Jad Houssein

Actn17

Clara Jimenez

Laurent Acl10

Thomas Dejaeghere

Muriel Pierre

Philippe Le Carin

Mickaël Masson

Development Courtois

Christophe Rodiet

Aerodynamic levitation Thomas Pierre

Benjamin Rémy

Alain Degiovanni

Thibaut Colinart

Patrick Glouannec

Acti20 Luiz

F S Ferreira

Leonardo A B Varon

Parameter estimation Helcio R B Orlande

Jad Acti19

Mickaël Houssein

Thomas Courtois

Gaëtan Pierre

Muriel Le Goïc

Carin

Dylan Acti18

Mickael Le Maux

Dylan Acti17

Thomas Acti3

Problématique Pierre

Actn26 Luiz

F S Feirrera

Thomas Os7

Philippe Pierre

Yves Le Masson

Thomas Os4

Andrzej Jannot

Kusiak

Nicolas Horny

Thierry Duvaut

Philippe Le Masson

Edouard Geslain

Physical properties characterization of liquid metals through aerodynamic levitation

Keywords: ACL21, ACTI19 ACL13, ACL14, ACL15, ACL16, ACL9, ACL15, ACTI9, ACTI13

and conductive transfers (Section 4.2). Indeed, classical aerogels are super insulating materials with thermal conductivity close to air thermal conductivity where radiativeconductive coupling occurs. At the same time, I continued to study hemp concrete by characterizing its millimetric component: the hemp shiv (Section 4.3). Literature did not mentioned properties of this cellulosic material. For this I developed the use of temperature-temperature analytical model and dedicated experiment to characterize those particles.

These first projects, developed in Section 4, allowed me to develop a systematic methodology in order to conduct scientific works: the setting up of experimental protocols from analytical models and sensitivity studies. Experimental protocol to estimate these properties is developed from a direct model. It is function of properties of interest (conductivity, diffusivity, thermal capacity) and minor others (thermal losses).

All are more or less related to each other, and they must be estimated. Sensitivity study is then necessary to be able to answer these two questions: i) what is the level of correlation between these properties? ii) is the correlation between properties time-dependent? By playing on input parameters of the direct model, answers to these questions define maximum duration of the experiment, dimensions of the sample, time and type of heating, what to measure and where to place the measuring probes… Once experimental protocol has been defined, estimation of parameters is tested from noisy data obtained from direct model. Estimation is carried out by inverse techniques; for example, minimizing the quadratic error between experimental temperature measured (observable) and that given by the direct model at the same location (least squares with Levenberg-Marquardt type optimization algorithm). There are several ways to develop direct models. I mainly work with analytical and semianalytical models. Most of the time, they have the advantage over numerical ones of operating faster. However, they are more limited because analytical solution to a problem is not always available, or by means of more difficult mathematical tools. The method of thermal quadrupole is very convenient for establishing multilayer analytical models. And addition of Laplace and Fourier integral transformations deals with transient, periodic, and multidirectional problems. However, the use of analytical model often requires to consider constant physical properties, thus constraining the experiment: input of energy such as the maximum temperature rise of the sample leads to variation in physical properties less than uncertainty in their estimation. mechanical, hydrodynamic, metallurgical, electromagnetism...), large number of physical properties is required and over a wide temperature range since the matter changes from the solid state to the liquid state and can reach its boiling point. However, the literature is almost mute from the liquid state and beyond. That is why PTR2 started in 2012 the development in the laboratory of high temperature apparatuses in order to characterize liquid metals: for example, a device of aerodynamic levitation of metal balls melted by a laser. The aimed properties are density, surface tension, viscosity, thermal diffusivity, radiative properties, and temperature. I participated and still participate to the estimation Table 3.1. PhD supervision.

First, I participated to the ANR project "Béton de chanvre" in 2008-2012 (Section 4.1) (Section 4.1). This project came within the scope of reducing the important energy consumption in the building. I worked on the definition and the characterization of the bio-sourced materials, the hemp concretes. Hemp concretes, considered heterogeneous, anisotropic, and porous material, are obtained by mixing a lime-based binder, hemp particles and water. They have the ability to breathe thanks to its high porosity: air more or less filled with water circulates there. The migrations of water and energy in the volume make its study complex.

In the continuity of working on insulating materials, the "composite" team of the IRDL, in collaboration with the INRA in Nantes, wanted to develop a bio-aerogel and it proposed me to characterize it and to determine the contribution of the inner radiative I became familiar with the characterization techniques of the hot-plate or hot-strip commonly used. But they remain difficult to perform serenely due to difficulty of controlling the instrumentation necessary for the experiment: thermal properties of heating element used for the thermal stress, contact resistance, heating element/sample contact surface, quantity of energy deposited, position of the thermocouples. Classical theoretical models have flux as input quantity and temperature as observable; they are flux-temperature models and impedance connects both of them. However, uncertainty on the flux actually absorbed by the sample is significant; the part absorbed can be very different from that emitted by the heating element. Therefore, I turned to the development of theoretical temperature-temperature models, which offer favorable outcome to limit the number of uncertain input data. For example, two thermocouples on either side of sample measure both input and the observable, therefore number of parameters to be estimated in the transmittance becomes limited to physical properties of the sample and the heat losses. Experimentally, measurements at ambient temperature by thermocouples have been used for characterization of bio-aerogels, where conductive radiative coupling is present (Section 4.2); but also, by infrared and visible thermography for, first, the study at room temperature of anisotropic materials of millimetric dimensions in order to characterize hemp shiv (Section 4.3), and, second, to estimate the thermal diffusivity of metals close to their melting point (Section 6.2). Indeed, part of PTR2 has been working for more than twenty years in collaboration with companies (ArcelorMittal, Schneider, EDF, CEA, Stellantis, Framatome, Naval Group) specializing in all types of welding or additive manufacturing. One of PTR2 contributions is to perform numerical simulations of these processes using commercial codes such as Comsol Multiphysics®. Industrial processes being highly multiphysical (thermal, of all these properties, but mainly the temperature and radiative properties (Section 5) and thermal diffusivity (Section 6).

Section 5 deals with the simultaneous estimation of temperature and emissivity of metals. Due to temperature that can up to 3 000 K, the disturbances of environment and atmosphere are negligible. The experience of my thesis dedicated to multispectral pyrometry enabled me to develop at IRDL in 2012 a six-wavelength pyrometer for estimating the temperature and emissivity of metals carried beyond their melting point (Section 5.1). This home-made six-wavelength pyrometer was first calibrated using a home-made blackbody. Radiative fluxes are then directly measured by the pyrometer, upon solid-liquid phase change of the metal balls. Temperature is estimated using the multispectral pyrometry, but it is difficult to address for several reasons. The system to be solved is underdetermined, it systematically includes one more unknown than equations. Indeed, radiative flux emitted by a surface depends on two parameters: temperature and emissivity. Thus, for N radiative fluxes measured at the same temperature at a given time, there are N + 1 unknowns: N emissivities and temperature (Section 5.2). One solution is to propose a spectral model of emissivity. The literature offers many physical and mathematical models, based on electromagnetic theory or experiment. Unfortunately, the behavior of the surface state is for the most part difficult to control satisfactorily from one experimental apparatus to another, but also from one experiment to another for a given apparatus. The unpredictability of emissivity increases with the dynamic of the experiment. In our case, we are exposed to the following issues: the temperature change of several thousands of kelvins in a few seconds, the diffusion, the presence of oxygen (even residual), the changes of state, the impurities. Therefore, establish a behavioral model of the emissivity believable is almost impossible with a good confidence. However, the low spectral range of the home-made multispectral pyrometer allows the use of simple polynomial models of maximum order 1. It is with this strong hypothesis that estimates of temperature and emissivity of metals have been established.

In addition, tests with different emissivity models, inverse techniques of different kinds were compared (Section 5.3): deterministic methods (least squares previously presented) and stochastic methods (different techniques of Bayesian inference). Only the wavelength dependence for the emissivity model is considered since an estimate is made for each time step, whether dynamic or static.

Experience acquired with pyrometry allowed me to work on thermal diffusivity estimation using radiative fluxes instead of direct temperatures, which is not possible due to the high temperatures of liquid metals (Section 6). First attempts focused on the feasibility of estimating thermal diffusivity of liquid metallic droplet during levitation (Section 6.1). Different direct models in spherical coordinates (Section 6.1) and Cartesian coordinates (Section 6.2) have been developed using inputs and observables of different nature: power of the laser, radiative fluxes measured by the six-wavelength pyrometer, by high speed camera... To sum up, my field of research is two-fold: i) the development of experimental protocols dedicated to the characterization of physical properties of materials; ii) the measurement of temperature and emissivity through multispectral pyrometry. Both themes joint themselves with the common use of inverse techniques, necessary tools as long as more than one parameter needs to be estimated when one experiment is performed.

This document tries to be the reflect of both aspects of my work, and mainly the research one. It is constructed as follows. My curriculum vitae is presented in Section 1. It concerns my university training since 2004 until my present position. Section 2 is the teaching I did in Nancy -as a monitor -, but especially in Lorient in the department named Métiers de la Transition et de l'Efficacité Énergétiques (ex Génie Thermique et Énergie) and at the Faculty of Science. My administrative responsibilities are also presented.

Section 3 deals with my scientific investments, supervisions, and publications. Then Sections 4 to 6 relate to my actual research works. They detail what was previously introduced, but I took care to present at the end of each section a synthetic conclusion and also a perspective. Finally, a general conclusion and perspectives Section proposes major axes of future works. Even if it is not conventional, I add a general nomenclature before Section 4. Each section has its own list of bibliographical references presented in this manner [First author Year].

1. Curriculum vitae 1.1. University training, 1998University training, -2007 After I first passed a two years diploma, DUT Mesures Physiques, I focused my formation in the field of mechanical engineering, and more specifically the heat transfers.

Between 2001 and 2004, I successively passed my bachelor and masters (Licence, Maîtrise, and DEA). Table 1.1 is a synthesis of my diplomas obtained at the university. Fourier, proposed me to study the measurement methods of the zenith intensity (Étude de méthodes de mesure de la luminance zénithale [START_REF] Jannot | Simplified estimation method for the determination of the thermal effusivity and thermal conductivity using a low cost hot strip[END_REF]. I shared my time between Grenoble and Besançon, one of the sunniest cities of France. The subject and its motivations are the following. The decrease of the ozone quantity on Earth imposed the creation of stations all around the Earth dedicated to the measurement of ground solar irradiance, in order to control its temporal evolution. IRSA laboratory belongs to an international network of stratosphere observation (Network for the Dectection of Atmosferic Chemical Change, NDACC). It possesses a measurement station composed of an ultraviolet spectroradiometer and a system able to record, treat, and convert raw data into irradiances. In the case of my internship, the station is equipped of a "barrel" in order to aim specifically the zenith and is used to determine zenith intensity. The objective of my internship was to test a method proposed by A. de La Casinière [de La Casinière 2003] to convert the raw data collected by the spectroradiometer into intensity. After calibration of the spectroradiometer equipped with the barrel, results were successfully obtained since experimental intensities were compared with those obtained by the resolution of the radiative transfer equation.

This internship was mostly experimental but I worked on the radiative transfer equation and the solar radiation. It allowed me to discover the principle of the photon counting with a spectroradiometer and its calibration procedure.

No communication has been proposed for this work. In this inverse problem, the direct model was based on the radiative transfer equation and the inversion was performed with the gradient technique. The work was only numerical. One of the possible applications is the detection of cancerous tumours in the human tissues.

The results of this internship were published according to the following references in Section 3.

3: [ACL2], [ACT5] [COM2].
2. Teaching and responsibilities

During my PhD

During my PhD, I did tutorials as a temporary worker the first year and then as a monitor for the next two years under the supervision of Professor Denis Maillet. Table 2.1 summarizes these lessons with a total volume of 193.5 hTD. The courses were given to first-year students of the engineering cycle of the l'École Nationale Supérieure en Génie des Systèmes Industriels (ENSGSI) and to first-year students of the preparatory cycle of the l'École Européenne d'Ingénieurs en Génie des Matériaux. At the same time, I held and hold the following pedagogical responsibilities:

• 2014/2017: director of studies, establishment of the timetables for the following students: MT2E1/MT2E2/MT2E2 apprentices;

• 2018/2021: Parcoursup referent for student recruitment;

• 2020/2023: director of studies, establishment of the timetables for the following students: MT2E1/MT2E2/MT2E2 apprentices.

I also take part at the UFR SSI energy specialty in various fields:

• since 2014/2015: Master degree level, radiative metrology (lecture, 4h)

• 2019/2020: level L3, fluid mechanics/energetics (practical works, 24h)

• regular monitoring of trainees (2 or 3 per year, Bachelor to Master degrees).

3. Scientific investments, supervision, and production

Scientific investments

At the national level, I am an active member of the French thermal society (SFT) and I participate in the High temperatures thematic (2013,2016,2023). In addition, for the national SFT congress, I review two or three articles annually.

I have never held a project of my own but I had and still am involved in some:

•

I participate in thematic schools of Thermal Measurements and Inverse

Techniques METTI (2005METTI ( , 2015METTI ( , 2019METTI ( and in project 2023)), for which I presented two dedicated workshops "measurement by the hot plane method" (9h) and "multispectral pyrometry" in 2019, and will also present in 2023 a lecture on the simultaneous estimation of temperature and emissivity in multispectral pyrometry.

At the international level, in 2008 I made my post-doctoral application at the University of Quebec at Chicoutimi (see Section 1.2). I experimented with the use of inverse techniques in the field of optical tomography.

Recently, I was invited from October 1 st to November 30 th , 2022 at the Federal University of Rio de Janeiro (UFRJ) thanks to a collaboration with Professor Helcio R. B. Application to aerodynamic levitation and laser heating.

Maxime Bastard 2015

Development of an apparatus dedicated to temperature measurement by multispectral pyrometry during a welding process. The studies presented in this section constitute a continuity in the setting of experimental protocols dedicated to the characterization of thermal properties of materials heated by Joule effect. Protocols depend on the nature of the material, but should conduct as much as possible to the use of analytical or semi-analytical solutions of the observable over time. Experimentally, the observable is delivered either through the thermoelectric effect (thermocouple) or by radiation (camera, pyrometer). Hemp concrete has a low bulk density (300 < ρ < 600 kg•m -3 ) and a high porosity ( > 65 %), and the dry thermal conductivity at ambient temperature is between 0.050 and 0.150 W•m -1 •K -1 . The knowledge of a single characteristic value may nevertheless not be sufficient to represent the thermal behaviour of the material since temperature and moisture content due to hygroscopic behaviour influence thermal properties during climatic changes: a significant amount of pores is filled with water, which thermal

Scientific production

ℬ quadrupole coefficient Bi Biot number C lightspeed, m•s -1 C specific heat, J•K -1 𝒞 quadrupole coefficient 𝐶 1 Planck's law constant, W•µm 4 •m -2 𝐶 2 Planck's law constant, µm•K 𝑐 𝑝 𝑚 heat capacity, J•kg -1 •K -1 D quadrupole coefficient
conductivity is 0.600 W•m -1 •K -1 , while that of air is 0.026 W•m -1 •K -1 . The consequence
of the presence of these three phases (solid matrix, air, and water) and the competition of their effects between them determine the effective thermal properties of the hemp concrete. So, the motivation for determination of the effective thermal conductivity of such a porous material as a function of temperature and moisture content appears very significant.

The experimental procedure is developed to assess temperature and relative humidity dependence of hemp concrete's thermal properties. The effective thermal conductivity and diffusivity of hemp concrete samples are estimated by inverse method and by using a hot-strip. The measured values of thermal conductivity are then compared with results from a predictive model based on Krischer's model.

The theoretical model

The hot-strip technique has been used to estimate both thermal conductivity and diffusivity of hemp concrete [START_REF] Jannot | [END_REF]]. It is a 2D transient model developed with the quadrupole formalism. Proposing semi-analytical solution, this formalism is very convenient in case of multilayer problem, including if necessary contact resistance, lumped body assumption, and semi-infinite medium for instance [Maillet 2000].

Considering only the sample, Figure 4.3 presents the sample (in white) and the hot-strip (in grey). There is a symmetrical plane in x = 0. The heat transfer is assumed diffusing only in the (x,y) plane. The characteristic dimensions of the sample are its halfwidth L and its half-thickness e, and for the hot-strip its half-width b. 𝑇(𝑥, 𝑒, 𝑡) = 0 (4.9)

For both temperature and flux, a double integral transform in space according to the x-direction (cosine) and time (Laplace) [Carslaw 1959][Ozisik 2000] where 𝑓(𝑥, 𝑦, 𝑡) is the initial function and 𝐹(𝛽 𝑛 , 𝑦, 𝑠) its transform, s the Laplace parameter, and X is either L or b, the upper limit of the cosine transform for the temperature and the flux, respectively. Solving Eq. (4.10) allows to get the following heat equation:

𝑑 2 𝜃 𝑑𝑦 2 = 𝑢 𝑛 2 𝜃 (4.11) 𝑢 𝑛 2 = 𝑠 𝛼 + 𝛽 𝑛 2 (4.12)
where 𝜃 is the transformed temperature and 𝛽 𝑛 are the roots of a transcendental equation obtained with the boundary conditions (4.5) and (4.6). The solution of Eq. (4.11) is:

𝜃 = 𝜙 𝑐𝑜𝑠ℎ(𝑢 𝑛 2 𝑦) + 𝜁 𝑠𝑖𝑛ℎ(𝑢 𝑛 2 𝑦) (4.13)
where 𝜙 and 𝜁 are constants.

Through the y-direction, experiment shows that the medium is multi-layered since the heating element, and the contact resistance between the latter and the sample cannot be neglected. Therefore, the quadrupole formalism and solutions of (4.13) allow to express the heat transfer [Maillet 2000] as follows:

[ 𝜃 0 Φ 0 ] = [ 1 0 𝐶 ℎ𝑠 𝑠 1 ] [ 1 𝑟 𝑐 0 1 ] [ 𝐴 𝑠 𝐵 𝑠 𝐶 𝑠 𝐷 𝑠 ] [ 𝜃 𝑒 Φ 𝑒 ] (4.14) [ 𝜃 0 Φ 0 ] = [ 𝒜 ℬ 𝒞 𝒟 ] [ 0 Φ 𝑒 ] (4.15)
The index for the temperatures and fluxes are for the locations y = 0 and e. The first square matrix on the right belongs to the half-hot-strip of heat capacity 𝐶 ℎ𝑠 = (𝜌𝑐 𝑝 𝑚 𝑒) ℎ𝑠 , the second one represents the contact resistance rc between the hot-strip and the sample, and the third one is the sample where:

𝐴 𝑠 =
𝐷 𝑠 = 𝐴 𝑠 (4.19)
The solution is at the location y = 0: it happens that the two first parameters of vector 𝛃 are not correlated, meaning that they can be estimated as presented in Figure 4.5. Note also that effusivity and heat capacity of the sample behave differently from contact resistance and heat capacity of the hot-strip. The sample is equipped of two type K thermocouples, one is located at the front face between the hot-strip and the sample, meaning at the location (0,0) according to Eq. Since experimental measurements of thermal properties as a function of temperature and relative humidity are very time consuming, development and application of predictive models are the most convenient method. In literature, many works present prediction of effective thermal conductivity of porous media as a function of the phase number (solid, liquid, gaseous), the porosity, the orientation, and the shape of the pores [Wang 2008]. In our case, Krischer [START_REF] Krischer | [END_REF]] proposed a weighted harmonic mean of series and parallel models (4.27)- (4.31). This model has been adapted over temperature T and water content w variations (4.32). • Quantities ks, kw and ka are the thermal conductivity of solid phase, moisture and dry air, respectively. Thermal conductivities of air and water are well-known [Sacadura 2000] and their use with temperature is under polynomial forms in the working temperature range.

𝜃 0 = Φ 0 (
𝑘 𝑒𝑓𝑓 =

•

Weighting parameter n ranges between 0 and 1. It is a common practice that n is the volume fraction of the medium disposed orthogonally to the heat flux direction.

•

Air volumetric fraction in a dry porous material is given by the total porosity ε (0 ≤ ε ≤ 1). In the case of penetration of liquid water, the filled part of the porous space is given by water saturation S (0 ≤ S ≤ 1).

•

Water saturation S in Eqs. (4.28) and (4.29) is given by Eq. (4.30).

•

Water content w is expressed by Eq. ( 4.31) over the entire relative humidity RH range by fitting Guggenheim-Anderson-de Boer model (GAB) [Guggenheim 1966] to experimental data: w0 = 0.0143 kg•kg -1 , C = 14.164 and K = 0.877. These parameters have been estimated by least-squares by minimizing the quadratic error between experimental water content and w from Eq. (4.31).

In Eq. (4.32), the unknown parameters are n, , and ks(T). First, they have been estimated using OLS with LM algorithm by minimizing quadratic error between experimental conductivities k and keff at dry state (S = 0). Then a global polynomial for ks(T) is deduced. -3 °C and 30 °C, since the experimental apparatus does not allow measurement of the mass evolution during experiments. However, a 0.6 % mass variation has been observed before and after experiments, which leads to a water content variation of 8 %. In parallel, error on thermal conductivity can be approached as the sum of the errors on temperature measurement (experimental noise of about 0.05 °C), on hot-strip dimensions (systematic error of 0.1 %) and on flux (errors of 0.5 %). As a consequence, error on thermal conductivity is estimated at 3 %. The use of hot-strip technique is easy to set, but it presents several drawbacks, and particularly with this type of heterogeneous porous material. The presence of instrumentation penalizes systematically the measurements. As mentioned previously, there are unknown parameters appearing in the theoretical expression of the temperature: the hot-strip heat capacity, the contact resistance between the hot-strip and the sample. I could also mention the presence of the thermocouple and the heat rate.

•

The Heat capacity of the hot-strip is difficult to estimate due to its own heterogeneity. Possibilities to estimate heat capacity of heating element exist but seem much appropriate with square as for hot-plane technique, with the use of semi-infinite models [Jannot 2018][Pierre 2019b].

•

The problem with the contact resistance is that it is systematically correlated with the sample thermal conductivity. However, for insulating material, which is the case in this study, contact resistance can be neglected if the sample thickness is large enough. Of course, the innocuousness of the contact resistance depends on the thickness of the sample and of its nature (from insulating to conductive).

•

The presence of thermocouple between the sample and the hot-strip can result in irregular and large value of contact resistance if the sample does not possess a certain flexibility, like rubber. With heterogeneous porous material, punctual measurement with a thermocouple is an issue, since the tip can be either in contact with the solid phase, or with a pore: measurement is rather local than representative of global surface of the sample, this issue is also encountered during measurements with the hot-wire technique [Sjöström 2014][Seng 2019].

• Finally, there is also an uncertainty on the heat rate delivered by the hot-strip. The heat dissipation along wires becomes negligible with increasing the surface of the hot-strip. But it remains an uncertainty of the contact surface with the sample. The real contact surface is not systematically the hot-strip surface, it can be smaller. A procedure of estimation of real contact surface is proposed in [Jannot 2018] with the help of a well-known material. But it may be questioned if this calibrated real surface is the same with another material, assuming that the pressure applied to maintain the whole experiment is controlled.

Parameter estimation with hot-strip technique is a practical manner to characterize homogeneous insulating materials preferentially with very weak apparent surface porosity, at least with a porosity weaker than the thermocouple dimensions.

Characterization of more complex materials such as hemp concrete is also possible but with a global view with the estimation of effective parameters. The control of rear face temperature allows to use semi-infinite or imposed temperature assumptions, and thus to not consider its heat losses. Among all unknown parameters that affect the estimation of those of interest, the main uncertainties come from the knowledge of the heat capacity of the heating element and the heat rate due to the surface contact.

The calculation time with the use of this analytical model for the parameter estimation through OLS with LM is of several tens of seconds. This analytical model is very convenient in our case since we have considered an effective view of the material, meaning a global approach but representative of the material in use. A more accurate study of the material would imply in most of the case the use of multiphysics numerical models and thus a non-negligible increase of the calculation time [Niezgoda 2012b] [Nguyen 2016][Bourdot 2017] [Bennai 2018].

