Mathematical model and method for solving the generalized neumann problem of heat exchange of a hemispherical body
DOI:
https://doi.org/10.34185/1562-9945-5-136-2021-15Keywords:
комплексний ряд Фур'є, крайова задача Неймана, інтегральне перетворення Лапласа, час релаксаціїAbstract
Currently, the issue of the distribution of temperature fields in the workpieces under the new method of heating, which is carried out by rotating the blanks in the magnetic field of a direct current, which is created in exciters with superconducting windings without knowledge of which it is impossible to implement its technical implementation with high technical and economic indicators, is not sufficiently studied. In this case, a small number of scientific works is devoted to the study of electromagnetic and thermal phenomena within the workpiece during its rotation in a magnetic field of direct current. Therefore, the task of developing mathematical methods for modeling the temperature fields in the workpieces with the induction heating of the metal with the use of innovative technology, the solution of which is dedicated to this work, is very relevant.
The purpose of the article is to construct a new generalized spatial mathematical model for calculating temperature fields in the workpieces in the form of a paraboloid of rotation, rotating with a constant angular velocity, taking into account the final rate of heat propagation in the form of the boundary value problem of mathematical physics, as well as finding solutions to the resulting boundary value problem.
For the first time, a mathematical model for calculating the temperature fields in a paraboloid of rotation has been constructed, taking into account the finite rate of propagation of the rotating heat in the form of a boundary value problem of mathematical physics for hyperbolic heat equations with boundary conditions of Neumann. An integral transformation for a two-dimensional finite space was constructed, with the use of which a temperature field was found in the form of convergent series in Fourier functions.
The solution of the generalized boundary value problem of heat exchange of the rotational paraboloid, which is rotated, taking into account the finiteness of the velocity of heat propagation, can be found in the modulation of temperature fields that arise during induction heating, which is carried out by rotating the blanks in a magnetic field of direct current created in the exciters with superconducting windings.
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