Hassan Flity

Mariam Abdo

Lucas Terrei

Zoubir Acem

Rabah Mehaddi

Paul Lardet

Gilles

Gilles Parent
email: gilles.parent@univ-lorraine.fr

3D-Modeling of thermal degradation of spruce wood under inert atmosphere

Keywords: Wood, Thermal degradation, 3D-Pyrolysis model, Thermogravimetric analysis, Cone calorimeter

published or not. The documents may come

Introduction

Wood is a material with interesting thermal and mechanical properties and its carbon footprint is one of the lowest among building materials [START_REF] Sikkema | A market inventory of construction wood for residential building in Europe -in the light of the Green Deal and new circular economy ambitions[END_REF][START_REF] Hildebrandt | The contribution of wood-based construction materials for leveraging a low carbon building sector in europe[END_REF]. Unfortunately, the fire behavior of wood limits its use in buildings [START_REF] Wang | Use of wood in green building: a study of expert perspectives from the UK[END_REF]. A better understanding and control of the fundamental processes of wood combustion, in particular the primary pyrolysis phase, can help to reduce the constraints on the use of wood as a building material. This topic is mainly studied at matter scale using TGA [START_REF] Koufopanos | Kinetic modelling of the pyrolysis of biomass and biomass components[END_REF][START_REF] Branca | Kinetics of the isothermal degradation of wood in the temperature range 528-708 K[END_REF][START_REF] Chen | Characteristics and kinetic study on pyrolysis of five lignocellulosic biomass via thermogravimetric analysis[END_REF] under inert atmosphere.

At this scale, the sample mass is very small (4 to 50 mg) and the sample is thermally thin, so heat transfer can be neglected and a 0D model can be used to predict the mass loss. Several mechanisms have been proposed in the literature to predict the mass loss and the mass loss rate obtained from TGA. These mechanisms can be divided into two main categories. Some studies consider the wood as a homogeneous material for which thermal degradation occurs according to a single step [START_REF] Wagenaar | Pyrolysis of biomass in the rotating cone reactor: modelling and experimental justification[END_REF], consecutive steps [START_REF] Terrei | Experimental and numerical multiscale study of spruce wood degradation under inert atmosphere[END_REF], parallel reactions [START_REF] Thunman | Combustion of wood particles -a particle model for eulerian calculations[END_REF] or a combination of parallel and consecutive reactions [START_REF] Benkorichi | Investigation of thermal degradation of pine needles using multi-step reaction mechanisms[END_REF]. Other authors consider that this description is not sufficient to predict the thermal degradation of wood, which should rather be considered as the sum of the degradation of its main components: hemicellulose, cellulose, lignin [START_REF] Branca | Critical evaluation of global mechanisms of wood devolatilization[END_REF][START_REF] Gunnar Grã¸nli | Thermogravimetric Analysis and Devolatilization Kinetics of Wood[END_REF]. Models of these two categories have advantages and drawbacks: more or less complex, more or less predictive. However, the results of all of them are satisfactory in 0D.

In the context of building fires, wood materials are thermally thick, requiring the study of pyrolysis at larger scales. Today, the cone calorimeter is the largest scale at which the thermal degradation is studied in an inert atmosphere [START_REF] Terrei | Experimental and numerical multiscale study of spruce wood degradation under inert atmosphere[END_REF][START_REF] Kashiwagi | Effects of external radiant flux and ambient oxygen concentration on nonflaming gasification rates and evolved products of white pine[END_REF]. At this scale, heat transfer controls the pyrolysis phenomena and must be taken into account. The multiscale approach is used to model the degradation at this scale i.e. the kinetic parameters are estimated at 0D and used in a 1D or 3D model coupled with heat transfer. Several tools can be used to model the thermal degradation of materials e.g. Fire Dynamic Simulator (FDS), GPYRO [START_REF] Lautenberger | Generalized pyrolysis model for combustible solids[END_REF], THERMAKIN [START_REF] Stanislav | Prediction of the burning rates of charring polymers[END_REF]. Bustamante [START_REF] Bustamante | Experimental and numerical investigation of the thermal decomposition of materials at three scales: application to polyether polyurethane foam used in upholstered furniture[END_REF] showed the inadequacy of FDS to simulate cone calorimeter tests with kinetic parameters calculated from TGA.

THERMAKIN cannot simulate 3D problems. Meanwhile, several studies have shown the ability of GPYRO to simulate materials degradation at a large scale. However, in GPYRO, thermal properties vary according to power laws and material properties are calculated according to mixture laws. The latter assumption is true for some properties such as the heat capacity, but it is not appropriate for the thermal conductivity. These features are two significant weaknesses of GPYRO. It is worth mentioning the studies conducted by Richter and Rein [START_REF] Richter | A multiscale model of wood pyrolysis in fire to study the roles of chemistry and heat transfer at the mesoscale[END_REF], as well as the study by Lautenberger and Fernandez-Pello [START_REF] Lautenberger | A model for the oxidative pyrolysis of wood[END_REF], who utilized the GPYRO model to simulate the pyrolysis and oxidation of wood under three different oxygen concentrations: 0 %, 10.5 %, and 21 %. The models were validated on the basis of the experimental results obtained by Kashiwagi et. al [START_REF] Kashiwagi | Effects of external radiant flux and ambient oxygen concentration on nonflaming gasification rates and evolved products of white pine[END_REF] on samples of white pine, considering constant thermal properties. The results were quite good and the remaining discrepancies were attributed to the uncertainty in the boundary conditions, the material properties and their variation with temperature (in particular the thermal diffusivity of wood and char) and the kinetic parameters of char oxidation.

It should be noted that Kashiwagi et al's experiments [START_REF] Kashiwagi | Effects of external radiant flux and ambient oxygen concentration on nonflaming gasification rates and evolved products of white pine[END_REF] were performed on 3.8 cm side cubic samples, so the heat transfer can hardly be considered 1D in this case, especially for the in-depth temperature at locations far from the surface of the sample (temperature was measured up to 15 mm deep).

The aim of this study was to simulate the thermal degradation of spruce wood at the cone calorimeter scale in an inert atmosphere, following the work of Terrei et al. [START_REF] Terrei | Experimental and numerical multiscale study of spruce wood degradation under inert atmosphere[END_REF], but now using measured wood thermal properties and an improved modeling including in particular a 3D heat transfer (instead of 1D). First, wood was characterized at the matter scale thanks to TGA tests as it is commonly done. These tests were carried out at different heating rates. The kinetic parameters involved in a global multi-reaction mechanism were estimated by fitting the mass loss and the mass loss rate obtained by the TGA. More originally, DSC measurements were performed to determine the heats of reaction corresponding to the different reactions taking place in the material during its degradation. In a second step, tests were carried out at the material scale in inert atmosphere using the cone calorimeter. At this scale, particular attention was paid to the determination of the thermal boundary conditions. For this purpose, tests with an inert material were carried out with the same experimental conditions as the wood tests. An inverse method using a 3D heat transfer model was applied to these tests to determine the boundary conditions. Finally, a 3D pyrolysis model was developed in PYTHON to simulate wood degradation at the cone scale. The model accounts for the anisotropic nature of wood. The kinetic parameters obtained from TGA and the boundary conditions obtained from the inert material tests were used in the 3D pyrolysis model. In addition, all properties of the model were measured experimentally except the thermal conductivity above 200 °C. The latter property was varied until the best agreement between numerical and experimental results was obtained. Since the model used is complex and most of its parameters have been measured experimentally, the thermal conductivity obtained should be reliable.

Matter scale

The aim of this section is to present the thermal degradation of spruce wood at the matter scale. For this purpose, the degradation of spruce wood in an inert atmosphere has been studied by TGA and DSC. At this scale, the spatial temperature gradients within the wood sample are negligible, thus only the kinetics of the chemical reactions occurring during pyrolysis is involved.

Thermogravimetric analysis

Dry solid wood samples with an initial mass varying between 4.5 and 5 mg were subjected to a temperature ramp from room temperature to 1000 °C at different heating rates of 5, 10 and 20 • C • min -1 . The tests were carried out under inert atmosphere using nitrogen with a flow rate of 80 mL•min -1 to avoid oxidation of the char and thus concentrate on the sole pyrolysis. The evolution of the sample mass as a function of the sample temperature, was continuously recorded. Fig. 1 shows the mean normalized mass loss and the mean normalized mass loss rate as a function of temperature for the different heating rates. For each heating rate, three repeatability tests were performed with a relative difference of approximately 1.3 %. For a given heating rate, the observed stages of decomposition of the spruce wood are as follows:

-Below 210 °C: the mass of the samples remains constant.

-At 210 °C: the degradation of spruce begins and the mass of spruce starts to decrease.

-Between 300 °C and 350 °C: there is a peak in the mass loss rate. This peak corresponds mainly to the degradation of hemicellulose as reported in several studies [START_REF] Chen | Characteristics and kinetic study on pyrolysis of five lignocellulosic biomass via thermogravimetric analysis[END_REF][START_REF] Terrei | Experimental and numerical multiscale study of spruce wood degradation under inert atmosphere[END_REF][START_REF] Branca | Critical evaluation of global mechanisms of wood devolatilization[END_REF].

-Between 350 °C and 420 °C: the main step of degradation occurs and a significant peak in mass loss rate is observed. During this stage, the wood looses more than 70 % of its initial mass. This step corresponds mainly to the degradation of cellulose [START_REF] Chen | Characteristics and kinetic study on pyrolysis of five lignocellulosic biomass via thermogravimetric analysis[END_REF][START_REF] Terrei | Experimental and numerical multiscale study of spruce wood degradation under inert atmosphere[END_REF][START_REF] Branca | Critical evaluation of global mechanisms of wood devolatilization[END_REF].

-After 420 °C: the mass loss rate decreases slightly until a plateau is reached. This is due to the degradation of a part of the lignin. In fact, lignin degradation occurs over a wide temperature range, from 200 °C to 900 °C [START_REF] Yang | Characteristics of hemicellulose, cellulose and lignin pyrolysis[END_REF][START_REF] Dhyani | A comprehensive review on the pyrolysis of lignocellulosic biomass[END_REF].

