Науковий вісник НГУ

Mathematical modeling and analysis of heat transfer in structures with foreign elements

• 
• 

User Rating:  / 0

PoorBest 

Details

Parent Category: 2026

Category: Content №1 2026

Created on 27 February 2026

Last Updated on 27 February 2026

Published on 30 November -0001

Written by V. Havrysh, L. Kolyasa

Hits: 2221

Authors:

V. Havrysh, orcid.org/0000-0003-3092-2279, Lviv Polytechnic National University, Lviv, Ukraine, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

L. Kolyasa*, orcid.org/0000-0002-9690-8042, Lviv Polytechnic National University, Lviv, Ukraine, , e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

* Corresponding author e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

повний текст / full article

Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu. 2026, (1): 034 - 042

https://doi.org/10.33271/nvngu/2026-1/034

Abstract:

Purpose. The study aims to develop linear and nonlinear mathematical models for determining temperature fields in isotropic three-dimensional media containing foreign thermoactive semi-through elements. This approach enhances the accuracy of temperature regime analysis under thermal loads and contributes to the improvement of design methods for devices whose individual units include foreign thermoactive elements.

Methodology. To obtain analytical-numerical solutions of linear and nonlinear boundary value problems of heat conduction, asymmetric unit functions were used. As a result, the thermal conductivity coefficient for a structure with a foreign semi-through cylindrical element is represented as a unified whole. This ensures the fulfillment of ideal thermal contact conditions at the interfaces of dissimilar materials in the structure, reducing the problem to solving a single heat conduction equation with discontinuous and singular coefficients. For the nonlinear boundary value problem, a linearizing function was introduced, allowing the transformation into a second-order linear partial differential equation with discontinuous and singular coefficients, and a quasi-linear boundary condition. By approximating the temperature as a function of spatial coordinates on the inclusion surface and the layer’s boundary surface using piecewise-constant functions, the nonlinear boundary value problem was fully linearized.

Findings. Linear and nonlinear mathematical models were developed to determine the temperature field and analyze thermal regimes in devices containing a foreign thermoactive semi-through inclusion. A linearizing function was proposed to simplify the nonlinear boundary value problem. Analytical-numerical solutions for both the linear and nonlinear heat conduction problems were obtained, allowing the determination of the temperature distribution as a function of spatial coordinates. Comparative analysis revealed a 7 % difference between results for constant and linearly varying thermal conductivity coefficients, explained by the small values of the thermal conductivity temperature coefficient for the selected construction materials.

Originality. A method for linearizing the nonlinear mathematical model of heat conduction was proposed. Analytical-numerical solutions to the corresponding linear and nonlinear boundary value problems were obtained in closed form. The use of asymmetric unit functions allowed for a correct mathematical description of heat transfer processes in media containing foreign thermoactive semi-through elements.

Practical value. The developed heat transfer mathematical models enable the assessment of media in terms of their thermal resistance, contributing to the improved performance of devices containing foreign thermoactive semi-through elements. This prevents overheating and extends their operational life. The results can be applied to practical problems of heat exchange and thermal insulation in industrial structures, including predicting temperature fields in mining equipment mechanisms, ventilation systems, and compressor stations. Implementing the proposed models improves the efficiency of ore extraction and processing, as well as reduces heat loss in industrial systems.

Keywords: temperature field, material thermal conductivity, structural thermal resistance, thermo-sensitive material, convective heat transfer

References.

1. Shah, N. V., Girfoglio, M., & Rozza, G. (2021). Thermomechanical modelling for industrial applications. arXiv. Retrieved from https://arxiv.org/abs/2108.13366

2. Wang, L., Chen, L., Yuan, F., Zhao, L., Li, Y., & Ma, J. (2023). Thermal Stress Analysis of Blast Furnace Hearth with Typical Erosion Based on Thermal Fluid-Solid Coupling. Processes, 11(2), 531.

