A Theoretical Heat Transfer Model for Unidirectional Solidification of Pure Metals on a Coated Sinusoidal Mold with Constant Boundary Temperature

  • Mehmet Hakan DemirEmail author
  • Faruk Yigit
Research Article - Mechanical Engineering


A theoretical model is presented in this study for investigating the heat transfer problem during solidification of pure metals on a coated sinusoidal mold. The previous works are extended by considering effects of both the sinusoidal coating layer’s properties and prescribed temperature boundary condition at the lower surface of the mold on the solidification process. The thermal diffusivities of the shell, coating and mold materials are assumed to be infinitely large, and it helps us to solve two-dimensional heat conduction problem analytically. The effects of the ratios between the thermal conductivities of coating, shell and mold materials, coating thickness and the amplitude ratios between the wavelengths of coating and mold surfaces on the solidification process are investigated in detail. Furthermore, the inverse design problem for the directional solidification process is briefly discussed. The results show that the thickness of the coating causes the decrease in the amplitude of the undulation at the solidification front for all cases considered. When the ratio between the amplitudes of both surface wavelengths of the mold is negative, the decrease in the ratio between thermal conductivities of the shell and coating materials causes more uniform growth in the shell regardless of the amplitudes of wavelength ratio between the surfaces of the coating layer. A bandwidth for thermal conductivity ratio between the shell and coating materials is determined depending on the process parameters for more uniform growth.


