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Bubble-mediated transfer of dilute gas in turbulence

Published online by Cambridge University Press:  14 June 2021

Palas Kumar Farsoiya Open the ORCID record for Palas Kumar Farsoiya [Opens in a new window] ,
Stéphane Popinet Open the ORCID record for Stéphane Popinet [Opens in a new window]  and
Luc Deike Open the ORCID record for Luc Deike [Opens in a new window]
Palas Kumar Farsoiya
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ08544, USA
Stéphane Popinet
Affiliation:
Institut Jean Le Rond d'Alembert, CNRS UMR 7190, Sorbonne Université, Paris75005, France
Luc Deike*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ08544, USA High Meadows Environmental Institute, Princeton University, Princeton, NJ08544, USA
*
†Email address for correspondence: [email protected]
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Abstract

Bubble-mediated gas exchange in turbulent flow is critical in bubble column chemical reactors as well as for ocean–atmosphere gas exchange related to air entrained by breaking waves. Understanding the transfer rate from a single bubble in turbulence at large Péclet numbers (defined as the ratio between the rate of advection and diffusion of gas) is important as it can be used for improving models on a larger scale. We characterize the mass transfer of dilute gases from a single bubble in a homogeneous isotropic turbulent flow in the limit of negligible bubble volume variations. We show that the mass transfer occurs within a thin diffusive boundary layer at the bubble–liquid interface, whose thickness decreases with an increase in turbulent Péclet number, $\widetilde {{Pe}}$. We propose a suitable time scale $\theta$ for Higbie (Trans. AIChE, vol. 31, 1935, pp. 365–389) penetration theory, $\theta = d_0/\tilde {u}$, based on $d_0$ the bubble diameter and $\tilde {u}$ a characteristic turbulent velocity, here $\tilde {u}=\sqrt {3}\,u_{{rms}}$, where $u_{{rms}}$ is the large-scale turbulence fluctuations. This leads to a non-dimensional transfer rate ${Sh} = 2(3)^{1/4}\sqrt {\widetilde {{Pe}}/{\rm \pi} }$ from the bubble in the isotropic turbulent flow. The theoretical prediction is verified by direct numerical simulations of mass transfer of dilute gas from a bubble in homogeneous and isotropic turbulence, and very good agreement is observed as long as the thin boundary layer is properly resolved.

JFM classification

Drops and Bubbles: Bubble dynamics Mass Transport: Coupled diffusion and flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 920 , 10 August 2021 , A34
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Berny, A., Deike, L., Séon, T. & Popinet, S. 2020 Role of all jet drops in mass transfer from bursting bubbles. Phys. Rev. Fluids 5 (3), 033605.CrossRefGoogle Scholar
Bird, R., Stewart, W. & Lightfoot, E. 2002 Transport Phenomena, 2nd edn. John Wiley & Sons.Google Scholar
Bothe, D. & Fleckenstein, S. 2013 A volume-of-fluid-based method for mass transfer processes at fluid particles. Chem. Engng Sci. 101, 283–302.CrossRefGoogle Scholar
Bothe, D., Koebe, M., Wielage, K., Prüss, J. & Warnecke, H.-J. 2004 Direct numerical simulation of mass transfer between rising gas bubbles and water. In Bubbly Flows, pp. 159–174. Springer.CrossRefGoogle Scholar
