Hussain Najmi

Eddy El Tabach

Khaled Chetehouna

Nicolas Gascoin

François Falempin

EFFECT OF FLOW CONFIGURATION ON DARCIAN AND FORCHHEIMER PERMEABILITIES DETERMINATION IN A POROUS COMPOSITE TUBE

Using Fuel Cell on board of aircraft imposes to extract light species (such as Hydrogen and light hydrocarbons) from the liquid fuel which is stored and used, possibly at temperatures where a fuel pyrolysis occurs. Natural porosity of composite material could be used to filtrate the selected species. Hence the permeability of the porous media becomes one of the key parameter to be accurately measured. It is often determined experimentally in laboratory with disc samples (outlet of the flow is achieved through the porous material) and normal flow. However, this configuration is far from the realistic one consisting of tubes (a main flow is found additionally to the one through the material, tangential permeability). Therefore, the effect of a second outlet on the Darcy's and Forchheimer's permeabilities characterization should be studied (despite the permeability is an intrinsic property of the material itself and it should not be dependent on the test apparatus). This paper focuses on a new way of using an existing test bench for the determination of Darcy's and Forchheimer '

I. Introduction

Using Fuel Cell on board of aircraft could be useful in the coming years for local energy production. The Solid Oxide Fuel Cell (SOFC) for example could be used, mostly if species like hydrogen and light hydrocarbons are available onboard. In the meantime, liquid fuel (stored for propulsion application when burned in the combustion chamber) may encounter temperature that pyrolyses it and that produces light species [START_REF] Wang | An experimental investigation on transpiration cooling of wedge shaped nose cone with liquid coolant[END_REF][START_REF] Gascoin | High temperature and pressure reactive flows through porous media[END_REF][3][START_REF] Dahm | Experimental and modelling investigation of the thermal decomposition of n-dodecane[END_REF][START_REF] Fau | Fuel pyrolysis through porous media: Coke formation and coupled effect on permeability[END_REF] . Since the higher the H 2 or light hydrocarbons content, the longer the life time of the fuel cell, using heavy fuel would decrease the life time of the fuel cell. Thus, a separation of light/heavy species is required. To do so, the natural porosity of the composite materials which are used onboard may be used to separate light from heavy species and thus to increase the H 2 or light hydrocarbons concentration by filtration process.

Ceramic Matrix Composites (CMC) are usually used as a porous material with controlled permeability [START_REF] Steinhauser | A new concept for thermal protection of all-mullite composites in combustion chambers[END_REF][START_REF] François | Current MBDA R&T Effort on Ram/Scramjet and Detonation Wave Engine[END_REF][START_REF] Langener | Experimental investigations of transpiration cooling applied to C/C material[END_REF][START_REF] Dahmen | Numerical boundary layer investigations of transpiration-cooled turbulent channel flow[END_REF] to sustain large heat load compared to metallic materials, while operating at much higher temperatures.

The species like H 2 , C 2 H 2 and C 2 H 4 can even get separated from CH 4 by means of the selective permeation property of the CMC porous media [START_REF] Gilbert | Transport properties of a mixture of permanent gases and light hydrocarbons through the polytetrafluoroethylene capillary tubes of a GP-100 gas extractor[END_REF] . Therefore, the fuel could get enriched with low ignition delay species in order to have a more efficient combustion. In the meantime, the filtrated film inside the combustion chamber would be generated with CH 4 mostly which is characterized by a high auto ignition delay compared to H 2 , C 2 H 2 and C 2 H 4 [START_REF] Abe | Progress in aircraft gas turbine engine development[END_REF][START_REF] Spadaccini | Ignition delay characteristics of methane fuels[END_REF] . Previous work shows that as the methane content increases there is increase in absorbance value which is clearly seen in IR spectrum around 3100 cm -1 13 . Hence the generated film would thus play a positive role of an additional wall protection by absorbing a maximum of Infrared Radiation from the flame.

