One-Dimensional Modelling of Drop Heating and Evaporation Under Steady Conditions

  • Gianpietro Elvio CossaliEmail author
  • Simona Tonini
Part of the Mathematical Engineering book series (MATHENGIN)


We have seen how modelling of drop evaporation implies the solution of a set of PDEs (momentum, energy and species conservation), which may represent a remarkable challenge, particularly when an analytical approach is chosen. The problem can be greatly simplified introducing some assumptions, like sphericity of the drop, constancy of the thermo-physical properties and steadiness.


  1. 1.
    Fuchs, N.A.: Vaporisation and Droplet Growth in Gaseous Media. Pergamon Press, London (1959)Google Scholar
  2. 2.
    Abramzon, B., Sirignano, W.A.: Droplet vaporization model for spray combustion calculations. Int. J. Heat Mass Tran. 32(9), 1605–1618 (1989)CrossRefGoogle Scholar
  3. 3.
    Tonini, S., Cossali, G.E.: On molar- and mass-based approaches to single component drop evaporation modelling. Int. Commun. Heat Mass 77, 87–93 (2016)CrossRefGoogle Scholar
  4. 4.
    Fuller, E.N., Schetter, P.D., Giddings, J.C.: New method for prediction of binary gas-phase diffusion coefficients. Ind. Eng. Chem. 58(5), 18–27 (1966)CrossRefGoogle Scholar
  5. 5.
    Wannier, G.H.: Statistical Physics. Wiley (1966)Google Scholar
  6. 6.
    Bernstein, H.J.: Power series for the temperature dependence of the heat capacity of gases. J. Chem. Phys. 24, 911–912 (1956)Google Scholar
  7. 7.
    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
  8. 8.
    Hirschfelder, J.O., Curtiss, C.F., Bird, R.B.: Molecular Theory of Gases and Liquids, 2nd edn. Wiley, New York (1964)zbMATHGoogle Scholar
  9. 9.
    Ferziger, J.H., Kaper, H.G.: Mathematical Theory of Transport Processes in Gases. North-Holland, Amsterdam (1972)Google Scholar
  10. 10.
    Wassiljewa, A.: Heat conduction in gas mixtures. Physikalische Zeitschrift 5(22), 737–742 (1904)zbMATHGoogle Scholar
  11. 11.
    Lindsay, A.L., Bromley, L.A.: Thermal conductivity of gas mixtures. Ind. Eng. Chem. 42, 1508–1511 (1950)CrossRefGoogle Scholar
  12. 12.
    Ebrahimian, V., Habchi, C.: Towards a predictive evaporation model for multi-component hydrocarbon droplets at all pressure conditions. Int. J. Heat Mass Transf. 54, 3552-3565 (15–16) (2011)Google Scholar
  13. 13.
    Labowsky, M.: A formalism for calculating the evaporation rates of rapidly evaporating interacting particles. Combust. Sci. Technol. 18, 145–151 (1978)CrossRefGoogle Scholar
  14. 14.
    Olver, F.W., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press (2010)Google Scholar
  15. 15.
    Cossali, G.E., Tonini, S.: Modelling the effect of variable density and diffusion coefficient on heat and mass transfer from a single component spherical drop evaporating in high temperature air streams. Int. J. Heat Mass Transf. 118, 628–636 (2018)CrossRefGoogle Scholar
  16. 16.
    Cossali, G.E., Tonini, S.: An analytical model of heat and mass transfer from liquid drops with temperature dependence of gas thermophysical properties. Int. J. Heat Mass Transf. 138, 1166–1177 (2019)CrossRefGoogle Scholar
  17. 17.
    Pruppacher, H.R., Pitter, R.L.: A semi-empirical determination of the shape of cloud and rain drops. J. Atmos. Sci. 28, 86–94 (1971)CrossRefGoogle Scholar
  18. 18.
    Chandrasekhar, S.: The oscillations of a viscous liquid globe. Proc. Lond. Math. Soc. 9, 141–149 (1959)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Eggers, J., Villermaux, E.: Physics of liquid jets. Reports on Progress in Physics (036601) (2008)Google Scholar
  20. 20.
    Sazhin, S.: Modelling of fuel droplet heating and evaporation: recent results and unsolved problems. Fuel 196, 69–101 (2017)CrossRefGoogle Scholar
  21. 21.
    Tonini, S., Cossali, G.E.: An analytical model of liquid drop evaporation in gaseous environment. Int. J. Therm. Sci. 57, 45–53 (2012)CrossRefGoogle Scholar
  22. 22.
    Sazhin, S.: Droplet and Sprays. Springer (2014)Google Scholar
  23. 23.
    Lian, Z.W., Reitz, R.D.: The effect of vaporization and gas compressibility on liquid jet atomization. At. Sprays 3(3), 249–264 (1993)CrossRefGoogle Scholar
  24. 24.
    Mashayek, F.: Dynamics of evaporating drops. Part I: formulation and evaporation model. Int. J. Heat Mass Transf. 44(8), 1517–1526 (2001)Google Scholar
  25. 25.
    Tonini, S., Cossali, G.E.: One-dimensional analytical approach to modelling evaporation and heating of deformed drops. Int. J. Heat Mass Transf. 9, 301–307 (2016)CrossRefGoogle Scholar
  26. 26.
    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

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© Springer Nature Switzerland AG 2021

Authors and Affiliations

  1. 1.Department of Engineering and Applied SciencesUniversity of BergamoDalmineItaly

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