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Review

Applications of Methods of Solving Inverse Heat Conduction Problems for Energy-Intensive Industrial Processes and Energy Conversion—Current State of the Art and Recent Challenges

by
Magda Joachimiak
* and
Damian Joachimiak
Faculty of Environmental Engineering and Energy, Institute of Thermal Energy, Poznan University of Technology, 60-965 Poznań, Poland
*
Author to whom correspondence should be addressed.
Energies 2026, 19(5), 1291; https://doi.org/10.3390/en19051291
Submission received: 26 January 2026 / Revised: 23 February 2026 / Accepted: 25 February 2026 / Published: 4 March 2026

Abstract

This paper presents methods and applications of inverse heat conduction problems (IHCPs) that are ill-posed in the Hadamard sense. The IHCP solution allows for the determination of boundary conditions in the form of heat flux or temperature in places where measurement is impossible or difficult to perform. The applications of IHCP solutions to energy-intensive industrial processes, such as heat treatment and thermochemical treatment, are described. Examples are given of determining boundary conditions on the inner surface of the wall of a power boiler and piston machine, as well as on the surface of a gas turbine blade. It is noted that the application of IHCP solutions to the above-mentioned issues often requires simplification of the computational model, in particular, the method of stabilising the inverse problem (IP). For this purpose, quasi-regularisation of IP and machine learning are currently used. Methods with stabilising properties and neural networks were identified as a challenging and interesting direction for the development of IHCP solutions.
Keywords: heat transfer; inverse heat conduction problem (IHCP); quasi-regularisation; regularisation; neural networks; boilers; gas turbines; heat treatment; thermochemical treatment heat transfer; inverse heat conduction problem (IHCP); quasi-regularisation; regularisation; neural networks; boilers; gas turbines; heat treatment; thermochemical treatment

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MDPI and ACS Style

Joachimiak, M.; Joachimiak, D. Applications of Methods of Solving Inverse Heat Conduction Problems for Energy-Intensive Industrial Processes and Energy Conversion—Current State of the Art and Recent Challenges. Energies 2026, 19, 1291. https://doi.org/10.3390/en19051291

AMA Style

Joachimiak M, Joachimiak D. Applications of Methods of Solving Inverse Heat Conduction Problems for Energy-Intensive Industrial Processes and Energy Conversion—Current State of the Art and Recent Challenges. Energies. 2026; 19(5):1291. https://doi.org/10.3390/en19051291

Chicago/Turabian Style

Joachimiak, Magda, and Damian Joachimiak. 2026. "Applications of Methods of Solving Inverse Heat Conduction Problems for Energy-Intensive Industrial Processes and Energy Conversion—Current State of the Art and Recent Challenges" Energies 19, no. 5: 1291. https://doi.org/10.3390/en19051291

APA Style

Joachimiak, M., & Joachimiak, D. (2026). Applications of Methods of Solving Inverse Heat Conduction Problems for Energy-Intensive Industrial Processes and Energy Conversion—Current State of the Art and Recent Challenges. Energies, 19(5), 1291. https://doi.org/10.3390/en19051291

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