The methodology for constructing a temperature profile in a flat insulation layer with a nonlinear dependence of thermal conductivity on temperature

About the authors:

Aleksey N. Chadaev, Postgraduate Student, Department of Food Production Equipment, Kazan National Research Technological University, Kazan, Russia

ac312@mail.ru, https://orcid.org/0009-0004-3180-5933

Andrey V. Dmitriev, Dr. Sci. (Tech.), Head of the Department Theoretical Foundations of Heat Engineering, Kazan State Power Engineering University; ieremiada@gmail.com; ORCID: 0000-0001-8979-4457

Rustem Ya. Bikkulov, Cand. Sci. (Tech.), Lecturer, Department of Engineering Graphics, Kazan State Power Engineering University, Kazan, Russia

bikkulov-ry@mail.ru, https://orcid.org/0000-0003-4783-147X

Oksana S. Dmitrieva, Dr. Sci. (Tech.), Associate Professor, Associate Professor of the Department of Food Production Equipment, Kazan National Research Technological University, Kazan, Russia

ja_deva@mail.ru, https://orcid.org/0000-0001-6221-0167

Abstract:

Accurate modeling of temperature fields in materials with nonlinear thermal conductivity — described by a second-degree polynomial — is essential for developing reliable thermal protection systems for high-temperature equipment. Unlike existing approximate methods, the scientific novelty of this work lies in the development of an analytical approach for calculating the steady-state one-dimensional temperature profile with a known quadratic dependence of thermal conductivity on temperature through solving a cubic equation using the Cardano’s method with a multi-stage procedure for verifying the roots for physical correctness. The purpose of the study is to create an algorithm for constructing a temperature profile with a known dependence of thermal conductivity on temperature to verify the inverse problems of determining the dependence of thermal conductivity on temperature using a known temperature profile. The method is based on energy balance and provides high accuracy in calculating the temperature profile within the insulation layer. The results confirm the effectiveness of the approach for porous corundum materials. The proposed method serves as a reliable tool for calibrating inverse algorithms in the design of thermal protection systems.

References:

Boiko, E. E., Byk, F. L., Ilyushin, P. V., & Myshkina L. S. (2024). The role of electric and thermal energy storage in energy supply systems. Electrical Stations, 5(1114), 50–57. https://doi.org/10.34831/EP.2024.1114.5.006 [In Russian]

Bondarenko, S. L., Litovskiy, E. Ya., Ivanov, A. B., Borovetskaya, A. I., & Pakhomov, E. P. (1986). Heat and thermal conductivity of corundum refractories based on hollow spheres in the temperature range 500–2000 K in various gaseous media. High Temperature Thermophysics, 24(3), 618–619. [In Russian]

Vatulyan, A. O., & Nesterov, S. A. (2019). On the peculiarities of solving the coefficient inverse problem of heat conduction for a two-part layer. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 19(4), 409–423. https://doi.org/10.18500/1816-9791-2019-19-4-409-423 [In Russian]

Gribov, A. F., Zhidkov, E. N., & Krasnov, I. K. (2019). On the numerical solution of the inverse problem of heat conduction with radiation. Mathematical Modeling and Computational Methods, 1(21), 43–53. https://doi.org/10.18698/2309-3684-2019-1-4353 [In Russian]

Zhukovskaya, A. E., & Khlebnikova, I. Yu. (2006). Comparison of Russian and foreign standards for methods of determining the heat conductivity of refractories. Refractories and Industrial Ceramics, 2, 38–43. [In Russian]

Zinurov, V. E., Chadaev, A. N., Dmitriev, A. V., Dmitrieva, O. S., Muginov, A. M., & Ziangirov, A. F. (2025). Theoretical and applied aspects of designing a high-temperature thermal energy storage system based on graphite material. Bulletin of South Ural State University. Series “Power Engineering”, 25(4), 71–81. https://doi.org/10.14529/power250409 [In Russian]

Kuznetsova, A. E., Skvortsova, M. P., & Stefanyuk, E. V. (2014). The inverse problem solution of heat conduction for the initial conditions identification of the boundary value problem. Vestnik of Samara State Technical University. Technical Sciences Series, 3(43), 155–162. [In Russian]

Tomochakov, M. M., & Berezovskaya, I. E. (2022). Analysis of the current state of the solution of inverse thermal conductivity problems and their application. Universum: Technical Sciences, 5-3(98), 39–44. https://doi.org/10.32743/UniTech.2022.98.5.13591 [In Russian]

Chirkin, V. S. (1968). Thermophysical Properties of Nuclear Engineering Materials. Atomizdat. [In Russian]

Ashby, M. F. (2016). Materials Selection in Mechanical Design. Butterworth-Heinemann.

Bigeard, A., Buchovecky, E., Mesnager, C., Bolore, D., Bouvard, D., & Jauffres, D. (2026). Modelling of sintering consolidation and thermal damage in a refractory ceramic. Journal of the European Ceramic Society, 46(3), 117893. https://doi.org/10.1016/j.jeurceramsoc.2025.117893

Kabanikhin, S. I., Hasanov, A., & Penenko, A. V. (2008). A gradient descent method for solving an inverse coefficient heat conduction problem. Numerical Analysis and Applications, 1, 34–45. https://doi.org/10.1134/S1995423908010047

Khare, S., Dell’Amico, M., Knight, C., & McGarry, S. (2012). Selection of materials for high temperature latent heat energy storage. Solar Energy Materials and Solar Cells, 107, 20–27. https://doi.org/10.1016/j.solmat.2012.07.020

Khare, S., Dell’Amico, M., Knight, C., & McGarry, S. (2013). Selection of materials for high temperature sensible energy storage. Solar Energy Materials and Solar Cells, 115, 114–122. https://doi.org/10.1016/j.solmat.2013.03.009

Li, Y., Zhou, F., Zhang, M., Deng, L., Chen, L., Mweemba, S., Yang, S., Shen, G., & Zhang, X. (2025). Enhanced thermal performance of corundum-mullite ceramics via in-situ formation of Al-12Si@Al2O3 microcapsules. Ceramics International, 51(24), Part A, 41512–41521. https://doi.org/10.1016/j.ceramint.2025.06.376

Manzolaro, M., Corradetti, S., Andrighetto, A., & Ferrari, L. (2013). A steady-state high-temperature method for measuring thermal conductivity of refractory materials. Review of Scientific Instruments, 84(5), 054902. https://doi.org/10.1063/1.4804258

Mohan, G., Venkataraman, M. B., & Coventry, J. (2019). Sensible energy storage options for concentrating solar power plants operating above 600 °C. Renewable and Sustainable Energy Reviews, 107, 319–337. https://doi.org/10.1016/j.rser.2019.01.062

Nazir, H., Batool, M., Osorio, F. J. B., Isaza-Ruiz, M., Xu, X., Vignarooban, K., Phelan, P., Inamuddin, & Kannan, A. M. (2019). Recent developments in phase change materials for energy storage applications: A review. International Journal of Heat and Mass Transfer, 129, 491–523. https://doi.org/10.1016/j.ijheatmasstransfer.2018.09.126

Šimonová, P. Š., Bezdička, P., & Pabst, W. (2026). Determination of the content, composition and density of the glass phase in silica refractories and its changes due to thermal cycling. Journal of the European Ceramic Society, 46(9), 118235. https://doi.org/10.1016/j.jeurceramsoc.2026.118235

Woodbury, K. A., Najafi, H., de Monte, F., & Beck, J. V. (2023). Inverse heat conduction problems. In K. A. Woodbury, H. Najafi, F. Monte, & J. V. Beck (Eds.), Inverse Heat Conduction. Book Chapter 1 (pp. 1–23). https://doi.org/10.1002/9781119840220.ch1