A two-phase Stefan problem with power-type temperature-dependent thermal conductivity. Existence of a solution by two fixed points and numerical results

Julieta Bollati; María F. Natale; José A. Semitiel; Domingo A. Tarzia; Julieta Bollati; María F. Natale; José A. Semitiel; Domingo A. Tarzia

doi:10.3934/math.20241029

AIMS Mathematics

2024, Volume 9, Issue 8: 21189-21211. doi: 10.3934/math.20241029

Previous Article Next Article

Research article Special Issues

A two-phase Stefan problem with power-type temperature-dependent thermal conductivity. Existence of a solution by two fixed points and numerical results

1.
Departamento de Matemática, Universidad Austral, Paraguay 1950, Rosario, S2000FZF, Argentina
2.
CONICET, Argentina

Received: 24 April 2024 Revised: 12 June 2024 Accepted: 20 June 2024 Published: 01 July 2024
MSC : 35R35, 80A22, 41A10, 35C05, 35K05

A one-dimensional two-phase Stefan problem for the melting of a semi-infinite material with a power-type temperature-dependent thermal conductivity was considered. The assumption of taking thermal parameters as functions of temperature found its basis in physical and industries applications, allowing for a more precise and realistic description of phase change processes. By imposing a Dirichlet condition at the fixed face, a theoretical and approximate study was developed. Through a similarity transformation, an equivalent ordinary differential problem was obtained from which an integral problem was deduced. The existence of at least one analytical solution was guaranteed by using the Banach fixed point theorem. Due the unavailability of an analytical solution, a heat balance integral method was applied, assuming a quadratic temperature profile in space, to simulate temperature variations and the location of the interface during the melting process. For constant thermal conductivity, results can be compared with the exact solution available in the literature to check the accuracy of the approximate method.
- Stefan problem,
- temperature-dependent thermal conductivity,
- similarity solution,
- Banach fixed point theorem,
- heat balance integral method
Citation: Julieta Bollati, María F. Natale, José A. Semitiel, Domingo A. Tarzia. A two-phase Stefan problem with power-type temperature-dependent thermal conductivity. Existence of a solution by two fixed points and numerical results[J]. AIMS Mathematics, 2024, 9(8): 21189-21211. doi: 10.3934/math.20241029

Related Papers:

Abstract

A one-dimensional two-phase Stefan problem for the melting of a semi-infinite material with a power-type temperature-dependent thermal conductivity was considered. The assumption of taking thermal parameters as functions of temperature found its basis in physical and industries applications, allowing for a more precise and realistic description of phase change processes. By imposing a Dirichlet condition at the fixed face, a theoretical and approximate study was developed. Through a similarity transformation, an equivalent ordinary differential problem was obtained from which an integral problem was deduced. The existence of at least one analytical solution was guaranteed by using the Banach fixed point theorem. Due the unavailability of an analytical solution, a heat balance integral method was applied, assuming a quadratic temperature profile in space, to simulate temperature variations and the location of the interface during the melting process. For constant thermal conductivity, results can be compared with the exact solution available in the literature to check the accuracy of the approximate method.