To avoid influence of the heat capacity and of the heat rate, an appropriate manner is the use of temperature-temperature model. The model presented in Eq. ( 4.21) is a fluxtemperature model, meaning that input is the heat rate and observable is the temperature.

In this case, the transfer function is called impedance. In the case of temperature- Nevertheless, even if convection is neglected due to the small diameter of pores, gas conduction and radiation occur in addition to solid conduction in aerogel [Ebert 2011[START_REF] Cuce | [END_REF]]. Experimentally, the greatest difficulty is to evaluate the contribution of each heat mechanism. Regarding the macroscopic approach, radiative flux that reaches a material surface can be reflected, absorbed, or transmitted along the thickness of the material [Incropera 2011][Xie 2013]. As aerogel is capable of absorbing, emitting, and scattering thermal radiation [Wei 2013], heat transfers in it are described by solving combined conductive (Fourier's law) and radiative transfer equations [Ozisik 1973].

Combined calculation of radiation and conduction heat transfer has been extensively studied through more or less sophisticated numerical models [START_REF] Kunc | [END_REF] First of all, the quadrupole formalism is used to solve conductive-radiative coupling within aerogel [Maillet 2000][Degiovanni 2002][Rémy 2006]. The aerogel thickness is large enough to be opaque. It is assumed that boundaries are opaque too and their effects negligible. In this case, radiative transfer is viewed like a pure diffusion process and can be modelled by a simple thermal resistance Rr = e/kr [Degiovanni 2002] (Figure 4.14 and Rosseland's model [Ozisik 1973]). Second of all, Jannot et al. [Jannot 2009] developed an apparatus dedicated to thermal characterization of opaque insulating material. This is a pulsed transient technique, where the sample is heated on its front face thanks to heating resistance and where two brass plates are placed on both front and rear faces of the sample. We have adapted this technique in our case with only one brass plate on the rear face of the sample (Figures 4.12 and 4.13). Moreover, the following assumptions and considerations are proposed:

•

The contact resistances between the heating element and the aerogel and between the aerogel and the brass are not taken into account.

•

The brass plate is assumed purely capacitive (𝐶 𝑏 in Eq. ( 4.33)) and not resistive.

•

The heat capacity of the heating element is not taken into account. Consequently, 1D quadrupole approach leads to express the whole system as follows and presented in Figure 4.14: (4.43) where s is the Laplace parameter, Rr = e/kr is the radiative resistance, kr is the radiative conductivity, e is the sample thickness,  is the phonic thermal diffusivity [André 1995], kc the thermal conductivity, and 𝜌𝑐 𝑝 𝑚 the thermal capacity of the sample. The boundary condition at z = e+eb is:

[ 𝜃 0 Φ 0 ] = [ 𝐴 𝑟𝑐 𝐵 𝑟𝑐 𝐶 𝑟𝑐 𝐷 𝑟𝑐 ] [ 1 0 𝐶 𝑏 𝑠 1 ] [ 𝜃 𝑒+𝑒 𝑏 Φ 𝑒+𝑒 𝑏 ] = [ 𝒜 ℬ 𝒞 𝒟 ] [ 𝜃 𝑒+𝑒 𝑏 Φ 𝑒+𝑒 𝑏 ] ( 4
Φ 𝑒+𝑒 𝑏 = ℎ𝜃 𝑒+𝑒 𝑏 (4.44)
where h is the heat transfer coefficient between the brass plate and the air. Jannot et al. [Jannot 2009] proposed an approach based on the expression of the transfer function 𝒯(𝑠) between temperature expressions at positions 0 and e+eb:

𝒯(𝑠) = 𝜃 𝑒+𝑒 𝑏 (𝑠) 𝜃 0 (𝑠) = 1 𝒜 + ℎℬ (4.45)
This theoretical expression is used here to estimate experimental temperature T exp (e+eb,t) versus experimental temperature T exp (0,t):

𝑇 𝑒𝑥𝑝 (𝑒 + 𝑒 𝑏 , 𝑡) = 𝑇 𝑒𝑥𝑝 (0, 𝑡) ⊗ ℒ -1 [𝒯(𝑠)] (4.46)
Beforehand, Figure 4.15 compares theoretical 1D models based on Eq. ( 4.33) and 3D models based on [Jannot 2010] expressing front and rear faces temperatures.

Dimensions and thermophysical properties values are the following:

• Brass plate: eb = 0.15 mm, 𝐶 𝑏 = 3.44 × 10 6 J•m -3 •K -1 ;

• Aerogel: e = 9 mm, kc = 20 mW•m -1 •K -1 , kr = 20 mW•m -1 •K -1 ,  = 1.33 mm 2 •s -1 ; • Heat transfer coefficient: h = 10 W•m -2 •K -1 .
Both models present the same thermograms during more than 200 s (only the first 100 s are plotted here) with mean residuals less than 10 -3 °C, which means that 1D model is enough for this study. The flux density 0 is also plotted. The Laplace numerical inversion, which allows to perform 0(t) = ℒ -1 [0(s)] used here [De Hoog 1970] gives satisfying results.

The plot of T th (e+eb,t) in Four parameters are unknown in the transfer function H(s) in Eq. ( 4.45): the thermal diffusivity , the thermal conductivity kc, the radiative conductivity kr, and the heat transfer coefficient h. These four parameters are estimated by inverse method, minimizing the sum of the quadratic difference between experimental values T exp (e+eb,t)

with those calculated by relation (4.46) with experimental value T exp (0,t). The minimization is realized by using LM method. Now sensitivity study is presented in Figure 4.17:

• All sensitivity curves have equivalent amplitudes and present possibility to estimate the four parameters independently.

•

The thermal diffusivity seems to be estimated properly even if a non-negligible correlation exists with the radiative conductivity.

•

The principal parameter correlated with the thermal conductivity seems to be the radiative conductivity.

•

The presence of the brass plate on the rear face decorrelates the conductivities and the heat transfer coefficient, which is not possible without it [Degiovanni 1977].

Indeed, due to its capacitive properties, the brass delays the development of the convective effects at the rear face.

• Direct contact between the heating element and the sample makes possible decorrelation between radiative and thermal conductivities. Radiative effects being quicker than conductive effects, temperature is at first sensitive to radiative effect. A series of simulations have been performed to define the optimal thickness of the brass plate, between 10 µm and 200 µm, considering the thickness (9 mm) of aerogel samples. Experimental measurements have been simulated with addition of a nullaverage normal noise b of 0.1 °C to T exp (0,t). Except for the variance var(), every curve present a minimum variance between 50 µm (var(kr)) and 150 µm approximately (var(h) and var(kc)). Experimentally, the brass plate has been chosen with a thickness of 150 µm in order to be not too flexible.

Results and discussions

Experimental apparatus is presented in Figures 4.12 and 4.13. Aerogel sample is placed between a heating element and a brass plate. Initially at z = 0, the heating element delivers a thermal flux 𝜑(𝑡) during a time step 𝜏. The pressure, which is not controlled, of the brass plate on the top is assured by several punctual PVC tips with a very low contact area. As aerogel is a very fragile material, the two type K thermocouples, which measure experimental temperatures T exp (0,t) and T exp (e +eb,t), are placed upward the heating element and on the upper face of the brass plate. Thermocouples characteristics are the following: 125 µm of diameter for the wire and 300 µm for the tip of the probe. Integration time dt is set at 100 ms (limitation of the temperature recorder).

Figure 4.19 plots the fitting of Eq. ( 4.46) on experimental rear face temperature with and without radiative transfers. Figure 4.20 shows that the reduced sensitivities 𝑋 𝛼 of thermal diffusivity seems weakly correlated with other parameters, but its amplitude is weaker. Both models (with and without radiative transfers) give satisfying fittings and close results of estimation as reported in Table 4.2 in terms of thermal conductivity. The sum of both thermal and radiative conductivities is 20 % lower than conductivity estimated with the model without radiation. Heat exchange coefficients are also approximately the same and consistent with typical values for natural convection.

However, estimation of thermal diffusivity gives a value 4 times higher without the radiative component in the model. This has already been observed and discussed in [Hahn 1997] [Niezgoda 2012a].

, mm•s -1 kc, mW•m -1 •K -1 kr, mW•m -1 •K -1 h, W•m -1 •K -1
with kr

 (mm•s -1 ) 1.12 ± 0.13 kc (mW•m -1 •K -1 ) 15.1 ± 1.0 kr (mW•m -1 •K -1 ) 14.2 ± 2.1 keff (mW•m -1 •K -1 ) 29.3 ± 3.0 36.2 ± 0.3 32.1 ± 0.5 h (W•m -2 •K -1 ) 8.8 ± 1.5 𝐶 𝑠 (kJ•m -3 •K -1 ) 13.5 ± 1.0 17.4 ± 1.5 𝑐 𝑝 𝑚 (J•kg -1 •K -1 ) 1 500 ± 100
As it was briefly mentioned,  = 660 m -1 for a temperature of T = 310 K, which is the mean temperature of the sample during experiments. In every case, opacity criterion  = e > 5 is fulfilled, supporting a posteriori the assumptions taken. Literature mentions also a specific spectral extinction coefficient K expressed in m 2 •kg -1 , that is to say / with  the density. In our case, K is equal to 44 m 2 •kg -1 , which is in good agreement with specific spectral extinction coefficient of aerogel found in literature [Wei 2011][Wei 2013].

𝑘 𝑟 𝑡ℎ = 16 3 𝑛 2 𝜎 𝑇 3 𝛽 (4.48) 1 𝛽 𝑅 = ∫ 1 𝛽 𝜆,𝑒𝑓𝑓 𝑑𝐿 0 (𝑇) 𝑑𝑇 𝑑𝜆 Δ𝜆 ∫ 𝑑𝐿 0 (𝑇) 𝑑𝑇 𝑑𝜆 ∞ 0 (4.49) 1 𝛽 𝑃 = ∫ 𝛽 𝜆,𝑒𝑓𝑓 𝐿 0 (𝑇)𝑑𝜆 Δ𝜆 ∫ 𝐿 0 (𝑇)𝑑𝜆 ∞ 0 (4.50) 𝜏 𝜆 ⊥ = 𝐼 𝜆 𝐼 𝜆 0 = 𝑒 -𝛽 𝜆,𝑒𝑓𝑓 𝑒 (4.51)

Conclusion

This section was dedicated to the estimation of thermal properties of bio-aerogel.

As both conductive and radiative transfers occur, they have been theoretically modelled through the quadrupole formalism, considering a simple resistance for the radiative transfer in parallel of conduction.

Parameter estimation has been performed through inverse method with the same procedure as in Section 4.1, but theoretical model is this time a temperature-temperature model, meaning that input data are the front face temperatures and observable are the rear face temperatures. This model is very practical since it diminishes drastically the number of unknowns due to instrumentation (Section 4.1.4). Indeed, there were four parameters to estimate, but three were of interest, the last one being the rear face heat exchange coefficient.

The knowledge of global behaviour of heterogeneous porous materials is very convenient in an engineering way for applications. Specific conduction-radiation coupling is a classical topic of research, but it is generally studied through the use of numerical simulations [START_REF] Kunc | [END_REF]][Ping 1989][Zeng 1995a][Zeng 1995b[START_REF] Heinemann | Radiation-conduction interaction: an investigation on silica aerogels[END_REF]][Asllanaj 2007[START_REF] Chandrasekar | [END_REF]] implying therefore high computational time.

However, the use of the quadrupole formalism dedicated to radiative transfers is very interesting. Since it offers possibility to run fast analytical models and to go more deeply in microscopic structure of the tested material.

In this experiment, aerogel sample was briefly heated, such as in the flash method [Degiovanni 1977][Degiovanni 2002]. As theoretical model is developed in the Laplace domain, the right form of the heat flux in the time domain should be verify: de Hoog numerical inversion technique has been used successfully. It exists several numerical inversion techniques from Laplace to time domains if no solution can be easily obtained:

de Hoog, Stehfest, Fourier.

As mentioned in Section 4.1, this experimentation was not only based on the use of an existing and validated technique, but it was an opportunity to improve and modify it for the necessity of the study. For this purpose, sensitivity studies are powerful tools, they allow:

• to observe that heat exchange coefficient is decorrelated from both thermal and radiative conductivities due the presence of the rear face brass plate;

• to decorrelate thermal conductivity and radiative conductivity with the removal of the first brass plate from [Jannot 2009];

• to define an objective criterion for definition of the thickness of the brass plate.

However, numerous experiments were necessary to obtain satisfying and reliable results. Many times, both thermal and radiative conductivities were correlated, and no sensitivity to the thermal diffusivity was observed. The great difficulty to control the thickness of the sample is a key issue due to its flexibility. The contact resistance between it and the brass plate that has been assumed first negligible should be reconsidered for thin sample. It is thus better to have thicker sample.

Finally, measurement with thermocouples placed on the sample is not appropriate, since their contact can cause a risk of deterioration of the sample. That is why they are placed on the heating element and on the brass plate, implying possible problem of thermal inertia. Measurement without contact can thus be a good alternative with the use of thermography, but it implies reconfiguration of the experiment -and a non-negligible increase of its financial cost. This type of visualization is presented in the next section with the use of infrared thermography in order to estimate in-plane thermal diffusivity of materials ranged between insulators to conductors. So far, experiments were dedicated to in-depth properties.

𝑇 ̅ (𝑧, 𝑡) = 𝑇 ̅ (𝑧 1 , 𝑡) ⊗ ℒ -1 [𝒯 1 (𝑧, 𝑠)] + 𝑇 ̅ (𝑧 2 , 𝑡) ⊗ ℒ -1 [𝒯 2 (𝑧,

The cylindrical model

The quadrupole formalism [Maillet 2000] expresses temperature along the rdirection between two coordinates r1 and r (Figure 4.24).

where I0, I1, K0, and K1 are the modified Bessel functions.

In time domain, temperature at radius r is linearly related to temperature at radius r1 as: [Carré 1990][Speight 2005]:

𝑇(𝑟, 𝑡) = 𝑇(𝑟 1 , 𝑡)⨂ℒ -1 [𝒯(𝑟,
• Polycarbonate: k = 0.200 W•m -1 •K -1 ,  = 1 000 kg•m -3 , 𝑐 𝑝 𝑚 = 1 250 J•kg -1 •K -1 ; • Balsa: k = 0.05 W•m -1 •K -1 ,  = 100 kg•m -3 , 𝑐 𝑝 𝑚 = 1 400 J•kg -1 •K -1 .
and for three types of surrounding conditions (vacuum, natural, and forced convection),

with the following values for global heat transfer coefficient:

• Vacuum: h = 2 W•m -2 •K -1 (see Eq. (4.68)); • Natural convection: h = 10 W•m -2 •K -1 ; • Forced convection: h = 40 W•m -2 •K -1 .
For natural and forced convection, typical values for air are chosen based on [Incropera 2011]. For natural convection, correlation for laminar flow along a vertical plate of 20 mm long at temperature surface of 313 K and surrounding air at 293 K is used given the experimental set-up. For forced convection, correlation for flat plate in parallel laminar flow is used assuming gas velocity of around 2 m•s -1 .

Results are presented in Figures 4.25 Reduced sensitivities are presented in the case of balsa and aluminum (k = 40 In this section, correlation coefficient 𝜎 𝛼𝐻 2 [START_REF] Beck | [END_REF]] has been studied to see correlation level between both parameters. "Parameters tend not to be correlated when 𝜎 𝛼𝐻 2 tends to 0, and on the contrary become more and more correlated when 𝜎 𝛼𝐻 2 tends to ± 1" [Rigollet 2019]. This parameter is studied versus two positions r1 and r, which are not as random as expected. Here is presented the case of choice of position r, and more precisely distance r -r1. In Quality of the estimation is also discussed through calculation of covariance 𝜎 𝛼𝐻 from covariance matrix (4.47) and of residuals. Convergence is not achieved for the four last combinations, due to null time derivative of the temperature. The six first cases are successful, covariance between thermal diffusivity and heat exchange coefficient is close to zero, which means that both parameters can be estimated separately. As experiment is performed in vacuum, valuable estimation of heat exchange seems to be for combinations 2 and 3, where both covariances are weak. In vacuum, theoretical linearized heat exchange coefficient is evaluated at h = 3.3 W•m -2 •K -1 using the following expression:

W•m -1 •K -1 ,  = 2 700 kg•m -3 , 𝑐 𝑝 𝑚 = 860 J•kg -1 •K -1 [Geslain 2018]) in Figures 4.27
ℎ = 4𝜀𝜎𝑇 3 (4.68)
where here T is in kelvin, 𝜀 is the emissivity of balsa. Thus, thermal diffusivity is estimated considering results from combinations 2 and 3. The present methodology is applied to five samples presented in Table 4.4 for three kinds of environment conditions in the chamber. Estimation of thermal diffusivity (Figure 4.38) of polycarbonate leads to an average value of same order of magnitude but higher than literature of 28 % [Speight 2005]. Estimation of thermal diffusivities of balsa shows a ratio of about 2.2 between parallel and orthogonal samples, which is consistent with results found in literature [Carré 1990][Speight 2005]. Same observation can be done for hemp shives, which values of thermal conductivity are also consistent with literature for cellulosic materials [Carré 1990] and less than 25 % higher compared to estimated values of [Dartois 2017]. Most of the time, covariance matrix tends to a null value, which makes considering that both parameters are uncorrelated and have been estimated properly. The only cases where covariance matrix is close to unity is for parallel balsa during measurements in the vacuum and with natural convection. It is probably because z-positions were not wellchosen; change them would help to get a better covariance and a stronger estimation. In For the second study, every infrared scene at each time step are recorded and then analyzed with the help of a Matlab ® program. As shown in Figure 4.42, the position r = 0 is the center of the copper spot and is defined through the position (x0, y0) in the infrared scene. Therefore, it is easy to know location of temperature T(r,t) for some given r radius and  angle, in (x, y) coordinate system, with the following relations:

Contrary to previous study which is valid for average temperatures, this one deals with temperatures of unique pixel, implying possible weak signal-to-noise ratio.

Conversion from pixel to distance is made possible thanks to the resolution (15 µm).

With PVC sample, for two given r1 and r positions, and for different angles ranged between 0 and /2 according to Eqs. The methodology is applied to some samples presented in Table 4.4, and Table 4.6 gathers all estimation results for the five samples. gives a consistent value for free convection [Incropera 2011].

For DC54 mild steel, and AA5182 aluminum alloy, in-plane thermal diffusivity estimation is consistent with literature. However, dispersion is important (particularly for aluminum alloy). Because of the high in-plane thermal diffusivity of metallic samples and the limited power supply, temperature increase could not be above 3 K for aluminum alloy (instead of 10 K). On the one side, this makes the choice of the position r1 more difficult. On the other side, modified Biot number H cannot be estimated, since observable is insensitive to it. Tests with aluminum show one limitation of the techniques considering the current size of the sample. A solution is to work with larger sample and supply a higher electric power to reach 10 K and have higher signal-to-noise ratio. Finally, balsa is tested. In Figure 4.46, temperature distribution is not homogeneous highlighting anisotropic structure of the material. Here, heat flux tends to flow preferentially along natural orientation of fibers. Except some points, all results are centered on a single value with a rather important standard deviation. The mean values have the same order of magnitude than literature data [Pierre 2019]. Nevertheless, it is slightly lower in longitudinal direction and higher in transverse direction. Therefore, thermal diffusivity ratio between both directions is only 1.3, compared to the ratio of 2.2 measured by guarded hot-plate technique [Carré 1990]. On the other hand, modified Biot number H ratio is equal to 2.5 assuming identical heat exchange coefficient in both directions.

Choice of r1 position is experimental and can vary from one sample to another. Of course, it must be placed after r0, meaning beyond the radius of the copper spot, but not too close to avoid some potential effects of contact resistance between the copper spot and the sample.

Heat exchange coefficient h is consistent with free convection values considering estimated values of H and thermal conductivity k of samples found in literature. Besides, real Biot number (he/2k) is calculated to verify the fin effect assumption. A value of h = 10 W•m -2 •K -1 is considered for aluminum and mild steel. Except for measurements with balsa with 𝜃 = 0°, where Bi = 0.09 ± 0.06, meaning close to the limit value of 0.1, the fin effect assumption is always verified.

Partial conclusion

Theoretical models proposed in these three publications allowed to estimate by inverse methods in-plane thermal diffusivity of different materials with the help this time of non-contact infrared thermography. Experimentally, infrared camera measured all temperatures necessary for the theoretical direct temperature-temperature model, which remains very convenient, since the unwanted parameters are reduced to one coefficient.

This coefficient is assimilated to a Biot number, since it depends on thermal conductivity, on the heat losses due to the fin effect, and on dimensions of the material.

Sensitivity study showed that influence of the Biot number decreases for weak heat losses (e. g. vacuum) and for conductive materials (e.g. aluminum). And once again, coefficient correlation matrix permitted to:

• discuss the choice of temperatures used as input data of observable;

• discuss quality of the estimation;

• give more precise information about minimum estimation time.

Generally, these experiments allowed to characterize samples of millimetric dimensions and even more, and also of insulating to conductive materials. It is required to consider a maximum temperature increase of 10 K, in order to assume temperatureindependent of thermal properties of the sample.

Characterization becomes more difficult with the decrease of conductivity, due to non-linearity of the heat losses. Measurements in vacuum is an appropriate substitution environment for insulating materials. On the contrary, characterization of conductive materials of small dimensions becomes difficult due to the quick dissipation of heat.

Samples of greater dimensions are required. examples of high temperatures with "high-temperature" mentioned in the title of the article: 2 000 °C [START_REF] Khalid | [END_REF]], 300 °C [Ikeuchi 1998], 200 °C [Smith 2000], and -173 °C [START_REF] Gömöry ; Gömöry | Characterization of High-Temperature Superconductors by AC Susceptibility Measurements[END_REF]]. I would suggest that the temperature of a surface is high as long as the surrounding can be ignored. Finally, the qualifying adjective multispectral is easier to understand, meaning at several wavelengths.

In the case where the temperature of the investigated surface is higher than the temperature of the surrounding, the theoretical radiative flux Φ 𝑖 𝑡ℎ received by a multispectral pyrometer is:

Φ 𝑖 𝑡ℎ = 𝐻 𝑖 ∫ 𝑓 𝑖 (𝜆)𝜀(𝜆, 𝜃, 𝜑, 𝑇)𝐼 0 (𝜆, 𝑇)𝑑𝜆 Δ𝜆 (5.1) where 𝑓 𝑖 (𝜆) is the spectral transfer function around the i th wavelength given by manufacturer data, 𝐻 𝑖 is an amplitude correction obtained after calibration with a blackbody (this point is discussed later), 𝜀(𝜆, 𝑇) is the emissivity depending on direction (𝜃, 𝜑), wavelength (𝜆), and temperature (T), 𝐼 0 is the intensity given by Planck's law (5.2)

or Wien's approximation (5.3) [Ozisik 1973][Sparrow 1978[START_REF] Siegel | [END_REF][START_REF] Modest | Radiative heat transfer[END_REF]].

The transfer function 𝐻 𝑖 (𝜆) is estimated through calibration with a blackbody, it considers spectral and geometrical aspects (i.e. solid angle, transmittance of the optics given by manufacturers) of radiative flux between the surface to the pyrometer. Since the measurement is never strictly monochromatic, the flux expressed in watt is obtained by integration in the narrow spectral domain Δ𝜆 around the i th wavelength. During experiments when the position of the pyrometer is fixed, in Eq. ( 5.4), the emissivity is independent of direction for every radiative flux measured at each wavelength. But this property depends on wavelength and temperature.