Boussinesq, J. 1905 Calcul du poivoir refroidissant des courants fluide. J. Math. Pure Appl. 60, 285–332.Google Scholar
Brackbill, J.U., Kothe, D.B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100 (2), 335–354.CrossRefGoogle Scholar
Cano-Lozano, J.C., Martinez-Bazan, C., Magnaudet, J. & Tchoufag, J. 2016 Paths and wakes of deformable nearly spheroidal rising bubbles close to the transition to path instability. Phys. Rev. Fluids 1 (5), 053604.CrossRefGoogle Scholar
Claassen, C.M.Y., Islam, S., Peters, E.A.J.F., Deen, N.G., Kuipers, J.A.M. & Baltussen, M.W. 2020 An improved subgrid scale model for front-tracking based simulations of mass transfer from bubbles. AIChE J. 66 (4), e16889.CrossRefGoogle Scholar
Clift, R., Grace, J.R. & Weber, M.E. 2005 Bubbles, Drops, and Particles. Courier Corporation.Google Scholar
Colombet, D., Legendre, D., Risso, F., Cockx, A. & Guiraud, P. 2015 Dynamics and mass transfer of rising bubbles in a homogenous swarm at large gas volume fraction. J. Fluid Mech. 763, 254–285.CrossRefGoogle Scholar
Darmana, D., Deen, N.G. & Kuipers, J.A.M. 2006 Detailed 3d modeling of mass transfer processes in two-phase flows with dynamic interfaces. Chem. Engng Technol. 29 (9), 1027–1033.CrossRefGoogle Scholar
Davidson, M.R. & Rudman, M. 2002 Volume-of-fluid calculation of heat or mass transfer across deforming interfaces in two-fluid flow. Numer. Heat Transfer 41 (3–4), 291–308.CrossRefGoogle Scholar
Deike, L. & Melville, W.K. 2018 Gas transfer by breaking waves. Geophys. Res. Lett. 45 (19), 10482–10492.CrossRefGoogle Scholar
Deike, L., Melville, W.K. & Popinet, S. 2016 Air entrainment and bubble statistics in breaking waves. J. Fluid Mech. 801, 91–129.CrossRefGoogle Scholar
Deising, D., Bothe, D. & Marschall, H. 2018 Direct numerical simulation of mass transfer in bubbly flows. Comput. Fluids 172, 524–537.CrossRefGoogle Scholar
Deising, D., Marschall, H. & Bothe, D. 2016 A unified single-field model framework for volume-of-fluid simulations of interfacial species transfer applied to bubbly flows. Chem. Engng Sci. 139, 173–195.CrossRefGoogle Scholar
Dodd, M.S. & Ferrante, A. 2016 On the interaction of Taylor length scale size droplets and isotropic turbulence. J. Fluid Mech. 806, 356–412.CrossRefGoogle Scholar
Dodd, M.S., Mohaddes, D., Ferrante, A. & Ihme, M. 2021 Analysis of droplet evaporation in isotropic turbulence through droplet-resolved dns. Intl J. Heat Mass Transfer 172, 121157.CrossRefGoogle Scholar
Elghobashi, S. 2019 a Direct numerical simulation of turbulent flows laden with droplets or bubbles. Annu. Rev. Fluid Mech. 51, 217–244.CrossRefGoogle Scholar
Elghobashi, S. 2019 b Direct numerical simulation of turbulent flows laden with droplets or bubbles. Annu. Rev. Fluid Mech. 51 (1), 217–244.CrossRefGoogle Scholar
Farsoiya, P., Popinet, S. & Deike, L. 2020 a Diffusion from static bubble. http://basilisk.fr/sandbox/farsoiya/static_bubble.c, [Online; accessed 02-Nov-2020].Google Scholar
Farsoiya, P., Popinet, S. & Deike, L. 2020 b Diffusion from rising bubble. http://basilisk.fr/sandbox/farsoiya/axi_rising_bubble.c, [Online; accessed 02-Nov-2020].Google Scholar
