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Thermal Radiative Properties of Complex Media:Theoretical Prediction Versus ExperimentalIdentificationJean-François Sacadura aa Institut National des Sciences Appliquées de Lyon, CETHIL, Villeurbanne, France

Available online: 31 Mar 2011

To cite this article: Jean-François Sacadura (2011): Thermal Radiative Properties of Complex Media: Theoretical PredictionVersus Experimental Identification, Heat Transfer Engineering, 32:9, 754-770

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Heat Transfer Engineering, 32(9):754–770, 2011Copyright C©© Taylor and Francis Group, LLCISSN: 0145-7632 print / 1521-0537 onlineDOI: 10.1080/01457632.2011.525140

Thermal Radiative Properties ofComplex Media: TheoreticalPrediction Versus ExperimentalIdentification

Jean-Francois SacaduraInstitut National des Sciences Appliquees de Lyon, CETHIL, Villeurbanne, France

In many engineering applications and natural phenomena, thermal radiation interacts with complex media composed ofdispersed phases that may be of different type: solid/solid, solid/gas, or liquid/gas. Most of them are semitransparent mediathat emit, absorb, and scatter thermal radiation. Heat transfer by combined radiation with conduction or convection in suchmedia is a problem of high practical importance, mostly in situations where radiation is a dominant mode. Improvement ofthermal performance of such materials or of the manufacturing processes that involve these media requires the availability ofefficient methods (i) for radiative transfer modeling, and (ii) to predict and to experimentally determine the thermophysicalproperties intended to feed the models. This paper is focused on radiative properties assessment. After a brief overview of thematerials and properties of interest, the emphasis is put on methodology of property investigation combining both theoreticalprediction and experimental identification. Examples related to different particulate media are presented, showing recentadvances and needs for further investigation.

INTRODUCTION

Thermal radiation is an important and even predominantmode of energy transfer in many engineering applications andnatural phenomena. Most of them involve semitransparent me-dia that may be not only absorbing/emitting materials but alsoscattering media due to their dispersed structure. Examples ofthese media are very common and numerous. They includesolid/solid systems like nonporous ceramics, surface pigmentedcoatings, or solid/gas systems (media composed of fibers, foams,or micro-sized powders, as most insulating materials are, tex-tiles, food products, fluidized and packed beds, combustors,catalytic reactors, and soot), or liquid/gas systems like bubblingmedia, and sprays. Examples are shown in Figure 1.

The author gratefully acknowledges Dr. Leonid Dombrovsky and Dr. Do-minique Baillis for the clarifying discussions and the material and informationprovided that have been very helpful in the preparation of this paper.

Address correspondence to Professor Jean-Francois Sacadura, Institut Na-tional des Sciences Appliquees de Lyon, CETHIL-UMR CNRS 5008, INSALyon–Bat. Sadi Carnot, 69621 Villeurbanne Cedex, France. E-mail: jean-francois.sacadura@insa-lyon.fr

Improving thermal performance of these media or efficiencyand quality of the manufacturing processes involving them re-quires the capability of modeling the radiative heat transfer in-side these media and the assessment of the radiative propertiesin order to feed the models. Owing to the significant progressesacquired in numerical methods over the two past decades, theprediction of heat transfer by combined radiation with conduc-tion or convection in most industrial applications is nowadayspossible through direct models handled by using commercialcodes in which the radiative transfer equation is commonlysolved by different families of methods, of which some are nu-merical, other ones semi-analytical [1]. A remaining problem isthe assessment of radiative properties. It is an important topicthat attracts continuously a great deal of interest from the radia-tive transfer community, as shown by the reviews of Viskantaand Menguc [2] and Baillis and Sacadura [3] dealing with ra-diation transfer in particulate media. This paper is focused ontheoretical prediction and experimental determination of semi-transparent media radiative properties. After a brief recall ofthe properties of interest, the emphasis is put on methods ofproperty investigation, both predictive and experimental. Someexamples are presented showing recent advances and needs for

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Figure 1 Examples of particulate media. Scanning electronic microscope (SEM) views of sample cuts.

further investigation. The aim of this overview is not to pro-vide a comprehensive reference list. Instead we prefer to focuson relevant works and reviews where the reader can find morereferences.

THEORETICAL BACKGROUND: RADIATIVETRANSFER EQUATION AND RADIATIVE PROPERTIES

At the basis of radiative transfer calculations is the choiceof a direct model that may be of a different kind and level ofsophistication according to several criteria: complexity of theproblem of interest (medium, radiation source, and boundaryconditions), aim of the calculation and accuracy expected, timeand effort allocated to the investigation, and also the experienceof the investigator. Modeling radiation transfer inside particulatemay be achieved by continuous or discrete approaches.

Continuous Approaches

The continuous approaches used in engineering applicationsare mostly based on the radiative transfer equation (RTE). Theparticulate medium is modeled as a random distribution of parti-cles and the scattering and absorption properties of the mediumare based on those of the discrete particles. In some cases theparticles are embedded in a host medium that may significantlycontribute to the absorption of the global medium. Independentscattering, also called far-field single-scattering approximation,is assumed, which means that single particles are random anduncorrelated and the average distances between neighboringparticles are greater than particle size and wavelength.

Practical data on scattering regimes are available in the lit-erature in terms of the particle size to wavelength ratio x =πd/λ, called the particle size parameter or diffraction param-eter, where d is the particle diameter and λ the wavelength,the ratio of particles clearance to wavelength c/λ, and fv, theparticle volume fraction, which is related to the ratio c/d. Itis commonly recognized (see the independent and dependentscattering map from Tien and Drolen [4]) that dependent scat-tering effects may be neglected when fv < 0.006 or c/λ > 0.5.

This is the case in most engineering applications, exceptingsituations involving very small agglomerated particles such assoot, nanosphere insulations, and other nanoparticles that arecurrently receiving an increasing interest. Dependent scatteringis also present in densely packed beds of particles larger thanwavelength.

