Оценка погрешности дискретной аппроксимации стационарной задачи радиационно-кондуктивного теплообмена в системе абсолютно черных стержней квадратного сечения

Никита [Nikita] Евгеньевич [E.] Крымов [Krymov]

doi:10.24160/1993-6982-2021-5-128-134

Никита [Nikita] Евгеньевич [E.] Крымов [Krymov]

DOI: https://doi.org/10.24160/1993-6982-2021-5-128-134

Keywords: radiant-and-conduction heat transfer, discrete approximation, error estimate

Abstract

Studying heat transfer processes in periodic media containing vacuum interlayers or cavities, heat through which is transferred by radiation, is of significant interest for applications. Direct numerical solution of such problems involves considerable computational efforts and becomes almost impossible for systems containing a large number of heat conducting elements, especially for 2D and 3D structures. Therefore, it is of issue to develop effective approximate solution methods for such problems.

This publication continues a series of studies on developing and substantiating special discrete and asymptotic approximations of radiant-and-conduction heat transfer problems in periodic systems of heat conducting elements separated by vacuum.

In this study, the stationary radiant-and-conduction heat transfer problem in a system of absolutely black square rods is considered. The sought quantity is the absolute temperature, which is found from the solution of the boundary-value problem for the stationary heat conduction equation with nonlinear nonlocal boundary conditions describing radiant heat transfer between the rods through vacuum interlayers.

A special discrete approximation of this problem leading to the system of linear algebraic equations with respect to the fourth power of the temperature is presented. The solution of this system as approximation of the mean temperature over the rod cross-section is described.

The discrete approximation error estimate as a function of the square rod side length (the small parameter of the problem) and the thermal conductivity coefficient has been obtained. The obtained error estimate proves applicability of the discrete approximation for materials with a high thermal conductivity coefficient.

Information about author

Никита [Nikita] Евгеньевич [E.] Крымов [Krymov]

Ph.D-student of Mathematical and Computer Modeling Dept., NRU MPEI, e-mail: krymovny@mpei.ru

References

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Для цитирования: Крымов Н.Е. Оценка погрешности дискретной аппроксимации стационарной задачи радиационно-кондуктивного теплообмена в системе абсолютно черных стержней квадратного сечения // Вестник МЭИ. 2021. № 5. С. 128—134. DOI: 10.24160/1993-6982-2021-5-128-134.
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Работа выполнена при поддержке: Министерства науки и высшего образования РФ (проект № FSWF-2020-0022)
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2. 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.
3. Kremkova A.A. Poludiskretnye i Asimptoticheskie Approksimatsii dlya Zadachi Radiatsionno-konduktivnogo Teploobmena v Periodicheskoy Strukture. Vestnik MEI. 2012;6:151—161. (in Russian).
4. Amosov A.A., Kremkova A.A. Otsenka Pogreshnosti Poludiskretnogo Metoda Resheniya Zadachi Radiatsionno-konduktivnogo Teploobmena v Dvumernoy Periodicheskoy Strukture. Vestnik MEI. 2013;6:22—36. (in Russian).
5. Amosov A.A., Maslov D.A. Poludiskretnye i Asimptoticheskie Approksimatsii Statsionarnoy Zadachi Radiatsionno-konduktivnogo Teploobmena v Dvumernoy Sisteme Plastin. Vestnik MEI. 2015;3:118—125. (in Russian).
6. Amosov A.A., Maslov D.A. Semidiscrete Approximations for the Stationary Radiative-conductive Heat Transfer Problem in the Two-dimensional System of Plates. Russ. J. Numer. Anal. Math. Modell. 2016;31(1):1—17.
7. Amosov A.A. Asymptotic Approximations for the Stationary Radiative-sonductive Heat Transfer Problem in a Two-dimensional System of Plates. Russ. J. Numer. Anal. Math. Modell. 2017;32(3):173—186.
8. Amosov A.A., Krymov N.E. On a Nonstandard Boundary Value Problem Arising in Homogenization of Complex Heat Transfer Problems. J. Math. Sci. 2020;244(3):357—377.
9. Amosov A.A., Krymov N.E. Discrete and Asymptotic Approximations for One Stationary Radiative-conductive Heat Transfer Problem. Russ. J. Numer. Anal. Math. Modell. 2020;35(3):127—141.
10. Amosov A.A., Krymov N.E. Error Estimate for Discrete Approximation of the Radiative-conductive Heat Transfer Problem in a System of Absolutely Black Rods. J. Math. Sci. 2020;251(6):773—786.
11. Amosov A.A. O Razreshimosti Zadachi Teploobmena Izlucheniem po Zakonu Stefana-Bol'tsmana. Vestnik MGU. Seriya «Vychislitel'naya Matematika i Kibernetika». 1980;3:18—26. (in Russian).
12. Amosov A.A. Stationary Nonlinear Nonlocal Problem of Radiative-conductive Heat Transfer Problem in a System of Opaque Bodies with Properties Depending on the Radiation Frequency. J. Math. Sci. 2010;164(3)L309—344.
13. Samarskiy A.A. Vvedenie v Teoriyu Raznostnykh Skhem. M.: Nauka, 1971. (in Russian).
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For citation: Krymov N.E. Estimating the Discrete Approximation Error in Solving the Stationary Radiant-and-Conduction Heat Transfer Problem in a System of Absolutely Black Square Rods. Bulletin of MPEI. 2021;5:128—134. (in Russian). DOI: 10.24160/1993-6982-2021-5-128-134.
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The work is executed at support: Ministry of Science and Higher Education of the Russian Federation (Project No. FSWF-2020-0022)