This selectivity property of CMC depends upon the permeability of the material or of the porous medium [START_REF] Pinnaua | Pure-and mixed-gas permeation properties of a microporous spirobisindanebased ladder polymer (PIM-1)[END_REF][START_REF] Baker | Permeability, permeance and selectivity: A preferred way of reporting pervaporation performance data[END_REF] . Over the last decade, it has been recognized that the separation factor for gas pairs varies inversely with the permeability of the more permeable gas of the specific pair. Indeed, an analysis of the literature data for binary gas mixtures from the list of He, H 2 , O 2 , N 2 , CH 4 , and CO 2 reveals this relationship for these mixtures [START_REF] Robeson | Correlation of separation factor versus permeability for polymeric membranes[END_REF][START_REF] Chenara | Gas permeation properties of commercial polyphenylene oxide and Cardo-type polyimide hollow fiber membranes[END_REF] . Hence the determination of the permeability and studying the factor affecting it is of prime importance. Performance criteria for the filtration media are high flux rate, stability at variable temperature, pressure and high selectivity even in the presence of other gases and contaminants (e.g. CO, CO 2 , CH 4 , H 2 O, H 2 S, etc.).

The filtration process in aerospace sector can be found in other fields of application such as geology [START_REF] Luquot | Hydro-dynamically controlled alteration of fractured Portland cements flowed by CO2-rich brine[END_REF][START_REF] Abdulgader | Hybrid ion exchange -Pressure driven membrane processes in water treatment: A review[END_REF] , petrochemical [START_REF] Tanakaa | Preparation of palladium and silver alloy membrane on a porous α-alumina tube via simultaneous electroless plating[END_REF][START_REF] Shamsabadi | Role of critical concentration of PEI in NMP solutions on gas permeation characteristics of PEI gas separation membranes[END_REF] or fuel cell. For the fuel cell application, several literature works show that filtration process through membranes is a simple technique which is characterised by a low cost and by a low energy consumption as well as a high purity of hydrogen separation [START_REF] Spillman | Economics of gas separation membranes[END_REF][START_REF] Lu | Nanoporous materials: science and engineering[END_REF][START_REF] Phair | Materials for separation membranes in hydrogen and oxygen production and future power generation[END_REF][START_REF] Chi | Transient dynamic of hydrogen permeation through a palladium membrane[END_REF][START_REF] Chi | Preparation of thin Pd membrane on porous stainless steel tubes modified by a twostep method[END_REF] . But in current fuel cells, the temperature remains relatively low or moderate.

In previous works, a high pressure test bench with one inlet and one outlet configuration has been developed [START_REF] Gascoin | High temperature and pressure reactive flows through porous media[END_REF][START_REF] Fau | Fuel pyrolysis through porous media: Coke formation and coupled effect on permeability[END_REF] . It has been tested on different type of metal and composite disks to determine the Darcian's and Forchheimer's permeabilities with inert and reactive fluids at high temperature and pressure. Similar experimental setup has been used by several authors to determine the permeability of same type of sample geometry or of bar type specimens [START_REF] Ilias | Preparation and characterization of composite membrane for high temperature gas separation[END_REF][START_REF] Tremblay | Gas permeability, diffusivity and solubility of nitrogen, helium, methane, carbon dioxide and formaldehyde in dense polymeric membranes using a new on-line permeation apparatus[END_REF][START_REF] Papaioannou | Uphill' permeation of carbon dioxide across a composite molten salt-ceramic membrane[END_REF] . Ilias et al. (1997) investigated the permeability and selectivity of hydrogen through a palladium membrane at high temperature and low pressure. Tremblay et al. (2006) determined the permeability, diffusivity and solubility of polydimethylsiloxane, polyisoprene, polyoctenamer, and polyurethane membranes for pure nitrogen, helium, carbon dioxide, methane and formaldehyde. Papaioannou et al. (2015) evaluated carbon dioxide permeability for sintered supports with molten carbonate dual-phase membranes and its performances were found to be stable for more than 200 h of operation. Instead of so many researches on the porous media, still we face a lack in evaluating the correct flow through porous media. Permeability of the material is mostly determined through experiments using disk sample in laboratory, with normal flow. The outflow through the porous media is measured to determine the permeability of the material. Nevertheless, when using experimentally measured values to numerically simulate realistic engineering configurations, a major disagreement is found regarding the estimation of through flow. Hence, several hypothesis are possible, among which: thermal heterogeneity in the system and particularly in the porous material, material heterogeneity at macroscale, estimation of porous medium properties. Since the lab configuration (the porous disc is inserted into a tube which imposes the entire fluid to flow through the porous medium) is not exactly similar to a realistic one, a dedicated work appears necessary. In real engine, a perpendicular flow to the main one occurs in the cooling tube (cf. Figure 1) and the through flow (secondary outlet) is only a small percentage of the fluid that flows into the pipe from the inlet to the main outlet. Hence, it makes necessary to study the effect of a second flow outlet on the determination of Darcian's and Forchheimer's permeabilities (despite the fact that permeability is often considered as a characteristic of the material itself [START_REF] Allaby | A Dictionary of Earth Sciences[END_REF] ). The aim of this paper is to determine Darcian's and Forchheimer's permeabilities of a permeable medium in a realistic configuration. Therefore, a porous tube made up of generic C/SiC composite is used as a porous medium and the effect of a second outlet on the Darcian's and Forchheimer's permeabilities is investigated. The next section is dedicated to the description of the experimental setup where different "Primary over Secondary flowrates" ratios (noticed PO/SO ratios) and three values of openings of the second outlet are studied. Two analytical methods for the determination of Darcian's and Forchheimer's permeabilities are given. The results of the porous composite tube permeabilities are presented and discussed in the third section of this paper.

