Skip to main content
SpringerLink
Log in

A Correlation for the Characteristic Velocity Ratio to Predict Hydrodynamics of Capillary Gas–Liquid Taylor Flow

  • Published:
Theoretical Foundations of Chemical Engineering Aims and scope Submit manuscript

Abstract

The ratio between bubble velocity and mean velocity of the two-phase flow is a key parameter in Taylor flow. Correlations for this characteristic velocity ratio that are valid over the entire range of attainable capillary numbers are missing so far. Here, we develop such a correlation for laminar gas–liquid Taylor flow in circular capillary channels. The proposed model is a two-parameter logistic function, which approaches the theoretical asymptotic limit of Bretherton at low capillary number. The correlation relies on prior known parameters such as channel diameter, fluid properties and gas/liquid volumetric flow rates only. In comparison with numerical and experimental data, it is accurate within ±5 and ±18%, respectively. The correlation should be useful to estimate various prior unknown hydrodynamics features of Taylor flow such as bubble velocity, mean liquid velocity, gas holdup, uniform liquid film thickness, bubble diameter, and streamline patterns in the liquid slug. The derived two-parameter logistic function may be useful to develop similar correlations for non-circular channels and liquid–liquid Taylor flow.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.

Similar content being viewed by others

REFERENCES

  1. Kreutzer, M.T., Kapteijn, F., Moulijn, J.A., and Heiszwolf, J.J., Multiphase monolith reactors: Chemical reaction engineering of segmented flow in microchannels, Chem. Eng. Sci., 2005, vol. 60, no. 22, p. 5895.

    Article  CAS  Google Scholar 

  2. Triplett, K.A., Ghiaasiaan, S.M., Abdel-Khalik, S.I., and Sadowski, D.L., Gas-liquid two-phase flow in microchannels—Part I: Two-phase flow patterns, Int. J. Multiphase Flow, 1999, vol. 25, no. 3, p. 377.

    Article  CAS  Google Scholar 

  3. Günther, A., Jhunjhunwala, M., Thalmann, M., Schmidt, M.A., and Jensen, K.F., Micromixing of miscible liquids in segmented gas-liquid flow, Langmuir, 2005, vol. 21, no. 4, p. 1547.

    Article  CAS  Google Scholar 

  4. Haase, S., Murzin, D.Y., and Salmi, T., Review on hydrodynamics and mass transfer in minichannel wall reactors with gas-liquid Taylor flow, Chem. Eng. Res. Des., 2016, vol. 113, p. 304.

    Article  CAS  Google Scholar 

  5. Abiev, R., Svetlov, S., and Haase, S., Hydrodynamics and mass transfer of gas-liquid and liquid-liquid Taylor flow in microchannels, Chem. Eng. Technol., 2017, vol. 40, no. 11, p. 1985.

    Article  CAS  Google Scholar 

  6. Angeli, P. and Gavriilidis, A., Hydrodynamics of Taylor flow in small channels: A review, Proc. Inst. Mech. Eng.,Part C, 2008, vol. 222, no. 5, p. 737.

    CAS  Google Scholar 

  7. Howard, J.A. and Walsh, P.A., Review and extensions to film thickness and relative bubble drift velocity prediction methods in laminar Taylor or slug flows, Int. J. Multiphase Flow, 2013, vol. 55, p. 32.

    Article  CAS  Google Scholar 

  8. Giavedoni, M.D. and Saita, F.A., The axisymmetric and plane cases of a gas phase steadily displacing a Newtonian liquid—A simultaneous solution of the governing equations, Phys. Fluids, 1997, vol. 9, no. 8, p. 2420.

    Article  CAS  Google Scholar 

  9. Magnini, M., Ferrari, A., Thome, J.R., and Stone, H.A., Undulations on the surface of elongated bubbles in confined gas-liquid flows, Phys. Rev. Fluids, 2017, vol. 2, no. 8, p. 084001.

    Article  Google Scholar 

  10. Giavedoni, M.D. and Saita, F.A., The rear meniscus of a long bubble steadily displacing a Newtonian liquid in a capillary tube, Phys. Fluids, 1999, vol. 11, no. 4, p. 786.

    Article  CAS  Google Scholar 

  11. Goldsmith, H.L. and Mason, S.G., Flow of suspensions through tubes. II. Single large bubbles, J. Colloid Sci., 1963, vol. 18, no. 3, p. 237.

