Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-18T20:17:15.227Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat

Part of: Parabolic equations and systems Thermodynamics and heat transfer Representations of solutions Miscellaneous topics - Partial differential equations General summability methods

Published online by Cambridge University Press:  15 June 2020

J. BOLLATI Open the ORCID record for J. BOLLATI [Opens in a new window]  and
D. A. TARZIA
J. BOLLATI
Affiliation:
Departamento Matemática - FCE, Universidad Austral-CONICET, Paraguay 1950, S2000 FZF, Rosario, Argentina, emails: jbollati@austral.edu.ar; dtarzia@austral.edu.ar
D. A. TARZIA
Affiliation:
Departamento Matemática - FCE, Universidad Austral-CONICET, Paraguay 1950, S2000 FZF, Rosario, Argentina, emails: jbollati@austral.edu.ar; dtarzia@austral.edu.ar
Get access Rights & Permissions [Opens in a new window]

Abstract

The work in this paper concerns the study of different approximations for one-dimensional one-phase Stefan-like problems with a space-dependent latent heat. It is considered two different problems, which differ from each other in their boundary condition imposed at the fixed face: Dirichlet and Robin conditions. The approximate solutions are obtained by applying the heat balance integral method (HBIM), the modified HBIM and the refined integral method (RIM). Taking advantage of the exact analytical solutions, we compare and test the accuracy of the approximate solutions. The analysis is carried out using the dimensionless generalised Stefan number (Ste) and Biot number (Bi). It is also studied the case when Bi goes to infinity in the problem with a convective condition, recovering the approximate solutions when a temperature condition is imposed at the fixed face. Some numerical simulations are provided in order to assert which of the approximate integral methods turns out to be optimal. Moreover, we pose an approximate technique based on minimising the least-squares error, obtaining also approximate solutions for the classical Stefan problem.

Keywords

Stefan problemvariable latent heatheat balance integral methodrefined heat balance integral methodexact solutions

MSC classification

Primary: 80A22: Stefan problems, phase changes, etc.
Secondary: 40C10: Integral methods 35R35: Free boundary problems 35K05: Heat equation 35C05: Solutions in closed form
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 2 , April 2021 , pp. 337 - 369
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

The present work has been partially sponsored by the Project PIP No. 0275 from CONICET-UA, Rosario, Argentina, by the Project ANPCyT PICTO Austral 2016 No. 0090 and by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement 823731 CONMECH.

