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Considerations on bubble fragmentation models

Published online by Cambridge University Press:  01 October 2010

C. MARTÍNEZ-BAZÁN ,
J. RODRÍGUEZ-RODRÍGUEZ ,
G. B. DEANE ,
J. L. MONTAÑES  and
J. C. LASHERAS
C. MARTÍNEZ-BAZÁN*
Affiliation:
Área de Mecánica de Fluidos, Departamento de Ingeniería Mecánica y Minera, Universidad Jaén, Campus de las Lagunillas, 23071 Jaén, Spain
J. RODRÍGUEZ-RODRÍGUEZ
Affiliation:
Área de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid, 28911 Leganés, Spain
G. B. DEANE
Affiliation:
Marine Physical Laboratory, Scripps Institution of Oceanography, La Jolla, CA 92093-0238, USA
J. L. MONTAÑES
Affiliation:
Departamento de Motopropulsión y Termofluidodinámica, Universidad Politécnica de Madrid, Plaza del Cardenal Cisneros 3, 28040 Madrid, Spain
J. C. LASHERAS
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego. La Jolla, CA 92093-0411, USA
*
†Email address for correspondence: [email protected]
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Abstract

In this paper we describe the restrictions that the probability density function (p.d.f.) of the size of particles resulting from the rupture of a drop or bubble must satisfy. Using conservation of volume, we show that when a particle of diameter, D0, breaks into exactly two fragments of sizes D and D2 = (D30−D3)1/3 respectively, the resulting p.d.f., f(D; D0), must satisfy a symmetry relation given by D22f(D; D0) = D2f(D2; D0), which does not depend on the nature of the underlying fragmentation process. In general, for an arbitrary number of resulting particles, m(D0), we determine that the daughter p.d.f. should satisfy the conservation of volume condition given by m(D0) ∫0D0 (D/D0)3f(D; D0) dD = 1. A detailed analysis of some contemporary fragmentation models shows that they may not exhibit the required conservation of volume condition if they are not adequately formulated. Furthermore, we also analyse several models proposed in the literature for the breakup frequency of drops or bubbles based on different principles, g(ϵ, D0). Although, most of the models are formulated in terms of the particle size D0 and the dissipation rate of turbulent kinetic energy, ϵ, and apparently provide different results, we show here that they are nearly identical when expressed in dimensionless form in terms of the Weber number, g*(Wet) = g(ϵ, D0) D2/30 ϵ−1/3, with Wet ~ ρ ϵ2/3D05/3/σ, where ρ is the density of the continuous phase and σ the surface tension.

JFM classification

Drops and Bubbles: Breakup/coalescence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 661 , 25 October 2010 , pp. 159 - 177
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Coulaloglou, C. A. & Tavlarides, L. L. 1977 Description of interaction processes in agitated liquid-liquid dispersions. Chem. Engng Sci. 32, 1289–1297.CrossRefGoogle Scholar
Deane, G. B. & Stokes, M. D. 2002 Scale dependence of bubble creation mechanisms in breaking waves. Nature 418, 839–844.CrossRefGoogle ScholarPubMed
Diemer, R. B. & Olson, J. H. 2002 A moment methodology for coagulation and breakage problems. Part 3. Generalized daughter distribution functions. Chem. Engng Sci. 57, 4187–4198.CrossRefGoogle Scholar
Hesketh, R. P., Etchells, A. W. & Russell, T. W. F. 1991 Bubble breakage in pipeline flow. Chem. Engng Sci. 46, 1–9.CrossRefGoogle Scholar
Hinze, J. O. 1955 Fundamentals of the hydrodynamics mechanisms of splitting in dispersion process. AIChE J. 1, 289–295.CrossRefGoogle Scholar
Kolev, N. I. 1993 Fragmentation and coalescence dynamics in multiphase flows. Exp. Therm. Fluid Sci. 6, 211–251.Google Scholar
Kolmogorov, A. N. 1949 On the breakage of drops in a turbulent flow. Dokl. Akad. Navk. SSSR 66, 825–828.Google Scholar
Konno, M., Aoki, M. & Saito, S. 1983 Scale effect on breakup process in liquid-liquid agitated tanks. J. Chem. Engng Japan 16, 312–319.CrossRefGoogle Scholar
Konno, M., Matsunaga, Y., Arai, K. & Saito, S. 1980 Simulation model for breakup process in an agitated tank. J. Chem. Engng Japan 13, 67–73.CrossRefGoogle Scholar
Lasheras, J. C., Eastwood, C., Martínez-Bazán, C. & Montañes, J. L. 2002 A review of statistical models for the break-up of an immiscible fluid immersed into a fully developed turbulent flow. Intl J. Multiph. Flow 28, 247–278.CrossRefGoogle Scholar
Luo, H. & Svensen, H. F. 1996 Theoretical model for drop and bubble breakup in turbulent dispersions. AIChE J. 42, 1225–1233.CrossRefGoogle Scholar
Martínez-Bazán, C., Montañes, J. L. & Lasheras, J. C. 1999a On the breakup of an air bubble injected into a fully developed turbulent flow. Part 1. Breakup frequency. J. Fluid Mech. 401, 157–182.CrossRefGoogle Scholar
Martínez-Bazán, C., Montañes, J. L. & Lasheras, J. C. 1999b On the breakup of an air bubble injected into a fully developed turbulent flow. Part 2. Size PDF of the resulting daughter bubbles. J. Fluid Mech. 401, 183–207.CrossRefGoogle Scholar
Melville, W. K. 1996 The role of surface-wave breaking in air-sea interaction. Annu. Rev. Fluid Mech. 18, 279–321.CrossRefGoogle Scholar
Prince, M. J. & Blanch, H. W. 1990 Bubble coalescence and break-up in air-sparged bubble columns. AIChE J. 36, 1485–1499.CrossRefGoogle Scholar
Rodríguez-Rodríguez, J. 2004 Estudio de la rotura de gotas y burbujas en flujos a altos números de Reynolds. PhD thesis, Universidad Carlos III de Madrid, Madrid, Spain.Google Scholar
Rodríguez-Rodríguez, J., Gordillo, J. M. & Martínez-Bazán, C. 2006 Breakup time and morphology of drops and bubbles in a high-Reynolds-number flow. J. Fluid Mech. 548, 69–86.CrossRefGoogle Scholar
Rodríguez-Rodríguez, J., Martínez-Bazán, C. & Montañes, J. L. 2003 A novel particle tracking and break-up detection algorithm: application to the turbulent break-up of bubbles. Meas. Sci. Technol. 14, 1328–1340.CrossRefGoogle Scholar
Tsouris, C. & Tavlarides, L. L. 1994 Breakage and coalescence models for drops in turbulent dispersions. AIChE J. 40, 395–406.CrossRefGoogle Scholar
Wang, T. F., Wang, J. F. & Jin, Y. 2003 A novel theoretical breakup kernel function for bubbles/droplets in turbulent flows. Chem. Engng Sci. 58, 4629–4637.CrossRefGoogle Scholar
Williams, F. A. 1985 Combustion Theory, 2nd edn. Addison-Wesley.Google Scholar
Zaccone, A., Gäbler, A., Maab, S., Marchisio, D. & Kraume, M. 2007 Drop breakage in liquid-liquid stirred dispersions: modelling of single drop breakage. Chem. Engng Sci. 62, 6297–6307.CrossRefGoogle Scholar