Semi-discrete and Asymptotic Approximations of the Non-stationary Problem of Radiation-conductive Heat Transfer in a System of Heat-conductive Shields with Alternating Thermophysical Properties
Abstract
Of significant interest for applications is a study of radiation-conductive heat transfer in periodic media containing vacuum layers or cavities, heat through which is transferred by radiation. Direct numerical solution of such problems involves significant computational costs and becomes practically unrealistic with a large number of heat-conducting elements, especially for 2D and 3D structures. Therefore, it is relevant to construct effective approximate methods for solving these problems, which, in particular, can be based on special averaging of the initial problems.
This article continues a series of works devoted to the construction and analysis of special discrete, semi-discrete and asymptotic approximations of problems involving complex (radiation-conductive) heat transfer in small-scale structures, consisting of a large number of heat-conducting elements separated by vacuum layers or cavities.
We consider a nonstationary problem that describes the heat propagation process in a system consisting of parallel layers of heat-conducting materials (shields) with alternating thermophysical properties, separated by vacuum layers. The function to be found is the absolute temperature . The heat transfer inside each of the shields is described by the heat conduction equation. At the interfaces between the shields, the conditions of radiation heat transfer between the adjacent shields are specified. At the shield system boundaries, the conditions desribung radiation heat transfer from the outer shields to the external environment are specified.
Three semi-discrete and two asymptotic approximations of the problem under consideration are proposed, which make it possible to find its approximate solutions. Each of the semi-discrete approximations represents the Cauchy problem for a system of nonlinear differential equations, where the unknowns are the approximations to the mean values of temperature on the -th shield. Each of the asymptotic approximations represents a nonlinear initial-boundary value problem for a partial differential equation with non-standard boundary conditions, the solution of which is considered as an approximation to the solution of the original problem.
The results of computational experiments are presented, which demonstrate fairly high accuracy of the proposed methods.
References
2. Бахвалов Н.С., Панасенко Г.П. Осреднение процессов в периодических средах. М.: Наука, 1984.
3. Allaire G., El-Ganaoui K. Homogenization of a Conductive and Radiative Heat Transfer Problem. Simulation with CAST3M // Proc. ASME Summer Heat Transfer Conf. San Francisco, 2005. Pp. 1—6.
4. El-Ganaoui K. Homogénéisation de Modéles de Transfers Thermiques et Radiatifs Dans le Coeur des Réacteur á Coloporteur Gaz. Palaiseau: Ecole Polytechnique, 2006.
5. Allaire G., El-Ganaoui K. Homogenization of a Conductive and Radiative Heat Transfer Problem // SIAM Interdiscip. J. 2009. V. 7(3). Pp. 1148—1170.
6. Habibi Z. Homogénéisation et Convergence á Deux Échelles lors Déchanges Thermiques Stationnaires et Transitoires, Application aux Coeurs de Réacteurs Nucléaires á Caloporteur Gaz. Palaiseau: Ecole Polytechnique, 2011.
7. Allaire G., Habibi Z. Second Order Corrector in the Homogenization of a Conductive-radiative Heat Transfer Problem // Discrete Contin. Dynam. Systems. Ser. B. 2013. V. 18(1). Pp. 1—36.
8. Allaire G., Habibi Z. Homogenization of a Conductive, Convective, and Radiative Heat Transfer Problem in a Heterogeneous Domain // SIAM J. Math. Anal. 2013. V. 45(3). Pp. 1136—1178.
9. Amosov A.A., Krymov N.E. Justification of Discrete and Asymptotic Approximations for the Complex Heat Transfer Problem // J. Math. Sci. 2022. V. 264(5). Pp. 489—513.
10. Амосов А.А., Гулин А.В. Полудискретные и асимптотические аппроксимации задачи переноса тепла в системе серых экранов при наличии излучения // Вестник МЭИ. 2008. № 6. С. 5—15.
11. Amosov A.A. Nonstationary Radiative-conductive Heat Transfer Problem in a Periodic System of Grey Heat Shields // J. Math. Sci. 2010. V. 169(1). Pp. 1—45.
