Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic spatial heat-active media subjected to internal local heat load, have been developed. In the case of a nonlinear boundary-value problem, the Kirchhoff transformation is applied, using which the original nonlinear heat conduction equation and nonlinear boundary conditions are linearized, and as a result, a linearized second-order differential equation with partial derivatives and a discontinuous right-hand side and partially linearized boundary conditions is obtained. For the final linearization of the partially linearized boundary conditions, the approximation of the temperature by the radial spatial coordinate on the boundary surface of the thermosensitive medium was performed by a piecewise constant function, as a result of which the boundary value problem was obtained completely linearized. To solve the linear boundary value problem, as well as the obtained linearized boundary value problem with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, an analytical relationship was obtained for determining the temperature distribution in this medium. Numerical analysis of temperature behavior as a function of spatial coordinates for given values of geometric and thermophysical parameters was performed. The influence of the power of internal heat sources and environmental materials on the temperature distribution was studied. To determine the numerical values of the temperature in the given structure, as well as to analyze the heat exchange processes in the middle of these structures, caused by the internal heat load, software tools were developed, using which a geometric image of the temperature distribution depending on the spatial coordinates was made. The developed linear and nonlinear mathematical models for determining the temperature field in spatial heat-active environments with internal heating testify to their adequacy to a real physical process. They make it possible to analyze such environments for their thermal stability. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and their elements, but also the entire structure.
1. Haopeng, S., Kunkun, X., & Cunfa, G. (2021). Temperature, thermal flux and thermal stress distribution around an elliptic cavity with temperature-dependent material properties. International Journal of Solids and Structures, 216, 136-144.
https://doi.org/10.1016/j.ijsolstr.2021.01.010
2. Zhang, Z., Zhou, D., Fang, H., Zhang, J., & Li, X. (2021). Analysis of layered rectangular plates under thermo-mechanical loads considering temperature-dependent material properties. Applied Mathematical Modelling, 92, 244-260.
https://doi.org/10.1016/j.apm.2020.10.036
3. Gong, J., Xuan, L., Ying, B., & Wang, H. (2019). Thermoelastic analysis of functionally graded porous materials with temperature-dependent properties by a staggered finite volume method. Composite Structures, 224, 111071.
https://doi.org/10.1016/j.compstruct.2019.111071
4. Demirbas, M. D. (2017). Thermal stress analysis of functionally graded plates with temperature-dependent material properties using theory of elasticity. Composites Part B: Engineering, 131, 100-124.
https://doi.org/10.1016/j.compositesb.2017.08.005
5. Ghannad, M., & Yaghoobi, M. P. (2015). A thermoelasticity solution for thick cylinders subjected to thermo-mechanical loads under various boundary conditions. International Journal of Advanced Design &Manufacturing Technology, 8(4), 1-12.
6. Yaghoobi, M. P., & Ghannad, M. (2020). An analytical solution for heat conduction of FGM cylinders with varying thickness subjected to non-uniform heat flux using a first-order temperature theory and perturbation technique. International Communications in Heat and Mass Transfer, 116, 104684.
https://doi.org/10.1016/j.icheatmasstransfer.2020.104684
7. Eker, M., Yarımpabuç, D., & Celebi, K. (2020). Thermal stress analysis of functionally graded solid and hollow thick-walled structures with heat generation. Engineering Computations, 38(1), 371-391.
https://doi.org/10.1108/EC-02-2020-0120
8. Bayat, A., Moosavi, H., & Bayat, Y. (2015). Thermo-mechanical analysis of functionally graded thick spheres with linearly time-dependent temperature. Scientia Iranica, 22(5), 1801-1812.
9. Evstatieva, N., & Evstatiev, B. (2023). Modelling the Temperature Field of Electronic Devices with the Use of Infrared Thermography. 13th International Symposium on Advanced Topics in Electrical Engineering (ATEE), Bucharest, Romania, pp. 1-5.
https://doi.org/10.1109/ATEE58038.2023.10108375
10. Haoran, L., Jiaqi, Y., & Ruzhu, W. (2023). Dynamic compact thermal models for skin temperature prediction of porta-ble electronic devices based on convolution and fitting methods, International Journal of Heat and Mass Trans-fer, 210, 124170.
https://doi.org/10.1016/j.ijheatmasstransfer.2023.124170
11. Vincenzo Bianco, Mattia De Rosa, Kambiz Vafai (2022). Phase-change materials for thermal manage-ment of electronic devices, Applied Thermal Engineering, Volume 214, 118839, ISSN 1359-4311.
https://doi.org/10.1016/j.applthermaleng.2022.118839
12. Mathew J., & Krishnan, S. (2021). A Review On Transient Thermal Management of Electronic Devices. Journal of Electronic Packaging, 144(1), 010801.
https://doi.org/10.1115/1.4050002
13. Havrysh, Vasyl, & Kochan, Volodymyr. (2023). Mathematical Models to Determine Temperature Fields in Heterogeneous Elements of Digital with Thermal Sensitivity Taken into Account. Proceedings of the 12 th IEEE International Conference on Intelligent Data Acguisition and Advanced Computing Systems: Technology and Applications, IDAACS' 2023, 2, pp. 983-991.
https://doi.org/10.1109/IDAACS58523.2023.10348875
14. Havrysh V. I., Kolyasa L. I., Ukhanska O. M., & Loik V. B. (2019). Determination of temperature fielde in thermally sensitive layered medium with inclusions. Naukovyi Visnyk Natsionalnoho Hirnychoho Universetety, 1, 94-100.
https://doi.org/10.29202/nvngu/2019-1/5
15. Havrysh, Vasyl, Koliasa, Liubov, & Vozna, Svitlana. (2021). Temperature field in a layered plate with local heating. International scientific journal "Mathematical modeling", 5(3), 90-94.
16. Zhou, Kun, Ding, Haohao, Steenbergen, Michael, Wang, Wenjian, Guo, Jun, & Liu, Qiyue (2021). Temperatute field and material response as a function of rail grinding parameters. Internation Journal of Heat and Mass Transfer, 175, 121366.
https://doi.org/10.1016/j.ijheatmasstransfer.2021.121366
17. Liu, Xu, Peng, Wei, Gong, Zhiqiang, Zhou, Weien, & Yao, Wen. (2022). Temperature Field Inversion of Heat-Source System via Physics-Informed Neurual Networks. Cornell University.
https://doi.org/10.1016/j.engappai.2022.104902
18. Kong, Qian, Jiang, Genshan, Liu, Yuechao, & Yu, Miao. (2020). Numerical and experimental study on temperature field reconstruction based on acoustic tomography. Applied Thermal Engineering, 170, 114720.
https://doi.org/10.1016/j.applthermaleng.2019.114720