    Article  CAS  Google Scholar 

  12. Olbricht, W.L. and Kung, D.M., The deformation and breakup of liquid-drops in low Reynolds-number flow through a capillary, Phys. Fluids A, 1992, vol. 4, no. 7, p. 1347.

    Article  CAS  Google Scholar 

  13. Lac, E. and Sherwood, J.D., Motion of a drop along the centreline of a capillary in a pressure-driven flow, J. Fluid Mech., 2009, vol. 640, p. 27.

    Article  Google Scholar 

  14. Suo, M. and Griffith, P., Two-phase flow in capillary tubes, J. Basic Eng., 1964, vol. 86, no. 3, p. 576.

    Article  CAS  Google Scholar 

  15. Wörner, M., A key parameter to characterize Taylor flow in narrow circular and rectangular channels, 7th International Conference on Multiphase Flow (ICMF 2010), Gainesville, Fla.: Univ. of Florida, 2010,

  16. Lizarraga-Garcia, E., Buongiorno, J., Al-Safran, E., and Lakehal, D., A broadly-applicable unified closure relation for Taylor bubble rise velocity in pipes with stagnant liquid, Int. J. Multiphase Flow, 2017, vol. 89, p. 345.

    Article  CAS  Google Scholar 

  17. Leung, S.S.Y., Gupta, R., Fletcher, D.F., and Haynes, B.S., Gravitational effect on Taylor flow in horizontal microchannels, Chem. Eng. Sci., 2012, vol. 69, no. 1, p. 553.

    Article  CAS  Google Scholar 

  18. Magnini, M., Khodaparast, S., Matar, O.K., Stone, H.A., and Thome, J.R., Dynamics of long gas bubbles rising in a vertical tube in a cocurrent liquid flow, Phys. Rev. Fluids, 2019, vol. 4, no. 2, p. 023601.

    Article  Google Scholar 

  19. Schwartz, L.W., Princen, H.M., and Kiss, A.D., On the motion of bubbles in capillary tubes, J. Fluid Mech., 1986, vol. 172, p. 259.

    Article  CAS  Google Scholar 

  20. Cherukumudi, A., Klaseboer, E., Khan, S.A., and Manica, R., Prediction of the shape and pressure drop of Taylor bubbles in circular tubes, Microfluid. Nanofluid., 2015, vol. 19, no. 5, p. 1221.

    Article  CAS  Google Scholar 

  21. Kurimoto, R., Nakazawa, K., Minagawa, H., and Yasuda, T., Prediction models of void fraction and pressure drop for gas-liquid slug flow in microchannels, Exp. Therm. Fluid Sci., 2017, vol. 88, p. 124.

    Article  CAS  Google Scholar 

  22. Abiev, R.S., Simulation of the slug flow of a gas-liquid system in capillaries, Theor. Found. Chem. Eng., 2008, vol. 42, no. 2, p. 105.

    Article  CAS  Google Scholar 

  23. Taylor, G.I., Deposition of a viscous fluid on the wall of a tube, J. Fluid Mech., 1961, vol. 10, no. 2, p. 161.

    Article  Google Scholar 

  24. Bretherton, F.P., The motion of long bubbles in tubes, J. Fluid Mech., 1961, vol. 10, no. 2, p. 166.

    Article  Google Scholar 

  25. Reinelt, D.A. and Saffman, P.G., The penetration of a finger into a viscous-fluid in a channel and tube, SIAM J. Sci. Stat. Comp., 1985, vol. 6, no. 3, p. 542.

    Article  Google Scholar 

  26. Shen, E.I. and Udell, K.S., A finite-element study of low Reynolds-number two-phase flow in cylindrical-tubes, J. Appl. Mech., 1985, vol. 52, no. 2, p. 253.

    Article  Google Scholar 

  27. Halpern, D. and Gaver, D.P., Boundary-element analysis of the time-dependent motion of a semiinfinite bubble in a channel, J. Comput. Phys., 1994, vol. 115, no. 2, p. 366.

    Article  Google Scholar 

  28. Martinez, M.J. and Udell, K.S., Boundary integral analysis of the creeping flow of long bubbles in capillaries, J. Appl. Mech., 1989, vol. 56, no. 1, p. 211.

    Article  CAS  Google Scholar 

  29. Cox, B.G., An experimental investigation of the streamlines in viscous fluid expelled from a tube, J. Fluid Mech., 1964, vol. 20, no. 2, p. 193.