The radiative properties are used in RTE, which governs theconservation of the radiation spectral intensity Iλ at wavelengthλ, and along a direction �. Excepting situations involving veryfast radiative sources like in laser pulse experiments, which arebeyond the scope of this paper, the RTE may be considered inits stationary form and written as:

d Iλ(�)

ds= − (σλ + κλ) Iλ

(�

) + κλ Ibλ

+ σλ

4π

∫4π

Iλ(�′)Pλ

(�′ → �

)d�′ (1)

where s is the space coordinate, σλ and κλ are the scattering andabsorption spectral volumetric coefficients, respectively, and Ibλ

is Planck’s blackbody emission function, which involves the realpart nλ of the refractive index mλ of the medium. Pλ

(�′ → �

)is the spectral scattering phase function and is proportional tothe probability that the radiation propagating in a direction �′

is scattered in a direction �. d�′ is an infinitesimal solid angleassociated to the direction �′. Note that, alternatively, the RTEmay be and is often expressed in terms of two other spectral vol-umetric parameters, the extinction coefficient βλ = κλ + σλ, andthe single-scattering albedo ωλ = σλ/βλ, instead of κλ and σλ.This option does not affect Pλ. Thus, the radiation properties ofthe pseudo-continuous medium are four: nλ, κλ, σλ (or βλ, ωλ),and Pλ

(�′ → �

), and even more, depending on the number of

shape parameters used in the phase function representation. Theclassical representation of the scattering phase function uses anexpansion in Legendre polynomials series.

One should note that a significant complexity is added tothe analysis in case of a nonisotropic medium, which meansthat its radiative properties depend on the direction of radiationincident onto each elemental volume of the medium. The phasefunction Pλ especially depends on both directions �′ and �. Animportant contribution to the subject is found in the pioneeringpapers by Houston and Korpela [5] and Lee [6–8] dealing with

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fiber radiative properties. When the medium may be assumedas isotropic the phase function is expressed in term of the angle� between the directions of incident and scattered radiation ata scattering location. In engineering applications the media areoften assumed as isotropic. However, even with this assumptionthe phase function generally varies with �. In this case it isreferred as anisotropic phase function.

As a number of particulate media exhibit anisotropic phasefunctions, often sharply peaked, this requires a high-order ex-pansion and, subsequently, an impressive number of coefficientsis needed to define the phase function from a polynomial seriesexpansion. Therefore it is generally preferred to use an approx-imate phase function based on smaller number of shape pa-rameters. Among useful models, the Henyey–Greenstein phasefunction, which only involves one shape parameter, referred asthe asymmetry parameter gλ , is perhaps the most popular:

PH G(�, gλ) = 1 − g2λ(

1 + g2λ − 2gλ cos �

)3/2 (2)

where � is the angle between the directions of incident andscattered radiation at a scattering point and

gλ = 1

2

∫ π

0Pλ(�) cos (�) sin (�) d� (3)

The PHG model enables one to represent scattering patternsexhibiting just one peak, even sharp, forward or backward di-rected. For more complex patterns, Nicolau et al. [9] proposeda weighted combination of two HG functions plus an isotropiccomponent:

PH G N (�, g1λ, g2λ, f1λ, f2λ) = f2λ [ f1λ PH G(�, g1λ)+(1 − f1λ)PH G(�, g2λ)] + (1 − f2λ)

(4)

where f 1λ and f 2λ are the weights associated to the scatteringcomponents PHG(�,g1λ) and [ f1λ PH G(�, g1λ) + (1 − f1λ)PH G(�, g2λ)], respectively. This model using four shape param-eters is capable of representing a scattering pattern involving aforward peak and a backward peak and an isotropic component.

For applications requiring the use of less complicated scat-tering phase functions, simpler approximations are available.Among other simple descriptions of scattering phase functions,when usable, are delta-M, linear anisotropic, and isotropic scal-ing approximations (see for instance Siegel and Howell [10]).The last one is also called the transport approximation, as it wasbeen first used in neutron transport theory. It is based on a phasefunction Pλ(µ0) = 1 − µλ + 2µλδ(1 − µ0) where µ0 = cos �,µλ = gλ, and δ is the Dirac function. This phase function is ade-quate for media exhibiting a forward scattering peak. Transportapproximation provides a reduction of the RTE to the form cor-responding to isotropic scattering, with a transport scatteringcoefficient σtr

λ , or a transport single-scattering albedo ωtrλ , and a

transport optical thickness τtrλ0 = βtr

λ L , instead of σλ or ωλ, andτλ0 = βλL , respectively. The forms of the scaled parametersdepend on the reduced model adopted for the solution of theRTE [11–13]). A considerable amount of work was achieved

by Dombrovsky [14–16] and by Dombrovsky and co-workers[17–19] aiming to develop analytical solutions that may be usedin rather simple way for both radiative heat transfer calculationsand radiative properties identification.

As an alternative to RTE, the radiation diffusion approxima-tion, which is similar to P1 approximation, is used by Petrov andco-workers as a direct model for the identification of radiativeproperties [20].

Scaling correlations applied to radiative properties were sug-gested by authors aiming to extend continuous approaches tosituations involving dependent scattering. This aspect is treatedlater (see Densely Packed Spherical Particle Beds section).

Discrete Approaches

In some cases like dense particulate media, when independentscattering may not be assumed, or for large-scale cellular media[21], the continuous approach of radiative heat transfer may notbe appropriate and discrete approaches based on the geometricaloptics approximation (see next section) are preferred. Amongthese, Monte Carlo methods are among the most popular [22].They were initially used as a reference model aimed to validateother dicretization based methods. Due to the increase of com-puting power and to the implementation of different techniquesof variance reduction aimed to save computation time, like theradiation reciprocity principle, for instance, the use of MC as asolving tool for problems of radiative transfer in participatingmedia is continuously increasing [23–26]. A recent review onthe application of MC methods to radiative transfer in semi-transparent media and on recent progress with reciprocal MCmethods is available in [27]. Nowadays, MC methods are com-monly used for theoretical calculations of radiative properties[28–32]. This aspect is developed in the next section.

RADIATIVE PARAMETERS DETERMINATIONMETHODS: THEORETICAL PREDICTION VERSUSEXPERIMENTAL IDENTIFICATION

The radiative properties of particulate media may be eithertheoretically predicted or experimentally determined. Theoret-ical prediction provides a better insight on the influence ofmorphology and optical constants of the particle and matrixmaterials on the radiative properties of the medium. Thus itopens tracks toward designing new material with customizedproperties. Conversely, experimental identification provides theknowledge of real material properties. Identification may alsobe used as a means to verify the validity of the choice of theparticle property prediction model. Nevertheless, owing to thenumber of parameters involved, radiative properties identifi-cation probably ranks among the most intricate identificationprocesses.

