Temperature field of transparent for radiation solid with absorbing spherical inclusion


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 the temperature field determining of a transparent to radiation solid containing absorbing inclusions. A hierarchy of simplified analogues of the basic mathematical model of the heat transfer process in the system under study was developed, including “refined capacitance model”, “concentrated capacity” model and “truncated concentrated capacitance model”. Each of the mathematical models of the hierarchy represents a mixed problem for a parabolic type partial differential equation with a specific boundary condition actually accounting for the presence of a spherical inclusion in the system under study.

Solutions for corresponding problems of non‑stationary heat conductivity under the impact of a radiation flow of constant power density on the subject under study were obtained applying Laplace integral transformation and the standard technique for Mellin integral calculating in an analytical closed form. The “concentrated capacitance model” based on the hypothesis of the extreme high heat conductivity of the absorbing inclusion was analyzed in detail. It was shown, that its realization allowed representing the solution of non‑stationary heat conductivity in the analytical form more convenient from the view point of both its practical application and obtaining applicability conditions of simplified analogs of the basic model.

Sufficient conditions are established at which satisfaction the simplified analogues of the basic model allow identify the temperature field of the analyzed system with a specified accuracy. The article presents theoretical evaluations of the possible error in determining the temperature field of an object under study while applying the simplified analogs of the basic model.

Keywords:

isotropic solid, laser radiation, absorbing spherical inclusion, temperature field, Laplace’s integrated transformation

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