Farsoiya, P., Popinet, S. & Deike, L. 2020 c Gas transfer from bubble in isotropic turbulence. https://doi.org/10.1103/APS.DFD.2020.GFM.V0012, [Online; accessed 19-Nov-2020].CrossRefGoogle Scholar
Farsoiya, P.K., Mayya, Y.S. & Dasgupta, R. 2017 Axisymmetric viscous interfacial oscillations–theory and simulations. J. Fluid Mech. 826, 797–818.CrossRefGoogle Scholar
Farsoiya, P.K., Roy, A. & Dasgupta, R. 2020 Azimuthal capillary waves on a hollow filament – the discrete and the continuous spectrum. J. Fluid Mech. 883, A21.CrossRefGoogle Scholar
Fleckenstein, S. & Bothe, D. 2015 A volume-of-fluid-based numerical method for multi-component mass transfer with local volume changes. J. Comput. Phys. 301, 35–58.CrossRefGoogle Scholar
Fortescue, G.E. & Pearson, J.R.A. 1967 On gas absorption into a turbulent liquid. Chem. Engng Sci. 22 (9), 1163–1176.CrossRefGoogle Scholar
Gumulya, M., Utikar, R.P., Pareek, V.K., Evans, G.M. & Joshi, J.B. 2020 Dynamics of bubbles rising in pseudo-2d bubble column: effect of confinement and inertia. Chem. Engng J. 405, 126615.CrossRefGoogle Scholar
Haroun, Y., Legendre, D. & Raynal, L. 2010 Volume of fluid method for interfacial reactive mass transfer: application to stable liquid film. Chem. Engng Sci. 65 (10), 2896–2909.CrossRefGoogle Scholar
Herlina, H. & Wissink, J.G. 2016 Isotropic-turbulence-induced mass transfer across a severely contaminated water surface. J. Fluid Mech. 797, 665–682.CrossRefGoogle Scholar
Herlina, H. & Wissink, J.G. 2019 Simulation of air–water interfacial mass transfer driven by high-intensity isotropic turbulence. J. Fluid Mech. 860, 419–440.CrossRefGoogle Scholar
Higbie, R. 1935 The rate of absorption of a pure gas into a still liquid during short periods of exposure. Trans. AIChE 31, 365–389.Google Scholar
van Hooft, J.A., Popinet, S., van Heerwaarden, C.C., van der Linden, S.J.A., de Roode, S.R. & van de Wiel, B.J.H. 2018 Towards adaptive grids for atmospheric boundary-layer simulations. Boundary-Layer Meteorol. 167 (3), 421–443.CrossRefGoogle ScholarPubMed
Jia, H., Xiao, X. & Kang, Y. 2019 Investigation of a free rising bubble with mass transfer by an arbitrary Lagrangian–Eulerian method. Intl J. Heat Mass Transfer 137, 545–557.CrossRefGoogle Scholar
Karn, A., Monson, G.M., Ellis, C.R., Hong, J., Arndt, R.E.A. & Gulliver, J.S. 2015 Mass transfer studies across ventilated hydrofoils: a step towards hydroturbine aeration. Intl J. Heat Mass Transfer 87, 512–520.CrossRefGoogle Scholar
Katul, G. & Liu, H. 2017 Multiple mechanisms generate a universal scaling with dissipation for the air-water gas transfer velocity. Geophys. Res. Lett. 44 (4), 1892–1898.CrossRefGoogle Scholar
Keeling, R.F. 1993 On the role of large bubbles in air-sea gas exchange and supersaturation in the ocean. J. Mar. Res. 51 (2), 237–271.CrossRefGoogle Scholar
Levich, V.G. 1962 Physicochemical Hydrodynamics. Prentice Hall.Google Scholar
Liang, J.-H., McWilliams, J.C., Sullivan, P.P. & Baschek, B. 2011 Modeling bubbles and dissolved gases in the ocean. J. Geophys. Res.: Oceans 116 (C3), C03015.Google Scholar
Loisy, A., Naso, A. & Spelt, P.D.M. 2017 Buoyancy-driven bubbly flows: ordered and free rise at small and intermediate volume fraction. J. Fluid Mech. 816, 94–141.CrossRefGoogle Scholar