    Article  Google Scholar 

  30. Thulasidas, T.C., Abraham, M.A., and Cerro, R.L., Flow patterns in liquid slugs during bubble-train flow inside capillaries, Chem. Eng. Sci., 1997, vol. 52, no. 17, p. 2947.

    Article  CAS  Google Scholar 

  31. Kececi, S., Wörner, M., Onea, A., and Soyhan, H.S., Recirculation time and liquid slug mass transfer in co-current upward and downward Taylor flow, Catal. Today, 2009, vol. 147S, p. S125.

    Article  CAS  Google Scholar 

  32. Aussillous, P. and Quere, D., Quick deposition of a fluid on the wall of a tube, Phys. Fluids, 2000, vol. 12, no. 10, p. 2367.

    Article  CAS  Google Scholar 

  33. Klaseboer, E., Gupta, R., and Manica, R., An extended Bretherton model for long Taylor bubbles at moderate capillary numbers, Phys. Fluids, 2014, vol. 26, no. 3, p. 032 107.

    Article  CAS  Google Scholar 

  34. Balestra, G., Zhu, L., and Gallaire, F., Viscous Taylor droplets in axisymmetric and planar tubes: From Bretherton’s theory to empirical models, Microfluid. Nanofluid., 2018, vol. 22, no. 6, p. 67.

    Article  CAS  Google Scholar 

  35. de Ryck, A., The effect of weak inertia on the emptying of a tube, Phys. Fluids, 2002, vol. 14, no. 7, p. 2102.

    Article  CAS  Google Scholar 

  36. Heil, M., Finite Reynolds number effects in the Bretherton problem, Phys. Fluids, 2001, vol. 13, no. 9, p. 2517.

    Article  CAS  Google Scholar 

  37. Han, Y. and Shikazono, N., Measurement of the liquid film thickness in micro tube slug flow, Int. J. Heat Fluid Flow, 2009, vol. 30, no. 5, p. 842.

    Article  CAS  Google Scholar 

  38. Langewisch, D.R. and Buongiorno, J., Prediction of film thickness, bubble velocity, and pressure drop for capillary slug flow using a CFD-generated database, Int. J. Heat Fluid Flow, 2015, vol. 54, p. 250.

    Article  Google Scholar 

  39. Kurimoto, R., Hayashi, K., Minagawa, H., and Tomiyama, A., Numerical investigation of bubble shape and flow field of gas–liquid slug flow in circular microchannels, Int. J. Heat Fluid Flow, 2018, vol. 74, p. 28.

    Article  Google Scholar 

  40. Fairbrother, F. and Stubbs, A.E., Studies in electro-endosmosis. Part VI. The “bubble-tube” method of measurement, J. Chem. Soc., 1935, p. 527.

  41. Liu, H., Vandu, C.O., and Krishna, R., Hydrodynamics of Taylor flow in vertical capillaries: Flow regimes, bubble rise velocity, liquid slug length, and pressure drop, Ind. Eng. Chem. Res., 2005, vol. 44, no. 14, p. 4884.

    Article  CAS  Google Scholar 

  42. Kreutzer, M.T., Kapteijn, F., Moulijn, J.A., Kleijn, C.R., and Heiszwolf, J.J., Inertial and interfacial effects on pressure drop of Taylor flow in capillaries, AlChE J., 2005, vol. 51, no. 9, p. 2428.

    Article  CAS  Google Scholar 

  43. Abiev, R.S., Bubbles velocity, Taylor circulation rate and mass transfer model for slug flow in milli- and microchannels, Chem. Eng. J., 2013, vol. 227, p. 66.

    Article  CAS  Google Scholar 

  44. Zuber, N. and Findlay, J.A., Average volumetric concentration in two-phase flow systems, J. Heat Transfer, 1965, vol. 87, no. 4, p. 453.

    Article  CAS  Google Scholar 

  45. Nicklin, D.J., Wilke, J.O., and Davidson, J.F., Two-phase flow in vertical tubes, Trans. Inst. Chem. Eng., 1962, vol. 40, p. 61.

    CAS  Google Scholar 

  46. Yuster, S.T., Theoretical considerations of multiphase flow in idealized capillary systems, Proceedings of the Third World Petroleum Congress. Section II. Drilling and Production, Schurink, H.B.J. et al., Eds., Leiden: E.J. Brill, 1951, p. 437.

