Use of Inverse Methods for Simultaneous Identification of Thermal Conductivity and Transfer Coefficients

Abstract:

An inverse problem involves determining unknown physical quantities denoted as u = (u₁, ..., u_np), which cannot be directly measured but need to be evaluated based on accessible measurements, represented as y = M(u), where M is a mathematical model. Solving such problems often requires mathematical techniques like differential equations or optimization methods such as least squares. Inverse problems can be well-posed (stable, unique solutions) or ill-posed (unstable or non-unique), with ill-posedness often resulting from poor experimental setups or measurement errors. This study addresses the identification of thermophysical parameters - specifically thermal conductivity and heat transfer coefficients—in a 2D steady-state diffusive medium. The proposed method uses a boundary element approach and an iterative descent algorithm to minimize a functional and identify the unknown parameters, validated through simulated thermograms. As a result, the use of sensitivity functions to weight the functional to be minimized makes it possible to avoid selection of the sensors according to the parameter to be identified.

• Periodical: Defect and Diffusion Forum (Volume 445)
• Pages: 225-234
• DOI: https://doi.org/10.4028/p-m3KISR
• Online since: December 2025
• Authors: Abdelkarim Maamar, Mohamed Bouanini, Eugenia Rossi di Schio, Paolo Valdiserri, Cesare Biserni*
• Keywords: Direct Modelling, Heat Exchange Coefficient, Heat Source, Inverse Problems, Simulated Temperatures, Thermal Conductivity

References

[1] M. N. Özişik, H-R.B. Orlande, Inverse Heat Transfer: Fundamentals and Applications. Taylor & Francis, 2000.

[2] A. Jahanbin, G. Semprini, A.N. Impiombato, C. Biserni, E. Rossi di Schio, Effects of the circuit arrangement on the thermal performance of double U-tube ground heat exchangers. Energies. 13 (2020) 3275.

DOI: 10.3390/en13123275

[3] C. Naldi, E. Zanchini, Full-Time-Scale Fluid-to-Ground Thermal Response of a Borefield with Uniform Fluid Temperature. Energies. 12 (2019) 3750.

DOI: 10.3390/en12193750

[4] V. Ballerini, E. Rossi di Schio, P. Valdiserri, C. Naldi, M. Dongellini, A Long-Term Dynamic Analysis of Heat Pumps Coupled to Ground Heated by Solar Collectors. Applied Sciences. 13 (2023) 7651.

DOI: 10.3390/app13137651

[5] E. Rossi di Schio, S. Lazzari, S., A. Abbati, Natural convection effects in the heat transfer from a buried pipeline. Applied Thermal Engineering. 102 (2016) 227-233.

DOI: 10.1016/j.applthermaleng.2016.03.140

[6] S.W. Rees, M.H. Adjali, Z. Zhou, M. Davies, H.R. Thomas, Ground heat transfer effects on the thermal performance of earth-contact structures. Renewable and Sustainable Energy Reviews. 4 (2000) 213-265.

DOI: 10.1016/s1364-0321(99)00018-0

[7] E.J. Nascimento, E. dos Santos Magalhães, L.E. dos Santos Paes, Estimation of thermal properties at high temperatures through the application of radial basis function interpolation in an inverse heat transfer problem. International Communications in Heat and Mass Transfer, 161 (2025), 108482.

DOI: 10.1016/j.icheatmasstransfer.2024.108482

[8] L.B. Dantas, H.R.B. Orlande, R.M. Cotta, An inverse problem of parameter estimation for heat and mass transfer in capillary porous media. International Journal of Heat and Mass Transfer. 46 (2003) 1587-1598.

DOI: 10.1016/s0017-9310(02)00424-6

[9] P. DuChateau, An inverse problem for the hydraulic properties of porous media. SIAM Journal on Mathematical Analysis. 28 (1997) 611-632.

DOI: 10.1137/s0036141095285673

[10] E. R. Di Schio, M. Celli, I. Pop, Buoyant flow in a vertical fluid saturated porous annulus: The Brinkman model. International Journal of Heat and Mass Transfer. 54 (2011) 1665-1670.

DOI: 10.1016/j.ijheatmasstransfer.2010.11.014

[11] J.V. Beck, B. Blackwell, C.R. St. Clair Jr., Inverse Heat Conduction: Ill-posed Problems, Wiley, 1985.

[12] X. Wang, I. Perreard, Bayesian inference for thermal diffusivity using transient experiments. International Journal of Heat and Mass Transfer. 171 (2021) 121050.

[13] B. Jin, J. Zou, A regularized total least squares method for a sideways heat equation. Inverse Problems. 25 (2009) 045009.

[14] G. Lorenzini, L.A.O. Rocha, C. Biserni, E.D. Dos Santos, L.A. Isoldi, Constructal design of cavities inserted into a cylindrical solid body. ASME Journal of Heat Transfer. 134 (2012) 071301.

DOI: 10.1115/1.4006103

[15] G. Lorenzini, C. Biserni, L.A.O. Rocha, Geometric optimization of isothermal cavities according to Bejan's theory. International Journal of Heat and Mass Transfer. 54 (2011) 3868-3873.

DOI: 10.1016/j.ijheatmasstransfer.2011.04.042

[16] L.A. O. Rocha, G. Lorenzini, C. Biserni, Y. Cho, Constructal design of a cavity cooled by convection. Int. J. Design and Ecodynamics. 5 (2010) 212-220.

DOI: 10.2495/dne-v5-n3-212-220

[17] G. Lorenzini, C. Biserni, A Vapotron Effect application for electronic equipment cooling. ASME Journal of Electronic Packaging. 125 (2003) 475-479.

DOI: 10.1115/1.1615796

[18] T. Mauder, J. Kůdela L. Klimeš, M. Zálešák, P. Charvát, Soft computing methods in the solution of an inverse heat transfer problem with phase change: A comparative study. Engineering Applications of Artificial Intelligence. 133, Part B (2024) 108229.

DOI: 10.1016/j.engappai.2024.108229

[19] C.A. Brebbia, J.C.J. Telles, L.C. Wrobel – Boundary Element Techniques. Theory and Applications in Engineering. Springer-Verlag,1984.

[20] D. Petit, D. Maillet, Estimation of thermal boundary conditions at the surface of a sample subjected to an oxygen-acetylene flame, French Thermal Engineering Conference, Thermal Engineering in Extreme Conditions, Bordeaux, France, January 2012.