López-Herrera, J.M., Ganan-Calvo, A.M., Popinet, S. & Herrada, M.A. 2015 Electrokinetic effects in the breakup of electrified jets: a volume-of-fluid numerical study. Intl. J. Multiphase Flow 71, 14–22.CrossRefGoogle Scholar
Maes, J. & Soulaine, C. 2020 A unified single-field volume-of-fluid-based formulation for multi-component interfacial transfer with local volume changes. J. Comput. Phys. 402, 109024.CrossRefGoogle Scholar
Magnaudet, J. & Calmet, I. 2006 Turbulent mass transfer through a flat shear-free surface. J. Fluid Mech. 553, 155–185.CrossRefGoogle Scholar
Marić, T., Kothe, D.B. & Bothe, D. 2020 Unstructured un-split geometrical volume-of-fluid methods–a review. J. Comput. Phys. 420, 109695.CrossRefGoogle Scholar
Marschall, H., Hinterberger, K., Schüler, C., Habla, F. & Hinrichsen, O. 2012 Numerical simulation of species transfer across fluid interfaces in free-surface flows using openfoam. Chem. Engng Sci. 78, 111–127.CrossRefGoogle Scholar
Mathai, V., Lohse, D. & Sun, C. 2020 Bubbly and buoyant particle–laden turbulent flows. Annu. Rev. Condens. Matter Phys. 11, 529–559.CrossRefGoogle Scholar
Maxworthy, T., Gnann, C., Kürten, M. & Durst, F. 1996 Experiments on the rise of air bubbles in clean viscous liquids. J. Fluid Mech. 321, 421–441.CrossRefGoogle Scholar
Moore, D.W. 1963 The boundary layer on a spherical gas bubble. J. Fluid Mech. 16 (2), 161–176.CrossRefGoogle Scholar
Moore, D.W. 1965 The velocity of rise of distorted gas bubbles in a liquid of small viscosity. J. Fluid Mech. 23 (4), 749–766.CrossRefGoogle Scholar
Mostert, W. & Deike, L. 2020 Inertial energy dissipation in shallow-water breaking waves. J. Fluid Mech. 890, A12.CrossRefGoogle Scholar
Overholt, M.R. & Pope, S.B. 1996 Direct numerical simulation of a passive scalar with imposed mean gradient in isotropic turbulence. Phys. Fluids 8 (11), 3128–3148.CrossRefGoogle Scholar
Perrard, S., Rivière, A., Mostert, W. & Deike, L. 2021 Bubble deformation by a turbulent flow. arXiv:2011.10548.CrossRefGoogle Scholar
Pope, S.B. 2001 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228 (16), 5838–5866.CrossRefGoogle Scholar
Popinet, S. 2015 A quadtree-adaptive multigrid solver for the Serre–Green–Naghdi equations. J. Comput. Phys. 302, 336–358.CrossRefGoogle Scholar
Popinet, S. 2018 Numerical models of surface tension. Annu. Rev. Fluid Mech. 50, 49–75.CrossRefGoogle Scholar
Popinet, S. & collaborators 2013–2020 Basilisk. http://basilisk.fr.Google Scholar
Rachih, A., Legendre, D., Climent, E. & Charton, S. 2020 Numerical study of conjugate mass transfer from a spherical droplet at moderate Reynolds number. Intl J. Heat Mass Transfer 157, 119958.CrossRefGoogle Scholar
Reichl, B.G. & Deike, L. 2020 Contribution of sea-state dependent bubbles to air-sea carbon dioxide fluxes. Geophys. Res. Lett. 1, e2020GL087267.Google Scholar
Risso, F. 2018 Agitation, mixing, and transfers induced by bubbles. Annu. Rev. Fluid Mech. 50, 25–48.CrossRefGoogle Scholar
Rivière, A., Mostert, W., Perrard, S. & Deike, L. 2021 Sub-hinze scale bubble production in turbulent bubble break-up. J. Fluid Mech. 917, A40.CrossRefGoogle Scholar
