Abstract
The vast majority of theoretical and numerical studies of buoyancy-thermocapillary convection use one-sided models [9, 19–22] that only consider the convection in the liquid phase, while ignoring transport through the gas phase and the phase change across the liquid-gas interface. The heat and mass transport in the gas phase are incorporated indirectly through boundary conditions at the liquid-gas interface, and the interface is assumed to be rigid (and, in most of the cases, flat). Such an approach might be justifiable for nonvolatile liquids, since air is a relatively poor conductor of heat, and for the volatile liquids at ambient (atmospheric) conditions when phase change is strongly suppressed. Indeed, the predictions of such models are for the most part consistent with experimental studies of volatile and nonvolatile fluids at ambient (atmospheric) conditions [1, 9–12]. However, for volatile liquids at reduced air pressure, phase change can lead to significant heat fluxes in the liquid layer due to the latent heat released or absorbed at the interface, and the interfacial mass flux (which defines the latent heat flux) cannot be computed reliably without a proper model of bulk mass transport in the gas phase. Therefore a two-sided model is required, where the heat and mass transport in both phases are modeled explicitly. Two-sided models have been formulated previously [49, 50, 82, 87–91]. These models, however, assume rather than compute the shape of the liquid-gas interface, employ extremely restrictive assumptions and/or use a very crude description of one of the two phases.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Y. Li, R.O. Grigoriev, M. Yoda, Experimental study of the effect of noncondensables on buoyancy-thermocapillary convection in a volatile low-viscosity silicone oil. Phys. Fluids 26, 122112 (2014)
D. Villers, J.K. Platten, Coupled buoyancy and marangoni convection in acetone: experiments and comparison with numerical simulations. J. Fluid Mech. 234, 487–510 (1992)
R.J. Riley, G.P. Neitzel, Instability of thermocapillary–buoyancy convection in shallow layers. Part 1. Characterization of steady and oscillatory instabilities. J. Fluid Mech. 359, 143–164 (1998)
H. Ben Hadid, B. Roux, Buoyancy- and thermocapillary-driven flows in differentially heated cavities for low-Prandtl-number fluids. J. Fluid Mech. 235, 1–36 (1992)
V.M. Shevtsova, A.A. Nepomnyashchy, J.C. Legros, Thermocapillary-buoyancy convection in a shallow cavity heated from the side. Phys. Rev. E 67, 066308 (2003)
A. Oron, S.H. Davis, S.G. Bankoff, Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69(3), 931–980 (1997)
K. Kafeel, A. Turan, Axi-symmetric simulation of a two phase vertical thermosyphon using Eulerian two-fluid methodology. Heat Mass Transf. 49, 1089–1099 (2013)
B. Fadhl, L.C. Wrobel, H. Jouhara, Numerical modelling of the temperature distribution in a two-phase closed thermosyphon. Appl. Therm. Eng. 60, 122–131 (2013)
F.M. White, Viscous Fluid Flow, 2nd edn. (McGraw-Hill, New York, 1991)
J. Zhang, S.J. Watson, H. Wong, Fluid flow and heat transfer in a dual-wet micro heat pipe. J. Fluid Mech. 589, 1–31 (2007)
H. Wang, J.Y. Murthy, S.V. Garimella, Transport from a volatile meniscus inside an open microtube. Int. J. Heat Mass Transf. 51, 3007–3017 (2008)
H. Wang, Z. Pan, S.V. Garimella, Numerical investigation of heat and mass transfer from an evaporating meniscus in a heated open groove. Int. J. Heat Mass Transfer 54, 3015–3023 (2011)
T. Kaya, J. Goldak, Three-dimensional numerical analysis of heat and mass transfer in heat pipes. Heat Mass Transf. 43, 775–785 (2007)
S. Chapman, T. Cowling, The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction, and Diffusion in Gases (Cambridge University Press, Cambridge, 1990)