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Theoretical Prediction

The theoretical prediction of the radiative properties of dis-perse media is based on solutions of Maxwell’s equations ap-plied to the electromagnetic wave/particle interaction, whichconsists of elastic scattering (that means without change ofwavelength) and absorption by the particle. An initial issue forthe problem was the simple theory for spherical particles muchsmaller than the radiation wavelength, derived by Rayleigh dur-ing the second part of the 19th century. Then Mie published in1908 a solution of Maxwell’s equations for an electromagneticwave traveling through a nonabsorbing medium hosting a singleembedded sphere. Mie’s scattering theory, which is also referredas Lorentz–Mie theory in recognition of Lorentz’s contributionto the field, remained for a long time the only issue availablefor the calculation of the radiative properties of dispersed me-dia. Details of derivation of Mie’s solution and data on radiativeproperties of various particles derived on the basis of Mie’stheory can be found in textbooks by Kerker [33], van de Hulst[34], and Bohren and Huffman [35]. Simpler solutions apply insome limiting regions depending on particle size parameter xand complex refractive index m values [36]:

x � 1 and x |m − 1| � 1 Rayleigh scattering.x � 1 and x |m − 1| � 1 Geometrical optics and diffraction

theory.|m − 1| � 1 and x |m − 1| � 1 Rayleigh–Debye–Gans

(RDG) scattering.x � 1 and |m − 1| � 1 Anomalous diffraction.

Mie’s theory was then extended to several situations such asnonspherical particles (among which are infinitely long cylin-ders under oblique incidence [37], coated [15], inhomogeneous[17, 38], externally multilayered particles [39], absorbing hostmedium [40–45]). Several cases are treated in papers by Dom-brovsky [14–16] and Dombrovsky and co-workers [17–19]. Arecent paper by Wriedt provides a useful overview on the subject[46].

For a medium that hosts a population of embedded particles,if the independent scattering is assumed, the contributions of thedifferent particles are simply additive and the radiation proper-ties of a monodispersed population are deduced from the prop-erties of a single particle multiplied by the number of particlesper unit volume. For polydispersed particles the single-particlesolution is averaged over the distribution of particle size, andin the case of fibrous media it should also be averaged over theparticle angular distribution. Important contributions to suchfiber properties calculations are found in papers by Lee and co-workers [47–50] and Jeandel and co-workers [51–53], amongothers. Thanks to its successive developments, Mie’s theory re-mains a most frequently used tool for particulate media propertycalculation.

Geometrical optics laws are used for the property deriva-tion through approaches assuming independent scattering. The

particle geometry is generally simplified: spheres, cylinders,and ellipsoids [54–57]. Another family of methods is based onMonte Carlo approaches that enable one to more easily accountfor the real morphology of particles and for some near-field de-pendent scattering effect, such as the so-called RDFI methodby Taine and co-workers [28, 29, 58, 59] that was successfullyused by Petrasch, Steinfeld, and co-workers [30, 31], and theapproach proposed by Coquard and Baillis for opaque particlebeds [32]. The last authors extended their approach to semi-transparent particles, as reported hereafter in section DenselyPacked Spherical Particle Beds.

Situations in which independent scattering may not be as-sumed cannot be adequately treated by Mie’s theory. This isthe case for densely packed beds of particles or particle aggre-gates like soot or nanoparticle thermal insulations, for instance.Nanoparticles, individually, satisfy the Rayleigh scattering ap-proximation. But they combine into chained aggregates (which,fortunately, may be described as mass fractals) that may containseveral dozens or hundreds of primary particles. The modelingof the absorption and scattering by these aggregates requires thatthe interaction of the electric field components of em waves scat-tered by each individual sphere be accounted for. Therefore, anumber of theoretical and numerical techniques for handling theelastic scattering by mutually interacting particles of differentsizes, including aggregates, started to be developed in the secondpart of the 20th century. More or less approximate techniques areavailable. Some are very simple, like the Rayleigh–Debye–Gansfractal approximation (RDG-FA) frequently used for soot ag-gregates [60–62]. Others, as in the pioneering work of Tienand co-workers [63–65], account for dependent effects througha near-field factor and a far-field coherent addition correctionthat are applied to the Mie’s solution. These corrections werederived by solving an equation proposed by Jones [66] for theinternal field of a particle interacting with several other neigh-boring particles. This solution, which is approximate, assumesa statistical pair-particle distribution function. Several pair dis-tribution functions are discussed in [63–65]. Methods based ondirect numerical solutions of Maxwell’s equations are receivingincreasing attention, due to their capability of handling particlesof complex shape that may be separated or aggregated. Numer-ical methods in electromagnetic theory are reviewed in a recentpaper by Kahnert [67]. Among the most currently used methodsare the so-called T-matrix [68–70] and discrete dipole approxi-mation (DDA) [71–74]. Note that a free DDA computing code,referred as DDSCAT, is available via the Internet [74]. Mucheffort has been expended in recent years by the electromagneticscattering community in the development of tools that may alsobe used in the radiative heat transfer field. Recent books byMishchenko and co-authors [75–77] and Mishchenko’s tutorialpaper [78] are valuable sources of information on the subject.A review by Manickavasagam and Menguc [79] is also avail-able. In a recent article by Dombrovsky, various theoreticalapproaches for the determination of particle radiative propertiesare overviewed [80].

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Figure 2 Different experimental configurations for radiative property identi-fication.

Experimental Determination

The experimental identification of the radiative properties isa stepped approach that requires appropriate choices of:

i. A direct model of radiative transfer and related solutionmodel, for instance, RTE solved by DOM or another samelevel of sophistication method, or simpler equations as trans-port equation or diffusion equation that may be solved bysemi-analytical methods.

ii. An experimental configuration.iii. A set of parameters to be identified/theoretically predicted

and the operating sequence.iv. A method of optimization.

Steps (i) to (iii) are obviously not independent. Making rightchoices is essential for a good identification strategy.

The experimental techniques available for the identificationof the radiative properties consist of spectral and directionalmeasurements of reflectance and/or transmittance. Emittancemeasurements have also been used.

The experimental configurations that may be used areschematically represented in Figure 2 (from Moura et al. [81]):

a. Collimated beam nearly normally incident onto the sample,bidirectional transmittance and reflectance measurements.

b. Collimated beam with different angles of incidence onto thesample, hemispherical transmittance measurements; hemi-spherical reflectance also possible for single nearly normalincidence direction.

c. Diffuse radiation onto the sample, bidirectional transmit-tance and reflectance measurements.

d. Measurements of bidirectional radiation flux emitted by ahot sample (self emission).

Figure 3 Bidirectional transmittance and reflectance setup.

A few examples of such facilities are displayed inFigures 3 to 5. Figure 3 shows an apparatus enabling spec-tral bidirectional transmittance and reflectance measurements,developed by Nicolau et al. [9] and Moura et al. [81] from aninitial concept by Sacadura et al. [82]. It consists basically ofa Fourier transform infrared (FTIR) spectrometer coupled to abidirectional attachment on which is the sample, with its frontsurface located on the rotation axis of the device, and the detectorstanding on the rotating arm. In situations where bidirectionaltransmittance measurements may be acquired with a relativelythin angular resolution in the neighborhood of the direction ofthe incident collimated beam, it provides a concentration ofexperimental data in a region, which is suitable for the identifi-cation of properties of forward peaked scattering materials [57].

Figure 4 provides a scheme of a golden-coated integratingsphere, also used as a FTIR spectrometer attachment, whichenables directional–hemispherical transmittance or reflectancemeasurements. This sphere is an item of commercial equip-ment in which the direction of incidence is fixed to a con-figuration nearly normal to the sample. One should note thatdirectional–hemispherical measurements are rather simple andfaster to acquire than bidirectional. They may be performed bya moderately trained operator. On the other hand, bidirectionaldata acquisition involves long and careful measurements pro-ceeded by a delicate alignment of the setup, both requiring ahighly experienced operator.

Figure 5 shows the facility developed by Lopes et al. [83]for spectral directional emission measurements at high temper-atures. Heating isothermally up to high temperature a samplethat may be a poor thermal conductor, without placing it insidea furnace, is a real challenge. A possible issue consists of simul-taneously heating with equal power incident fluxes both sidesof a thin sample, by using lasers or arc lamps. This was adoptedby Lopes et al. [83, 84] in the facility shown on Figure 5.

Making appropriate choices in the identification approach,mostly for steps (i) and (ii), requires some previous knowledgeof the material composition and structure (the latter one maybe acquired from microscopic observation, x-ray tomography,and other material investigation techniques) and of the behav-ior of the radiation parameters under investigation, in particular

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Sample

Detector

Incident beam

Baffle

Sample or Reference

DetectorBaffle

Reference

Incident beam

Figure 4 Integrating sphere FTIR attachment enabling measurement of: (left) directional–hemispherical transmittance and (right) directional–hemisphericalreflectance.

their rate of anisotropy, which may be obtained from the theo-retical prediction models of radiation properties. Therefore, inpractice both approaches, identification and theoretical predic-tion, are in fact frequently combined and may be considered ascomplementary.

The most commonly used experimental configurations con-sist of spectral measurements of bidirectional transmittanceand/or directional–hemispherical reflectance and transmittance.Directional–hemispherical measurements are performed withcommercial integrating sphere attachments for spectrometers,which are designed to operate at a fixed direction of incidenceof radiation, generally nearly normal to the sample. Emittancemeasurements have also been used [85, 86]. Several methods

enable one to achieve the identification, such as Gauss lineariza-tion, the Levenberg–Marquardt method, and genetic algorithms.Reviews on the subject are given by Baillis and Sacadura [3] andSacadura and Baillis [87]. The last reference provides detaileddescription of the parameter identification procedure based onGauss linearization, as well as of the experimental setups usedthat are illustrated here in Figures 3 and 4.

Whatever will be the identification procedure adopted, oneshould remember some principles that are summarized here:

The search is spectral, as the experimental tool of propertyinvestigation is the spectral radiation traveling through a samplein a narrow band around a given wavelength. The interactionof radiation with the material structure varies with the ratio of

Figure 5 High-temperature emittance setup of Lopes et al. [83].

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wavelength to structure size. This means that the contribution ofmicro- or nanostructure of the material to the radiative propertiesvaries along the wavelength spectrum. Therefore, the choice ofan adequate physical model depends on the wavelength rangeof interest. As shown in the preceding section, some physi-cal models are only adequate for limited regions of the ther-mal radiation spectrum. As a consequence, different physicalmodels may be necessary to cover the whole wavelength rangeof interest. Different experimental configurations may also beneeded; in a spectral range where the sample is not too muchoptically thick, transmittance may be used. Conversely, config-urations based on reflectance or emittance should be preferredfor wavelength ranges in which the sample is optically thickand the radiation measured mostly comes from a thin surfacelayer.

The smaller are the number of parameters to identify, the eas-ier will be the operation. The set of parameters governing theRTE is nλ , κλ , σλ (or βλ, ωλ), and Pλ

(�′ → �

), that is, three

plus the number of scattering phase function parameters. Gener-ally this is too many, owing to the ill-posed nature of the inverseproblem to be solved and the uncertainty (noise) that affects themeasurements, mostly the bidirectional ones. Thus, the proba-bility of success of the identification depends on a well-suitedstrategy. Usually nλ is available from separate measurementsand is assumed as known, but this may not be the case for somematerials. In order to reduce further the number of parametersto identify, one can combine theoretical prediction for some ofthem with experimental identification for others. In some casesand for given wavelength ranges the absorption of the mate-rial may be mostly due to the matrix and the scattering due tothe imbedded particles, or conversely. This suggests the possi-bility of determining separately the absorption and scatteringproperties.

One should note that the choice of (βλ, ωλ) instead of (κλ,σλ) as identification parameters is not without consequence. Al-though the first pair is frequently used, an identification basedon (κλ ,σλ) may be more convenient as it would retrieve directlyand separately the absorption and the scattering parameters. Thismay have a consequence on the issue of the identification pro-cess. We believe that this subject requires further investigation.

Pλ is generally the most difficult to identify due to its possi-ble complexity, which may require bidirectional measurements.But if some knowledge is available on the material and its mor-phology (thanks, for instance, to scanning electron microscopy[SEM]), the scattering phase function may be theoretically es-timated from one the above listed methods. If the shape of Pλ

is compatible with a reduced model (requiring no more thanone shape parameter), its identification may be tried with an in-creasing probability of success. Note that if the design problemfor which the radiative properties are needed may be solved bya simplified RTE model, like the transport model, for instance,this means that a very simple definition of Pλ—isotropic in thiscase—may be used for the properties identification. However, ineach case it is convenient to get an estimation of the inaccuracyintroduced by the simplification. On the other hand, one should

also estimate the experimental errors as they may drasticallylimit the use of more sophisticated models.

EXAMPLES OF DISPERSED MEDIA AND PROPERTYDETERMINATION

Anisotropic Media: Fibers

Most fibrous materials, like highly porous glass-wool insu-lation, for instance, show an anisotropic structure resulting innonisotropic radiative properties. This is a source of complexityfor property identification which:

• Requires experiments that allow one to vary the direction ofincidence of radiation onto the sample (specially developedlaboratory setups).

• Does not allow the use of simplified direct models of RTE.

Nevertheless, when the diameter and orientation distributionsof the fibers as well as the refractive index of the material areavailable, which is often the case, the radiative properties maybe computed from theoretical models based on an extension ofMie’s theory [33, 37]. Several papers by Lee and Cunnington[49, 50] and Jeandel and co-workers [51–53] provide a goodbackground on the subject. In such situations the experimentalidentification of radiative properties should be rather consideredas a validation means for the theoretical models established onthe basis of an independent scattering assumption. Identificationmay also be useful when there is not enough data available fora theoretical computation of properties. As for sake of simplic-ity their identification is generally based on the assumption ofisotropic properties, this may result in some inaccuracy whenproperties determined from normal incidence experiments arethen used in design calculations involving diffuse irradiation ofthe material [14].

However, the question is, “When and for what purpose aredetailed anisotropic properties really necessary?” Clearly, evenin highly anisotropic materials, after traveling some depth in-side the medium, anisotropy of radiative properties is attenuateddue to an averaging effect from multiple scattering. As practicalapplications of insulating materials concern relatively thick lay-ers, this means that predictions and/or identification of radiativeproperties based on simple models such as the transport model,for instance, or on experiments that do not allow varying thedirection of incidence of radiation, probably are satisfactory formost engineering applications. A few situations, however, mayrequire more detailed investigation of the real anisotropy of theproperties. The radiative behavior of thin surface layers of ma-terial submitted to high external incident fluxes and/or internalradiative source profile showing important variation close to thesurface layer are among these situations.

Radiative properties of fibrous media used as thermal in-sulation, mostly low-density media, have long been investi-gated through both theoretical modeling [6–8, 47–49, 88] and

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experimental identification [9, 82, 88, 89]. These authors mostlyused numerical solutions of the RTE by the discrete ordi-nates method combined with inverse approaches based on theGauss linearization method to determine each of the unknownparameters.

The work of Moura et al. [81] provides an attempt at sys-tematic methodology for this kind of media. The four experi-mental configurations shown in Figure 2 are discussed. In thefirst three cases, the emission term is not considered in theRTE since the experiments use modulated radiation incidentonto the sample, combined with a phase-sensitive detectiontechnique. The direct model is solved by a differential dis-crete ordinates method for a one-dimensional plane slab withor without azimuthal symmetry. The phase function model is acombination of two Henyey–Greenstein (HG) functions cou-pled with an isotropic component as suggested by Nicolauet al. [9] (Eq. 3). A study of condition number, as a functionof the optical thickness, is carried out in order to determinethe ability of each configuration to identify radiative properties.For each configuration and sample thickness l, the direct modeldescribed earlier in which a set of data is assumed for the ma-terial radiative properties (τo = βl, ω, g1, f1, g2, f2) is used tocalculate the theoretical transmittances, reflectances, or emit-tances, all referred as Tt(θ,ϕ). The experimental bidirectionaltransmittances and reflectances, Ted(θ,ϕ) for an incident radia-tion, the experimental hemispherical transmittance Teh, and theexperimental directional emittance εed(θ,ϕ), are defined by thefollowing expressions:

Ted(θ,ϕ) = i(θ,ϕ)

iodωo

Teh =∫ 2π

0 i(θ,ϕ) cos θd�

iodωo

εed(θ,ϕ) = i(θ,ϕ)

ib(5)

where i is the transmitted, reflected, or emitted intensity,io is the intensity of the beam incident onto the samplewithin a solid angle dωo, and ib is the blackbody emissionintensity.

For the cases (a) and (b), dωo depends on the experimentaldevice; for case (c), dωo is the half-hemisphere (2π).

The identification of a radiative parameter χk is based on aminimization of the quadratic error between the measured, Ten,and calculated, Ttn, transmittances, reflectances, and emittancesover the measurements:

F(χk) =N∑

n=1

[Tt,n(χk) − Te,n

]2k = 1, . . . , K (6)

where K is the number of parameters to be identified.In cases (a), (c), and (d) the summation, Eq. (6), is per-

formed over the different bidirectional measurements. In case(b) the summation is over the hemispherical transmittance and

Table 1 Radiative property data used in [81]

Radiative properties

ω 0.95g1 0.84f1 0.9g2 –0.6f2 0.95

reflectance measurements for different angles of incidence ontothe sample.

The method adopted in the reported work to achieve thisminimization is the Gauss linearization method that minimizesF(χk) by setting to zero the derivatives with respect to each ofunknown parameters χk . As the system is nonlinear, an iterativeprocess is performed over m iterations [9]. It involves a matrixS that is composed of the sensitivity coefficient products, calcu-lated from the theoretical model and does not directly depend onthe experimental values. This matrix, S, can be used in the sen-sitivity analysis to verify possible linear dependences betweenthe sensitivity coefficients calculated for each parameter. Thecalculation of a condition number CN of this matrix can be usedto determine the degree of ill-posedness of the identificationproblem [90],

CN(S) = ‖S−1‖ · ‖S‖ (7)

where the norm ‖S‖ is calculated from the elements Sk’,k, as:

‖S‖ = maxk ′=1,K

K∑k=1

Sk ′,k

Of course, the larger the condition number is, the more ill-conditioned the system is: Small changes in the measurementsresult in very large change in the solution vector, i.e., the in-crements �χk . It is then almost impossible to simultaneouslydetermine all of the unknown parameters. Poor conditioning oc-curs when at least two of the sensitivity coefficients are quasi-linearly dependent or when at least one is very small or verylarge compared to the other.

