Steady-State Temperature Field in a Separator System Featuring Active Thermal Protection with Anisotropic Properties

We stated and solved the problem of determining the steady-state temperature field of a system simulated by a wall separating two different media. One of the wall surfaces has a thermally active layer that functions according to the feedback principle and features an anisotropic coating subjected to local heating while undergoing heat exchange with the environment. We show that the sought-after temperature field is an additive composition of two independent components, the first of which is exclusively a function of the heat exchange intensity between the separated media and the boundary surfaces of the system under consideration, and the second is a function of the heat flow power density affecting the separator system for the temperatures of the media separated being zero. We used integral transform methods in an analytically closed form to find solutions to respective steady-state heat conductivity problems. The results obtained confirm the existence of a previously found temperature field "drift" effect in an anisotropic material displaying general anisotropy of its properties

References

[1] Polezhaev Yu.V., Yurevich F.B. Teplovaya zashchita [Heat protection]. Moscow, Energiya Publ., 1976.

[2] Zarubin V.S. Raschet i optimizatsiya teploizolyatsii [Calculation and optimization of heat isolation]. Moscow, Energoatomizdat Publ., 1991.

[3] Polezhaev Yu.V., Shishkov A.A. Gazodinamicheskie ispytaniya teplovoy zashchity [Gas-dynamic tests of heat protection]. Moscow, Promedek Publ., 1992.

[4] Galitseyskiy B.M., ed. Teplovaya zashchita lopatok turbin [Heat protection of turbine blades]. Moscow, MAI Publ., 1996.

[5] Zinchenko V.I. Matematicheskoe modelirovanie sopryazhennykh zadach teploobmena [Mathematical simulation of heat exchange adjoint problems]. Tomsk, TGU Publ., 1985.

[6] Grishin A.M., Golovanov A.N., Zinchenko V.I., et al. Matematicheskoe i fizicheskoe modelirovanie teplovoy zashchity [Mathematical and physical simulation of heat protection]. Tomsk, TGU Publ., 2011.

[7] Formalev V.F., Kuznetsova E.L. Teplomassoperenos v anizotropnykh telakh pri aerogazodinamicheskom nagreve [Heat and mass transfer in anisotropic bodies during aerogasdynamic heating]. Moscow, MAI-Print Publ., 2010.

[8] Formalev V.F. Teploperenos v anizotropnykh tverdykh telakh. Chislennye metody, teplovye volny, obratnye zadachi [Heat transfer in anisotropic solids. Numerical methods, heat waves, reverse problems]. Moscow, FIZMATLIT Publ., 2015.

[9] Formalev V.F., Kolesnik S.A. Matematicheskoe modelirovanie aerogazodinamicheskogo nagreva zatuplennykh anizotropnykh tel [Mathematical simulation of gas-dynamic heating of blunt anisotropic objects]. Moscow, MAI Publ., 2016.

[10] Lykov A.V. Teoriya teploprovodnosti [Heat conduction theory]. Moscow, Vysshaya shkola Publ., 1967.

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

[12] Formalev V.F. Teploprovodnost’ anizotropnykh tel. Analiticheskie metody resheniya zadach [Heat conduction of anisotropic bodies. Analytical methods for problems solving]. Moscow, FIZMATLIT Publ., 2014.

[13] Anatychuk L.I. Termoelementy i termoelektricheskie ustroystva [Thermal elements and thermoelectric devices]. Kiev, Naukova dumka Publ., 1979.

[14] Attetkov A.V., Volkov I.K., Tverskaya E.S. Thermosetting pad as a means of controlled influence on the temperature field construction. Izvestiya RAN. Energetika [Proceedings of the Russian Academy of Sciences. Power Engineering], 2002, no. 4, pp. 131--141 (in Russ.).

[15] Attetkov A.V., Volkov I.K., Tverskaya E.S. Baseline model of heat transfer in screened half-space with a thermosetting layer. Izvestiya RAN. Energetika [Proceedings of the Russian Academy of Sciences. Power Engineering], 2009, no. 2, pp. 147--155 (in Russ.).

[16] Attetkov A.V., Volkov I.K., Tverskaya E.S. Mathematical modeling of the process of heat transfer in a shielded half-space with a thermoactive spacer under external thermal action. J. Eng. Phys. Thermophy., 2008, vol. 81, iss. 3, pp. 588--597. DOI: https://doi.org/10.1007/s10891-008-0070-z

[17] Attetkov A.V., Volkov I.K., Tverskaya E.S. Temperature field of a screened wall with a thermoactive lining on exposure to an axisymmetric thermal effect. J. Eng. Phys. Thermophy., 2009, vol. 82, no. 5, art. no. 940. DOI: https://doi.org/10.1007/s10891-009-0276-8

[18] Attetkov A.V., Volkov I.K., Tverskaya E.S. Temperature field in the multi-layer halfspace under non-ideal thermal contact between layers. Izvestiya RAN. Energetika [Proceedings of the Russian Academy of Sciences. Power Engineering], 2010, no. 3, pp. 83--91 (in Russ.).

[19] Volkov I.K., Tverskaya E.S. Optimum thickness of screened wall with a feedback heat-activated padding. Nauka i obrazovanie: nauchnoe izdanie MGTU im. N.E. Baumana [Science and Education: Scientific Publication], 2012, no. 5 (in Russ.). DOI: http://dx.doi.org/10.7463/0512.0396333

[20] Attetkov A.V., Volkov I.K., Tverskaya E.S. Mathematical models hierarchy of thermal field forming process in system "isotropic plate-thermal active pad-anisotropic coating". Teplovye protsessy v tekhnike, 2013, vol. 5, no. 5, pp. 224--228 (in Russ.).

[21] Attetkov A.V., Volkov I.K., Tverskaya E.S. Stationary temperature field of a cooled orthotropic wall with a thermal active layer and anisotropic coating, under the influence of external heat flux. Izvestiya RAN. Energetika [Proceedings of the Russian Academy of Sciences. Power Engineering], 2013, no. 5, pp. 136--145 (in Russ.).

[22] Attetkov A.V., Volkov I.K. Temperature field of construction with active heat protection with anisotropic coating. Izvestiya RAN. Energetika [Proceedings of the Russian Academy of Sciences. Power Engineering], 2013, no. 6, pp. 125--136 (in Russ.).

[23] Attetkov A.V., Volkov I.K. Ready-state temperature field of a system with active heat protection. Teplovye protsessy v tekhnike, 2014, vol. 6, no. 2, pp. 81--86 (in Russ.).

[24] Attetkov A.V., Volkov I.K. Peculiarities of formation of the temperature field in a system with an active heat shield. Izvestiya RAN. Energetika [Proceedings of the Russian Academy of Sciences. Power Engineering], 2014, no. 3, pp. 69--81 (in Russ.).

[25] Negoita C.V. Management applications of system theory. Bazel and Stuttgart, 1979.

[26] Dеsоеr С.А., Vidуаsаgаr М. Fееdbаск sуstеm: inрut--оutрut рrореrtiеs. Academic Press, 1975.

[27] Koshlyakov N.S., Gliner E.B., Smirnov M.M. Uravneniya v chastnykh proizvodnykh matematicheskoy fiziki [Partial differential equations of mathematical physics]. Moscow, Vysshaya shkola Publ., 1970.

[28] El’sgolts L.E. Differentsial’nye uravneniya i variatsionnoe ischislenie [Differential equations and variational calculus]. Moscow, Nauka Publ., 1969.

[29] Sneddon I. Fourier transforms. New York, Dover Publ., Inc., 1951.

[30] Bellman R. Introduction to matrix analysis. McGraw-Hill, 1970.