  47. Russell, T.W.F. and Charles, M.E., The effect of the less viscous liquid in the laminar flow of two immiscible liquids, Can. J. Chem. Eng., 1959, vol. 37, no. 1, p. 18.

    Article  CAS  Google Scholar 

  48. Makuch, K., Gorce, J.-B., and Garstecki, P., Non-wetting droplets in capillaries of circular cross-section: Scaling function, Phys. Fluids, 2019, vol. 31, no. 4, p. 043 102.

    Article  CAS  Google Scholar 

  49. Cox, B.G., On driving a viscous fluid out of a tube, J. Fluid Mech., 1962, vol. 14, no. 1, p. 81.

    Article  Google Scholar 

  50. Dupont, J.-B., Legendre, D., and Fabre, J., Motion and shape of long bubbles in small tube at low Re-number, International Conference on Multiphase Flow (ICMF 2007), Sommerfeld, M., Ed., Leipzig, 2007.

  51. Gottschalk, P.G. and Dunn, J.R., The five-parameter logistic: A characterization and comparison with the four-parameter logistic, Anal. Biochem., 2005, vol. 343, no. 1, p. 54.

    Article  CAS  Google Scholar 

  52. Asadolahi, A.N., Gupta, R., Leung, S.S.Y., Fletcher, D.F., and Haynes, B.S., Validation of a CFD model of Taylor flow hydrodynamics and heat transfer, Chem. Eng. Sci., 2012, vol. 69, no. 1, p. 541.

    Article  CAS  Google Scholar 

  53. Butler, C., Cid, E., and Billet, A.M., Modelling of mass transfer in Taylor flow: Investigation with the PLIF-I technique, Chem. Eng. Res. Des., 2016, vol. 115, p. 292.

    Article  CAS  Google Scholar 

  54. Carroll, R.M. and Gupta, N.R., Inertial and surfactant effects on the steady droplet flow in cylindrical channels, Phys. Fluids, 2014, vol. 26, no. 12, p. 122102.

    Article  CAS  Google Scholar 

  55. Khodaparast, S., Magnini, M., Borhani, N., and Thome, J.R., Dynamics of isolated confined air bubbles in liquid flows through circular microchannels: An experimental and numerical study, Microfluid. Nanofluid., 2015, vol. 19, no. 1, p. 209.

    Article  CAS  Google Scholar 

  56. Soares, E.J., Thompson, R.L., and Niero, D.C., Immiscible liquid–liquid pressure-driven flow in capillary tubes: Experimental results and numerical comparison, Phys. Fluids, 2015, vol. 27, no. 8, p. 082105.

    Article  CAS  Google Scholar 

  57. Abiev, R.S. and Dymov, A.V., Modeling the hydrodynamics of slug flow in a minichannel for liquid-liquid two-phase system, Theor. Found. Chem. Eng., 2013, vol. 47, no. 4, p. 299.

    Article  CAS  Google Scholar 

  58. Keskin, Ö., Wörner, M., Soyhan, H.S., Bauer, T., Deutschmann, O., and Lange, R., Viscous co-current downward Taylor flow in a square mini-channel, AlChE J., 2010, vol. 56, no. 7, p. 1693.

    Article  CAS  Google Scholar 

Download references

ACKNOWLEDGMENTS

The author thanks Prof. R. Abiev for the invitation to contribute to this special issue. While the idea for the present study dates back to the ICMF conference in 2010, the research would have probably never been completed and published without this invitation. In addition, the author thanks G. Balestra for providing numerical data on request and R. Kurimoto, D.R. Langewisch and J. Buongiorno for additional information related to their publications. The useful comments of the anonymous referees are also acknowledged.

Author information

Authors and Affiliations

  1. Karlsruhe Institute of Technology (KIT), Institute for Catalysis Research and Technology, 76131, Karlsruhe, Germany

    Martin Wörner

Authors
  1. Martin Wörner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Wörner.

Additional information

Special issue: “Two-phase flows in microchannels: hydrodynamics, heat and mass transfer, chemical reactions”. Edited by R.Sh. Abiev

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Martin Wörner A Correlation for the Characteristic Velocity Ratio to Predict Hydrodynamics of Capillary Gas–Liquid Taylor Flow. Theor Found Chem Eng 54, 3–16 (2020). https://doi.org/10.1134/S0040579520010236

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040579520010236

Keywords:

Search

Navigation