The investigation was carried out for an optical thickness τ0

ranging from 0.1 to more than 20 and a set of data for the otherradiative properties shown in Table 1.

These data correspond to the phase function represented inFigure 6, which is both forward and backward peaked.

This detailed investigation provided useful information onthe capabilities and weaknesses of the different experimentalconfigurations and inversion strategies for handling the radiativeparameters identification.

Independent scattering assumed for the direct models ofradiative transfer in fiber media is realistic for low-densityinsulation. But this may not be acceptable for high-densityfibrous materials, such as those used for industrial fur-nace insulation. An issue was recently proposed by Co-quard and Baillis [91], who extended successfully to dense fi-brous media a model based on geometrical optics and Monte

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Figure 6 Phase function with (f1 = 0.9, g1 = 0.84, f2 = 0.95, g2 = –0.6).

Carlo approach previously developed for beds of sphericalparticles.

Due to the growing difficulty of the radiative transfer mea-surements when samples are heated, most experimental identi-fications of radiative property of fibrous media are performedat room temperature even for materials that may be used forhigh-temperature insulation. Recent reports by Zhang et al. [92]and Zhao et al. [93] are focused on the determination of high-temperature properties of fibrous insulation. Instead of radia-tion measurements, this approach is based on effective thermalconductivity obtained from thermal conduction setup allowingboth steady-state and transient thermal tests on large samples(180 mm diameter, 20 mm thickness). An inverse approach ac-counting for the combined conduction and radiation heat trans-fer inside the sample is used to simultaneously estimate three“gray” radiative properties (extinction, albedo, coefficient of alinear assumed phase function) plus a thermal conductivity dueto the gas and solid conduction. Obviously, the radiative prop-erties determined by this approach are global and depend on thetemperature profile inside the medium.

Recent contributions from Tagne and Baillis [94, 95] andDombrovsky et al. [96] were dedicated to the development ofmethods of property investigation based on transport approxi-mation or isotropic scaling that provide simple analytical meansof property identification. The last group of authors developeda modified two-flux solution for the transport model aiming toaccount for collimated irradiation of the medium, as is the casein some specific radiative property identification experiments.They also investigated the limits of this solution by comparisonwith numerical solutions obtained from high-order DOM cal-culations. Application to fibrous media was examined by Tagneand Baillis [95]. Owing to their features, these simplified modelsare obviously not able to retrieve properties like the phase func-tion shown in Figure 6 that displays both forward and backwardpeaks. But the real question is: Do the transport extinction co-efficient and transport albedo experimentally retrieved by usingsuch simplified models provide sufficiently accurate predictions

of radiative transfer if the predictions are achieved through thesame models? If the goal is just to provide a calculation of thedivergence of the radiative heat flux to be input in the energyequation, as is the case for a number of engineering applications,the response probably is “yes.” Conversely, if a good estimateof the radiative flux is expected, more accurate determination ofthe radiative properties may be necessary.

Bubbles, Hollow Microspheres in A Host SemitransparentMedium

A number of recent studies were motivated by bubbles gener-ated during the industrial glass melting process or similar struc-tures with bubbles or hollow microspheres hosted in a semitrans-parent matrix, as in some advanced thermal insulation materials.One of the first theoretical investigations on radiative propertiesof glass with bubbles is due to Fedorov and Viskanta [97], whoused the anomalous diffraction approximation. Then Pilon andViskanta [98] performed calculations based on the same approx-imation. A Mie’s solution was then proposed by Dombrovsky[15]. For moderately anisotropic properties, including the phasefunction, the use of transport model for identification is not socritical. Therefore, it was frequently used in recent works deal-ing with experimental identification of the radiation propertiesof fused quartz containing bubbles [19, 99]. The last group ofauthors, Dombrovsky et al. [19] proposed an improved identifi-cation procedure of the infrared properties of this material fromexperimental measurements of directional–hemispherical trans-mittance and reflectance and by using the transport approxima-tion. A modification of the two-flux approximation accountingfor the Fresnel interface reflections at sample interfaces wasused to analytically solve the transport equation [96]. Note thatthis study also provided an identification of the volume fractionof bubbles. The error of this modified two-flux approximationcombined to the transport model of phase function has beenestimated for an arbitrary scattering medium by comparisonwith numerical calculations using the discrete ordinates methodcombined to Henyey–Greenstein phase function. It appears tobe not greater than 5% in the most important range of problemparameters. Again, measurements of hemispherical reflectanceand transmittance performed on fused quartz with bubbles andradiation properties identified from these measurements wereused by Randrianalisoa et al. [100] as tools for comparingpredictions from CMT, FFA, and NFA models. The transportmodel was also used to identify the properties of porous zirconiaceramics [17].

Foams

Optimizing the thermal performance of foams is currently animportant industrial challenge. A considerable effort of investi-gation has been concentrated on the subject over the decade; seefor instance references [55–57, 101, 102], among others. These

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investigations covered not only the radiative transfer throughthe structure but also the solid and conductive heat transfer,as heat transfer inside foam involves different coupled mecha-nisms. For other dispersed materials the radiative properties maybe theoretically predicted and/or identified from FTIR measure-ments using techniques described in earlier sections. Polymeror metallic foams are structured into cells. SEM photos of thesefamilies of foams show reticular or cellular structures. In retic-ular structures the cells are open, the material is concentratedon struts and junctures, and the interstitial gas is the same asthe external environment, which is typically air. Cellular struc-tured foams have closed cells whose walls are thin membranes.Struts and junctures appear at cell edges and corners as mem-brane thickenings. The internal gas may be different from airand may contribute to increase the insulation effect. Accord-ing to the bulk material used and to the manufacturing process,foams show more or less complex morphology and cells mayhave a single- or double-scale structure. Good descriptions ofinsulating foams as well as recent reviews on the topic may befound, for instance, in [102, 103]. As the mean diameter of cellsis generally larger than the wavelength range of interest, thecalculation of radiation properties may be performed by usinggeometric optics and/or Mie theory to describe the interactionof radiation with the elements of the foam structure.