Overall, the experimental results show that wood degradation follows the same trend regardless of the heating rate. However, Fig. 1 (b) shows a strong influence of the heating rate on the intensities of the mass loss rate peaks. Thus, the higher the heating rate, the higher the maximum mass loss rate. Fig. 1 (b) also shows that as the heating rate increases, the different mass loss rate peaks occur at higher temperatures. This behavior has also been observed in several studies [START_REF] Benkorichi | Investigation of thermal degradation of pine needles using multi-step reaction mechanisms[END_REF][START_REF] Branca | Critical evaluation of global mechanisms of wood devolatilization[END_REF]. Indeed, when the heating rate is low, the temperature within the sample increases uniformly. On the other hand, when the heating rate is high, the temperature within the sample is no longer uniform and the temperature in the core of the wood is not exactly that of the reactor [START_REF] Gunnar Grã¸nli | Thermogravimetric Analysis and Devolatilization Kinetics of Wood[END_REF].

Kinetic model

Based on the experimental results, the model used to predict the wood degradation is a global multi-reaction model. In this model, dry wood is considered to be composed of three pseudo-solids A 1 , A 2 and A 3 , with initial mass fractions equal to α 1 , α 2 and α 3 respectively. These pseudo-components degrade simultaneously and independently according to first-order kinetic laws. A 1 degradation only leads to a gaseous constituent G 1 , the decomposition of A 2 leads to a gaseous constituent G 2 and a char constituent C 2 .

The decomposition of A 3 leads to a gaseous constituent G 3 and a char constituent C 3 .

Further details of this model and the theory behind it will be published in a future paper which is currently under review. The reaction scheme adopted is therefore as follows:

A 1 k 1 -→ G 1 (Gaz) A 2 k 2 -→ G 2 (Gaz) + γ 2 .C 2 (Solid) A 3 k 3 -→ G 3 (Gaz) + γ 3 .C 3 (Solid)
Where: γ 2 and γ 3 are the stoichiometric coefficients. K 1 , K 2 and K 3 are the rate coefficients of the reactions which follow an Arrhenius law (K i =K 0i .exp(-E i /RT)). K 0i and E i are the pre-exponential factor and the activation energy of the reaction i, respectively, T is the temperature and R is the ideal gas constant. The mass balance of A 1 , A 2 , A 3 , C 2 and C 3 are respectively:

dm A 1 dt = -K 1 .m A 1 (1) dm A 2 dt = -K 2 .m A 2 (2)
dm A 3 dt = -K 3 .m A 3 ( 3
)
dm C 2 dt = γ 2 .K 2 .m A 2 ( 4
)
dm C 3 dt = γ 3 .K 3 .m A 3 (5)
Where

m A 1 , m A 2 , m A 3 , m C 2
M (t) = m A 1 (t) + m A 2 (t) + m A 3 (t) + m C 2 (t) + m C 3 (t) (6)
The kinetic parameters of the model i.e. pre-exponential factors (K 01 , K 02 and K 03 ), activation energies (E 1 , E 2 and E 3 ) and stoichiometric coefficients (α 1 , α 2 , α 3 , γ 2 and γ 3 )

were estimated by simultaneously fitting the mass loss and the mass loss rate curves obtained by TGA. Levenberg-Marquardt algorithm [START_REF] Marquardt | An Algorithm for Least-Squares Estimation of Nonlinear Parameters[END_REF] was used to minimize the following objective function:

S = N i=1 [M exp (t i ) -M mod (t i )] 2 + δ N i=1 dM exp dt (t i ) - dM mod dt (t i ) 2 (7)
Where: N is the number of experimental points; M exp (t i ) and dMexp dt (t i ) are the measured mass and mass loss rate, respectively; M mod (t i ) and dM mod dt (t i ) are the corresponding numerical quantities. As the magnitude of the mass loss is extremely higher than the mass loss rate, the sum of the mass loss rate is multiplied by a factor (δ) to increase the weight of the mass loss rate in the optimization. Several constraints were applied on the parameters during the optimization. Firstly, as there is a compensation effect between the pre-exponential factor and the activation energy [START_REF] Gong | A Review on Flaming Ignition of Solid Combustibles: Pyrolysis Kinetics, Experimental Methods and Modelling[END_REF], the algorithm could converge toward different sets of parameters. Thus, the activation energy was constrained to vary between 60 and 200 kJ • mol -1 which is the order of magnitude of this parameter in the literature. Table 1 shows the values of the kinetic parameters obtained by optimization.

Fig. 2 shows that the model generally well predicts the normalized mass loss and the normalized mass loss rate obtained by TGA. A slight discrepancy is observed above 400 °C. This can be attributed to a delayed degradation of the third component A 3 . In addition, this kinetic mechanism, with only three first order reactions, can be considered too simple to account for all the reactions that occur at high temperature.

Differential scanning calorimetry

The differential scanning calorimeter used in this study is the SETARAM DSC 111.

This instrument has been modified and equipped with an original fixed-bed setup to ensure better control of sample scanning. Prior to performing the tests, the instrument Table 1. The kinetic parameters obtained by optimisation. , ranging from room temperature to 600 °C, under an inert atmosphere using nitrogen at a flow rate of 20 mL • min -1 . The samples tested were cylindrical with a diameter of 3.4 mm and a height of 6.5 mm. These samples were oven dried at 105°C for 24 hours prior to testing. Note that Q exp total is the heat flux measured by the DSC, corrected by the instrument baseline and normalized by the initial mass of the sample and the heating rate. Q exp total was assumed to be the sum of an effective heat capacity (sensible heat flux Q s ) and the heat of reaction Q r . The effective heat capacity is assumed to be as follows [START_REF] Rath | Heat of wood pyrolysis[END_REF]:

Reaction K 0i (s -1 ) E i (kJ • mol -1 ) α i (-) γ i (-) (1
C p (T ) = (1 -X(T )).C p,wood (T ) + X(T ).C p,char (T ).(γ 2 + γ 3 ) (8)
Where:

(γ 2 + γ 3 ) is the final char yield, C p,wood (T ) is the heat capacity of wood, C p,char (T )
is the heat capacity of char and X(T) is the mass conversion defined by:

X(T ) = m 0 -m(T ) m 0 -(γ 2 + γ 3 ) (9)
Where m 0 and m(T ) are the initial mass and the mass of the sample at the temperature T calculated from the kinetic model, respectively. Therefore, the normalized heat flux measured by DSC can be modeled as:

Q num total (T ) = Effective heat capacity (Cp(T )) (1 -X(T ))C p,wood (T ) + X(T )C p,char (T )(γ 2 + γ 3 ) + Heat of reactions (Qr) 1 β 3 i=1 dm A i dt (T )H i α i (10)
Where: H i is the heat of reaction i, and β is the heating rate. Finally, the heats of reactions H 1 , H 2 , and H 3 were estimated by minimizing the sum of the quadratic error between Q num total and Q exp total . Fig. 3 (a) shows that the three reactions used to predict the mass loss at the macro scale were unable to predict the heat flux measured by DSC.

Therefore, a fourth reaction was added, and the reaction scheme became as follows:

A 1 k 1 -→ G 1 (Gaz) + H 1 A 2 k 2 -→ G 2 (Gaz) + γ 2 .C 2 (Solid) + H 2 A 3 k 3 -→ G 3 (Gaz) + γ 3 .C 3 (Solid) + H 3 C 2 (Solid) k 4 -→ C 4 (Solid) + H 4 Where H 1 , H 2 , H 3 ,
and H 4 are the respective heats of reaction 1, 2, 3, and 4. Note that in the fourth reaction, there is only heat production and no mass loss. In this reaction, the char C 2 is converted into another char C 4 with the same mass but a different morphology.

In this case, the heat of reactions Q r in Eq. 10 becomes: The experimental data is the average of three repeatability tests.

Q r = 1 β 3 i=1 dm A i dt H i α i + dm C 2 dt H 4 α 2 γ 2 (11)

Cone calorimeter scale

This section is dedicated to the study of spruce wood degradation under inert atmosphere at the cone calorimeter scale. At this scale, heat transfer cannot be neglected.

Materials and methods

The aim of this work was to model the thermal degradation of spruce wood in an inert atmosphere at the cone calorimeter scale, following the work of Terrei et al. [START_REF] Terrei | Experimental and numerical multiscale study of spruce wood degradation under inert atmosphere[END_REF]. These . Further details about the experimental protocol can be found in [START_REF] Terrei | Experimental and numerical multiscale study of spruce wood degradation under inert atmosphere[END_REF].

Since several physical and chemical phenomena occur during wood experiments, it is difficult to characterize the thermal boundary conditions for the sample inside a chamber, in particular considering that the sapphire window is heated up during the experiment.

Therefore, tests with an inert material were performed inside the chamber under the same conditions as the wood tests i.e. same heat fluxes, argon flow rate and sample holder. In this case, only heat transfer occurs and it is easier to characterize the boundary conditions such as convection and radiative heat losses. The chosen inert material was low density calcium silicate (Silcal 1100 ® ) since its thermal properties are close to those of spruce wood: the specific heat is around 1000 J • kg -1 • K -1 and the thermal conductivity varies between 0.08 and 0.2 W • m -1 • K -1 depending on the temperature [START_REF] Jannot | An improved model for the parallel hot wire: Application to thermal conductivity measurement of low density insulating materials at high temperature[END_REF]. Twelve thermocouples were also embedded along the centerline of the calcium silicate sample at the same depths as the wood samples. Moreover, an additional thermocouple was fixed at the rear face of the sample.

3D heat transfer model and estimation of the boundary conditions

A 3D heat transfer model was developed in PYTHON environment. The heat transfer balance described by eq. 12 was numerically solved using a finite volume method. For temporal discretization, an explicit scheme was used.

ρC p (T ) ∂T ∂t = ∇ • λc (T )∇T ( 12
)
λc is the thermal conductivity tensor. For calcium silicate, it reduces to a scalar because it is isotropic, but this will not be the case for wood. The boundary conditions are given in Fig. 4. Where: h front and h lateral are the convective heat transfer coefficients for

ϕ = π ϵ 1 (λ)ϵ 2 (λ) ϵ 1 (λ) + ϵ 2 (λ) -ϵ 1 (λ)ϵ 2 (λ) I bb λ (T 1 ) -I bb λ (T 2 ) -Tr 1 (λ)ϵ 2 (λ)I bb λ (T 2 ) dλ (13)
subscript 1 refers to the sapphire and subscript 2 to the sample (here the calcium silicate).

I bb λ (T ) is the blackbody intensity (Planck function). The first term corresponds to the radiative heat exchange between the sapphire surface and the sample (if sapphire is hotter than the sample, this term is positive and corresponds to an energy gain, otherwise it is negative and it is an energy loss). The second term corresponds to the radiative heat flux emitted by the sample and transmitted through the sapphire window. The integral has been approximated by the rectangular method with a ∆λ = 0.2 µm interval.