Roghair, I. 2012 Direct numerical simulations of hydrodynamics and mass transfer in dense bubbly flows. PhD thesis, Eindhoven University of Technology.Google Scholar
Rosales, C. & Meneveau, C. 2005 Linear forcing in numerical simulations of isotropic turbulence: physical space implementations and convergence properties. Phys. Fluids 17 (9), 095106.CrossRefGoogle Scholar
Ruth, D., Mostert, W., Perrard, S. & Deike, L. 2019 Bubble pinch-off in turbulence. Proc. Natl Acad. Sci. USA 116 (51), 25412–25417.CrossRefGoogle ScholarPubMed
Sander, R. 2015 Compilation of Henry's law constants (version 4.0) for water as solvent. Atmos. Chem. Phys. 15 (8), 4399–4981.CrossRefGoogle Scholar
Sato, T., Jung, R.-T. & Abe, S. 2000 Direct simulation of droplet flow with mass transfer at interface. Trans. ASME J. Fluids Engng 122 (3), 510–516.CrossRefGoogle Scholar
Scapin, N., Costa, P. & Brandt, L. 2020 A volume-of-fluid method for interface-resolved simulations of phase-changing two-fluid flows. J. Comput. Phys. 407, 109251.CrossRefGoogle Scholar
Scardovelli, R. & Zaleski, S. 1999 Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31 (1), 567–603.CrossRefGoogle Scholar
Schumacher, J., Sreenivasan, K.R. & Yeung, P.K. 2005 Very fine structures in scalar mixing. J. Fluid Mech. 531, 113–122.CrossRefGoogle Scholar
Standart, G. 1964 The mass, momentum and energy equations for heterogeneous flow systems. Chem. Engng Sci. 19 (3), 227–236.CrossRefGoogle Scholar
Tanguy, S., Sagan, M., Lalanne, B., Couderc, F. & Colin, C. 2014 Benchmarks and numerical methods for the simulation of boiling flows. J. Comput. Phys. 264, 1–22.CrossRefGoogle Scholar
Taqieddin, A. 2018 Modeling of bubbles hydrodynamics and mass transfer in electrochemical gas-evolving systems. PhD thesis, Northeastern University.Google Scholar
Theofanous, T.G., Houze, R.N. & Brumfield, L.K. 1976 Turbulent mass transfer at free, gas-liquid interfaces, with applications to open-channel, bubble and jet flows. Intl J. Heat Mass Transfer 19 (6), 613–624.CrossRefGoogle Scholar
Treybal, R.E. 1980 Mass Transfer Operations, McGraw-Hill Chemical Engineering Series, vol. 466. McGraw-Hill.Google Scholar
Tryggvason, G., Scardovelli, R. & Zaleski, S. 2011 Direct Numerical Simulations of Gas–Liquid Multiphase Flows. Cambridge University Press.Google Scholar
Weiner, A. & Bothe, D. 2017 Advanced subgrid-scale modeling for convection-dominated species transport at fluid interfaces with application to mass transfer from rising bubbles. J. Comput. Phys. 347, 261–289.CrossRefGoogle Scholar
Whitman, W.G. 1923 The two-film theory of gas absorption. Chem. Metall. Engng 29, 146–148.Google Scholar
Wolfram Research, Inc. 2020 Mathematica, Version 12.1. https://www.wolfram.com/mathematica, Champaign, IL, USA.Google Scholar
Woolf, D.K. & Thorpe, S.A. 1991 Bubbles and the air-sea exchange of gases in near-saturation conditions. J. Mar. Res. 49 (3), 435–466.CrossRefGoogle Scholar
Yang, L., Peters, E.A.J.F., Fries, L., Harshe, Y.M., Kuipers, J.A.M. & Baltussen, M.W. 2020 Direct numerical simulation of mass transfer and mixing in complex two-phase systems using a coupled volume of fluid and immersed boundary method. Chem. Engng Sci. 5, 100059.Google Scholar