Theoretical Approaches to the Heating of an Extensive Homogeneous Plate with Convective Cooling

Paweł Jabłoński (), Tomasz Kasprzak, Sławomir Gryś () and Waldemar Minkina
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Paweł Jabłoński: Faculty of Electrical Engineering, Czestochowa University of Technology, Al. Armii Krajowej 17, PL-42-200 Czestochowa, Poland
Tomasz Kasprzak: Faculty of Electrical Engineering, Czestochowa University of Technology, Al. Armii Krajowej 17, PL-42-200 Czestochowa, Poland
Sławomir Gryś: Faculty of Electrical Engineering, Czestochowa University of Technology, Al. Armii Krajowej 17, PL-42-200 Czestochowa, Poland
Waldemar Minkina: Faculty of Science & Technology, Jan Dlugosz University in Czestochowa, Al. Armii Krajowej 13/15, PL-42-200 Częstochowa, Poland

Energies, 2025, vol. 18, issue 17, 1-28

Abstract: The article presents a mathematical description of the thermal phenomena occurring both inside and on the surfaces of a homogeneous plate subjected to an external heat flux on one side. Analytical formulae for thermal excitation, with a given duration and constant power, are derived, enabling the determination of temperature increases on both the heated and unheated surfaces of the plate under specific heat transfer conditions to the surroundings. Convective heat transfer, with individual heat transfer coefficients on both sides of the slab, is considered; however, radiative heat loss can also be included. The solution of the problem obtained using two methods is presented: the method of separation of variables (MSV) and the Laplace transform (LT). The advantages and disadvantages of both analytical formulae, as well as the impact of various factors on the accuracy of the solution, are discussed. Among others, the MSV solution works well for a sufficiently long time, whereas the LT solution is better for a sufficiently short time. The theoretical considerations are illustrated with diagrams for several configurations, each representing various heat transfer conditions on both sides of the plate. The presented solution can serve as a starting point for further analysis of more complex geometries or multilayered structures, e.g., in non-destructive testing using active thermography. The developed theoretical model is verified for a determination of the thermal diffusivity of a reference material. The model can be useful for analyzing the method’s sensitivity to various factors occurring during the measurement process, or the method can be adapted to a pulse of known duration and constant power, which is much easier to implement technically than a very short impulse (Dirac) with high energy.

Keywords: heat transfer; non-destructive evaluation (NDE); thermal diffusivity; one-dimensional analytical model; Laplace transform (search for similar items in EconPapers)
JEL-codes: Q Q0 Q4 Q40 Q41 Q42 Q43 Q47 Q48 Q49 (search for similar items in EconPapers)
Date: 2025
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