Heat transfer processes in a solid with absorbing inclusion while the laser radiation impact

Аuthors

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

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

*e-mail: fn2@bmstu.ru
**e-mail: kseniyagaydaenko@gmail.com

Abstract

The article considers the problem of determining of temperature field of the isotropic solid with spherical inclusion while laser radiation impact. The analyzed mathematical model of the heat transfer process in the system under study is based on the hypothesis on the possibility of equating the average integrated temperature of the inclusion to its contact boundary temperature, i.e. on realization of the idea of the “cumulative capacitance”. It represents a mixed problem for the linear equation in partial derivatives of the second order parabolic type with specific edge condition, actually accounting for the absorbing inclusion presence in the system. The approximate analytical method, employing integral representation of the considered heat transfer problem, was proposed. The idea of solution representation in the form of the key small-parameter expansion of the mathematical model being realized, namely, the simplex of similarity of physical properties of the system being analyzed with further employing of mixed Fourier transform on the spatial variable for obtaining the sought-after functions, lies in the basis of the above said method. The article demonstrates that the proposed method is the most convenient in terms of its practical realizability while performing parametrical analysis of the temperature field of the object under study. The obtained results were used for theoretical evaluation of the absorbing scenic inclusion impact on temperature field being formed while the object under study was affected by the radiation of constant power density.

Keywords:

isotropic solid, laser radiation, absorbing spherical inclusion, temperature field, mixed Fourier integral transform

References

1. Karslou G., Eger D. Teploprovodnost’ tvyordyhtel [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. Assovsky I.G.Fizika goreniya i vnutrennyaya ballistika [Combustion physics and internal ballistics]. Moscow: Nauka, 2005. 357 p. In Russ.

6. Chernai A.V. On the mechanism of ignition of condensed secondary explosives by a laser pulse. Combustion, Explosion, and Shock Waves, 1996, vol. 32, no. 1, pp. 8–15.

7. Burkina R.S., Morozova E.Y., Tsipilev V.P.Initiation of a reactive material by a radiation beam absorbed by optical heterogeneities of the material. Combustion, Explosion, and Shock Waves, 2011, vol. 47, no. 5, pp. 581–590.

8. Kriger V.G., Kalenskii A.V., Zykov I.Y., Nikitin A.P., Zvekov A.A. Heat-transfer processes upon laser heating of inert-matrix-hosted inclusions. Thermophysics and Aeromechanics,2013, vol. 20, no.3, pp. 367–374.

9. Aduev B.P., Anan’eva M.V., Zvekov A.A., Kalenskii A.V., Kriger V.G., Nikitin A.P. Miro-hotspot model for the laser initiation of explosive decomposition of energetic materials with melting taken into account. Combustion, Explosion, and Shock Waves, 2014, vol. 50, no. 6, pp. 704–710.

10. Kalensky A.V., Zvekov A.A., Nikitin A.P. Mikrochagovaya model’ s uchetom zavisimosti koeffitsienta effektivnosti pogloshheniya lazernogo impul’sa ot temperatury [Micro-focal model taking into account the dependence of the efficiency coefficient of a laser pulse absorption on temperature]. Khimicheskaya fizika – Chemical Physics, 2017, vol. 36, no. 4, pp. 43–49. In Russ.

11. Attekov A.V., Gaydayenko K.A.Protsessy teploperenosa v tverdom tele so sfericheskim ochagom razogreva [Processes of heat transfer in a solid with a spherical heating center]. Saarbrücken: LAP Lambert Academic Publishing, 2017. 52 с. In Russ.

12. Zarubin V.S., Kuvyrkin G.N.Matematicheskie modeli mekhaniki i ehlektrodinamiki sploshnoj sredy [Mathematical models of mechanics and electrodynamics of continua]. Moscow: Publishinghouse Bauman, 2008. 512 p.In Russ.

13. Pudovkin M.А., Volkov I.K.Kraevye zadachi matematicheskoj teorii teploprovodnosti v prilozhenii k raschetam temperaturnykh polej v neftyanykh plastakh pri zavodnenii [Boundary-value problems of the mathematical theory of heat conduction in application to calculations of temperature fields in oil reservoirs in water flooding].Kazan: Publishinghouse of Kazan University, 1978. 188 p. In Russ.

14. Attekov A.V., Volkov I.K., Gaidaenko K.A.Protsessy teploperenosa v prozrachnom dlya izlucheniya tverdom tele s pogloshhayushhim sfericheskim vklyucheniem [Heat transfer processes in a radiation-transparent solid with absorbing spherical inclusion]. Trudy Sed’moj Rossijskoj natsional’noj konferentsii po teploobmenu – Proceedings of the Seventh Russian National Conference on Heat Exchange, Moscow, 2018, vol. 3, pp. 7–11. In Russ.

15. Ladyzhenskaya O.А., Solonnikov V.А., Ural’tseva N.N. Linejnye i kvazilinejnye uravneniya parabolicheskogo tipa[Linear and quasilinear equations of parabolic type]. Moscow, Nauka, 1967. 736 p. In Russ.

16. Nayfeh A.H. Introduction to Perturbation Techniques. Wiley, New York, 1981. 536 p. (Russ. ed. Nayfeh A. Vvedenie v metody vozmushhenij. Moscow: Mir, 1984. 535 p.).

17. Naimark M.A. Linejnye differentsial’nye operatory [Linear Differential Operators]. Moscow: Nauka, 1969. 528 p. In Russ.

18. Volkov I.K., Kanatnikov A.N. Integral’nye preobrazovaniya i operatsionnoe ischislenie [Integral transformations and operational calculus]. Moscow: Publishinghouse Bauman, 2015. 228 p. In Russ.