Lucas Terrei
email: lucas.terrei@univ-lorraine.fr

Guillaume Gerandi

Hassan Flity

Virginie Tihay-Felicelli

Zoubir Acem

Gilles Parent

Paul-Antoine Santoni

Experimental and numerical multi-scale study of spruce wood degradation under inert atmosphere

Keywords: Wood degradation, TGA, Cone calorimeter, GPYRO

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Wood is a complex material studied in fire safety sciences with various topics such as building fires. The modeling of building fires requires more and more detailed physical models for which the knowledge of the thermal degradation of combustible materials is a key point for a correct modeling. This topic is mainly studied at matter scale using thermogravimetric experiments (TGA) with low heating rates generally lower than 60 °C.min -1 [START_REF] Leroy | Thermal degradation of fuels: DSC and TGA studies[END_REF][START_REF] Branca | A unified mechanism of the combustion reactions of lignocellulosic fuels[END_REF]. However, these values are not representative of those encountered in building fires, which are generally at least five times higher. A multiscale approach is therefore necessary to validate the model at a larger scale where the heating conditions are closer to those involved in building fires. This essential step can correspond to experiments performed under a cone calorimeter [START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF] or under a Single Burning Item [START_REF] Gossiaux | Small-scale single burning item test for the study of the fire behavior of building materials[END_REF].

In the frame of building fires, wood materials are thermally thick and often used as beam, cladding, facade or landscaping. The study of wood degradation is performed with a lot of experimental setups (TGA, Cone calorimeter, Single Burning Item) and modelled with numerous software (CFD Codes, simplified pyrolysis models) [START_REF] Babrauskas | Development of the cone calorimeter-a bench-scale heat release rate apparatus based on oxygen consumption[END_REF][START_REF] Lautenberger | Generalized pyrolysis model for combustible solids[END_REF][START_REF] Bryden | Modeling thermally thick pyrolysis of wood[END_REF][START_REF] Mierlo | The Single Burning Item (SBI) Test Method -a Decade of Development and Plans for the Near Future[END_REF].

Multiscale studies are also used to predict the thermal degradation of building materials across scales [START_REF] Vermesi | Pyrolysis and spontaneous ignition of wood under transient irradiation: experiments and a-priori predictions[END_REF][START_REF] Fateh | Multi-scale modeling of the thermal decomposition of fire retardant plywood[END_REF] since heating rates involved in building fires are much higher at actual scale. The experiments performed at a material scale often use a cone calorimeter device. Many authors use it on the basis of the standardized configuration (ISO 5660) in horizontal orientation [START_REF] Vermesi | Pyrolysis and spontaneous ignition of wood under transient irradiation: experiments and a-priori predictions[END_REF][START_REF] Delichatsios | Piloted ignition times, critical heat fluxes and mass loss rates at reduced oxygen atmospheres[END_REF][START_REF] Gerandi | Multi-scale simulations of the thermal degradation of thin wooden plates: evaluation of two kinetic mechanisms[END_REF] to assess several aspects of the fire behavior like ignition time, critical heat flux for ignition, mass loss rate or in-depth temperature. However, this configuration is not always representative of the real scale where many building materials are oriented vertically and are submitted to a horizontal incident heat flux. For that reason, some authors studied the fire behavior of wood with the cone calorimeter in vertical configuration [START_REF] Kashiwagi | Effects of external radiant flux and ambient oxygen concentration on nonflaming gasification rates and evolved products of white pine[END_REF][START_REF] Terrei | Experimental tools applied to ignition study of spruce wood under cone calorimeter[END_REF].

These studies revealed important differences in the ignition time and critical heat flux, for which the values are considerably higher with the cone calorimeter in vertical configuration [START_REF] Tsai | Orientation effect on cone calorimeter test results to assess fire hazard of materials[END_REF][START_REF] Linteris | Modeling solid sample buming[END_REF] than in horizontal one. Some studies [START_REF] Tsai | Orientation effect on cone calorimeter test results to assess fire hazard of materials[END_REF][START_REF] Bartlett | A review of factors affecting the burning behaviour of wood for application to tall timber construction[END_REF] have attributed these difference to changes in the flow of pyrolysis gases.

The pyrolysis of wood is a complex phenomenon which is modeled in the literature by different mechanisms. Some models consider parallel steps where the degradation of the three main constituents of wood (hemicellulose, cellulose and lignin) is studied separately leading to char [START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF]. Some models consider also the activation of each constituent before leading to tar and char. This leads to models involving nine steps [START_REF] Vermesi | Pyrolysis and spontaneous ignition of wood under transient irradiation: experiments and a-priori predictions[END_REF][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]. This kind of model was used by Richter el al.

and Vermesi et al. to model efficiently the degradation of different woods. Successive reaction models are also available and can take into account the interaction between each constituent inside the wood by the apparition of intermediate condensed phases [START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF]. Among them, parallel and lumped models provide efficient results whatever the kind of wood studied [START_REF] Fateh | Kinetic and mechanism of the thermal degradation of a plywood by using thermogravimetry and Fourier-transformed infrared spectroscopy analysis in nitrogen and air atmosphere[END_REF][START_REF] Branca | Global interinsic kinetics of wood oxidation[END_REF][START_REF] Branca | Critical evaluation of global mechanisms of wood devolatilization[END_REF]. Recently, a comparison between these models showed a better performance for the lumped mechanism with successive reactions to study the wood degradation [START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF]. Moreover, these three steps can be gathered in only one to simplify the modeling. This method provides efficient results although less performing than the previous mechanisms with three steps [START_REF] Gerandi | Multi-scale simulations of the thermal degradation of thin wooden plates: evaluation of two kinetic mechanisms[END_REF][START_REF] Cancellieri | WinGPYRO : a software platform for kinetic study of forest fuels[END_REF][START_REF] Benkorichi | Investigation of thermal degradation of pine needles using multi-step reaction mechanisms[END_REF]. Finally, the choice of the number of steps of the pyrolysis process often depends on the aim of the study, as for instance knowing the impact of the chemical effects of the constituents or for being further implemented in a CFD code for fire engineering application. In both cases, the different models always provide satisfying results at matter scale using TGA experiments. At material scale, many authors have studied the thermal degradation of wood under a cone calorimeter in oxidative atmosphere [START_REF] Fateh | Multi-scale modeling of the thermal decomposition of fire retardant plywood[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]. This study was less often performed under inert atmosphere [START_REF] Kashiwagi | Effects of external radiant flux and ambient oxygen concentration on nonflaming gasification rates and evolved products of white pine[END_REF]. The study of Kashiwagi et al. provides experimental data on mass loss rates and in-depth temperatures for two heat fluxes: 25 and 40 kW .m -2

