Basic model of heat transfer process in a solid with spherical inclusion absorbing penetrating radiation


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

Mathematical model of the heat transfer process in a transparent to radiation solid with absorbing spherical inclusion fills am important place in theoretical studies on the problem of laser initiation of explosive decomposition in energy materials. Difficulties, encountered while searching for analytical solution of the non-stationary heat transfer problem represented by this model, are well known. A possible way to overcome them is associated with adoption of various kinds of assumptions, leading to the original (basic) mathematical model replacement with its simplified analogues, followed by determining the range of possible application of each. The simplified analogue of the basic model, based on the standard assumption on the thermal contact ideality in the system being radiated and the hypothesis on the “uttermost large thermal conductivity of the absorbing inclusion”, which practically means the possibility of the thermally thin body model realization, represents theoretical and meaningful practical interest. This model represents a mixed problem in second order partial derivatives of parabolic type with specific boundary condition, in fact accounting for the presence of the absorbing inclusion in the system. The presented work analyzes the basic mathematical model of heat transfer process in the system under study. The model being realized represents a mixed problem for the system of two equations in second order partial derivatives of the parabolic type with non-stationary spatially uniform internal source of heat. The article proposes analytical method, based on the idea of the problem solution representation in the images space of the integral Laplace transform in the form of the two functions, one of which characterizes the flow impact mode being analyzed, while the second is presentable as a sum of the uniformly convergent functional series. Solution of the non-stationary thermal conductivity problem, being considered, was obtained in a closed form using the well-known theorems of operational calculus.

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