Therefore, in the case of a pyrometer able to separate the received flux in N separated quasi-monochromatic fluxes, for a given surface at temperature T, the system to be solved is the following [Pierre 2022]:

Φ 1 𝑡ℎ ≈ 𝐹 1 𝜀(𝜆 1 , 𝜃, 𝜑, 𝑇) 𝐼 0 (𝜆 1 , 𝑇)

(5.7) ⋮ Φ 𝑁 𝑡ℎ ≈ 𝐹 𝑁 𝜀(𝜆 𝑁 , 𝜃, 𝜑, 𝑇) 𝐼 0 (𝜆 𝑁 , 𝑇)

where there is systematically N equations and N + 1 unknowns: N radiative properties of emission and the temperature. The system is thus ill-posed since underdetermined, whatever the temperature of the investigated surface is constant or variable.

Simultaneous estimation of temperature and emissivity is now a centenary problem [Campbell 1925] and publications dealing with this are numerous, but there is no absolute methodology unanimously accepted. A common way to circumvent this underestimation problem is the use of a spectral emissivity model defined with a restricted number of 𝑁 ̃ -1 coefficients. Therefore, the minimum number of wavelengths is 𝑁 ̃. Model describing emissivity is unknown a priori, so it should be a pragmatic choice considering the nature of the targeted material and the working spectral range of the pyrometer. Besides the problem of emissivity model, the contradictory debates concern two points: i) the number of wavelengths; ii) and their position, that is discussed in Section 5.2. Introducing photon energy Eq. ( 5.8), a monochromatic photonic flux density expression can be obtained Eq. ( 5.9). The sensor being able to investigate surface with emissivity, the emitted photonic flux is given by Eq. ( 5.10).

𝐸 𝜆 = ℎ 𝑃 𝑐 𝜆 (5.8)

𝓅 𝜆 0 = 𝐼 0 (𝜆, 𝑇) 𝐸 𝜆 (5.9)

𝒫 𝑖 𝑡ℎ ≈ 𝐹 𝑖 𝜀(𝜆 𝑖 , 𝜃, 𝜑, 𝑇)𝓅 𝜆 0 (𝜆 𝑖 , 𝑇) (5.10)

The use of photonic flux by spontaneous emission of the surface is not as widespread as radiative flux (the stimulated photonic emission is not concerned here).

This is an alternative solution when the signal-to-noise ratio is too weak during radiative fluxes measurements. This generally concerns UV-visible range and therefore implies mostly applications where the temperatures exceed 1 000 °C [Hervé 1988][Hervé 1991][Ohji 1995[START_REF] Hervé | [END_REF][START_REF] Pierre | [END_REF]]. There are two advantageous aspects by working with shorter wavelengths: i) there is a high sensitivity on the left increasing part of Planck's law; ii) emissivity tends to show less variation. The detector is assumed perfect with a quantum efficiency of 100 %. • A six-wavelength pyrometer developed in laboratory: [0.48 µm -1.55 µm];

• A high-speed camera Phantom VEO-E 310L: [0.4 µm -1.0 µm];

• A narrow-band pyrometer Lumasense IS6: [0.7 µm -1.1 µm];

• A bichromatic pyrometer Lumasense ISR6: 0.90 µm and 1.05 µm;

• A bichromatic pyrometer Optris CTratio: 1.45 µm and 1.75 µm.

The six-wavelength pyrometer has been developed for simultaneous temperature (5.11) has been proposed by [Bauer 1971] and [Hartmann 2009]; Eq. ( 5.12) is taken from [Pierre 2007] -demonstration of this relation was given as an exercise proposed to In Eqs. (5.11) and (5.12), 𝜀 𝑤 is the wall emissivity, meaning emissivity of the inner of the blackbody. In both cases, using aforementioned dimensions of the blackbody, apparent emissivity is 0.996 and 0.974, respectively, with Eq. ( 5.11) and Eq. (5.12).

First it is necessary to detail the components of the pyrometer. According to Figure 5.12, it is composed of optical lenses (of transmittivity 𝜏 𝑙 ), collimators (𝜏 𝑐 ), optical fibre of the blackbody is 𝜀 𝑎 (5.19), (5.11), and (5.12). Figure (5.15) presents the evolution of the six radiative fluxes measured by the six-wavelength pyrometer and the temperature given by both commercial pyrometers, which give the same temperatures 𝑇 𝑒𝑥𝑝 (5.20).

The recording was only possible during the cooling stage of the blackbody due to the perturbation of the laser during the heating period. Indeed, the wavelength of the laser of 1.070 µm disturbs all the pyrometers when it is switched on. Evolutions of Fi do not present an immediate constant value. When the laser is turned off, it takes about 2 s to have the blackbody that behaves like a lumped body.

In Figures 5.16 (5.21).

Figure 5.17. Estimation of F2 over time with Eq.

(5.21). (5.21). (5.21).

Finally, considering the last stable values of the estimated Fi coefficients, statistical treatment is operated to calculate average value and standard deviation of the Fi. Figures 5.22 to 5.27 and Table 5.1 gather final results of the six transfer functions. The characteristic wavelength 𝜆 𝑖 of every flux determined through Eq. ( 5.6) and Eqs. (5.13)- (5.18) are also presented in Table 5.1. Intensity 𝐼 0 (𝑇) expressed in Eq. ( 5.23) is calculated from Eq. ( 5.24) considering

Planck's laws 𝐼 𝜆 0 (𝑇) and 𝐼 𝜆 0 0 (𝑇), where 𝜆 0 is the characteristic wavelength of the camera in its working spectral range Δ𝜆 and 𝑐 0 a coefficient. The combination of Eqs. (5.22)- (5.24) allows to express explicitly the temperature:

𝑇 = ℎ 𝑐 𝑘 𝜆 0 𝑙𝑛 [ 𝑐 0 𝑘 𝜀 𝜏 10 -𝐷 𝑆 𝐺𝑆 -𝑑 + 1] -1
(5.25)

Parameters 𝑐 0 and 𝜆 0 are first determined either using Wien's approximation by linearization (5.26) or Planck's law by minimization (5.27) considering a fixed temperature range. The choice of the temperature modifies the values of 𝑐 0 and 𝜆 0 when using Wien's approximation, but not with Planck's law due to the product T that increases. For example, in Figure 5.28, evolution of both parameters is plotted considering a fixed minimal temperature of 1 000 K and a changing maximal temperature between 1 500 K and 2 500 K, which represents approximately the temperature range of the experiments performed and presented in Section 6. It has been said in Section 5.1 that the system to solve in multispectral pyrometry is underdetermined since there is always N + 1 unknown and N equations (5.7). This problem disappears when an emissivity model is chosen: the minimum number of wavelengths corresponds to the number of parameters of the emissivity model plus the temperature. For example, for a first order polynomial emissivity model, a minimum of three wavelengths is required.

Manufacturers propose monochromatic, bispectral, monoband, multispectral, hyperspectral pyrometers… The selection of the pyrometer is sometimes difficult, it depends on the measured temperature range. It is also possible to develop a home-made pyrometer, as it was the case in my PhD [Pierre 2007] and during the PhD of L. Dejaeghere [Dejaeghere 2016]. In this last case, the choice of the wavelengths has been done

regarding the following criteria:

• temperature uncertainty;

• fluxes ratio.

Let express the simplest case of bispectral pyrometry assuming Wien's approximation, the temperature and its uncertainty are explicitly given by:

𝑇 = 𝐶 2 (𝜆 2 -1 -𝜆 1 -1 ) 𝑙𝑛 [ Φ 1 𝑒𝑥𝑝 Φ 2 𝑒𝑥𝑝 𝐹 2 𝐹 1 𝜀 𝜆 2 𝜀 𝜆 1 ( 𝜆 2 𝜆 1 ) -5 ]
(5.28)

𝑒 𝑇 𝑇 = 𝑒 𝜆 1 𝜆 1 |5 - 𝐶 2 𝜆 1 𝑇 - 𝜆 1 𝜀 𝑑𝜀 𝑑𝜆 | + 𝑒 𝜆 2 𝜆 2 |5 - 𝐶 2 𝜆 2 𝑇 - 𝜆 2 𝜀 𝑑𝜀 𝑑𝜆 | + 𝑒 Φ 1 𝑒𝑥𝑝 Φ 1 𝑒𝑥𝑝 + 𝑒 Φ 2 𝑒𝑥𝑝 Φ 2 𝑒𝑥𝑝 𝐶 2 𝑇 | 1 𝜆 1 - 1 𝜆 2 | (5.29) 𝜀 𝜆 2 = 𝜀 𝜆 1 + 𝑑𝜀 𝑑𝜆 (𝜆 2 -𝜆 1 ) (5.30)
Eq. ( 5.30) is first order Taylor's series on emissivity at two positions 𝜆 1 and 𝜆 2 , and 𝐶 2 = ℎ𝑐/𝑘 𝐵 . The principle of bispectral pyrometry is to assume a known emissivity ratio.

This ratio has a higher probability to be equal to one if wavelengths are getting close to each other. However, experimental radiative fluxes have also a higher probability to become equivalent considering the noise of the signals. Moreover, the relative uncertainty given by Eq. ( 5.29) tends to infinity if 𝜆 2 gets closer to 𝜆 1 . Therefore, in order not to penalize the temperature uncertainty in Eq. ( 5.30), the following criterion has been imposed [START_REF] Pierre | [END_REF]]:

𝐶 2 𝑇 | 1 𝜆 1 - 1 𝜆 2 | > 1 (5.31) Δ𝜆 > 𝑇𝜆 1 𝜆 2 𝐶 2 (5.32)
The criterion (5.32) proposes a minimum distance between two consecutive wavelengths. If this criterion is respected between 𝜆 1 and 𝜆 2 , for multispectral pyrometry, it is also respected between 𝜆 1 and 𝜆 3 ... However, between 𝜆 1 and 𝜆 3 , the emissivity ratio is likely to be different. Figure 5.30 presents example of evolution of Δ𝜆 versus 𝜆 1 and for different temperatures.

The second criterion presented is based on the radiative fluxes ratio 𝛼 in the system (5.33)-( 5.36) [START_REF] Pierre | [END_REF]]. Still starting from the bispectral case assuming

Wien's approximation, and the transfer functions and emissivity ratios equal to one in Eq.

(5.28), let x be the wavelength ratio:

𝑥 * 𝑒 𝑥 * = 𝛾(𝜆 1 ) (5.33) 𝛾(𝜆 1 ) = - 𝐶 2 5𝜆 1 𝑇 1 𝛼 1/5 𝑒 - 𝐶 2 5𝜆 1 𝑇 (5.34) 𝑥 * = - 𝐶 2 5𝜆 1 𝑇 𝑥 (5.35) 𝑥 -5 𝑒 - 𝐶 2 𝜆 1 𝑇 (1-𝑥) = 𝛼
(5.36) For a given temperature and the first wavelength 𝜆 1 , Figure 5.31 plots the evolution of the wavelength ratio versus the signal ratio 𝛼. Signals are noiseless, but it would be interesting to include this specificity since it is of particular importance for a signal ratio close to unity. Note as a verification that for 𝛼 = 1, 𝜆 1 /𝜆 2 = 1 too.

As an example, Table 5.2 presents a sum up of numerical results in order to check the criteria of Eq. ( 5.32) considering the six-wavelength pyrometer [START_REF] Dejaeghere | [END_REF]].

Only five wavelengths are here considered: 𝜆 1 to 𝜆 5 , and the wavelength of interest is 𝜆 4 .

For three temperatures T, we can observe that the distance criterion Δ𝜆 is respected between wavelength 𝜆 4 and the other ones except between 𝜆 4 and 𝜆 5 , which is too weak implying an amplification of the temperature uncertainty (5.29) with a denominator inferior to 1. The second point is the fluxes ratio 𝛼, which is generally quite high, meaning greater than 2, most of the time, except for 𝛼 43 and 𝛼 54 , which are both close to 1 for a temperature of 3 273 K. At this temperature, the wavelength of maximal emission is 0.885 µm. As these three wavelengths are around the maximum of Planck's law, the signal variations are very low. Therefore, assuming that 𝜀 𝜆3 /𝜀 𝜆4 and 𝜀 𝜆4 /𝜀 𝜆5 are known, bichromatic estimation is not recommended with ratios 𝛼 43 and 𝛼 54 for T = 3 273 K.

Moreover, it would be not necessary to use 𝜆 4 and 𝜆 5 in multispectral pyrometry at T = 3 273 K as long as the emissivity model is linear. Furthermore, one should be careful that a system does not become underdetermined with less wavelengths than parameters to estimate (the emissivity coefficients and the temperature).

Multispectral wavelength selection

From those two previous criteria, the most subjective one remains the fluxes ratio, From a given temperature, the first wavelength is estimated considering 𝛃 = 𝑇 thanks to Eq. ( 5.41) reduced to a single value 𝜎 𝑇 2 by minimising Eq. (5.40). Considering one single radiative flux Φ 1 and T = 2 273 K, Figure 5.32 plots its evolution versus the wavelength (5.37), meaning Wien's approximation, and its derivative over temperature, 𝜕Φ 1 /𝜕𝑇, which maximum has the same wavelength as the minimum of 𝜎 𝑇 = 𝑓(𝜆). Maximum of Wien's approximation is for 𝜆 𝑚 𝑇 = 𝐶 2 /5, with 𝜆 𝑚 the wavelength of maximal emission, here equals to 1.266 µm for T = 2 273 K. Maximum of the derivative corresponds to the wavelength 𝜆 1𝑜𝑝𝑡 = 1.055 µm, which is the first selected wavelength.

Φ 𝑡ℎ = 𝜀(𝜆)𝐶 1 𝜆 -5 𝑒 - 𝐶 2 𝜆𝑇
(5.37)

𝜀(𝜆) = ∑ 𝑎 𝑖-2 𝜆 𝑖-2 𝑁 𝑖=2 (5.38) 𝛃 = [𝑇, 𝑎 0 … , 𝑎 𝑁-1 ] 𝑇 (5.39) 𝐽 = ∑|Φ 𝑖 𝑡ℎ (𝛃) -Φ 𝑖 𝑒𝑥𝑝 | 2 𝑁 𝑖=1 (5.40) cov(𝛽) = [ 𝜎 𝑇 2 cov(𝑎 0 , 𝑇 ) cov(𝑇, 𝑎 0 ) 𝜎 𝑎 0 2 … cov(𝑇, 𝑎 𝑛 ) ⋱ cov(𝑎 0 , 𝑎 𝑛 ) ⋮ ⋮ ⋱ ⋮ cov(𝑎 𝑛 , 𝑇 ) cov(𝑎 𝑛 , 𝑎 0 ) … 𝜎 𝑎 𝑛 2 ] = (𝑋 𝑇 𝑋) -1 𝜎 𝑛𝑜𝑖𝑠𝑒 2 (5.41) 𝑋 = [ 𝜕Φ 1 𝜕𝑇 𝜕Φ 2 𝜕𝑇 𝜕Φ 1 𝜕𝑎 0 𝜕Φ 2 𝜕𝑎 0 … 𝜕Φ 1 𝜕𝑎 𝑛 … 𝜕Φ 1 𝜕𝑎 𝑛 ⋮ ⋮ ⋮ ⋮ 𝜕Φ 𝑛 𝜕𝑇 𝜕Φ 𝑛 𝜕𝑎 0 … 𝜕Φ 𝑛 𝜕𝑎 𝑛 ]
(5.42)

The selection of 𝜆 2𝑜𝑝𝑡 , once 𝜆 1𝑜𝑝𝑡 is known, considers the covariance matrix with a dimension 2 × 2, Eq. ( 5.38) with N = 2 and 𝛃 = [𝑇 𝑎 0 ] 𝑇 .

Partial conclusion

We presented in Section 5.2 methodologies to select wavelengths for a given temperature assuming a faithful emissivity model. Some of these procedures of selection have the advantage to be based on objective criteria and have the possibility to consider technical and experimental limitations, such as the spectral range of the detector.

The techniques of selection are relevant with the condition that the temperature of the investigation surface does not vary a lot during application. Indeed, the use of pyrometer with chosen fixed wavelengths is limitative and would become inefficient when the signal to noise ratio is too low.

The selection of the wavelengths is based on the choice of a characteristic temperature, which must be an average if the temperature is not constant along the experiment. Therefore, it is important to have a good prior of the temperature range of the application. Consequently, if it becomes too large, selected wavelengths would not be that optimal and the temperature uncertainty would increase.

Working with a reduced number of wavelengths has the advantage to have a short calculation time compared with measurements with a greater number of wavelengths, but it becomes limited when the temperature range becomes too large. The next section is dedicated to different techniques I experimented for the temperature emissivity estimation based on ordinary least squares or Bayesian inference. Experiments have been performed with the six-wavelength pyrometer. Whatever the relation used to express the theoretical flux, Eqs. ( 5.2) or ( 5.3), the signal is experimentally contaminated by an additive random noise ei. Thus, let's define the observable Yi as following:

𝑌 𝑖 = 𝜀 𝑖 (𝑇)Φ 𝑖 0,𝑡ℎ (𝑇) + 𝑒 𝑖 (5.43) where Φ 𝑖 0,𝑡ℎ (𝑇) is the blackbody radiative flux. Moreover, Section 5.3.4 presents temperature emissivity estimation through Bayesian technique using linear approximation. Therefore, assuming Wien's approximation (5.3) and introducing mean values 𝜆 𝑖 , 𝜀 𝑖 and Fi for the wavelength, the emissivity, and the transfer function (see Section 5.1.1), respectively, in each spectral band i, the flux in the band i can thus be simplified to:

Φ 𝑖 𝑡ℎ (𝑇) = 𝐹 𝑖 𝜀 𝑖 𝐶 1 𝜆 𝑖 -5 𝑒 - 𝐶 2 𝜆 𝑖 𝑇
(5.44)

Eq. ( 5.44) can be linearized by taking the logarithm function. The proposed method here implies to use experimental flux 𝑌 𝑖 according to Planck's law. In order not to amplify the standard deviation on the estimated temperature uncertainties are supposed Gaussian with known zero-mean and a covariance matrix, additive, and independent from the unknown parameters. Thus, the likelihood function can be written:

𝜋(𝐘|𝛃) ∝ 𝑒𝑥𝑝 {- 1 2 [𝐘 -𝛆⨂𝚽 𝟎,𝒕𝒉 (𝑇)] 𝑇 𝛀 -1 [𝐘 -𝛆⨂𝚽 𝟎,𝒕𝒉 (𝑇)]} (5.51)
where 𝛀 is the measurement error covariance matrix and ⨂ denotes the element-wise product. Regarding the density of an a priori distribution of the parameters, 𝛀 is assumed independent and given by Eq. ( 5.52), where 𝜋(𝛆) and 𝜋(T) are the probability densities of the emissivity and the temperature, respectively:

𝜋(𝛃) = 𝜋(𝛆)𝜋(𝑇) (5.52)

Multispectral pyrometry in linear approximation

In the case of the linear approximation, Eq. ( 5.45) can be rewritten as: 5.55) where 𝐈 𝐦𝐦 is the identity matrix of size m × m. Thus, Eq. ( 5.51) becomes, considering Eq.

𝐘 = 𝐗𝛃 + 𝐞′ (5.53) 𝐗 = [𝐈 𝐦𝐦 -𝛍 𝒎𝟏 ] (5.54) 𝛍 = [𝜇 1 𝜇 2 … 𝜇 𝑚 ] 𝑇 (
(5.52) and the linear hypothesis:

𝜋(𝛃|𝐘) ∝ 𝑒𝑥𝑝 {- 1 2 [(𝐘 -𝐗𝛃) 𝑇 𝛀 -1 (𝐘 -𝐗𝛃) + (𝛃 -𝛃 𝒑𝒓𝒊𝒐𝒓 ) 𝑇 𝐖 -1 (𝛃 -𝛃 𝒑𝒓𝒊𝒐𝒓 )]} (5.56)
where 𝐖 is the error matrix of the parameters. The maximum a posteriori (MAP)

estimator is then obtained when the derivative of (5.56) is null with respect to the parameter vector. Introducing the MAP estimator and the posterior covariance matrix, Eq. ( 5.56) becomes:

𝜋(𝛃|𝐘) ∝ {- 1 2 [(𝛃 -𝛃 ̂𝑀𝐴𝑃 ) 𝑇 𝚪 𝛃|𝐘 -𝟏 (𝛃 -𝛃 ̂𝑀𝐴𝑃 )]} (5.57)
In relation (5.57), 𝛽 ̂𝑀𝐴𝑃 is the MAP estimator (5.58) and 𝚪 𝛃|𝐘 is the posterior covariance matrix (5.59). They are given by [Krapez 2019]:

𝛃 ̂𝑀𝐴𝑃 = 𝛃 𝐩𝐫𝐢𝐨𝐫 + 𝐖𝐗 𝐓 (𝐗𝐖𝐗 𝐓 + 𝛀) -𝟏 (𝐘 -𝐗𝛃 𝐩𝐫𝐢𝐨𝐫 ) (5.58)

𝚪 𝛃|𝐘 = 𝐖 -𝐖𝐗 𝐓 (𝐗𝐖𝐗 𝐓 + 𝛀) -1 𝐗𝐖 (5.59)

Multispectral pyrometry in non-linear estimation

In the non-linear case, the theoretical fluxes are calculated from Eqs. ( 5.2) and

(5.4), that is with Planck's law. The 𝛃 vector of the parameter is the same as in Table 5.3.

A sampling method based on the Markov Chain Monte Carlo (MCMC) method is used in this work. The Metropolis-Hastings algorithm was applied to generate samples of the posterior distribution based on the likelihood given by Eq. (5.51). This algorithm begins with the selection of a proposal distribution 𝑝(𝛃 * , 𝛃 𝑗-1 ), which is used to define a new candidate *, given the current state 𝛃 𝑗-1 of the Markov Chain. Once the proposal distribution is defined, the Metropolis-Hastings sampling algorithm can be implemented by repeating the following steps:

1-Sample a candidate * from the proposal distribution 𝑝(𝛃 * , 𝛃 𝑗-1 ).

2-Calculate the acceptance:

𝛼 = 𝑚𝑖𝑛 [1, 𝜋(𝛃 * |𝐘), 𝑝(𝛃 * , 𝛃 𝐣-𝟏 ) 𝜋(𝛃 𝑗-1 |𝐘), 𝑝(𝛃 𝑗-1 , 𝛃) ] (5.60)
3-Generate a random value U uniformly distributed between 0 and 1.

4-If U ≤ 𝛼, set 𝛃 𝑗 = 𝛃 * ; otherwise, set 𝛃 𝑗 = 𝛃 𝑗-1 .

5-Return to step 1 to generate the sequence {𝛃 1 , 𝛃 2 … , 𝛃 𝑛 }.

Temperature emissivity estimation with multispectral pyrometry 115

In this manner, a sequence is generated to represent the posterior distribution, and inference on this posterior distribution is obtained from inference on the samples {𝛽 1 , 𝛽 2 … , 𝛽 𝑛 }. Note that the values 𝛽 𝑖 must be rejected as long as the chain has not converged.

Multispectral pyrometry using particle filters algorithm

A state estimation problem is solved, that is more consistent with the sequential estimation of the dynamic (time-dependent) vector 𝛃 𝑘 , k = 1,… M, being the index of the time step. State estimation problems may be written in the form of evolution and observation models, which are modelled as stochastic processes [START_REF] Arulampalam | A tutorial on particle filters for on-line non-linear/non-gaussian Bayesian tracking[END_REF]][Ristic 2004][Kaipio 2006[START_REF] Ozisik | [END_REF]]. The state variables are given by the vector 𝛃 𝑘 (see Table 5.3). The observed quantities, that is, the available measurements at each time tk, are the fluxes from each photodetector arranged in the following vector:

𝚽 𝑘 𝑒𝑥𝑝 = [Φ 𝑘,1 𝑒𝑥𝑝 , … , Φ 𝑘,𝑁 𝑒𝑥𝑝 ]
𝑇 . We further assume that the state evolution model and the observation model are given by the general functions 𝐟 𝑘 and 𝐠 𝑘 , respectively, that is:

𝛃 𝑘 = 𝐟 𝑘 (𝛃 𝑘-1 , 𝐯 𝑘-1 ) for 𝑘 = 1, … , 𝑀 (5.61)
𝚽 𝑘 = 𝐠 𝑘 (𝛃 𝑘 , 𝐧 𝑘 ) for 𝑘 = 1, … , 𝑀 (5.62)
where 𝚽 𝑘 is the model prediction of the measurements 𝚽 𝑘 𝑒𝑥𝑝 , while 𝐯 and 𝐧 represent the noises in the state evolution model and in the observation model, respectively.

Although different kinds of problems can be considered with the evolutionobservation models described above [START_REF] Arulampalam | A tutorial on particle filters for on-line non-linear/non-gaussian Bayesian tracking[END_REF]][Ristic 2004 In the SIR algorithm, the importance density 𝜋(𝛃 𝑘 𝑗 |𝛃 𝑘-1 𝑗 ) is chosen as the prior density that is obtained from the evolution model. Since the importance density is independent of the information conveyed by the measurements 𝚽 𝑘 𝑒𝑥𝑝 , the state space is explored without any information about the measurements at the current time. Hence, this algorithm can be inefficient and sensible to outliers [START_REF] Arulampalam | A tutorial on particle filters for on-line non-linear/non-gaussian Bayesian tracking[END_REF]][Ristic 2004][Kaipio 2006[START_REF] Ozisik | [END_REF]]. The SIR algorithm can be summarized by the steps presented in Table 5.4, as applied to the system evolution from tk-1 to tk [START_REF] Arulampalam | A tutorial on particle filters for on-line non-linear/non-gaussian Bayesian tracking[END_REF]][Ristic 2004][Kaipio 2006[START_REF] Ozisik | [END_REF]]. Step 1

For j = 1,...,Q draw new particles 𝛃 𝑘 𝑗 from the prior density 𝜋(𝛃 𝑘 𝑗 |𝛃 𝑘-1 𝑗 ) and then use the likelihood density to calculate the corresponding weights 𝑤 𝑘 𝑗 = 𝜋(𝚽 𝑘 𝑒𝑥𝑝 |𝛃 𝑘 𝑗 ) .

Step 2

Normalize the weights, that is, for j

= 1,...,Q let 𝑤 𝑘 𝑗 = 𝑤 𝑘 𝑗 ∑ 𝑤 𝑘 𝑗 𝑄 𝑗=1 ⁄ .
Step 3

Resample the particles as follows:

Construct the cumulative sum of weights (CSW) by computing 𝑐 𝑗 = 𝑐 𝑗-1 + 𝑤 𝑘 𝑗 for j = 1,...,Q with 𝑐 0 = 0.

Let j = 1 and draw a starting point 𝑢 1 from the uniform distribution 𝒰(0, 𝑄 -1 )

For j * = 1,...,Q Move along the CSW by making 𝑢 𝑗 * = 𝑢 1 + 𝑄 -1 (𝑗 * -1)

While 𝑢 𝑗 * > 𝑐 𝑗 make j = j + 1

Assign sample 𝛃 𝑘 𝑗 * = 𝛃 𝑘 𝑗 Assign uniform weights to the samples, that is, 𝑤 𝑘 𝑗 * = 𝑄 -1

The Auxiliary Sampling Importance Resampling particle filter algorithm (ASIR)

The Auxiliary Sampling Importance Resampling (ASIR) algorithm is an extension of the SIR algorithm, in which an attempt is made to overcome the loss of diversity by performing the resampling step at time tk-1 with the available measurement at time tk [START_REF] Arulampalam | A tutorial on particle filters for on-line non-linear/non-gaussian Bayesian tracking[END_REF]][Ristic 2004 ) and the ASIR algorithm may not be as effective as the SIR algorithm. The ASIR algorithm can be summarized by the steps presented in Table 5.5, as applied to the system evolution from tk-1 to tk [START_REF] Arulampalam | A tutorial on particle filters for on-line non-linear/non-gaussian Bayesian tracking[END_REF]][Ristic 2004[START_REF] Ozisik | [END_REF]]. Step 2

Normalize the weights, that is, for j = 1,...,Q let

𝑤 𝑘 𝑗 = 𝑤 𝑘 𝑗 ∑ 𝑤 𝑘 𝑗 𝑄 𝑗=1 ⁄ .
Step 3

Resample the particles as follows:

Construct the cumulative sum of weights (CSW) by computing 𝑐 𝑗 = 𝑐 𝑗-1 + 𝑤 𝑗 𝑘 for j = 1,...,Q with 𝑐 0 = 0.

Let j = 1 and draw a starting point 𝑢 1 from the uniform distribution 𝒰(0, 𝑁 -1 )

For j * = 1,...,Q Move along the CSW by making 𝑢 𝑗 * = 𝑢 1 + 𝑄 -1 (𝑗 * -1)

While 𝑢 𝑗 * > 𝑐 𝑗 make j = j + 1

Assign sample 𝛃 𝑘 𝑗 * = 𝛃 𝑘 𝑗 Assign parent 𝑖 𝑗 = 𝑗

Step 4

For j =1,..., Q draw new particles 𝛃 𝑘 𝑗 from the prior density 𝜋(𝛃 𝑘 𝑗 |𝛃 𝑘-1 𝑗 ), using the parent 𝑖 𝑗 , and use the likelihood density to calculate the corresponding weights

𝑤 𝑘 𝑗 = (𝚽 𝑘 𝑒𝑥𝑝 |𝛃 𝑘 𝑗 )/(𝚽 𝑘 𝑗 |𝛃 k 𝑗 )
Step 5

Normalize the weights, that is, for j

= 1,...,Q let 𝑤 𝑘 𝑗 = 𝑤 𝑘 𝑗 ∑ 𝑤 𝑘 𝑗 𝑄 𝑗=1
⁄ .

Implementation of the particle filters algorithm

For the implementation of the particle filter algorithms described above, the observation model is given by: Φ 𝑘,𝑖 𝑡ℎ ≡ 𝐹 𝑖 𝜀(𝜆 𝑖 , 𝑇 𝑘 )𝐼 0 (𝜆 𝑖 , 𝑇 𝑘 ) + 𝑒 𝑘 𝑖 (5.64) where 𝑒 𝑘 𝑖 is the additive measurement uncertainty, which is modeled as an independent Gaussian variable, with zero mean and constant standard deviation. Based on the calibration process, the standard deviation of the measurement errors was fixed as 𝜎 = 100 nW. Thus, the likelihood function is given by:

𝜋(𝚽 𝑘 𝑒𝑥𝑝 |𝛃 𝑘 ) ∝ 𝑒𝑥𝑝 {- 1 2 ∑ [Φ 𝑘,𝑖 𝑒𝑥𝑝 -Φ 𝑘,𝑖 𝑡ℎ ] 2 𝜎 2 𝑁 𝑖=1 } (5.65)
The evolution model for the temperature is based on the calibration process of the measurement system with the blackbody (see [Pierre 2022] for details). We denote 𝜃 𝑖 (Φ 𝑏 0 ) as the function that fits the temperature with respect to the blackbody spectral flux Φ 𝑏 0 , which was measured with detector i in the calibration process (see Section 5.1.3,Figure 5.15). A first order Taylor´s series approximation for the time variation of the temperature gives:

𝑇 ̂𝑖(𝑡 𝑘 ) ≈ 𝑇 ̂𝑖(𝑡 𝑘-1 ) + 𝜕𝜃 𝑖 (𝑡 𝑘-1 ) 𝜕Φ 𝑏 0 (Φ 𝑘,𝑖 𝑒𝑥𝑝 -Φ 𝑘-1,𝑖 𝑒𝑥𝑝 ) (5.66)
where 𝑇 ̂𝑖(𝑡 𝑘 ) is the temperature at time 𝑡 𝑘 estimated with the calibration curve of detector i, that is, 𝜃 𝑖 (Φ 𝑏 0 ). Note in Eq. ( 5.66) that the time variations of the fluxes measured during the experiment were assumed equal to the time variations of the fluxes measured in the calibration process with the blackbody. The evolution model for temperature was then obtained as:

𝑇(𝑡 𝑘 ) = 𝑇 ̃(𝑡 𝑘 ) + 𝑒 𝑘,𝑇 (5.67)
where 𝑒 𝑘,𝑇 is a Gaussian random variable with zero mean and standard deviation of 0.05𝑇 ̃(𝑡 𝑘 ) and:

𝑇 ̃(𝑡 𝑘 ) = 1 𝑁 ∑ 𝑇 ̂𝑖(𝑡 𝑘 ) 𝑁 𝑖=1
(5.68) Such an evolution model for the temperature is independent of the emissivities, which were time-dependent with a random walk model for the parameters 𝑎 0 and 𝑎 1 (see Table 5.3), that is:

𝑎 0,𝑘 = 𝑎 0,𝑘-1 + 𝑒 𝑎 0 ,𝑘 (5.69)
𝑎 1,𝑘 = 𝑎 1,𝑘-1 + 𝑒 𝑎 1 ,𝑘 (5.70) where the random variables 𝑒 𝑎 0 ,𝑘 and 𝑒 𝑎 1 ,𝑘 are also Gaussian, with zero means and standard deviations of 0.01𝑎 0,𝑘-1 and 0.01𝑎 1,𝑘-1 , respectively.

Since the initial distributions of the state variables were unknown, samples were initially generated at time t = 0 from the following uniform distributions: 1 000 K ≤ 𝑇 ≤ 2 500 K, -2 μm -1 ≤ 𝑎 1 ≤ +2 μm -1 and 0 ≤ 𝑎 0 ≤ 1. The sample that resulted in the maximum likelihood at t = 0 was selected and then the particles were generated from a

Gaussian distribution centered at this sample with a relative standard deviation of 10 -2 .

Results and discussion

All the results, due to the multiple techniques presented in Table 5.3, are not presented in this section. As already mentioned in introduction of Section 5.3.4, tests have been performed on niobium and mild steel, but only results with niobium are presented.

First, Figures 5.38 As the results are not satisfying, greybody assumption should be relieved, which means that one should solve Eqs. (5.47)- (5.48) by introducing an emissivity ratio 𝜉 43 . Estimation of this parameter is only possible using a known data, such as the solidification plateau.

This estimation is based on the experimental data, obtained during the cooling period, when the bispectral commercial pyrometer measures temperature Tbis. This minimization is based on a high level of confidence in the temperature given by the commercial bispectral pyrometer Tbis (black dashed line in Figure 5.36). Indeed, the measured temperature during the solidification plateau has been estimated at (2 731 ± 12) K (i.e. 0.5 %), which is consistent with literature [Incropera 2002], and its emissivity ratio has been set to 1.025. Then the principle is to minimize the mean of the residuals (5.71) between estimated temperature Tij and Tbis during a short time interval (between 5.9 s and 6.9 s).

1 𝑁 ∑[𝑇 𝑖𝑗 (𝑡 𝑘 ) -𝑇 𝑏𝑖𝑠 (𝑡 𝑘 )] 2 𝑁 𝑘=1
(5.71)

Consequently, 43 = 1.087 and the results of emissivity and temperature obtained for this value are presented in Figure 5.38. Even if the emissivity ratio has been evaluated

during the short time interval of solidification, the obtained value has been applied to the whole duration of the experiment. Even if literature mentions a rather constant emissivity in the liquid state [Seifter 2001], this is a major assumption. 5. Temperature emissivity estimation with multispectral pyrometry 122 However, the use of constrains developed in Section 5.2.2 with Eqs. (5.37)-(5.42) allows to estimate correctly the temperature using a proper choice of wavelengths such as in Figure 5.40: 𝜆 2 = 0.530 µm, 𝜆 3 = 0.680 µm, 𝜆 5 = 0.940 µm, and 𝜆 6 = 1.055 µm.

Constant, linear, and parabolic emissivity models have also been tested and it appears that, in the spectral range of the pyrometer, the linear model gets relevant temperature due to the knowledge of solidification plateau temperature.

Finally, estimated emissivity at the wavelength 𝜆 3 = 0.680 µm is presented in 5.3) were assigned with uniform priors in the intervals [-2 , 0], and [0 , 2],

respectively. Temperature was also assigned with a uniform prior in the interval between 5. Temperature emissivity estimation with multispectral pyrometry 125 273 K and 4 000 K. Markov chains were started with parameters with the same priors as previously (of good quality for the temperature and bad quality for the emissivity). The chains were run with 30 000 states and the 15 000 first states were neglected for the computation of the posterior statistics (burn-in period). Figures 5.47 and 5.48 show the good fitting between the experimental and estimated flux and temperature, respectively.

The 2.5 % and 97.5 % quantiles have also been calculated and presented in these figures, for the verification of the estimated fluxes and temperature versus the experimental ones. The last estimation technique is the Bayesian inference with particle filters. The variation with temperature of the emittance at the wavelength of sensor 3, obtained with SIR algorithm and the measurements of detectors 2 to 6, is presented in Simultaneous estimation of temperature and emissivity can be performed through different experimental apparatuses: mono-, bispectral pyrometers, (home-made) multispectral pyrometer, or spectroradiometer (e.g. for hyperspectral pyrometry). It seems more interesting to have the maximum of information, that is using the maximum of wavelengths and thus reconstruct a part of Planck's law in the spectral range of the detector. However, the increase of wavelengths increases also the calculation time for estimation.

If it is possible, for applications where temperature varies slightly (but what does slightly mean?) the selection of a limited number of wavelengths would be sufficient, as it is the case with the six-wavelength pyrometer, or even with commercial pyrometers listed in Section 5.1.3. In the extreme case, for a given temperature, the selected wavelengths must belong to the useful spectral range, meaning between 0.5𝜆 𝑚 and 5𝜆 𝑚 , where 𝜆 𝑚 is the wavelength of maximum emission for the temperature T. The signal to noise ratio would be acceptable and the radiative measurements for each wavelength exploitable.

For applications where the temperature variation exceeds several thousands of kelvins, measurements with a spectroradiometer would be more relevant. These apparatuses are appropriate since they can cover spectral range of about several micrometres and some can make possible hyperspectral measurements: less than ten nanometres between two consecutive wavelengths. In this case, each sequence of measurement at a time tk represents a non-negligible part of Planck's law, where the observable is the radiative flux and the variable the wavelength.

In every case, calibration remains mandatory using reference temperature (e.g., blackbody, melting point).

The major issue is emissivity: what is well-known is that emissivity is not.

Behaviour of emissivity of a surface during experiment is more or less unpredictable, since it depends on many factors: wavelength, temperature, direction, surface roughness, impurities. So, literature presents researches who try to consider emissivity, or to skip it, temperature being the parameter of interest most of the time. And a bad choice for an emissivity model can lead to important uncertainty regarding the temperature estimation.

All the methodology of estimation tested and presented in Section 5.3 are confronted to the problem of emissivity model and on the need of good priors and prediction of temperature and emissivity. I suppose neural networks will be also confronted to the same problem.

The major perspective to all the studies presented in Section 5 is to investigate the behaviour of this parameter. First, the knowledge of this behaviour would lay on a cautious study of the microscopic aspect of the surface (roughness, impurities). Moreover, for anyone who works with radiative transfers, it is a necessity to read reference books dealing with them: [START_REF] Siegel | [END_REF][START_REF] Modest | Radiative heat transfer[END_REF]], mainly (unfortunately, I only recently start to read these books). [START_REF] Siegel | [END_REF][START_REF] Modest | Radiative heat transfer[END_REF]] of recent editions have several chapters dedicated to radiative properties from both theoretical and experimental points of view. For example, basics on electromagnetic theory state relations to predict radiative properties considering ideal, meaning optically smooth surface. But departures between predictive properties and measured ones exist and are caused by "surface roughness, surface contamination, impurities, and crystal-structure modification by surface

working" [START_REF] Siegel | [END_REF]]. However, trends are most of the time respected and satisfied numerous numbers of applications. During experiment for the estimation of thermal diffusivity in Section 6.1.2, the sixwavelength pyrometer would be placed next to the laser. It is not shown in Figure 6.1.

Theoretical model of the sphere in the levitation apparatus and results

The proposed methodology is based on the front-face flash method with laser excitation and contactless transient flux measurements available from a multispectral pyrometer [START_REF] Lamien | [END_REF]]. However, the parameter estimation presented at the end of this Section is based on synthetic data and not from experimental ones. Based on Figure 6.2, the mathematical formulation of the heat conduction problem in dimensionless form in spherical coordinates (r, θ, φ), with transformed polar angle µ = cos θ and associated initial and boundary conditions, is given by: 6.16) and the subscript ref denotes reference value for the material being characterized. These reference values can be taken from literature for similar materials. Sref is a reference surface, given by the area of the spherical cap observed by the pyrometer around the symmetry axis. In the formulation of Eq. ( 6.2), boundary heat source term 𝑓(𝜇, 𝜏, 𝜇 0 , 𝜏 ℎ )

𝑆 𝑟𝑒𝑓 = 2𝜋𝑅 2 (1 -𝜇 𝑠𝑝𝑜𝑡 ) (
represents the spatio-temporal profile of the laser excitation written in terms of transformed polar angle µ and of dimensionless time τ. The functional form of 𝑓(𝜇, 𝜏, 𝜇 0 , 𝜏 ℎ ) is written in terms of the product of functions 𝑔(𝜇, 𝜇 0 ) and 𝑢(𝜏, 𝜏 ℎ ),

representing the spatial and temporal profiles, respectively. These functions are given as:

𝑔(𝜇, 𝜇 0 ) = exp [-2 (1 -𝜇 2 ) (1 -𝜇 0 2 ) ] (6.17)
for the spatial distribution, while the temporal profile is either a square pulse represented by: 6.18) or an exponential decaying pulse:

𝑢(𝜏, 𝜏 ℎ ) = 1 2 {1 + erf[𝜙(𝜏 -𝜏 ↑ )]} - 1 2 {1 + erf[𝜙(𝜏 -𝜏 ↓ )]} (
𝑢(𝜏, 𝜏 ℎ ) = 𝜏 2 4𝜏 ℎ 2 exp (2 - 𝜏 𝜏 ℎ ) (6.19)
Figure 6.2. Scheme of the metallic ball.

6. Characterization of thermal properties by non-contact techniques 140

In the above equations, ϕ is a smoothing parameter, while subscripts " ↑" and " ↓" refer to the ascent and descent times of the square pulse. For solution of the complete model given by Eq. ( 6.1), finite element method was used through COMSOL Regarding the inverse problem, the objective is the estimation of thermal diffusivity of the solid metal from measured fluxes by the multispectral pyrometer. The measurements of the flux obtained at surface S1 are therefore more appropriate for the estimation of the parameters of interest, because the sensitivity coefficients of the parameters of interest present large magnitudes.

• Regarding the effect of the temporal profile of the incident laser pulse on the sensitivity coefficients, a square pulse and an exponentially decaying pulse were examined (Eqs. (6.18) and (6.19)). Figure 6.9 presents the sensitivity coefficients for the case of an exponentially decaying pulse (P = 30 W, 𝜏 ℎ = 15 ms), while the case of a square pulse is given in Figure 6.6. In these figures the case of a square pulse presents sensitivity coefficients of the same magnitude with those of the exponentially decaying pulse.

Moreover, one can note in Figure 6.9 that thermal diffusivity and thermal conductivity are strongly correlated in the case of the exponentially decaying pulse. It is also clear from these figures that thermal diffusivity and thermal conductivity tend to be uncorrelated once the heating is stopped. For the solution of inverse problem of simultaneous estimation of thermal diffusivity and thermal conductivity of the steel sphere, prior probability densities associated to thermal diffusivity and thermal conductivity given in Table 6.1 were used. The first 1 000 states of Markov chain were discarded (burn-in period) for the computation of statistics. One can note in these figures that information provided by measurements has resulted in a considerable reduction of the initial uncertainties on thermal diffusivity and thermal conductivity. Moreover, the most likely values of these parameters are concentrated around their exact values. where 𝛼 = 𝑘/(𝜌𝑐 𝑝 𝑚 ) is the thermal diffusivity, k is the thermal conductivity, 𝜌 is the density, 𝑐 𝑝 𝑚 is the specific heat, r0 is the heat flux radius, Tinit is the initial temperature, and 𝑇 ∞ is the room temperature including the gas and the enclosure temperatures, that are assumed constant regarding the large size of the enclosure, the short experiment duration (less than 30s) and a laser beam focusing only on the centre of the sample. Convective and radiative heat exchanges are assumed linear and described by the coefficients h0 and he at locations z = 0 and e, respectively.

Solution of this problem is given in Laplace domain using the quadrupole formalism [Maillet 2000]. For sake of clarity, the mathematical formulation is detailed in Appendix A of [START_REF] Houssein | [END_REF]]. The expressions of temperatures at locations z = 0 and z = e are:

𝜃 0 (𝑟, 𝑠) = ∑ (𝐴 + ℎ 𝑒 𝐵)(Φ 0 + 𝑌) -𝑋(𝐶 + ℎ 𝑒 𝐷) (𝐶 + ℎ 𝑒 𝐷) + ℎ 0 (𝐴 + ℎ 𝑒 𝐵) The 2D average temperatures 𝜃 ̃0(𝑠) and 𝜃 ̃𝑒(𝑠) considering an area of radius 𝑟 𝑎 centered around r = 0 can be obtained knowing that, from Eq. ( 6.29) and Eq. ( 6.30) that:

1 𝜋𝑟 𝑎 2 ∫ 2𝜋𝐽 0 (𝛽 𝑛 𝑟)𝑟𝑑𝑟 𝑟 𝑎 0 = 2𝐽 1 (𝛾 𝑛 𝑟 𝑎 ) 𝛾 𝑛 𝑟 𝑎 (6.33)
From Eq. ( 6.29) and Eq. ( 6.30), once removed the radial dependency, the 1D temperature expressions become:

𝜃 0 (𝑠) = (𝐴 + ℎ 𝑒 𝐵)(Φ 0 + 𝑌) -𝑋(𝐶 + ℎ 𝑒 𝐷) (𝐶 + ℎ 𝑒 𝐷) + ℎ 0 (𝐴 + ℎ 𝑒 𝐵) (6.34)

𝜃 𝑒 (𝑠) = Φ 0 + 𝑌 + ℎ 𝑒 𝑋 (𝐶 + ℎ 𝑒 𝐷) + ℎ 0 (𝐴 + ℎ 𝑒 𝐵) (6.35) In this case X and Y are computed in the same way but without the dr integration.

The experimental originality of this work lays on the simultaneous visualization of both rear and front faces of the sample with the same camera. Therefore, a model is presented here where the front face temperature is the input data and the rear face temperature the where 𝐵𝑖 = ℎ𝑒/𝑘 is the Biot number and ℒ -1 represents inverse Laplace transform obtained through numerical procedure such as presented in [Stehfest 1970][de Hoog 1982]. This procedure is also used to express the temperature from Laplace space to the time space for Eqs. (6.29)-( 6.35).

Figure 6.17 shows that there is no significant difference between 1D (6.29) and 2D (6.34) front face temperatures, except at the very first time due to Laplace numerical inversion. On the contrary, there is a difference between 1D and 2D models for the rear face.

In Eq. ( 6.36), the transmittances between brackets are functions of the parameter of interest: thermal diffusivity 𝛼 and Biot number Bi. The estimation procedure is based on minimization of the quadratic error between experimental and theoretical rear face temperatures (6.37) with an ordinary least square procedure coupled with a Levenberg-Marquardt optimization [START_REF] More | The Levenberg-Marquardt: Implementation And Theory[END_REF]]. In Eq. ( 6.37), 𝛃 = [𝛼, 𝐵𝑖] 𝑇 and the subscript j = 1,… ∑[𝑻 𝒆𝒙𝒑 (0, 𝑒, 𝑡 𝑘 ) -𝑻 𝒕𝒉 (0, 𝑒, 𝑡 𝑘 , 𝛃)] 2 𝑀 𝑘=1 (6.37)

𝐗 𝛃 = 𝛃 𝜕𝐓 𝐭𝐡 (0,0, 𝑡) 𝜕𝛃 (6.38) Figure 6.18 presents the evolution of 𝐗 𝛃 over time calculated with the same input parameters as in Figure 6.17. Both evolutions of 𝐗 𝛃 behave differently. Thus, thermal diffusivity and Biot number can be estimated with a good level of confidence. The amplitude of 𝑋 α is higher than X Bi 's, which becomes constant very quickly, meaning that transmittivities in Eq. ( 6.36) tend to become insensitive to Biot number a short time during the laser pulse.

Results and discussion

Example of estimation is presented in Figure 6.19 for pure iron and stainless steel 304. The video recorded by the experimental apparatus is used to get temperature rise curve at the centre zone of each heat spot by converting gray level into temperature.

Estimated temperature versus time at the rear surface fits well with experimental rear surface temperature, with a highest variation in the residuals less than 1 K. As the residuals mean is close to zero, and for a sake of clarity, residuals in Figure 6.19 are shifted of a value equal to the initial rear face temperature. 6.20 and 6.21 show the diffusivity estimation results from stainless steel 304 and iron for a temperature range between 1 300 K and 1 600 K. Results are compared with data from literature [Kim 1975][Touloukian 1977a][Mills 2002]. For stainless steel 304, the estimated diffusivity values lie within the 10 % error range proposed by [Touloukian 1977a] while for iron samples, the diffusivity values are more scattered but still close from data published in literature. In the case of alloys, we suppose that small differences in the chemical components proportion may influence thermophysical characterization results.

Two major error sources influence our results. The first one is the uncertainty related to the sample thickness, a property which is rather hard to measure for very thin samples. The second one is related to the assumed radiative properties, i.e. the emittance, of iron samples used to produce the initial signal-temperature calibration curve (see Section 5.1.3.) and therefore it leads to temperature measurement errors. To address the first error source, uncertainty in the thickness value, first we compute thickness variation induced by thermal expansion at 1 400 K. The coefficients of thermal expansion are taken from [Touloukian 1977b]. For stainless steel 304, a thickness variation of 5 µm leads to a maximum variation in the estimated diffusivity of 9.8 %. On 6. Characterization of thermal properties by non-contact techniques 156 the other side, a 10 % variation in the assumed emittance value during calibration, leads to an average temperature difference of 13 K and 14 K for iron and steel. In fact, for such a small temperature variation, it can be assumed that any change in the thermophysical properties is negligible, and therefore the high-speed thermography technique can be efficiently used to measure temperature in a laser flash experiment. At the same time, extra caution must be kept while measuring the thickness of the sample, as a very small variation in this value can significantly shift the estimation results.

In order to further test robustness of the proposed method, we cross check estimation results for iron and steel in Table 6.2, using the proposed analytical solution.

With the 1D quadrupole model, estimated parameter vector consists of two dimensionless numbers: Fourier and Biot with 𝐹𝑜 = 𝛼/𝑒 2 . The advantage of Fourier number is to estimate thermal diffusivity regardless the thickness, a parameter of great influence and which is hard to measure in our case. Setting the thickness value to 230 µm diffusivity estimated by the 1D quadrupole method is smaller than the value given by the 2D model. In fact, for the same heat flux condition on the front face and to get the same heat transfer rate for both models, i.e. the same rear face temperature, the diffusivity estimated by the 1D model is smaller, given that the latter does not account for the radial heat losses. The originality of the experimental configuration with the mirrors and the visible high-speed camera allowed the possibility to observe simultaneously both sides of the plate. The use of the high-speed camera also allowed to record temperature fields at a frequency of 1 000 frames per second. This speed was a necessity if we wanted to see the in-depth heat propagation with a plate of about 200 µm of thickness.

Parameter estimation has been successfully performed by ordinary least-squares coupled with Levenberg-Marquardt algorithm by minimization of the quadratic error between experimental and theoretical temperatures. Estimated thermal diffusivity for the tested metals is consistent with data encountered in literature.

Conclusion

Section 6 was mainly dedicated to estimation of thermal diffusivity of solid metallic samples at temperature close to the melting point. This subject was a first step to a more general project in characterization of liquid metal as described in introduction of Section for about twenty years mainly for metals, less for dielectrics since they are less of an issue a priori. The approach must be theoretical, experimental, and pragmatic at the same time.

For that purpose, in the framework of the Carnot Art contribution, a collaborative project between IRDL and LAMIH (Université de Bourgogne) will be submitted for the specific study of emissivity as a function of roughness; the measurements will be made at room temperature initially. Abstract: Two major themes are presented. The first axis concerns the characterization of the thermophysical properties of materials of different natures by setting up experimental protocols from analytical models and sensitivity studies. The definition of an experimental protocol requires the development of a direct analytical or numerical model. The analytical model can strongly constrain the experiments, but the calculation times are usually shorter than with the numerical model, which is comfortable for parameter estimation by inverse techniques. In addition, the use of temperaturetemperature model reduces the number of unknown parameters compared to more conventional fluxtemperature models.

The second axis concerns the simultaneous estimation of temperature and emissivity by multispectral pyrometry. It is based on an initially underdetermined problem and very often leads to a dead end. Indeed, the unpredictable nature of the emissivity of a surface does not allow to define it theoretically in the form of a direct model.

The common point between the two themes is the use of inverse techniques for the estimation of the parameters of interest. Among all the existing ones, several techniques have been used and sometimes compared: deterministic techniques of the least-squares type (OLS and MLE) and stochastic techniques of Bayesian type (MCMC, MAP, particle filter).

Introduction

  Between 2004 and 2007, I made my PhD at the LEMTA, a laboratory of the Université de Lorraine. The subject of my PhD dealt with the microscale temperature measurement by bispectral pyrometry using photon counting in the UV-visible spectral range. This work was mainly experimental and I acquired skills in radiative transfer for the theoretical part, in pyrometry for the experiment part, and also in inverse techniques since the temperature was estimated simultaneously with the emissivity. Between January and July 2008, I made a post-doctoral application at the University of Quebec in Chicoutimi, in the CURAL laboratory. My acquired skills during my PhD in radiative transfers and inverse techniques allow me to work on infrared tomography for the detection of radiative properties in semi-transparent medium. This work was exclusively numerical but the aimed applications were the detection of cancerous tumours in the human tissues. Since September 2008, I had a position of Associate Professor at the Université Bretagne Sud, and I share equally my time between teaching in the department Métiers de la Transition et de l'Efficacité Énergétiques (ex Génie Thermique et Énergie) of the Institut Universitaire Technologique and research in the Institut de Recherche Dupuy de Lôme (IRDL). I am a member of the team Assemblages multi-matériaux, one of the five research teams of the IRDL. Its activities are focused on the development, the characterization, and the modelling of processes dedicated to the assembly of materials of different natures [PTR2].

References[

  photon counting.Mostly experimental, a great part of my work was dedicated to the development of an optical bench and of a high temperature blackbody of small dimensions. This work also was the opportunity to use diverse experimental apparatuses: infrared camera, quantum detectors, FTIR spectroradiometer, for example. Nevertheless, my PhD had been the opportunity to discover the quadrupole methods, to perform parameter estimation with the use of inverse techniques(Levenberg-Marquardt), and develop numerical model with the help of MatLab or FlexPDE. Some results of my PhD are presented in Section 5 of this manuscript. In addition, the results obtained along my PhD have been published according to the following

  and 2.2 attempt to summarize the courses given during the period 2008-2023, according to the following presentation where the numbers in parentheses are the hours in front of students: discipline (lecture/tutorial/practical work). The colour code is blue for the lecture, orange for the tutorial, and green for the practical work. Moreover, Figures 2.3 and 2.4 summarize the hours done in the period 2008/2023.

Figure 2 .

 2 Figure 2.3 shows the distinction between the service due (in blue) and the additional hours (in orange).Figure 2.4 shows the distinction between the hours done in front of students and the hours equivalent to administrative responsibilities and others.

Figure 2 . 1 .

 21 Figure 2.1. Disciplines and hours in BUT1 according to the following presentation: discipline (lecture/tutorial/practical work). Blue: lecture, orange: tutorial, green: practical work.

Figure 2 . 2 .

 22 Figure 2.2. Disciplines and hours in BUT2 according to the following presentation: discipline (lecture/tutorial/practical work). Orange: tutorial, green: practical work.

Figure 2

 2 Figure 2.3. Evolution of my annual teaching.

Figure 2 . 4 .

 24 Figure 2.4. Distinction between hours in front of students (blue) and hours due to responsibilities (orange).

•

  In 2008, I first participated in the ANR project (research national agency) Béton de chanvre, supervised by Professor Patrick Glouannec of the Université Bretagne Sud. Between 2021 and 2023, three unsuccessful attempts have been made to work on simultaneous temperature emissivity estimation through ANR project. I am responsible for the IRDL of this ANR project in partnership with several French laboratories and companies (IUSTI, ITheMM, CERTES, THEMACS). The proposed project, supervised by Professor Thierry Duvaut of the Université de Reims Champagne-Ardenne. • Still in 2023, Associate Professor Mickaël Courtois of the Université Bretagne Sud proposed an ANR project dedicated to the thermal characterization of physical properties of liquid metals. If it is accepted, I will be involved in this project. I co-supervised two thesis through collaborations with ArcelorMittal (thesis of E. Geslain, 2015-2018) and with Mapepas research chair (thesis of L. Dejaeghere, 2012-2016). I realized deliveries for Orolia, company specialized in the development of emergency beacons: evaluation by infrared thermography of beacon surface temperature increase caused by internal lithium battery thermal runaway. I organized a working day dedicated to the model reduction (December 9 th 2021) in collaboration with LMEE from Université d'Evry.

Figure 3

 3 Figure 3.1. Number of publications since 2006.

Figure 3 . 2 .

 32 Figure 3.2. Cumulated number of publications since 2006.

4 .

 4 photonic flux, ph•s -1

Figures 4 .

 4 Figures 4.1 and 4.2 present focused and tomographic views of hemp concrete.

Figure 4 . 1

 41 Figure 4.1 Focused view of a hemp concrete [Pierre 2014] (approximate scale 1:2).

Figure 4 .

 4 Figure 4.2. Tomographic view of a hemp concrete [Pierre 2014] (approximate scale 1:2).

Figure 4 .

 4 Figure 4.3. Scheme of the sample with the hot-strip.

  leads to: 𝐹(𝛽 𝑛 , 𝑦, 𝑠) = ∫ ∫ 𝑓(𝑥, 𝑦, 𝑡)𝑐𝑜𝑠(𝛽 𝑛 𝑥)𝑒 -

Figure 4 . 4 .

 44 Figure 4.4. Reduced sensitivity of the parameters given by Eq. (4.22).

Figure 4 . 5 . 4 .

 454 Figure 4.5. Reduced sensitivity of the parameters given by Eq. (4.23).

( 4 .

 4 21), and one at the rear face (0,e). Heating is stopped once the temperature at the position (0,e) increases, as it can be seen in Figure 4.6 at t = 200 s. Estimation results are presented in Figure 4.7.

Figure 4 . 6 .

 46 Figure 4.6. Front face and rear face experimental temperatures.

Figure 4 .

 4 Figure 4.7. Experimental points and adjusted model (4.21) over time and residuals for given room temperature and relative humidity.

Figure

  Figure 4.8 shows estimation of heat capacity 𝜌𝑐 𝑝 𝑚 , which varies linearly with water

Figure 4 .

 4 Figure 4.8. Heat capacity versus water content for both samples and comparison with theoretical model.

Figure 4 .

 4 Figure 4.9. Conductivity versus relative humidity (on the left) and temperature for both samples.

Figures 4 .

 4 Figures 4.10and 4.11 present predicted conductivities of both hemp concrete samples as a function of moisture content w and experimental points. Thermal conductivity calculations have been done for a water content supposed constant between

Figure 4 .

 4 Figure 4.10. Krischer's model adjustment on experimental thermal conductivities versus water content for sample 1.

Figure 4 .

 4 Figure 4.11. Krischer's model adjustment on experimental thermal conductivities versus water content for sample 2.

  temperature model, it is a transmittance. It can be adapted with the hot-strip technique, and we propose to present it in the next sections dedicated to the characterization of conductive and radiative properties of bio-aerogels in Section 4.2 and of various insulating and conductive materials in Sections 4.3 and 4.4, and of high temperature metals in Section 6. 4.2. Estimation of the conductive and radiative properties of bio-aerogel with a transient pulsed technique [Pierre 2017] 4.2.1. Context This study presents thermal characterization by inverse method of a bio-aerogel with the use of apparatus based on a pulsed technique and a simple theoretical 1D model. Parameters of interest are solid and gas thermal conductivity, radiative effect through a radiative conductivity, and thermal diffusivity. Aerogels consist of highly porous materials with very low density and large specific surface. Aerogels' low thermal conductivity at ambient atmospheric pressure makes them good candidate of nanoporous heat insulation materials. Currently, usual techniques, such as conductometer [Nguyen 2014], guarded hot-plate [Shi 2013], hot-disk [Grishechko 2013], hot-wire [Rudaz 2014], allow assessing only an effective thermal conductivity.

Figure 4 .

 4 Figure 4.12. Diagram of the experimental set-up.

Figure 4 .

 4 Figure 4.13. Picture of the experimental setup. Dimensions of the aerogel sample (in white): e = 9 mm and S = 80 × 80 mm 2 .

  .33)  = (0,s) and e+eb = (e+eb,s) correspond, in the Laplace domain, to theoretical temperatures at positions z = 0 and z = e+eb, where experimental temperatures are measured. The first square matrix is the conductive-radiative matrix Mrc and the second one is the purely capacitive matrix of the brass plate. The terms of the conductiveradiative matrix are the followings (details in Appendix A in [Pierre 2017]):

Fig. 4 .

 4 Fig. 4.14. Quadrupole network of the experimentation.

Figure 4 .

 4 Fig. 4.15. Theoretical 1D and 3D front and rear faces temperatures calculated from equations (4.34) to (4.43) and flux density 0.

Figure 4 .

 4 Figure 4.17. Reduced sensitivities of the transmittance function ℒ -1 [𝐻(𝑠)] over time.

Figure 4 .

 4 Figure 4.18. Evolution of the variance of each parameter function of the thickness of the brass plate.

Figure 4 .𝑣𝑎𝑟

 4 18 presents the evolution of the variances (4.47) for each parameter estimations [Orlande 2011]. 4. Characterisation of thermal properties by heating contact techniques 56

Figure 4 .

 4 Figure 4.19. Experimental and fitted theoretical rear face temperatures with and without radiation.

Figure 4

 4 Figure 4.20. Reduced sensitivity considering three parameters: diffusivity, thermal conductivity and heat exchange coefficient.

Figure 4 .

 4 21 presents experimental rear face temperatures versus time. A slight slope change is observable, probably due to transition between radiative and conductive effects, and has been correctly fitted by theoretical curve, and already observed in previous studies [André 1995][Niezgoda 2011]. A final point concerns the comparison between estimated experimental value of the radiative conductivity kr and theoretical expression of the Rosseland approximation 𝑘 𝑟 𝑡ℎ (4.49) where  = 5.67 × 10 -8 W•m -2 •K -4 is the Stefan-Boltzmann constant. This expression allows to give an estimation of a mean extinction coefficient  which can be defined according to Eqs. (4.49) and (4.50) representing, respectively, R and P Rosseland's and the Planck's mean extinction coefficients [Ozisik 1973]. Experimentally, as aerogel scatters the heat flux and is not mainly absorbing, this is an effective mean extinction coefficient eff, which has been measured with a spectroradiometer working between 1 = 2 µm and 2 = 25 µm for a sample of 4 mm of thickness. Spectral normal direct transmittivity has been measured at first (4.51), then Rosseland's and Planck's mean extinction coefficients (Figure 4.22) have been compared with values found in the literature. From Eqs. (4.49) and (4.50) and experimental data, calculations of R and P give 591 m -1 and 664 m -1 , respectively. These approximated values are satisfying since they can be compared with the mean extinction coefficient obtained from estimation of radiative conductivity (4.48):

Figure 4

 4 Figure 4.21. Experimental points: observation of a break between radiative and conduction heat transfers around 8 s.

Figure 4

 4 Figure 4.22. Spectral effective, Planck's and Rosseland's mean extinction coefficients of the aerogel versus the wavelength in the IR range.

4. 3 .Figure 4 .

 34 Figure 4.23. Sketch of the sample.

  and 4.26 for random positions z1 = 2 mm, z = 3 mm, and z2 = 5 mm. The solid lines concern temperature sensitivity to thermal diffusivity and dashed line to modified Biot number, and circle , star  , and triangle ▼ m -2 •K -1 . In each case, it is shown that estimation of both parameters is possible. As expected, temperature becomes less sensitive to modified Biot number, when the latter decreases.

Figure 4 .

 4 Figure 4.25. Reduced temperature sensitivities for thermal diffusivity and modified Biot number versus time obtained for polycarbonate for three types of environment condition.

Figure 4 .

 4 Figure 4.26. Reduced temperature sensitivities for thermal diffusivity and modified Biot number versus time obtained for balsa for three types of environment condition.

  and 4.28, respectively. Both figures show different behaviors between sensitivity to in-plane thermal diffusivity and to modified Biot number, which allows to predict that estimation of both parameters seems possible. In Figure4.27, for balsa, sensitivity to modified Biot number H decreases slowly, but in Figure4.28 transfer function seems to be insensitive to H in the case of aluminum, which presumes that its estimation would be difficult.

Figure 4 .

 4 Figure 4.27. Reduced sensitivity curves to 𝛼 and H in the case of the balsa.

Figure 4 .

 4 Figure 4.28. Reduced sensitivity curves to 𝛼 and H in the case of the aluminum.

Figure 4 .

 4 27 for the balsa and r -r1 = 1 mm 𝜎 𝛼𝐻 2 = -0.245, and in Figure 4.28 for the aluminum and r -r1 = 2 mm 𝜎 𝛼𝐻 2 = 0.192.

Figure 4 .

 4 29 shows for balsa and aluminum theoretical evolution of 𝜎 𝛼𝐻 2 versus distance r -r1. It is shown that the former has a minimum value, corresponding to a minimum correlation between inplane thermal diffusivity and modified Biot number.

Figure 4 .

 4 Figure 4.29. Comparison between evolution of correlation 𝜎 𝛼𝐻 2 coefficient for balsa and aluminum.

Figure 4 .

 4 Figure 4.30. Experimental apparatus: vacuum chamber and IR camera.

Figure 4 .

 4 Figure 4.31. Experimental apparatus: sample and heating element.

Figure 4 .

 4 Figure 4.32. View of experimental set-up.Figure 4.33. View of the copper spot

Figure 4 .

 4 Figure 4.32. View of experimental set-up.Figure 4.33. View of the copper spot embedded in the balsa sample.

Figure 4 .

 4 Figure 4.35 plots both evolutions of the five average temperatures (solid lines) at these locations and their temperature derivatives (dashed lines) versus time: 𝜕𝑇 ̅ (𝑧, 𝑡) 𝜕𝑡 (4.67)

Figure 4 .

 4 Figure 4.35. Evolution of th temperature rises and their time derivatives at zpositions versus time.

Figure 4 .

 4 Figure 4.36. Example of estimation and results in the case of balsa in vacuum.

Figure 4 .

 4 Figure 4.37. Evolution of estimated thermal diffusivity over estimation time in the case of balsa sample during experiment in vacuum.

Figure 4 .

 4 Figure 4.36 presents estimation plots for combination 3. The upper figure plots temperature evolutions versus time and residuals. The lower figure plots reduced sensitivities of estimated temperature 𝑇 ̅ (𝑧, 𝑡) versus estimated parameters. As previously shown in Figure 4.26, sensitivity of temperature to heat exchange coefficient in vacuum is very weak compared to sensitivity to thermal diffusivity. Last point concerns estimation time. Figure 4.37 plots evolution of estimation of thermal diffusivity versus estimation time. Obviously, it is necessary to have a minimum time of estimation, since estimated values tend to a constant one: a minimum estimation time of 10 s is required. This observation can be related to sensitivity of time derivative (4.68), which tends to zero after 10 s after reaching its maximum as shown in Figure (4.22).

Figures 4 .

 4 38 to 4.41 gather all estimation results for the five samples.

Figure 4 .

 4 Figure 4.38. Thermal diffusivity results for all samples with their uncertainties.

Figure 4 .

 4 Figure 4.39. Volumetric heat capacity results for all samples with their uncertainties.

Figure 4 .

 4 Figure 4.40. Thermal conductivity results for all samples with their uncertainties.

Figure 4 .

 4 Figure 4.41. Heat exchange coefficients for three experimental conditions with their uncertainties.

Figure 4 .

 4 Figure 4.39, two close values for volumetric heat capacities appear for balsa, on the one

Figure 4 .

 4 Figure 4.42. Representation of infrared scene measured for mild steel and of relation between Cartesian and polar coordinates. Scale on the right is temperature expressed in degrees Celsius.

Figure 4 .

 4 Figure 4.43. Experimental temperature variations of PVC for angles  = 0, /6, /4, /3, /2 and for r1 = 2 mm (the highest) and r = 3 mm.

Figure 4 . 4

 44 44 presents correlation coefficient 𝜎 𝛼𝐻 2 as a function of the distance r -r1 for r1 = 2 mm and r varying until maximum distance filmed by infrared camera (r1 -r = 3 mm), which varies with experiments. Figure = 0.7 mm. It shows that for r -r1 < 0.70 mm, covariance varies to large extent with values close to 1: fitting of theoretical model on experimental temperature is unsuccessful and estimated values have no meaning. For r -r1 > 0.70 mm, covariance increases slightly from 0.45 to 0.7 and theoretical model fits on experimental temperatures: large possibilities exist to estimate the properties. An Example of temperature estimation during test with PVC for 𝜃 = 0 is presented in Figure 4.45, for given position r1 = 2 mm and r1 -r = 0.7 mm when covariance 𝜎 𝛼𝐻 2 starts to have low value. Residuals are constantly around zero. Estimation is extended for several positions r1. For a given r1, position r is searched to reach minimum of 𝜎 𝛼𝐻 2 over the distance r -r1 and, then, estimated parameters are recorded.

Figure 4 .

 4 Figure 4.44. Evolution of the variance 𝜎 𝛼𝐻 2 versus distance r -r1 for  = 0 and r1 = 2 mm.

Figure 4 .

 4 Figure 4.45. Experimental and estimated temperatures during test with PVC.

Figure 4 .

 4 Figure 4.46. Infrared scene of balsa with fibers oriented in vertical direction ( = /2). Scale on the right is temperature expressed in degrees Celsius.

Fin

  effect assumption allows to focus only on one face of the sample, and thus the use of infrared camera. This apparatus offers many advantages. It allows a global visualization of the sample and the possibility to record thermal images over time for the post-treatment with a sufficient frequency.The microscopic G1 lens gives access to scene with a resolution of 15 µm, meaning that one pixel covers a surface of 225 µm 2 : a surface about 50 times smaller than with type K thermocouple of 125 µm of diameter presented in Section 4.2.3. But signal is more impacted by the noise (Figure4.45). It should be advantageous to consider average temperature given by 50 pixels instead of one pixel such as in Figure4.34.It was previously mentioned that a maximum temperature increase should not exceed 10 K to assume physical properties independent of temperature and thus make possible analytical solution. Measurements in a maximum 10 K temperature range gives the possibility to use digital level information of the camera instead of temperature, as long as there exists linear evolution between them such as presented in Figure4.47 for IR camera used for these studies. If the initial temperature of the sample is the room temperature, it is not necessary to mention emissivity of the sample in the setting of the camera.

Figure 4 .

 4 Figure 4.47. Sensitivity of the infrared camera in the range [25 °C -32 °C].

  and (5.3), hP = 6.662 × 10 -34 J•s, kb = 1.38 × 10 -23 J•K -1 , and c = 3 × 10 8 m•s -1 are Planck's constant, Boltzmann's constant, and light speed, respectively[Ozisik 1973][Sparrow 1978[START_REF] Siegel | [END_REF][START_REF] Modest | Radiative heat transfer[END_REF]].It is common to simplify Eq. (5.1) by Eqs.(5.4)-(5.6) in order to help the calculation and be less time consuming[Pierre 2023].Φ 𝑖𝑡ℎ ≈ 𝐹 𝑖 𝜀(𝜆 𝑖 , 𝜃, 𝜑, 𝑇) 𝐼 0 (𝜆 𝑖 , 𝑇)

  5.1.2. The radiative and the photonic fluxes In terms of observable, two types of signals are possible in multispectral pyrometry: radiative flux expressed in watt and photonic flux expressed in photons•s -1 . The former is measured generally by photon detectors, which have a working spectral range between 1.50 µm and 40.00 µm (Figure 1 [Audaire 2000]) "absorbing the energy of the incident radiation with their electrons by producing free charge carriers" [Modest 2003]. The latter is measured through photomultiplier tubes (PMTs) of working spectral range between 0.15 µm and 2.00 µm (Figure 5.2 [Audaire 2000]): "they detect light by making use of the external photoelectric effect or namely the phenomenon in which electrons are released into the vacuum when light strikes a metal or semiconductor in a vacuum" [www.hamamatsu.com].

Figure 5 .

 5 Figure 5.1. Specific detectivity of infrared detectors over the wavelength and photon energy [Audaire 2000].

Figure 5 .

 5 Figure 5.3 presents evolution of spectral photonic flux 𝑃 𝜆 0 over temperature emitted by a surface s = 𝜋𝜆 2 /4 and for different values of wavelength 𝜆 of measurement.

Figure 5 .

 5 Figure 5.3. Photonic flux emitted from a surface of the size of the detection wavelength versus temperature.

Figure 5 . 4 .

 54 Figure 5.4. Photonic fluxes measured at three wavelengths and for a blackbody temperature of 1 000 K [Pierre 2008].

Figure 5 . 5 .

 55 Figure 5.5. Blackbody dedicated to high temperature measurements [Pierre 2008].

Figure 5 Figure 5 . 7 .

 557 Figure 5.6. Statistical distributions of photonic fluxes with Gaussian's law [Pierre 2008].

Figure 5 .

 5 Figure 5.8. Up: temperature estimation T and thermocouple temperature Tc versus time; down: temperature error eT [Pierre 2016].

Figure 5 .

 5 Figure 5.9. Optical experimental bench general view [Pierre 2008].

Figure 5

 5 Figure 5.10. Optical experimental bench [Pierre 2008].

  emissivity estimation[Dejaeghere 2016][START_REF] Dejaeghere | [END_REF]][Pierre 2022][Pierre 2023][START_REF] Lamien | [END_REF]] (Figure5.12). The high-speed camera is used at the same time for the observation of the sample[START_REF] Houssein | [END_REF]] (Section 6) and for the measurement of its temperature. The last three pyrometers are strictly used to indicate temperature of liquid sample, and when it is necessary as references. The operation of these three pyrometers is not presented here, but they have been calibrated by manufacturer. It has been a requirement too to calibrate the six-wavelength pyrometer and the high-speed camera: the former with a blackbody, the latter with the help of the solidification temperature of pure iron.The calibration of the six-wavelength pyrometer has been performed by replacing the nozzle of the levitation by a blackbody (red rectangle in Figure5.11). The blackbody is made of carbon and its dimensions are: 8.00 mm of height, hole radius of r = 0.75 mm and depth of ℓ = 4.00 mm (Figure5.13). Considering these dimensions, literature proposes two ways to calculate the apparent emissivity 𝜀 𝑎 of the open upper surface: Eq.

5 .

 5 Figure 5.11. Aerodynamic levitation apparatus.

Figure 5 .

 5 Figure 5.12. Sketch of the six-wavelength pyrometer.

(

  𝜏 𝑜𝑓 ), dichroic mirrors (of transmittivity 𝜏 𝑚 and reflectivity 𝜌 𝑚 ) splitting radiative fluxes for a given cut-off wavelength 𝜆 𝑐 when receiving a radiative flux with an angle of 45°, quasi-monochromatic spectral filters (𝜏 𝑓 ), and silicon sensors working in specific narrow spectral band. The value of the central wavelength of the six quasi-monochromatic spectral filters are 0.480 µm, 0.530 µm, 0.680 µm, 0.850 µm, 0.940 µm, and 1.550 µm(Figure 5.12

Figure 5 .

 5 Figure 5.14 presents the evolution of the six spectral transfer functions (5.13)-(5.18) over wavelength according to manufacturer data. It appears that transfer function 𝑓 1 (𝜆) has a very low value: this is due to the dichroic mirror m1, which has a transmittivity 𝜏 𝑚1 close to zero before 0.500 µm. Secondly, these spectral transfer functions are used to determine a representative wavelength of the filter 𝜆 𝑖 according to Eq.(5.6). But this transfer function 𝑓 𝑖 (𝜆) is not enough, since it does not consider geometric aspects of radiative transfer such as view factors. Only calibration with a blackbody makes possible the determination of one definitive transfer function specific to every optical path, called Fi in Eqs.(5.4) and(5.5).

Figure 5

 5 Figure 5.13. Views of the blackbody for the levitation apparatus.

Figure 5 .

 5 Figure 5.14. Evolution of the six transfer functions according to manufacturer data.

Figure 5 .Figure 5 .

 55 Figure 5.15. Evolutions of the six-wavelength pyrometer fluxes and of the mono-and bichromatic pyrometers temperatures over time during calibration with the blackbody in the levitation apparatus.

  to 5.18 mainly, it is noticeable that the values of Fi oscillate. This is probably due to natural convective effects. Indeed, the blackbody is the only heated part in the inert chamber(Figure 5.11). As the surrounding is at room temperature, large 5. Temperature emissivity estimation with multispectral pyrometry 99 temperature difference between the surrounding and the blackbody induces air movement and thus natural convection at the upper surface of the blackbody, which is detected by the pyrometers.

Figure 5 .

 5 Figure 5.16. Estimation of F1 over time with Eq.

Figure 5 .

 5 Figure 5.18. Estimation of F3 over time with Eq.(5.21).

Figure 5

 5 Figure 5.19. Estimation of F4 over time with Eq.(5.21).

Figure 5 .

 5 Figure 5.20. Estimation of F5 over time with Eq.

Figure 5 .

 5 Figure 5.21. Estimation of F6 over time with Eq.

Figure 5

 5 Figure 5.22. Statistical treatment of F1.

Figure 5

 5 Figure 5.23. Statistical treatment of F2.

Figure 5

 5 Figure 5.24. Statistical treatment of F3.

Figure 5 .

 5 Figure 5.25. Statistical treatment of F4.

Figure 5 .

 5 Figure 5.26. Statistical treatment of F5.

Figure 5 .

 5 Figure 5.27. Statistical treatment of F6.

  last parameters to estimate to complete the calibration are k and d in Eq.(5.25). Experimentally, the protocol consists in measuring the greyscale signals SGS at the melting/solidification plateaus of pure metals assuming known their emissivity, coefficient 𝜏 and for different values of D. The tested metal is iron (Tfusion = 1 811 K) and its emissivity is assumed 0.3 during the phase changes[Watanabe 2003]. Eight measurements have been performed and linear evolution of SGS versus X is presented in Figure5.29. Therefore, using equation mentioned in Figure5.29, temperature T can directly be calculated from Eq. (5.25) for a given emissivity. 5. Temperature emissivity estimation with multispectral pyrometry 102 5.1.4. Partial conclusion This section was dedicated to the presentation of basics in multispectral pyrometry and of different types of fluxes: radiative or photonic, and calibration. Considering temperature measurement with a multispectral pyrometer with N fluxes measured at N different wavelengths, by principle the problem is always ill-posed since there are always N equations and N + 1 unknowns: one temperature and N emissivities.And it comes out that, once identified and estimated the maximum of uncertainty sources, the most important problem is the knowledge of the emissivity. This parameter is naturally present in each radiative flux and happens to be unpredictable most of the time, depending on numerous other parameters. The next section is dedicated to the selection of the wavelengths in the case where pragmatic choice of the emissivity model is made to set multispectral pyrometry.

Figure 5 .

 5 Figure 5.28. Values of coefficients 𝑐 0 and 𝜆 0 for different temperature ranges.

Figure 5 .

 5 Figure 5.29. Calibration curve of the highspeed camera.

Figure 5 .

 5 Figure 5.30. Evolution of the wavelength distance Δ𝜆 = 𝜆 2 -𝜆 1 versus 𝜆 1 for different temperatures.

Figure 5 .

 5 Figure 5.31. Evolution of the wavelength ratio versus signal ratio 𝛼 for a given temperature of 1 273 K.

  which depends on the signal-to-noise ratio of each radiative flux. The works of Rodiet et al. are interesting, since they try to bring objectivity for the selection of the wavelengths with the use of a minimization of the variance on the estimated temperature in the correlation matrix of the estimated parameters (5.41)-(5.42) by ordinary least squares (5.40) [Rodiet 2016]. They start with linear Wien's approximation (5.37) with 𝐶 1 = 2𝜋ℎ𝑐 2 . The wavelengths are progressively estimated assuming first a blackbody emissivity (N = 1 and 𝛃 = 𝑇) for the first wavelength 𝜆 1𝑜𝑝𝑡 , then a grey body emissivity (N = 2 and 𝛃 = [𝑇 𝑎 0 ] 𝑇 ), then linear emissivity (N = 3 and 𝛃 = [𝑇 𝑎 0 𝑎 1 ] 𝑇 ), and finally a parabolic emissivity (N = 4 and 𝛃 = [𝑇 𝑎 0 𝑎 1 𝑎 2 ] 𝑇 ), for, respectively, the three next ones 𝜆 2𝑜𝑝𝑡 , 𝜆 3𝑜𝑝𝑡 , and 𝜆 4𝑜𝑝𝑡 , in a narrow spectral band imposed by infrared camera.

Figure 5 .

 5 33 plots 𝜎 𝑇 over the wavelength, which is minimal for two wavelengths: 983 µm. For example, the second value can be rejected since it is beyond the spectral range of the detector, even if 𝜎 𝑇 is smaller at this position than the other one.Thus 𝜆 2𝑜𝑝𝑡 = 𝜆 2𝑜𝑝𝑡 1 = 0.725 µm. Procedure is repeated for the two last wavelengths and presented inFigures 5.34 and 5.35. Several possibilities are offered and the use of the previous criteria presented in Eq. (5.32) and Eqs. (5.33)-(5.36) can help the choice.

Figure 5 .

 5 Figure 5.32. Intensity and its spectral derivative over the wavelength. Derivative of the flux gives position of the first wavelength.

Figure 5 .

 5 Figure 5.33. Standard deviation on temperature 𝜎 𝑇 from Eq. (5.41) versus wavelength for the estimation of the parameter of the constant emissivity model.

Figure 5 .

 5 Figure 5.34. Standard deviation on temperature 𝜎 𝑇 from Eq. (5.41) versus wavelength for estimation of the second parameter of the linear emissivity model.

Figure 5 .

 5 Figure 5.35. Standard deviation on temperature 𝜎 𝑇 from Eq. (5.41) versus wavelength for estimation of the third parameter of the parabolic emissivity model.

5. 3 .

 3 The simultaneous temperature emissivity estimation 5.3.1. Presentation of the experiment and some previous theoretical considerations Several techniques to estimate simultaneously temperature and emissivity have been tested during multispectral pyrometry measurements: deterministic ordinary least squares (OLS) [Pierre 2022], and statistic Bayesian MCMC [Pierre 2022][Pierre 2023], and Bayesian particle filters [Lamien 2023]. All of them have been tested with the same experiment with niobium and a mild steel. Only results with niobium are presented here.

Figure 5 .

 5 Figure 5.36. Niobium sample fluxes and commercial bispectral pyrometer temperature

  Figure 5.36 plots all the radiative fluxes recorded along one experiment.5. Temperature emissivity estimation with multispectral pyrometry 110Commercial bispectral pyrometer is also used during experiment and gave the temperature (black dashed line in Figure5.36), only during the cooling stage since the laser wavelength disturbs this pyrometer. This temperature is considered as a reference since the solidification plateau temperature is respected for pure niobium at 𝑇 𝑓 = 2 750 K.

Figure 5 .

 5 Figure 5.37 shows at the same time the same radiative fluxes and the laser heating sequence. Both time scales are not synchronized. There are two sequences for the laser heating: one of 140 W during 1 s and a second one of 30 W during 6 s. Considering the radiative fluxes, measurements for only five wavelengths are presented (wavelength 𝜆 1 = 0.480 µm has been omitted due to its very weak signal). Melting and solidification plateaus and the undercooling are visible. The cooling starts when the laser is off.

Figure 5 .

 5 Figure 5.37. Simultaneous presentation of the laser heating sequence and the radiative fluxes measured with five wavelengths on six (wavelength 𝜆 1 = 0.480 µm has been skipped due to its very weak signal).

  multispectral methods, constrains presented in Eqs.(5.31)-(5.36) are used[Rodiet 2013[START_REF] Pierre | [END_REF][START_REF] Rodiet | [END_REF]][Rodiet 2016]. The objective is to minimize the cost function(5.49), where N is the number of considered wavelengths, Φ 𝑖 𝑡ℎ is the theoretical flux, and 𝛃 = [𝑇, 𝑎, 𝑏] 𝑇 is the parameter vector of the emissivity model. ) is the likelihood function or the probability density of measurements with the parameters 𝛃 given, 𝜋(𝛃) is the a priori density of the unknown or uncertain parameters, and 𝜋(𝐘) is the marginal probability density of the measurements, which plays the role of a normalization constant. As for Eqs.(5.42) and (5.45), measurement

  ][Kaipio 2006][START_REF] Ozisik | [END_REF]], here we deal with the filtering problem where the state variables 𝛃 𝑘 are sequentially estimated with the measurements 𝚽 𝑘 𝑒𝑥𝑝 at each time tk. The state estimation problem is solved within the Bayesian framework of statistics. Thus, we apply Bayes´ theorem[START_REF] Arulampalam | A tutorial on particle filters for on-line non-linear/non-gaussian Bayesian tracking[END_REF]][Ristic 2004][Kaipio 2006[START_REF] Ozisik | [END_REF]] (seeSection 5.3.4).Even though conceptually simple, the sequential estimation of the posterior probability density, 𝜋(𝛃 𝑘 |𝚽 𝑘 𝑒𝑥𝑝 ), only allows an analytical treatment if the evolution and observation models are linear and with Gaussian additive uncertainties. In this case, the optimal solution of the state estimation problem is obtained with the classical Kalman filter [Arulampalam 2001][Ristic 2004][Kaipio 2006][Ozisik 2021]. On the other hand, for the case of nonlinear and/or non-Gaussian models, such as in the present work, Sequential Monte Carlo Methods, also known as Particle Filters, are robust stochastic tools that can be generally applied to obtain samples of the posterior distribution at each time tk. The goal of the particle filter is to draw a set of independent and identically distributed samples (particles) {𝛃 𝑘 𝑗 } 𝑗=1 𝑄 that approximates 𝜋(𝛃 𝑘 |𝚽 𝑘 𝑒𝑥𝑝 ), by starting from a set {𝛃 0 𝑗 } 𝑗=1 𝑄 at time t0 = 0. Since it is not possible to sample directly from the posterior probability density, particle filters make use of importance sampling followed by an update step in order to approximate the posterior probability density [Arulampalam 2001][Ristic 2004][Kaipio 2006][Ozisik 2021]. The posterior probability density of the state variables 𝛃 𝑘 at time tk is then approximated through a weighted Monte Carlo realization from the importance density in terms of Q particles, i.e.: 𝑗 is the weight of the j th particle at the time t k with ∑ 𝑤 𝑘 𝑗 𝑄 𝑗=1 = 1 and 𝛿 is the Dirac Delta function. The most popular particle filter algorithms are the Sequential Importance Resampling (SIR) and the Auxiliary Sequential Importance Resampling (ASIR) [Arulampalam 2001][Ristic 2004][Kaipio 2006][Ozisik 2021]. Both are described next. The Sampling Importance Resampling particle filter algorithm (SIR)

  to 5.41 are dedicated to bi-and multispectral estimation using OLS. We first analyze the solution of Eqs. (5.37)-(5.40) using two radiative fluxes Φ 3 𝑒𝑥𝑝 and Φ 4 𝑒𝑥𝑝 by considering 𝜀 3 = 𝜀 4 , for different initial guesses of temperature and emissivity, by comparison with literature values of emissivity at the melting temperature.

5 .Figure 5 .

 55 Figure 5.39 shows temperature results after multiple multispectral combinations. No constrains have been imposed. It appears clearly that results are very different from one combination to the other, and the most credible one is for the combination using the radiative fluxes Φ 1 𝑒𝑥𝑝 , Φ 2 𝑒𝑥𝑝 , Φ 4 𝑒𝑥𝑝 , and Φ 5 𝑒𝑥𝑝 , due to the knowledge of the solidification plateau temperature.

Figure 5 .

 5 Figure 5.38. Bispectral temperature emissivity estimation with Φ 3 𝑒𝑥𝑝 and Φ 4 𝑒𝑥𝑝 .

Figure 5 .

 5 Figure 5.39. Estimated temperature with the combinations of four signals with OLS.

Figure 5

 5 Figure 5.40. Estimated temperatures by multispectral method using Eqs. (5-37)-(5.42).

Figure 5

 5 Figure 5.41. Estimated emissivity 3 (3 = 0.680 µm) versus temperature and comparison with literature data at approximatively the same wavelength [Touloukian 1970][Cezairliyan 1970][Seifer 2001].

Figure 5 .

 5 Figure 5.41 versus temperature and compared with data from literature. In solid state, emissivity tends to decrease from 0.6 around 1 200 K to 0.3 until the melting plateau, and are closer to Touloukian's value than Seifter's value [Seifer 2001][Touloukian 1970][Cezairliyan 1970]. However, in solid state, these emissivity and temperature are estimated values from fluxes recorded after 9 s (Figure 5.36), so they must be considered very carefully, since fluxes are lower than 50 nW, which is at the lower limit of confidence of the sensor. Then in liquid state, emissivity is constant and just below 0.3 whereas other values are rather between 0.3 and 0.4. There is obviously a strong dispersion between measurements and literature values, and they are in the range 0.20 -0.40. Now parameter estimation is performed according to Bayesian method linear approximation with fluxes Φ 3 𝑒𝑥𝑝 and Φ 4 𝑒𝑥𝑝 . Theoretical fluxes are linearized, as presented in Eqs. (5.45)-(5.46). Estimations are performed during the first part of the cooling, mainly during the solidification plateau. Three parameters are considered: temperature and two emissivities. Tests are performed with different levels of confidence, with priors of good or bad quality. In the first stage, we concentrated on the solidification plateau. Since the phasechange temperature is well-known, we introduced a temperature prior of small uncertainty, 𝑇 𝑝𝑟𝑖𝑜𝑟 = (2 750 ± 30) K whereas emissivity priors were given with a high uncertainty, namely 𝜀 3 𝑝𝑟𝑖𝑜𝑟 = 𝜀 4 𝑝𝑟𝑖𝑜𝑟 = 0.30 ± 0.15. Figures 5.42 and 5.43 show normalized probability densities for the three parameters. On the solidification plateau, mean and standard-deviation of emissivities are 𝜀 3 = 0.261 ± 0.023 and 𝜀 4 = 0.288 ± 0.020 and for temperature T = (2 748 ± 29) K. Method has then been extended to the whole duration of the experiment by levering the previous results. As a matter of fact, since we have now a better confidence into the emissivity values, we considered as new priors 𝜀 3 𝑝𝑟𝑖𝑜𝑟 = 0.260 ± 0.023 and

Figure 5 .

 5 Figure 5.42. A priori and estimated emissivities 𝜀 4 and 𝜀 5 at solidification plateau.

Figure 5 .

 5 Figure 5.43. A priori and estimated temperature at solidification plateau.

Figure 5 .Figure 5 .

 55 Figure 5.44. MAP estimator of temperature (in blue) with plus or minus one standarddeviation (dashed blue) and bispectral pyrometer (dashed black).

Figure 5 .

 5 Figure 5.45. Linear parameter model (MAP estimation) First step -Normalized probability density function of emissivity at t = 6.53 s, in five wavelengths for niobium. Dashed line: prior probability; solid line: MAP estimation.

Figure 5 .

 5 Figure 5.46. Linear parameter model (MAP estimation) Second step -Solid lines: Mean values of emissivities at 0.53 µm (purple) and 1.55 µm (red).Dashed lines: 99 % confidence interval.

Finally, Figures 5 .

 5 49 and 5.50 present the states of the Markov chains for both the temperature and emissivity at t = 6.7 s, that is, during the solidification plateau. They clearly show that the estimation tends to a rather constant value after 10 000 states. The histograms plotted for values after 15 000 states are not of Gaussian shapes, especially as regards the temperature. The prior for the temperature was centred at values at previous time with a constant standard deviation of 300 K, while the prior for the emissivity was considered as uniform in the interval [0, 1]. The estimated values for the solidification plateau are 𝑇 = (2 747 ± 2) K, 𝜀 34 = 0.303 ± 0.002.

Figure 5 .

 5 Figure 5.47. Estimated and experimental fluxes.

Figure 5

 5 Figure 5.48. Estimated and experimental temperatures.

Figure 5 .

 5 Figure 5.50. Markov chain states for the emissivity estimation.

Figures 5 .

 5 Figures 5.51 and 5.52 present the temperatures estimated with the SIR algorithm by using the measurements of sensors 2 to 6 and 3 to 6. The temperatures measured with the bichromatic pyrometer are also presented in these figures. Similar results are presented in Figures 5.53 and 5.54 for the ASIR algorithm. Results obtained with the measurements of other combinations of detectors used in the inverse analysis were similar to those shown by Figures 5.51 to 5.54, except with the measurements from detectors 3, 4 and 6 when the temperature residuals were larger than 300 K. The estimated temperatures shown by Figures 5.51 to 5.54 were quite consistent, despite the fact that they involved different combinations of measurements and algorithms used for the solution of the state estimation problem.Furthermore, the estimated temperatures show an excellent agreement with those measured by the bichromatic pyrometer that were used as a reference. In fact, the amplifications of Figures 5.51 and 5.52 (temperatures estimated with the measurements of detectors 2 to 6 by using the SIR and ASIR algorithms, respectively) in the region where the bichromatic pyrometer measurements were available reveal that the estimated 95 % credible intervals of temperature included the measurements. Also, the means of the estimates were within the pyrometer measurement uncertainties.

Figure 5 .

 5 Figure 5.51. Temperatures estimated with the SIR algorithm for niobium by using the measurements of detectors: (a) 2, 3, 4, 5 and 6; (b) 3, 4, 5 and 6.

Figure 5 .

 5 Figure 5.52. Temperatures estimated with the SIR algorithm for niobium by using the measurements of detectors: (a) 2, 3, 4, 5 and 6; (b) 3, 4, 5 and 6.

Figure 5 .

 5 Figure 5.53. Temperatures estimated with the ASIR algorithm for niobium by using the measurements of detectors: (a) 2, 3, 4, 5 and 6; (b) 3, 4, 5 and 6.

Figure 5 .

 5 Figure 5.54. Temperatures estimated with the ASIR algorithm for niobium by using the measurements of detectors: (a) 2, 3, 4, 5 and 6; (b) 3, 4, 5 and 6.

Figure 5 .

 5 Figure 5.55. This figure shows that there is a hysteresis of the estimated emissivity values as the sample was heated and then cooled. However, the emissivities shown in Figure 5.55, including the observed hysteresis, are quite consistent with values encountered in the literature [Seifer 2001][Touloukian 1970][Cezairliyan 1970].

Figure 5 .

 5 Figure 5.55. Variation of the emissivity (or emittance) of niobium at the wavelength of detector 3 estimated with the measurements of detectors 3 to 6 and the SIR algorithm.

Figure 6 .

 6 Figure 6.1. Scheme of the complete levitation apparatus.

Multiphysics® 4 .

