Analytical solution of the dirichlet generalized boundary problem of heat-exchange in the finite cylinder with homogeneous layers

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Authors:

M.H. Berdnyk, Candidate of Physics and Mathematics, Associate Professor, National Mining University, Senior Lecturer of the Computer Software Systems Department, Dnipro, Ukraine, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.">This email address is being protected from spambots. You need JavaScript enabled to view it.

Abstract:

Purpose. To develop a new generalized 3D mathematic model for calculating temperature fields in the solid finite cylinder with homogeneous layers in the form of the mathematical physics boundary problem for hyperbolic equations by the Dirichlet conditions (temperature on the cylinder surface is a continuous function of the coordinate axis), and to solve the obtained boundary problem.

Methodology. Use of the known Laplace and Fourier integral transformations and application of the new integral transformation to the space with homogeneous layers.

Findings. A nonstationary temperature field in the rotating double-layer solid finite cylinder in the cylindrical coordinate system with taking into account finite velocity of the heat conduction was defined. Heat-transfer properties of the cylinder in either layer are constant at an ideal heat contact between the layers while no internal sources of the heat are available. At the initial moment of time, the cylinder temperature is constant, and temperature on the outside surface of the cylinder is known.

Originality. It is the first a mathematical 3D model for calculating temperature fields in the rotating double-layer solid finite cylinder has been created in the form of the physicomathematical boundary problem for the heat conduction hyperbolic equations by the Dirichlet conditions and with taking into account finite velocity of the heat conduction. A new integral transformation was created for the space with homogeneous layers, with the help of which it became possible to present a temperature field in the finite cylinder with homogeneous layers in the form of convergence orthogonal series by Bessel and Fourier functions.

Practical value. The obtained analytical solution of the generalized boundary problem of heat exchange in the rotating double-layer cylinder, which takes into account the known time period of the heat-conduction relaxation, can be used for detecting temperature fields, which occur in different technical systems (forming rolls, satellites, turbines, etc.).

References

1. Berdnyk, M. H., 2014. Mathematical modelling for generalized three-dimensional energy balance equation for a rotating solid cylinder. In: O. M. Kiseliova, ed., 2014. Issues of applied mathematics and mathematical modelling. Dnipropetrovsk: Lira, рр. 26–35.

2. Yachmeniov, V. O. and Nikolenko, V. V., 2016. Calculation of temperature fields for the compound semi-infinite solid taking into consideration generalized Fourier law. Bulletin of the National Technical University “KhPI”, Series: Power and Heat Engineering Processes and Equipment, 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. Markovych, B. M., 2010. Equations of mathematical physics. Lviv: Vydavnytstvo Lvivskoii politehniky.

5. Lopushanska, G.P., Lopushanskyi, A.О. and Miaus, O.М., 2014. Fourier’s, Laplace’s transforms: synthesis and application. Lviv: lnu. Ivan Franko.

 

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ISSN (print) 2071-2227,
ISSN (online) 2223-2362.
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