The temperature field of the cylinder and shell of elliptical sections under boundary conditions of the third kind


Аuthors

Kanareykin A. I.

Sergo Ordzhonikidze Russian State University for Geological Prospecting, Moscow, 117997, Russia

e-mail: kanareykins@mail.ru

Abstract

The processes of heat exchange and the resulting mass transfer currently play an important role both in the technical field and in nature. The temperature regime of the environment directly depends on this group of processes. In addition, this group of processes determines the working process in various kinds of technological installations. This leads to the active development of the theory of heat transfer, especially in recent decades. The active development of the theory is also conditioned by the needs of such spheres of human activity as cosmonautics, thermal power engineering, and nuclear energy.

Thus, thermal processes and phenomena are the most common after mechanical motion. The discovery of processes and laws of thermal phenomena made it possible to effectively use them in the development of practical technical solutions, as well as in the design of heat engines and specialized installations.

The main provisions of the theory of heat and mass transfer have been developed for a long time. In the mathematical theory of thermal conductivity, an important place is occupied by studies of heat transfer processes in solids with a cylindrical channel, the surface of which is subject to a given mode of thermal action. Separately, heating elements made in the form of a cylinder can be distinguished. Such ele-ments have great mechanical strength and are widely used in heat exchangers. They can be both electrical and fuel-generating elements of a nuclear reactor.

Elliptical tube heat exchangers occupy a special place. Elliptical tube heat exchangers are increasingly being used in various industrial products. Their peculiarity lies in the fact that by manipulating the length change of the semi-axes of the ellipse, it is possible to obtain accurate analyses of stationary thermal conductivity problems for a very wide range of shape changes: from a cylinder (the semi-axes of the ellipse are equal) to a thin plate (one of the semi-axes significantly exceeds the other).

However, the temperature distribution in a body of elliptical cross-section under given boundary conditions has not been sufficiently studied. The paper calculates the temperature field of the cylinder and the elliptical cross-section shell surrounding it under boundary conditions of the fourth kind. It is assumed that the thermal contact between the solid–coating system is ideal. The boundary conditions are set by the law of heat exchange between the shell surface and the environment. The ellipses in question are confocal. At the same time, the task itself is stationary. The solution is found analytically when moving to the elliptical coordinate system. From the solution, a ratio is obtained for calculating the temperature distribution over the surface of an elliptical channel. The proposed mathematical model and its solution algorithm are intended for engineering calculations of heat exchangers and fuel rods of nuclear reactors using elliptical channels.

Keywords:

heat transfer, temperature field, elliptical cross section, boundary conditions of the third kind, boundary conditions of the fourth kind, elliptic integral

References

  1. Kanareikin AI. Temperature field distribution in a fuel element with an elliptical cross-section. Nauchnye trudy Kaluzhskogo gosudarstvennogo universiteta im. K.Eh. Tsiolkovskogo, seriya: estestvennye nauki, 2016: 230–231. (In Russ.).

  2. Tatsiy RM, Pazen OY, Vovk SY et al. Simulation of heat transfer process in a multilateral cylindrical shell taking into account the internal heat sources. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu. 2020; (3):27–32. (In Russ.).

  3. Vidin YuV, Kazakov RV, Zlobin VS. Heat transfer process in a two-layer cylindrical body. Izvestiya vysshikh uchebnykh zavedenii. Problemy ehnergetiki. 2018; 20(11–12):93–98. (In Russ.).

  4. Eremin AV, Stefanyuk EF, Kurganova OY et al. A generalized function in heat conductivity problems for multi-layer structures with heat sources. J. Mach. Manuf. Reliab. 2018;47:249–255.

  5. Tatsii RM, Pazen OY. Direct (classical) method of calculation of the temperature field in a hollow multilayer cylinder. J. Eng. Phys. Thermophys. 2018;91:1373–1384.

  6. Singh S, Jain PK. Rizwan-uddin. Analytical solution to transient heat conduction in polar coordinates with multiple layers in radial direction. Int J Therm Sci. 2008; 47:261–273.

  7. Jain PK, Singh S. Rizwan-uddin, Analytical solution to transient asymmetric heat condution in a multilayers annulus. J. Heat Transfer. 2009;131:1–7.

  8. Jain PK, Singh S, Rizwan-uddin. An exact analytical solution for two-diemsional, unsteady, multilayer heat conduction is spherical coordinates. Int. J. Heat Transfer. 2010;53:2133–2142.

  9. Norouzi M, Delouei A, Seilsepour M. A general exact solution for heat conduction in multilayer spherical composite laminates. Compos. Struct. 2013. Vol. 106. 288–295.

  10. Dalir N, Nourazar SS. Analytical solution of the problem on the tree-dimensional transient heat conduction in a multilayer cylinder. J. Eng. Phys. Thermophys. 2014;87: 89–97.

  11. Haji-Sheikh A, Beck JV. Temperature solution in multi-dimensional mutli-layer bodies. Int. J. Heat Mass Transf. 2002;45:1865–1877.

  12. Kanareikin AI. Temperature distribution in a body of elliptical cross-section with an internal heat source under boundary conditions of the first kind. Vestnik Kaluzhskogo universiteta, seriya: estestvennye nauki. 2020; (2):74–76. (In Russ.).

  13. Kanareykin AI. Energy calculation of the temperature field of an elliptical body without internal heat sources under boundary conditions of the third kind. IOP Conf. Series: Earth and Environmental Science. 2022;(1045). DOI: 10.1088/1755-1315/1045/1/012068

  14. Kanareykin A. Temperature distribution in an elliptical body with an internal heat source with partial adiabatic isolation. E3S Web of Conferences. 2021;258. DOI: 10.1051/e3sconf/202125809071

  15. Kanareikin AI. Temperature distribution in a body of elliptical cross-section without internal heat sources under boundary conditions of the third kind. Naukosfera. 2022; (7–2):96–100. (In Russ.).

  16. Kanareikin AI. Temperature distribution in a body of elliptical cross-section with an internal heat source under boundary conditions of the third kind. Vestnik Kaluzh-skogo universiteta. 2021;(2 (51)):107–109. DOI: 10.5407 2/18192173_2021_2_107

  17. Kanareikin AI. Temperature distribution in a body of elliptical cross-section with internal steady-state heat sources under boundary conditions of the third kind. Thermal processes in engineering. 2021;13(5):226–229. DOI: 10.34759/TPT-2021-13-5-226-229

  18. Dhakate T, Varghese V, Khalsa L. An analytical solution for transient asymmetric heat conduction in a multilayer elliptic annulus and its associated thermal stresses. Int. J. Math. And Appl. 2018;6(1):29–42.

  19. Fathi M. Heat Conduction within elliptic enclosure with internal heat generation. ERJ Engineering Research Journal. 2020;43(2):93–98.  DOI: 10.21608/erjm.2020.83837

  20. Gołebiowski J, Zareba M. The Stationary Thermal Field in a Multi-layer Elliptic Cylinder. Appl. Sci. 2023;13. DOI: 10.3390/app13042354

  21. Kanareikin A.I. Heat transfer between a cylindrical heating element and its sheath under boundary conditions of the fourth kind. Vestnik Mezhdunarodnoi akademii kholoda. 2023;(3):68–73. DOI: 10.17586/1606‑4313‑2023‑22‑3-68–73 (In Russ.).

mai.ru — informational site of MAI

Copyright © 2009-2026 by MAI