The authors propose a simultaneous solution to one-parameter heat-conduction problems for multilayer structures of basic (simple) geometric shapes with the direct (classical) method of investigation of the process of heat transfer. The presence of internal heat sources in the layers of such a structure and of ideal thermal contact between the layers with boundary conditions of the third kind on their bounding surfaces is assumed. In this connection, the authors solve the heat-conduction equation with a parameter having the meaning of the shape factor of a body. The reduction method, the concept of quasi derivatives, the present-day theory of systems of linear differential equations, the method of separation of variables, and a modified Fourier method of eigenfunctions are used as a basis for the solution scheme. A model example of numerical calculation of the temperature field in five-layer structures (rectangular wall, a hollow cylinder, and a hollow sphere) is given.
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References
R. M. Tatsii and O. Yu. Pazen, General boundary-value problems for the heat-conduction equation with piecewisecontinuous coefficients, J. Eng. Phys. Thermophys., 89, No. 2, 357–368 (2016).
R. M. Tatsii and O. Yu. Pazen, Direct (classical) method of calculation of the temperature field in a hollow multilayer cylinder, J. Eng. Phys. Thermophys., 91, No. 6, 1373–1384 (2018).
R. M. Tatsii, M. F. Stasyuk, and O. Yu. Pazen, Direct method of calculation of a temperature field in a multilayer hollow spherical structure, Vestn. Kokshetausk. Tekh. Univ. KChS MVD Respubliki Kazakhstan, No. 1 (29), 9–20 (2018).
R. Kushnir and Yu. Protsyuk, Temperature fields in canonically shaped layered bodies with a linear temperature dependence of thermal conductivities, Mashinovedenie, No. 1, 13–18 (2009).
V. N. Eliseev and T. V. Borovkova, Generalized analytical method of calculation of a stationary temperature field in bodies of simple geometric shape, Vestn. MGTU im. N. É. Baumana, Ser. Mashinostroenie, No. 1, 46–57 (2014).
V. N. Eliseev, V. A. Tovstonog, and T. V. Borovkova, Algorithm of solution of a generalized nonstationary-heatconduction problem in bodies of simple geometric shape, Vestn. MGTU im. N. É. Baumana, Ser. Mashinostroenie, No. 1, 112–128 (1017).
R. M. Tatsii, M. F. Stasyuk, and O. Yu. Pazen, Investigation of the processes of mass transfer in multilayer structures of basic geometric shapes with account of internal heat sources, Proc. Int. Sci.-Pract. Conf. "Information Technologies and Computer Simulation," 14–19 May 2018, Ivano-Frankovsk (2018), pp. 379–385.
V. V. Shevelev, Stochastic model of heat conduction with heat sources or sinks, J. Eng. Phys. Thermophys., 92, No. 3, 614–624 (2019).
A. V. Luikov, Heat Conduction Theory [in Russian], Vysshaya Shkola, Moscow (1967).
R. M. Tatsii and O. Yu. Pazen, Direct method of calculation of a nonstationary temperature field under fire conditions, Pozh. Bezopasnost’, No. 26, 135–141 (2015).
G. N. Kuvyrkin, A. V. Zhuravskii, and I. Yu. Savel’eva, Mathematical modeling of chemical vapor deposition of material on a curvilinear surface, J. Eng. Phys. Thermophys., 89, No. 6, 1374–1379 (2016).
R. M. Tatsii, M. F. Stasyuk, V. V. Mazurenko, and O. O. Vlasii, Generalized Quasi-Differential Equations [in Russian], Izd. L′vovsk. Gos. Univ. Bezopasnosti Zhiznedeytel′nosti, L′vov (2017).
R. M. Tatsii, T. I. Ushak, and O. Yu. Pazen, General third boundary-value problem for the heat-conduction equation with piecewise-continuous coefficients and internal heat sources, Pozh. Bezopasnost′, No. 27, 120–126 (2015).
V. Ya. Arsenin, Methods of Mathematical Physics [in Russian], Nauka, Moscow (1974).
A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian],
R. M. Tatsii, M. F. Stasyuk, and O. Yu. Pazen, Concept of quasi derivatives in mathematical-modeling problems, Proc. IV All-Ukr. Sci.-Pract. Conf. "Scientific Activity as a Way of Forming Professional Competencies of a Future Specialist," 1–2 December 2016, Sumy (2016), pp. 144–147.
Actions on structures, Part 1−2. General actions — Actions on structures exposed to fire. Eurocode 1 . The European Union per Regulation 305/2011, Directive 98/34/EC, Directive 2004/18/EC.
R. M. Tatsii, M. F. Stasyuk, and O. Yu. Pazen, Application of differential equations with pulse action to solution of boundary heat-conduction problems, Abstracts of Papers VI B. V. Vasilishin All-Ukr. Conf. "Nonlinear Problems of Analysis," 26–28 September 2018, Ivano-Frankovsk (2018), pp. 58–60.
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Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 94, No. 2, pp. 313–325, March–April, 2021.
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Tatsii, R.M., Stasyuk, M.F. & Pazen, O.Y. Direct Method of Calculating Nonstationary Temperature Fields in Bodies of Basic Geometric Shapes. J Eng Phys Thermophy 94, 298–310 (2021). https://doi.org/10.1007/s10891-021-02302-z
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DOI: https://doi.org/10.1007/s10891-021-02302-z