Densely Packed Spherical Particle Beds

Beds of particles are commonly used in many industrial ap-plications. Due to the weak clearance between particles, mostlyin packed beds, the scattering phenomena are typically in thedomain of dependent scattering. The modeling of radiative prop-erties of beds of spherical particles has received much attention.Among other methods, correlations have been used by Kamiuto[104] and Kamiuto et al. [105], and scaling factors by Kavianyand Singh [106], and Singh and Kaviany [107], to approximatedependent scattering effects. Dombrovsky derived approximatemodels for the radiation properties of packed hollow micro-spheres. In one of these models the clearances are treated asrandomly oriented spherical particles [16]. Coquard and Baillis[32] developed a Monte Carlo approach to predict the radiativeproperties of beds of large opaque particles that may be diffuselyor specularly reflecting spheres. The bed is generated as a setof randomly positioned particles. The porosity is an adjustableparameter enabling one to represent particles from those that arevery distant to touching ones. The procedure uses a large num-ber of rays starting from random points in the bed. Each ray istracked along its path in the bed; it may undergo several reflec-tions and partial reflections at the surface of the particles beforeleaving the computation domain. All these phenomena are ac-counted for, and this provides, for a great number of rays, theradiative properties of the bed. This model, for which results arein good agreement with experimental measurements, is a newpromising tool to predict radiative properties of beds of spheresindependently from particle clearance. It also contributes to the

assessment of the limits of independent scattering theories. ThenCoquard and Baillis [108] extended this model to beds of semi-transparent spherical particles. The material inside each particleis assumed to be an absorbing and scattering medium. Indepen-dent scattering from particles is assumed. This model may behelpful to investigate the radiative behavior of complex partic-ulate media. However, due to the time required by Monte Carlocalculations it does not seem appropriate for identification. Theso-called radiation distribution function identification (RDFI)method is a Monte Carlo technique recently developped byTancrez and Taine [28], and extended by Zeghondy et al. [29],for the determination of radiative properties of high-porositymedia made of solid particles embedded in a transparent fluid.The radiation propagates through the fluid medium and is par-tially reflected and absorbed when it hits a particle surface,according to the reflection model adopted and related laws. Theradiative properties are identified from cumulated distributionfunctions of radiation free path inside the medium, and cumu-lated probabilities of absorption and directional reflection bythe particle–wall interfaces. The strength of this method is thatit only requires the knowledge of the real medium morphologyand local radiative properties at the particle/host medium inter-face. This method was successfully applied to several porousmaterials, such as mullite foam, [58], rod bundles (for radiativetransfer simulation in a nuclear reactor core [59]), and packedbeds of reticulated porous ceramics that may be used as radiantabsorbers for high-temperature solar thermochemical processes[30, 31].

Agglomerated Nanoparticles

Due to their promise for future applications, a growing in-terest is observed for the understanding of thermal radiationmechanisms involving nanoparticles that are generally agglom-erated in more or less chained clusters, as is the case for sootor nano-powder thermal insulations. For other media, the in-vestigation of radiative properties involves experimental iden-tification and/or theoretical prediction, with the latter requiringone to account for possible dependent scattering effects. Re-visiting the models of Tien and co-workers [63–65], Prasherderived two approximate solutions for Rayleigh regime thataccount for multiple scattering, or multiple plus dependent scat-tering, respectively [109]. Both derivations are based on theso-called Percus–Yevick pair distribution function, consideredthe most successful distribution for capturing the effects ofdepending scattering. The last approximation, referred as thequasi-crystalline-approximation (QCA), captures much betterthe right physics than the independent scattering approximation.In recent works the DDA method is used for radiative propertycalculation. Lallich et al. [110, 111] investigated the radiativeproperties of silica nanoporous matrices that are componentsof superinsulating materials. The samples contain agglomer-ates of approximately 80 to 200 primary particles of diameterranging from 7 to 14 nm. The experimental identification of

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Figure 7 Aggregated particles studied by microwave scattering analogy andreal soot aggregate: (a) cubic agglomerate of 27 spheres, (b) soot-wise chainedagglomerate of 74 spheres, and (c) real soot agglomerate, TEM photo, from Xuet al. [113].

the transport extinction coefficient and transport albedo wasachieved from spectral directional–hemispherical transmittanceand reflectance measurements, by using the transport equationas the direct model and the Newton–Raphson algorithm forthe inverse problem. The authors also calculated theoreticallythe same properties by several methods: (i) Mie’s theory, (ii)Mie’s theory with three different dependent scattering correc-tions from Tien and co-workers and Prasher, and (iii) DDA,and they compared the identified with the theoretical results.As expected, Mie’s theory based on a proper representative sizeof the agglomerate appeared as predictive enough of radiativeproperties for wavelengths beyond 1 µm. For lower wavelengths(250 nm to 1 µm) the independent scattering model as well asthe dependent scattering corrected models failed to match theexperimentally identified parameters. Conversely, DDA predic-tions showed a better agreement with the experimental results.Note that in this work still in progress the DDA calculations werebased on computer-generated aggregates rather than on the realaggregate structure as it may be investigated by transmissionelectronic microscopy (TEM) or other means of structure inves-tigation. Another recent work deserves to be noticed [112]. Itdeals with the use of microwave analogy to experimentally char-acterize aggregated particles. It is not easy to perform light scat-tering measurements on arbitrarily shaped micro- and nanoscaleparticles due to the impossibility of tailoring them, but operat-ing with larger particles, and subsequently larger wavelength,makes it possible, through analogy between thermal radiationand microwave ranges. Magnitude and phase of the field scat-tered by aggregates of dielectric spheres of diameter 1.59 cm(Figure 7) were both (which is original) measured and the resultswere successfully compared with theoretical predictions fromDDA and T-matrix models [113]. This analogy seems to be apromising tool to investigate not only the scattering behavior ofaggregated particles, especially small-sized ones, but also thevalidity of radiative property identification strategies.

OPTIMIZATION ALGORITHMS

The reader may be disappointed by the lack of informa-tion reported in this paper on the selection of optimization

algorithms aimed to the radiation property identification. Sev-eral algorithms were used for that purpose, and sometimes com-pared. Some are deterministic (Gauss, Levenberg–Marquardt[114], conjugate gradient, and Tikhonov’s regularization); otherones are statistical, like genetic or particle collision algo-rithms, for instance, or hybrid methods coupling an evolution-ary/stochastic approach with a deterministic one [115, 116].In some works, two or three arbitrarily selected methods arecompared in terms of accuracy/computational effort, but a lackof justification or strategy of choice is generally observed. Inthe case of identification of temperature, heat flux, radiativesource, or conductive properties for semitransparent materials,a number of works are available in the literature that comparethe efficiencies of different identification algorithms [117–119].But their conclusions are not extensive for the radiative prop-erty identification, which certainly remains a subject of furtherinvestigation.