Solidification Coating Phase change Linear perturbation method Metal casting 


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  1. 1.
    Evans, G.W.: A note on the existence of a solution to a problem of Stefan. Q. Appl. Math. 9, 185–193 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Douglas, J.: A uniqueness theorem for the solution of a Stefan problem. Proc. Am. Math. Soc. 8, 402–408 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Crank, J.: Free and Moving Boundary Problem. Oxford University Press, Oxford (1984)zbMATHGoogle Scholar
  4. 4.
    Hill, J.M.: One-Dimensional Stefan Problems: An Introduction, Longman Scientific and Technical. Wiley, New York (1987)Google Scholar
  5. 5.
    Caldwell, J.; Kwan, Y.Y.: Nodal integral and enthalpy solution of one-dimensional Stefan problem. J. Math. Sci. 13(2), 99–109 (2002)MathSciNetGoogle Scholar
  6. 6.
    Caldwell, J.; Savovic, S.; Kwan, Y.Y.: Nodal integral and finite difference solution of one-dimensional Stefan problem. J. Heat Transfer 125, 523–527 (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Vynnycky, M.; Mitchell, S.L.: On the numerical solution of a Stefan problem with finite extinction time. J. Comput. Appl. Math. 276, 98–109 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mitchell, S. L.; M.Vynnycky, M.: On the numerical solution of two-phase Stefan problems with heat-flux boundary conditions. J. Comput. Appl. Math. 264, 49–64 (2014)Google Scholar
  9. 9.
    Kutluay, B.; Bahadir, A.R.; Ozdes, A.: The numerical solution of one-phase classical Stefan problem. J. Comput. Appl. Math. 81, 135–144 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Yang, H.; He, Y.: Solving heat transfer problems with phase change via smoothed effective heat capacity and element-free Galerkin methods. Int. Commun. Heat Mass. 37, 385–392 (2010)CrossRefGoogle Scholar
  11. 11.
    Zabaras, N.; Mukherjee, S.: An analysis of solidification problem by the boundary element method. Int. J. Numer. Methods Eng. 24, 1879–1900 (1987)CrossRefzbMATHGoogle Scholar
  12. 12.
    Vu, T.V.; Truong, A.V.; Hoang, N.T.B.; Tran, D.K.: Numerical investigations of solidification around a circular cylinder under forced convection. J. Mech. Sci. Technol. 30(11), 5019–5028 (2016)CrossRefGoogle Scholar
  13. 13.
    Pedroso, R.I.; Domoto, 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, 1816–1819 (1973)CrossRefGoogle Scholar
  14. 14.
    Huang, C.L.; Shih, Y.P.: Shorter communications: perturbation solution for planar solidification of a saturated liquid with convection at the wall. Int. J. Heat Mass Transfer 18, 1481–1483 (1975)CrossRefGoogle Scholar
  15. 15.
    Pedroso, R.I.; Domoto, G.A.: Perturbation solutions for spherical solidification of saturated liquids. Int. J. Heat Mass Transfer 95, 42–46 (1973)Google Scholar
  16. 16.
    Stephan, K.; Holzknecht, B.: Perturbation solutions for solidification problems. Int. J. Heat Mass Transfer 19, 597–602 (1976)CrossRefGoogle Scholar
  17. 17.
    Font, F.: A one-phase Stefan problem with size-dependent thermal conductivity. Appl. Math. Model. 63, 172–178 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dragomirescu, F.I.; Eisenschmidt, K.; Rohde, C.; Weigand, B.: Perturbation solutions for the finite radially symmetric Stefan problem. Int. J. Therm. Sci. 104, 386–395 (2016)CrossRefGoogle Scholar
  19. 19.
    Yigit, F.: Approximate analytical solution of a two-dimensional heat conduction problem with phase-change on a sinusoidal mold. Appl. Therm. Eng. 28, 1196–1205 (2008)CrossRefGoogle Scholar
  20. 20.
    Yigit, F.: Perturbation solution for solidification of pure metals on a sinusoidal mold surface. Int. J. Heat Mass Transfer 50, 2624–2633 (2007)CrossRefzbMATHGoogle Scholar
  21. 21.
    Yigit, F.: One-dimensional solidification of pure materials with a time periodically oscillating temperature boundary condition. Appl. Math. Comput. 217, 6541–6555 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Demir, M.H.; Yigit, F.: Early time perturbation solution of solidification on a coated sinusoidal mould of finite thickness. Adv. Mater. Process. Technol. 1(3–4), 327–337 (2015)Google Scholar
  23. 23.
    Vu, T.V.; Tryggvason, G.; Homma, S.; Wells, J.C.: Numerical investigations of drop solidification on a cold plate in the presence of volume change. Int. J. Multiph. Flow 76, 73–85 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Vu, T.V.; Nguyen, C.T.; Khanh, D.T.: Direct numerical study of a molten metal drop solidifying on a cold plate with different wettability. Metals 8, 47–56 (2018)CrossRefGoogle Scholar
  25. 25.
    Caldwell, J.; Kwan, Y.: Numerical methods for one-dimensional Stefan problems. Commun. Numer. Methods Eng. 20, 535–545 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hu, H.; Argyropoulos, S.A.: Mathematical modelling of solidification and melting: a review. Modell. Simul. Mater. Sci. Eng. 4, 371–396 (1996)CrossRefGoogle Scholar
  27. 27.
    Demir, M.H.; Yigit, F.: Effect of coating material on the growth instability in solidification of pure metals on a coated planar mold of finite thickness. Int. J. Solids Struct. 99, 12–27 (2016)CrossRefGoogle Scholar
  28. 28.
    Demir, M.H.; Yigit, F.: Thermoelastic stability analysis of solidification of pure metals on a coated planar mold of finite thickness. Metall. Mater. Trans. B 48(2), 966–982 (2017)CrossRefGoogle Scholar
  29. 29.
    Anyalebechi, P.N.: Undulatory solid shell growth of aluminum alloy 3003 as a function of the wavelength of a grooved mold surface topography, In: Anyalebechi, P. (ed.) Materials Processing Fundamentals, TMS (The Minerals, Metals and Materials Society), pp. 31–47 (2007)Google Scholar

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© King Fahd University of Petroleum & Minerals 2019

Authors and Affiliations

  1. 1.Department of Mechatronics EngineeringIskenderun Technical UniversityHatayTurkey
  2. 2.Undersecretariat for Defence IndustriesPresidency of the Republic of TurkeyAnkaraTurkey

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