 4 3b commercial package. For the purpose of speeding-up the inverse problem solution, a reduced model is proposed. This reduced model involves linear boundary conditions, with a combined heat transfer coefficient that takes into account heat losses by radiation and convection. Analytical solution of the heat conduction problem is obtained through classical integral transform technique [Ozisik 1993]. In order to reduce the importance of nonhomogeneous boundary condition on the convergence of the series solution, solution of the problem (not presented here) is defined as the sum of solutions of a quasi-steady state heat conduction problem with nonhomogeneous boundary condition (filtering problem) and a transient heat conduction problem with homogeneous boundary conditions (filtered problem) by assuming [Cotta 1997][Cotta 2017]: 𝑇 * (𝑟 * , 𝜇, 𝜏) = 𝑇 𝐻 * (𝑟 * , 𝜇, 𝜏) + 𝑇 𝐹 * (𝑟 * , 𝜇, 𝜏) (6.20) Direct problem, associated with the above physical problem consists in determining the temperature distribution at discrete angular positions on the surface seen by the pyrometer and the corresponding fluxes collected by the multispectral pyrometer, from the knowledge of initial, boundary conditions, heat source, geometry, and thermophysical properties of the solid metal sample (Figures 6.3 and 6.4).

Figure 6 .

 6 Figure 6.3. Surfaces observed by the pyrometer.

Figure 6 . 4 .

 64 Figure 6.4. Surfaces observed by the pyrometer.

Figure 6 . 9 .

 69 Figure 6.9. Effect of the temporal profile of the pulse on sensitivity coefficients for an exponentially decaying pulse.

Figure 6 .

 6 Figure 6.10 presents the states of Markov chains for thermal conductivity and thermal diffusivity. One can note a rapid convergence of the states of Markov chain to an equilibrium distribution around the exact values of the sought parameters.

Figures 6 .

 6 Figures 6.11 and 6.12 compare prior distributions of the parameters and the posterior distributions obtained from MCMC samples. The histograms of the marginal posteriors for thermal diffusivity and thermal conductivity are also shown in this figure.

Figure 6

 6 Figure 6.11 Scatter distribution of the samples representing: prior distribution with associated marginal (a).

Figure 6 .

 6 Figure 6.12. Scatter distribution of the samples representing: the posterior distribution with associated marginal (b).

Figure 6 .

 6 Figure 6.15. Gray-scaled front and rear faces heat field.

Figure 6 .

 6 Figure 6.16. Temperature map (K) of the front and rear faces heat field.

  observable[Jannot 2009][START_REF] Maillet | [END_REF]][Jannot 2018[START_REF] Jannot | [END_REF]][Pierre 2022]. This temperature-temperature model allows to get rid of the knowledge of the heat flux and the front face heat exchange coefficient. From the quadrupole formalism, link between both front and rear face temperatures is given by Eq. (6.36), whatever the 1D or 2D situations considered: 6. Characterization of thermal properties by non-contact techniques 153 𝑇 𝑡ℎ (0, 𝑒, 𝑡) = 𝑇 𝑒𝑥𝑝 (0,0, 𝑡)⨂ℒ -1

Figure 6 .

 6 Figure 6.17. Temperatures versus time at front (z = 0) and rear (z = e) faces of the sample: 1D (cross) and 2D (circle).

Figure 6 .

 6 Figure 6.18. Evolution of 𝐗 𝛃 over time.

Figure 6 .

 6 Figure 6.19. Estimated temperature profile and residuals curve over time.

Figures

  Figures 6.20 and 6.21 show the diffusivity estimation results from stainless steel

Figure 6

 6 Figure 6.20. Thermal diffusivity of stainless steel 304 and comparison with literature.

Figure 6 .

 6 Figure 6.21. Thermal diffusivity for pure iron and comparison with literature.

  Thickness e is assumed to be 230 µm and thermal conductivity values are taken from literature (𝑘 𝐹𝑒 = 33.1 W•m -2 •K -1 and 𝑘 𝑆𝑡𝑒𝑒𝑙 = 31.7 W•m -2 •K -1 ). For the 2D quadrupole, this coefficient is 1 380 W•m -2 •K -1 and 998 W•m -2 •K -1 for iron and steel, respectively. The large value of the convection coefficient is primarily due to the important radiative heat 6. Characterization of thermal properties by non-contact techniques 157 transfer at high temperature and in a second place to the uncontrolled high-speed argon flow into the inert chamber. Moreover, the method has shown a small sensitivity to Biot number variation. 6.2.4. Partial conclusion Section 6.2 presented the other study dedicated to estimation of thermal diffusivity of solid metallic plates at temperature close to the melting point when submitted to laser heating. The direct model was a temperature-temperature model with two parameters to estimate: thermal diffusivity and heat losses. Sensitivity study showed decorrelation between parameters.

6 .

 6 For that, two experimental apparatuses have been developed: the first one heats with a laser millimetric ball once in aerodynamic levitation; the second one still heats by laser a fixed horizontal plate of about 200 µm. Both apparatuses have the possibility to heat the samples above their melting point.The first study dealing with levitated balls showed that estimation of thermal diffusivity is possible under the condition that all the unwanted parameters present in the theoretical (inappropriate) flux-flux direct model are known: heat flux, absorption coefficient, view-factor… On the contrary, the second study, which uses this time a temperature-temperature direct model, succeeded to estimate thermal diffusivity, the number of unwanted parameters being reduced to one. As said in Section 6.1.4, regarding the first study, a possibility to reduce the number of unknown parameters would be the development of a flux-flux model or a temperature-temperature model. The latter would be possible if emissivity of the metallic ball is known, which remains a difficult task as discussed in Section 5.Inverse techniques used for estimation were different (OLS and Bayesian inference) but successful in both cases, which lets a variety of choices and possible comparisons.However, the development of temperature-temperature models would not be possible without the technical quality of the experimental apparatuses. Experimentally, the use of several mirrors offers the possibility to see different scenes of the same sample at the same time and with the same measurement instrument (the high-speed camera).The physical nature of the input and of the observable is thus the same: here the radiative flux. This principle has already been tested and presented in Section 4 with materials at room temperature with the help of thermocouples. Now in Section 6, the sample temperatures being very high, the use of a laser heating proscribed thermocouples, which are substituted by visible or infrared camera. Therefore, a step of calibration of the camera has been added. I have presented a synthesis of my activities as Associate Professor, where readers can find detailed conclusions at the end of Sections 4 to 6. Concerning my teaching activities, I wish to pursue and diversify my involvement in terms of teaching and administrative responsibilities. Regarding research, these last years have been an opportunity for me to develop my collaboration at different local, national, and international levels; and I also want to follow up on this investment by getting more involved through projects. I am about to co-direct a post-doctoral internship at UFRJ (Brazil) with Professor Helcio R. B. Orlande on the subject of understanding the development of microorganisms by flash method and inverse techniques. My previous stay in Rio de Janeiro was also an opportunity to meet Liviu Nicu, CNRS research director and director of the CNRS South America section: several possibilities for financial supports over different durations exist. But the construction of a collaborative project is still only at the maturation stage.Perspectives regarding thermophysical characterization developed in Section 4 concern a less macroscopic approach to transport phenomena with a more rigorous consideration of the internal structure of materials. Literature provides many works on this aspect with the development of predictive models and multiphase and multiconstituent behaviors, as for hemp concretes or aerogels. The use of analytical models taking into account directional thermal conductivity seems more complicated.Finite difference numerical models will take over, always in the spirit of developing direct temperature-temperature models. From an experimental point of view, the in-progress thesis work of J. Houssein [ACL16] (on liquid metals) proposes a promising experimental method for bidirectional visualization of the material tested by a set of mirrors and visible thermography. As the temperature in this case is close to ambient, the measurement would be made by infrared thermography, which would be able to observe the propagation of the heat flow in all directions, an ideal situation to verify and quantify the anisotropy.The first part of Section 6 has shown the limitation of presented flux-flux model concerning the estimation of thermal conductivity and diffusivity of levitated balls of metals at temperature close to liquid state. Correlation between the aforementioned parameters of interest and the unknown ones makes impossible their proper estimation.Therefore, one interesting perspective would be to reformulate the direct fluxtemperature model by a new temperature-temperature or flux-flux model. The number of parameters to estimate is then reduced to three: thermal conductivity, thermal diffusivity, and convective and radiative losses. The main issue deals with the calibration of the pyrometer and the experiment. As the ball turns around its vertical axis, it is necessary that the heat propagation in the ball is axisymmetric with a heating at the top of the ball.My implication with the characterization of the physical properties at temperature around and above the melting point continues with the thesis of J. Houssein (defense in 2024) for thermal diffusivity; but also, with ANR project proposed by Associate Professor Mickaël Courtois (see Section 3) for the estimation of heat capacity and latent heat of liquid metal through levitation technique.For the axis of research developed in Section 5 about multispectral pyrometry, efforts should be made on models of knowledge of emissivity according to the following dependencies: wavelength, roughness and surface impurities. Different methods of simultaneous estimation of temperature-emissivity couple have shown satisfaction, but their success is conditioned by the most reliable model of emissivity. If the emissivity model is satisfying, then the number and the position of the wavelengths should become less of a concern when the measurements are made by a spectroradiometer, which has a wide choice of wavelengths. A first look at the state of the art shows that some remarkable works on the study of oxidation kinetics or the effects of roughness have been undertaken

  Deux axes majeurs sont présentés dans ce document. Le premier axe concerne la caractérisation des propriétés thermophysiques de matériaux de natures différentes par la mise en place de protocoles expérimentaux à partir de modèles analytiques et d'étude de sensibilité. La définition d'un protocole expérimental nécessite le développement d'un modèle direct analytique ou numérique. Le modèle analytique peut contrainte fortement les expériences, mais les temps de calcul sont le plus souvent plus courts qu'avec le modèle numérique, ce qui est confortable pour l'estimation de paramètre par techniques inverses. De plus, l'usage de modèle températuretempérature diminue le nombre de paramètres inconnus par rapport à des modèles plus classiques flux-température. Le second axe concerne l'estimation simultanée de la température et de l'émissivité par pyrométrie multispectrale. Elle repose sur un problème initialement sous-dimensionné et mène bien souvent dans une impasse puisque le caractère imprévisible de l'émissivité d'une surface ne permet pas de la définir théoriquement sous la forme d'un modèle direct. Le point commun entre les deux thématiques est l'usage de techniques inverses pour l'estimation des paramètres d'intérêt. Parmi toutes celles existantes, plusieurs techniques ont été utilisées et parfois comparées : les techniques déterministes de type moindres-carrés (OLS et MLE) et les techniques stochastiques de type bayésien (MCMC, MAP, filtre de particule).

Table 1

 1

		.1. Synthesis of my diplomas.
	1998	Baccalauréat STL Physique, Chalon-sur-Saône (71).
	1998-2001 DUT Mesures Physiques, Université de Bourgogne, Le Creusot (71).
	2001-2003	Licence/Maîtrise IUP Transferts Thermiques, Faculté des Sciences de Reims (51).
	2003-2004	DEA Énergétique Physique, Institut National Polytechnique de Grenoble (INPG), Grenoble (38).

The first internship dedicated to research was during my DEA in Grenoble in 2004. M. Alain Chiron de La Casinière (associate professor), from the IRSA laboratory (Interaction entre le Rayonnement Solaire et l'Atmosphère) of the Université Joseph

Table 2 .

 2 1. Synthesis of classes taught during my PhD.

			2004/2005	2005/2006	2006/2007
	Heat transfers	TD TP	16 40	20 40	10 40
	Mechanics	TD		7.5
	Materials	TP			20
	2.2. Teaching at the Université Bretagne Sud

I do most of my teaching in the department Métiers de la Transition et de l'Efficacité Énergétiques (MTE2, ex Génie Thermique et Énergie, GTE) of the Institut Universitaire Technologique in Lorient, one of the components of Université Bretagne Sud. Figures 2.1

  Orlande from the University of Rio de Janeiro, Brazil. I worked on the temperature emissivity estimation using different inverse techniques. During this journey, I was able

	Present situation: R&D Project Manager in Physico
	Chemistry at Huddle Corp
	Associated publications: ACL8, ACTI10
	3.2. Scientific supervision
	Tables 3.1 to 3.3 gather all my supervision, respectively, for PhD students (4

to present this work during a seminar at UFRJ and as an invited lecturer at the Brazilian national thermal congress ENCIT2022 in Bento Gonçalves [INV1]. I had the opportunity to give course (8h) during my stay. This course was delivered to students in Master degree of preparation for their PhD and dealt with radiative metrology and temperature/emissivity estimation by multispectral pyrometry. Florian Pradelle, Associate professor in the department of mechanical engineering of the Pontifícia Universidade Católica do Rio de Janeiro (Brazil), proposed me to present next November a video conference for graduated students. I have reviewed 6 articles over the period 2016-2023 for several international journals: International journal of thermophysics, Heat transfer engineering, Building performance, Applied thermal engineering, Inverse problems in science engineering. I also reviewed, for the title of Doctor Europaeus, the PhD manuscript of Andrea Morcerino, Università di Parma, Italy: Inverse heat transfer approach for heat exchanger characterization, February 12, 2019, work supervised by Professor Fabio Bozzoli. defended PhD, 1 in progress), Master Degree students, and Under graduation degree students. The name of all the supervisors are mentioned for the PhD supervision.

Table 3 .

 3 2. Master degree supervision.

	Student	Year		Work
	Mika Ranaivo	2022	Measurement of the viscosity of liquid metals by aerodynamic levitation.
	Thibault Faisandier 2021	Modeling of a process dedicated to the thermal characterization.
	Clémantyne Aubry 2019	Simultaneous estimation of temperature and emissivity by least-squares and bispectral pyrometry.
			Developments of algorithms based on least-squares
	Maxime Capp	2018	dedicated to the simultaneous estimation of temperature and emissivity for the thermal diffusivity estimation of
			liquid metals during aerodynamic levitation
			Radiative	measurements	through	multispectral
	Dylan Le Maux	2017	pyrometry. Application to liquid metals during
			aerodynamic levitation.
	Florian Gesbert	2016	Thermal characterization of materials through random heating -application to insulating materials.

Table 3 .

 3 3. Under graduation degree supervision.

	Student	Year	Work
	Tom Guehenneux 2023	Surface tension of liquid metal measurements through aerodynamic levitation.
	Maxime Lumeau 2019	Characterization of thermal properties of bituminous materials.
	Thomas Lenice	2018	High temperature characterization of materials.

Table 3 .

 3

4 gathers all the publications since 2006. The classification has been made according to the HCERES proposition (the link is given in the legend of Table 3.4). Moreover, Figures 3.1 and 3.2 present the progression of the published articles since 2006, only ACL, C-ACTI, and C-ACTN are presented.

Table 3 .

 3

	4. Number of publications since 2006 (classified according to the
	HCERES).
	International publications (ACL)	21
	International communications (C-ACTI)	20
	National communications (C-ACTN)	26
	Invited conference (C-INV)	1
	Book chapters (COS)	8
	Seminars and conferences (COM)	8
	Other publications (AP)	2

  Dylan Le Maux, et al., Mesure de tension de surface de métaux à l'état liquide par lévitation aérodynamique, Actes du congrès de thermique SFT, Belfort, 9 -12 juin 2020. ACTN23 Édouard Geslain, et al., Caractérisation des conditions de contact pour le soudage par point, 24 ème congrès de mécanique, Brest, 26 -30 août

	la détermination de la conductivité thermique de matériaux bio-
	sourcés, Actes du congrès de thermique SFT, Gérardmer, 28 -31 mai
	2013.
	ACTN12 Thibaut Colinart, et al., Étude expérimentale et numérique des
	caractéristiques thermiques d'une paroi multicouche de béton de 2019. chanvre, Actes du congrès de thermique SFT, Bordeaux, 29 mai -1
	ACTN22 Dylan Le Maux, et al., Mesure de la masse volumique de métaux à l'état juin 2012.
	liquide par lévitation aérodynamique, Actes colloque de thermique ACTN11 Christophe Rodiet, et al., Optimisation du nombre de longueur SFT, Nantes, 3 -6 juin 2019. d'onde pour la mesure de température par la méthode
	multispectrale sur des matériaux opaques à émissivité variable dans ACTN21 Bernard Lamien, et al., Développement d'un dispositif expérimental la gamme Infrarouge-Ultraviolet, Actes du congrès de thermique pour la caractérisation de métaux autour du point de fusion par lévitation aérodynamique, Actes colloque de thermique SFT, Pau, 29 SFT, Bordeaux, 29 mai -1 juin 2012.
	mai -1 er juin 2018. ACTN10 Thibaut Colinart, et al., Vers une définition de l'état sec d'agro-
	matériaux à base de chanvre, XXX e Rencontres de l'AUGC-IBPSA,
	Chambéry, 6-8 juin 2012.
	du congrès de thermique SFT, La Rochelle, 26 -29 mai 2015.
	ACTN14 Laurent Dejaeghere, et al., Développement d'un banc expérimental
	dédié à la caractérisation de propriétés thermophysiques de métaux
	à hautes températures -Validation de la mesure de température par
	méthode multispectrale, Actes du congrès de thermique SFT, Lyon, 3

Pierre, H. R. B. Orlande, Philippe Le Masson, Dylan Le Maux, M. Courtois, and Christophe Rodiet, Sequential estimation of high temperatures of liquid metals by using particle filter methods, High Temperatures -High Pressures, Vol. 52, pp 1-23, 2023. ACTN24 ACTN20 Édouard Geslain, et al., Mesure de température de contact lors d'une opération de soudage par point, Actes colloque de thermique SFT, Pau, 29 mai -1 er juin 2018. ACTN19 Thomas Pierre, et al., Définition d'une méthodologie pour l'estimation de la diffusivité thermique de métaux solides et liquides par méthode bayésienne, Actes colloque de thermique SFT, Marseille, 30 mai -2 juin 2017. ACTN18 Édouard Geslain, et al., Visualisation par thermographie infrarouge du développement de l'échauffement lors du soudage par résistance par point, Actes du colloque de thermique SFT, Marseille, 30 mai -2 juin 2017. ACTN17 Édouard Geslain, et al., Caractérisation des résistances de contact électrique entre tôles d'acier revêtues : application au soudage par résistance par point d'un assemblage dissymétrique de 3 tôles, Actes du colloque de thermique SFT, Marseille, 30 mai -2 juin 2017. ACTN16 Thomas Pierre, et al., Développement d'un dispositif expérimental de soudage MIG équipé d'un pyromètre multispectral -Estimation de la température et de l'émissivité autour du bain fondu, Actes du colloque de thermique SFT, Toulouse, 31 mai -3 juin 2016. ACTN15 Laurent Dejaeghere, et al., Influence de l'atmosphère environnante pour des mesures de hautes températures par voie radiative, Actes ACTN9 Thomas Pierre, Thibaut Colinart, Caractérisation thermo-hydrique d'agro-matériaux à base de chanvre, Actes du congrès de thermique SFT, Perpignan, 24 -27 mai 2011, p. 979-984. ACTN8 Christophe Rodiet, et al., Mesure de température par méthode multispectrale, Actes du congrès de thermique SFT, Perpignan, 24 -27 mai 2011, p. 495-500. ACTN7 Christophe Lanos, et al., Construire et réhabiliter : vers quelles solutions d'isolation ? colloque Matériaux -Génie civil et habitat du futur, Nantes, Octobre 2010. ACTN6 Patrick Glouannec, et al., Influence de la formulation et du procédé de mise en oeuvre sur les propriétés thermique, mécanique et hydrique du béton de chanvre, colloque Matériaux -Génie civil et habitat du futur, Nantes, Octobre 2010. ACTN5 Olivier Balima, et al., Modèle éléments finis de type moindres carrés pour des applications en Tomographie Optique, Actes du congrès de Thermique SFT, Vannes, 26 -29 mai 2009, p. 259-264. ACTN4

  Pierre, et al., Méthode multi-spectrale avec un réseau de diffraction pour la mesure de la température dans les ultraviolets,

			Nomenclature
			Thermique, Valenciennes, 31 mai -3 juin 2022.
	2019 COM6 Latin letters	Philippe Le Masson, Thomas Pierre, A development of an estimation
	A	method of a surface heat flux as a discrete time function, METTI 7 quadrupole coefficient
	𝒜	Advanced School, Porquerolles, France, sept. 29 th -Oct. 4 th 2019. quadrupole coefficient
	2016 COM5 b half width, m Thomas Pierre, et al., Présentation d'un dispositif hautes températures dédié l'estimation de la diffusivité thermique de métaux à l'état fondu, Séminaire Matériaux et Fluides à Hautes Températures 2016, Groupe B quadrupole coefficient
			thématique SFT « Hautes Températures », 20 septembre 2016, Paris.
	2015 COM4	Édouard Geslain, et al., Observation par caméra thermique du
			développement de la soudure au sein d'assemblages d'aciers
			hétérogènes lors du soudage par résistance par point, CCRS « Soudage
			par résistance », 2 avril 2015, Villepinte.
	Other publications (OP)
	2015 OP2	Édouard Geslain, et al., Mise au point d'un dispositif expérimental pour
			la visualisation par caméra thermique du développement du noyau
			lors du soudage par résistance par point, Institut de Soudure, juillet-
			août 2015, p.39-45.
	2008 OP1	Thomas PIERRE, et al., Mesure de hautes températures par méthode Actes du congrès de thermique SFT, Toulouse, 3 -6 juin 2008, p. 379-384. multi-spectrale et statistique, Revue de l'Électricité et de
		ACTN3 Thomas Pierre, et al., Mesure de hautes températures par méthode l'Électronique 10, p. 34-39, 2008.
			multi-spectrale et statistique dans la gamme UV-visible, Actes du 4 e
			Colloque Interdisciplinaire en Instrumentation -C2I 2007, Nancy, 17
			-19 octobre 2007, p. 94-103.

ACTN2

Thomas Pierre, et

al., Mesure de la température à l'échelle microscopique par méthode multi-spectrale et statistique dans la mesure de température par thermographie infrarouge à l'échelle microscopique, poster « Work in progress », Société Française de COM3 Édouard Geslain, et al., Observation par caméra thermique du développement de la soudure au sein d'assemblages d'aciers hétérogènes lors du soudage par résistance par point, Journée thématique SFT « Transferts aux interfaces », 11 mars 2015, Paris. 2014 COM2 Thomas Pierre, et al., Développement d'un banc à haute température -Mesure de température par pyrométrie multispectrale, Séminaire Méthodes de mesures et de caractérisation des matériaux à hautes températures, Groupe thématique SFT « Hautes Températures », Paris, 13 février 2014. 2008 COM1 Thomas Pierre, Mesure de la température à l'échelle microscopique dans la gamme ultraviolet-visible, Les midis du CURAL, Université du Québec à Chicoutimi, Québec, Canada, 19 juin 2008.

Table 4 .

 4 1. General information from my published works.

			[Pierre 2014]	[Pierre 2017]	[Pierre 2019a]	[Pierre 2020]	[Pierre 2022]	[Pierre 2019b]
	Material	Insulating Conducting	x	x		x	x x	x
	Observable	Thermocouple Camera	x	x		x	x	x
	Principal flux	In-depth	x	x				x
	direction	In-plane				x	x
	Model	Flux-temp. Temp.-temp.	x	x		x	x	x
	4.1. Estimation of thermal properties of insulating bio-sourced material with the hot-strip
	technique and influence of the temperature and humidity [Pierre 2014]

Table 4

 4 Mostly used in building, hemp concrete is made from vegetal aggregates such as hemp shives (or hemp hurds) and a lime-binder. With the appropriate proportion of hemp and binder, hemp concrete can cover different uses.

	.1 details different types of information concerning the nature of the
	investigated samples, of the observable, the principle direction of the flux, and the type of
	model. The advantages and the drawbacks of all this information are discussed in each
	further section, leading to a global reflection of experimental protocol for a given material
	and using appropriate tools.
	4.1.1. Context
	This study is dedicated to the thermal characterization of bio-sourced hemp
	concrete and to propose a predictive thermal conductivity model functions of the
	temperature and the water content.

