Multiple Inversion Techniques with Multispectral Pyrometry for the Estimation of Temperature and Emissivity of Liquid Niobium and 100c6 Steel

Abstract

The article presents the estimation of temperature and emissivities of niobium and 100c6 steel around their melting points using techniques based on the minimization of the least squares norm or on Bayesian inference. Two models were considered for the spectral emissivity, including a linear variation with respect to the wavelength and independent spectral values. The measured data used for the solution of the parameter estimation problem were radiative fluxes collected from a five-wavelength pyrometer. The experimental apparatus used in this work was designed for the characterization of metal samples with sizes of few millimeters at high temperatures, combining aerodynamic levitation and laser heating. The use of the linear spectral emissivity model provided quite consistent results with the ordinary least squares estimator and with stochastic simulations using the Metropolis-Hastings algorithm. Similarly, the model of independent spectral emissivities resulted in accurate emissivities, but it required an informative prior for temperature, such as the known melting point of the metal sample. Therefore, whatever the inverse method used, a priori supplementary information in terms of emissivities or temperatures are needed due to the temperature-emissivity correlation and the unknow emissivity behavior.

Acknowledgements

Helcio Orlande acknowledge the support provided by CNPq and FAPERJ, agencies of the Brazilian and Rio de Janeiro state governments, as well as Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brazil (CAPES) - Finance Code 001. Helcio Orlande would also like to acknowledge the financial support and hospitality of Institut de Recherche Dupuy De Lôme – Université Bretagne Sud, particularly during his visit in May 2022. The support of the French Embassy in Brazil and FAPERJ for the visit of Thomas Pierre to COPPE/UFRJ (October and November, 2022) is greatly appreciated.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Thomas Pierre

Thomas Pierre is an Associate Professor at the Université Bretagne Sud in Lorient since 2008. He obtained his Ph.D. degree in 2007. He works in the CNRS laboratory IRDL on the use of inverse methods for the estimation of physical properties of different materials ranging from insulators to good conductors. He works on experiments and protocols involving analytical models. His works are also focused on the simultaneous estimation of high temperatures and emittances of opaque surfaces by multispectral pyrometry, using deterministic and probabilistic techniques.

Jean-Claude Krapez

Jean-Claude Krapez is a Senior Research Scientist at ONERA, the French Aerospace Lab. He received his Ph.D. in 1991. He joined ONERA in 1992 after previous employment at Industrial Materials Institute, NRC, Boucherville, Canada. He has been working on thermal methods for nondestructive testing of materials and remote sensing. More specifically, his research includes modeling of heat transfer by conduction and radiation in materials and structures with scope including characterization of multilayer or graded structures, defect detection and characterization in metallic and composite materials, thermal and infrared modeling of the environment (natural and manmade) for an interpretation and an inversion of the infrared images obtained by airplane or drone remote sensing (multispectral radiometry), analytical modeling of energy transfer (heat diffusion, optical scattering, advection-diffusion in the atmosphere boundary layer) in media presenting graded properties.

Helcio Rangel Barreto Orlande

Helcio Rangel Barreto Orlande obtained his B.S. in Mechanical Engineering from the Federal University of Rio de Janeiro (UFRJ) in 1987 and his M.S. in Mechanical Engineering from the same university in 1989. After obtaining his Ph.D. in Mechanical Engineering in 1993 from North Carolina State University, he joined the Department of Mechanical Engineering at UFRJ, where he is currently a Professor and was the department head during 2006 and 2007. His research areas of interest include the solution of inverse heat and mass transfer problems, as well as the use of numerical, analytical and hybrid numerical-analytical methods of solution of direct heat and mass transfer problems. He is the coauthor of 4 books and more than 370 papers in major journals and conferences. He is a member of the Executive Committee of the International Center for Heat and Mass Transfer and a Delegate in the Assembly for International Heat Transfer Conferences. He serves in the editorial boards of the journals Heat Transfer Engineering, High Temperatures - High Pressures, Archives of Thermodynamics and International Journal of Thermal Sciences.

Christophe Rodiet

Christophe Rodiet is an Associate Professor at the Institute of Thermic, Mechanics and Materials (ITheMM) at the Université de Reims Champagne-Ardenne at Reims. His areas of expertise are associated with heat transfer, such as multispectral thermography, thermal characterization of multilayer anisotropic materials (bulks or thin materials), source estimation, as well as inverse methods.

Philippe Le Masson

Philippe Le Masson is a Doctor-ès-Sciences (1991), Professor at the University Bretagne Sud of Lorient. He works in the Laboratory IRDL - UMR CNRS 6027, where he is the director. His research interests include modeling of heat transfer, theory and practice of inverse heat transfer problems, measurements with thermocouples, thermal characterizations of the heat transfer during process. He is a member of the SFT (Société Française de Thermique) and works in the French group METTI (Metrologie Thermique et Techniques Inverses). He works more specially on the modeling (fluid mechanics, interaction process-material) and the instrumentation by thermocouples of the assembling operations.

Dylan Le Maux

Dylan Le Maux is a research engineer in the CNRS laboratory IRDL at the Université Bretagne Sud at Lorient. During his thesis, which he defended in 2020, he developed an experimental device for the measurement of thermophysical properties of liquid metals at high temperature. This work led to results on density, surface tension, and emissivity, which has been published in several journals. He is now pursuing this experimental work and is also developing a numerical model of laser welding with filler wire in collaboration with ArcelorMittal.

Mickaël Courtois

Mickael Courtois is an Associate Professor at the Université Bretagne Sud in Lorient since 2015. He works in the CNRS laboratory IRDL on multiphysics numerical modeling of metal fusion processing like welding and metal additive manufacturing and on molten metal characterization (density, surface tension, evaporation…) with a dedicated aerodynamic levitation apparatus able to reach very high temperatures.

Bernard Lamien

Bernard Lamien is Doctor of Sciences in Mechanical Engineering (2015) by the Federal University of Rio de Janeiro. He was a Postdoctoral Researcher of the Federal University of Rio de Janeiro and of the Université Bretagne Sud at Lorient. He is currently an Assistant Professor at Ecole Polytechnique de Ouagadougou, Burkina Faso, and is also in charge of the direction of internship and continuing education. He is broadly interested on the use of inverse problems solutions methodologies for parameter and function estimations in heat and mass transfer problems. He is familiar with Bayesian Inverse Problems, as well as with statistical model reduction techniques.