3. Zhang, Z., Sun, Y., Cao, X., Xu, J., & Yao, L. (2024). A slice model for thermoelastic analysis of porous functionally graded material sandwich beams with temperature-dependent material properties. Thin-Walled Structures, 198, 111700. https://doi.org/10.1016/j.tws.2024.111700

4. 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

5. Peng, X., Li, X., Gong, Z., Zhao, X., & Yao, W. (2022). A deep learning method based on partition modeling for reconstructing temperature field. International Journal of Thermal Sciences, 182, 107802. https://doi.org/10.1016/j.ijthermalsci.2022.107802

6. Ren, Y., Huo, R., Zhou, D., & Zhang, Z. (2023). Thermo-mechanical buckling analysis of restrained columns under longitudinal steady-state heat conduction. Iranian Journal of Science and Technology – Transactions of Civil Engineering, 47(3), 1411-1423. https://doi.org/10.1007/s40996-022-01020-7

7. Breukelman, H. J., Santofimia, M. J., & Hidalgo, J. (2023). Dataset of a thermal model for the prediction of temperature fields during the creation of austenite/martensite mesostructured materials by localized laser treatments in a Fe-Ni-C alloy. Data in Brief, 48, 109110. https://doi.org/10.1016/j.dib.2023.109110

8. Zhang, W., Wu, M., Du, S., Chen, L., Hu, J., & Lai, X. (2023). Modeling of steel plate temperature field for plate shape control in roller quenching process. IFAC-PapersOnLine, 56(2), 6894-6899. https://doi.org/10.1016/j.ifacol.2023.10.493

9. Khan, Z. H., Khan, W. A., Ibrahim, S. M., Mabood, F., & Huang, Z. (2024). Effects of thermal boundary conditions on Stokes’ second problem. Results in Physics, 60, 107662. https://doi.org/10.1016/j.rinp.2024.107662

10.      Evstatieva, N., & Evstatiev, B. (2023). Modelling the temperature field of electronic devices with the use of infrared thermography. 13 ^th International Symposium on Advanced Topics in Electrical Engineering, (pp. 1-5). IEEE. https://doi.org/10.1109/ATEE58038.2023.10108375

11.      Haoran, L., Jiaqi, Y., & Ruzhu, W. (2023). Dynamic compact thermal models for skin temperature prediction of portable electronic devices based on convolution and fitting methods. International Journal of Heat and Mass Transfer, 210, 124170. https://doi.org/10.1016/j.ijheatmasstransfer.2023.124170

12.      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.

13.      Song, H., Song, K., & Gao, C. (2019). Temperature and thermal stress around an elliptic functional defect in a thermoelectric material. Mechanics of Materials, 130, 58-64. https://doi.org/10.1016/j.mechmat.2019.01.008

14.      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

15.      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

16.      Wang, H., & Qin, Q. (2019). Thermal analysis of a functionally graded coating/substrate system using the approximated transfer approach. Coatings, 9(1), 51. https://doi.org/10.3390/coatings9010051

17.      Zhang, Q., Song, H., & Gao, C.-F. (2023). The 3-D problem of temperature and thermal flux distribution around defects with temperature-dependent material properties. Thermal Science, 27(5 Part B), 3903-3920. https://doi.org/10.2298/TSCI221003028Z

18.      Havrysh, V. I., Kolyasa, L. I., Ukhanska, O. M., & Loik, V. B. (2019). Determination of temperature field in thermally sensitive layered medium with inclusions. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu, (1), 94-100. https://doi.org/10.29202/nvngu/2019-1/5

19.      Havrysh, V. I. (2017). Investigation of temperature fields in a heat-sensitive layer with through inclusion. Materials Science, 52(4), 514-521.

20.      Havrysh, V. I., & Kosach, A. I. (2012). Boundary-value problem of heat conduction for a piecewise homogeneous layer with foreign inclusion. Materials Science, 47(6), 773-782. https://doi.org/10.1007/s11003-012-9455-4

21.      Gavrysh, V., Tushnytskyy, R., Pelekh, Y., Pukach, P., & Baranetskiy, Y. (2017). Mathematical model of thermal conductivity for piecewise homogeneous elements of electronic systems. 14 ^th International Conference The Experience of Designing and Application of CAD Systems in Microelectronics – Proceedings, (pp. 333-336). IEEE. Retrieved from https://ieeexplore.ieee.org/document/7916146

• < Prev
• Next >