The Influence of Curvature on the Modelling of Droplet Evaporation at Different Scales

  • Grazia LamannaEmail author
  • Gianpietro Elvio Cossali
  • Simona Tonini
Conference paper
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 121)


The evaporation of liquid drops in stagnant gaseous environment is modelled, accounting for the effect of drop curvature and size at the macro- and microscopic scales. At the macro-scale level, the validity of the conjectured dependence of the local fluxes on the drop surface curvature is analysed. Analytical solutions to the gas-phase conservation equations for five drop shapes (sphere, oblate and prolate spheroids and inverse oblate and prolate spheroids), under uniform Dirichlet boundary conditions, are used to calculate the local vapour and heat fluxes. The analysis shows that in general non-dimensional fluxes do not solely depend on local curvature, but possibly the effect of the whole drop shape must be taken into account. At the micro-scale level, the equilibrium vapour pressure at a convex curved surface is higher than that at a flat surface, thus leading to a considerable enhancement of the evaporation rate for nanometre sized droplets. To model the increase in equilibrium vapour pressure, we consider the Kelvin correction. Our analysis shows that the Kelvin correction is strictly required for droplet radii below 20 Å, as typically encountered for modelling the growth of critical clusters in phase transition processes initiated by homogeneous nucleation. At these conditions, it is mandatory to consider also the repartition of molecules in the different phases, in order to prevent a significant overestimation of the equilibrium vapour pressure.


