Temperature field of a solid transparent to radiation with thermally thin absorbing inclusion in the form of the globular layer

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

A »micro-focal model» of heat transfer in the two-phase material with spherical inclusions absorbing penetrating radiation occupies an important place in the problem of energy materials explosive decomposition laser initiation. The difficulties emerging while searching for the analytical solution of the corresponding mixed problem for the system of equations in parabolic type second order partial derivatives even in the simplest situation of the presence of a single absorbing inclusion in the material transparent to radiation are well known. A possible way of their overcoming is associated with accepting assumptions of one sort or another, leading to the initial (basic) mathematical model replacement by its simplified analogues. The article formulates the problem of determining a temperature filed of an isotropic solid with an inclusion in the form of a spherical layer absorbing penetrating radiation. The analyzed mathematical model of the heat transfer in the system under study is being based on the hypothesis that the absorbing inclusion is thermally thin. In other words, it is based on the idea of »concentrated capacitance» and represents a mixed problem for the system of two equations in the second order partial derivatives of parabolic type with specific conjugation condition, accounting for the absorbing inclusion presence in the system. The article suggests analytical methods based on the idea of representing the solution of the considered non-stationary heat transfer problem in the Laplace integral transformations image domain in the form of two functions product. One of the functions characterizes the analyzed mode of the radiation flux impact, while the other is being representable as a sum of the uniformly convergent series. Solution of the considered problem of non-stationary heat transfer was obtained in analytical closed form employing well known theorems of operation calculus.

Keywords:

isotropic solid, laser radiation, globular absorbing inclusion, temperature field, Laplace integral transformation

References

1. Karslou G., Eger D. Teploprovodnost’ tverdyh tel (Thermal conductivity of solids). Moscow, Nauka, 1964, 488 p. In Russ.
2. Lykov A.V. Teoriya teploprovodnosti (Theory of thermal conductivity). Moscow, Vysshaya shkola, 1967, 600 p. In Russ.
3. Kartashov E.M. Analiticheskie metody v teorii teploprovodnosti tverdyh tel (Analytical methods in the theory of thermal conductivity of solids), Moscow, Vysshaya shkola, 2001, 550 p. In Russ.
4. Formalev V.F. Teploprovodnost’ anizotropnyh tel. Analiticheskie metody resheniya zadach (Thermal conductivity of
5. anisotropic bodies. Analytical methods for solving problems). Moscow, Fizmatlit, 2014, 312 p. In Russ.
6. Assovskii I.G. Fizika goreniya i vnutrennyaya ballistika (Gorenje physics and internal ballistics). Moscow, Nauka, 2005, 357 p. In Russ.
7. Chernai A.V. O mekhanizme zazhiganiya kondensirovannykh vtorichnykh VV lazernym impul’som. Fizika goreniya i vzryva, 1996, vol. 32, no. 1, pp. 11–19. In Russ.
8. Burkina R.S., Morozova E.Yu., Tsipilev V.P. Initsiirovanie reaktsionnosposobnogo veshchestva potokom izlucheniya pri ego pogloshchenii opticheskimi neodnorodnostyami veshchestva. Fizika goreniya i vzryva, 2011, vol. 47, no. 5, pp. 95–105. In Russ.
9. Kriger V.G., Kalenskii A.V., Anan’eva M.V., Zvekov A.A., Zykov I.Yu. Fiziko-khimicheskie osnovy mikroochagovoi modeli vzryvnogo razlozheniya energeticheskikh materialov. Izv. Vuzov. Fizika, 2013, vol. 56, no. 9-3, pp. 175–180. In Russ.
10. Aduev B.P., Anan’eva M.V., Zvekov A.A., Kalenskii A.V., Kriger V.G., Nikitin A.P. Mikroochagovaya model’ lazernogo initsiirovaniya vzryvnogo razlozheniya energeticcheskikh materialov s uchetom plavleniya. Fizika goreniya i vzryva, 2014, vol. 50, no. 6, pp. 92–99. In Russ.
11. Kalenskii A.V., Zvekov A.A., Nikitin A.P. Mikroochagovaya model’ s uchetom zavisimosti koeffitsienta effektivnosti pogloshcheniya lazernogo impul’sa ot impul’sa. Himicheskaya fizika, 2017, vol. 36, no.4, pp. 43–49. In Russ.
12. Kalenskii A.V., Gazenaur I.V., Zvekov A.A., Nikitin A.P. Kriticheskie usloviya initsiirovaniya reaktsii v TENe pri lazernom nagreve svetopogloshchayushchikh nanochastits. Fizika goreniya i vzryva, 2017, vol. 53, no.2, pp. 107–117. In Russ.
13. Attetkov A.V., Volkov I.K., Gaidaenko K.A. Protsessy teploperenosa v prozrachnom dlya izlucheniya tverdom tele s pogloshchayushchim sfericheskim vklyucheniem. Trudy VII Rossijskoi nacional’noi konferencii po teploobmenu, Moscow, 2018, vol. 3, pp. 7–11. In Russ.
14. Attetkov A.V., Volkov I.K., Gaidaenko K.A. Temperaturnoe pole prozrachnogo dlya izlucheniya tverdogo tela s pogloshchayushchim sfericheskim vklyucheniem. Teplovye processy v tehnike, 2018, vol.10, no. 5–6, pp. 256–264. In Russ.
15. Attetkov A.V., Volkov I.K., Gaidaenko K.A. Protsessy teploperenosa v tverdom tele s pogloshchayushchim vklyucheniem pri vozdeistvii lazernogo izlucheniya. Teplovye
16. processy v tehnike, 2019, vol. 11, no. 5, pp. 216–221. In Russ.
17. Attetkov A.V., Volkov I.K., Gaidaenko K.A. Protsessy teploperenosa v tverdom tele s pogloshchayushchim pronikayushchee izluchenie vklyucheniem v vide sharovogo sloya. Teplovye processy v tehnike, 2020, vol.12, no. 1, pp. 18–24. In Russ.
18. Attetkov A.V., Volkov I.K., Gaidaenko K.A. Avtomodel’nye protsessy teploperenosa v prozrachnom dlya izlucheniya tverdom tele s pogloshchayushchim vklyucheniem v vide sharovogo sloya. Teplovye processy v tehnike, 2020, vol. 12, no. 5, pp. 219–224. In Russ.
19. Koshlyakov N.V., Gliner E.B. Smirnov M.M. Uravneniya v chastnykh proizvodnykh matematicheskoi fiziki (Partial differential equations of mathematical physics). Moscow, Vysshaya shkola, 1970, 480 p. In Russ.
20. Lykov A.V. Teplomassoobmen: Spravochnik (Heat and mass transfer: Reference). Moscow, Jenergija, 1978, 708 p. In Russ.
21. Pudovkin M.A., Volkov I.K. Kraevye zadachi matematicheskoi teorii teploprovodnosti v prilozhenii k raschetam temperaturnyx polei v neftyanyx plastax pri zavodnenii (Boundary value problems of the mathematical theory of thermal conductivity in application to calculations of temperature fields in oil reservoirs during flooding). Kazan’, Kazanskii universitet, 1978, 188 p. In Russ.
22. El’sgol’ts L.E. Differentsial’nye uravneniya i variatsionnoe ischislenie (Differential equations and calculus of variations). Moscow, Nauka, 1969, 424 p. In Russ.
23. Ditkin V.A., Prudnikov A.P. Spravochnik po operatsionnomu ischisleniyu (Handbook of Operational Calculus). Moscow, Vysshaya shkola, 1965, 468 p. In Russ.