References

[1]	J. Bollati, A. Briozzo, M. Natale, Analytical solution for a cylinder glaciation model with variable latent heat and thermal, Int. J. Nonlin. Mech., 150 (2023), 104362. https://doi.org/10.1016/j.ijnonlinmec.2023.104362 doi: 10.1016/j.ijnonlinmec.2023.104362
[2]	J. Bollati, D. Tarzia, Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat, Eur. J. Appl. Math., 32 (2020), 337–369. https://doi.org/10.1017/S0956792520000170 doi: 10.1017/S0956792520000170
[3]	L. Bougoffa, A. Khanfer, On the solutions of a phase change problem with temperature-dependent thermal conductivity and specific heat, Results Phys., 19 (2020) 103646. https://doi.org/10.1016/j.rinp.2020.103646 doi: 10.1016/j.rinp.2020.103646
[4]	L. Bougoffa, A. Khanfer, Solution of non-classical Stefan problem with nonlinear thermal coefficients and a Robin boundary condition, AIMS Math., 6 (2021), 6569–6579. https://doi.org/10.3934/math.2021387 doi: 10.3934/math.2021387
[5]	L. Bougoffa, R. Rach, A. Mennouni, On the existence, uniqueness, and new analytic approximate solution of the modified error function in two-phase Stefan problems, Math. Method. Appl. Sci., 44 (2021), 10948–10956. https://doi.org/10.1002/mma.7457 doi: 10.1002/mma.7457
[6]	J. Bollati, M. Natale, J. Semitiel, D. Tarzia, Existence and uniqueness of solution for two one-phase Stefan problems with variable thermal coefficients, Nonlinear Anal.-Real, 51 (2020), 103001. https://doi.org/10.1016/j.nonrwa.2019.103001 doi: 10.1016/j.nonrwa.2019.103001
[7]	J. Bollati, J. Semitiel, M. Natale, D. Tarzia, Existence and uniqueness of the p-generalized modified error function, Electron. J. Differ. Eq., 2020 (2020), 1–11. https://ejde.math.txstate.edu/Volumes/2020/35/bollati.pdf
[8]	J. Bollati, M. Natale, J. Semitiel, D. Tarzia, Exact solution for non-classical one-phase Stefan problem with variable thermal coefficients and two different heat source terms, Comput. Appl. Math., 41 (2022), 1–11. https://doi.org/10.1007/s40314-022-02095-8 doi: 10.1007/s40314-022-02095-8
[9]	L. Bougoffa, S. Bougouffa, A. Khanfer, An analysis of the one-phase Stefan problem with variable thermal coefficients of order p, Axioms, 12 (2023), 497. https://doi.org/10.3390/axioms12050497 doi: 10.3390/axioms12050497
[10]	A. Kumar, A. Singh, R. Rajeev, A freezing problem with varying thermal coefficients and convective boundary condition, Int. J. Appl. Comput. Math., 6 (2020), 148. https://doi.org/10.1007/s40819-020-00894-3 doi: 10.1007/s40819-020-00894-3
[11]	A. Kumar, A. Singh, R. Rajeev, A moving boundary problem with variable specific heat and thermal conductivity, J. King Saud Univ.-Sci., 32 (2020), 384–389. https://doi.org/10.1016/j.jksus.2018.05.028 doi: 10.1016/j.jksus.2018.05.028
[12]	A. Kumar, A. Singh, R. Rajeev, A Stefan problem with temperature and time dependent thermal conductivity, J. King Saud Univ.-Sci., 32 (2020), 97–101. https://doi.org/10.1016/j.jksus.2018.03.005 doi: 10.1016/j.jksus.2018.03.005
[13]	J. Bollati, A. Briozzo, Stefan problems for the diffusion-convection equation with temperature-dependent thermal coefficients, Int. J. Nonlin. Mech., 134 (2021), 103732. https://doi.org/10.1016/j.ijnonlinmec.2021.103732 doi: 10.1016/j.ijnonlinmec.2021.103732
[14]	V. Cregan, J. Williams, T. Myers, Contact melting of a rectangular block with temperature-dependent properties, Int. J. Therm. Sci., 150 (2020), 106218. https://doi.org/10.1016/j.ijthermalsci.2019.106218 doi: 10.1016/j.ijthermalsci.2019.106218
