Bubbles in Media of Infinite or Restricted
Extent", Aiche J, Vol. 6, pp. 281-288, (1960).
[60] Davies, R., and Taylor, G.I., "The Mechanics of Large Bubbles Rising through Extended
Liquids and through Liquids in Tubes", Proceedings of the Royal Society of London
Series A Mathematical and Physical Sciences, Vol. 200, pp. 375-390, (1950).
[61] Vecer, M.,Lestinsky, P.,Wichterle, K., and Ruzicka, M., "On Bubble Rising in
Countercurrent Flow", International Journal of Chemical Reactor Engineering, Vol. 10,
pp. 1-21, (2012).
[62] Mendelson, H.D., "The Prediction of Bubble Terminal Velocities from Wave Theory",
AICHE Journal, Vol. 13, pp. 250-253, (1967).
[63] Liu, N.,Yang, Y.,Wang, J., Ju, B., Brantson, E.T., Tian, Y., Dong, Y., and Mahlalela, B.,
"Experimental Investigations of Single Bubble Rising in Static Newtonian Fluids as a
Function of Temperature Using a Modified Drag Coefficient", Natural Resources
Research, Vol. 29, pp. 2209–2226, (2020).
[64] Sun, B.,Guo, Y.,Wang, Z.,Yang, X.,Gong, P.,Wang, J., and Wang, N., "Experimental
Study on the Drag Coefficient of Single Bubbles Rising in Static non-Newtonian Fluids
in Wellbore", Journal of Natural Gas Science and Engineering, Vol. 26, pp. 867-872,
[65] Arkhipov, V.A.,Vasenin, I.,Tkachenko, A., and Usanina, A., "Unsteady Rise of a Bubble
in a Viscous Fluid at Small Reynolds Numbers", Fluid Dynamics, Vol. 50, pp. 79-86,
[66] Schiller, L., and Naumann, Z., "A Drag Coefficient Correlation", Z Ver Deutsch Ing, Vol.
77, pp. 318-320, (1935).
[67] Ishii, M., and Chawla, T., "Local Drag Laws in Dispersed Two-phase Flow", Nasa
Sti/Recon Technical Report N, Vol. 80, pp. 79-105, (1979).
[68] Turton, R., and Levenspiel, O., "A Short Note on the Drag Correlation for Spheres",
Powder technology, Vol. 47, pp. 83-86, (1986).
[69] Rodi, W., and Fueyo, N., "Engineering Turbulence Modelling and Experiments 5
Proceedings of the 5th International Symposium on Engineering Turbulence Modelling and
Measurements", Mallorca, Spain, 16–18 September, (2002).
[70] Kurimoto, R.,Hayashi, K., and Tomiyama, A., "Terminal Velocities of Clean and Fully-
Contaminated Drops in Vertical Pipes", International Journal of Multiphase Flow, Vol.
49, pp. 8-23, (2013).