The temperature of the sapphire window was measured as a function of the exposure time (the sapphire window is heated up during the experiment) by an infrared camera (FLIR SC7300 LW) equipped with a 7.9 µm optical bandpass filter. At this wavelength sapphire is opaque and almost black (0.98 emissivity). The radiation emitted by the cone and reflected by the sapphire surface was subtracted from the measured radiation to only measure the radiation emitted by the sapphire window, following the procedure described in [START_REF] Acem | Surface temperature of carbon composite samples during thermal degradation[END_REF].

The heat capacity of the calcium silicate was measured as a function of temperature in [START_REF] Jannot | An improved model for the parallel hot wire: Application to thermal conductivity measurement of low density insulating materials at high temperature[END_REF] and was found to be equal to: C p (T ) = 439 + 83 log 10 (T + 273) where T is the temperature in °C. The density of calcium silicate was measured and found to be 240 kg•m -3 .

The thermal conductivity (λ c ) was assumed to vary with the temperature according to a second degree polynomial:

λ c (T ) = λ c1 T 2 + λ c2 T + λ 3 (14)
Where λ c1 , λ c2 and λ c3 are three constants to be estimated. An inverse method was applied on the temperature measurements obtained from the inert atmosphere calcium silicate tests at 35 and 45 kW • m -2 using this model to characterize the boundary conditions.

The objective function described by eq. 15 was minimized using the Levenberg-Marquardt algorithm.

S = N i=1 [T exp (t i ) -T mod (t i )] 2 (15)
Where: N is the number of experimental points, T exp (t i ) and T mod (t i ) are respectively the measured temperature and the temperature calculated by the 3D heat transfer model. The minimization was performed using the two heat fluxes simultaneously to estimate h f ront and the thermal conductivity of the calcium silicate. Although the thermal conductivity of the calcium silicate was measured as function of temperature in Jannot et al.'s study [START_REF] Jannot | An improved model for the parallel hot wire: Application to thermal conductivity measurement of low density insulating materials at high temperature[END_REF], this property was kept as unknown in order to check the reliability of the parameters obtained by optimisation. Thus, an estimated thermal conductivity by inverse method close to the thermal conductivity measured experimentally by the hot wire technique [START_REF] Jannot | An improved model for the parallel hot wire: Application to thermal conductivity measurement of low density insulating materials at high temperature[END_REF] indicates that the convective heat transfer coefficient obtained by optimisation is reliable.

h lateral was supposed to be equal to 10 W • m -2 • K -1 . In fact, a sensitivity analysis showed that the measured temperatures are not sensitive to h lateral , which makes sens because of the insulating role of the calcium silicate sample holder. It is therefore not possible to estimate this parameter by inverse method.

3D pyrolysis model and thermo-physical properties of wood

The 3D heat transfer model was coupled with the kinetic model in order to predict the thermal degradation of spruce wood in an inert atmosphere at the cone calorimeter scale.

The following assumptions were made for the 3D pyrolysis model:

-Heat and mass transfer in the gas phase were not considered in the model.

-Effect of shrinking, cracking and swelling were neglected.

The degradation of wood occurs following the kinetic scheme presented in the matter scale section. The heat capacity of spruce wood was measured between 30 and 100 °C in 10 °C increments using a DSC SETARAM microdSc3 instrument. The heat capacity obtained is C p,wood (T ) = 1131 + 4.67T where T is in °C. The latter correlation was extrapolated for the higher temperatures. The heat capacity of char was also measured between 30 and 600 °C and the heat capacity obtained is C p,char (T ) = 693 + 3.44T . The effective heat capacity was calculated as in equation 8. More details on the measurement technique of the heat capacity will be published in a future paper. Therefore, the energy conservation equation is as follow:

∂ ∂t (ρ.C p .T ) = ∇ • λc (T )∇T + k 1 .ρ 1 .H 1 + k 2 .ρ 2 .H 2 + k 3 .ρ 3 .H 3 + k 4 .ρ C 2 .H 4 (16)
Where: ρ is the bulk density calculated by The emissivity of the sample surface was measured as a function of the exposure time to the cone heater [START_REF] Terrei | Experimental and numerical multiscale study of spruce wood degradation under inert atmosphere[END_REF]. In the present model, the mean of the latter measurements was used i.e. 0.9 and the absorptivity was considered equal to the emissivity, since wood is almost grey and the surface chars quite rapidly, and char is even greyer than wood. The thermal conductivity of spruce wood was measured between 30 and 160 °C in the two directions parallel and perpendicular to the fiber direction [START_REF] Flity | Thermal conductivity parallel and perpendicular to fibers direction and heat capacity measurements of eight wood species up to 160 °C[END_REF] using the anisotropic parallel hot wire technique [START_REF] Jannot | Apparent thermal conductivity measurement of anisotropic insulating materials at high temperature by the parallel hot-wire method[END_REF]. The latter measurements show that the transverse thermal conductivity

ρ = ρ 1 + ρ 2 + ρ 3 + ρ C 2 + ρ C 3 + ρ C 4 ; ρ 1 , ρ 2 , ρ 3 , ρ C 2 ,
increase from 0.096 W • m -1 • K -1 at 20 °C to 0.110 W • m -1 • K -1 at 160 °C. Meanwhile,
the longitudinal thermal conductivity was found to be almost independent of temperature and equal to 0.345

W • m -1 • K -1 .
It is worth noting that several studies have shown that the radial and tangential thermal conductivity of wood are similar [START_REF] Lagüela | Thermal conductivity measurements on wood materials with transient plane source technique[END_REF][START_REF] Hankalin | On thermal properties of a pyrolysing wood particle[END_REF][START_REF] Korkut | The transverse thermal conductivity coefficients of wild cherry wood heat-treated using the thermowood method[END_REF][START_REF] Kol | The Transverse Thermal Conductivity Coefficients of Some Hardwood Species Grown in Turkey[END_REF][START_REF] Dupleix | Measuring the thermal properties of green wood by the transient plane source (TPS) technique[END_REF].

Therefore, the thermal conductivities used in the model below the onset of pyrolysis were

λ c⊥ = 0.102 W • m -1 • K -1
in the transverse direction (radial and tangential) and λ c∥ = 0.345 W • m -1 • K -1 in the longitudinal direction. Above 200 °C, the model assumes that the longitudinal thermal conductivity remains constant and the transverse thermal conductivity depends only on temperature according to the following law:

λ c⊥ = 0.102 + λ 1 .(T -200) + λ 2 .(T -200) 2 (17)
Where T is the temperature in °C, λ 1 and λ 2 were chosen in order to give a calculated mass loss, a calculated mass loss rate and calculated in-depth temperatures similar to those obtained from the cone calorimeter tests on spruce under inert atmosphere. The thermal conductivity tensor is therefore

λc =   λ c⊥ 0 0 0 λ c⊥ 0 0 0 λ c∥   (18)
x-axis is horizontal parallel to the exposed surface, y-axis is parallel to the cone heat flux and z-axis is vertical (thus wood fibers are oriented vertically), as shown in Fig. 4.

Finally, a perfect thermal contact was considered between the wood sample and the calcium silicate sample holder and the thermal properties of the latter material were taken from the work of Jannot et al. [START_REF] Jannot | An improved model for the parallel hot wire: Application to thermal conductivity measurement of low density insulating materials at high temperature[END_REF]. The temperature measured at the rear face of the sample (50 mm) is higher than the temperature at 40 mm depth at the beginning of the test, indicating that the atmosphere inside the chamber is heated during the tests. The numerical and the experimental temperatures are in good agreement. The root-mean-square error obtained simultaneously for the two tests is 7 °C and the normalized root-mean-square error (NRMSE) is approximately 1.2 %. Fig. 6 shows the comparison between the estimated thermal conductivity and the thermal conductivity measured by the hot wire technique [START_REF] Jannot | An improved model for the parallel hot wire: Application to thermal conductivity measurement of low density insulating materials at high temperature[END_REF]. The two conductivies look very similar, almost linear with temperature, same slope, there is only a slight shift between them. The mean deviation between the estimated and the measured thermal conductivity is approximately 11.6 %. This is satisfactory considering i) the temperature measurement uncertainties (thermocouple accuracy and thermocouple position uncertainty), ii) that the inherent uncertainty of the hot wire technique used to measure the thermal conductivity is about 10 % [START_REF] Jannot | An improved model for the parallel hot wire: Application to thermal conductivity measurement of low density insulating materials at high temperature[END_REF] and iii) that the heat flux applied to the front of the sample was assumed to be homogeneous (which is not exactly the case 1 ). This is 1 consideration of a non-uniform heat flux will be investigated in the near future very reassuring for the reliability of the results. The convective heat transfer coefficient

Results and discussion

Calcium silicate tests for estimation of boundary conditions

obtained is h f ront = 31 W • m -2 • K -1 .
Fig. 6. Comparison between the estimated and the measured thermal conductivity of the calcium silicate [START_REF] Jannot | An improved model for the parallel hot wire: Application to thermal conductivity measurement of low density insulating materials at high temperature[END_REF].

Wood degradation at the cone calorimeter scale

As shown in the section 4.1, all properties of wood and calcium silicate were measured except thermal conductivity of wood/char above 200 °C. In addition, the boundary conditions were estimated from experiments with an inert material. Probably the convective

Where T is in °C. It is interesting to note that the TGA analysis showed that spruce wood loses more than 75 % of its initial weight between 200 and 400 °C i.e more than 89 % of the degradation was already occured below this temperature. Therefore, although there is a bias between experimental and numerical in-depth temperature at high temperatures, the model predicts the mass loss and MLRPUA well at the different heating fluxes. The discrepancy between numerical and experimental temperatures at high temperatures could be caused by several reasons. Firstly, at high temperatures the wood turns to char, its porosity becomes very high and consequently the thermocouple is no longer in good contact with the material. The second reason is that the model considers a homogeneous medium, whereas when wood chars, cracks appear and the model no longer applies. Finally, the main reason is that although the atmosphere was inerted, smoldering combustion, i.e. char oxidation, can occur at a low rate because the oxygen content was not exactly zero.

However, temperature and MLR predictions obtained with the model proposed in this paper have been significantly improved compared to our previous work [START_REF] Terrei | Experimental and numerical multiscale study of spruce wood degradation under inert atmosphere[END_REF].

Conclusion

In this work, a 3D heat transfer and pyrolysis model for spruce wood was developed at the cone calorimeter scale under inert atmosphere. First, the thermal degradation of spruce wood was studied by TGA under inert atmosphere between ambient temperature and 1000 °C at different heating rates (5, 10 and 20 K • min -1 ). A global three-reaction mechanism was used to predict the mass loss and the mass loss rate. The kinetic parameters were estimated by fitting TGA data. The numerical and experimental results were in good agreement at this scale, where the wood sample is thermally thin and temperature gradients are negligible. Moreover, DSC measurements were carried out under an inert atmosphere at a heating rate of 10 K • min -1 . It was found that the kinetic scheme with three reactions is unable to predict the heat flux measured by DSC. A fourth reaction was therefore added. Based on the DSC and ATG measurements, as well as previous measurements of the heat capacity of wood and char, the heat of pyrolysis of spruce wood was modeled as a function of temperature. At a larger scale, cone calorimeter tests were then carried out with an inert material in an inert atmosphere at 35 and 45 kW • m -2 .

The aim was to characterize the heat transfer boundary conditions for tests performed on wood samples under the same conditions. An inverse method was applied to the indepth temperatures within a low density calcium silicate sample using a 3D heat transfer model to estimate the convective heat coefficients and thermal conductivity of that sample. The results showed that the thermal conductivity of calcium silicate obtained by optimization was in good agreement with the thermal conductivity measured by the hot wire technique. Therefore, the other parameters obtained from optimization, such as the convective heat transfer coefficient, can be considered reliable. Finally, a 3D pyrolysis model was developed by coupling the kinetic model, whose parameters were obtained from TGA, with a 3D heat transfer model. The boundary conditions estimated from the calcium silicate tests were fixed. All properties of wood were measured experimentally such as emissivity, absorptivity, heat capacity of wood until 100 °C, heat capacity of char until 600 °C, thermal conductivity in transverse and longitudinal direction until 160 °C.

The heats of reaction were also determined using the DSC technique. The only property that remained unknown was the thermal conductivity above 200 °C. This latter property was varied until a good fit was obtained between the numerical mass loss, mass loss rate and in-depth temperatures and their experimental counterparts. The results obtained with the 3D pyrolysis model developed and presented in this work comply very satisfactorily with the experimental results and the thermal conductivity obtained above 200 °C sounds physical.

Figure captions

Fig. 1 .

 1 Fig. 1. Mean normalized mass loss and mean normalized mass loss rate as a function of temperature for the different heating rates: 5, 10 and 20 • C • min -1 .

  and m C 3 are respectively the mass of the components A 1 , A 2 , A 3 , C 2 and C 3 . Eqs. 1, 2, 3, 4 and 5 were solved numerically to determine the mass of each component at the time t. Then, the total mass of the sample M(t) is calculated by:

Fig. 3 (Fig. 3 .

 33 Fig.3 (b) shows that the kinetic scheme with 4 reactions predicts the experimental data very well. The following heats of reaction were obtained:H 1 = -37.5kJ • kg -1 , H 2 = 124.6kJ • kg -1 , H 3 = 0, and H 4 = -88kJ • kg -1 .The kinetic parameters for the fourth reaction are: K 04 = 1.1 × 10 5 s -1 and E 4 = 100kJ • mol -1 . The kinetic parameters of reactions (1), (2), and (3) remain the same as the parameters presented in Table1.

  authors carried out cone calorimeter experiments with spruce wood samples (0.1 × 0.1 × 0.05 m 3 ) oriented vertically. During the tests, the samples were placed in a stainlesssteel chamber. Continuous argon injection at a low flow rate maintained an oxygen concentration less than 3 % inside the chamber to prevent char oxidation. The sides and rear of the chamber were cooled down by water heat exchangers. The heat flux coming from the cone passes through a sapphire window placed on the front face of the chamber.Wood samples were placed in a calcium silicate sample holder (3 cm thick) to reduce lateral convection losses. The mass loss of the samples was measured by a weighing scale with 0.01 g accuracy. Twelve K-type 0.1 mm diameter thermocouples were embedded in the samples at different depths[START_REF] Hildebrandt | The contribution of wood-based construction materials for leveraging a low carbon building sector in europe[END_REF][START_REF] Koufopanos | Kinetic modelling of the pyrolysis of biomass and biomass components[END_REF][START_REF] Chen | Characteristics and kinetic study on pyrolysis of five lignocellulosic biomass via thermogravimetric analysis[END_REF][START_REF] Terrei | Experimental and numerical multiscale study of spruce wood degradation under inert atmosphere[END_REF][START_REF] Benkorichi | Investigation of thermal degradation of pine needles using multi-step reaction mechanisms[END_REF][START_REF] Gunnar Grã¸nli | Thermogravimetric Analysis and Devolatilization Kinetics of Wood[END_REF][START_REF] Lautenberger | Generalized pyrolysis model for combustible solids[END_REF][START_REF] Bustamante | Experimental and numerical investigation of the thermal decomposition of materials at three scales: application to polyether polyurethane foam used in upholstered furniture[END_REF][START_REF] Lautenberger | A model for the oxidative pyrolysis of wood[END_REF][START_REF] Dhyani | A comprehensive review on the pyrolysis of lignocellulosic biomass[END_REF][START_REF] Hankalin | On thermal properties of a pyrolysing wood particle[END_REF] and 40 mm), along the centerline of the samples, following the technique developed in Terrei et al.'s study [24]. Tests were performed at 35 and 45 kW • m -2

Fig. 4 .

 4 Fig. 4. Boundary conditions of a sample exposed to the cone calorimeter.

ρ C 3 3 and C 4

 334 and ρ C 4 are the mass of respectively the constituents A 1 , A 2 , A 3 , C 2 , C per unit volume. The thermal boundary conditions are given in Fig. 4 and the convective heat transfer coefficients are those estimated from the calcium silicate tests.

Fig. 5

 5 Fig. 5 shows the experimental in-depth temperatures at 35 and 45 kW • m -2 and their comparison with the numerical temperatures.

( 2 Fig. 5 .

 25 Fig. 5. Comparison between numerical and experimental in-depth temperature within the calcium silicate sample at two different heat fluxes.

( 2 Fig. 7 .

 27 Fig. 7. Comparison between numerical and experimental mass loss and MLRPUA for wood tests at different heat fluxes.

Fig. 8 .

 8 Fig. 8. Comparison between numerical and experimental in-depth temperature within the wood samples at different heat fluxes.

Fig. 7

 7 Fig.8shows the comparison between numerical and experimental in-depth temperatures for the two heat fluxes. Below 400 °C, numerical and experimental temperatures comply very well. However, above 400 °C the pyrolysis model underestimates the temperature.

Fig. 1 : 3 Fig. 2 : 6 Fig. 3 : 8 Fig. 4 : 9 Fig. 5 : 12 Fig. 6 : 13 Fig. 7 : 13 Fig. 8 :

 132638495126137138 Fig.1: Mean normalized mass loss and mean normalized mass loss rate as a function of temperature for the different heating rates: 5, 10 and 20 • C • min -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Fig.2: Comparison between the numerical and experimental results for different heating rates. A 1 , A 2 , and A 3 are the pseudo-components of wood, while C 2 and C 3 are two char constituents. . . . . . . . . . . . 6

  the same experimental conditions as the wood tests, i.e., the same heating rate, argon flow rate, and temperature program. Prior to each wood test, an empty tube test was performed to establish the baseline of the apparatus. The tests were performed at a heating rate of 10 K • min-1

	)	7.1 × 10 8	123	0.17	-
	(2)	1.3 × 10 12	173	0.61	0.1
	(3)	552	81	0.22	0.59

(a) Mass loss at 5 • C • min -1 (b) Mass loss rate at 5 • C • min -1 (c) Mass loss at 10 • C • min -1 (d) Mass loss rate at 10 • C • min -1 (e) Mass loss at 20 • C • min -1 (f) Mass loss rate at 20 • C • min -1 Fig. 2. Comparison between the numerical and experimental results for different heating rates. A 1 , A 2 , and A 3 are the pseudo-components of wood, while C 2 and C 3 are two char constituents.

was calibrated (temperature and sensitivity) using the melting temperature and enthalpy 190 performed under

Acknowledgements

LCPP (Laboratoire Commun de la Prefecture de Police) and LRGP (Laboratoire Réactions et Génie des Procédés) are fully acknowledged for TGA measurements and DSC measurements respectively.


Miltiadis C Filippou
email: miltiadis.filippou@intel.com

Vasiliki Lamprousi

Jafar Mohammadi

Mattia Merluzzi

Elif Ustundag Soykan

Tamas Borsos

Nandana Rajatheva

Nuwanthika Rajapaksha

Luc Le Magoarou

Pietro Piscione

Andras Benczur

Quentin Lampin

Guillaume Larue

Dani Korpi

Pietro Ducange

Alessandro Renda

Hamed Farhadi

Johan Haraldson

Leonardo Gomes Baltar

Emilio Calvanese Strinati

Panagiotis Demestichas

Emrah Tomur

Pervasive Artificial Intelligence in Next Generation Wireless: The Hexa-X Project Perspective

Keywords: connecting intelligence, 6G networks, sustainability, trustworthiness, ML for air interface design, edge AI, explainable AI

The European 6G flagship project Hexa-X has the objective to conduct exploratory research on the next generation of mobile networks with the intention to connect human, physical and digital worlds with a fabric of technology enablers. Within this scope, one of the main research challenges is the ambition for beyond 5G (B5G)/6G systems to support, enhance and enable real-time trustworthy control by transforming Artificial Intelligence (AI) / Machine Learning (ML) technologies into a vital and trusted tool for large-scale deployment of interconnected intelligence available to the wider society. Hence, the study and development of concepts and solutions enabling AI-driven communication and computation co-design for a B5G /6G communication system is required. This paper focuses on describing the possibilities that emerge with the application of AI & ML mechanisms (with emphasis on ML) to 6G networks, identifying the resulting challenges and proposing some potential solution approaches.

Introduction

Mobile communications systems have aggressively developed during the last 50 years, starting from analog cellular technology, then shifting to digital cellular technology in the 1990s and further evolving after 2000 towards third-(3G) and fourth-generation (4G) mobile technology devising packet switching technology and offering higher data throughput and voice quality. Until 4G, most applications were personal subscriber-oriented. In 2019, the global deployment of the fifth-generation (5G) technology for cellular networks started; this time, the focus was on applications serving vertical industries, besides further enhancing personal consumer experience.

Recently, research on B5G/ 6G technology has kicked-off worldwide influenced by the advent of technology trends, e.g., network softwarization and virtualization for flexible utilization of (virtual) network resources [START_REF] Chowdhury | A survey of network virtualization[END_REF], Multi-access Edge Computing (MEC), a key technology enabling a rich computing environment at the edge of an access network [START_REF] Giust | Multi-access edge computing: An overview of ETSI MEC ISG[END_REF], and the steady growth of network and application data generation with corresponding analytics opportunities. Apart from technological trends, societal needs have also triggered the need for 6G research; such needs refer to the digital inclusion of the remote and vulnerable, service trustworthiness and sustainability, expressed in terms of energy consumption, carbon footprint and cost. The European Flagship 6G project Hexa-X [START_REF]Hexa-x deliverable D1.2 -expanded 6G vision, use cases and societal values -including aspects of sustainability, security and spectrum[END_REF] has identified six fundamental challenges: i) connecting intelligence involving wide use of Machine Learning (ML) models across the network, ii) "network of networks", i.e., from indoor to wide area networks iii) sustainability, iv) global service coverage, v) extreme experience and vi) trustworthiness. In this paper, we address the challenges i) and vi) by concentrating on two design directions: a) "ML for air interface design" and b) "Communications for more efficient AI/ML" [START_REF]Hexa-x deliverable D4.1 -AI-driven communication & computation co-design: Gap analysis and blueprint[END_REF].

The rest of the paper is organized as follows: Section 2 attempts to reply to the question of why ML is needed in 6G networks. Section 3 explores ML-driven wireless transceiver design approaches, as well as ML-driven radio resource management approaches for high communication performance and resilience to radio condition changes. Section 4 elaborates on networks for high-performing, sustainable and trustworthy AI. A detailed description of technical enablers elaborated in Sections 3 and 4 appear in [START_REF]Hexa-x deliverable D4.1 -AI-driven communication & computation co-design: Gap analysis and blueprint[END_REF], unless otherwise referenced. Section 5 presents main challenges of applying AI/ML in 6G and, finally, the paper concludes with Section 6.

Connecting Intelligence -why do we need ML in 6G?

Algorithm deficiency occurs when a (network) design task is well defined, nonetheless, the complexity of obtaining a well-performing algorithm is prohibitively high. In wireless communications, problems of allocating radio resources, e.g., power, time, frequencies, antennas fall under this category. On the other hand, model deficiency characterizes network (channel) and device (hardware) conditions that cannot be summarized by a universally applicable formal model. To tackle these deficiencies, ML is foreseen to be integral part of 6G, which can be attributed to several technology enablers. Computing capacity, is becoming more and more ubiquitous, calling for smart allocation of various AI workloads depending on computing cost, device constraints, or how data intensive these workloads are. As AI/ML gained focus in several industries, the usage of dedicated hardware solutions has increased to support and accelerate both training and inference tasks. The characteristics of structures that are hard or cannot be represented by an explicit mathematical ruleset, can often be modeled by ML models by learning from large datasets available. The value of data is already well recognized, and numerous data sources are made accessible internally within systems, as well as externally for business and open use. Intelligence will be distributed across agents in self-governed sub-systems of network functions (i.e., interconnected functional building blocks within a network infrastructure) and applications which will then interact with other sub-systems. Trusted and efficient communication support among these entities is an essential enabler for scaling out AI capabilities. The recent expansion in the volume of accessible compute and data resources resulted in several major breakthroughs in the areas of AI algorithms and architectures, which brings an explosion of AI/ML services and applications.

Some representative new 6G use cases relevant to ML-driven air interface design are the ones of "merged reality game/work", "interacting and cooperative mobile robots" and "flexible manufacturing". These use cases share some "conventional" communication Key Performance Indicators (KPIs), such as bit rate, latency, communication outage probability, some new ML related KPIs, i.e., convergence speed, flexibility, data quality, and complexity gain and some environmental characteristics that need to be considered, e.g., coverage, connection density and positioning accuracy. Relevant "soft goals" (also termed after as Key Value Indicators -KVIs) are generalizability, deployment flexibility, distributed learning with frugal AI, trustworthiness and resistance to adversarial attacks. Accordingly, some 6G use cases relevant to in-network learning are the three mentioned above, plus the ones of "digital twins for manufacturing", "immersive smart city", "AI partners", and some new enabling services, i.e., AI-as-a-Service, Compute-as-a-Service (CaaS) and AI-assisted Vehicle-to-Everything. The KVIs related to innetwork learning are explainability, fairness, data economy and model complexity. Finally, the KPIs related to in-network learning are latency, AI agent density, interpretability level, network energy efficiency and inferencing accuracy. For a more detailed view, please refer to [START_REF]Hexa-x deliverable D1.2 -expanded 6G vision, use cases and societal values -including aspects of sustainability, security and spectrum[END_REF][START_REF]Hexa-x deliverable D4.1 -AI-driven communication & computation co-design: Gap analysis and blueprint[END_REF].

ML for better wireless communications

ML-based techniques for physical layer design, i.e., at bit-level transmission between wireless devices, show great promise for enhancing the spectral and energy efficiency of future 6G communication systems. They can introduce more flexibility, increase robustness against hardware impairments, facilitate joint optimization of transmitter and receiver, as well as reduce required signaling overhead. In the following, we investigate a few applications of ML for wireless communications.

From hardware impairment mitigation to low complexity decoder: transceiver design by ML

Joint learning of transmitter and receiver improves spectral efficiency in various scenarios [START_REF] Korpi | Waveform learning for reduced out-of-band emissions under a nonlinear power amplifier[END_REF].

The key aspect is finding the proper balance between prior knowledge/restricted tasks and the freedom to learn several tasks jointly. A particularly interesting solution involves learning only the digital signal modulation scheme (constellation) on the transmitter side, while jointly learning the full receiver, thereby discarding the need to use pilot signals [START_REF] Korpi | Waveform learning for reduced out-of-band emissions under a nonlinear power amplifier[END_REF]. Neural Networks (NNs), due to their universal function approximation properties, demonstrate resilience against hardware impairments, be learning how to compensate for the the hardware imperfections.

For instance, the transmitter can be trained to produce a waveform that results in 5-10 dB less out-of-band emissions caused by a nonlinear power amplifier [START_REF] Korpi | Waveform learning for reduced out-of-band emissions under a nonlinear power amplifier[END_REF]. At the receiver side, it has also been shown that a context-aware receiver can be used to compensate the impact of hardware impairment including oscillator phase noise at highband transmissions [START_REF] Farhadi | Machine learning empowered context-aware receiver for highband transmission[END_REF].

On the channel and environment awareness side, ML has demonstrated impeccable performance in a few areas. As for channel estimation, NNs inspired by the optimal Minimum Mean Square Error (MMSE) solution [START_REF] Chen | Part II: Multi-dimensional iterative ML-based channel estimation for B5G[END_REF] demonstrated an excellent trade-off between complexity and accuracy. The benefit of using NNs for the task of matrix inversion, which is the most computationally expensive part of the MMSE estimator, as shown in [START_REF] Chen | Part II: Multi-dimensional iterative ML-based channel estimation for B5G[END_REF] is twofold: smaller computational complexity and lower sample complexity, in comparison to the MMSE estimator. In [START_REF] Magoarou | Online unsupervised deep unfolding for MIMO channel estimation[END_REF], an NN designed via the deep unfolding method is initialized with a dictionary of steering vectors, which is then trained online in an unsupervised way, yielding great performance improvements. Mitigating dynamic blockage effects which occur due to movements, is crucial when using the higher frequencies in 6G. Sensing can enable situational awareness for the wireless system and the LiDAR appears as a strong candidate for this [START_REF] Marasinghe | LiDAR aided human blockage prediction for 6G[END_REF]. In the context of massive multi-antenna transmission with, ML has recently been applied to predict appropriate precoders based on the location of the targeted user. An NN comprising random Fourier features in order to allow high spatial frequencies to be learned has been proposed and trained on data containing the locations of users and their channels [START_REF] Magoarou | Deep learning for location based beamforming with NLOS channels[END_REF].

As for forward error correction, ML applications have already demonstrated significant improvements in mechanisms such as Turbo codes, Low-Density Parity-Check (LDPC) codes or polar codes. The code-agnostic, low-complexity weighted belief propagation inspired NN decoder demonstrated in [START_REF] Larue | Blind neural belief propagation decoder for linear block codes[END_REF] and developed in Hexa-X is a promising candidate to improve the decoding performance. This approach opens new perspectives for the design of codes suitable for low-complexity and fully explainable decoders, particularly suited to the needs of constrained 6G devices.

Applying ML-based solutions to enhance the communications at the physical layer allows for lower sample complexity for estimation tasks, higher accuracy for detection tasks, and better coping with non-linearity stemming from hardware imperfections. It becomes apparent that the future of air interface design involves, if not entirely, partially NN-based solutions.

AI-driven radio resource allocation & management

Radio resource management (RRM) often involves computationally complex sub-optimal solutions. RRM becomes even more challenging when dealing with 6G envisioned requirements. In the following, we present some applications of ML for RRM that can achieve higher performance. In [START_REF] Rajapaksha | Deep learning-based power control for cell-free massive MIMO networks[END_REF], unsupervised learning is used for uplink power control in a cell-free massive antenna system to achieve max-min user fairness. It has a reduced computational complexity, while keeping tight bounds to the conventional solution. As wireless networks are most representable using graphs, Graph NNs (GNNs) can be used for RRM tasks in supervised or unsupervised manner. Furthermore, Access Point (AP) selection in cell-free network is formulated as a link prediction on a graph, where, given the measurements of known set of APs from a wireless device, a GNN predicts the candidate APs which can serve the device. Here, the GNN leverages the knowledge of static correlated fading, due to static environmental features to predict the APs. Certain RRM tasks rely on the knowledge of the relative location of devices and base stations e.g., cell search and hand-over. Additionally, a computationally efficient version of channel charting has recently been proposed, relying on a distance measure in the channel domain that is insensitive to small-scale fading and has been shown to perform well on other tasks, such as channel mapping and positioning . Incorporating timestamp information can greatly improve channel charting, as it allows using contrastive learning methods, such as triplet loss.

Given a large number of heterogeneous sensors connected to Federated Learning (FL) nodes at the network edge, load balancing is necessary to remedy potential hot spots and data diversity to ensure quality balance for the federated learners. By dynamic load rebalancing, we reconnect sensors to nodes in the radio network if load is uneven or some nodes receive insufficient variety of data for serving local models. Another challenge in RRM, related to the inferencing capability and energy efficiency of edge AI, is prioritizing learning data transmissions per a data importance criterion. Challenges lie in the joint presence of channel uncertainty. e.g., interference, noise, and data uncertainty, which is measured by entropy. State-of-the-art works, such as [START_REF] Liu | Wireless data acquisition for edge learning: Data-importance aware retransmission[END_REF] mostly concentrate on time as a resource to be optimized per a data significance criterion, however, more parameters may be considered leading to frugal over-the-air learning.

The optimization problems describing RRM are computationally intensive. The research work in Hexa-X demonstrates that techniques from ML can reduce optimization complexity by leveraging from structure in data.

Networks for high-performing, sustainable & trustworthy AI

Enabling edge AI in 6G networks calls for a holistic view of several KPIs/KVIs entailing energy efficiency, latency, trustworthiness, etc. In this section, we present the main technological enablers, which we envision to enable in-network AI in 6G, with a deeper look into such targets.

Joint communication/computation co-design for efficient Edge AI/ML

The convergence of communication and computing for edge AI will play a key role in 6G, as a means of enabling both lean network orchestration and an efficient platform for AI services. First of all, enabling in-network AI functionalities will need to support discovery and selection of the network node(s), such as a user device or an edge cloud server, capable of processing a workload (e.g., an ML model or any application-related processing load) with high performance, security and energy efficiency. The CaaS concept aims to offer such capabilities, to enable services, such as the offloading of computations from end devices to the discovered nodes (e.g. nearby edge cloud servers). To this end, to reduce the communication burden due to frequent exchange of data and/or local models, semantic and goal-oriented communications have been identified as two (possibly inter-playing) technical solutions, as their focus is on the actual interpretation of the receiver, also according to the effectiveness achieved in performing a task, rather than on error free transmissions.

To further improve efficiency, emerging AI architectures need to be explored, due to the radically different requirements on the communication infrastructure. Biologically inspired NNs, like Spiking NNs [START_REF] Tavanaei | Deep learning in spiking neural networks[END_REF] are often mentioned as the next AI generation, where neurons communicate with spikes within a single device or over wireless channels. The sparsity of the events enables highly energy-efficient operations. These events can be timing sensitive, so the communication medium must keep this relation with high fidelity, while highly localized functions in neuromorphic systems may benefit from network features for local communications.

Changing perspective towards AI for automated radio network control, a promising solution is the concept of Explainable AI (XAI), whose goal is to investigate tools and techniques aimed at opening the so-called opaque (or black-box) models (e.g., Deep NNs -DNNs) or at devising intrinsically interpretable and accurate models (e.g., rule based systems). This concept can be used to explain and distinguish the effect of network configuration and user device load to network performance KPIs. The main technical difficulty is that network configuration (e.g., policy of allocating radio resources) affects the user device load, while the load itself has a more direct effect on the network KPIs. Therefore, XAI will primarily find the importance of the load and explain the predicted KPI based on load, mostly ignoring the explanation of how network configuration and control affects the KPI. The vision is to combine game [START_REF] Frye | Asymmetric shapley values: incorporating causal knowledge into model-agnostic explainability[END_REF] and information theoretic measures. From a network resource orchestration standpoint, centralized AI solutions [START_REF] Foukas | Network slicing in 5G: Survey and challenges[END_REF] suffer from orchestration reaction time, a crucial issue for 6G use cases with strict latency requirements. AI-based distributed orchestration at the network edge is a promising solution for automatic and efficient up/down scaling of User Plane Network Function instances at the edge. Distributed Network Data Analytics Functions (NWDAFs) [START_REF]3GPP[END_REF] and other data sources at the same edge location would feed the AI algorithm for its (re)training, thus reducing the reaction time and the amount of sensitive data sent to the cloud, however with a limit on inference accuracy, due to the scarcity of edge resources. Indeed, the additional computational cost of training the models in each participating node is a challenge to be addressed. Some of the techniques aimed at minimum communication costs and efficient knowledge sharing are FL, blockchain and approaches that combine both. To ensure resource efficiency, privacy, learning quality, and resilience, when deploying AI/ML methods in large scale wireless networks, different resource management and knowledge sharing methods should be evaluated.

Privacy, security & trust-enhancing enablers for in-network AI/ML

Besides efficiency related KPIs/KVIs, trustworthiness is a requirement to be addressed throughout the ML pipeline, to ensure data protection and privacy, and avoid attacks affecting models robustness. In particular, distributed training agents may behave malevolent and pose model poisoning threat in collaborative learning settings. Certain attacks can be applied to the inference phase [START_REF] Melis | Inference attacks against collaborative learning[END_REF], such as membership inference, model extraction, or model inversion. To mitigate these attacks, the first step is to implement fundamental security and privacy solutions that constitute a first layer of defense, e.g., implementing proper identity management and access control methodologies for unauthorized data access attacks. Implementing basics will resolve well-known issues, but a second layer, with an ML/AI specific mitigation approach, is needed to handle residual threats. Advanced privacy enhancing technologies can be used to prevent those that cannot be solved via basic methodologies.

In addition to this, advanced mechanisms searching for vulnerabilities are needed, especially in distributed and real-time operation contexts. In this case, data sanitization could rely on techniques, such as anomaly detection, removal of negative impact (elimination of training data with negative impact on the accuracy), training with micromodels (to reduce the risk of attacks), and usage of Generative Adversarial Networks (GANs). The effect of data poisoning (i.e., injection of malicious points in a dataset) in FL, can spread across the network, where the model is distributed. Leveraging on the GAN paradigm, realistic samples resembling adversarial data are generated and used in the training, obtaining FL models immunity to the attack. Furthermore, removing biased data from the original data set is a fundamental aspect to improve accuracy. In 5G, this issue may occur in NWDAF [START_REF]3GPP[END_REF] which is composed of two logical components: the Analytics Logical Function (AnLF), performing inference, and the Model Training Logical Function (MTLF), for training the ML model, which can be then consumed on-demand by the AnLF. The data source of MTLF could be biased, leading to potential unfair decisions by the AnLF. To this end, a continuous verification and an unfairness mitigator help avoiding bias and unfair decisions, respectively, the former for detecting sources of unfairness from the AnLF, while the latter for mitigating biased decisions taken by the MTLF. In such a direction, AI systems must also meet explainability requirements, as already mentioned in Section 4.1. However, most of existing FL solutions, conceived for collaborative training of ML models in a privacy preserving fashion, disregard the explainability requirement and only comprise models optimized via Stochastic Gradient Descent, e.g., DNNs. The Federated XAI (FED-XAI) vision is about devising methods and approaches compliant with both FL and XAI paradigms and it specifically targets the collaborative learning of inherently explainable models.

Which are the challenges of applying AI/ML in 6G?

The Hexa-X 6G vision involves generation, extraction, collection and processing of data from distributed network entities for different functionalities to enhance network performance. These functionalities introduce challenges from different perspectives ranging from fairness in data processing and inference, processing sensitive data in compliance with the regulations, distributed trust and ownership to sustainability. The distributed nature of data resources and the diversity of the use cases creates a huge amount of data with different granularity ranging from telemetry data, application specific data to management data. More sensitive data will not only be transported by 6G, but also be processed within the 6G system. Therefore, not only following the privacy-by-design approach, but also developing advanced privacy enhancing mechanisms enabling distributed and collaborative computations will become inevitable.

Decentralized and disjoint AI-driven 6G network functions are expected to be deployed as cloud services. The ownership of the 6G components, AI capabilities, and the data being processed may belong to different parties. In addition, the underlying cloud infrastructure may or may not be operated by those parties handling 6G functionalities. Thus, the diversity of the ownership and the control in the architecture create complex trust relationships. AI life-cycle comprising training, development, and deployment of the model and operating and managing the AI service should be carefully designed taking these complexities into account. In such a distributed learning environment involving network domains of different levels of trust, the generalization capability of an ML model may be affected, due to introduced limitations to data ingestion from different geographical areas where different data use regulations may be in effect. From a standardization point of view, a challenge is to specify interoperable interfaces facilitating AI agent discovery and selection, while, from a technology standpoint, the challenge is enhancing ML model generalization capability by not violating personal and corporate data regulations. Sustainability is another fundamental aspect of the 6G vision [START_REF]Hexa-x deliverable D1.2 -expanded 6G vision, use cases and societal values -including aspects of sustainability, security and spectrum[END_REF]. Since data traffic grows at high pace, also considering the pervasive deployment of computing resources needed to enable AI, along with the unavoidable network densification, ever higher efficiency is required to limit network energy consumption, thus keeping a positive balance between direct and indirect effects of the ICT industry on the global carbon footprint. Sustainable AI in 6G poses unprecedented challenges in terms of hardware re-usability, lean architectural design and operations, as well as higher reliance on renewable energy sources.

Conclusions

In this paper, we have introduced the main motivations, envisioned opportunities and incurred challenges of widely using AI (with emphasis on ML) in future 6G mobile communications systems, per the vision of the EU 6G Flagship project Hexa-X. First, we have introduced several technical enablers for designing the 6G air interface using data-centric techniques, applicable to wireless transceivers for high performance signal transmission and reception, aiming to offer wireless device cost, energy and complexity reductions and deal with wireless transceiver hardware impairments. Enablers for efficient radio resource and interference management have also been elaborated to address large-scale transmission problems suffering algorithm deficiencies. Then, methodologies for joint communication and computation system co-design have been also described to address engineering trade-offs for different collaborative learning structures. Privacy, security and trust-enhancement enablers for in-network AI/ML have been also elaborated. Finally, challenges of applying AI/ML in 6G networks have been detailed, calling for innovation by both AI and wireless communications communities regarding both aspects of "learning to communicate" and "communicating to learn" in 6G cellular networks.

Acknowledgments

This work has been partly funded by the European Commission through the H2020 project Hexa-X (Grant Agreement no. 101015956).


Shaolin Liu

Azita Ahmadi-Senichault

Cyril Levet

Jean Lachaud

EXPERIMENTAL INVESTIGATION OF HEAT TRANSFER IN CALCARB : ONE OR TWO TEMPERATURE MODEL ?