References

Alexiades, V. & Solomon, A. D. (1993) Mathematical Modeling of Melting and Freezing Processes, Hemisphere Publishing Corp., Washington.Google Scholar
Bollati, J., Semitiel, J. & Tarzia, D. (2018) Heat balance integral methods applied to the one-phase Stefan problem with a convective boundary condition at the fixed face. Appl. Math. Comput. 331, 1–19.Google Scholar
Bollati, J. & Tarzia, D. (2018) Explicit solution for the one-phase Stefan problem with latent heat depending on the position and a convective boundary condition at the fixed face. Commun. Appl. Anal. 22, 309–332.Google Scholar
Bollati, J. & Tarzia, D. (2018) 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, 1–15.CrossRefGoogle Scholar
Bollati, J. & Tarzia, D. (2018) One-phase Stefan problem with a latent heat depending on the position of the free boundary and its rate of change. Electron. J. Diff. Equations 2018-10, 1–12.Google Scholar
Cannon, J. R. (1984) The One-Dimensional Heat Equation, Addison-Wesley, Menlo Park, California.CrossRefGoogle Scholar
Dorian, T. (2014) Spatial and Temporal Variability of Latent Heating in the Tropics Using TRMM Observations. Master of science thesis, University of Misconsin-Madison.Google Scholar
Goodman, T. (1958) The heat balance integral methods and its application to problems involving a change of phase. Trans. ASME 80, 335–342.Google Scholar
Gottlieb, H. (2002) Exact solution of a Stefan problem in a nonhomogeneous cylinder. Appl. Math. Lett. 15, 167–172.CrossRefGoogle Scholar
Gupta, S. C. (2003) The Classical Stefan Problem. Basic Concepts, Modelling and Analysis, Elsevier, Amsterdam.Google Scholar
Hristov, J. (2009) The heat-balance integral method by a parabolic profile with unspecified exponent: analysis and benchmark exercises. Therm. Sci. 13, 27–48.Google Scholar
Hristov, J. (2009) Research note on a parabolic heat-balance integral method with unspecified exponent: an entropy generation approach in optimal profile determination. Therm. Sci. 13, 49–59.Google Scholar
Lunardini, V. J. (1991) Heat Transfer with Freezing and Thawing, Elsevier Science Publishers B. V. Google Scholar
McConnell, T. (1991) The two-sided stefan problem with a spatially dependent latent heat. Trans. Am. Math. Soc. 326, 669–699.CrossRefGoogle Scholar
Mitchell, S. L. (2012) Applying the combined integral method to one-dimensional ablation. Appl. Math. Modell. 36, 127–138.CrossRefGoogle Scholar
Mitchell, S. L. & Myers, T. (2010) Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions. Int. J. Heat Mass Transfer 53, 3540–3551.CrossRefGoogle Scholar
Primicerio, M. (1970) Stefan-like problems with space-dependent latent heat. Meccanica 5, 187–190.CrossRefGoogle Scholar
Ribera, H. & Myers, T. (2016) A mathematical model for nanoparticle melting with size dependent latent heat and melt temperature. Microfluid Nanofluid 20-147, 1–13.CrossRefGoogle Scholar
Ribera, H., Myers, T. & Mac Davette, M. (2019) Optimising the heat balance integral method in spherical and cilyndrical stefan problems. Appl. Math. Comput. 354, 216–231.Google Scholar
Sadoun, N., Si-ahmed, E. & Colinet, J. (2006) On the refined integral method for the one-phase stefan problem with time-dependent boundary conditions. Appl. Math. Modell. 30, 531–544.CrossRefGoogle Scholar
Salva, N. N. & Tarzia, D. A. (2011) Explicit solution for a Stefan problem with variable latent heat and constant heat flux boundary conditions. J. Math. Anal. Appl. 379, 240–244.CrossRefGoogle Scholar
Solomon, A. D. (1979) An easily computable solution to a two-phase Stefan problem. Solar Energy 33, 525–528.CrossRefGoogle Scholar
Swenson, J., Voller, V., Paola, C., Parker, G. & Marr, J. (2000) Fluvio-deltaic sedimentation: a generalized stefan problem. Eur. J. Appl. Math. 11, 433–452.CrossRefGoogle Scholar
Tarzia, D. A. (2017) Relationship between Neumann solutions for two-phase Lamé-Clapeyron-Stefan problems with convective and temperature boundary conditions. Therm. Sci. 21-1, 187–197.CrossRefGoogle Scholar
Tarzia, D. A. (2000) A bibliography on moving-free boundary problems for heat diffusion equation. The Stefan problem, MAT-Serie A 2, 1–297.CrossRefGoogle Scholar
Tarzia, D. A. (2011) Explicit and approximated solutions for heat and mass transfer problems with a moving interface. In: El-Amin, M. (editor), Advanced Topics in Mass Transfer, Ch. 20, In Tech, Rijeka, pp. 439–484.Google Scholar
Voller, V. R., Swenson, J. & Paola, C. (2004) An analytical solution for a Stefan problem with variable latent heat. Int. J. Heat Mass Transfer. 47, 5387–5390.CrossRefGoogle Scholar
Wood, A. S. (2001) A new look at the heat balance integral method. Appl. Math. Modell. 25, 815–824.CrossRefGoogle Scholar
Zhou, Y., Bu, W. & Lu, M. (2013) One-dimensional consolidation with a threshold gradient: a Stefan problem with rate-dependent latent heat. Int. J. Numer. Anal. Methods Geomechan. 37, 2825–2832.CrossRefGoogle Scholar
Zhou, Y., Hu, X., Li, T., Zhang, D. & Zhou, G. (2018) Similarity type of general solution for one-dimensional heat conduction in the cylindrical coordinate. Int. J. Heat Mass Transfer 119, 542–550.CrossRefGoogle Scholar
Zhou, Y., Shi, X. & Zhou, G. (2018) Exact solution for a two-phase Stefan problem with power-type latent heat. J. Eng. Math. 110, 1–13.CrossRefGoogle Scholar
Zhou, Y., Wang, Y. & Bu, W. (2014) Exact solution for a Stefan problem with a latent heat a power function of position. Int. J. Heat Mass Transfer 69, 451–454.CrossRefGoogle Scholar
Zhou, Y. & Xia, L. (2015) Exact solution for Stefan problem with general power-type latent heat using Kummer function. Int. J. Heat Mass Transfer 84, 114–118.CrossRefGoogle Scholar