The mathematic model of and method for solving the Neumann generalized heat-exchange problem for empty isotropic rotary body

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Authors:

M. H. Berdnyk, Cand. Sc. (Phys.-Math.), Assoc. Prof., orcid.org/0000-0003-4894-8995, National Mining Uversity, Dnipro,Ukraine, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Abstract:

Purpose. To build a new generalized mathematic 3D model for calculating temperature fields in the empty isotropic rotary body, which rotates with constant angular velocity, with the known equations of generating lines, which takes into account finite velocity of the heat conductivity and is made in the form of boundary problem of mathematical physics for hyperbolic equations of the heat conductivity, as well as to find solutions for the obtained boundary problem.

Methodology. Usage of known integral transforms of Laplace and Fourier and developed new integral transformation for 2D finite space.

Findings. A non-stationary temperature field was found in the empty isotopic body, which rotates with constant angular velocity around the axis OZ, taking into account finite velocity of the heat conductivity under condition that heat-transfer properties of the body are constant, and no internal sources of the heat are available. At the initial moment of time, the body temperature is constant, and values of heat flows on the outside surface of the body are known.

Originality. It is the first generalized 3D mathematic model, which is created for calculating temperature fields in the empty isotropic rotary body, which is restricted by end surfaces and lateral surface of rotation and rotates with constant angular velocity around the axis OZ, with taking into account finite velocity of the heat conductivity in the form of the Neumann problem. In this work, an integral transformation was formulated for the 2D finite space, with the help of which a temperature field in the empty isotropic rotary body was determined in the form of convergence series by the Fourier functions.

Practical value. A non-stationary temperature field was found in the empty isotopic body taking into account finite velocity of the heat conductivity. The obtained solution can be used for modeling temperature fields, which occur in different technical systems (satellites, forming rolls, turbines, and others).

References.

1. Berdnyk, M. G., 2014. Mathematic modeling of a 3-dimentional integrating task of heat transfer of a rotating solid cylinder. Pytannia prykladnoii matematyky i matematychnoho modeliuvannia, рр. 26‒35.

2. Yachmeniov, V. O. and Nikolenko, V. V., 2016. Calculation of temperature fields in a compound semi-infinite body taking into account basic Fourier law. Visnyk NTU HPI. Seriia: Energetychni ta teplotehnichni procesy i ustatkuvannia, 10(1182), рр. 61–65.

3. Povstenko, Y., 2013. Time-fractional heat conduction in an infinite medium with a spherical hole under Robin boundary condition. Fract. Calc. Appl. Anal., 16, рр. 356–369.

4. Lopushanska, G. P., Lopushanskyi, А. О. and Мia­us, О. М., 2014. Fourier, Laplace, synthesis and application. Lviv: LNU. Ivan Franko.

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ISSN (print) 2071-2227,
ISSN (online) 2223-2362.
Journal was registered by Ministry of Justice of Ukraine.
Registration number КВ No.17742-6592PR dated April 27, 2011.

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