Wärme - und Stoffübertragung

, Volume 16, Issue 2, pp 105–111 | Cite as

Planar solidification of a warm flowing-liquid under different boundary conditions

  • R. V. Seeniraj
  • T. K. Bose


Planar solidification of a warm flowing liquid with the convective heat transfer to the growing solid layer, has been analysed for the boundary conditions of constant temperature, constant heat flux and convective heat flux at the surface respectively. The mathematical formulation of the problem resulted in a coupled set of two differential equations in temperature and solid thickness as function of position, time and the problem parameters. Analytical expressions for the temperature distribution within the growing solid layer, the rate of solidification and the solidification time are obtained. The perturbation techniques employed here is simple and straight forward in contrast with the earlier techniques. Good agreement between the experimental results and the present solutions is obtained for the convective heat flux boundary condition. The results of this analysis are useful in the design and analysis of experiments dealing with freezing/melting in one dimension. The role of the parameter Stefan number which is small for phase change materials, is discussed in context with the storage of thermal energy.


Heat Transfer Heat Flux Convective Heat Transfer Phase Change Material Problem Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



thickness of wall, m


Biot Number


specific heat, J/kg/K


freezing-front speed, m/sec


convective heat transfer coefficient, W/m2/K


thermal conductivity of solid, W/m2/K


latent heat of fusion, W/kg


heat flux, case (b), W/m2


dimensionless frozen layer thickness


modified Stefan Number, case (b)


temperature, K


time, s


dimensionless temperature


position coordinate in frozen layer measured from wall, m


thickness of frozen layer at steady state, m


dimensionless coordinate


natural logarithm

Greek symbols


thermal diffusivity of solid, m2/s


flux parameter, case (b)


Stefan number

\(\bar \varepsilon \)

dimensionless thickness of solid layer, case (b)


density of solid, kg/m3


dimensionless time





freezing front




steady state




liquid phase of solidifying substance

Gefrieren einer warmen, strömenden Flüssigkeit in ebener Front unter verschiedenen Bedingungen


Der Gefriervorgang einer warmen, strömenden Flüssigkeit mit konvektivem Wärmeübergang zur ebenen wachsenden Phasengrenze wurde mit den Randbedingungen konstante Temperatur, konstanter Wärmestrom und konvektiver Wärmestrom an der Oberfläche untersucht. Man erhält zwei gekoppelte Differentialgleichungen für die Temperatur und die Dicke der festen Schicht als Funktion des Ortes, der Zeit und der Problemparameter. So gelangt man zu analytischen Ausdrücken für die Temperaturverteilung in der festen Schicht, der Geschwindigkeit und die Zeit der Erstarrung. Die hier angewandte Störungsrechnung ist einfach und, im Gegensatz zu älteren Verfahren, explizit. Für konvektive Übertragung als Randbedingung erhält man gute Übereinstimmung zwischen Experiment und Theorie. Diese Ergebnisse sind wichtig für Entwurf und Auswertung von experimentellen Resultaten mit Frieren und Schmelzen in einer Ebene. Der Einfluß der Stefan-Zahl, die für in Frage kommende Stoffe klein ist, wird im Zusammenhang mit Wärmespeicherung untersucht.


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  1. 1.
    Carslaw, H. S.; Jaeger, J. C.: Conduction of heat in solids 2nd ed., Oxford: Clarendon Press 1959Google Scholar
  2. 2.
    Muehlbauer, J. C.; Sunderland, J. E.: Heat conduction with freezing or melting. Appl. Mech. Rev. 18 (1965) 951Google Scholar
  3. 3.
    Bankoff, S. G.: Heat conduction or diffusion with change of phase. Chem. Eng. 5 (1964) 75–150Google Scholar
  4. 4.
    Libby, P. A.; Chen, S.: The growth of deposited layer on a cold surface. Int. J. Heat Mass Transfer. 8 (1965) 395–402Google Scholar
  5. 5.
    Lapadula, C.; Mueller, W. K.: Heat conduction with solidification and a convective boundary condition at the freezing front. Int. J. Heat Mass Transfer. 9 (1966) 702–705Google Scholar
  6. 6.
    Beaubouef, R. T.; Chapman, A. J.: Freezing of fluids in forced flow. Int. J. Heat Mass Transfer. 10 (1967) 1581–1588Google Scholar
  7. 7.
    Savino, J. M.; Siegel, R.: An analytical solution for solidification of a moving warm liquid onto an isothermal cold wall. Int. J. Heat Mass Transfer. 12 (1969) 803–809Google Scholar
  8. 8.
    Goodling, J. S.; Khader, M. S.: Results of the numerical solution for outward solidification with flux boundary conditions. J. Heat Transfer. 97 (1975) 307–309Google Scholar
  9. 9.
    Savino, J. M.; Siegel, R.: Experimental and analytical study of the transient solidification of a warm liquid flowing over a chilled flat plate. NASA TO D-4015 (1967)Google Scholar
  10. 10.
    Stephan, K.: Influence of heat transfer on melting and solidification in forced flow. Int. J. Heat Mass Transfer. 12 (1969) 199–214Google Scholar
  11. 11.
    Huang, C. L.; Shih, Y. P.: Perturbation solutions of planar diffusion-controlled moving-boundary problems. Int. J. Heat Mass Transfer. 18 (1975) 689–695Google Scholar
  12. 12.
    Shamsundar, N.; Sparrow, E. M.: Effect of density change on multidimensional conduction phase change. J. Heat Transfer. 98 (1976) 550–557Google Scholar
  13. 13.
    Weinbaum, S.; Jiji, L. M.: Singular perturbation theory for melting or freezing in finite domains initially not at the fusion temperature. J. Appl. Mech. 44 (1977) 25–30Google Scholar
  14. 14.
    Shanks, D.: Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34 (1955)Google Scholar
  15. 15.
    Seeniraj, V.: Planar solidification of a saturated liquid under different boundary conditions. To be published in the Ind. J. of Technol.Google Scholar
  16. 16.
    Pedroso, R. I.; Domoto, G. A.: Inward spherical solidification — solution by the method of strained coordinates. Int. J. Heat Mass Transfer. 16 (1973) 1037–1043Google Scholar
  17. 17.
    Pedroso, R. I.; Damoto, G. A.: Exact solution by perturbation method for planar solidification of a saturated liquid with convection at the wall. Int. J. Heat Mass Transfer. 16 (1973) 1816–1819Google Scholar
  18. 18.
    Pedroso, R. I.; Damoto, G. A.: Perturbation solutions for spherical solidification of saturated liquids. J. Heat Transfer. 95 (1973) 42–46Google Scholar
  19. 19.
    Pedroso, R. I.; Damoto, G. A.: Technical note on planar solidification with fixed wall temperature and variable thermal properties. J. Heat Transfer 95 (1973) 553–555Google Scholar
  20. 20.
    Lock, G. S. H.: On the use asymptotic solutions to plane icewater problems. J. Glaciol. 8 (1969) 285Google Scholar
  21. 21.
    Lock, G. S. H.: A study of one dimensional ice formation with special reference to periodic growth and decay. Int. J. Heat Mass Transfer. 12 (1969) 1343–1352Google Scholar
  22. 22.
    Lock, G. S. H.: On the perturbation solution of the ice-water layer problem. Int. J. Heat Mass Transfer. 14 (1971) 643–644Google Scholar
  23. 23.
    Jiji, L. M.: On the application of perturbation to freeboundary problems in radial systems. J. Franklin. Inst. 289 (1970) 282Google Scholar
  24. 24.
    Yan, M. M.; Huang, P. N. S.: Perturbation solutions to phase change problem subject to convection and radiation. J. Heat Transfer. 101 (1979) 96–100Google Scholar
  25. 25.
    Cole, J. D.: Perturbation methods in applied mathematics. New York: Blaisdell 1968Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • R. V. Seeniraj
    • 1
  • T. K. Bose
    • 1
  1. 1.Department of Aeronautical EngineeringIndian Institute of TechnologyMadrasIndia

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