New Relations for Analytical Solution of the Local Non-Equilibrium Heat Transfer

Authors: Kartashov E.M.  Published: 14.12.2023
Published in issue: #6(111)/2023  
DOI: 10.18698/1812-3368-2023-6-4-24

Category: Mathematics and Mechanics | Chapter: Differential Equations and Mathematical Physics  
Keywords: generalized mathematical models, locally non-equilibrium heat transfer, regions with a moving boundary


The paper develops generalized model representations of the locally non-equilibrium heat transfer in terms of the theory of non-stationary heat conduction for the hyperbolic type equations. The model simultaneously includes three coordinate systems: 1) Cartesian coordinates--a massive body bounded by a flat surface (one-dimensional case); 2) spherical coordinates--a massive body with an internal spherical cavity (central symmetry); 3) cylindrical coordinates--a massive body with an internal cylindrical cavity (radial heat flow). Three types of intensive heating (cooling) are considered: 1) temperature; 2) thermal; 3) media heating. Examples of locally non-equilibrium heat transfer of a wave nature are provided taking into account the finite speed of heat propagation. The wave character is expressed by the Heaviside step function in analytical solution to the basic problems for the partially bounded regions. Isochrones for the temperature functions were constructed. It is shown that the temperature profile is having discontinuity on the surface of the traveling wave front. This leads to a delay in the heat outflow beyond the discontinuity boundary, which is a characteristic feature in analytical solutions to the wave equations. Functional constructions of analytical solutions to basic problems of the locally non-equilibrium heat transfer are presented in a region with the moving boundary, it is a practically new case in analytical thermophysics. The presented material has wide practical application

Please cite this article in English as:

Kartashov E.M. New relations for analytical solution of the local non-equilibrium heat transfer. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2023, no. 6 (111), pp. 4--24 (in Russ.). DOI: https://doi.org/10.18698/1812-3368-2023-6-4-24


[1] Kartashov E.M. New operational relations for mathematical models of local nonequilibrium heat transfer. Russian Technological Journal, 2022, vol. 10, no. 1, pp. 7--18 (in Russ.). DOI: https://doi.org/10.32362/2500-316X-2022-10-1-68-79

[2] Kartashov E.M. Analiticheskie metody v teorii teploprovodnosti tverdykh tel [Analytical methods of heat conduction theory of solid body]. Moscow, Vysshaya shkola Publ., 2001.

[3] Kartashov E.M., Kudinov V.A. Analiticheskie metody teorii teploprovodnosti i ee prilozheniy [Analytical methods of heat conduction theory and its applications]. Moscow, URSS Publ., 2018.

[4] Lykov A.V. Teoriya teploprovodnosti [Theory of thermal conductivity]. Moscow, Vysshaya shkola Publ., 1967.

[5] Zarubin V.S. Inzhenernye metody resheniya zadach teploprovodnosti [Engineering methods for solving heat conduction problems]. Moscow, Energoatomizdat Publ., 1983.

[6] Tikhonov A.N., Samarskiy A.A. Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow, Nauka Publ., 1977.

[7] Formalev V.F. Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow, URSS Publ., 2021.

[8] Sobolev S.L. On hyperbolic heat-mass transfer equation. Int. J. Heat Mass Transf., 2018, vol. 122, pp. 629--630. DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2018.02.022

[9] Kudinov I.V., Kudinov V.A. Mathematical simulation of the locally nonequilibrium heat transfer in a body with account for its nonlocality in space and time. J. Eng. Phys. Thermophy., 2015, vol. 88, no. 2, pp. 406--422. DOI: https://doi.org/10.1007/s10891-015-1206-6

[10] Kudinov V.A., Eremin A.V., Kudinov I.V. The development and investigation of a strongly non-equilibrium model of heat transfer in fluid with allowance for the spatial and temporal non-locality and energy dissipation. Thermophys. Aeromech., 2017, vol. 24, no. 6, pp. 901--907. DOI: https://doi.org/10.1134/S0869864317060087

[11] Kirsanov Yu.A., Kirsanov A.Yu. About measuring the thermal relaxation time of solid body. Izvestiya RAN. Energetika [Proceedings of the Russian Academy of Sciences. Power Engineering], 2015, no. 1, pp. 113--122 (in Russ.).

[12] Sinkevich O.A., Semenov A.M. Solution of the Boltzmann equation by expanding the distribution function with several time and coordinate scales in the Enskog series in Knudsen parameter. Tech. Phys., 2003, vol. 48, no. 10, pp. 1221--1225. DOI: https://doi.org/10.1134/1.1620111

[13] Maxwell J.C On the dynamical theory of gases. Phil. Trans. Royal. Soc., 1967, vol. 157, no. 1, pp. 49--88. DOI: https://doi.org/10.1098/rstl.1867.0004

[14] Lykov A.V. Teploprovodnost i diffuziya v proizvodstve kozhi, zameniteley i drugikh materialov [Heat conduction and diffusion in the production of leather, substitutes and other materials]. Moscow, Gizlegprom Publ., 1941.

[15] Cattaneo C. Sulla Conduzione de Calore. Atti del. Seminario Matematico e fisico dela Universita di Modena, 1948, vol. 3, pp. 3--21.

[16] Vernotte P. Les paradox de la theorie continue de l’equation de la chaleur. C. R. Acad. Sci., 1958, vol. 246, no. 22, pp. 3154--3155.

[17] Kartashov E.M. Analytical solutions of hyperbolic heat-conduction models. J. Eng. Phys. Thermophy., 2014, vol. 87, no. 5, pp. 1116--1125. DOI: https://doi.org/10.1007/s10891-014-1113-2

[18] Fok I.A. Reshenie odnoy zadachi teorii diffuzii po metodu konechnykh raznostey i prilozhenie ego k diffuzii sveta [Solving the problem of diffusion theory by the finite difference method and its application to light scattering]. Leningrad, GOI Publ., 1926.

[19] Davydov B.I. Diffusion equation with molecular velocity. DAN SSSR, 1935, no. 2, pp. 474--475 (in Russ.).

[20] Predvoditelev A.S. Problemy teplo- i massoperenosa [Problems of heat and mass transfer]. Moscow, Energiya Publ., 1970.

[21] Zarubin V.S., Kuvyrkin G.N. Matematicheskie modeli termomekhaniki [Mathematical models of thermomechanics]. Moscow, FIZMATLIT Publ., 2002.

[22] Attetkov A.V., Volkov I.K. Temperature field of anisotropic half-space with movable boundary at local thermal impact in conditions of heat exchange with outer ambient. Teplovye protsessy v tekhnike [Thermal Processes in Engineering], 2018, vol. 10, no. 1-2, pp. 56--61 (in Russ.).

[23] Attetkov A.V., Belyakov N.S., Volkov I.K. Influence of boundary mobility on temperature field of solid body with cylindrical channel under non-stationary conditions of heat exchange with environment. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2006, no. 1 (20), pp. 31--40 (in Russ.).