Keywords: One and two temperature model, Inverse analysis method, Experimental investigation, Heat transfer, Carbon fiber preform

Thermal protection systems (TPS) are used to ensure acceptable temperatures for the outer surface of space vehicles during atmospheric entry. Carbon fiber preforms, such as Calcarb, are widely used in TPS materials due to their small weight and low thermal conductivity. The main question addressed in this work is the choice of the macroscopic model allowing to accurately describe coupled heat and mass transfer in Calcarb. For this purpose, both one and twotemperature (1T& 2T) models are used. In the coupling between the two macroscopic equations for each phase, the 2T model features a heat transfer coefficient (hv) that is a subject of numerous studies. However, the correlations in these studies are not suitable for Calcarb due to its properties such as small characteristic length (fiber diameters of 50 µm), anisotropic micro-structure and low thermal conductivity (0.23W/(m K)). An experimental investigation was conducted based on the transient single-blow technique (TSBT) to obtain the temperature fields inside Calcarb. To investigate the influence of the anisotropic structure of the materials on hv, different experiments were performed by changing the orientation of the sample (Through-Thickness (TT), In-Plane (IP)). The gas flow inside Calcarb is considered compressible and laminar. The velocity field is modeled by Darcy's law. Temperature-dependent thermophysical properties are implemented in the numerical simulations. The effective parameters, that is, the thermal conductivities and heat transfer coefficients, are obtained by carrying out an inverse analyses using 1T and 2T models. The method adopted allows automatic optimization of parameters by coupling the Open Source optimization software Dakota with the porous material analysis toolbox software PATO. The results show that the value of hv in the 2T model is around 10 9 W/(m 3 • K). Gas and solid temperatures are in local thermal equilibrium within Calcarb in the conditions of the experiment and in this case the use of a 2T model is not necessary. It remains important to take into account the fact that the effective thermal conductivity in the 1T model is altered by gas flow rate due to the dispersion mechanism.

INTRODUCTION

Carbon fiber preforms are widely used in the industry as insulators in high temperature furnaces. At the end of the 1990's, NASA used a rigid carbon felt called FiberForm, produced by Fiber Materials Inc., to develop a new generation of low density heat shield to protect space vehicles during hypersonic atmospheric entry [START_REF] Tran | Phenolic impregnated carbon ablators (PICA) as thermal protection systems for discovery missions[END_REF]. This new class of materials called Phenolic Impregnated Carbon Ablators (PICA) has flawlessly been used since then: Stardust (NASA, 2006) [START_REF] Stackpoole | Post-flight evaluation of stardust sample return capsule forebody heatshield material[END_REF][START_REF] Lachaud | Multiscale approach to ablation modeling of phenolic impregnated carbon ablators[END_REF], Mars Science Laboratory (NASA, 2012) [START_REF] Edquist | Mars science laboratory heat shield aerothermodynamics: Design and re-construction[END_REF][START_REF] Meurisse | Multidimensional material response simulations of a full-scale tiled ablative heatshield[END_REF], Dragon vehicles (SpaceX, since 2012) [START_REF] Stackpoole | Postflight evaluation of PICA and PICA-X : Comparisons of the stardust SRC and Space-X Dragon 1 forebody heatshield materials[END_REF]. The European Space Agency and ArianeGroup have developed ASTERM [START_REF] Bouilly | Asterm: Maturation of a new low density ablative material[END_REF] based on Calcarb, produced by Mersen. FiberForm and Calcarb are both made of chopped carbon fibers of millimeter length and of about 10 micrometer in diameter [START_REF] Tran | Phenolic impregnated carbon ablators (PICA) as thermal protection systems for discovery missions[END_REF][START_REF] Triantou | Thermomechanical performance of an ablative/ceramic composite hybrid thermal protection structure for re-entry applications[END_REF] as shown in Fig. 1. During the manufacturing process, the carbon fibers tend to align according to the compression plane resulting in anisotropic properties. The direction perpendicular to this plane is defined as "Through-Thickness" (TT) and that parallel as "In-Plane" (IP), they are defined in Fig. 2. The thermal conductivity ratio between TT and IP directions is of about two [START_REF] Panerai | Experimental measurements of the permeability of fibrous carbon at high-temperature[END_REF]. For permeability this ratio has been found to lay around 1.8 [START_REF] Borner | High temperature permeability of fibrous materials using direct simulation monte carlo[END_REF].

The knowledge of the distribution of the temperature in the gas and solid phases in Carbon fiber felts is required for a comprehensive understanding of heat transfer [START_REF] Xu | Analytical considerations of local thermal nonequilibrium conditions for thermal transport in metal foams[END_REF] and to reduce design uncertainties [START_REF] Wright | Sizing and margins assessment of the mars science laboratory aeroshell thermal protection system[END_REF][START_REF] Wright | An overview of technology investments in the NASA entry systems modeling project[END_REF]. The heat transfer process in porous materials can either be studied under the assumption of local thermal equilibrium (LTE) or local thermal non-equilibrium (LTNE) [START_REF] Xu | Analytical considerations of local thermal nonequilibrium conditions for thermal transport in metal foams[END_REF][START_REF] Chikh | Analytical solution of non-Darcian forced convection in an annular duct partially filled with a porous medium[END_REF][START_REF] Shugard | A simple model of gas flow in a porous powder compact[END_REF][START_REF] Lu | Thermal analysis on metal-foam filled heat exchangers. part i: Metalfoam filled pipes[END_REF]. In LTE models, it is assumed that the average temperatures of the solid and fluid phases are equal. There are many studies regarding incompressible non-reacting flows [START_REF] Nield | Convection in porous media[END_REF]. In a recent work a generic LTE model was proposed for porous reactive materials submitted to high temperatures [START_REF] Lachaud | A generic local thermal equilibrium model for porous reactive materials submitted to high temperatures[END_REF], the energy conservation can be written as in Eq.1, [START_REF] Lachaud | A generic local thermal equilibrium model for porous reactive materials submitted to high temperatures[END_REF][START_REF] Puiroux | Non-equilibrium theories for macroscale heat transfer: ablative composite layer systems[END_REF]. The generic LTE model needs to be upgraded in the local disequilibrium case, such as possibly for the pyrolysis of PICA [START_REF] Scoggins | Pyrolysis mechanism of PICA[END_REF]. The two-temperature thermal transport equations for the LTNE model in the case of compressible and reacting flows are given in Eq.2 and 3.

[(ϵ i ρ i c p,i ) ∂ t T ] -∂ x • (k eff • ∂ x T ) = - i∈[1,Np] h i ∂ t (ϵ i ρ i ) -∂ t (ϵ g ρ g h g -ϵ g p) + ∂ x • (ϵ g ρ g h g v g ) + ∂ x • k∈[1,Ng] (Q k ) + µϵ 2 g K -1 s • v g • v g (1) i∈[1,Np] [(ϵ i ρ i c p,i ) ∂ t T s ] + i∈[1,Np] h i ∂ t (ϵ i ρ i ) = ∂ x • (k s,eff • ∂ x T s ) + h v (T g -T s ) (2)
(ϵ g ρ g c p,g ) ∂ t T g -∂ t (ϵ g p) + j∈[1,Ns] (h j ∂ t (ϵ g ρ g y j ) + ∂ x •Q j +∂ x •(ϵ g ρ g h g v g ) = ∂ x •(k g,eff ∂ x T g )+h v (T s -T g ) (3)
In the Eq. 2 and 3, there are three effective properties, h v , k s,eff and k g,eff . The purpose of this article is to determine k eff in the LTE model and k s,eff , k g,eff , h v in the LTNE model and analyze the anisotropic properties of Carbon fiber felts. The remainder of the paper is organized as follows. In Section 2, the experimental setup and test procedure are presented. In section 3, a two-dimensional geometric model is generated. An optimization problem for minimizing the error between measured and predicted temperatures is introduced. In section 4, the value of k eff in the LTE model and k s,eff , k g,eff , h v in the LTNE model for different inlet gas flow rates in two directions are presented. The results obtained by solving the LTE model and the LTNE model were compared. The conclusions of the paper (Section 5) provide an outlook on the determinations of the k s,eff , k g,eff , h v for complex porous media when correlations are not available.

EXPERIMENT METHOD

As already mentioned, the determination of the effective properties through an inverse method requires experimental data. The experiments could be conducted in steady state or in transient conditions [START_REF] Fuller | Measurement and interpretation of the heat transfer coefficients of metal foams[END_REF][START_REF] Kamiuto | Heat transfer correlations for open-cellular porous materials[END_REF][START_REF] Xia | Experiment on the convective heat transfer from airflow to skeleton in open-cell porous foams[END_REF][START_REF] Ando | An accurate experimental determination of interstitial heat transfer coefficients of ceramic foams using the single blow method[END_REF]. In the former, the sample is heated by the wall to a constant temperature. Cold gas is injected at the inlet of the sample and its temperature is measured at specific positions as it heats up. The transient method considered in this work is called the transient single blow technique (TSBT) [START_REF] Xia | Experiment on the convective heat transfer from airflow to skeleton in open-cell porous foams[END_REF][START_REF] Gürüf | Experimental based numerical approach for determination of volumetric heat transfer coefficients of modified graphite foams[END_REF]. In this method, the gas is heated upwind of the sample. It is this gas that heats the sample. The temperature profile of the sample is recorded as a function of time. For the steady state method, the macroscopic solid temperature was affected by the heat source from outside of the channel and gas inside the sample, which is not suitable for the study of heat convection inside porous materials. Therefore, we selected the TSBT in this work.

Experimental system and test procedure

A schematic of the experimental system that we developed and an enlarged view of the test section (with thermocouples shown) is presented in Fig. 3. It consists of a gas tank, the mass flow controller, the heating equipment, the test section, and measuring device.

In the experiment, the first step is to set the flow rate of inlet gas. This value is controlled and measured by the mass flow controller (Bronkhorst F-201CV-020-AAD-11-Z). The gas is heated with a heat exchanger made of Calcarb. The hot gas flows trough the sample. The red points in Fig. 3 indicate the positions of thermocouples (type K, 0.25 mm sheath diameter). The inlet and outlet gas temperatures, tube surface temperature and the solid temperature inside the sample are measured. The thermocouples are connected to a display data logger (Pico Technology TC-08) that records the temperature at a time steps of 1.0 s. The mass flow rates of inlet gas vary

Experimental uncertainty analysis

There are two types of experimental uncertainties: the first one is the direct measurement error (caused by experimental devices), and the second one is the error arising from the measurement process. In this work, the range and uncertainty due to the experimental device can be seen in Table .1. Here, errors from the measurement process are chiefly due to the positioning of the thermocouples. In the flow1 and flow2 regions (defined in the simulation section), the uncertainties δ 1 and δ 2 on thermocouple positions are 0.01% and 0.2% respectively. In the sample region, the uncertainties δ 3 is 0.5%. Based on the uncertainty analysis, the uncertainties on h v , k s,eff and k g,eff is smaller than 12.1 %.