Table 4

 4

.2 presents all results obtained by estimation and comparison with measurements with other techniques. It shows that, with the radiative component, estimated thermal diffusivity and thermal conductivity lead to a value of heat capacity in accordance with the one obtained with calorimetry. Table 4.2. Values of estimated parameter with and without radiation.

Table 4

 4

	apparatus of this study	guarded hot plate	hot plate calorimeter

.3. Experimental results of thermal properties of aerogels.

  Two experimental temperatures are required to perform an estimation with the relation (4.63): observable 𝑇 𝑒𝑥𝑝 (𝑟, 𝑡) and input temperature 𝑇 𝑒𝑥𝑝 (𝑟 1 , 𝑡). Estimation procedure is identical with parameters to estimate (4.66).

	4. Characterisation of thermal properties by heating contact techniques
	4.3.3.2. The cylindrical model
	66

Table 4 .

 4 The acquisition frequency is 50 Hz and the tests last 30 s. Sample is warmed with a heating element covered by a thin aluminum foil, on which test sample is stuck. Aluminum foil is used to avoid radiation of the heating element to the sample. Type K thermocouple is placed at the rear side of the sample only to control initially the temperature given by The second experimental set-up, developed to correspond to the theoretical model, is presented inFigures 4.32 and 4.33. A thin sample is heated on its center by a heating source. The heating source is a cylindrical copper spot of about r0 = 1 mm radius embedded in the sample and heated by Joule effect through electric current coming from the sample rear face. The electric wires are maintained with conductive stick, which should be let dry at least for 24 hours. Then the heating element is switched on few seconds and turned off when a maximum temperature increase of 10 K is reached in the sample or when semi-infinite assumption is no more valid. All tested samples are gathered in Table4.4. 4. Synthesis of the materials tested for the Cartesian and the cylindrical models.

	the camera, then it is removed. A continuous heating is applied until sample surface
	temperature has increased by a maximum of about 20 K. Three types of surrounding are
	considered in the stainless steel chamber: vacuum, natural convection, and forced

4.30 and 4.31

present global view of experimental apparatus composed of a stainless steel chamber with a sapphire window in which the heated sample is placed, and of an IR camera. Vacuum can be performed within the chamber. IR camera is a Flir X6580sc model working in the range [1.5 µm -5.1 µm] and equipped with a 15 µm resolution microscopic lens. convection. It is noteworthy that the air velocity is not measured and the Reynolds number cannot be calculated.

Table 5

 5

	gathers estimation results for the ten possible combinations when balsa is
	in vacuum. Once thermal diffusivity is estimated, the thermal conductivity is calculated
	using values of density 𝜌 and heat capacity 𝑐 𝑝 𝑚 from measurements performed in our
	laboratory with a micro calorimeter for each sample [Pierre 2019a]. Finally, combination
	of Eq. (4.57) and (4.3) is used to calculate heat exchange coefficient h.

Table 4 .

 4 5. Estimation results for all possible combinations in the case of balsa in vacuum.

	combination	zi/z/zj mm/mm/mm	convergence	𝜎 𝛼𝐻	𝛼 mm 2 •s -1	h W•m -2 •K -1	covariance matrix of residuals
	1	0/0.99/1.77	yes	0.047	0.304	0.7	0.022
	2	0/0.99/2.90	yes	0.0008 0.407	2.9	0.032
	3	0/0.99/4.14	yes	0.104	0.467	3.7	0.017
	4	0/1.77/2.90	yes	0.115	0.228	9.4	0.197
	5	0/1.77/4.14	yes	0.097	0.378	9.2	0.185
	6	0/2.90/4.14	yes	0.125	0.319	10.9	0.230
	7	0.99/1.77/2.90	no	1	x	x	x
	8	0.99/1.77/4.14	no	1	x	x	x
	9	0.99/2.90/4.14	no	1	x	x	x
	10	1.77/2.90/4.14	no	1	x	x	x

Table 4 . 5

 45

		balsa ⊥	balsa //	PVC	DC54	AA5182
	r1 (mm)	2.00	2.00	2.00	2.00	2.00
	rmin (mm)	3.15	3.10	3.15	4.65	6.00
	 est ±  (mm 2 •s -1 )	0.97 ± 0.27 1.20 ± 0.38 0.22 ± 0.03	19 ± 4	49 ± 17
	 literature (mm 2 •s -1 )	0.48 ± 0.02 0.95 ± 0.03 0.20 ± 0.03	18.70*	53.6* ± 1.4*
	H est ± H (10 3 m -2 )	90 ± 30	21.5 ± 6.9	31.0 ± 4.5	-	-
	h (W•m -2 •K -1 )	4.1 ± 2.1	4.5 ± 1.9	7.8 ± 2.0	-	-
	k (4.65)	0.09 ± 0.06 0.02 ± 0.03 0.03 ± 0.02	< 10 -3 **	< 10 -3 **
	For PVC, estimated in-plane thermal diffusivity 𝛼 has a mean value similar to
	reference. Regarding modified Biot number H, distribution is more uniform and scattered,
	which may be due to the low sensitivity of this parameter. Nevertheless, from thermal

. Estimation results for the four samples (* interpolated value from laboratory measurements, ** for h = 10 W•m -2 •K -1 ). conductivity value of PVC found in literature, calculation of heat exchange coefficient h

  (𝜆) = 𝜏 𝑙 𝜏 𝑐 𝜏 𝜆𝑓 𝜏 𝑐 𝜌 𝑚1 𝜏 𝑓6 = 𝜏 𝑖𝑛𝑝𝑢𝑡 𝜌 𝑚1 𝜏 𝑓6(5.13)

	𝑓 5 (𝜆) = 𝜏 𝑖𝑛𝑝𝑢𝑡 𝜏 𝑚1 𝜏 𝑚2 𝜏 𝑓5	(5.14)
	𝑓 4 (𝜆) = 𝜏 𝑖𝑛𝑝𝑢𝑡 𝜏 𝑚1 𝜌 𝑚2 𝜌 𝑚3 𝜏 𝑓4	(5.15)
	𝑓 3 (𝜆) = 𝜏 𝑖𝑛𝑝𝑢𝑡 𝜏 𝑚1 𝜌 𝑚2 𝜏 𝑚3 𝜏 𝑚4 𝜏 𝑓3	(5.16)
	𝑓 2 (𝜆) = 𝜏 𝑖𝑛𝑝𝑢𝑡 𝜏 𝑚1 𝜌 𝑚2 𝜏 𝑚3 𝜌 𝑚4 𝜏 𝑚5 𝜏 𝑓2	(5.17)
	𝑓 1 (𝜆) = 𝜏 𝑖𝑛𝑝𝑢𝑡 𝜏 𝑚1 𝜌 𝑚2 𝜏 𝑚3 𝜌 𝑚4 𝜌 𝑚5 𝜏 𝑓1	(5.18)

). Each component has its own transfer function over the wavelength given by manufacturers and the global spectral function of each optical path is: 97 𝑓 6

Table 5 .

 5 1. Characteristic wavelengths 𝜆i and transfer functions Fi estimation.

	i	1	2	3	4	5	6
	Average Fi × 10 9	33.060 2.279	3.899	2.502	3.249	3.592
	Standard deviation 𝜎 𝐹 𝑖 × 10 9 0.330	0.011	0.013	0.005	0.005	0.009
	Average 𝜆i (µm)	0.483	0.532	0.680	0.851	0.940	1.554

Table 5

 5

	.2. Example of numerical results of radiative fluxes ratio and distance between
	wavelengths in the case of the six-wavelength pyrometer [Dejaeghere 2018] and
	presented in Section 5.1.3. Green cells represent cases where the criteria are satisfied
	contrary to red ones.
	T (K)	𝜆 4 = 0.850 µm	|𝜆 4 -𝜆 1 | = 370 nm	|𝜆 4 -𝜆 2 | = 320 nm	|𝜆 4 -𝜆 3 | = 170 nm	|𝜆 5 -𝜆 4 | = 90 nm
		Eq. (5.32)	𝛼 41 (5.36)	𝛼 42	𝛼 43	𝛼 54
	1 273	64 nm	1 622	289	9	2,2
	2 273	115 nm	17	8,5	2,1	1,2
	3 273	164 nm	3,1	2,2	1,2	1,0

Table 5

 5

.2 shows that the criteria are temperature-dependent and the selection of wavelengths depends on the desired temperature range. The use of the six wavelengths (with the complete pyrometer!) is not systematic for large temperature variation. 5. Temperature emissivity estimation with multispectral pyrometry 106

Table 5

 5 Temperature emissivity estimation with multispectral pyrometry 111 study is dedicated to multispectral pyrometry with the same experimental data, also using Bayesian inference with particle filter[Pierre 2023[START_REF] Lamien | [END_REF]]. Table5.3 explains the theoretical model used: Planck's law or Wien's approximation; the emissivity model: constant, linear, even independent; what parameters to estimate; and their prior for Bayesian inference: Gaussian, normal.

.3. A first study concerns bispectral pyrometry using OLS, Bayesian inference using Maximum a Posteriori (MAP), and Bayesian inference with Monte Carlo Markov Chains (MCMC) [Pierre 2022]. The second 5.

Table 5 .

 5 𝑒 𝑖 ′ differs from 𝑒 𝑖 because of the linearization. 3. Synthesis of the techniques used for simultaneous temperature emissivity estimation[Pierre 2022][Pierre 2023[START_REF] Lamien | [END_REF]]. The bispectral method with ordinary least squares Fundamentally, the bispectral method consists in estimating temperature by using two radiative fluxes measured at wavelengths 𝝀 𝒊 and 𝝀 𝒋 . Regarding the spectral emissivity values, one of them is estimated in the same time as temperature, say 𝜺 𝒊 at 𝝀 𝒊 , whereas a functional relationship is assumed between 𝜺 𝒋 (at 𝝀 𝒋 ) and 𝜺 𝒊 . One possible functional relationship is 𝜺 𝒋 /𝜺 𝒊 = 𝝃 𝒋𝒊 where 𝝃 𝒋𝒊 is a predetermined constant. In the classical "greybody" approximation constant 𝝃 𝒋𝒊 is set to one, which, to be satisfactory, requires that wavelengths 𝝀 𝒊 and 𝝀 𝒋 be not too far from each other (see Section 5.2.1).

	This gives the

′ ] 𝑻 and in Eq.

(5.45)

µi is defined by Eq.

(5.46)

, Tref is an arbitrary reference temperature used for scaling the temperature, 𝑒 𝑖 ′ is a Gaussian additive experimental error affecting the observables 𝑌 𝑖 ′ , and m is the length of the time step.

Table 5 .

 5 4. Sampling Importance Resampling (SIR) algorithm[START_REF] Arulampalam | A tutorial on particle filters for on-line non-linear/non-gaussian Bayesian tracking[END_REF]][Ristic 2004][Kaipio 2006[START_REF] Ozisik | [END_REF]].

  ][Ozisik 2021]. It is then expected that the optimal

	importance density	𝜋(𝛃 𝑘 𝑗 |𝛃 𝑘-1 𝑗 , 𝚽 𝑘 𝑒𝑥𝑝 ) can be better represented [Arulampalam
	2001][Ristic 2004]. The resampling is based on some point estimate 𝝁 𝑘 𝑗 that characterizes
	𝜋(𝛃 𝑘 𝑗 |𝛃 𝑘-1 𝑗	), which can be the mean or simply a sample of this density. If the state
	evolution model noise is small, then 𝜋(𝛃 𝑘 𝑗 |𝛃 𝑘-1 𝑗 ) is well characterized by 𝝁 𝑘 𝑗 , so that the
	weights 𝑤 𝑘 𝑗 are more even and the ASIR algorithm is less sensitive to outliers than the SIR
	algorithm. On the other hand, if the state evolution model noise is large, then the single
	point estimate 𝝁 𝑘 𝑗 in the state space may not characterize well 𝜋(𝛃 𝑘 𝑗 |𝛃 𝑘-1 𝑗

Table 5 .

 5

	5. Auxiliary Sampling Importance Resampling (ASIR) algorithm [Arulampalam
	2001][Ristic 2004][Ozisik 2021].
	Step 1
	For j = 1,...,Q draw new particles 𝛃 𝑘 𝑗 from the prior density 𝜋(𝛃 𝑘 𝑗 |𝛃 𝑘-1 𝑗 ) and then
	calculate some characterization 𝝁 𝑘 𝑗 of 𝛃 𝑘 𝑗 , given 𝛃 𝑘-1 𝑗	. Then use the likelihood density to
	calculate the corresponding weights 𝑤 𝑘 𝑗 = 𝜋(𝚽 𝑘 𝑒𝑥𝑝 |𝛃 𝑘 𝑗 )𝑤 𝑘-1 𝑘 .

  6. Characterization of thermal properties by non-contact techniquesAs already mentioned in Introduction of this document, part of PTR2 has been working for more than twenty years in collaboration with companies specializing in all

	6. Characterization of thermal properties by non-contact techniques
	6.1. The estimation of the thermal diffusivity of metals at temperature close to the melting
	point with the aerodynamic levitation
	First part of this section concerns the definition of an experiment of aerodynamic
	types of welding or additive manufacturing. One of PTR2 contributions is to perform levitation to estimate both thermal conductivity and diffusivity through Bayesian
	numerical simulations of welding processes using commercial codes such as Comsol inference, where the theoretical model is a flux-temperature one [Lamien 2019]. The
	Multiphysics®. These industrial processes being highly multiphysical (thermal, levitation apparatus, briefly introduced in Section 5, is well-detailed in [Le Maux 2022].
	mechanical, hydrodynamic, metallurgical, electromagnetism...), large number of physical Note that I also participate to the estimation with this apparatus of hydrodynamic
	properties is required (viscosity, density, surface tension, heat capacity, conductivity...) properties when matter is at liquid state (density, surface tension, and viscosity) but it is
	and over a wide temperature range since matter changes from solid state to liquid state not presented in this manuscript. However, readers may refer to the following references:
	and can reach its boiling point. However, literature is almost mute from liquid state and [Le Maux 2019][Le Maux 2022][Le Maux 2023].
	beyond.
	Since 2016, we have developed two high temperature apparatuses. The first one 6.1.1. Presentation of the levitation apparatus
	deals with metallic balls aerodynamically levitated, the second one with fixed horizontal The experimental device is presented in Figure 6.1 (see also [Le Maux 2019]). Its
	plates with of thickness less than a millimetre. In every case, heating is assured by laser, legend using capital letters in brackets is indicated below. An argon hydrogen mix Ar +
	radiative measurements by infrared and visible fast cameras. 2.5 %vol. H2 is blown through a 60° convergent-divergent nozzle to levitate an
	So far, we are able to determine two physical parameters with these apparatuses: approximately 30 mg sample (about 2 mm of diameter for iron alloys) (A). A first
	density and surface tension (thesis of D. Le Maux [Le Maux 2019][Le Maux 2022]). And numerical flowmeter Brooks SLA5850S controls the lift gas (B). The levitation part is set
	we are working on estimation of viscosity [Le Maux 2023] and on thermal diffusivity in a vacuum chamber equipped with borosilicate viewports (C). The upper window is
	(post-doctoral application of B. Lamien [Lamien 2019], thesis of J Houssein [Houssein dedicated to the heating laser (IPG Photonics YLR 300/3000 QCW at 1 070 nm) (D) and
	2023]). pyrometric measurement. The heating laser has a 280 W maximal power in continuous
	Section 6 deals with the use of pyrometers and cameras during the mode. It is equipped with a 500 mm focalization optic slightly defocused to make the
	characterization of physical properties of materials and is divided in two sub-Sections. aimed spot about 2 mm. A set of mirrors is placed at the upper window and at a lateral
	Section 6.1 is dedicated to estimation of thermal diffusivity of metals at solid state but one (E) in order to deport the upper and lateral views of the sample to a high-speed
	near the melting point [Lamien 2019]. Second 6.2 deals with the other experimental camera (F) Phantom VEO-E 310L (see its calibration in Section 5.1.3). A backlighting laser
	configuration. This time thermal diffusivity estimation is performed through comparison (300 mW, 532 nm diode laser with a beam expander) (G) is placed at the opposite lateral
	between a temperature-temperature analytical model and experimental measurements window. The sample is heated up by the laser until it becomes liquid, its temperature is
	[Houssein 2023]. measured by the pyrometer previously detailed in Section 5.1.3.
	136

Table 6 .

 6 1: Prior probability densities used in the computation of the approximation error.

	Property / Parameter	Dimensionless	Prior	Mean	Standard
		Parameter	Distribution		deviation
	Thermal diffusivity	𝛼 *	Gaussian	𝛼 ̅ * = 1.09𝛼 *	0.15𝛼 ̅ *
	Thermal conductivity	𝑘 *	Gaussian	𝑘 ̅ * = 1.09𝑘 *	0.15𝑘 ̅ *
	Irradiance Radiative Biot number	𝑞 0 * 𝐵𝑖 𝑟𝑎𝑑	Gaussian Gaussian	𝑞 ̅ 0 * = 𝑞 0 * 𝐵𝑖 ̅̅̅ 𝑟𝑎𝑑 = 1.05𝐵𝑖 𝑟𝑎𝑑	* 0.05𝑞 ̅ 0 0.05𝐵𝑖 ̅̅̅ 𝑟𝑎𝑑
	Absorptivity at λlaser = 1 060 nm	𝜅 𝜆,𝑙𝑎𝑠𝑒𝑟	Gaussian	𝜅̅ 𝜆,𝑙𝑎𝑠𝑒𝑟 = 𝜅 𝜆,𝑙𝑎𝑠𝑒𝑟	0.05𝜅̅ 𝜆,𝑙𝑎𝑠𝑒𝑟
	Normal spectral emissivity at λfilter Amplitude correction Form factor Convective Biot number	𝜀 𝜆,𝑝𝑦𝑟𝑜 𝑓 𝑖 𝐹 𝑑1-2 𝐵𝑖 𝑐𝑣	Gaussian Gaussian Gaussian Gaussian	𝜀̅ 𝜆,𝑝𝑦𝑟𝑜 = 1.05𝜀 𝜆,𝑝𝑦𝑟𝑜 𝑓 ̅ 𝑖 = 1.05𝑓 𝑖 𝐹 ̅ 𝑑1-2 = 1.05𝐹 𝑑1-2 𝐵𝑖 ̅̅̅ 𝑐𝑣 = 1.05𝐵𝑖 𝑐𝑣	0.05𝜀̅ 𝜆,𝑝𝑦𝑟𝑜 0.1𝑓 ̅ 0.1𝐹 ̅ 𝑑1-2 0.1𝐵𝑖 ̅̅̅

𝑐𝑣 Figure 6.10. States of Markov Chain for thermal diffusivity and for thermal conductivity.

  𝜃 0 , 𝜃 𝑒 , and Φ 0 are, respectively, the expression of temperature at positions z = 0 and e and the heat flux 𝜑 0 following Laplace and Hankel integral transforms. The coefficients A, B, C and D define the quadrupole matrix[Maillet 2000]. They depend on the Laplace parameter s, the sample thickness e, and the thermophysical properties of the material. 𝐽 𝜈 is the Bessel function of first kind of order . X and Y are given by:

	∞ 𝜃 𝑒 (𝑟, 𝑠) = ∑ 𝑛=1	Φ 0 + 𝑌 + ℎ 𝑒 𝑋 (𝐶 + ℎ 𝑒 𝐷) + ℎ 0 (𝐴 + ℎ 𝑒 𝐵)	𝐽 0 (𝛾 𝑛 𝑟) 𝑅 2 2 𝐽 1 2 (𝛾 𝑛 𝑅)	(6.30)
	𝑋 =	(𝑇 𝑖𝑛𝑖𝑡 -𝑇 ∞ ) 𝑠	[𝑐𝑜𝑠ℎ(𝜔 𝑛 𝑒) -1]	𝑅𝐽 1 (𝛾 𝑛 𝑅) 𝛾 𝑛	(6.31)
	𝑌 =	𝜌𝑐 𝑝 𝑚 𝜔 𝑛	(𝑇 𝑖𝑛𝑖𝑡 -𝑇 ∞ )𝑠𝑖𝑛ℎ(𝜔 𝑛 𝑒)	𝑅𝐽 1 (𝛾 𝑛 𝑅) 𝛾 𝑛	(6.32)
	∞ 𝑛=1			𝐽 0 (𝛾 𝑛 𝑟) 𝑅 2 2 𝐽 1 2 (𝛾 𝑛 𝑅)	(6.29)

Table 6 .

 6 2. Results of estimation for iron and steel using three different direct models.On the side, it is possible to compute a global heat exchange coefficient h, accounting for both convection and radiation heat transfers from estimated Biot number.

			Iron (T = 1 385 K)		Steel (T = 1 502 K)
	Method	𝛼 (mm 2 •s -1 )	Bi × 10 -3	Fo	testim (s)	𝛼 (mm 2 •s -1 )	Bi × 10 -3	Fo	testim (s)
	1D	7.30	9.45	138	5.7	5.52	7.15 104.3	5
	2D	7.45	9.58	140.8	13	5.86	7.24 110.7 18

ACTN13 Thibaut Colinart, et al., Prise en compte des transferts de masse dans

However, the lack of knowledge and/or uncertainties on parameters appearing in the mathematical formulation, such as thermal conductivity of the solid metal, its total hemispherical emissivity, heat losses and laser parameters (irradiance, spot size, pulse width), must be taken into account. A sensitivity analysis is later performed to determine whether these parameters can be estimated together with the parameters of interest.

The sensitivity analysis is performed with the complete forward model and the effects of:

• the laser beam size incident on the steel ball;

• the temporal profile of the pulse;

• and the position of the surface seen by the pyrometer (surface S1 centred on θ1 = 0°, surface S2 centred on θ2 = 20°, surface S3 centred on θ3 = 60°) as illustrated in Figures 6.3 and 6.4, were investigated. The parameters to estimate are the following:

Note that 𝑓 𝑖 is HMi in Figures 6.5 to 6.9. In order to perform sensitivity analysis, the maximum temperature increase of the heated surface was set to 50 K to have a reasonable approximation of the assumption of constant physical properties with respect to temperature variation. The sensitivity coefficients computed for the different wavelengths of the multispectral pyrometer presented the same behaviour, with the largest flux magnitudes obtained for the wavelength 𝜆 = 1 550 nm. Therefore, the analysis below is presented only for this wavelength. Moreover, for the case under study, flux measurements obtained at the wavelength 𝜆 = 1 550 nm and 𝜆 = 940 nm are the only useful information, since the lower measured threshold power of the multispectral pyrometer is 50 nW.

• The influence of the incident laser beam size on the sensitivity coefficients is examined for beam diameters of 1 mm and 2.6 mm, for square pulses with P = 13 W and 𝜏 ℎ = 3 ms, as well as for P = 30 W and 𝜏 ℎ = 15 ms. These values were chosen so that the maximum temperature increase of the heated surface was limited to 50 K. The analysis of the sensivitity coefficients suggests the possibility of simultaneously estimating the thermal diffusivity and the thermal conductivity of the steel ball. However, the remaining model parameters have to be known, since they are correlated and the magnitudes of their sensitivity coefficients are one order larger or of the same order of those for the parameters of interest.

• For the analysis of the effect of the position of the surface where the measurements are taken by the pyrometer on the sensitivity coefficients, a square pulse of power P = 30 W, with pulse duration 𝜏 ℎ = 15 ms is considered, with the laser beam diameter of 2.6 mm. Figures 6.5 In order to maximize the area of the homogeneous temperature zone, and therefore reduce the radial temperature gradients, the initial heating stage is done at a relatively low power (70 W) and takes up to 30 seconds. The duration of the pulse is about 10 ms at a power of 270 W. The inert chamber contains four gold coated mirrors (ThorLabs ME1-M01) that are symmetrically arranged around the sample, so that at the faces of the prismatic mirror, we get a simultaneous view of both faces of the sample.