Abstract
Recently, in Tarzia (Thermal Sci 21A:1–11, 2017) for the classical two-phase Lamé–Clapeyron–Stefan problem an equivalence between the temperature and convective boundary conditions at the fixed face under a certain restriction was obtained. Motivated by this article we study the two-phase Stefan problem for a semi-infinite material with a latent heat defined as a power function of the position and a convective boundary condition at the fixed face. An exact solution is constructed using Kummer functions in case that an inequality for the convective transfer coefficient is satisfied generalizing recent works for the corresponding one-phase free boundary problem. We also consider the limit to our problem when that coefficient goes to infinity obtaining a new free boundary problem, which has been recently studied in Zhou et al. (J Eng Math 2017. https://doi.org/10.1007/s10665-017-9921-y).
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Abbreviations
- \(d_\mathrm{l}\), \(d_\mathrm{s}\) :
-
Diffusivity coefficient (m\(^2\)/s)
- \(h_0\) :
-
Coefficient that characterizes the heat transfer in condition (1.6) [kg/(\(^{\circ }\)Cs\(^{5/2}\))]
- \(k_\mathrm{l}\),\(k_\mathrm{s}\) :
-
Thermal conductivity [W/(m \(^{\circ }\)C)]
- s :
-
Position of the free front (m)
- t :
-
Time (s)
- \(T_{\infty }\) :
-
Coefficient that characterizes the bulk temperature in condition (1.6) [\(^{\circ }\)C/s\(^{\alpha /2}\)]
- \(T_i\) :
-
Coefficient that characterizes the initial temperature of the material in condition (1.7), [\(^{\circ }\)C/m\(^{\alpha }\)]
- x :
-
Spatial coordinate (m)
- \(\alpha \) :
-
Power of the position that characterizes the latent heat per unit volume (dimensionless)
- \(\gamma \) :
-
Coefficient that characterizes the latent heat per unit volume [kg/(s\(^2\)m\(^{\alpha +1}\))]
- \(\nu \) :
-
Coefficient that characterizes the free interface (dimensionless)
- \(\eta \) :
-
Similarity variable in expression (2.1) (dimensionless)
- \(\Psi _\mathrm{l},\Psi _\mathrm{s}\) :
-
Temperature (\(^{\circ }\)C).
- l :
-
Liquid phase
- s :
-
Solid phase
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Acknowledgements
The present work has been partially sponsored by the Projects PIP No. 0534 from CONICET-UA and ANPCyT PICTO Austral 2016 No. 0090, Rosario, Argentina.
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Bollati, J., Tarzia, D.A. Exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition at the fixed face. Z. Angew. Math. Phys. 69, 38 (2018). https://doi.org/10.1007/s00033-018-0923-z
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DOI: https://doi.org/10.1007/s00033-018-0923-z