THERMAL SCIENCE

International Scientific Journal

Thermal Science - Online First

online first only

Non parabolic interface motion for the one-dimensional Stefan problem: Dirichlet boundary conditions

ABSTRACT
Over a finite one dimensional specimen containing two phases of a pure substance, it has been shown that the liquid-solid interface motion exhibits parabolic behavior at small time intervals. We study the interface behavior over a finite domain with homogeneous Dirichlet boundary conditions for large time intervals, where the interface motion is not parabolic due to finite size effects. Given the physical nature of the boundary conditions, we are able to predict exactly the interface position at large time values. These predictions, that to the best of our knowledge, are not found in the literature, were confirmed by using the heat balance integral method of Goodman and a non-classical finite difference scheme. Using heat transport theory, it is shown as well, that the temperature profile within the specimen is exactly linear and independent of the initial profile in the asymptotic time limit. The physics of heat transport provides a powerful tool that is used to fine tune the numerical methods. We also found that in order to capture the physical behavior of the interface, it was necessary to develop a new non classical finite difference scheme that approaches asymptotically to the predicted interface position. We offer some numerical examples where the predicted effects are illustrated, and finally we test our predictions with the heat balance integral method and the non classical finite difference scheme by studying the liquid-solid phase transition in Aluminum.
KEYWORDS
PAPER SUBMITTED: 2015-11-14
PAPER REVISED: 2016-04-17
PAPER ACCEPTED: 2016-04-26
PUBLISHED ONLINE: 2016-05-08
DOI REFERENCE: https://doi.org/10.2298/TSCI151114098H
REFERENCES
  1. Tarzia, D.A., Explicit and Approximated Solutions for Heat and Mass Transfer Problems with a Moving Boundary, in: Advanced Topics in Mass Transfer (Ed. M.El-Amin), Rijeka, Croatia, 2011, pp. 439-484
  2. E. Javierre-Pérez, Literature Study: Numerical Problems for Solving Stefan Problems. Report No. 03-16, Delf University of Technology, Delft, Netherlands, 2003
  3. Javierre, E., et. al., Comparison of Numerical Models for One-Dimensional Stefan Problems, J. Comput. Appl. Math., 192(2006), 2, pp. 445-459
  4. Mitchell, S.L., Vynnycky M., On the Numerical Solution of Two-Phase Stefan Problems with Heat-Flux Boundary Conditions, J. Comput. Appl. Math., 264(2014), pp. 49-64
  5. Mitchell, S.L., Vynnycky M., Finite-Difference Methods with Increased Accuracy and Correct Initialization for One-Dimensional Stefan problems, Appl. Math. Comput., 215(2009), 4, pp. 1609-1621.
  6. Tadi, M., A Four-Step Fixed-Grid Method for 1D Stefan Problems, J. Heat Transf., 132(2010), 11, pp. 114502-114505
  7. Wu, Zhao-Chun, Wand, Qing-Cheng, Numerical Approach to Stefan Problem in a Two-Region and Limited Space, Therm. Sci., 16(2012), 5, pp. 1325-1330
  8. Savovic, S., Caldwell, J., Numerical solution of Stefan problem with time-dependent boundary conditions by variable space grid method, Therm. Sci., 13(2009), 4, pp. 165-174
  9. Savovic, S., Caldwell, J., Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions, Int. J. Heat Mass Tran., 46(2003), 15, pp. 2911-2916
  10. Caldwell, J., et. al., Nodal integral and finite difference solution of one-dimensional Stefan problem, J. Heat Trans-T. ASME, 125(2003), 3, pp. 523-527
  11. T.R. Goodman, Application of Integral Methods to Transient Nonlinear Heat Transfer, Advances in Heat Transfer, Academic Press, New York, 1964
  12. Fraguela, A., et. al., An approach for the Identification of Diffusion Coefficients in the Quasi-Steady State of a Post-Discharge Nitriding Process, Math. Comput. Simulat., 79(2009), 6, pp. 1878-1894
  13. Mitchell, S.L., Myers T. G., Application of Standard and Refined Heat Balance Integral Methods to One Dimensional Stefan problems, SIAM Rev., 52(2010), 1, pp. 57-86
  14. Sadoun, N., et. al., On the Goodman Heat Balance Integral Method for Stefan Like Problems: Further Considerations and Refinements, Therm. Sci., 13(2009), 2, pp. 81-96
  15. Sadoun, N.,et. al., On Heat Conduction with Phase Change: Accurate Explicit Numerical Method, J. Appl. Fuid Mech., 5(2012), 1, pp. 105-112
  16. Sadoun, N., et. al., On the Goodman Heat-Balance Integral Method for Stefan Like-Problems, Therm. Sci., 13(2009), 2, pp. 81-96
  17. Wu, Z., et. al., A Novel Algorithm for Solving the Classical Stefan Problem, Therm. Sci., 15(2011), suppl 1., pp. 39-44
  18. Yvonnet, J., et. al., The Constrained Natural Element Method (C-NEM) for Treating Thermal Models Involving Moving Interfaces, Int. J. Therm. Sci., 44(2005), 6, pp. 559-569
  19. Fasano, A., Primicerio, M., General Free Boundary Problems for the Heat Equation, I, J. Math. Anal. Appl., 57(1977), 3, pp. 694-723
  20. Fasano, A., Primicerio, M., General Free Boundary Problems for the Heat Equation, II, J. Math. Anal. Appl., 58(1977), 1, pp. 202-231