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

  • Research Article - Mechanical Engineering
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Abstract

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.

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References

  1. Evans, G.W.: A note on the existence of a solution to a problem of Stefan. Q. Appl. Math. 9, 185–193 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  2. Douglas, J.: A uniqueness theorem for the solution of a Stefan problem. Proc. Am. Math. Soc. 8, 402–408 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  3. Crank, J.: Free and Moving Boundary Problem. Oxford University Press, Oxford (1984)

    MATH  Google Scholar 

  4. Hill, J.M.: One-Dimensional Stefan Problems: An Introduction, Longman Scientific and Technical. Wiley, New York (1987)

    Google Scholar 

  5. Caldwell, J.; Kwan, Y.Y.: Nodal integral and enthalpy solution of one-dimensional Stefan problem. J. Math. Sci. 13(2), 99–109 (2002)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  11. Zabaras, N.; Mukherjee, S.: An analysis of solidification problem by the boundary element method. Int. J. Numer. Methods Eng. 24, 1879–1900 (1987)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. Stephan, K.; Holzknecht, B.: Perturbation solutions for solidification problems. Int. J. Heat Mass Transfer 19, 597–602 (1976)

    Article  Google Scholar 

  17. Font, F.: A one-phase Stefan problem with size-dependent thermal conductivity. Appl. Math. Model. 63, 172–178 (2018)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  20. Yigit, F.: Perturbation solution for solidification of pure metals on a sinusoidal mold surface. Int. J. Heat Mass Transfer 50, 2624–2633 (2007)

    Article  MATH  Google Scholar 

  21. Yigit, F.: One-dimensional solidification of pure materials with a time periodically oscillating temperature boundary condition. Appl. Math. Comput. 217, 6541–6555 (2011)

    MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  25. Caldwell, J.; Kwan, Y.: Numerical methods for one-dimensional Stefan problems. Commun. Numer. Methods Eng. 20, 535–545 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hu, H.; Argyropoulos, S.A.: Mathematical modelling of solidification and melting: a review. Modell. Simul. Mater. Sci. Eng. 4, 371–396 (1996)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

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Authors and Affiliations

  1. Department of Mechatronics Engineering, Iskenderun Technical University, 31200, Hatay, Turkey

    Mehmet Hakan Demir

  2. Undersecretariat for Defence Industries, Presidency of the Republic of Turkey, 06420, Ankara, Turkey

    Faruk Yigit

Authors
  1. Mehmet Hakan Demir
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  2. Faruk Yigit
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Correspondence to Mehmet Hakan Demir.

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Demir, M.H., Yigit, F. A Theoretical Heat Transfer Model for Unidirectional Solidification of Pure Metals on a Coated Sinusoidal Mold with Constant Boundary Temperature. Arab J Sci Eng 44, 5825–5837 (2019). https://doi.org/10.1007/s13369-019-03736-7

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  • DOI: https://doi.org/10.1007/s13369-019-03736-7

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