Mathematical model of and method for solving the Neumann generalized heat-exchange problem for a cylinder with homogeneous layers
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- Last Updated on Wednesday, 06 September 2017 17:06
- Published on Wednesday, 06 September 2017 17:06
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Authors:
M.H.Berdnyk, Cand. Sc. (Phys.-Math.), Assoc. Prof., State Higher Educational Institution “National Mining University”, Dnipro, Ukraine, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
Abstract:
Purpose. To develop a new generalized mathematical model of temperature distribution in the homogeneous-layer cylinder in the form of the physicomathematical boundary-value problem for the heat conduction equation, and to solve the obtained boundary-value problem.
Methodology. To use the known Laplace and Fourier finite integral transformations, and to apply the obtained new integral transformation to the homogeneous-layer space.
Findings. A non-stationary temperature field in the solid circular cylinder with outside radius R in the polar coordinate system (r, ), layers of which were homogeneous to the direction of the polar radius r, and which rotated with a constant angular velocity around the axis OZ, was defined with taking into account finite velocity of the heat conduction. The heat-transfer properties of each of the layers do not depend on the temperature in case of ideal contact between the layers and if no internal sources of the heat are available. At the initial moment of time, the cylinder temperature is constant G_{0}, and heat flow G()on the outside surface of the cylinder is known.
Originality. The mathematical model of the distribution of the temperature field in a piecewise homogeneous cylinder in the form of the Neumann boundary-value problem for the hyperbolic heat conduction equation was developed for the first time. 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 solid homogeneous-layer circular cylinder in the form of convergence orthogonal series by Bessel and Fourier functions.
Practical value. The obtained solution of the generalized boundary-value problem of heat exchange in the rotating cylinder with taking into account finite velocity of the heat conduction can be used for modeling temperature fields occurred in different technical systems (satellites, forming rolls, turbines, etc.).
References
1. Kushnir, R.M. and Popovych, V.S., 2013. On defining steady thermostressed state of multilayer structures at high-temperature heating. Bulletin of Taras Shevchenko National University of Kyiv. Ser.: Fiz.-mat. Nauky, 3, pp. 42–47.
2. Rakocha, I.I. and Popovych, V.S., 2014. Mathematical modeling and definition of thermostressed state of a two-component thermosensitive cylinder under complex heating. Prykladni problemy mehaniky i matematyky, 12, pp. 69‒77.
3. Kalyniak, B.M., 2015. Determining the temperature field and thermomechanical characteristics of a material which ensure zero radial stress in a long hollow cylinder inhomogeneous in the radial direction. Reports of the National Academy of Sciences of Ukraine, 6, pp. 46‒55.
4. Berdnyk, M.G., 2014. Mathematical modeling of three-dimensional generalized problem of heat transfer solid cylinder which rotates. In: Questions of applied mathematics and mathematical modeling, pp. 26‒35.
5. Lopushanska, G.P., Lopushanskyi, А.О. and Мiaus, О.М., 2014. Fourier, Laplace, synthesis and application. Lviv: LNU. Ivan Franko.
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