[15]	A. Kumar, A. Singh, R. Rajeev, A Stefan problem with moving phase change material, variable thermal conductivity and periodic boundary condition, Appl. Math. Comput., 386 (2020), 125490. https://doi.org/10.1016/j.amc.2020.125490 doi: 10.1016/j.amc.2020.125490
[16]	T. Nauryz, Nonlinear Stefan problem for one-phase generalized heat equation with heat flux and convective boundary condition, Res. Square, 2022. https://doi.org/10.21203/rs.3.rs-2004382/v2
[17]	T. Nauryz, S. Kharin, Existence and uniqueness for one-phase spherical Stefan problem with nonlinear thermal coefficients and heat flux condition, Int. J. Appl. Math., 35 (2022), 645–659. http://dx.doi.org/10.12732/ijam.v35i5.2 doi: 10.12732/ijam.v35i5.2
[18]	S. Cho, J. Sunderland, Phase-change problems with temperature-dependent thermal conductivity, J. Heat Transf., 96 (1974), 214–217. https://doi.org/10.1115/1.3450167 doi: 10.1115/1.3450167
[19]	A. Ceretani, N. Salva, D. Tarzia, An exact solution to a Stefan problem with variable thermal conductivity and a Robin boundary condition, Nonlinear Anal.-Real, 40 (2018), 243–259. http://dx.doi.org/10.1016/j.nonrwa.2017.09.002 doi: 10.1016/j.nonrwa.2017.09.002
[20]	T. Goodman, The heat balance integral methods and its application to problems involving a change of phase, Trans. ASME, 80 (1958), 335–342. https://doi.org/10.1115/1.4012364 doi: 10.1115/1.4012364
[21]	A. Wood, A new look at the heat balance integral method, Appl. Math. Model., 25 (2001), 815–824. https://doi.org/10.1016/S0307-904X(01)00016-6 doi: 10.1016/S0307-904X(01)00016-6
[22]	J. Bollati, M. F. Natale, J. A. Semitiel, D. A. Tarzia, Approximate solutions to the one-phase Stefan problem with non-linear temperature-dependent thermal conductivity, Chapter 1, In Heat Conduction: Methods, Applications and Research, J. Hristov – R. Bennacer (Eds.), Nova Science Publishers, Inc., 2019, 1–20.
[23]	J. Bollati, M. Natale, J. Semitiel, D. Tarzia, Integral balance methods applied to non-classical Stefan problems, Therm. Sci., 24 (2020), 1229–1241. https://doi.org/10.2298/TSCI180901310B doi: 10.2298/TSCI180901310B
[24]	J. Bollati, J. Semitiel, D. Tarzia, Heat balance integral methods applied to the one-phase Stefan problem with a convective boundary condition at the fixed face, Appl. Math. Comput., 331 (2018), 1–19. https://doi.org/10.1016/j.amc.2018.02.054 doi: 10.1016/j.amc.2018.02.054
[25]	G. Garguichevich, C. Sanziel, D. Tarzia, Comparison of approximate methods for the determination of thermal coefficients through a phase-change problem, Int. Commun. Heat Mass Tran., 12 (1985), 451–464. https://doi.org/10.1016/0735-1933(85)90039-9 doi: 10.1016/0735-1933(85)90039-9
[26]	N. Sadoun, E. Si-Ahmed, J. Colinet, On the refined integral method for the one-phase Stefan problem with time-dependent boundary conditions, Appl. Math. Model., 30 (2006), 531–544. https://doi.org/10.1016/j.apm.2005.06.003 doi: 10.1016/j.apm.2005.06.003
[27]	D. Tarzia, A variant of the heat balance integral method and a new proof of the exponentially fast asymptotic behavior of the solutions in heat conduction problems with absorption, Int. J. Eng. Sci., 28 (1990), 1253–1259. https://doi.org/10.1016/0020-7225(90)90073-R doi: 10.1016/0020-7225(90)90073-R
[28]	J. Hristov, Integral-balance method with transmuted profiles: Concept, examples, and emerging problems, J. Comput. Appl. Math., 416 (2022), 114547. https://doi.org/10.1016/j.cam.2022.114547 doi: 10.1016/j.cam.2022.114547