CONCLUDING REMARKS

As a closure of the overview presented in this paper, thefollowing concluding remarks may be outlined:

• The determination of radiative parameters may be shared invariable proportions between theoretical prediction and ex-perimental identification, with the choice being specific toeach particular situation, depending on the properties to bedetermined. Theoretical prediction may be considered as aend method or, alternatively, as a first approach to an exper-imental identification, and this is valid for the whole or justpart of the properties set.

• Theoretical prediction and experimental identification areboth based on a physical model (direct model). Its choicein terms of sophistication should be appropriate owing to thefinal use of the parameters that are sought and to the accuracyof the data available for the model nurturing.

• A number of physical models are available allowing the de-termination of radiative properties with more or less detailsand accuracy, from the classical theory of Mie and its de-velopments or simplified variants, to modern developmentsof electromagnetic scattering theory that lend themselves totreating media hosting particles of arbitrary shapes, or inho-mogeneous particles, that may mutually interact—in a word,to account for more complex situations. Major advances wereseen in the past decade in the development of methods forsolving the electromagnetic scattering problem and free soft-ware packages are available in some specific websites.

• A similar diversity is also observed for the radiative transferequation forms used for the property identification.

• It is not mandatory, even perhaps not convenient, to use asophisticated direct model and/or RTE form when one is notable to feed the form with data of appropriate accuracy, orwhen the application that has motivated the property searchmay be satisfied with a simpler set of properties. In spite of the

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availability of physical models that now make it possible todescribe very complex particulate media, these models maynot be tractable for identification due to the extensive com-putational time required. Therefore, in a number of practicalsituations the use of simpler models based on reduced defini-tion of the phase function, such as the transport model, maybe a valuable issue.

• Important progress is observed during the last two decadesin the methodology of determination of radiative properties,in both theoretical prediction and experimental identificationfor several kinds of particulate media. The literature reportsan amount of related experience that may be very useful as astarting point for finding a methodology suited to a specificmedium.

• Further research is needed on the optimal choice and sequenceof identification parameters to be adopted, such as the choiceof the couple (κλ , σλ) instead of (βλ, ωλ).

• Identification of properties requires the choice of an optimiza-tion method. Although different methods have been used forradiative properties identification, some deterministic, otherones stochastic, or hybrid, a lack is observed of methodologyaimed to operate an “optimal choice of the algorithm of op-timization” well matched to radiative properties sought andrelated experimental configurations.

• An important need is still observed in the assessment of ra-diative properties at high temperatures. Due to the difficultyof development and operation of experimental setups that canwork at high temperatures to provide spectral and directionalradiation measurements, most current experimental facilitiesonly work at room temperature. Owing to the growing impor-tance of the design of materials intended to efficiently workat moderate to high temperatures, such as thermal insulations,the development of such facilities and techniques is a remain-ing challenge.

NOMENCLATURE

CN condition number of matrix Sf iλ weights associated to a phase function, i = 1, 2F function to be minimizedgλ spectral asymmetry parameter of a phase functioni, Iλ spectral intensity of radiation, W m−3 sr−1

Ibλ spectral intensity of blackbody radiation, W m−3 sr−1

K number of parameters to be identifiedm complex index of refractionn refractive index (real part of m)N number of data used in the identification processp, P spectral phase functions space coordinate, mS matrix of sensitivity coefficientsSi,j element of matrix ST spectral transmittance, or reflectance, or emittance

x particle size parameter

Greek Symbols

βλ spectral extinction coefficient, m−1

ε spectral emittanceθ, ϕ polar, azimuthal angle, rad� scattering angle, radκλ spectral absorption volumetric coefficient, m−1

λ wavelength, m−1

µ0 = cos(�)σλ spectral scattering volumetric coefficient, m−1

τλ spectral optical coordinate, m−1

ωλ spectral albedo = σλ / βλ

ω, � solid angle, sr

Subscripts

b blackbodyd directionale experimental (measured data)h hemisphericalt theoretical (data calculated from a model)HG Henyey–GreensteinHGN Henyey–Greenstein–Nicolauλ spectral (at a given wavelength, per unit wavelength)

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Jean-Francois Sacadura is Emeritus Professor ofMechanical Engineering and Thermal Sciences atINSA Lyon (National Institute of Applied Sciences)in Lyon, France. He received his mechanical engi-neering degree in 1963 from INSA Lyon, his Ph.D. in1969 from the University of Lyon, and the doctor-es-sciences physiques degree from INSA Lyon in 1980.He started his academic career in 1963 as a teach-ing and research assistant at INSA Lyon, where hereached the position of full professor of mechanical

engineering with tenure in 1983. He has been an emeritus professor since 2005.Dr. Sacadura was dean of the master and PhD programs in thermal engineeringat INSA Lyon (1998–2005) and director of the Thermal Science Centre of Lyon(CETHIL) from 1997 to 2003. His research interests include thermal radia-tion and combined mode of heat transfer, thermophysical properties, particulatemedia, semitransparent media, high temperatures, modeling and experimen-tal characterization by direct and inverse methods, and radiative–convectivedesign of water-spray fire protections for oil and chemical plants. He has co-authored more than 120 refereed publications, and is co-editor and/or co-authorof four books. He serves on several editorial boards and is co-chief-editor ofHigh Temperatures–High Pressures (2008–). He is a member of the boards ofSociete Francaise de Thermique (SFT, 1970–, chairman 1997–1999), Associa-tion Francaise de Mecanique (AFM), European Conference in ThermophysicalProperties (ECTP, chairman 1993–1996), Eurotherm Committee (founding andhonorary member), and ICHMT Scientific Council and Executive Commit-tee. His efforts in engineering education and research have been recognizedwith several awards: Chevalier des Palmes Academiques, 1986; Prix Academ-INSA as an outstanding PhD promotor and advisor, 1989; Officier des PalmesAcademiques, 1998; and Commandeur des Palmes Academiques, 2005 (PalmesAcademiques are French Ministry of Education awards for performance inhigher education and research).

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