R.W. Schunk, Mathematical structure of transport equations for multispecies flows. Rev. Geophys. 15(4), 429–445 (1977)
H. Grad, On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2(4), 331–407 (1949)
B.B. Hamel, Two-fluid hydrodynamic equations for a neutral, disparate-mass, binary mixture. Phys. Fluids 9, 12 (1966)
M.J. Moran, H.N. Shapiro, Fundamentals of Engineering Thermodynamics (Wiley, New York, 2004)
C.L. Yaws, Yaws’ Handbook of Thermodynamic and Physical Properties of Chemical Compounds (Electronic Edition): Physical, Thermodynamic and Transport Properties for 5,000 Organic Chemical Compounds (Knovel, Norwich, 2003)
C.L. Yaws, Yaws’ Thermophysical Properties of Chemicals and Hydrocarbons (Electronic Edition) (Knovel, Norwich, 2009)
R. Reid, J. Prausnitz, B. Poling, The Properties of Gases and Liquids. Chemical Engineering Series (McGraw-Hill, New York, 1987)
K.E. Grew, T.L. Ibbs, Thermal Diffusion in Gases. Cambridge Monographs on Physics (Cambridge University Press, Cambridge, 1952)
L. Landau, E. Lifshitz, Fluid Mechanics (Elsevier, Amsterdam, 2013)
R.B. Bird, Dynamics of Polymeric Liquids (Wiley, New York, 1977)
J. Hirschfelder, C. Curtiss, R. Bird, Molecular Theory of Gases and Liquids. Structure of Matter Series (Wiley, New York, 1954)
O.L. Flaningam, Vapor pressures of poly(dimethylsiloxane) oligomers. J. Chem. Eng. Data 31, 266–272 (1986)
C.R. Wilke, Viscosity equation for gas mixtures. J. Chem. Phys. 18, 517–519 (1950)
W. Sutherland, LII. The viscosity of gases and molecular force. Philos. Mag. Ser. 5 36(223), 507–531 (1893)
F. White, Heat and Mass Transfer. Addison-Wesley Series in Mechanical Engineering (Addison-Wesley, Reading, MA, 1988)
R.W. Schrage, A Theoretical Study of Interface Mass Transfer (Columbia University Press, New York, 1953)
G. Wyllie, Evaporation and surface structure of liquids. Proc. R. Soc. Lond. 197, 383–395 (1949)
R. Rudolf, M. Itoh, Y. Viisanen, P. Wagner, Sticking probabilities for condensation of polar and nonpolar vapor molecules, in Nucleation and Atmospheric Aerosols, ed. by N. Fukuta, P.E. Wagner (A. Deepak Publishing, Hampton, 1992), pp. 165–168
G. Balekjian, D.L. Katz, Heat transfer from superheated vapors to a horizontal tube. AIChE J. 4(1), 43–48 (1958)
J. Klentzman, V.S. Ajaev, The effect of evaporation on fingering instabilities. Phys. Fluids 21, 122101 (2009)
S. Kjelstrup, D. Bedeaux, Non-Equilibrium Thermodynamics of Heterogeneous Systems (World Scientific, Singapore, 2008)
H. Struchtrup, S. Kjelstrup, D. Bedeaux, Temperature-difference-driven mass transfer through the vapor from a cold to a warm liquid. Phys. Rev. E 85, 061201 (2012)
C.A. Ward, R.D. Findlay, M. Rizk, Statistical rate theory of interfacial transport. I. Theoretical development. J. Chem. Phys. 76, 5599 (1982)
C.A. Ward, G. Fang, Expression for predicting liquid evaporation flux: statistical rate theory approach. Phys. Rev. E 59, 429–440 (1999)
G. Fang, C.A. Ward, Temperature measured close to the interface of an evaporating liquid. Phys. Rev. E 59, 417–428 (1999)
V.K. Badam, V. Kumar, F. Durst, K. Danov, Experimental and theoretical investigations on interfacial temperature jumps during evaporation. Exp. Thermal Fluid Sci. 32, 276–292 (2007)
M. Bond, H. Struchtrup, Mean evaporation and condensation coefficients based on energy dependent condensation probability. Phys. Rev. E 70, 061605 (2004)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Qin, T. (2017). Mathematical Model. In: Buoyancy-Thermocapillary Convection of Volatile Fluids in Confined and Sealed Geometries. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-61331-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-61331-4_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-61330-7
Online ISBN: 978-3-319-61331-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)