. The temperature measurements include only three thicknesses: 5, 10 and 20 mm, which provides few information for the temperature profile evolution and the virgin/char progression rate. This study was the basis of many numerical investigations to establish a pyrolysis model. Jia et al. [START_REF] Jia | Numerical simulation of the mass loss process in pyrolizing char materials[END_REF] established a model coupling the progression of the char/virgin front in the thermal pyrolysis model in one and threedimensions. Lautenberger and Fernandez-Pello used it to validate their Generalized Pyrolyis Model For Combustible Solids [START_REF] Lautenberger | A model for the oxidative pyrolysis of wood[END_REF]. This one provides efficient mass loss rate and temperature predictions for the various heat fluxes investigated despite some differences. This model was recently used to investigate the thermal degradation of different building materials such as plywood or wood cladding [START_REF] Fateh | Multi-scale modeling of the thermal decomposition of fire retardant plywood[END_REF][START_REF] Lautenberger | A model for the oxidative pyrolysis of wood[END_REF] under different atmospheres (air and nitrogen) and according to various conditions: sample thickness, heat flux, wood species, reactional mechanisms [START_REF] Lautenberger | Generalized pyrolysis model for combustible solids[END_REF][START_REF] Fateh | Multi-scale modeling of the thermal decomposition of fire retardant plywood[END_REF][START_REF] Richter | The effect of chemical composition on the charring of wood across scales[END_REF]. Other models also exist in literature as Thermakin to study the thermal decomposition of wood at the material level scale [START_REF] Stoliarov | Two-dimensional model of burning for pyrolyzable solids[END_REF].

Recently, a study performed by Gong et al. under inert atmosphere showed a good scalability of a kinetic model developed at matter scale with TGA experiments to represent the thermal degradation of OSB for Microscale Combustion Calorimetry Tests [START_REF] Gong | Development of a pyrolysis model for oriented strand board. Part I: kinetics and thermodynamics of the thermal decomposition[END_REF].

In summary, there are many studies that aim to predict the wood degradation exposed to a constant heat flux. The models deal with more or less complexities and are in quite good agreement with the experimental data. Char oxidation is a complex phenomenon which is induced by the presence of oxygen and may lead to discrepancies between experiments and prediction. It is therefore interesting to reduce the complexity of the phenomena involved in order to correctly predict the degradation of wood without the presence of oxygen. The devolatilization steps correspond to the highest mass loss of wood. Therefore, this simplification allows to focus on the pyrolysis reactions and to ensure a better prediction of this phenomenon. On the other hand, experiments at the cone calorimeter scale in inert atmosphere are still limited, as well as the associated measurements, mass loss rate and few in-depth temperatures. Moreover, it was shown recently that in depth temperatures usually made by implementing mm-size diameter sheathed thermocouples from the rear side of the sample can be strongly underestimated [START_REF] Terrei | In-depth wood temperature measurement using embedded thin wire thermocouples in cone calorimeter tests[END_REF]. In addition, due to the mass and stiffness of such thermocouples, measuring at the same time the mass loss than temperatures is not possible.

The aim of this study is to improve the predictions of the mass loss rate and the temperatures inside the wood when submitted to different heat fluxes under inert atmosphere. For that purpose, a multiscale approach was adopted to assess the thermal degradation of spruce wood. For each scale (matter and material), a set of experiments and simulations was performed. At matter scale, thermogravimetric experiments were conducted to determine the best set of kinetic parameters for a three successive step kinetic model. At material scale, a dedicated experimental device allowing to expose a sample to a constant heat flux under an inert atmosphere (to avoid oxidation reactions) was set up. This one includes a cone calorimeter in vertical configuration and a chamber where argon was injected. Spruce wood samples were exposed to three constant heat fluxes. The in-depth temperatures were simultaneously with the mass loss thanks to twelve thin wire thermocouples embedded in the samples [START_REF] Terrei | In-depth wood temperature measurement using embedded thin wire thermocouples in cone calorimeter tests[END_REF]. This new technique had shown a more reliable temperature measurement. Experimental results were compared to a predicting model of wood degradation using GPYRO. The reactional mechanism was set up in GPYRO with the kinetic parameters previously determined thanks to TGA experiments. Finally, numerical mass loss and temperatures were compared to the experimental data.

Materials and methods

This part presents the experimental setups used to study spruce wood degradation under inert atmosphere at matter and material scales.

Wood sample

The study was carried out using square shaped spruce (Picea Abies) wood plates of 100 × 100 × 50 mm 3

. This kind of wood is typical for wooden construction. As the thermo-physical properties of wood vary with moisture content, dry samples were used. The wooden plates were therefore oven dried at 105 °C during 24 h to remove the moisture according to the European Norm NF EN ISO 18134. This method of drying does not impact the pyrolysis of wood.

The average density was measured at 360 kg .m -3 . As some studies showed that the surface emissivity of wood varied when the sample is exposed to heat [START_REF] Försth | Absorptivity and its dependence on heat source temperature and degree of thermal breakdown[END_REF][START_REF] Boulet | On the influence of the sample absorptivity when studying the thermal degradation of materials[END_REF][START_REF] Chaos | Spectral aspects of bench-scale flammability testing: application to hardwood pyrolysis[END_REF], a specific work was conducted to characterize it. Consequently, the experiments were performed in order to assess the emissivity evolution of spruce wood surface with time and heat flux.

The samples were exposed to two heat fluxes (38 and 59 kW .m -2 ) with a cone calorimeter oriented vertically for different prescribed times. After these times, the samples were promptly cooled down with liquid nitrogen to stop the degradation process. The sample emissivity was then measured using an integrating sphere coupled to a Fourier Transform Infrared spectrometer. Fig. 1 presents the variation of the emissivity of wood sample surface according to the exposure time to the cone heater.

alt-text: Fig. 1 Fig. 1 i Images are optimised for fast web viewing. Click on the image to view the original version.

The emissivity of the virgin wood surface was measured at 0.83 (blue zone). After being exposed to the heat flux, the emissivity of the sample firstly decreases until reaching a minimum of 0.67 which is illustrated by the red zone. In this zone, some pyrolysis gases are released and are associated to a transformation of the virgin wood at the surface. After, the emissivity increases up to a value close to 0.95 corresponding to the transition from wood to char. These experiments showed that the wood emissivity could vary notably at the beginning of the exposure to the heat flux. The decrease of emissivity can be explained by both the complex internal multiple reflections causing by the formation of the char [START_REF] Boulet | On the influence of the sample absorptivity when studying the thermal degradation of materials[END_REF] and the formation of viscous liquid tars on the wood surface which increase the reflectivity at the beginning of the degradation process. This large change should be considered in order to model accurately the wood degradation since this parameter intervenes in the resolution of the heat transfer in the material.

TGA and cone calorimeter experiments

The experiments at matter scale were performed using a thermogravimetric analyzer (TGA) TGA/DSC 3+ from Mettler Toledo under an inert atmosphere of argon with a flow rate of 30 mL .min -1 . The samples were ground into powder and were oven-dried at 105 °C for 24 h before each experiment in order to remove the moisture. They were then heated from the ambient temperature to 900 °C with three heating rates: 10, 20 and 30 °C.min -1

. For each heating rate, two tests were carried out to ensure a good repeatability of the experiments. The initial dry mass of spruce wood was 10.1 ± 0.2 mg.

At material scale, the degradation of the wood sample was studied with a cone calorimeter oriented vertically and put on a sliding table. This latter simplified the setup of the sample before the exposition to the heat flux. More details about the modified cone calorimeter device are given in the work of Terrei et al. [START_REF] Terrei | Study of wood self-extinguishment with a double sliding cone calorimeter[END_REF]. For the purpose of studying the wood degradation without oxygen in the atmosphere, a stainless-steel chamber was built in order to place a sample inside, in vertical orientation. This chamber was placed in front of the cone heater to expose wood samples to prescribed heat fluxes. Fig. 2 presents the experimental setup. The wood samples were exposed to three heat fluxes (38, 49 and 59 kW .m -2 ) during 600 s.

Emissivity evolution of spruce wood samples in function of the exposure time for two heat fluxes.

alt-text: Fig. 2 Fig. 2 i Images are optimised for fast web viewing. Click on the image to view the original version.

The chamber had a rectangular parallelepiped shape (dimensions: 240 mm width, 220 mm height and 150 mm depth) with 1.5 mm stainless-steel thickness. A circular opening (80 mm diameter) at the top of the chamber allowed to drain off the pyrolysis gases. The oxygen in the chamber was removed by a constant injection of argon at a flow rate of 10 L .min -1 -1 . The injection was performed at the bottom of the chamber. The argon is a heavy gas, so it was trapped in the chamber and was little entrained by the pyrolysis gases coming from the sample. The injection was done through the rear side of the chamber using a flexible tube (blue pipe in Fig. 2). The gas contained in the enclosure was sampled from the upper part of the rear face of the chamber so as to control the oxygen content in the enclosure. A Testo 350

Gas analyzer was used for this purpose. A test started when the percentage of oxygen in the chamber was stabilized around 3%. During all the tests, the oxygen concentration never exceeded 6% and was lower than this value most of the time. The stainless-steel enclosure was designed to only place the sample inside the chamber through the rear face consisting of a sliding door. As shown in Fig. 2, the chamber was raised so as to place the load cell below the lower part. The scale was therefore not constrained (thermally or mechanically) contrary to conventional controlled atmosphere chamber like the one provided by the supplier of the cone calorimeter. The load cell sampling rate was 1 Hz and its precision was 0.1 g. Concerning the exposed side, a sapphire window, with 20 cm diameter and 5 mm thickness, was fixed in order to free pass the heat flux of the cone while preventing the injected argon from escaping.

This system made it possible to maintain a controlled atmosphere during the tests without disturbing the continuous measurement of temperatures (sample surface and in-depth) and of the mass loss. The presence of the sapphire window and the chamber constrained to adjust the distance between the sample and the cone heater at 45 mm. As the total transmission of the window is around 60%, the cone temperature had to be increased to obtain the required heat flux.

The radiative heat flux coming from the cone was controlled before each test with a Schmidt-Boelter fluxmeter (Medtherm). Given the distance between the sample and the heating cone, the flux received by the wood sample is not uniform. A Monte Carlo Simulation has been carried out to evaluate the heat flux non-uniformity. The results show that heat flux heterogeneity is ±10% on 95% of the sample surface and ±5% on 60% of the sample surface. The lateral and back sides of the steel chamber were cooled down by a water circuit fixed on each side. For each test, the wood sample was placed in a specific sample holder machined with low density calcium silicate plates (silcal 1100, 20 mm thickness, 0.1 W .m -1 K -1 thermal conductivity) to limit the thermal interactions with the borders. This sample holder insulated the lateral sides of the sample while the back face was left in the ambient atmosphere.

The in-depth temperatures were measured with twelve thin (0.1 mm diameter) wire thermocouples embedded in the spruce wood samples. The implantation consists in cutting the wood panel perpendicularly to the exposed surface and 0.2 mm depth and width square-grooves were machined at the desired location. Then, wire K-type thermocouples composed of alumel and chromel cables were welded by autogenous electric welding and put in the square-grooves.

Ten thermocouples were placed every 2 mm from 2 to 20 mm depth from the exposed surface, one at a 30 mm depth and one at a 40 mm depth. Finally, wood panels were glued with Melamine-urea-formaldehyde (MUF). This method was developed according to a specific machining detailed in Ref. [START_REF] Terrei | In-depth wood temperature measurement using embedded thin wire thermocouples in cone calorimeter tests[END_REF]. Since the thermocouple wires used are very supple, it made it possible to simultaneously measure the temperature and the mass loss during the tests as demonstrated in Ref. [START_REF] Terrei | In-depth wood temperature measurement using embedded thin wire thermocouples in cone calorimeter tests[END_REF].

3 Results and discussion This section presents the experimental and numerical results obtained at matter scale. Fig. 3 shows the experimental and numerical mass and conversion rate evolutions with temperature for the three heating rates tested: 10, 20 and 30 °C.min -1

. The experimental data correspond to the average of two tests for which the relative difference was less than 1%. Table 1 presents the kinetic parameters obtained by optimization with the Levenberg-Marquardt algorithm as well as the mean performance parameter.

alt-text: Fig. (e, f).

i Images are optimised for fast web viewing. Click on the image to view the original version.

alt-text: i The table layout displayed in this section is not how it will appear in the final version. The representation below is solely purposed for providing corrections to the table. To preview the actual presentation of the table, please view the Proof.

The experimental thermal degradation of spruce wood follows the same trend whatever the heating rate and is described below:

At the end of the thermal degradation, it remained 21% of the initial mass in average, calculated with the three heating rates. This value, corresponding to the char yield, was in the range of those currently observed in the literature [START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF][START_REF] Fateh | Kinetic and mechanism of the thermal degradation of a plywood by using thermogravimetry and Fourier-transformed infrared spectroscopy analysis in nitrogen and air atmosphere[END_REF][START_REF] Yang | Characteristics of hemicellulose, cellulose and lignin pyrolysis[END_REF][START_REF] Wang | Thermal decomposition behavior and kinetics for pyrolysis and catalytic pyrolysis of Douglas fir[END_REF].

The comparison between the experimental data and the numerical predictions showed a good agreement for the three heating rates tested. Indeed, we obtained a value of the Fit parameter of 1.53, 1.16 and 1.29% for 10, 20 and 30 °C.min -1 respectively, leading to a mean Fit of 1.33%. Over the temperature range studied, the mean relative error was lower than 3.5%. The main difference occurred around 350 °C and 600 °C where the most significant peak of conversion rate took place followed by a plateau. Whatever the heating rate, the maximal difference between the experimental and predicted conversion rate was around 25%. This underestimation was due to the kinetic mechanism considering successive steps as well as the objective function chosen for the optimization of the kinetic parameters that only assessed the error on the degree of conversion leading to slight discrepancies on DTG curves. This optimization was performed with the three heating rates simultaneously. For this reason, we observed higher discrepancies on the MLR. These differences are also reported in the literature when using optimization algorithm method [START_REF] Gerandi | Multi-scale simulations of the thermal degradation of thin wooden plates: evaluation of two kinetic mechanisms[END_REF][START_REF] Fateh | Kinetic and mechanism of the thermal degradation of a plywood by using thermogravimetry and Fourier-transformed infrared spectroscopy analysis in nitrogen and air atmosphere[END_REF][START_REF] Tihay | Influence of oxygen concentration on the kinetics of cellulose wadding degradation[END_REF].

In order to show the different steps of the thermal degradation, Fig. 3 presents the mass of each species and the gases conversion rate predicted by the kinetic mechanism for all heating rates. The first reaction began at 250 °C and is completed at 400 °C. During this reaction, the mass of DW was completely converted after 400 °C. The decomposition of DW started at 250 °C leading to the beginning of the formation of DW

1 . The maximum conversion rate took place between 314 °C and 337 °C according to the heating rate. This maximum meant that the intermediary species DW 1 was completely formed. The second reaction occurred between 300 °C and 425 °C with a maximum between 356 °C and 381 °C. As for the first reaction, the higher the heating rate the greater the temperature at which the maximum conversion rate occurred. This tendency was in accordance with the literature [START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF][START_REF] Fateh | Kinetic and mechanism of the thermal degradation of a plywood by using thermogravimetry and Fourier-transformed infrared spectroscopy analysis in nitrogen and air atmosphere[END_REF][START_REF] Branca | Critical evaluation of global mechanisms of wood devolatilization[END_REF]. While this reaction took place, DW 1 began to decrease until 400 °C which led to the formation of the second intermediary species DW 2 . Finally, the third reaction took place between 350 °C and 600 °C. The maximum peak occurred in a range between 378 °C and 405 °C. This reaction led to the formation of char while DW 2 is consumed. The char was completely formed at 600 °C.

Concerning the kinetic parameters obtained by the optimization, the activation energy of the first reaction was equal to 127.6 kJ .mol -1 and the pre-exponential factor was equal to 21.8 s -1 . This couple of values was lower than those obtained by Wu and Dollimore [START_REF] Wu | Kinetic studies of thermal degradation of natural cellulosic materials[END_REF] for the same wood species with an activation energy of 163.7 kJ .mol -1 and Kinetic parameters of spruce wood.

Reaction (-)

(s -1 At 250 °C, the decomposition began; • Between 250 °C and 300 °C, a first peak of conversion rate occurred. This one mainly corresponded to the degradation of hemicellulose [START_REF] Yang | Characteristics of hemicellulose, cellulose and lignin pyrolysis[END_REF];

) (kJ.mol -1 ) (-) (%) 1
• Around 400 °C, a second reaction took place leading to the most significant peak of conversion rate.

The mass loss was around 70% after this step. This stage mainly corresponded to the degradation of cellulose [START_REF] Yang | Characteristics of hemicellulose, cellulose and lignin pyrolysis[END_REF];

• Beyond 400 °C, the conversion rate decreased drastically and was then followed by a weak plateau until the end of the degradation. This step mainly coincided with the degradation of lignin [START_REF] Yang | Characteristics of hemicellulose, cellulose and lignin pyrolysis[END_REF]. The position of each peak was in agreement with the experiments performed by Wang et al. [START_REF] Wang | Thermal decomposition behavior and kinetics for pyrolysis and catalytic pyrolysis of Douglas fir[END_REF] for the thermal degradation of Douglas Fir under nitrogen atmosphere.

• a pre-exponential factor equal to 24.4 s -1

. It was also higher than the value obtained by Branca and Di Blasi [START_REF] Branca | Global interinsic kinetics of wood oxidation[END_REF] for the thermal degradation of Douglas Fir under air atmosphere with a value of 106 kJ .mol -1 and a pre-exponential factor of 17.9 s -1 . These differences may be attributed to the atmosphere of degradation where reactions under inert atmospheres generally take place at slightly higher temperatures [START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF][START_REF] Fateh | Kinetic and mechanism of the thermal degradation of a plywood by using thermogravimetry and Fourier-transformed infrared spectroscopy analysis in nitrogen and air atmosphere[END_REF]. However, the stoichiometric coefficient was in accordance with that of Branca and Di Blasi [START_REF] Branca | Global interinsic kinetics of wood oxidation[END_REF]. For the two other reactions, the activation energy varied between 139.2 and 152.1 kJ .mol -1 and the pre-exponential factors between 19.2 and 21.3 s -1 . These couple of values were lower than those obtained for oak and eucalyptus [START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF]. These differences can be explained by the change of the degradation atmosphere as mentioned above, but also by the change of the type of wood. Spruce is indeed a softwood while oak and eucalyptus are hardwoods. This leads to differences in the composition of the wood (percentage of hemicellulose, cellulose and lignin) and therefore to the thermal degradation.

The reaction orders vary from 0.90 to 3.00. For the first reaction, the value of 1.48 obtained for spruce wood is close to those obtained for plywood, oak, eucalyptus, Douglas Fir or Beech [START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF][START_REF] Fateh | Kinetic and mechanism of the thermal degradation of a plywood by using thermogravimetry and Fourier-transformed infrared spectroscopy analysis in nitrogen and air atmosphere[END_REF][START_REF] Branca | Global interinsic kinetics of wood oxidation[END_REF]. The reaction order of the second reaction is also consistent with the literature values that currently varied from 0.90 to 2.70. For the third reaction, a value of 3.00 was found by optimization. This value is consistent with the literature although slightly higher [START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF][START_REF] Fateh | Kinetic and mechanism of the thermal degradation of a plywood by using thermogravimetry and Fourier-transformed infrared spectroscopy analysis in nitrogen and air atmosphere[END_REF] (between 2.38 and 2.70). This high value allowed a good modeling of the plateau observed in the conversion rate of Fig. 3 from 400 °C.

Finally, the product of the stoichiometric coefficients gives a value of 21%, corresponding to the char yield numerically predicted. This value is equal to the experimental one as mentioned above.

Cone calorimeter results

Thermophysical properties

Thermophysical properties of each condensed phase, i.e. thermal conductivity, coefficient, heat capacity, coefficient, and emissivity, are presented in Table 2. The thermal conductivity of dry wood at 27 °C was measured with a hot wire method [START_REF] Assael | Historical evolution of the transient hot-wire technique[END_REF] and was find to be equal to 0.108 W .m -1 K -1 . For the dry wood, the coefficient was set to 0.60, which is in accordance with the result for Pine wood [START_REF] Lautenberger | A model for the oxidative pyrolysis of wood[END_REF]. For the heat capacity, the initial value was is set to 1500 J .kg -1 K -1 . This parameter also varied with temperature with a coefficient equal to 1. According to the spectrometry measurement, the emissivity of the dry wood is equal to 0.83.

For the condensed phases DW 1 and DW 2 , we assumed that each thermo-physical property excepted emissivity was equal to the value of dry wood and char, respectively. Each one followed the same variation with temperature, as it was already performed by Gerandi et al. [START_REF] Gerandi | Multi-scale simulations of the thermal degradation of thin wooden plates: evaluation of two kinetic mechanisms[END_REF]. For the emissivity, the spectrometry measurements were used to define the input values for DW 1 and DW 2 . We considered that, following the results obtained in TGA (see Fig. 3), the condensed phase DW 1 and DW 2 appeared around 200 °C and 350 °C, respectively. These temperatures corresponded to an exposure time between 50 and 100 s on Fig. 1, which induced an emissivity value of 0.7 for DW 1 and DW 2 .

alt-text: Table 2 Table 2 Thermo-physical properties of each condensed phase.

Condensed phase/Thermo-physical property

(kg.m -3 For the thermal properties of char, literature values were used. We fixed a value of heat capacity equal to 1100 J .kg -1 K -1 and a value of 0.08 W .m -1 K -1 for the thermal conductivity [START_REF] Bryden | Modeling thermally thick pyrolysis of wood[END_REF][START_REF] Richter | The effect of chemical composition on the charring of wood across scales[END_REF]. We considered also that only the thermal conductivity of char varied with temperature with a coefficient equal to 1.4 [START_REF] Fateh | Multi-scale modeling of the thermal decomposition of fire retardant plywood[END_REF]. Accordingly with Fig. 1 , the emissivity of char was set to 0.95, which complies with the value used in Ref. [START_REF] Gupta | Specific heat and thermal conductivity of softwood bark and softwood char particles[END_REF]. The density of each intermediary species was calculated according to equation 8b equation 8b. It led to a final value equal to 75 kg .m -3 for the char. This value is in the same magnitude than those previously obtained for oak (136 kg .m -3

) (W.m -1 .K -1 ) (-) (J.kg -1 .K -1 ) (-) (
) and timber (111 kg .m -3

) degradations [START_REF] Gerandi | Multi-scale simulations of the thermal degradation of thin wooden plates: evaluation of two kinetic mechanisms[END_REF][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].

Experimental and numerical temperature results

This section shows experimental and predicted results concerning the spruce wood degradation at the material-level scale. Fig. 4 presents the in-depth temperatures measured with thin wire thermocouples embedded in the wood samples for tests at 38 and 59 kW .m -2 . For each heat flux, three tests were performed and errors bars correspond to the standard deviation. The thermocouple implantation was based on a specific method detailed in Ref. [START_REF] Terrei | In-depth wood temperature measurement using embedded thin wire thermocouples in cone calorimeter tests[END_REF] which ensured an accurate temperature measurement. Indeed, repeatability experiments carried out with this technique presented in

Ref. [START_REF] Terrei | In-depth wood temperature measurement using embedded thin wire thermocouples in cone calorimeter tests[END_REF] showed that the average scattering between 0 and 600 s was around 20 °C at 2 mm depth. This error decreased towards 5 °C at 8 mm depth. The in-depth temperature evolutions were similar to those observed in Terrei et al. [START_REF] Terrei | In-depth wood temperature measurement using embedded thin wire thermocouples in cone calorimeter tests[END_REF] with a fast increase of the temperature for thermocouples close to the exposed surface (from 2 mm to 6 mm depth). Indeed, after 300 s, the temperature reached 500 °C and 620 °C at 2 mm depth for tests at 38 and 59 kW .m -2 respectively. The mean standard deviation was generally less than 25 °C which showed a good repeatability of the experiments. The temperatures reached inside the wood sample were different according to the thickness. For both heat fluxes, the temperature for depths greater than 16 mm was less than 200 °C. According to the experiments in TGA, the degradation of spruce began from 200 °C. Therefore, the wood remained virgin for thicknesses greater than 16 mm leading to weak mass losses. Indeed, only 32% of the slab had heated during the exposure to the three heat fluxes.

From these experimental results, the heating rates were calculated at different thicknesses across the experiment. For

38 kW .m -2
, the heating rates reached values higher than 50 °C.min -1 up to 6 mm depth during a time lower than 200 s. Then, the heating rates were in the range of those studied in TGA experiments. For 59 kW .m -2 , the heating rates inside the wood are higher than 50 °C.min -1 during 300 s above 10 mm. From this thickness, the heating rate remained constant to this value up to the end of the experiments. These values meant that for this heat flux, the heating is higher than in TGA along the whole experiment. This result could explain the less efficiency of the model to represent the MLRPUA for this heat flux (Fig. 4f). The kinetic parameters were determined in TGA with experiments between 10 and 30 °C.min -1 , that corresponded to lower values than those obtained by exposing a spruce slab to a heat flux of 59 kW .m -2 .

alt-text: Fig. 4 Fig. 4 i Images are optimised for fast web viewing. Click on the image to view the original version. , below 500 °C, the GPYRO predictions tended to slightly overestimate the experimental measurements whatever the depth. As for 38 kW .m -2 , the temperature predictions were better in depth than on surface. From 8 mm, the difference between the experimental and the predicted temperature did not exceed 60 °C. At 4 mm, GPYRO underestimated the temperature of about 100 °C at the end of the experiments for both heat fluxes. The discrepancies observed should come from two reasons. The discrepancies observed could come from three reasons. Firstly, the evolution with the temperature of the thermal properties of wood may be overestimated by the model especially for the thermal conductivity and the heat capacity.

Especially, in GPYRO, the temperature is extracted from the sensible enthalpy given by the following equation [START_REF] Lautenberger | Generalized pyrolysis model for combustible solids[END_REF]:

where is the sensible enthalpy, the heat capacity of each condensed phase and the temperature datum used to establish an absolute value of enthalpy (taken as 200 K). In this equation, if the heat capacity is underestimated, the predictions on the temperature should be affected and the in-depth temperatures should be inaccurate. In addition, for high heat fluxes, the progression of the char layer is very fast leading to a thicker insulation layer. The temperatures distant from the surface are consequently overestimated. Finally, the differences can also be due to the cracks in the char layer (Fig. 6). These cracks enhance the radiative heating inside the sample leading to a larger increase of the temperature. This observation could explain the gap of 100 °C observed at the end of the experiments for a thickness of 4 mm. (

) 1
i Images are optimised for fast web viewing. Click on the image to view the original version.

In order to visualize the evolution of the thermal degradation inside the wood sample, the temperature profiles are plotted in Fig. 7 Snapshot of a spruce wood sample after an experiment under the cone calorimeter.

Images are optimised for fast web viewing. Click on the image to view the original version. . For tests at 59 kW .m -2 , this difference was also visible at 480 s. At 240 s, the prediction was only 40 °C lower than the experiments. These discrepancies were also reported in the literature for the temperature close to the exposed surface of the sample whatever the atmosphere used to study the thermal degradation [START_REF] Richter | The effect of chemical composition on the charring of wood across scales[END_REF][START_REF] Weng | Predicting the pyrolysis of wood considering char oxidation under different ambient oxygen concentrations[END_REF]. Under 6 mm depth, the predicted temperature was very close to the measured one with a maximum difference of about 18 °C. These predictions were better than those reported in the literature for an equivalent heat flux [START_REF] Lautenberger | A model for the oxidative pyrolysis of wood[END_REF][START_REF] Richter | The effect of chemical composition on the charring of wood across scales[END_REF][START_REF] Weng | Predicting the pyrolysis of wood considering char oxidation under different ambient oxygen concentrations[END_REF]. For tests at 59 kW .m -2 , temperature profiles were slightly overestimated from 6 mm depth with a maximum variation of 70 °C and 30 °C at 240 s and 480 s respectively. These differences were visible up to 12 mm depth. Then, the predicted temperatures were close to the experiments for the two heat fluxes. At 480 s of exposure, the temperatures reached 100 °C at 18 mm for 38 kW m -2 and at 20 mm for 59 kW .m -2 . The temperature did not exceed 48 °C at 40 mm depth for 38 kW .m -2 , which meant that wood remained virgin at this depth. The same behavior was observed for tests at 59 kW .m -2 .

The char front was determined from temperature measurements and simulations taking 300 °C as reference temperature [START_REF] Terrei | In-depth wood temperature measurement using embedded thin wire thermocouples in cone calorimeter tests[END_REF][START_REF] Njankouo | Experimental study of the charring rate of tropical hardwoods[END_REF][START_REF] Lizhong | Predicting charring rate of woods exposed to timeincreasing and constant heat fluxes[END_REF]. Indeed, this comparison with the 300 °C isotherm measured with embedded thermocouples in Ref. [START_REF] Terrei | In-depth wood temperature measurement using embedded thin wire thermocouples in cone calorimeter tests[END_REF] showed a good agreement. The symbols in Fig. 8 show the experimental positions obtained as a function of time for the different heat fluxes. The experimental position of the char front can be approximated by a power law (dashed line in Fig. 8) whatever the heat flux. These approximate curves are close to the curves obtained from the GPYRO simulations (continuous line in Fig. 8) for 38 kW/m 2 . The same results were indeed obtained at 6 and 8 mm for the experiments and the simulations. However, the increase of the heat flux entailed an overestimation of the char location with GPYRO. For example, at 59 kW .m -2 , there was a difference of 1 minone minute to reach 6 mm depth and the deviation increased with the time. The charring rate (Fig. 8b) between the experiments and GPYRO calculated by the char front position were very similar. The charring rate decreased at the beginning of the test to a constant value between 0.72 and 0.78 mm .min -1 according to the imposed heat flux. The mean charring rates (between 300 and 600 s) for each heat flux were detailed in Table 3. The values increased with the heat flux from 0.66 to 0.76 mm .min -1 . The predicted and the experimental mean charring rate were very close. These results were lower than those determined by Lizhong et al. and Tran and White [46,[START_REF] Tran | Burning rate of solid wood measured in a heat release rate calorimeter[END_REF] for which the values were approximately two times higher with experiments performed under air atmosphere.

alt-text: Fig. Images are optimised for fast web viewing. Click on the image to view the original version.

alt-text: Fig. 8 Fig. 8 Images are optimised for fast web viewing. Click on the image to view the original version.

Experimental and numerical mass loss results

This section shows experimental and predicted results concerning the spruce wood degradation at the material-level scale. The behavior is similar for the three heat fluxes tested. Regarding the MLRPUA, it increased until reaching a maximum value for the three heat fluxes. We obtained 4.5 (±0.1) g.m -2 .s -1 for 38 kW .m -2 , 6.5 (±0.5) g.m -2 .s -1 for 49 kW .m -2 and 7.9 (±0.7) g.m -2 .s -1 for 59 kW .m -2 . Then, the MLRPUA slightly decreased until a quasi-steady state. The final MLRPUA is equal to 2.7 g .m -2 s -1 for 38 kW .m -2 , 3.9 g .m -2 s -1 for 49 kW .m -2 and 5.5 g .m -2 s -1 for 59 kW .m -2 . These observations were in good agreement with Richter et al. [START_REF] Richter | The effect of chemical composition on the charring of wood across scales[END_REF]. The degradation was studied until 600 s as the main dynamic changes occurred during this period, that is to say the maximum mass loss rate and the quasi-steady state. Consequently, with the large thickness (5 cm) of the samples, the remaining mass at the end of the experiments was important with final values greater than 79% whatever the heat flux. Fig. 6 presents a photograph of a sample after a cone calorimeter test performed at 59 kW .m -2 . We can clearly see that thick layer of char developed during the degradation but no ashes are visible. This indicates that char combustion was not significant and thus that only pyrolysis reactions occurred during these experiments.

Comparison between experiments and GPYRO model for the char front (a) and the charring rate (b).

alt-text: The table layout displayed in this section is not how it will appear in the final version. The representation below is solely purposed for providing corrections to the table. To preview the actual presentation of the table, please view the Proof.

In order to compare the predicted and the experimental mass loss and MLRPUA, Table 3 presents various parameters of the thermal degradation such as MLRPUA max corresponding to the maximal value reached by the MLRPUA, MLRPUA mean which represents the mean value of thethe the quasi-steady state reached by the MLRPUA between 300 and 600 s, the remaining mass and the normative root mean squared error (NRMSE) between experimental and predicted MLRPUA calculated as follows:

alt-text: Fig. 9 (2)

In a general way, the mass and the MLRPUA were in good agreement whatever the heat flux. The growth of the heat flux from 38 to 59 kW .m -2 induced an additional mass loss of 7%. The best predictions were obtained for tests performed at 38 and 49 kW .m -2 . For these heat fluxes, the difference between predicted and experimental MLRPUA was less than 10.8% highlighting an accurate prediction. For 59 kW .m -2 , the experimental and the predicted mass loss rates followed the same trend. However, two differences were observed. The first one concerned the mass loss peak which is slightly higher and occurs earlier for the prediction. The second one was related to underestimation of the quasi-steady state. As a result, the differences between them were slightly high with a NRMSE of 23.1%.

This performance factor showed therefore a good agreement between the predictions and the experimental data. For tests at 38 kW .m -2 , the predicted MLRPUA increased until 4.6 g .m -2 s -1 in 100 s. Then, the MLRPUA decreased slightly to 2.7 g .m -2 s -1 after 600 s. This behavior was very similar to the experimental one. Indeed, the predicted and experimental values of MLRPUA max and MLRPUA mean were very close with differences lower than 5%. For tests at 49 kW .m -2 , the global predicted MLRPUA was in good agreement with the experimental one. The predicted MLRPUA mean was indeed equal to 4.6 g .m -2 s -1 and corresponded to the experimental one. The predicted value obtained for MLRPUA max was within the range of the experimental data. Finally, the most significant difference between the predictions and the experimental data appeared for 59 kW .m -2 . However, evolutions between experimental and numerical MLRPUA were similar. The intensity of the predicted MLRPUA was close to the experimental one with a value of 8.6 g .m -2 s -1 . However, the peaks appeared at different times, 60 s later for the predicted one. The MLRPUA decreased stronger than the experimental one. This tendency led to discrepancies of the MLRPUA mean with a numerical value of 5.9 g .m -2 s -1 which was 17% lower than the experimental value. The differences observed for the highest heat fluxes (49 and 59 kW .m -2 ) were probably due to the overestimation of the temperatures from the minor thicknesses leading to an overestimation of the predicted MLRPUA since the beginning of the degradation.

Conclusion

An experimental and numerical multiscale study of the thermal degradation of spruce wood was performed under inert atmosphere obtained by a constant injection of argon. Two scales were studied experimentally and numerically: the matter scale and the material scale. First, an experimental campaign with a thermogravimetric analyzer was carried out with three heating rates (10, 20 and 30 °C.min -1

) measuring the mass loss. A reactional mechanism consisting of three steps was defined and the kinetic parameters associated were determined by minimization with the Levenberg-Marquardt algorithm. The results showed efficient predictions in agreement with experimental data whatever the heating rate. Secondly, the wood degradation at material scale was studied with a cone heater in vertical orientation. A specific stainless-steel chamber was designed in order to perform tests in inert atmosphere. Spruce wood samples were exposed to three heat fluxes [START_REF] Wang | Thermal decomposition behavior and kinetics for pyrolysis and catalytic pyrolysis of Douglas fir[END_REF][START_REF]OpenFOAM: User guide: OpenFOAM®: open source CFD : documentation[END_REF] and 59 kW .m -2 ) during 600 s. The mass loss was recorded with a load cell disposed outside the steel chamber to limit its mechanical and thermal constraints. The in-depth temperatures were measured by thin wire thermocouples embedded into the wood samples. Finally, the thermal degradation was studied using a one-dimensional pyrolysis model, GPYRO, at the cone scale. This model used the kinetic parameters determined at matter scale and the thermal properties of spruce wood. The results predicted by the model were in good agreement with the experimental measurements concerning the mass loss, the MLRPUA, the temperatures and the charring rates.

Finally, this study brings more knowledge on the thermal degradation of thermally thick wood plates. The use of argon as inert gas and the position in vertical orientation of the cone heater are original ways to investigate this topic. Then, classical determination method of the kinetic parameters was used to represent the thermal degradation at matter scale.

The scalability of these parameters was then tested using GPYRO and an efficient modeling was obtained. These results are encouraging and will be completed by studies under oxidative atmosphere to model the char oxidation with and without the coupling with flaming.

Uncited References [START_REF] Yang | Characteristics of hemicellulose, cellulose and lignin pyrolysis[END_REF].[Instruction: The reference [START_REF] Yang | Characteristics of hemicellulose, cellulose and lignin pyrolysis[END_REF] can be deleted]

CRediT authorship contribution statement

3. 1

 1 Thermogravimetric resultsa) Cone calorimeter with the steel chamber b) Front side of the steel chamber.

3

 3

Fig. 3

 3 Fig. 3

Fig. 5

 5 Fig. 5 presents the experimental and predicted temperature evolutions at 4, 8, 10, 12 and 16 mm depth for tests at 38, 48 and 59 kW .m -2heat fluxes. On the whole, the experimental and predicted in-depth temperatures were in good agreement for each heat flux. The consecutive curves were ordered according to the thermocouple depth and their spacing is rather regular. For tests at 38 kW .m -2 , the predictions close to the surface were in good agreement with the experimental data up to 330 °C. Then, the GPYRO predictions slowed down more rapidly leading to a difference of 110 °C at the end of the experiments for 4 mm depth. Temperature predictions were better for large depths (i.e. from 8 mm) than on the surface. For the highest heat fluxes, i.e. 49 and 59 kW .m -2

  In-depth temperature and heating rates evolutions measured by embedded wire thermocouples located from 2 mm to 40 mm depth to the exposed surface for tests at 38 kW .m -2(a and c) and 59 kW .m -2(b and d).

  for the lowest and the highest heat flux tested, i.e. at 38 and 59 kW .m -2 for 240 and 480 s exposure times. The temperatures are higher for 59 kW .m -2 than 38 kW .m -2 whatever the depth. Fig. 7a indicates similar temperature profiles at same exposure time. The major difference was located close to the exposed surface (2 mm) where the temperature between the experiments and GPYRO differed by 100 °C at 240 and 480 s for tests at alt-text: Fig. 5

Fig. 5

 5 Fig. 5

Fig. 6

 6 Fig. 6

7

 7

Fig. 7

 7 Fig. 7

Fig. 9

 9 presents the comparison of the mass (in percent) and the Mass Loss Rate Per Unit of Area (MLRPUA) for tests performed at 38, 49 and 59 kW .m -2 . For each heat flux, experimental data are an average of three tests. The error bars correspond to the standard deviation between the tests. Experimental standard deviations for the mass loss are less than 2% which indicates a good repeatability of the tests. The thermal degradation of spruce began rapidly after the positioning of the heating cone in front of the sample by the sliding system. The onset of wood degradation took place, at the latest, 20 s after the beginning of the experiment. Then, the mass began to decrease consistently during 600 s.

Fig. 9 i

 9 Fig. 9

Table

Table 1

 1

Table 3

 3

Table 3

 3 Parameters evaluating the performance of the kinetic mechanism at the material-level scale. Experimental (Exp) and Numerical (Num) values are presented in this table.

	Heat flux (kW.m -2	) Data			38		49	59
								Exp	Num	Exp	Num	Exp	Num
	Final mass (%)							86 ± 0.7	86	85 ± 0.7	84	79 ± 1.9	82
	MLRPUA max (g.m -2	.s -1	)		4.9 ± 0.1	4.6	6.5 ± 0.5	6.8	7.9 ± 0.7	8.6
	MLRPUA mean (g.m -2	.s -1	)	2.9 ± 0.2	3.4	4.7 ± 0.1	4.6	5.9 ± 0.5	6.9
	NRMSE (%)							8.9		10.8	23.1
	(mm.s -1	)				0.72 ± 0.04	0.66	0.77 ± 0.02	0.73	0.78 ± 0.02	0.76

Acknowledgments

The authors thank the LEMTA's technical staff, Jean-Yves Morel, Sébastien Lejeune, Jérémy Bianchin, for their contribution during samples machining and instrumentation.

Conceptualization, Supervision. Paul-Antoine Santoni: Supervision.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A

This section presents the mathematical modeling of the thermal degradation of spruce wood at matter scale and at material scale.

A.1. Reactional mechanism

Based on the TGA analysis presented in section and on previous works on thermal degradation of wood [START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF][START_REF] Gerandi | Multi-scale simulations of the thermal degradation of thin wooden plates: evaluation of two kinetic mechanisms[END_REF][START_REF] Branca | Critical evaluation of global mechanisms of wood devolatilization[END_REF], a kinetic mechanism including three steps was developed. As the degradation was studied under inert atmosphere, only pyrolysis reactions occurred. A lumped mechanism consisting of successive reactions was used since it has been shown it provides the best representation of the chemical and physical phenomena throughout the heating [START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF][START_REF] Gerandi | Multi-scale simulations of the thermal degradation of thin wooden plates: evaluation of two kinetic mechanisms[END_REF].

Consequently, the kinetic mechanism was developed as follows considering that dry wood (DW) is transformed into two intermediary species (DW 1 and DW 2 ) before leading to char. Each reaction lead to gases production that were not measured in this study:

where , and are the stoichiometric coefficients. The degree of conversion was defined considering the following relation:

where is the mass, the initial dry mass and the final mass corresponding to the remaining mass. The reaction rates of these stages follow Arrhenius laws and correspond to:

Where (k [START_REF] Leroy | Thermal degradation of fuels: DSC and TGA studies[END_REF][START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF]) represents the reaction rate of the kth reaction, is the reaction order, the pre-exponential factor, the activation energy, the perfect gas constant and the temperature. In this equation, is equal to 1.

The gases conversion rate of the three steps mechanism is respectively given by:

The total conversion rate was given by:

The kinetic parameters, i.e. the order reactions, the pre-exponential factors, the activation energies and the stoichiometric coefficients were estimated by minimization with TGA experiments using the Levenberg-Marquardt algorithm. This method is currently used for the determination of the kinetic parameters [START_REF] Gerandi | Multi-scale modeling of the degradation of thermally thin wood plates[END_REF][START_REF] Tihay | Influence of oxygen concentration on the kinetics of cellulose wadding degradation[END_REF]. The following objective function, based on the evaluation of the mean square error between the experimental and the simulated degrees of conversion, was used to define the best set of kinetic parameters:

where is the number of points equal to 950 for each heating rate. The exponents and correspond to the experimental and calculated degrees of conversion respectively. The data obtained with the three heating rates were used simultaneously for the minimization. The performance of the kinetic mechanism was evaluated thanks to the normative squared root of the squared difference between the experimental and the calculated degrees of conversion:

A.2. Simulation of the cone calorimeter experiments

At material scale, as the wood sample had a large thickness, the heat and mass transfers are coupled and cannot be dissociated as can be done at matter scale using TGA. CFD codes as FDS or FIREFOAM [START_REF] Mcgrattan | Fire Dynamics Simulator User's Guide[END_REF][START_REF]OpenFOAM: User guide: OpenFOAM®: open source CFD : documentation[END_REF] can be used for studying coupled phenomena during thermal degradation of wood. However, simplified models like GPYRO for wood [START_REF] Gerandi | Multi-scale simulations of the thermal degradation of thin wooden plates: evaluation of two kinetic mechanisms[END_REF][START_REF] Lautenberger | A model for the oxidative pyrolysis of wood[END_REF][START_REF] Richter | The effect of chemical composition on the charring of wood across scales[END_REF] or THERMAKIN for polymeric materials [START_REF] Stoliarov | Prediction of the burning rates of charring polymers[END_REF][START_REF] Stoliarov | Two-dimensional model of burning for pyrolyzable solids[END_REF] can also be used. Both models are specifically used when only the degradation of the solid phase is studied without simulating the flame retroaction. In this work, Generalized

Pyrolysis Model for Combustible Solids (GPYRO) was chosen to study the thermal degradation of spruce wood under inert atmosphere. GPYRO solves the main conservation equations (mass, chemical species, energy and momentum) in the solid phase considering M distinct condensed phase species. These species changed into N gas species as the wood sample is heated and degraded progressively. More details on these equations and the parameters involved are presented in the work of Lautenberger and Fernandez-Pello [START_REF] Lautenberger | Generalized pyrolysis model for combustible solids[END_REF].

Thermal degradation of spruce wood was represented by a one-dimensional model. This configuration is currently used to model cone calorimeter experiments because the temperature gradient along the thickness of wood is more important than those found along the other directions of the space [START_REF] Fateh | Multi-scale modeling of the thermal decomposition of fire retardant plywood[END_REF][START_REF] Gerandi | Multi-scale simulations of the thermal degradation of thin wooden plates: evaluation of two kinetic mechanisms[END_REF][START_REF] Lautenberger | A model for the oxidative pyrolysis of wood[END_REF]. Moreover, the lateral sides of the sample were insulated using a sample holder made of 20 mm thick low density calcium silicate plate all around the sample edges.

This considerably limited the heat losses through the edges. Both reasons justified the use of a 1D approximation to model our experiments. This 1D formulation implies that only heat and mass transfers are considered along the thickness of the sample. The sample did not shrink or swell during the heating. Only heterogeneous reactions between solids and gases were considered. They convert a reactant B k into a product C k following the same scheme than equations (1a) to (1c) representing the steps of the kinetic mechanism. They are described as follows:

where is a N × K matrix specified by the user to define the coefficients and The matrix coefficient is equal to 1 for the gases emitted during the pyrolysis and 0 for others. The atmosphere of degradation was set with 94% of inert gases and 6% of oxygen in GPYRO. The destruction rate of each condensed phase species B k by the kth reaction is given by: where is the degree of conversion of the condensed phase , the formation rate of the condensed phase and the mass fraction of the condensed phase i. The symbol means that equations A.9a, A.9b and A.9c contain an integral evaluated numerically as a summation.

A.2.1. Calculation domain, initial and boundary conditions

The calculation domain is derived from the size of the samples which is 100 × 100 × 50 mm 3

. As a 1D formulation is adopted for this study, only the thickness of the plate is considered. Consequently, the size of the calculation domain is equal to 50 mm. 501 nodes were implemented along this thickness, which correspond to a grid size of 0.1 mm, except for the two boundary cells with a size of 0.05 mm. This value represents the best compromise between time calculation and convergence of the results.

The conservation equations require boundaries conditions to be numerically solved. Three boundaries conditions were applied for the exposed face and the back face of the wooden plate, associated to the condensed phase energy conservation, the gas phase chemical species conservation and the gas phase momentum conservation. They are described as follows for the front face of the plate:

And for the back face of the plate:

where is the average emissivity of spruce wood, is the incident heat flux, and are the convective heat transfer coefficients for the front and the back face of the plate respectively. is the ambient temperature equal to 27 °C, the molecular diffusion coefficient of the species j, the heat capacity of the gases equal to 1000 J .kg

is the mean porosity of the wood sample and is the ambient pressure equal to 1.013 × 10 5

Pa. The diffusive mass flux of gaseous species at the front face is approximated using the heat/mass transfer analogy. This is a strong assumption that requires further studies to test this similitude between heat and mass transfers in this configuration.

For the experiments at matter scale, the argon flow rate was set to 10 L .min -1 what corresponded to a slight current of argon on the sample oriented vertically. This configuration corresponds to a problem of natural convection at the surface of the plate. The convective heat transfer coefficient was therefore set to 12 W .m -2 K -1 . This value is currently found in the literature for these kinds of experiments [START_REF] Gerandi | Multi-scale simulations of the thermal degradation of thin wooden plates: evaluation of two kinetic mechanisms[END_REF]. The back face of the plates is not insulated. We assumed that the same behavior occurred for convective losses at this face and led to a convective heat transfer coefficient equal to 12 W .m -2 K -1 .

A.2.2. Thermo-physical properties

To solve the conservation equations, GPYRO requires the knowledge of effective thermal properties. It is assumed that the thermal conductivity and heat capacity of every condensed phase varied with the temperature as follows [START_REF] Lautenberger | Generalized pyrolysis model for combustible solids[END_REF]:

where is the reference temperature equal to 27 °C, and the initial values at for the conductivity and the heat capacity respectively, and two exponents.

The mean effective heat capacity and thermal conductivity were then weighted by the mass fraction and the volume fraction respectively as follows:

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