II. Material and Methods

Permeation experiments and test methodology

The bench used here is composed of a porous tube connected to gas injection system. Porous tube is made up of C/SiC composite [START_REF] Charles | Analysis of Generalized Pattern Searches[END_REF] having a square cross section area of 1 cm 2 and a length of 1 m. This tube has one inlet and two outlets (the main outlet "outlet-1" is noticed as Primary Outlet -P.O-and the one through the permeable medium "outlet-2" is called Secondary Outlet -S.O-, as seen on Figure 1). The pressure and the mass flow rate are measured on each end of the porous tube. The inlet is located at the left and the outlet-2 is all along the tube although outlet-1 is at extreme right. The injection system is capable of handling liquid as well as gaseous fuels with a high pressure pump (80 bar, 0.5 g.s -1 ). In this study, nitrogen gas is used as fluid. Four sensors to monitor transient variations of mass flowrates, pressure and temperature are connected to a data acquisition system (Keithley 2700, 1 Hz, 16 bits, 48 channels). The mass flowmeter with a range of 0 to 3 g/s is placed at inlet in order to measure mass flowrate of fluid entering into the porous media. An additional Low flow-Brooks glass tube variable area flowmeter is used at P.O to estimate the through flowrate of S.O by substracting the P.O flowrate from the inlet one. This Brooks flowmeter is calibrated for maximum flow of 25 mg/s for the nitrogen. Pressure transducer with a range of 0 to 60 bar is placed at inlet side after the mass flowmeter in order to measure the pressure of fluid (Pinlet) whereas both outlets (P.O and S.O) are kept at atmospheric pressure (1 bar). Flow from the primary outlet is controlled by manually operated valve where three opening percentage values (0%, 50% and 100%) are selected. Hence, this imposes different PO/SO ratios depending on the pressure conditions. Figures 2 and3 illustrate respectively a photograph of the developed permeation experimental setup and a schematic overview of the porous composite tube. ISO 4022 and P 2 methods for the determination of fluid permeabilities are followed for all the tests which are performed at room temperature and repeated at least three times.

Mathematical procedures of Darcian's and Forchheimer's permeabilities determination

Accurate description of fluid flow behavior in the porous media is essential and is described in several ways in various literature studies [START_REF] Hassanizadeh | General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow[END_REF][START_REF] Martins | The effects of non-Darcy flow in propped hydraulic fractures, SPE 20790[END_REF][START_REF] Zeng | A Criterion for Non-Darcy Flow in Porous Media[END_REF] . Usually, Darcy's law depicts fluid flow behavior in porous media [START_REF] Garanzha | Validation of non-Darcy well models using direct numerical simulation[END_REF] . According to Darcy's law, the pressure gradient is linearly proportional to the fluid velocity in the porous material. This linear relationship is valid only for very small flow velocities or low pressure gradients, typically for pore Reynolds number lower than one [START_REF] Hassanizadeh | High velocity flow in porous media[END_REF] . As the Reynolds number (Re) approaches to a critical value (i.e. turbulent regime), the relationship becomes nonlinear but it gets quadratic due to inertial force. The one-dimensional Darcy equation can be written as:

D dP V dx K    ( 1
)
where P is the pressure, x is the direction of fluid flow, µ is the fluid viscosity, V is the superficial velocity, and K D is the darcian permeability. The transient behavior in porous media is described by Navier-Stokes equations whereas Brinkman's equation below is used to describe the macroscopic fluid flow in large range flow regimes:

2 D F P V V L K K      (2)
where inlet outlet P P P    is the pressure drop through the porous medium, L is the external mean sample thickness, ρ is the inlet density (with respect to inlet pressure), K D and K F are the Darcy's and Forchheimer's terms. The right term of Eq. ( 2) is composed of two parts, one related to the Darcy's law for low velocity regime filtration (Darcian flows) and the quadratic one is related to the Stokes's law (non-Darcian flows) which takes into account the inertial effects related with flow resistance.

The Stokes equation is also called as Forchheimer's equation [START_REF] Tully | Fluid Flow and Heat Transfer in a Liquid Rocket Fuel Injector[END_REF][START_REF] Murthy | Thermal Dispersion Effects on Non-Darcy Convection over a Cone[END_REF] . Numerous other formulations of the Brinkman's equation can be found [START_REF] Kim | Fully discrete mixed finite element approximations for non-darcy flows in porous media[END_REF][START_REF] Choi | Permeability modeling of fibrous media in composite processing[END_REF][START_REF] Martin | Simulation of pyrolysis gas within a thermal protection system[END_REF][START_REF] Valdes-Parada | On the effective viscosity for the Darcy-Brinkman equation[END_REF] . Large range of flow regimes through porous media can also be described by using power laws [START_REF] Rathish | Free convection in a non-Darcian wavy porous enclosure[END_REF] and cubic laws [START_REF] Aulisa | Geometric framework for modeling nonlinear flows in porous media, and its applications in engineering[END_REF][START_REF] Nguyen | Mapping of foam mobility in porous media[END_REF][START_REF] Pazos | Compan Calculation of Klinkenberg permeability, slip factor and turbulence factor of core plugs via nonlinear regression[END_REF] .

In order to compute Darcian's and Forchheimer's permeabilities from the measured parameters on the experimental bench, the ISO 4022 norm urges to use the modified form of the above equation: 11

D F P V L V K K       (3)
The dynamic viscosity, the density and the fluid velocity in the model given by Eq. ( 3) are calculated at the mean value of the inlet and the outlet pressures. This linear model (called ISO model) can be easily used for the experiments but not in case of Computational Fluid Dynamics codes or engineering which mostly rely on welldefined upstream conditions. It make necessary to use another method based only on inlet condition. To overcome this shortcoming, another method [START_REF] Langener | Experimental investigation of transpiration cooling with subsonic and supersonic flows at moderate temperature levels[END_REF] which based on the use of a quadratic model (called P 2 model) is considered as a second method for the determination of Darcian's and Forchheimer's permeabilities. The P² model is written as:

2 2 2 2 inlet outlet inlet inlet inlet inlet D F P P V V P L P L K K        (4)
with inlet inlet m V A   and inlet inlet P rT  
where r is the gas constant and T is room temperature.

Using both models, the determination procedure considered here can be stated as the following bound constrained minimization problem:

      exp exp minimize subject to with l u th J J              (5) where   , D F K K
  is the decision variables vector that will be identified, l  , u  are respectively the lower and the upper bounds of these variables,  denotes the usual Euclidean norm, th  is the theoretical pressure loss given by ISO model (Eq. 3) or P² model (Eq. 4) and exp  is the experimental one.

A direct search algorithm provided by the ''Global Optimization'' Toolbox of Matlab® is used to solve the bound constrained optimization problem of relation (5). Unlike traditional optimization methods which need gradient or higher derivatives information to search for an optimal point, this optimization algorithm does not require any information about the gradient of the objective function. This direct search algorithm can be used to solve problems with no differentiable objective function or not even continuous, it searches a set of points around the current point, looking for a point where the objective function value is lower than the one at the current point [START_REF] Kolda | Optimization by direct search: new perspectives on some classical and modern methods[END_REF][START_REF] Charles | Analysis of Generalized Pattern Searches[END_REF] .