12. Amosov A.A. Semidiscrete and Asymptotic Approximations for the Nonstationary Radiative-conductive Heat Transfer Problem in a Periodic System of Grey Heat Shields // J. Math. Sci. 2011. V. 176(3). Pp. 361—408.
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Для цитирования: Амосов А. А., Николаев А.А. Полудискретные и асимптотические аппроксимации нестационарной задачи радиационно-кондуктивного теплообмена в системе теплопроводящих экранов с чередующимися теплофизическими свойствами // Вестник МЭИ. 2024. № 6. С. 154—163. DOI: 10.24160/1993-6982-2024-6-154-163
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Результаты работы получены в рамках выполнения государственного задания Минобрнауки России (проект № FSWF-2023-0012)
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Конфликт интересов: авторы заявляют об отсутствии конфликта интересов
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1. Bakhvalov N.S. Osrednenie Processa Peredachi Tepla v Periodicheskikh Sredakh pri Nalichii Izlucheniya. Differencial'nye Uravneniya. 1981;17;10:1765—1773. (in Russian).
2. Bakhvalov N.S., Panasenko G.P. Osrednenie Processov v Periodicheskikh Sredakh. M.: Nauka, 1984. (in Russian).
3. Allaire G., El-Ganaoui K. Homogenization of a Conductive and Radiative Heat Transfer Problem. Simulation with CAST3M. Proc. ASME Summer Heat Transfer Conf. San Francisco, 2005:1—6.
4. El-Ganaoui K. Homogénéisation de Modéles de Transfers Thermiques et Radiatifs Dans le Coeur des Réacteur á Coloporteur Gaz. Palaiseau: Ecole Polytechnique, 2006.
5. Allaire G., El-Ganaoui K. Homogenization of a Conductive and Radiative Heat Transfer Problem. SIAM Interdiscip. J. 2009;7(3):1148—1170.
6. Habibi Z. Homogénéisation et Convergence á Deux Échelles lors Déchanges Thermiques Stationnaires et Transitoires, Application aux Coeurs de Réacteurs Nucléaires á Caloporteur Gaz. Palaiseau: Ecole Polytechnique, 2011.
7. Allaire G., Habibi Z. Second Order Corrector in the Homogenization of a Conductive-radiative Heat Transfer Problem. Discrete Contin. Dynam. Systems. Ser. B. 2013;18(1):1—36.
8. Allaire G., Habibi Z. Homogenization of a Conductive, Convective, and Radiative Heat Transfer Problem in a Heterogeneous Domain. SIAM J. Math. Anal. 2013;45(3):1136—1178.
9. Amosov A.A., Krymov N.E. Justification of Discrete and Asymptotic Approximations for the Complex Heat Transfer Problem. J. Math. Sci. 2022;264(5):489—513.
10. Amosov A.A., Gulin A.V. Poludiskretnye i Asimptoticheskie Approksimacii Zadachi Perenosa Tepla v Sisteme Serykh Ekranov pri Nalichii Izlucheniya. Vestnik MEI. 2008;6:5—15. (in Russian).
11. Amosov A.A. Nonstationary Radiative-conductive Heat Transfer Problem in a Periodic System of Grey Heat Shields. J. Math. Sci. 2010;169(1):1—45.
12. Amosov A.A. Semidiscrete and Asymptotic Approximations for the Nonstationary Radiative-conductive Heat Transfer Problem in a Periodic System of Grey Heat Shields. J. Math. Sci. 2011;176(3):361—408
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For citation: Amosov A.A., Nikolaev A.A. Semi-discrete and Asymptotic Approximations of the Non-stationary Problem of Radiation-conductive Heat Transfer in a System of Heat-conductive Shields with Alternating Thermophysical Properties. Bulletin of MPEI. 2024;6:154—163. (in Russian). DOI: 10.24160/1993-6982-2024-6-154-163
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The Results of the Work were Obtained within the Framework of the State Assignment of the Ministry of Education and Science of the Russian Federation (Project No. FSWF-2023-0012
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Conflict of interests: the authors declare no conflict of interest
