Three layers thermal protection system modeling by Riemann–Liouville fractional derivative
Abstract This paper introduces a mathematical model of a thermal protection system incorporating the Riemann–Liouville fractional derivative. The system is considered as a three-layer structure, where the temperature distribution in the first two layers follows the classical heat conduction equation...
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Nature Portfolio
2025-07-01
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| Series: | Scientific Reports |
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| Online Access: | https://doi.org/10.1038/s41598-025-10302-9 |
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| author | Rafał Brociek Edyta Hetmaniok Damian Słota |
| author_facet | Rafał Brociek Edyta Hetmaniok Damian Słota |
| author_sort | Rafał Brociek |
| collection | DOAJ |
| description | Abstract This paper introduces a mathematical model of a thermal protection system incorporating the Riemann–Liouville fractional derivative. The system is considered as a three-layer structure, where the temperature distribution in the first two layers follows the classical heat conduction equation. In contrast, the third layer, characterized by its porous nature, is modeled using a fractional-order heat conduction equation. The thermal contact resistances between the layers are taken into account. The external surface is subjected to a boundary condition of the second kind, incorporating an aerothermal heat flux derived from NASA Langley Research Center data, while the internal surface is governed by a Dirichlet boundary condition. Additionally, the temperature-dependent material properties are considered. A significant contribution of this study is the development of a numerical scheme for a three-layer thermal protection system model, in which one layer is porous and described using the Riemann–Liouville fractional derivative. The proposed approach allows for accurate simulation of heat conduction in systems with complex material structures. The influence of the fractional derivative order $$\beta$$ on the temperature profile was investigated, showing that variations in $$\beta$$ significantly affect the thermal response of the system. Furthermore, a mesh refinement study was conducted to assess the impact of spatial discretization on the numerical results. These findings establish the model as a valuable tool for computer simulations and provide a basis for further development and optimization of mathematical and computational approaches in the analysis of thermal protection systems. |
| format | Article |
| id | doaj-art-a2c463928ead41cc851440d2ad505321 |
| institution | Kabale University |
| issn | 2045-2322 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | Nature Portfolio |
| record_format | Article |
| series | Scientific Reports |
| spelling | doaj-art-a2c463928ead41cc851440d2ad5053212025-08-20T04:02:46ZengNature PortfolioScientific Reports2045-23222025-07-0115111510.1038/s41598-025-10302-9Three layers thermal protection system modeling by Riemann–Liouville fractional derivativeRafał Brociek0Edyta Hetmaniok1Damian Słota2Department of Artificial Intelligence Modelling, Silesian University of TechnologyDepartment of Mathematical Methods in Technology and Computer Science, Silesian University of TechnologyDepartment of Mathematical Methods in Technology and Computer Science, Silesian University of TechnologyAbstract This paper introduces a mathematical model of a thermal protection system incorporating the Riemann–Liouville fractional derivative. The system is considered as a three-layer structure, where the temperature distribution in the first two layers follows the classical heat conduction equation. In contrast, the third layer, characterized by its porous nature, is modeled using a fractional-order heat conduction equation. The thermal contact resistances between the layers are taken into account. The external surface is subjected to a boundary condition of the second kind, incorporating an aerothermal heat flux derived from NASA Langley Research Center data, while the internal surface is governed by a Dirichlet boundary condition. Additionally, the temperature-dependent material properties are considered. A significant contribution of this study is the development of a numerical scheme for a three-layer thermal protection system model, in which one layer is porous and described using the Riemann–Liouville fractional derivative. The proposed approach allows for accurate simulation of heat conduction in systems with complex material structures. The influence of the fractional derivative order $$\beta$$ on the temperature profile was investigated, showing that variations in $$\beta$$ significantly affect the thermal response of the system. Furthermore, a mesh refinement study was conducted to assess the impact of spatial discretization on the numerical results. These findings establish the model as a valuable tool for computer simulations and provide a basis for further development and optimization of mathematical and computational approaches in the analysis of thermal protection systems.https://doi.org/10.1038/s41598-025-10302-9Thermal protection systemRiemann–Liouville fractional derivativeAerothermal heat flux |
| spellingShingle | Rafał Brociek Edyta Hetmaniok Damian Słota Three layers thermal protection system modeling by Riemann–Liouville fractional derivative Scientific Reports Thermal protection system Riemann–Liouville fractional derivative Aerothermal heat flux |
| title | Three layers thermal protection system modeling by Riemann–Liouville fractional derivative |
| title_full | Three layers thermal protection system modeling by Riemann–Liouville fractional derivative |
| title_fullStr | Three layers thermal protection system modeling by Riemann–Liouville fractional derivative |
| title_full_unstemmed | Three layers thermal protection system modeling by Riemann–Liouville fractional derivative |
| title_short | Three layers thermal protection system modeling by Riemann–Liouville fractional derivative |
| title_sort | three layers thermal protection system modeling by riemann liouville fractional derivative |
| topic | Thermal protection system Riemann–Liouville fractional derivative Aerothermal heat flux |
| url | https://doi.org/10.1038/s41598-025-10302-9 |
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