III. Results and Discussion

As mentioned in previous sections, the Darcian's and Forchheimer's permeabilities of porous composite tube are determined by two different approaches (ISO method and P 2 method) and the influence of different secondary outlet openings is investigated. To achieve this determination, ISO method considers the evolution of the term

P L V   as a function of V  
whereas the P 2 method considers the evolution of P L  versus inlet V . Figure 5 presents these experimental evolutions for three different P.O openings (0%, 50% and 100%). The curves are typical and very different because in case of turbulent flow the ISO method gives a linear trend while for P² method quadratic one is observed. However in case of a purely laminar flow (Darcian regime), the ISO method gives a horizontal line and a linear trend is observed for the P² method. In Figure 4 (left), it can be easily seen that for maintaining the same mass flow rate through the porous media the required inlet pressure increases slightly as the P.O opening increases. It is obviously because of the fact that as the P.O opens it creates a new pressure drop along the tube leading to a diminution in driving force. To overcome this, the pressure at inlet side should be increased. However, in order to have the same pressure drop across the tube for all three cases (0%, 50% and 100%) the inlet fluid velocity required is to be slightly lower in later case than the former which can be clearly observed in Figure 4 (right). It is also noted that as pressure drop over material thickness ratio reaches the value of 5×10 7 Pa.m -1 there is no much change in inlet fluid velocity which shows the existence of the turbulent regime expressed by the Forchheimer term. This stagnation regime due to turbulent flow (slight velocity increase for major pressure drop increase) can be illustrated by the permeation images given as example in Figure 5 for fully closed P.O case. Initially at atmospheric pressure, there is no permeation as there is no driving force. Then having vaporised water and soap on the outer surface of the tube, when the inlet pressure increases, the soap bubble formation on the tube surfaces increases along the length of the porous tube. Bubbles appear near the inlet first and the inlet pressure increases, bubbles are found further along the tube. In addition, once inlet pressure reaches the value of 3 bar, the permeation occurs from all the four faces of the tube and become more uniform as it reaches a value of 4 bar. A semi-quantitative post-treatment of the images of the bubbles formation is proposed in Figure 6 where the locally estimated S.O flowrate is divided by the entire S.O flowrate. Hence permeation distribution is given in percentage of locally permeating flow in order to express how the through flow is distributed over the entire tube length. It is seen that the central part presents a deficit in permeation which is due to the production process of the tube itself which required some structural parts (around 0.15 m, 0.30 m to 0.60 m and around 0.75 m). These structural additional parts are in the same material but the higher local thickness impacts the permeation locally. For pressure of about 4 bar, the permeation tends to be uniform while a major part of the through flow is found upstream for lower inlet pressure. As described in earlier section, a direct search algorithm is used to minimize quadratic errors between experimental data and the functions of Eq. (3) and Eq. ( 4). ISO method (related to Eq. 3) and P 2 method (related to Eq. 4) are used to determine the Darcian's and Forchheimer's permeabilities for different P.Oopenings (0%, 50% and 100%) which are given in Table 1. A comparison between the two methods (ISO and P 2 ) shows that Darcian's permeabilities determined by P 2 method for three different P.O openings (0%, 50% and 100%) are slightly higher than Darcian's permeabilities determined by ISO method. The Forchheimer's permeabilities are very close to each other for both the methods in all three P.O openings. Determination of K D by ISO method leads to a discrepancies of 2.2% and 17.2% for respectively 50% and 100% open cases. However for K F coefficients, relative gaps of 10.8% and 6.9% is found respectively for 50% and 100% P.O opening. On the other hand, K D determination by P 2 method leads to relative errors of 13% and 31% for respectively 50% and 100% open cases whereas relative changes of 7% and 10.9% are observed for K F terms. The permeability levels obtained in this paper are in quite agreement with those of literature for this kind of composite material (C/SiC). Indeed, Gascoin et al. (2013) [START_REF] Gascoin | Comparison of Two Permeation Test Benches and of Two Determination Methods for Darcy's and Forchheimer's Permeabilities[END_REF] Even if measuring accurately these permeability levels is not so easy, we can see a small increase of the Darcy permeability while the tangential flow increase.