[29]	S. Mitchell, T. Myers, Application of standard and refined heat balance integral methods to one-dimensional Stefan problems, SIAM Rev., 52 (2010), 57–86. https://doi.org/10.1137/080733036 doi: 10.1137/080733036
[30]	F. Mosally, A. Wood, A. Al-Fhaid, An exponential heat balance integral method, Appl. Math. Comput., 130 (2002), 87–100. https://doi.org/10.1016/S0096-3003(01)00083-2 doi: 10.1016/S0096-3003(01)00083-2
[31]	J. Hristov, The heat-balance integral method by a parabolic profile with unspecified exponent: Analysis and benchmark exercises, Therm. Sci., 13 (2009), 27–48. https://doi.org/10.2298/TSCI0902027H doi: 10.2298/TSCI0902027H
[32]	J. Hristov, Research note on a parabolic heat-balance integral method with unspecified exponent: An entropy generation approach in optimal profile determination, Therm. Sci., 13 (2009), 49–59. https://doi.org/10.2298/TSCI0902049H doi: 10.2298/TSCI0902049H
[33]	J. Hristov, On a non-linear diffusion model of wood impregnation: Analysis, approximate solutions, and experiments with relaxing boundary conditions, Advances in Mathematical Modelling, Applied Analysis and Computation, Springer, Singapore, 2023, 25–53. https://doi.org/10.1007/978-981-19-0179-9_2
[34]	S. Mitchell, Applying the combined integral method to one-dimensional ablation, Appl. Math. Model., 36 (2012), 127–138. https://doi.org/10.1016/j.apm.2011.05.032 doi: 10.1016/j.apm.2011.05.032
[35]	S. Mitchell, Applying the combined integral method to two-phase Stefan problems with delayed onset of phase change, J. Comput. Appl. Math., 281 (2015), 58–73. https://doi.org/10.1016/j.cam.2014.11.051 doi: 10.1016/j.cam.2014.11.051
[36]	S. Mitchell, N. McInerney, S. O'Brien, Approximate solution techniques for the sorption of a finite amount of swelling solvent in a glass polymer, Appl. Math. Model., 92 (2021), 624–650. https://doi.org/10.1016/j.apm.2020.11.018 doi: 10.1016/j.apm.2020.11.018
[37]	S. Mitchell, T. Myers, Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions. Int. J. Heat Mass Tran., 53 (2010), 3540–3551. https://doi.org/10.1016/j.ijheatmasstransfer.2010.04.015 doi: 10.1016/j.ijheatmasstransfer.2010.04.015
[38]	S. Mitchell, T. Myers, Application of heat balance integral methods to one-dimensional phase change problems, Appl. Math. Comput., 2012 (2012), 1–22. https://doi.org/10.1155/2012/187902 doi: 10.1155/2012/187902
[39]	S. Mitchell, B. O'Brien, Asymptotic and numerical solutions of a free boundary problem for the sorption of a finite amount of solvent into a glassy polymer, SIAM J. Appl. Math., 74 (2014), 697–723. https://doi.org/10.1137/120899200 doi: 10.1137/120899200
[40]	A. Gonzalez, D. Tarzia, Determination of unknown coefficients of a semi-infinite material through a simple mushy zone model for the two-phase Stefan problem, Int. J. Eng. Sci., 34 (1996), 799–817. https://doi.org/10.1016/0020-7225(95)00107-7 doi: 10.1016/0020-7225(95)00107-7
[41]	B. Dhage, Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem, Nonlinear Anal., 70 (2009), 2485–2493. https://doi.org/10.1016/j.na.2008.03.033 doi: 10.1016/j.na.2008.03.033
[42]	B. Dhage, Some characterization of nonlinear first order differential equations of unbounded intervals, Differ. Equat. Appl., 2 (2010), 151–162. dx.doi.org/10.7153/dea-02-10 doi: 10.7153/dea-02-10
[43]	S. Myers, Banach spaces of continuous functions, Ann. Math., 49 (1948), 132–140. https://doi.org/10.2307/1969119 doi: 10.2307/1969119

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)