Mathematical modelling of heat transfer processes in a two-phase material with inclusions in the form of a ball layer absorbing penetrating radiation. 1. Hierarchy of simplified analogs of the heat transfer basic model


Аuthors

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

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

«Micro-focal» model of the heat transfer process in a two-phase material with spherical inclusions absorbing penetrating radiation occupies an important place in the problem of laser initiation of the energy materials explosive decomposition. Difficulties occurring while obtaining analytical solution to the corresponding mixed problem for the system of two equations in second order partial derivatives of the parabolic type even in the simplest situation of the presence of the single absorbing inclusion of spherical shape in the material transparent to radiation are well known. A possible way to overcome them is associated with adoption of various kinds of assumptions, leading to the initial (basic) mathematical model replacement with its simplified analogs with further definition of the range of possible application of each of them.

Absorbing inclusions of other geometric forms represent theoretical and significant practical interest while heat transfer processes studying in the analyzed two-phase material. Earlier, the authors formulated and solved the problem of determining the temperature field of the material transparent for radiation with absorbing inclusions in the form of the ball layer. The presented results demonstrate that while parametric analysis performing of the temperature field under study significant technical difficulties, stipulated by the complex nature of dependence of the kernel and specter of eigenvalues of the developed integral transformation on the parameters of the initial (basic) mathematical model my occur.

A hierarchy of simplified analogs of the basic mathematical model of the heat transfer process in a two-phase material with radiation-absorbing inclusions in the form of a ball layer has been developed, including a «refined model of concentrated capacitance», a «concentrated capacitance» model and a «truncated model of concentrated capacitance». Each of the mathematical models of the hierarchy assumes thermal isolation of the outer boundary of the ball layer and is a mixed problem for an equation in partial second-order derivatives of the parabolic type with a specific boundary condition that actually accounts for the presence of absorbing inclusions in the two-phase material.

Sufficient conditions, at which satisfaction the simplified analogues of the basic mathematical model allowed identifying the temperature field of the two-phase material with the specified accuracy were determined.

Keywords:

two-phase material, laser radiation, absorbing inclusions in the form of a ball layer, temperature field

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