It may depend on the way of computing the density (here taken at entry conditions, the pressure drop can increase between the different PO levels, changing the local density to be theoretically taken into account).

In order to have better understanding of above statement and deeper insight of P.O influence on Darcian's and Forchheimer's term contributions on pressure losses, Figure 7 presents the evolutions of these contribution as well as their ratio as functions of pore Reynolds number. The pore Reynolds number is computed for metallic samples by using the relation /

IV. Conclusion

In many applications from aerospace to fuel cell, the permeability of materials is a key parameter that is generally determined experimentally. Lab test conditions (1 inlet/1outlet) are not always exactly those of realistic conditions of use of these materials (1 inlet/2outlets). Consequently, the present work intended to investigate how realistic configuration plays a role on Darcian and Forchheimer permeabilities determination. Additionally, two different procedures (ISO method and P 2 method) of permeability determination were compared. A C/SiC composite tube is used as a porous medium in this paper. It is found that determination procedures (ISO method and P 2 method) gives almost same results when the primary outlet is fully closed for both Darcian's and Forchheimer's permeabilities. After performing the tests for different P.O openings, it is observed that there is no much change in Forchheimer's permeability but there is a slight increase in the Darcian's one as the P.O opening increases. In order to understand more clearly the above statement, pore Reynolds number has been estimated and Darcian's and Forchheimer's contributions are plotted with respect to Reynolds number. It shows that the Darcian's contribution decreases whereas the Forchheimer's contribution remains more or less the same as P.O opening increases. These slight variations are mostly related to the experimental accuracies and further studies will be necessary to conclude on the best way to use the Brinkman equation in case of tangential permeation on a long tube, first under ambient temperature conditions and then in presence of temperature gradients. These experimental values obtained under realistic test conditions (2 outlets) are in agreement with existing data obtained under lab test conditions (1 outlet).

In future works, mathematical backgrounds will be developed to correlate Darcian's and Forchheimer's permeabilities to the selectivity of the porous materials since this would play a role in the fuel enrichment by separating H 2 from CH 4 for example. After that, the spatial selectivity distribution of several binary mixtures will be investigated for different variable parameters such as the type of porous media, the fluid temperature, the pressure and the time.

  s permeabilities of C/SiC porous composite tube by taking two outlets into account. Operating parameters (temperature, pressure and mass flow rate) are measured for three different configurations: i) secondary outlet (S.O) is 0% open ii) S.O is 50% open and iii) S.O is 100% open. Then Darcy's and Forchheimer's permeabilities are computed by ISO and P 2 methods using a direct search algorithm. Obtained results from different methods are compared and discussed. They are in agreement with the literature data which guarantees the reliability of the test bench and of related measures. Nomenclature Acronym CMC = Ceramic Matrix Composite PO/SO ratio = ratio of the flowrate from the Primary Outlet over the one from the Secondary Outlet P

Figure 1 :

 1 Figure 1: Overview of realistic flows through porous medium in transpiration cooling system (tangential permeation).

Figure 2 :

 2 Figure 2: Photograph of the permeation test bench.

Figure 3 :

 3 Figure 3: A schematic overview of the C/SiC porous composite tube.

Figure 4 .

 4 Figure 4. Evolutions of pressure loss in terms of ρV/µ (left) and inlet fluid velocity (right).

Figure 5 .

 5 Figure 5. Step wise permeation images with respect to change in pressure and mass flowrate.

Figure 6 .

 6 Figure 6. Permeation profile of the composite tube for 0% open case.

  area and g d is the grain diameter. Let us notice that the grain diameter 3material is computed from a correlation between thickness, overall porosity and grain diameter deduced from the literature data (Gascoin 2011). It is observed from the set of plotted data that for porous composite tube, Darcian's term is always lower than the Forchheimer's term for the three different P.O openings (0%, 50% and 100%) showing that the flow is always in a turbulent regime. There is a decrease in Darcian's contribution as P.O opening increases and a relatively very low change is observed for Forchheimer's contribution. The Darcy's pressure term reaches its limit around unity for all cases (0%, 50% and 100% open) as seen in a previous work for different porous media(Gascoin 2011).

Figure 7 .

