Temperature field of anisotropic separation wall of two different media un-der local thermal effect

Аuthors

Attetkov A. V.^*, Volkov I. K.

Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia

*e-mail: fn2@bmstu.ru

Abstract

Mathematical model of the temperature field forming in an anisotropic separation wall of two different media exposed to local thermal effects is proposed. It is shown that the temperature field of the studied object is the sum of the two additive components. The first additive component is being determined from the solution of the problem of determining the temperature field of the isotropic wall under conditions of convective heat exchange with two different media in the absence of an external thermal effect on the studied object. Analytical solution of the non-stationary heat conduction problem under consideration was obtained by applying the Laplace integral transformation. The second independent additive component of the temperature field formed due to the impact of non-stationary thermal flow on the anisotropic wall, when its initial temperature concurs with the temperatures of the external separation media, was identified. Solution of the corresponding problem of non-stationary thermal conductivity was found employing two-dimensional exponential Fourier transform and Laplace integral transform in analytical closed form. The obtained results confirm the previously observed “drift” effect of the temperature field in an anisotropic material with anisotropy properties of the general form.

Keywords:

anisotropic separation wall, local thermal effect, temperature field, integral transformations

References

1. Karslou G., Eger D. Teploprovodnost’ tvyordyh tel [Thermal conductivity of solids]. M.: Nauka, 1964. 488 p. In Russ.

2. Lykov A.V. Teoriya teploprovodnosti [Theory of heat conductivity]. Moscow: Vysshaya shkola, 1967. 600 p. In Russ.

3. Kartashov E.M. Analiticheskie metody v teorii teploprovodnosti tvyordyh tel [Analytical methods in the theory of the thermal conductivity of solids]. M.: Vysshaya shkola, 2001. 552 p. In Russ.

4. Formalev V.F. Teploprovodnost’ anizotropnyh tel. Analiticheskie metody resheniya zadach [Thermal conductivity of anisotropic bodies. Analytical methods for solving problems]. M.: Fizmatlit, 2014. 312 p.

5. Formalev V. F. Heat and mass transfer in anisotropic bodies. High Temperature, 2001,vol. 39, iss. 5, pp. 753–774.

6. Formalev V. F., Reviznikov D.L. Chislennye metody [Numerical methods] Moscow: Fizmatlit, 2004. 400 p. In Russ.

7. Formalev V.F. Teploperenos v anizotropnyh tvyordyh telah. Chislennye metody, teplovye volny, obratnye zadachi [Heat transfer in anisotropic solids. Numerical methods, heat waves, inverse problems]. M.: Fizmatlit, 2015. 280 p.

8. Formalev V.F., Kolesnik S.A. Matematicheskoe modelirovanie aehrogazodinamicheskogo nagreva zatuplyonnyh anizotropnyh tel [Mathematical modeling of aerogasdynamic heating of blunted anisotropic bodies]. M.: Izd-vo MAI, 2016. 160 p.

9. Formalev V.F., Tyukin O.A. Investigation of three-dimensional unsteady heat conduction in anisotropic bodies based on ananalytical solution. High Temperature, 1998, vol. 36, no. 2, pp. 222–229.

10. Formalev V.F., Kolesnik S.A. Аnaliticheskoe reshenie vtoroj nachal’no-kraevoj zadachi anizotropnoj teploprovodnosti [Analytic solution of second initial boundary problem of anisotropic heat conduction]. Matematicheskoe modelirovanie – Mathematical modeling, 2001, vol. 13, no. 7, pp. 21–25. In Russ.

11. Аttetkov А.V., Volkov I.K. Temperaturnoe pole anizotropnoj okhlazhdaemoj plastiny, nakhodyashhejsya pod vozdejstviem vneshnego teplovogo potoka [Temperature field of anisotropic cooled plate under the influence of external heat flow]. Izv. RАN. Energetika – Proceedings of the Russian Academy of Sciences. Power Engineering, 2012, no. 6, pp. 108–117. In Russ.

12. Аttetkov А.V., Volkov I.K. Temperaturnoe pole okhlazhdaemoj izotropnoj plastiny s anizotropnym pokrytiem, nakhodyashhejsya pod vozdejstviem vneshnego teplovogo potoka [Temperature field of a cooled isotropic plate with anisotropic covering under influence of external heat flow]. Teplovye protsessy v tekhnike – Thermal processes in engineering, 2013, vol. 5, no. 2, pp. 50–58. In Russ.

13. Аttetkov А.V., Volkov I.K. Temperaturnoe pole anizotropnogo poluprostranstva, podvizhnaya granitsa kotorogo nakhoditsya pod vozdejstviem vneshnego teplovogo potoka [Temperature field of an anisotropic half-space with movable boundary being under influence of external heat flux]. Teplovye protsessy v tekhnike – Thermal processes in engineering, 2015, vol. 7, no. 2, pp. 73–79. In Russ.

14. Аttetkov А.V., Volkov I.K. Temperaturnoe pole anizotropnogo poluprostranstva, podvizhnaya granitsa kotorogo soderzhit plenochnoe pokrytie [Temperature field of anisotropic half-space, the moving boundary of which contains a film coating]. Izv. RАN. Energetika – Proceedings of the Russian Academy of Sciences. Power Engineering, 2015, no. 3, pp. 39–49. In Russ.

15. Formalev V.F., Kolesnik S.A. Analytical investigation of heat transfer in an anisotropic band with heat fluxes assigned at the boundaries. Journal of Engineering Physics and Thermophysics, 2016, vol. 89, no. 4, pp. 975–984.

16. АttetkovА.V., Volkov I.K. Temperaturnoe pole anizotropnogo poluprostranstva s podvizhnoj granitsej, obladayushhej termicheskitonkim pokrytiem, pri ego nagreve vneshnej sredoj [Temperature field of the anisotropic half-space with the moving boundary, which has a thermally thin coating, heated by convection]. Teplovye protsessy v tekhnike – Thermal processes in engineering, 2016, vol. 8, no. 8, pp. 378–384. In Russ.

17. Formalev V.F., Kolesnik S.A., Kuznetcova E.L., Selin I.A. Аnaliticheskoe issledovanie teploperenosa v teplozashhitnykh kompozitsionnykh materialakh s anizotropiej obshhego vide pri proizvol’nom teplovom potoke [Heat transfer analytical investigation in heat protective composites with general type anisotropy under arbitrary heat loading]. Mekhanika kompozitsionnykh materialov i konstruktsij – Mechanics of composite materials and structures, 2017, vol. 23, no. 2, pp. 168–182. In Russ.

18. Formalev V.F., Kolesnik S.A., Kuznetsova E.L. Time-dependent heat transfer in a plate with anisotropy of general form under the action of pulsed heat sources. High Temperature, 2017, vol. 55, no. 5, pp. 761–766.

19. АttetkovА.V., Volkov I.K. Kvazistatsionarnoe temperaturnoe pole anizotropnoj sistemy s podvizhnoj granitsej, nagrevaemoj sredoj s ostsilliruyushhej temperaturoj [Quasistationary temperature field of the anisotropic system with a moving boundary of the heated environment with temperature oscillating]. Izv. RАN. Energetika – Proceedings of the Russian Academy of Sciences. Power Engineering, 2017, no.5, pp. 144–155. In Russ.

20. Koshlyakov N.S., Gliner E.B., Smirnov M.M. Uravneniya v chastnykh proizvodnykh matematicheskoj fiziki [Partial differential equations of mathematical physics]. Moscow: Vysshaya shkola, 1970. 712 p. In Russ.

21. Pekhovich A.I., Zhidkih V.M. Raschyot teplovogo rezhima tvyordyh tel [Calculation of the thermal regime of solids]. L.: Energiya, 1968. 304 p. In Russ.

22. El’sgol’c L.E. Differencial’nye uravneniya i variacionnoe ischislenie [Differential equations and calculus of variations]. Moscow: Nauka, 1969. 424 p. In Russ.

23. Sneddon I. Preobrazovaniya Fur’e [Fourier transforms]. M.: Izd-vo inostr. lit., 1955. 668 p. In Russ.

24. Bellman R. Vvedenie v teoriyu matric [Introduction to the theory of matrices]. M.: Nauka, 1969. 368 p. In Russ.

25. Budak B.M., Fomin S.V. Kratnye integraly i ryady [Multiple integrals and series]. Moscow: Nauka, 1965. 608 p. In Russ.

26. Bejtmen G., Erdeji A. Tablicy integral’nyh preobrazovanij. Preobrazovaniya Fur’e, Laplasa, Mellina [Tables of integral transformations. Fourier, Laplace, Mellin transformations]. M.: Nauka, 1969. 334 p. In Russ.