PARAMETER ESTIMATION METHOD

Mathematical model

The Reynolds number based on the pore diameter (d p ) is defined as, Re = ρgudp µg . When the inlet gas mass flow rate caries from 1.92e-5 kg/s to 7.63e-5 kg/s, Re varies from 1.12 to 4.4. The upper limit of validity of Darcy's law lies around Re = 5 ∼ 10. [START_REF] Majid | High velocity flow in porous media[END_REF][START_REF] Sobieski | Darcy's and forchheimer's laws in practice. part 1. the experiment[END_REF]. The flow inside the carbon fiber felt can be modeled by Darcy's law. The governing equations are given by the LTNE model which consists of the mass conservation equation, Eq.4 and the two-phase energy conservation equations, Eq.2, 3. To solve implicitly Eq.2, 3, it is convenient to express it in terms of temperature. Eq.2 and 3 can be simplified as follows

∂ ∂t ε M p RT g + ∇ • M p RT g - K s µ (∇p) = 0 (4) ∂ ∂t (ε(ρc p ) g T g ) + ∂ ∂t (εp) + ∇ • ((ρc p ) g uT g ) -u∇p = ∇ • (k g,eff • ∇T g ) + h v (T s -T g ) ∂ ∂t ((1 -ε)(ρc p ) s T s ) = ∇ • (k s,eff • ∇T s ) + h v (T g -T s ) (5)
The two-dimensional compressible two-temperature Darcy's scale equations are solved using the Porous material Analysis Toolbox based on OpenFOAM (PATO) [START_REF] Lachaud | A generic local thermal equilibrium model for porous reactive materials submitted to high temperatures[END_REF] that integrates chtMultiRegionFoam of OpenFOAM 7 [START_REF] Renze | Simulation of conjugate heat transfer in thermal processes with open source cfd[END_REF] to simulate the gas flow.

Geometric model and simulation results

The geometric model established is presented in Fig. 4. It has same length as the experimental device. This model consists of 5 regions, flow1, sample, flow2, thermocouple and tube. It is a 2D axi-symmetric geometry and the wedge mesh is used in this model.

Thermal properties of gas and solid phase

The thermophysical properties (k g , k s ,cp g ,cp s ) are tabulated as a function of temperature for solid and gas. The effective thermal conductivities k eff in the LTE model (Eq.1) and k s,eff , k g,eff in the LTNE model (Eq.5) are expressed as,

k ef f,|| = c1 • k s , k ef f,⊥ = c2 • k s k s,ef f,|| = c3 • k s , k s,ef f,⊥ = c4 • k s k g,ef f,|| = εk g + c5, k g,ef f,⊥ = εk g + c6 (6)
The values of k s,eff in the case of no gas flow are measured in TT and IP directions respectively by using transient Plane Sources technique (Hot Disk, TPS 3500) [START_REF] Silas E Gustafsson | Transient plane source techniques for thermal conductivity and thermal diffusivity measurements of solid materials[END_REF]. At room temperature, these values are 0.212 and 0.399 W/(m • K) in TT and IP directions respectively. For the same type of fiber materials (FiberForm), the mean thermal conductivity obtained in the literature [START_REF] Panerai | Micro-tomography based analysis of thermal conductivity, diffusivity and oxidation behavior of rigid and flexible fibrous insulators[END_REF] are 0.167 and 0.392 W/(m • K) in the TT and IP directions.

Pressure, velocity and temperature fields in the full domain

Fig. 5, 6 and 7 show the pressure, velocity and temperature fields in the full domain. Fig. 8 shows the temperature fields for the fluid and solid phases inside the sample. The pressure remains constant in the flow1 and flow2 regions, and in the sample region, the pressure changes are in agreement with Darcy's law.

The average gas velocity in the tube is calculated from the volumetric mass flow rate. The gas velocity distribution in the tube is in agreement with Poiseuille's law. The gas velocity inside the sample v g is equal to the Darcy velocity u divided by the porosity ε. The gas velocity profile can be considered unidirectional within the sample.

In the sample region, the temperature difference between the macroscopic average gas and solid temperatures T g and T s is negligible. It shows that in this case, when the value of h v is large enough (h v ≈ 10 9 w/m 3 • K), a rapid heat transfer between the gas and the solid takes place and local thermal equilibrium is reached.

Optimization process

In this part, we coupled the Open Source optimization software Dakota [START_REF] Adams | Dakota, a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis: Version 6.15 user's manual[END_REF] with PATO to perform parameter optimization. We possess three data sequences {T i g } n i=0 , {T i s1 } n i=0 , {T i s2 } n i=0 that are the measured outlet gas temperature data and solid temperature data collected at every time step in experiment. A minimization problem for the relative error between the measured and predicted temperature can be defined as:

S = 1 3 ( 1 n n i=1 T i g,num -T i g T i g 2 + 1 n n i=1 T i s1,num -T i s1 T i s1 2 + 1 n n i=1 T i s2,num -T i s2 T i s2 2 ) (7)
The gradient descent algorithm and genetic algorithm were used in the Dakota software. For faster and more stable convergence, we also apply the adaptive nonlinear least-squares algorithm (ANLSA) from the nl2sol package available in Dakota, which is more suitable to solve an unconstrained nonlinear minimization problem [START_REF] Dennis | An adaptive nonlinear least-squares algorithm[END_REF].

RESULTS AND DISCUSSIONS

Determination of k eff in LTE model

Both samples (TT and IP directions) are tested in the experimental set-up at different volumetric flow rates from 1.0 L/min to 4.0 L/min. The corresponding mass flow rate of inlet gas qm varies from 1.9e-5 kg/s to 7.63e-5 kg/s. The transient gas temperature data T1 is fitted into a secondary order polynomial using least square, which we use as boundary condition. Fig. 9 shows the influence of 3 variables (h v , c5, c6) on the error S. The gas flow direction is TT direction in this case. The values of c3 and c4 are set to 0.053 and 0.099. The corresponding values of k s,ef f in the flow direction and perpendicular to the flow direction are 0.212 and 0.399 W/(m 3 • K). The colour bar indicates the value of S. The genetic algorithm was used in this optimization process. The bounding value of c5, c6 and h v are set between 0 to 0.5, 0 to 0.3 and 10 6 to 10 11 . When the value of S is below The comparison between the predicted and measured outlet gas temperatures T4, solid temperature T5 and T2 in IP and TT directions at different velocity are shown in Fig. 11 and Fig. 12. In the LTE model, the gas and solid temperature inside the sample are the same. In these simulations, the Fig. 10. The influence of h v on the error S value of k eff are derived from the optimal solution. These values were summarized in Table .2. As the table shows, the inverse analysis method has a higher sensitivity in the component perpendicular to the flow direction. The components in the flow direction have low sensitivity and only a range of data is obtained.

Comparisons with LTNE model

For the case of gas flowing through the carbon fibre felts, the fluid and solid phases are at same temperature due to a sufficiently large h v . In this part, simulations are conducted using the LTNE model for comparsion with the LTE model. The value of k g,eff in radial direction is calculated according to Eq.8.

k g,eff = k eff -k s,eff (8)
The value of k g,eff and square root of the relative error (S2) between LTNE and LTE model were summarized in Table .3. The definition of S2 is similar to that of S in Eq.7. In all simulations, the value of S2 is less than 0.05%. So the LTE model can be used to solve the heat transfer problems inside the Carbon fiber felts.

CONCLUSIONS

We have developed an original experimental set-up to characterize heat and mass transfer in carbon fiber materials. It The results on Calcarb show that the value of hv in the LTNE model is around 10 9 W/(m 3 • K). Gas and solid temperatures are in LTE within Calcarb in the conditions of the experiment and in this case the use of a LTNE model is not necessary.

The effective thermal conductivity k eff in the LTE model is altered by the gas flow rate due to the dispersion mechanism. In the LTNE model, k eff is split into two terms, k g,eff and k s,eff . The error S2 difference obtained by solving the LTE model and the LTNE model is less than 0.05 %. Preliminary investigation showed that k g,eff in the LTNE model increases exponentially with increasing Péclet number, that is, increasing flow velocity. The dispersion mechanism in Calcarb should be further investigated.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Scanning electron micrography (SEM) images of FiberForm and Calcarb

Fig. 3 .

 3 Fig. 3. Schematic diagram of the experimental system and an enlarged view of the test section with thermocouples

Table 1 .

 1 The uncertainty analysis δ in the experimental Range Uncertainty Temperature sensor -270 • C to +1820 • C ± 0.2% (Pico Technology) K-type thermocouple -100 • C to + 800 • C ± 0.5 % Flow controller 0 to 4 L/min ± 0.5 % The placement error z=± 2mm 0.1% (flow1, δ 1 ) The placement error z=± 2mm 0.2% (flow2, δ 2 ) The placement error z=± 0.5mm 0.5% (sample, δ 3 )

Fig. 4 .

 4 Fig. 4. The geometric model of tube

Fig. 5 .Fig. 6 .

 56 Fig. 5. Pressure p distributions obtained by LTNE models (Pa)

Fig. 7 .Fig. 8 .

 78 Fig. 7. Gas temperature T distributions in the flow region obtained by CHT foam solver (K)

Fig. 9 .

 9 Fig. 9. The influence of 3 variables on the error S, qm=7.63e-5 kg/s

Fig. 11 .Fig. 12 .

 1112 Fig. 11. The comparison of predicated and measured temperature in IP direction.

Table 2 .

 2 The value of effective conductivity k eff and relative error S used in LTE model in different mass flow rates

Table 3 .

 3 The value of k g,eff and relative error S1 between LTE and LTNE models in different mass flow rates was used to measure the properties of Calcarb. Inverse analysis was performed by coupling PATO and Dakota. It proved to be a robust method to determine h v , k g,eff , k s,eff for the LTNE model or k eff for the LTE model for complex porous media when correlations are not available.

	Flow	qm	c3	c4	c6	k s,eff	k g,eff	S2
	direction (kg/s)				(radial) (radial)
	IP	1.9e-5	0.099 0.053 0.079 0.212	0.107	1.57e-4
		3.82e-5 0.099 0.053 0.097 0.212	0.127	3.21e-4
		5.73e-5 0.099 0.053 0.157 0.212	0.184	3.91e-4
		7.63e-5 0.099 0.053 0.183 0.212	0.21	4.04e-4
	TT	1.9e-5	0.053 0.099 0.021 0.399	0.047	1.21e-4
		3.82e-5 0.053 0.099 0.028 0.399	0.055	3.14e-4
		5.73e-5 0.053 0.099 0.061 0.399	0.088	2.78e-4
		7.63e-5 0.053 0.099 0.117 0.399	0.144	3.73e-4

2nd International Conference on Flight Vehicles, Aerothermodynamics and Re-entry Missions & Engineering (FAR) 19 -23 June 2022. Heilbronn, Germany

2nd International Conference on Flight Vehicles, Aerothermodynamics and Re-entry Missions & Engineering (FAR)