 7 Figure 7. Permeation contributions plotted versus pore Reynolds number for different secondary outlet openings in porous composite tube.

Table 1 . Darcian's and Forchheimer's permeability values by two methods at different secondary outlet openings for C/SiC composite material.

 1 estimated Forchheimer's permeabilities between 10 -10 and 10 -14 m.

	Primary outlet	ISO method	P 2 method
	opening
	percentage	K D ± ΔK D (m 2 )	K F ± ΔK F (m)	K D ± ΔK D (m 2 )	K F ± ΔK F (m)
	0%	(1.35± 0.3)×10 -14	(1.23± 0.01)×10 -12	(1.40± 0.3)×10 -14	(1.22± 0.009)×10 -12
	50%	(1.38± 0.1)×10 -14	(1.11± 0.006)×10 -12	(1.61± 0.2)×10 -14	(1.14± 0.006)×10 -12
	100%	(1.63± 0.4)×10 -14	(1.15± 0.03)×10 -12	(2.04± 0.6)×10 -14	(1.10± 0.03)×10 -12

PhD student, CE Group, PRISME Laboratory, hussain.najmi@insa-cvl.fr.

Associate Professor, DMS Group, PRISME Laboratory, eddy.el-tabach@univ-orleans.fr.

Tenure Professor, CE Group PRISME Laboratory, khaled.chetehouna@insa-cvl.fr.

Full Professor, CE Group, PRISME Laboratory, nicolas.gascoin@insa-cvl.fr, Senior AIAA Member.

Chief Engineer, Hypersonic Programs MBDA, francois.falempin@mbda-systems.com, Senior AIAA Member.

Zhao, L., Wang, J., Ma, J., Lin, J., Peng, J., Qu, D., Chen, L., An experimental investigation on transpiration cooling under supersonic condition using a nose cone model, International Journal of Thermal Sciences, Vol. 84, pp. 207-213, 2014.

Acknowledgments

This work as been performed under the support of Airbus Group and MBDA.


Eddy El Tabach
email: eddy.el-tabach@univ-orleans.fr.

Khaled Chetehouna
email: khaled.chetehouna@insa-cvl.fr.

Nicolas Gascoin
email: nicolas.gascoin@insa-cvl.fr

Florian Gaschet
email: florian.gaschet@insa-cvl.fr.

Guillaume Fau
email: guillaume.fau@insa-cvl.fr.

Modeling the spatio-temporal evolution of permeability during coking of porous material

Transpiration cooling is one of the most efficient cooling techniques, but one which generates complex phenomena that are difficult to model, and this all the more in that a reactive fluid such as an endothermic fuel is used. Above a certain temperature, such fuel is pyrolysed and, thanks to its endothermic behaviour, this ensures the active cooling of the hot walls of the combustion chamber. However, one of the consequences of this thermal decomposition is the unwanted formation of coke which blocks the porous material (both on the surface and in the interior). This gradual blocking reduces the material's permeability and thus the efficiency of the cooling system. Modelling the permeability distribution of porous materials is thus a key parameter in better understanding transpiration cooling. The present article shows several models intended to estimate the variation in time and space of the permeability of a material (stainless steel) during its coking. The fluid circulating in this porous material is n-dodecane that is maintained at a high temperature. Following a presentation of the measurement device and the measured experimental data of the mean permeability, two categories of model are studied, notably discontinuous mesh models (with 2 and 3 meshes) and continuous analytical models (linear and exponential). The results obtained show that discontinuous models with 2 and 3 meshes are very close in measuring the temporal evolution of the thickness of the coked zone of the porous material. They also revealed that the exponential model is more appropriate than the linear model in estimating the spatiotemporal evolution of the permeability. Additionally, the evolution of the coking rate in the porous material was determined as a function of time and the results show behaviour similar to that indicated in the literature. Lastly, the average Darcy permeability was linked to the mass of coke deposit in the porous material, the result of which reveals a quasi-linear decrease.

