International Scientific Journal


In this work, we study the liquid-solid interface dynamics for large time intervals on a 1-D sample, with homogeneous Neumann boundary conditions. In this kind of boundary value problem, we are able to make new predictions about the interface position by using conservation of energy. These predictions are confirmed through the heat balance integral method of Goodman and a generalized non-classical finite difference scheme. Since Neumann boundary conditions imply that the specimen is thermally isolated, through well stablished thermodynamics, we show that the interface behavior is not parabolic, and some examples are built with a novel interface dynamics that is not found in the literature. Also, it is shown that, on a Neumann boundary value problem, the position of the interface at thermodynamic equilibrium depends entirely on the initial temperature profile. The prediction of the interface position for large time values makes possible to fine tune the numerical methods, and given that energy conservation demands highly precise solutions, we found that it was necessary to develop a general non-classical finite difference scheme where a non-homogeneous moving mesh is considered. Numerical examples are shown to test these predictions and finally, we study the phase transition on a thermally isolated sample with a liquid and a solid phase in aluminum.
PAPER REVISED: 2016-11-29
PAPER ACCEPTED: 2016-12-03
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2017, VOLUME 21, ISSUE Issue 6, PAGES [2699 - 2708]
  1. E. Javierre-Pérez, Literature Study: Numerical Problems for Solving Stefan Problems. Report No. 03-16, Delf University of Technology, Delft, Netherlands, 2003
  2. Javierre, E., et. al., Comparison of Numerical Models for One-Dimensional Stefan Problems, J. Comput. Appl. Math.,192(2006), 2, pp. 445-459
  3. 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
  4. 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
  5. Tadi, M., A Four-Step Fixed-Grid Method for 1D Stefan Problems, J. Heat Transf., 132(2010), 11, pp. 114502-114505
  6. 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
  7. Esen, A., Kutluay, S., A numerical solution of the Stefan problem with a Neumann-type boundary condition by enthalpy method, Appl. Math. Comput., 148(2004), 2, pp. 321-329
  8. Caldwell, J., Kwan Y.Y., Numerical methods for one-dimensional Stefan problems, Commun. Numer. Meth. Engng., 20(2004), 7, pp. 535-545
  9. T.R. Goodman, Application of Integral Methods to Transient Nonlinear Heat Transfer, Advances in Heat Transfer, Academic Press, New York, 1964
  10. 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
  11. 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
  12. Sadoun, N., et. al., On the Goodman Heat-Balance Integral Method for Stefan Like-Problems, Therm. Sci., 13(2009), 2, pp. 81-96
  13. Hernández, E.M., et. al., Non Parabolic Interface Motion for the One-Dimensional Stefan Problem: Dirichlet Boundary Conditions, Therm. Sci., OnLine-First 2016. DOI: 10.2298/TSCI151114098H.

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence