Shaolin Liu
email: shaolin.liu@u-bordeaux.fr

Azita Ahmadi-Senichault

Cyril Levet

Jean Lachaud

Experimental investigation on the validity of the local thermal equilibrium assumption in ablative-material response models

Keywords:

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Carbon fiber felts are widely used in the industry as insulators in high-temperature furnaces. At the end of the 1990s, NASA used a rigid carbon felt called Fiber-Form, produced by Fiber Materials Inc., to develop a new generation of low-density ablative materials 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 ablative 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 reconstruction[END_REF][START_REF] Meurisse | Multidimensional material response simulations of a full-scale tiled ablative heatshield[END_REF], Mars 2020 (NASA, 2021) [START_REF] Mahzari | Development and Sizing of the Mars[END_REF][START_REF] Meurisse | Equilibrium model for the ablation response of silicone-coated pica[END_REF], Dragon vehicles (SpaceX, since 2012) [START_REF] Stackpoole | Post-flight 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 micrometers in diameter [START_REF] Tran | Phenolic impregnated carbon ablators (PICA) as thermal protection systems for discovery missions[END_REF][START_REF] Triantou | Thermo-mechanical performance of an ablative/ceramic composite hybrid thermal protection structure for re-entry applications[END_REF][START_REF] Jambunathan | Prediction of gas transport properties through fibrous carbon preform microstructures using direct simulation monte carlo[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 referred to as "Through-Thickness" (TT) and that parallel as "In-Plane" (IP); they are shown in Fig. 2. The thermal conductivity ratio between IP and TT directions is of about two [START_REF] Panerai | Experimental measurements of the permeability of fibrous carbon at high-temperature[END_REF]. When possible, the anisotropy is used to optimize the design of thermal protection systems (TPS) by placing the TT direction perpendicular to the vehicle's surface. This obviously helps reducing heat transfer towards the internal structure but it is also helpful in diffusing heat away from hot spots (stagnation point, shoulders) thanks to the higher IP-direction conductivity. Additionally, carbon preforms are impregnated with a low-density phenolic polymer to improve thermal protection by reducing internal radiation, by acting as a heat sink through endothermic pyrolysis processes, and by blowing pyrolysis gases in the boundary layer that partially block the incoming heat flux [START_REF] Lachaud | Porous-material analysis toolbox based on openfoam and applications[END_REF]. The velocity of the pyrolysis gases is of the order of 1 m/s at peak heating [START_REF] Meurisse | Multidimensional material response simulations of a full-scale tiled ablative heatshield[END_REF]. Detailed TPS material response models, taking into account this complex physics, have been developed since the 1950s, with strong progress in the 1960s during the design of the Apollo heat shield [START_REF] Bartlett | An analysis of the coupled chemically reacting boundary layer and charring ablator, part 1 summary report[END_REF][START_REF] Moyer | An analysis of the coupled chemically reacting boundary layer and charring ablator. part 2-finite difference solution for the in-depth response of charring materials considering surface chemical and energy balances[END_REF]. They are currently being improved and adapted to new generations of materials by an active community [START_REF] Lachaud | A short review of ablative material response models and simulation tools[END_REF] with the goal of reducing design uncertainties [START_REF] Wright | Sizing and margins assessment of 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 generalization of the use of porous fibrous materials raises new questions regarding the validity of the inherited hypotheses, such as [START_REF] Lachaud | A generic local thermal equilibrium model for porous reactive materials submitted to high temperatures[END_REF][START_REF] Li | Application and analysis of flip mechanism in the melting process of a triplex-tube latent heat energy storage unit[END_REF] or local thermal nonequilibrium (LTNE) [START_REF] Xu | Analytical considerations of local thermal non-equilibrium 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: Metal-foam filled pipes[END_REF][START_REF] Du | Numerical studies on a fin-foam composite structure towards improving melting phase change[END_REF]. In LTE models, it is assumed that the average temperatures of the solid and gas phases are equal. Florio [START_REF] Florio | A study of the effects of the assumption of local-thermal equilibrium on the overall thermally-induced response of a decomposing, glass-filled polymer composite[END_REF] and Puiroux [START_REF] Puiroux | Non-equilibrium theories for macroscale heat transfer: ablative composite layer systems[END_REF][START_REF] Puiroux | Macro-scale nonequilibrium heat transfer in ablative layer composite layers[END_REF] demonstrated the validity of the local thermal equilibrium assumption for dense charring materials. However, Scoggins et al. [START_REF] Scoggins | Pyrolysis mechanism of pica[END_REF] showed that the local thermal equilibrium assumption was invalid for PICA using available literature correlations to estimate the heat transfer coefficient between the solid phase and the gas flowing through the pores [START_REF] Scoggins | Pyrolysis mechanism of pica[END_REF]. Unfortunately, correlations for materials similar to PICA were not available in the literature and the major recommendation of Scoggins et al. [START_REF] Scoggins | Pyrolysis mechanism of pica[END_REF] was to perform experiments to measure the heat transfer coefficient in low-density carbon fiber materials. As such studies are still unavailable, the objective of this work is to contribute to filling this gap by investigating experimentally convective heat transfer in Calcarb. The problem is however more complex than it appears because the flow of gas through porous media also modifies their effective thermal conductivity through a process known as dispersion [START_REF] Kaviany | Principles of heat transfer in porous media[END_REF].

Nomenclature

To introduce the studied problem, let us start by presenting a generic model that was proposed for porous reactive materials submitted to high temperatures under the assumption of local thermal equilibrium. This model implements the physics encountered in materials submitted to very high temperatures in different applications such as thermal protection systems of space vehicles, porous materials submitted to fire, or biomass in thermochemical processes of biofuel production [START_REF] Lachaud | A generic local thermal equilibrium model for porous reactive materials submitted to high temperatures[END_REF]. For the purpose of this work we may ignore pyrolysis and chemistry terms and the mass and energy conservation equations write

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝜕 𝜕𝑡 ( 𝜀 𝑔 𝑀⟨𝑝⟩ 𝑔 𝑅⟨𝑇 ⟩ ) + ∇ ⋅ ( -𝑀⟨𝑝⟩ 𝑔 𝑅⟨𝑇 ⟩ 𝐊 𝜇 𝑔 ⋅ ∇⟨𝑝⟩ 𝑔 ) = 0 ( 𝜀 𝑔 𝜌 𝑔 𝑐 𝑝,𝑔 + 𝜀 𝑠 𝜌 𝑠 𝑐 𝑝,𝑠 ) 𝜕⟨𝑇 ⟩ 𝜕𝑡 + ( 𝜀 𝑔 𝜌 𝑔 𝑐 𝑝,𝑔 + 𝜀 𝑠 𝜌 𝑠 𝑐 𝑝,𝑠 ) ⟨𝐯 𝐠 ⟩ 𝐠 ⋅ ∇⟨𝑇 ⟩ = ∇ ⋅ (𝐤 𝐞𝐟𝐟 ⋅ ∇⟨𝑇 ⟩) (1)
where the first equation is the gas mass conservation written in terms of gas pressure (the gas velocity is substituted with Darcy's law) and the second equation is the energy conservation. The gas volume fraction 𝜀 𝑔 is equal to the porosity of the porous medium. 𝐊 and 𝐤 𝐞𝐟 𝐟 denote the permeability and effective thermal conductivity tensors respectively. ⟨𝑝⟩ 𝑔 and ⟨𝐯 𝐠 ⟩ 𝑔 are the intrinsic average pressure and velocity of the gas. ⟨𝑇 ⟩ denotes the superficial average temperature of the porous material (that is equal for both phases). The superficial and intrinsic phase averages of any quantity 𝜑 𝑖 associated to the 𝑖-phase are respectively given by

⟨𝜑 𝑖 ⟩ = 1 𝑉 ∫ 𝑉 𝑖 𝜑 𝑖 𝑑𝑉 , ⟨𝜑 𝑖 ⟩ 𝑖 = 𝜀 -1 𝑖 ⟨𝜑 𝑖 ⟩ = 1 𝑉 𝑖
∫ 𝑉 𝑖 𝜑 𝑖 𝑑𝑉 . In these relationships, 𝑉 𝑖 is the volume of the 𝑖-phase contained within the averaging volume 𝑉 which is a representative elementary volume of the porous medium.

In Eq.1, the permeability 𝐊 can be determined by experimental measurements or numerical simulations [START_REF] Scandelli | Computation of the permeability tensor of non-periodic anisotropic porous media from 3d images[END_REF]. The effective conductivity, 𝐤 𝐞𝐟 𝐟 , is known to be bounded by the arithmetic (𝐤 𝐞𝐟 𝐟 = (𝜀 𝑔 𝑘 𝑔 +𝜀 𝑠 𝑘 𝑠 )𝐈) and the harmonic (𝐤 𝐞𝐟 𝐟 = (𝜀 𝑠 ∕𝑘 𝑠 + 𝜀 𝑔 ∕𝑘 𝑔 ) -1 𝐈) averages of the solid and gas conductivities. Correlations for granular porous materials have been proposed in the literature [START_REF] Zamel | Estimating effective thermal conductivity in carbon paper diffusion media[END_REF][START_REF] Tavman | Effective thermal conductivity of granular porous materials[END_REF][START_REF] Abdulagatova | Effect of temperature and pressure on the thermal conductivity of sandstone[END_REF], amongst which the most simple and often used one that reads 𝐤 𝐞𝐟 𝐟 = 𝑘 𝑔 𝜀 𝑔 𝑘 𝑠 𝜀 𝑠 𝐈 corresponds to the geometric average. In these equations, 𝐤 𝐞𝐟 𝐟 is treated as a spherical tensor and the effect of the gas flow on the effective thermal conductivity is not captured.

A theoretical expression of 𝐤 𝐞𝐟 𝐟 was later derived using upscaling techniques such as the volume-averaging method [START_REF] Kaviany | Principles of heat transfer in porous media[END_REF][START_REF] Quintard | Two-medium treatment of heat transfer in porous media: numerical results for effective properties[END_REF][START_REF] Quintard | One-and two-equation models for transient diffusion processes in two-phase systems[END_REF][START_REF] Hsiao | Modified effective thermal conductivity due to heat dispersion in fibrous porous media[END_REF]. As shown in Eq.2, besides the arithmetic average, two additional terms are involved: the tortuosity term that accounts for the microstructure of the material and the dispersion term that accounts for gas flow effects. These two terms may be numerically estimated by solving a closure problem on a periodic unit cell representative of the structure [START_REF] Quintard | Two-medium treatment of heat transfer in porous media: numerical results for effective properties[END_REF][START_REF] Quintard | One-and two-equation models for transient diffusion processes in two-phase systems[END_REF][START_REF] Quintard | Introduction to heat and mass transport in porous media[END_REF][START_REF] Chevalier | On the effect of the internal gas flow velocity on the thermal conductivity of porous thermal protection systems[END_REF].

𝐤 𝐞𝐟𝐟 = ( 𝜀 𝑔 𝑘 𝑔 + 𝜀 𝑠 𝑘 𝑠 ) 𝐈+ 𝑘 𝑔 -𝑘 𝑠 𝑉 ∫ 𝐴 𝑔𝑠 𝐧 𝑔𝑠 𝐛 𝑔 𝑑𝐴- ( 𝜌𝑐 𝑝 ) 𝑔 ⟨ ṽ𝑔 𝐛 𝑔 ⟩ (2)
where 𝐯 𝑔 , ⟨𝐯 𝐠 ⟩ 𝐠 , and ṽ𝑔 =𝐯 𝑔 -⟨𝐯 𝑔 ⟩ 𝑔 are respectively the gas velocity within the pores, the intrinsic phase average velocity, and the deviation velocity. The closure variable 𝐛 𝑔 is the vector field that maps ∇⟨𝑇 𝑔 ⟩ 𝑔 onto T𝑔 in the LTE model, T𝑔 = 𝐛 𝐠 ⋅ ∇⟨𝑇 𝑔 ⟩ 𝑔 and the LTE condition is represented by

⟨𝑇 ⟩ = ⟨𝑇 𝑔 ⟩ 𝑔 = ⟨𝑇 𝑠 ⟩ 𝑠 .
The generic LTE model needs to be upgraded when local thermal equilibrium is no longer valid, as explained in the previous paragraph. The mass and energy conservation equations for the LTNE model in their simplest form [START_REF] Quintard | Two-medium treatment of heat transfer in porous media: numerical results for effective properties[END_REF] in the case of compressible and non reacting flows write

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝜕 𝜕𝑡 ( 𝜀 𝑔 𝑀⟨𝑝⟩ 𝑔 𝑅⟨𝑇 𝑔 ⟩ 𝑔 ) + ∇ ⋅ ( -𝑀⟨𝑝⟩ 𝑔 𝑅⟨𝑇 𝑔 ⟩ 𝑔 𝐊 𝜇 𝑔 ⋅ ∇⟨𝑝⟩ 𝑔 ) = 0 𝜕 𝜕𝑡 ( 𝜀 𝑠 𝜌 𝑠 𝑐 𝑝,𝑠 ⟨𝑇 𝑠 ⟩ 𝑠 ) = ∇ ⋅ (𝐤 𝐬,𝐞𝐟𝐟 ⋅ ∇⟨𝑇 𝑠 ⟩ 𝑠 ) + ℎ 𝑣 ( ⟨𝑇 𝑔 ⟩ 𝑔 -⟨𝑇 𝑠 ⟩ 𝑠 ) 𝜕 𝜕𝑡 ( 𝜀 𝑔 𝜌 𝑔 𝑐 𝑝,𝑔 ⟨𝑇 𝑔 ⟩ 𝑔 ) + ∇ ⋅ (𝜌 𝑔 𝑐 𝑝,𝑔 ⟨𝐯 𝐠 ⟩ 𝐠 ⟨𝑇 𝑔 ⟩ 𝑔 ) = ∇ ⋅ (𝐤 𝐠,𝐞𝐟𝐟 ⋅ ∇⟨𝑇 𝑔 ⟩ 𝑔 ) + ℎ 𝑣 ( ⟨𝑇 𝑠 ⟩ 𝑠 -⟨𝑇 𝑔 ⟩ 𝑔 ) (3)
where 𝑐 𝑝,𝑖 and 𝐤 𝐢,𝐞𝐟𝐟 respectively denote the heat capacity and effective thermal conductivity tensor of the 𝑖 phase. ⟨𝑇 𝑖 ⟩ 𝑖 and ⟨𝐯 𝐠 ⟩ 𝐠 denote the intrinsic phase average temperature for the 𝑖phase and the intrinsic gas average velocity. ℎ 𝑣 denotes the volumetric heat transfer coefficient between the gas stream and the sample. There are three effective properties, ℎ 𝑣 , 𝐤 𝐬,𝐞𝐟𝐟 and 𝐤 𝐠,𝐞𝐟 𝐟 to be determined. In many research works, regardless of the experimental conditions, simple expressions of 𝐤 𝐬,𝐞𝐟𝐟 and 𝐤 𝐠,𝐞𝐟𝐟 as a function of gas and solid phase conductivities and the porosity of the homogeneous and isotropic porous medium (metal foams, ceramic foams) have been considered [START_REF] Zamel | Estimating effective thermal conductivity in carbon paper diffusion media[END_REF][START_REF] Abdulagatova | Effect of temperature and pressure on the thermal conductivity of sandstone[END_REF][START_REF] Lin 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][START_REF] Hwang | Measurement of Interstitial Convective Heat Transfer and Frictional Drag for Flow Across Metal Foams[END_REF]. 𝐤 𝐢,𝐞𝐟𝐟 are treated as scalars and are given by

𝐤 𝐬,𝐞𝐟𝐟 = 𝜀 𝑠 𝑘 𝑠 𝐈 𝐤 𝐠,𝐞𝐟𝐟 = 𝜀 𝑔 𝑘 𝑔 𝐈 (4)
A more detailed formulation of 𝐤 𝐬,𝐞𝐟𝐟 , 𝐤 𝐠,𝐞𝐟𝐟 and ℎ 𝑣 in the LTNE model obtained by the volume-averaging method are given as follows [START_REF] Kaviany | Principles of heat transfer in porous media[END_REF][START_REF] Quintard | Two-medium treatment of heat transfer in porous media: numerical results for effective properties[END_REF][START_REF] Degroot | Closure of non-equilibrium volume-averaged energy equations in high-conductivity porous media[END_REF] [START_REF] Quintard | Two-medium treatment of heat transfer in porous media: numerical results for effective properties[END_REF]. Compared to Eq.2, Eq.5 splits the effective conductivity of the material into a gas and a solid contribution and includes a heat exchange term. Quintard et al. [START_REF] Quintard | Two-medium treatment of heat transfer in porous media: numerical results for effective properties[END_REF][START_REF] Quintard | One-and two-equation models for transient diffusion processes in two-phase systems[END_REF] constructed pore-scale closure problems and solved them on a model porous medium composed of arrays of cylinders to study the variations of the effective properties as a function of the parameters of the problem. The results show that the effective thermal conductivities 𝐤 𝐢,𝐞𝐟𝐟 and the volumetric heat transfer coefficient ℎ 𝑣 in the LTNE model vary as a function of the Peclet number 𝑃 𝑒 (𝜌 𝑔 𝑐 𝑝,𝑔 𝐮𝑙 𝑔 ∕𝑘 𝑔 ), the ratio of the solid thermal conductivity to the gas thermal conductivity and the gasphase volume fraction 𝜀 𝑔 . The determination of these tensors is not independent of ℎ 𝑣 , therefore all these properties should be sought simultaneously.

The heat transfer coefficient may be determined either by pore-scale numerical simulation in representative geometries [START_REF] Kuwahara | A numerical study of interfacial convective heat transfer coefficient in two-energy equation model for convection in porous media[END_REF][START_REF] Teruel | Entrance effect on the interfacial heat transfer and the thermal dispersion in laminar flows through porous media[END_REF][START_REF] Nakayama | A note on the confusion associated with the interfacial heat transfer coefficient for forced convection in porous media[END_REF] or experimentally [START_REF] Lin 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][START_REF] Hwang | Measurement of Interstitial Convective Heat Transfer and Frictional Drag for Flow Across Metal Foams[END_REF]. In the first method, the volume averaging theory is often applied to periodic structures, such are arrays of cylinders, to obtain ℎ 𝑣 [START_REF] Kuwahara | A numerical study of interfacial convective heat transfer coefficient in two-energy equation model for convection in porous media[END_REF][START_REF] Teruel | Calculation of the interfacial heat transfer coefficient in porous media employing numerical simulations[END_REF]. With the progress of the resolution of computed microtomography and computer resources, pore-level numerical simulations are becoming a realistic approach to determine ℎ 𝑣 [START_REF] Wu | Numerical simulation of convective heat transfer between air flow and ceramic foams to optimise volumetric solar air receiver performances[END_REF][START_REF] Petrasch | Tomography based determination of permeability, dupuit-forchheimer coefficient, and interfacial heat transfer coefficient in reticulate porous ceramics[END_REF][START_REF] Scandelli | Volumetric heat transfer coefficient: Numerical evaluation and introductory analysis to anisotropic effect[END_REF]. Concerning the experimental method, the progress in optimisation algorithms and computer resources has allowed the general usage of inverse methods to infer the parameters with more accuracy [START_REF] Lin Xia | Experiment on the convective heat transfer from airflow to skeleton in open-cell porous foams[END_REF][START_REF] Hwang | Measurement of Interstitial Convective Heat Transfer and Frictional Drag for Flow Across Metal Foams[END_REF][START_REF] Fu | Measurement and correlation of volumetric heat transfer coefficients of cellular ceramics[END_REF][START_REF] Ando | An accurate experimental determination of interstitial heat transfer coefficients of ceramic foams using the single blow method[END_REF]. Using these two approaches, numerous correlations were obtained for the heat transfer coefficient ℎ 𝑣 for a large variety of porous materials, like metal foams [START_REF] Lin Xia | Experiment on the convective heat transfer from airflow to skeleton in open-cell porous foams[END_REF][START_REF] Hwang | Measurement of Interstitial Convective Heat Transfer and Frictional Drag for Flow Across Metal Foams[END_REF][START_REF] Xiao | Effect of metal foam on improving solid-liquid phase change in a multichannel thermal storage tank[END_REF], ceramic foams [START_REF] Wu | Numerical simulation of convective heat transfer between air flow and ceramic foams to optimise volumetric solar air receiver performances[END_REF][START_REF] Ando | An accurate experimental determination of interstitial heat transfer coefficients of ceramic foams using the single blow method[END_REF] and graphite foams [START_REF] Gürüf | Experimental based numerical approach for determination of volumetric heat transfer coefficients of modified graphite foams[END_REF] under diverse experimental conditions. A summary of the most widely used correlations and of their validity ranges is proposed in Table 1. However, none of the available correlations are suitable for carbon fiber felt due to its complex anisotropic microstructure.

In this context, the purpose of this article is to determine the effective conductivities and the heat transfer coefficients needed to inform the LTE and the LTNE models for Calcarb. In Section 2, the experimental facility and the test procedure are presented. In Section 3, the numerical inverse analysis method developed to infer the intrinsic parameters is described. It is based on a multi-objective optimization method to minimize errors between measured and predicted data. In Section 4, the experimental results and the estimated values of the LTE and LTNE parameters are presented and discussed. In section 5, a conclusion provides recommendations regarding the choice of a convective heat transfer model for Calcarb.

Experimental method

The experiments may be conducted using either a steady state [START_REF] Kamiuto | Heat transfer correlations for opencellular porous materials[END_REF][START_REF] Fuller | Measurement and interpretation of the heat transfer coefficients of metal foams[END_REF] or a transient regime [START_REF] Lin Xia | Experiment on the convective heat transfer from airflow to skeleton in open-cell porous foams[END_REF][START_REF] Hwang | Measurement of Interstitial Convective Heat Transfer and Frictional Drag for Flow Across Metal Foams[END_REF][START_REF] Fu | Measurement and correlation of volumetric heat transfer coefficients of cellular ceramics[END_REF][START_REF] Liu | Design and assessments on a hybrid pin fin-metal foam structure towards enhancing melting heat transfer: An experimental study[END_REF] approach. With the steady state approach, the sample is heated up and maintained at a given temperature. Cold gas is flown through the sample. The gas temperature is measured at the inlet and at the outlet of the sample. It is also measured within the sample when possible, that is, when the pores are large enough to allow measuring the gas temperature. The heat transfer coefficient is obtained by the analysis of the spatial evolution of the gas temperature. In the transient method instead, hot gas is used to heat a cold sample. The time evolution of the gas temperature is recorded upwind and downwind of the sample. This method is called the transient single blow technique (TSBT) [START_REF] Lin 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]. This method provides more information as one may measure the increase of temperature of the solid, which brings robustness for the inverse analysis. Last, but not least, as the solid and gas temperatures now both vary in space, one may infer simultaneously their effective thermal conductivities. The experimental facility presented in the following section is based on this method.

Experimental system and test procedure

A schematic drawing of the experimental setup that we have developed and an enlarged view of the test section, where thermocouple positions are shown, are displayed in Fig. 3. The setup consists of a nitrogen gas inlet, a mass flow controller, a heat exchanger to heat the gas, and an insulated test section that contains the sample.

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) with a range of 1 to 4 liters/min, that is, a mass flow rate of Nitrogen ranging from 1.92×10 -5 kg/s to 7.68×10 -5 kg/s. The magnitude of the Darcy velocity 𝑢 varies from 0.21 m/s to 0.84 m/s at room temperature (𝑇 0 = 20 • C). The intrinsic gas velocity ⟨𝑣 𝑔 ⟩ 𝑔 inside the sample varies from 0.233 m/s to 0.932 m/s. When the gas temperature 𝑇 𝑔 rises to 80 • C, 𝑣 𝑔 varies from 0.27 m/s to 1.08 m/s as the gas density 𝜌 𝑔 decreases from 1.15 kg/m 3 to 0.99 kg/m 3 . This velocity range is consistent with the order of magnitude of 1 m/s of the pyrolysis gas velocity encountered in the TPS application, previously mentioned in the introduction.

The gas flows through a heat exchanger made of a Calcarb plug heated by a Ni-Cr wire wrapped around a quartz tube, as shown in Fig. 3. The Ni-Cr wire is heated by Joule effect with a continuous power supply (Velleman LABPS3020). The red points in Fig. 3 indicate the positions of the thermocouples (type K, 0.25 mm sheath diameter). The inlet and outlet gas temperatures, the tube surface temperature, and the temperature inside the sample are measured. The thermocouples are connected to a display data logger (Pico Technology TC-08) that records the temperature with a time step of 1.0 s. To measure the anisotropic properties of the materials, different experiments were performed by changing the orientation of the sampling (sample 1: Through-Thickness (TT) and sample 2: In-Plane (IP)). We performed X-ray scans for both samples to determine the exact location of the thermocouples (Fig. 4) in view of the inverse analysis. The sample dimension and the architectural properties of Calcarb are listed in Table 2. Due to the manufacturing process, there are clusters of fibers made of five to ten fibers. The mean diameter of the fiber clusters has been shown to be the most relevant characteristic length to compute the Reynolds number as it triggers the formations of eddies in the pores [START_REF] Scandelli | Computation of the permeability tensor of non-periodic anisotropic porous media from 3d images[END_REF]. It will be used in the modeling section.

Experimental uncertainty analysis

There are two types of experimental uncertainties: the first one is the direct measurement error (caused by the experimental device), and the second one is the error arising from the measurement process. In this work, the ranges and uncertainties of the different sensors are reported in Table 3. Errors from the temperature measurement process are chiefly due to the positioning of the thermocouples in the flow region, upwind and downwind of the sample. In the sample, there is no uncertainty on the position as it is precisely measured with X-ray scans before testing. In the flow, the maximum temperature difference is 0.4 K within 2 mm of the front and rear of the measurement point. So the maximum resulting uncertainty on the temperature measurement in the flow regions is found to be 0.1%.

Parameter estimation method

Due to the complexity of the problem, we used numerical inverse analysis to estimate the quantities of interest. In the first subsection, we present the numerical model, and in the second subsection we detail the optimization algorithm used to infer the parameters.

Numerical model

For the purpose of parameter estimation, it is necessary to model the whole test section represented in Fig. 3. The numerical model is sketched in Fig. 5. It consists of a two-dimensional axisymmetric geometry where the different regions are identified as follows : flow1 (upwind flow field), sample (porous sample), flow2 (downwind flow field), tube (quartz tube), and thermocouple (downwind thermocouple). The choice of a twodimensional axisymmetric geometry is discussed and justified in the Appendix.A.

Mathematical model and numerical implementation

The flow regions (flow1 and flow2) are modeled with the transient compressible laminar Navier-Stokes equations (conservation of mass, momentum, and energy). For the dense solid regions (tube, thermocouple), transient heat conduction is considered (energy conservation in a solid). In the porous sample region (sample), we consider either the LTE model (Eq.1) or the LTNE model (Eq.3). The Reynolds number based on the cluster diameter (𝑑 𝑐𝑙 ) is defined as, 𝑅𝑒 = 𝜌 𝑔 𝑢𝑑 𝑐𝑙 ∕𝜇 𝑔 . Considering the gas velocity and density ranges provided in the previous section, 𝑅𝑒 is found to vary from 0.95 to 4.2. In a previous study, it was proved that Darcy's law remains valid with an error under 1% for 𝑅𝑒 < 5 [START_REF] Scandelli | Computation of the permeability tensor of non-periodic anisotropic porous media from 3d images[END_REF]; hence, in this work, we will use Darcy's law. The coupling between the different regions is done by considering the conservation of mass and the continuity of temperatures and heat fluxes at the interfaces. The detailed expressions of the boundary conditions and of the initial conditions for the system of Eq.1 (LTE model) and the system of Eq.3 (LTNE model) are provided in Table 4 and Table 5 respectively. To simplify the notations, ⟨𝑇 𝑔 ⟩ 𝑔 , ⟨𝑇 𝑠 ⟩ 𝑠 , and ⟨𝑇 ⟩, will be denoted by 𝑇 𝑔 , 𝑇 𝑠 and 𝑇 respectively. In what follows, 𝑇 is the temperature in the open flow regions (flow1 and flow2), 𝑇 𝑔 and 𝑇 𝑠 are the gas and solid temperatures in the sample and 𝑇 𝑡 is the temperature in the tube and thermocouple regions. The solid thermal conductivity in the tube and thermocouple regions are denoted by 𝑘 𝑡 . The imposed external boundary conditions are given by a secondorder polynomial fitting of the experimental data.

The numerical model was implemented in finite volumes in 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] -where it is made available in open access. A multiblock approach is used, that is, in each region a different set of equations is solved at each time step with appropriate boundary conditions, computed from the mass and energy balances at the interfaces. The dense solid and flow region fields are computed using the conjugate heat transfer solver (chtMultiRegionFoam) of OpenFOAM 7 [START_REF] Renze | Simulation of conjugate heat transfer in thermal processes with open source cfd[END_REF]. For the sample region, the pressure equation is solved semi-implicitly using first order schemes in time (Euler) and space (Gauss linear). The pressure gradient term is implicit and the other terms are explicit. The same approach is used for the energy equations, where the temperature terms are implicit and other quantities explicit. The equations are solved in series, with appropriate mesh refinement and time steps to

Thermal properties of Calcarb and of Nitrogen

The effective conductivity of Calcarb in static conditions was measured in IP and TT directions using the Transient Plane Sources technique (Hot Disk, TPS 3500) [START_REF] Gustafsson | Transient plane source techniques for thermal conductivity and thermal diffusivity measurements of solid materials[END_REF]. At room temperature, these values are (𝑘 𝑒𝑓 𝑓 ,𝐼𝑃 ,𝑠𝑡𝑎𝑡𝑖𝑐 =0.399) and (𝑘 𝑒𝑓 𝑓 ,𝑇 𝑇 ,𝑠𝑡𝑎𝑡𝑖𝑐 =0.212) W/(m⋅K) with an uncertainty of 5 % . The effective conductivity tensor 𝐤 𝐞𝐟𝐟 ,𝐬𝐭𝐚𝐭𝐢𝐜 at room temperature may be defined in cylindrical coordinates as shown in Eq.6, where 𝑧 represents the TT direction. For the same type of fiber materials (FiberForm), the mean thermal conductivity values 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 0.392 and 0.167 W/(m⋅K) in the IP and TT directions. In these materials, as well as in our measurement, the thermal conductivity value in the IP direction is nearly twice the value obtained in the TT direction.

𝐤 𝐞𝐟 𝐟

Fig. 6 shows the variations as a function of temperature of the nitrogen gas thermal conductivity 𝑘 𝑔 and heat capacity 𝑐 𝑝,𝑔 , and of the effective thermal conductivity 𝑘 𝑒𝑓 𝑓 ,𝐼𝑃 ,𝑠𝑡𝑎𝑡𝑖𝑐 , 𝑘 𝑒𝑓 𝑓 ,𝑇 𝑇 ,𝑠𝑡𝑎𝑡𝑖𝑐 and heat capacity 𝑐 𝑝,𝑠 of the Calcarb sample. Presented data for the gas thermal conductivity 𝑘 𝑔 and heat capacity 𝑐 𝑝,𝑔 were obtained from the literature [START_REF] Stephan | Viscosity and thermal conductivity of nitrogen for a wide range of fluid states[END_REF]. The effective thermal conductivities 𝑘 𝑒𝑓 𝑓 ,𝐼𝑃 ,𝑠𝑡𝑎𝑡𝑖𝑐 and 𝑘 𝑒𝑓 𝑓 ,𝑇 𝑇 ,𝑠𝑡𝑎𝑡𝑖𝑐 were measured at 291 K (room temperature), 301 K, 311 K, and extended linearly to 370 K. The solid heat capacity 𝑐 𝑝,𝑠 of the Calcarb sample were obtained from a recently published work [START_REF] Torres-Herrador | Determination of heat capacity of carbon composites with application to carbon/phenolic ablators up to high temperatures[END_REF].

When considering the dispersion effect due to the flow of gas [START_REF] Esence | A review on experience feedback and numerical modeling of packed-bed thermal energy storage systems[END_REF][START_REF] Quintard | Two-medium treatment of heat transfer in porous media: numerical results for effective properties[END_REF], the effective conductivity 𝐤 𝐞𝐟𝐟 used in the LTE model (Eq.1) and presented in Eq.2 may be defined in cylindrical coordinates as shown in Eq.7, where 𝑘 𝑑𝑖𝑠,1 and 𝑘 𝑑𝑖𝑠,2 represent the two components of dispersion term 𝐤 𝐝𝐢𝐬 in the horizontal and transverse flow directions. For the purpose of carrying out the inverse analysis, we use coefficients 𝑐1 and 𝑐2 multiplied by the effective conductivity of Calcarb under static conditions to represent the components of effective conductivity. The solid effective thermal conductivity 𝐤 𝐬,𝐞𝐟𝐟 , and gas effective thermal conductivity 𝐤 𝐠,𝐞𝐟𝐟 used in the LTNE model (Eq.3) and presented in Eq.5 may be sought as shown in Eq.8 and Eq.9, where 𝑐3 ⋅ 𝑘 𝑔 and 𝑐4 ⋅ 𝑘 𝑔 represent the two components of tortuosity term 𝐤 𝐭𝐨𝐫 , 𝑐5 and 𝑐6 indicate the two components of dispersion term 𝐤 𝐝𝐢𝐬 . The parameters 𝑐1 to 𝑐6 will be optimized as described in section 3.2.

𝐤

𝐤 𝐠,𝐞𝐟𝐟 =

[ 𝜀 𝑔 𝑘 𝑔 +𝑐3⋅𝑘 𝑔 +𝑐5 0 0 0 𝜀 𝑔 𝑘 𝑔 +𝑐3⋅𝑘 𝑔 +𝑐5 0 0 0 𝜀 𝑔 𝑘 𝑔 +𝑐4⋅𝑘 𝑔 +𝑐6 ] (𝑟,𝜃,𝑧) (9)

Optimization process

An inverse analysis method was used to infer the intrinsic parameters from experimental measurements to minimize errors between measured and predicted data. We possess four data sequences {𝑇 1 𝑖 } 𝑛 𝑖=0 , {𝑇 4 𝑖 } 𝑛 𝑖=0 , {𝑇 2 𝑖 } 𝑛 𝑖=0 , {𝑇 5 𝑖 } 𝑛 𝑖=0 that correspond to the measured temperatures collected at every time step indicated by the index 𝑖, respectively at the inlet, at the outlet and inside the sample (see Fig. 3). The transient inlet gas temperature data 𝑇 1 is fitted into a second-order polynomial using least squares, and it is used as a boundary condition. The objective function 𝑆 for the optimization processes is defined as the average of the root mean squared relative error between

𝑆 = 1 3 ⎛ ⎜ ⎜ ⎝ √ √ √ √ 1 𝑛 𝑛 ∑ 𝑖=1 ( 𝑇 4 𝑖 𝑛𝑢𝑚 -𝑇 4 𝑖 𝑇 4 𝑖 ) 2 + √ √ √ √ 1 𝑛 𝑛 ∑ 𝑖=1 ( 𝑇 2 𝑖 𝑛𝑢𝑚 -𝑇 2 𝑖 𝑇 2 𝑖 ) 2 (10)
+ √ √ √ √ 1 𝑛 𝑛 ∑ 𝑖=1 ( 𝑇 5 𝑖 𝑛𝑢𝑚 -𝑇 5 𝑖 𝑇 5 𝑖 ) 2⎞ ⎟ ⎟ ⎠ (11)
The minimization of 𝑆 was performed with the Design Analysis Kit for Optimization and Terascale Applications (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]. A combination of the global optimization method DIviding RECTangles algorithm (NCSU DIRECT) [START_REF] Gablonsky | Direct version 2.0 userguide technical report[END_REF] and of the local optimization method Adaptive nonlinear leastsquares algorithm (NL2SOL) [START_REF] Dennis | An adaptive nonlinear least-squares algorithm[END_REF] were used. NCSU DIRECT, as a global optimization algorithm, always find the global optimum of a function within a given domain given enough function evaluations and highly sensitive stopping criteria. However, this exhaustive search can be computationally expensive to reach the minimum [START_REF] Gablonsky | Direct version 2.0 userguide technical report[END_REF]. Therefore, if final convergence becomes slow, NL2SOL is used to speed-up convergence within the sub-region

Results and discussions

Results of measured and predicted temperatures are reported in this section. In the first subsection, the volumetric heat transfer coefficient ℎ 𝑣 used in the LTNE model is determined and the pressure, velocity, and temperature fields predicted with the LTNE model are presented. The second subsection focuses on assessing the validity of the LTE model in the conditions of the experiment.

Determination of the volumetric heat transfer coefficient ℎ 𝑣 used in the LTNE model

Fig. 8 presents a typical comparison of the measured and predicted results, where measured results are represented by dots, and predicted results obtained by solving the LTNE model are represented by solid and dashed lines. The recorded inlet gas temperature, sample temperature at two locations (T2, T5), and outlet gas temperature are a function of time. The temperature data presented is the subset used in the optimization process. For the predicted results, solid temperatures are represented with solid lines and gas temperatures with dashed lines. The procedure for solving ℎ 𝑣 is then presented.

According to the model presented in the previous section, there are five parameters to optimize : ℎ 𝑣 , 𝑐3, 𝑐4, 𝑐5 and 𝑐6. Studies have indicated that the tortuosity terms represented by 𝑐3 ⋅ 𝑘 𝑔 and 𝑐4 ⋅ 𝑘 𝑔 in 𝐤 𝐠,𝐞𝐟𝐟 are significantly smaller than the dispersion terms represented by 𝑐5 and 𝑐6 [START_REF] Degroot | Closure of non-equilibrium volume-averaged energy equations in high-conductivity porous media[END_REF][START_REF] Quintard | Two-medium treatment of heat transfer in porous media: numerical results for effective properties[END_REF]. One can also notice that when summing the effective conductivity of the solid (Eq.8) and the effective conductivity of the gas (Eq.9), the tortuosity terms cancel each other. Due to the combination of these two properties, the sensitivity on 𝑐3 and 𝑐4 becomes very small when approaching local thermal equilibrium (that -we will see -is reached in our case). Therefore, in what follows, we set the values of 𝑐3 and 𝑐4 to zero and run the optimization on three parameters: ℎ 𝑣 , 𝑐5, and 𝑐6.

As a large number of simulations were carried out, a representative case (1m/s in the TT-direction) was selected to demonstrate the optimisation procedure and to present the twodimensional simulation results. In order to choose minimum and maximum values of the parameters to be optimized, orders of magnitude are sought from the literature. We used the correlations provided by Essence et al. [START_REF] Esence | A review on experience feedback and numerical modeling of packed-bed thermal energy storage systems[END_REF] for packed beds and obtained estimates of the dispersion terms of about 0.1 W∕(m ⋅ K) and of the volumetric heat transfer coefficient ℎ 𝑣 of the order of 10 7 W∕(m 3 ⋅K) for a Reynolds number of 4. Taking advantage of the global optimization ability of the NCSU DIRECT algorithm, we respectively set 𝑐5, 𝑐6, and ℎ 𝑣 from 0 to 0.5 W∕(m ⋅ K), 0 to 0.5 W∕(m ⋅ K), and 10 6 to 10 11 W∕(m 3 ⋅K). Convergence (𝑆 < 0.001) was reached after 115 iterations of the global optimization algorithm (in this case local optimisation was not necessary). Fig. 9 shows the influence of the three variables (ℎ 𝑣 , 𝑐5 and 𝑐6) on the error 𝑆. The grey scale indicates the value of 𝑐6, and the size of points indicates the value of 𝑆. As shown in Fig. 9, when 𝑆 is below 0.0025, the values of 𝑐6 are distributed between 0.05 to 0.17 W∕(m ⋅ K), the values of 𝑐5 are concentrated around on 0.115 W∕(m ⋅ K), and the values of ℎ 𝑣 span four orders of magnitude ranging from 10 7 to 10 11 W∕(m 3 ⋅K). To analyse further the behavior of the error when varying ℎ 𝑣 , we set 𝑐5 to 0.115 W∕(m ⋅ K) and 𝑐6 to the average of the range obtained above, that is, 0.11 W∕(m ⋅ K). As shown in Fig. 10, 𝑆 displays an asymptotic behavior. Beyond a critical value of ℎ 𝑣 of 10 8 W∕(m 3 ⋅K), a change in ℎ 𝑣 does not affect the error. This is due to the fact that local thermal equilibrium is reached. The temperature predictions for ℎ 𝑣 = 10 8 W∕(m 3 ⋅K) are plotted in Fig. 8; the predicted solid and gas temperatures overlap at the thermocouple positions. To present further the overall behavior of the sample, the color maps of the simulation results are now presented.

Figs. 11, 12 and 13 show the pressure, velocity, and temperature fields in the test section at 100 s. Fig. 14 shows the temperature fields inside the sample region at 100 s. The pressure gradient in the flow1 and flow2 regions is very small, whereas in the sample region the pressure gradient is driven by the permeability of the sample.

In the flow1 and flow2 regions the gas flow is laminar (𝑅𝑒 = 𝜌 𝑔 𝑢𝐷∕𝜇 𝑔 = 1.17×0.84×0.01∕0.000019 = 517) and the velocity distribution is in agreement with Poiseuille's law where it is fully developed (entrance of flow1, exit of flow2). The gas velocity inside the sample ⟨𝐯 𝐠 ⟩ 𝐠 is equal to the Darcy velocity 𝐮 divided by the porosity 𝜀 𝑔 . The gas velocity profile can be considered unidirectional in the sample.

In the sample region, the difference between the local average gas and solid temperatures 𝑇 𝑔 and 𝑇 𝑠 is negligible. This indicates that in this case, for which the value of ℎ 𝑣 is large (ℎ 𝑣 = 10 8 W∕m 3 K), heat transfer between the gas and the solid phases is important and local thermal equilibrium is reached.

The same methodology was applied for the other studied cases (flow rates from 𝑞𝑚=1.9×10 -5 kg/s to 𝑞𝑚=7.68×10 -5 kg/s for both TT and IP directions) and the same conclusions were reached, that is, ℎ 𝑣 is greater than or equal to 10 8 , in the conditions of the experiment (no chemical reactions, unity Reynolds number) and the local thermal equilibrium assumption holds.

Validation of the LTE model

The objective of this section is to infer the effective thermal conductivity 𝐤 𝐞𝐟𝐟 and to validate the LTE model. Experimental data are presented using dots in Fig. 15 and Fig. 16. As shown in Eq.7, there are two dimensionless coefficients 𝑐1 and 𝑐2 to optimize to infer the effective thermal conductivity 𝐤 𝐞𝐟𝐟 . They are respectively linked to 𝑘 𝑒𝑓 𝑓 ,𝐼𝑃 and 𝑘 𝑒𝑓 𝑓 ,𝑇 𝑇 . As mentioned in Subsection 3.2, the global optimization algorithm (NCSU DIRECT) and the local optimization algorithm (NL2SOL) are used. To guide the choice of the minimum and maximum for the parameters to optimize, we used the correlations provided by Quintard et al. [START_REF] Quintard | One-and two-equation models for transient diffusion processes in two-phase systems[END_REF] for packed beds and obtained estimates of the maximum dispersion terms of about 0.1 W∕(m K) for a Reynolds number of 4, that is, 0.25⋅𝑘 𝑒𝑓 𝑓 ,𝐼𝑃 ,𝑠𝑡𝑎𝑡𝑖𝑐 or 0.47 ⋅𝑘 𝑒𝑓 𝑓 ,𝑇 𝑇 ,𝑠𝑡𝑎𝑡𝑖𝑐 . The effective conductivity equals the effective conductivity in static conditions plus the dispersion terms. As a first step, we perform global optimization with the NCSU DIRECT algorithm with ranges for 𝑐1 and 𝑐2 between 0.1 and 3. During the optimization process, if the NCSU DIRECT algorithm reaches the convergence criterion (𝑆 < 10 -3 ), the optimisation ends. If the global optimisation fails to converge ((𝑆 𝑛+1 -𝑆 𝑛 )∕𝑆 𝑛 < 10 -5 ), the NL2SOL algorithm should be manually defined within a subregion which is obtained from the NCSU DIRECT, such that we can eventually achieve the final convergence.

Fig. 17 shows the optimization results of (𝑐1, 𝑐2) in the IP and TT directions for the different mass-flow rates of the study. The errors 𝑆 obtained from the NCSU DIRECT and NL2SOL algorithms are respectively represented with squares and dots. For the optimization in the IP direction as shown in Fig. 17 (left), 𝑐1 and 𝑐2 are respectively associated with the component of the effective thermal conductivity in the horizontal and transverse flow directions. When the error 𝑆 < 0.001, the values of 𝑐2 are distributed between 1.45 and 2.03. More specifically, for the four mass-flow rates, they are respectively 1.50 ± 0.05, 1.59 ± 0.05, 1.85 ± 0.05, and 1.98 ± 0.05. However, the sensitivity is smaller for 𝑐1 because there is less temperature gradient in the flow direction. For the optimization in the TT direction as shown in Fig. 17 (right), oppositely, 𝑐1 and 𝑐2 are respectively associated with the transverse and horizontal flow directions. When the error 𝑆 < 0.001, the values of 𝑐1 span from 1.05 to 1.34. Precisely, they are respectively 1.08 ± 0.03, 1.12 ± 0.03, 1.19 ± 0.03, and 1.31 ± 0.03. Similarly with the previous case, the sensitivity of 𝑐2 is small. To sum-up, when the error 𝑆 converges to the level of 10 -4 , the ranges of the effective thermal conductivity from the optimization solution are given in Table 6. The effective thermal conductivity is found to increase with the Peclet number 𝑃 𝑒 as expected from the theory presented in the introduction.

Fig. 15 and Fig. 16 show the comparisons between the measured and predicted temperatures in the IP and TT directions with different mass-flow rates. The solid line represents the predicted temperature curve, obtained by minimizing the error 𝑆. Taking Fig. 15 (d) as an example, the values of the error 𝑆 equals 8.01 × 10 -4 , which is the average of 8.20 × 10 -4 , 7.53 × 10 -4 , and 8.30 × 10 -4 corresponding to the three positions: T2, T5, and the outlet point. For reference, the temperature difference between the predicted and measured values is around 0.25 K at the outlet point. This implies that the LTE model may be used with a good level of accuracy to model materials such as Calcarb in the conditions of our experiments, that is, for non reactive flows and Reynolds numbers up to 4.

Conclusions

The objective of this work was to study coupled heat and mass transfer in Calcarb, a carbon fiber preform used as insulator in many applications, and, in particular, in TPS materials. The TPS community has recently been questioning the validity of the local thermal equilibrium hypothesis and was lacking experimental data to conclude. In this work we designed a new experimental facility, based on the transient single blow technique, to determine the heat transfer coefficient ℎ 𝑣 of low density porous materials. Experiments were conducted with nitrogen for Reynolds numbers representative of the TPS application (1 to 4) for both in-plane (IP) and through-thickness (TT) orientations. Parameter estimations were carried out using numerical inverse analysis. Local thermal equilibrium (LTE) and local . This implies that the LTE model may be used with a good level of accuracy to model materials such as Calcarb in the conditions of our experiments. One should note that although the LTNE model is not necessary in this case, the value of ℎ 𝑣 determined in this work is useful for porous media applications such as ablative heat-shield design, where chemical non-equilibrium may challenge local thermal equilibrium [START_REF] Scandelli | Two-temperature ablative material response model with application to stardust and msl atmospheric entries[END_REF]. Pioneering works, such as the study of Scoggins et al. [START_REF] Scoggins | Pyrolysis mechanism of pica[END_REF], may now be revisited in the light of this newly available data.

A. Appendix. Validation of the two-dimensional axisymmetric model in the IP direction

In this work, the numerical model consists of a twodimensional axisymmetric geometry. For the flow in the TT direction, the radial direction is IP and the sample has isotropic properties in the whole radial plane. It is adapted to use a two-dimensional axisymmetric geometry to represent the threedimensional geometry. However, for the flow in the IP direction, the radial direction contains both TT and IP, and the sample has anisotropic properties in the whole radial plane. A threedimensional simulation is performed to validate the simulation results of the two-dimensional axisymmetric geometry on the 4.

The mesh refinement and time steps for the three-dimensional model are the same as the two-dimensional axisymmetric model mentioned in Appendix B. For the value of 𝑐1 and 𝑐2, the optimal results obtained in section 4.2 are used, that is, 𝑐1 = 1.0, 𝑐2 = 1.98. To analyze further the temperature difference between the three-dimensional and two-dimensional axisymmetric numerical model, we plot the temperature field in three lines, that is, line1 (in the 𝑦𝑧 plane and along the 𝑧-direction, the coordinates of the starting point are (0, 0, 0)), line2 (in the 𝑥𝑦 plane and along the 𝑦-direction, the coordinates of the starting point are (0, 0, 0.021)), line3 (in the 𝑥𝑦 plane and along the 𝑥direction, perpendicular to line2.) The positions of line1, line2, and line3 are shown in Fig. 18. As shown in Fig. 19, the maximum temperature difference between the three-dimensional and twodimensional axisymmetric simulations on line1 is 0.1 K, and on line2 is 0.3 K. For line 2 and line 3 in the three-dimensional simulation, the maximum temperature difference between them is 0.3 K. We can assume that the numerical results of the threedimensional and two-dimensional axisymmetric models agree well on line 2 where the thermocouples are placed. This verifies that the two-dimensional axisymmetric model is suitable for this study, even when the flow direction is IP.

Figure 1 :Figure 2 :

 12 Figure 1: Scanning electron micrography (SEM) images of FiberForm (left) and Calcarb (right)

  (a) experimental setup (b) test section part

Figure 3 :

 3 Figure 3: Schematic diagram of the experimental system and enlarged view of the test section showing the thermocouple identification numbers

Figure 4 :

 4 Figure 4: X-ray images of the thermocouple positions inside sample 1 (TT direction) and sample 2 (IP direction)

3 ,

 3 𝑢 = 0 𝑝 = 𝑝, 𝑢 = 𝜀 𝑔 ⟨𝑣 𝑔 ⟩ 𝑔 sample-flow1 interface sample-tube interface sample-flow2 interface sample 𝑇 = 𝑇 0 = 𝑝, 𝜀 𝑔 ⟨𝑣 𝑔 ⟩ 𝑔 = 𝑢 𝜕𝑝 𝜕𝑟 = 0, 𝜕⟨𝑣 𝑔 ⟩ 𝑔 𝜕𝑟 = 0 𝑝 = 𝑝, 𝜀 𝑔 ⟨𝑣 𝑔 ⟩ 𝑔 = 𝑢 flow2-sample interface flow2-tube interface flow2𝑢 = 𝜀 𝑔 ⟨𝑣 𝑔 ⟩ 𝑢 = 0 𝑝 = 𝑝, 𝑢 = 𝜀 𝑔 ⟨𝑣 𝑔 ⟩ 𝑔 sample-flow1 interface sample-tube interface sample-flow2 interface sample 𝑇 𝑔 = 𝑇 0 𝑘 𝑔𝑇 𝑠 = 𝑇 0 𝑝 = 𝑝, 𝜀 𝑔 ⟨𝑣 𝑔 ⟩ 𝑔 = 𝑢 𝜕𝑝 𝜕𝑟 = 0, 𝜕⟨𝑣 𝑔 ⟩ 𝑔 𝜕𝑟 = 0 𝑝 = 𝑝, 𝜀 𝑔 ⟨𝑣 𝑔 ⟩ 𝑔 = 𝑢 interface tube-sample interface tube-flow2 interface tube 𝑇 𝑡 = 𝑇 0 𝑇 𝑡 is a function of 𝑘 𝑇 6, 𝑇 7, 𝑇 8

Figure 5 :

 5 Figure 5: Two-dimensional axisymmetric numerical model of the test section

  (a) gas thermal conductivity (b) gas heat capacity (c) effective thermal conductivity in IP direction (d) effective thermal conductivity in TT direction (e) solid heat capacity

Figure 6 :

 6 Figure 6: Thermal conductivity and heat capacity as a function of temperature for Nitrogen and for the solid phase of Calcarb

Figure 7 :

 7 Figure 7: The steps of optimization process

Figure 8 :

 8 Figure 8: The comparison of predicted and measured temperature in TT direction (𝑞𝑚 = 7.68 × 10 -5 kg/s).

Figure 9 :

 9 Figure 9: The influence of three variables on the error 𝑆, 𝑞𝑚=7.68×10 -5 kg/s

Figure 10 :Figure 11 :

 1011 Figure 10: The influence of ℎ 𝑣 on the error 𝑆

Figure 12 :

 12 Figure 12: Velocity magnitude 𝑢 (m/s) predicted with the LTNE model

Figure 13 :Figure 14 :

 1314 Figure 13: Gas temperature 𝑇 (K) predicted with the LTNE model

Figure 15 :

 15 Figure 15: Comparison of predicted and measured temperatures in IP direction with the LTE model.

Figure 16 :

 16 Figure 16: Comparison of predicted and measured temperatures in TT direction with the LTE model.

Figure 17 :Figure 18 :Figure 19 :

 171819 Figure 17: The optimization results in IP direction (left) and TT direction (right) in the LTE model

𝐤 𝐬,𝐞𝐟𝐟 = 𝜀 𝑠 𝑘 𝑠 𝐈 + 𝑘 𝑠 𝑉 ∫ 𝐴 𝑔𝑠 𝐧 𝑠𝑔 𝐛 𝐬𝐬 𝑑𝐴 𝐤 𝐠,𝐞𝐟𝐟 = 𝜀 𝑔 𝑘 𝑔 𝐈 +

  𝐛 𝐢𝐢 , and 𝑠 𝑖 are the vector and scalar fields that maps ∇⟨𝑇 𝑖 ⟩ 𝑖 onto T𝑖 in the LTNE model, T𝑔 = 𝐛 𝐠𝐠 ⋅ ∇⟨𝑇 𝑔 ⟩ 𝑔 -𝑠 𝑔 (⟨𝑇 𝑔 ⟩ 𝑔 -⟨𝑇 𝑠 ⟩ 𝑠 ) + ..., and T𝑠 = 𝐛 𝐬𝐬 ⋅ ∇⟨𝑇 𝑠 ⟩ 𝑠 -𝑠 𝑠 (⟨𝑇 𝑠 ⟩ 𝑠 -⟨𝑇 𝑔 ⟩ 𝑔 ) + ...

				],
	⎧
	⎪ ⎪ ⎩ ⎨	ℎ 𝑣 =	𝑉 ∫ 𝐴 𝑔𝑠 𝑘 𝑔	𝐧 𝑔𝑠 ∇𝑠 𝑔 𝑑𝐴 𝑘 𝑔 𝑉 ∫ 𝐴 𝑔𝑠	𝐧 𝑔𝑠 𝐛 𝐠𝐠 𝑑𝐴 -𝜌 𝑔 𝑐 𝑝,𝑔	⟨	ṽ𝑔 𝐛 𝐠𝐠	⟩	(5)
	where the closure variables

Table 1

 1 Empirical correlations for the heat transfer coefficient 1𝑃 𝑟 1∕3 𝑅𝑒 0.6 𝑁𝑢 𝑠 = ℎ 𝑠 ⋅ 𝑑 𝑝𝑎𝑟 ∕𝑘 𝑔 , 𝑅𝑒 = 𝜌 𝑔 𝑑 𝑝𝑎𝑟 𝑢∕𝜇 𝑔 1.83 ×10 7 beds 1 ≤ 𝑅𝑒 ≤ 10 4 𝑔 𝑃 𝑟 1∕3 𝑅𝑒 0.61 𝑁𝑢 𝑣 = ℎ 𝑣 ⋅ 𝑑 2 𝑝 ∕𝑘 𝑔 , 𝑅𝑒 = 𝜌 𝑔 𝑑 𝑝 𝑢∕𝜇 𝑔

	Investigators	Media	Correlation	Remarks	Predicted ℎ 𝑣 in
					(W/(m 3 ⋅K)) for 𝑅𝑒 =1
	[54] 𝑁𝑢 𝑠 = 2 + 1.[40] packed metal 𝑁𝑢 𝑣 = 0.34𝜀 -2		4.47 ×10 6
		foam		20 ≤ 𝑅𝑒 ≤ 10 3 , 0.87 ≤ 𝜀 𝑔 ≤ 0.97	(extrapolated for 𝑅𝑒 = 1)
	[42]	metal	𝑁𝑢 𝑣 = 0.3248𝑅𝑒 0.601	𝑁𝑢 𝑣 = ℎ 𝑣 ⋅ 𝑑 2 𝑝 ∕𝑘 𝑔 , 𝑅𝑒 = 𝜌 𝑔 𝑑 𝑝 𝑢∕𝜇 𝑔	3.89 ×10 6
		foam		1900 ≤ 𝑅𝑒 ≤ 7800, 𝜀 𝑔 = 0.95	(extrapolated for 𝑅𝑒 = 1)
	[55]	ceramic	𝑁𝑢 𝑣 = 0.638𝑅𝑒 0.42	𝑁𝑢

𝑣 = ℎ 𝑣 ⋅ 𝑑 2 𝑝 ∕𝑘 𝑔 , 𝑅𝑒 = 𝜌 𝑔 𝑑 𝑝 𝑢∕𝜇 𝑔 7.49 ×10 6 foam 24 ≤ 𝑅𝑒 ≤ 91, 𝑑 𝑝 = 0.29𝑚𝑚 (extrapolated for 𝑅𝑒 = 1) [56] open cellular 𝑁𝑢 𝑣 = 0.124(𝑅𝑒𝑃 𝑟) 0.791 𝑁𝑢 𝑣 = ℎ 𝑣 ⋅ 𝑑 2 𝑝 ∕𝑘 𝑔 , 𝑅𝑒 = 𝜌 𝑔 𝑑 𝑝 𝑢∕𝜇 𝑔 1.14 ×10 6 foam 1 ≤ 𝑅𝑒 ≤ 1000 Current work carbon not applicable 𝑅𝑒 = 𝜌 𝑔 𝑑 𝑐𝑙 𝑢∕𝜇 𝑔 > 1 ×10 8 fiber felt (asymptotic behavior) 1 ≤ 𝑅𝑒 ≤ 4 (local thermal equilibrium)

Table 2

 2 Dimension and structural properties of the Calcarb samplesSample dimensions (mm) Porosity fiber diameter (𝜇m) cluster diameter (𝜇m ) fiber density (𝑘𝑔∕𝑚 3 )

	diameter × length	𝜀 𝑔	𝑑 𝑓	𝑑 𝑐𝑙	𝜌 𝑠
	10 × 17.4	0.9	10	80	1600
				Table 3
		The uncertainty analysis in the experimental measurements
				Range	Uncertainty
	Temperature sensor (Pico Technology)	-270 • C to +1820 • C ± 0.2% of the reading
	K-type thermocouple		-100 • C to + 800 • C	± 0.5 %
	Volume mass flow controller		0 to 4 L/min	± 0.5 % of the reading
	Temperature measurement in the flow region 310 K to 360 K	0.1%
	(due to the position of the thermocouple)

  ,𝐬𝐭𝐚𝐭𝐢𝐜 , room temperature =

		[ 𝑘 𝑒𝑓 𝑓 ,𝐼𝑃 ,𝑠𝑡𝑎𝑡𝑖𝑐	0			0	]
		0		𝑘 𝑒𝑓 𝑓 ,𝐼𝑃 ,𝑠𝑡𝑎𝑡𝑖𝑐		0
		0		0	𝑘 𝑒𝑓 𝑓 ,𝑇 𝑇 ,𝑠𝑡𝑎𝑡𝑖𝑐	(𝑟,𝜃,𝑧)
	=	[ 0.399±5% 0 0	0 0.399±5% 0	0 0 0.212±5%	]	(𝑟,𝜃,𝑧)

Table 6

 6 The value of 𝐤

	IP	1.9×10 -5	0.7 0.399±0.11	0.319±0.01	1.57×10 -4
		3.82×10 -5 1.4 0.399±0.12	0.339±0.01	5.17×10 -4
		5.73×10 -5 2.1 0.399±0.13	0.396±0.01	8.51×10 -4
		7.68×10 -5 2.8 0.399±0.19	0.422±0.01	8.01×10 -4
	TT	1.9×10 -5	0.7 0.432±0.01	0.318±0.11	4.50×10 -4
		3.82×10 -5 1.4 0.448±0.01	0.318±0.11	7.85×10 -4
		5.73×10 -5 2.1 0.475±0.01	0.360±0.11	8.80×10 -4
		7.68×10 -5 2.8 0.523±0.01	0.360±0.11	6.93×10 -4

𝐞𝐟𝐟 , and error 𝑆 between LTE model and measured values for different mass-flow rates Flow 𝑞𝑚 𝑃 𝑒 LTE 𝑆 direction (kg/s) 𝑘 𝑒𝑓 𝑓 ,𝐼𝑃 (W∕(m ⋅ K)) 𝑘 𝑒𝑓 𝑓 ,𝑇 𝑇 (W∕(m ⋅ K))

Acknowledgements

This work was supported by the China Scholarship Council (CSC) program (No. 201907040064).

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.

B. Appendix. Mesh and time convergence

A structured mesh was generated using the blockMesh application of OpenFoam. The mesh was refined at the near-wall and near-thermocouple regions. The final mesh is shown in Fig. 20.

Mesh dependency was tested in flow and radial directions for the conditions presented in subsection 4.1. The mesh sensitivity analysis is plotted in Fig. 21. While time-step independence is shown in Fig 22. As can be seen from Fig. 21, when the number of cells increases in the gas flow direction from 5.29 ×10 4 to 6.97 ×10 4 and in the radial direction from 4.32×10 4 to 5.02 ×10 4 , there is no noticeable variation in the value of the volume averaged solid temperature 𝑇 𝑠 in the porous sample. Therefore, 5.29 ×10 4 cells in the gas flow direction and 4.32 ×10 4 cells in the radial direction are acceptable for the numerical simulations. Adjustable time steps with a user-set maximum value are used in the simulations. When the maximum time step decreases from 1×10 -4 s to 8×10 -5 s, there is no noticeable evolution in the value of volume averaged solid temperature 𝑇 𝑠 . Therefore, the maximum time step is set to 1×10 -4 s. The error in Fig. 21

where the index n+1 indicates the numerical simulation with more mesh refinement and a smaller max time step respectively.


Mathilde Marchand

Yves-Marie Saint-Drenan

Laurent Saboret

Claire Thomas

Lucien Wald

Lucien Wald Eval- Uating

Evaluating the spatial and temporal variations of the performance of CAMS Radiation Service and HelioClim-3 databases of surface irradiation in Germany

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Evaluating the spatial and temporal variations of the performance of CAMS Radiation Service and HelioClim-3 databases of surface irradiation in Germany

Mathilde Marchand 1 , Yves-Marie Saint-Drenan 2 , Laurent Saboret 1 , Claire Thomas 1 and Lucien Wald 2 1 Transvalor, Sophia Antipolis, 06904, France 2 MINES ParisTech, PSL Research University, OIE -Center Observation, Impacts, Energy, Sophia Antipolis, 06904, France Correspondence to: Mathilde Marchand (mathilde.marchand@transvalor.com) Satellite-derived databases of the surface solar irradiance (SSI) have become an essential source of information for various applications in solar energy. Assessing the accuracy of these data by comparison with reference in-situ measurements is therefore ever gaining in importance. Several authors have reported that performances of a given database differ from one site to another depending on the geographical region, topography, orography, climate, viewing angle from the satellite… A good understanding of the spatial and temporal variation of the SSI estimation error is key to allow end-user to have an appropriate level of expectation on the accuracy of this data. This knowledge can also be very important for the further developments of the algorithms.

The present work contributes to this objective by extending the validation works carried out in the last years for numerous regions (Europe, Brazil, Egypt, Arabic Peninsula, Morocco and The Netherlands) to Germany. We consider two databases: the CAMS Radiation Service version 3 (abbreviated as CAMS-Rad) and the HelioClim-3 version 5 (abbreviated as HC3v5) that are widely used by academics and practitioners.

The present communication focuses on several stations located in Germany operated by the Deutscher Wetterdienst (DWD). They are spread over the country, thus allowing the study of the spatial consistency of the performance of each database. Measurements of 10 min means of global irradiance made by pyranometers (CM11 and CM21) and SCAPP set publicly available by the DWD for the period 2010-2018 (9 years) have been used for the validation. Measurements were quality-checked using the method described by Roesch et al. (2011).

Satellite-derived SSI estimates were collected from the SoDa web site (www.soda-pro.com) for the same locations and same instants of measurements for both databases. CAMS-Rad uses the Heliosat-4 method with different inputs: the clear-sky radiation is evaluated using Copernicus Atmosphere Monitoring Service (CAMS) information on the aerosol, ozone and water vapour contained in the atmosphere, the cloud attenuation is considered using cloud optical properties retrieved every 15 min from Meteosat imagery using APPOLO. The second database is the HelioClim-3v5 that is derived from Meteosat images using the Heliosat-2 method, McClear and CAMS products.

For each database, standard error metrics are computed at each station. A particular attention is paid in the presentation of the validation results to evaluate the effects of different parameters such as e.g. the solar elevation and the clearness index on the error. A focus of this work is laid on the consistency of the errors with space and time.