A mathematical model of heat exchange analysis between an isotropic two-layer plate heated ba point heat source concentrated on the conjugation surfaces of layers and the environment has been developed. To do this, using the theory of generalized functions, the coefficient of thermal conductivity of the materials of the plate layers is shown as a whole for the wholesystem.Given this, instead of two equations of thermal conductivity for each of the plate layers and the conditions of ideal thermal contact, one equation of thermal conductivity ingeneralized derivatives with singular coefficients is obtained between them. To solve the boundary value problem of thermal conductivity containing this equation and boundary conditions on the boundary surfaces of the plate, the integral Fourier transform was used and as a result an analytical solution of the problem in images was obtained. An inverse integral Fourier transform was applied to this solution, which made it possible to obtain the final analytical solution of the original problem. The obtained analytical solution is presented in the form of an improper convergent integral. According to Simpsons method, numerical values of this integral are obtained with a certain accuracy for given values of layer thickness, spatial coordinates, specific power of a point heat source, thermal conductivity of structural materials of the plate and heat transfer coefficient from the boundary surfaces of the plate. The material of the first layer of the plate is copper, and the second is aluminum. Computational programs have been developed to determine the numerical values of temperature in the given structure, as well as to analyze the heat exchange between the plate and the environment due to different temperature regimes due to heating the plate by a point heat source concentrated on the conjugation surfaces. Using these programs, graphs are shown that show the behavior of curves constructed using numerical values of the temperature distribution depending on the spatial coordinates. The obtained numerical values of temperature indicate the correspondence of the developed mathematical model of heat exchange analysis between a two-layer plate with a point heatsource focused on the conjugation surfaces of the layersand the environment, the real physical process.
[1] Carpinteri, A., & Paggi, M. (2008). Thermoelastic mismatch in nonhomogeneous beams. Journal of Engineering Mathematics, 61(2–4), 371–384. https://doi.org/10.1007/s10665-008-9212-8
[2] Havrysh, V. I., & Fedasjuk, D. V. (2012). Modelling of temperature regimes in piecewise-homogeneous structures. Lviv: Publishing house of Lviv Politechnic National University, 176 p.
[3] Havrysh, V. I., Baranetskiy, Ya. O., & Kolyasa, L. I. (2018). Investigation of temperature modes in thermosensitive non-uniform elements of radioelectronic devices. Radio electronics, computer science, management, 3(46), 7–15. https://doi.org/10.15588/1607-3274-2018-3-1
[4] Havrysh, V. I., Kolyasa, L. I., & Ukhanka, O. M. (2019). Determination of temperature field in thermally sensitive layered medium with inclusions. Scientific Bulletin of the National Chemical University, 1, 94–100. https://doi.org/10.29202/nvngu/2019-1/5
[5] Kikoin, I. K. (Ed.). (1976). Tablitcy fizicheskikh velichin. Moscow: Atomizdat, 1008 p. [In Russian].
[6] Koliano, Iu. M. (1992). Metody teploprovodnosti i termouprugosti neodnorodnogo tela. Kyiv: Scientific thought, 280 p. https://doi.org/10.1192/bjp.161.2.280b
[7] Korn, G., & Korn, T. (1977). Spravochnik po matematike dlia nauchnykh rabotnikov i inzhenerov. Moscow: Science, 720 p. [In Russian].
[8] Nemirovskii, Iu. V., & Iankovskii, A. P. (2007). Asimptoticheskii analiz zadachi nestatcionarnoi teploprovodnosti sloistykh anizotropnykh neodnorodnykh plastin pri granichnykh usloviiakh pervogo i tretego roda na litcevykh poverkhnostiakh. Mathematical methods and physical and mechanical fields, 50(2), 160–175. [In Russian].
[9] Noda, N. (1991). Thermal stresses in materials with temperature-dependent properties. Applied Mechanics Reviews, 44, 383–397. https://doi.org/10.1115/1.3119511
[10] Otao, Y., Tanigawa, O., & Ishimaru, O. (2000). Optimization of material composition of functionality graded plate for thermal stress relaxation using a genetic algorithm. Journal of Thermal Stresses, 23, 257–271. https://doi.org/10.1080/014957300280434
[11] Podstrigach, Ia. S., Lomakin, V. A., & Koliano, Iu. M. (1984). Termouprugost tel neodnorodnoi struktury. Moscow: Science, 368 p. [In Russian].
[12] Tanigawa, Y., & Otao, Y. (2002). Transient thermoelastic analysis of functionally graded plate with temperature-dependent material properties taking into account the thermal radiation. Nihon Kikai Gakkai Nenji Taikai Koen Ronbunshu, 2, 133–134. https://doi.org/10.1299/jsmemecjo.2002.2.0_133
[13] Tanigawa, Y., Akai, T., & Kawamura, R. (1996). Transient heat conduction and thermal stress problems of a nonhomogeneous plate with temperature-dependent material properties. Journal of Thermal Stresses, 19(1), 77–102. https://doi.org/10.1080/01495739608946161
[14] Turii, O. (2008). Neliniina kontaktno-kraiova zadacha termomekhaniky dlia oprominiuvanoi dvosharovoi plastyny, ziednanoi promizhkovym sharom. Fizyko-matematychne modeliuvannia ta informatsiini tekhnolohii, 8, 118–132. [In Ukrainian].
Yangian, Xu., & Daihui, Tu. (2009). Analysis of steady thermal stress in a ZrO2/FGM/Ti-6Al-4V composite ECBF plate with temperature-dependent material properties by NFEM. 2009-WASE Int. Conf. on Informa. Eng., Vol. 2–2, (pp. 433–436). https://doi.org/10.1109/ICICTA.2009.842