  1. 1.
    Sazhin, S.S.: Droplet and Sprays. Springer (2014)Google Scholar
  2. 2.
    Sazhin, S.S., Shishkova, I.N., Al Qubeissi, M.: A self-consistent kinetic model for droplet heating and evaporation. Int. J. Heat Mass Transf. 93, 1206–1217 (2016)CrossRefGoogle Scholar
  3. 3.
    Polikarpov, A.Ph., Graur, I.A., Gatapova, E.Ya., and Kabov, O.A.: Kinetic simulation of the non-equilibrium effects at the liquid-vapor interface. Int. J. Heat Mass Transfer 136, 449–456 (2019)Google Scholar
  4. 4.
    Zhakhovsky, V.V., Kryukov, A.P., Levashov, V.Y., Shishkova, I.N., Anisimov, S.I.: Mass and heat transfer between evaporation and condensation surfaces: atomistic simulation and solution of Boltzmann kinetic equation. Proc. Natl. Acad. Sci. 116(37), 18209–18217 (2018)CrossRefGoogle Scholar
  5. 5.
    Chakraborty, S., Qiao, L.: Molecular investigation of sub-to-supercritical transition of hydrocarbon mixtures: multi-component effect. Int. J. Heat Mass Transf. 145, 118629 (2019)CrossRefGoogle Scholar
  6. 6.
    Xiao, G., Luo, K.H., Ma, X., Shuai, S.: A molecular dynamics study of fuel droplet evaporation in sub- and supercritical conditions. Proc. Combust. Inst. 37(3), 3219–3227 (2019)CrossRefGoogle Scholar
  7. 7.
    Maxwell, J.C.: Diffusion, 9th edn. Ency, Brit (1877)Google Scholar
  8. 8.
    Sazhin, S.S.: Modelling of fuel droplet heating and evaporation: recent results and unsolved problems. Fuel 196, 69–101 (2017)CrossRefGoogle Scholar
  9. 9.
    Abramzon, B., Sirignano, W.A.: Droplet vaporization model for spray combustion calculations. Int. J. Heat Mass Transf. 32(9), 1605–1618 (1989)CrossRefGoogle Scholar
  10. 10.
    Qian, J., Law, C.K.: Regimes of coalescence and separation in droplet collision. J. Fluid Mech. 331, 59–80 (1997)CrossRefGoogle Scholar
  11. 11.
    Jeng, S.M., Deng, Z.: Numerical simulation of deformed droplet dynamics and evaporation. Recent. Adv. Spray Combust.: Spray Combust. Meas. Model. Simul. 2, 305–328 (1996)Google Scholar
  12. 12.
    Mashayek, F.: Dynamics of evaporating drops. Part I:formulation and evaporation model. Int. J. Heat Mass Transfer 44, 1517–1526 (2001)Google Scholar
  13. 13.
    Lian, Z.W., Reitz, R.D.: The effect of vaporization and gas compressibility on liquid jet atomization. Atomization Sprays 3(3), 249–264 (1993)CrossRefGoogle Scholar
  14. 14.
    Tonini, S., Cossali, G.E.: An exact solution of the mass transport equations for spheroidal evaporating drops. Int. J. Heat Mass Transf. 60, 236–240 (2013)CrossRefGoogle Scholar
  15. 15.
    Tonini, S., Cossali, G.E.: One-dimensional analytical approach to modelling evaporation and heating of deformed drops. Int. J. Heat Mass Transfer 9, 301–307 (2016)CrossRefGoogle Scholar
  16. 16.
    Imaoka, R.T., Sirignano, W.A.: Transient vaporisation and burning in dense droplet spray. Int. J. Heat Mass Transf. 48, 4354–4366 (2005)CrossRefGoogle Scholar
  17. 17.
    Cossali, G.E., Tonini, S.: An analytical model of heat and mass transfer from liquid drops with temperature dependence of gas thermo-physical properties. Int. J. Heat Mass Transf. 138, 1166–1177 (2019)CrossRefGoogle Scholar
  18. 18.
    Thomson, W.: On the equilibrium of vapour at a curved surface of liquid. Philos. Mag. 4, 448–452 (1871)CrossRefGoogle Scholar
  19. 19.
    Gibbs, J.W.: The Scientific Papers of. J. Willard Gibbs, vol. 1, pp. 55–353. Woodbridge, Ox Bow (1993)Google Scholar
  20. 20.
    Elliott, J.A.W.: On the complete kelvin equation. Chem. Eng. Educ. 35, 274–278 (2001)Google Scholar
  21. 21.
    Kaptay, G.: The gibbs equation versus the Kelvin and the Gibbs- Thomson equations to describe nucleation and equilibrium of nano-materials. J. Nanosci. Nanotechnol. 12, 1–9 (2012)CrossRefGoogle Scholar
  22. 22.
    Kuz, V.A.: A vapor pressure equation for droplets. Langmuir 9, 3722–3723 (1993)CrossRefGoogle Scholar
  23. 23.
    Nguyen-Schäfer, H., Schmidt, J.P.: Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers. Springer (2014)Google Scholar
  24. 24.
    Tonini, S., Cossali, G.E.: Effect of local surface curvature on heating and evaporation of deformed droplets. In: DIPSI Workshop 2018-Droplet Impact Phenomena & Spray Investigations, UniversitGoogle Scholar
  25. 25.
    Moon, P., Spencer, D.E.: Field Theory Handbook, 2nd edn. Springer-Verlag, Berlin (1988)Google Scholar
  26. 26.
    Quan, S., Lou, J., Schmidt, D.P.: Modeling merging and breakup in the moving mesh interface tracking method for multiphase flow simulations. J. Comput. Phys. 228(7), 2660–2675 (2009)CrossRefGoogle Scholar
  27. 27.
    Goldman, R.: Curvature formulas for implicit curves and surfaces. Comput. Aided Geom. Des. 22, 632–658 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Langmuir, I.: The dissociation of hydrogen into atoms. Part II: calculation of the degree of dissociation and the heat of formation. J. Am. Chem. Soc. 37, 417–458 (1915)Google Scholar
  29. 29.
    Sone, Y., Onishi, Y.: Kinetic theory of evaporation and condensation - Hydrodynamic equation and slip boundary condition. J. Phys. Soc. Jpn. 44(6), 1981–1994 (1978)CrossRefGoogle Scholar
  30. 30.
    Young, J.B.: The condensation and evaporation of liquid droplets at arbitrary Knudsen number in the presence of an inert gas. Int. J. Heat Mass Transf. 36(11), 2941–2956 (1993)CrossRefGoogle Scholar
  31. 31.
    Luijten, C.C.M.: Nucleation and Droplet Growth at High Pressure. Ph.D. Thesis, Technische Universiteit Eindhoven (1998)Google Scholar
  32. 32.
    Peeters, P., Pieterse, G., van Dongen, M.E.H.: Multi-component droplet growth. II. A Theoretical Model. Phys. Fluids 16(7), 2575–2586 (2004)zbMATHGoogle Scholar
  33. 33.
    Kobayashi, K., Kazumasa, H., Kon, M., Sasaki, K., Watanabe, M.: Molecular dynamics study on evaporation and reflection of monatomic molecules to construct kinetic boundary condition in vapor-liquid equilibria. Heat Mass Transf. 52(9), 1851–1859 (2016)CrossRefGoogle Scholar
  34. 34.
    Kobayashi, K., Kon, M., Watanabe, M.: Kinetic boundary condition in vapor-liquid two-phase system during unsteady net evaporation/condensation. Eur. J. Mech. B. Fluids 64, 81–92 (2017)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Lemmon, E.W., Bell, I.H., Huber, M.L., McLinden, M.O.: NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 10.0. National Institute of Standards and Technology (2018)Google Scholar
  36. 36.
    Onishi, Y.: The spherical-droplet problem of evaporation and condensation in a vapour-gas mixture. J. Fluid Mech. 163, 171–194 (1986)CrossRefGoogle Scholar
  37. 37.
    Chernyak, V.G., Margilevskiy, A.Ye.: The kinetic theory of heat and mass transfer from a spherical particle in a rarefied gas. Int. J. Heat Mass Transfer 32(11), 2127–2134 (1989)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Grazia Lamanna
    • 1
    Email author
  • Gianpietro Elvio Cossali
    • 2
  • Simona Tonini
    • 2
  1. 1.Institute of Aerospace Thermodynamics (ITLR)University of StuttgartStuttgartGermany
  2. 2.Department of Engineering and Applied SciencesUniversity of BergamoBergamoItaly

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