Nomenclature

Latin letters e = Sample thickness e j [m] = Thickness of each layer j K = Hydraulic conductivity tensor

I. Introduction

n aerobic vehicules for hypersonic flights (i.e. a flight speeds exceeding Mach 5), the combustion chamber of the continuous flow engine is subjected to extreme heat stress. This results in difficulties in ensuring the mechanical strength of the combustion chamber. To overcome this problem, one technique found in the literature consists in injecting a fluid onto the porous walls of this combustion chamber so as to cool them. When the cooling fluid is nonreactive, the material's permeability and porosity have constant values. However, this is not be the case when the cooling fluid is an endothermic fuel used to supply the combustion chamber. Transporting a single fluid to ensure supply as well as cooling of the combustion chamber enables aerobic vehicles for hypersonic flight to be made lighter. The fuel circulating around the porous walls of the combustion chamber is subjected to high temperature gradients leading to its thermal decomposition [START_REF] Gascoin | Characterisation of coking activity during supercritical hydrocarbon pyrolysis[END_REF] . This pyrolysis results in the formation of coke (particles of carbon) on the surface of the porous material which subsequently gradually blocks up the material over time and space. This leads to a certain spatiotemporal distribution of the porosity and permeability in this porous structure which gradually becomes heterogeneous. Modelling the evolution of the permeability of porous materials is thus of great interest in better understanding the decrease in efficiency of the active transpiration cooling system [START_REF] Gascoin | High temperature and pressure reactive flows through porous media[END_REF][START_REF] El Tabach | Neural-Network Metamodelling for the Prediction of the Pressure Drop of a Fluid Passing Through Metallic Porous Medium[END_REF] .

Different studies can be found in the literature related to the determination of the variability of local permeabilities (or of the transmissivity, if two-dimensional flow problems are to be studied) of porous media [START_REF] Prochazka | Time dependent changes of material properties of FRC due to intensive heating[END_REF] . The great majority of the studies were performed in the hydrogeological field to describe the heterogeneity of the soil [START_REF] El Tabach | Use of artificial neural network simulation metamodelling to assess groundwater contamination in a road project[END_REF][START_REF] Tarek | Reservoir Engineering Handbook-Ch. 4: Fundamentals of rock properties[END_REF] . Various methods can be found the simplest of which use algebraic means: arithmetic, geometric and harmonic.

The latter enable a mean permeability to be determined from local permeabilities. Using a deterministic approach, where the field of permeability K D (x) is sufficiently simple (as is the case, for example, of stratified media that are composed of even layers of permeability juxtapositioned in space) and whose boundary conditions are assumed to be known, exact analytical solutions do exist. In the more general case, the diffusivity equation div(K.grad(h)=0 is solved numerically. The analytical resolution of this equation is, indeed, possible in the case of uniform flows in stratified media. The two most known results are the equivalent permeability of a flow that is perpendicular or parallel to the strata. If the layers (or strata) are in parallel with respect to the flow direction then the mean permeability is the arithmetic mean of the local permeabilities. If the layers are in series with respect to the flow direction, then the mean permeability is the harmonic mean of the local permeabilities [START_REF] Dell'arciprete | Comparison of three geostatistical methods for hydrofacies simulation: a test on alluvial sediments[END_REF] . When the flow is uniform and is both perpendicular and parallel to the strata, the mean permeability is the geometric mean of the local permeabilities [START_REF] De Marsily | Dealing with spatial heterogeneity[END_REF] .

In addition to those in the hydrogeological field, there are studies in the literature relating to the spatio-temporal variation of the permeability in porous materials [START_REF] Uh | Determining spatial distributions of permeability[END_REF][START_REF] Martin | Impact of vertical porosity distribution on the permeability of pervious concrete[END_REF] . Uh and Watson (2011) determined the spatial evolution of the I permeability in a bone sample by using Magnetic Resonance Imagery (MRI) and by solving an inverse problem. Martin III et al. (2014) studied the impact of a vertical porosity distribution on the overall permeability of a pervious concrete. They determined by experimentation the evolution of the permeability as a function of the depth of the material by using an image analysis technique described in an earlier work [START_REF] Martin | Using image analysis to measure the porosity distribution of a porous pavement[END_REF] .

In previous works [START_REF] Gascoin | High temperature and pressure reactive flows through porous media[END_REF][START_REF] Fau | Fuel pyrolysis through porous media: Coke formation and coupled effect on permeability[END_REF] , we experimentally determined the mean permeability by using the Brinkman equation. Several porous materials through which both reactive and nonreactive fluids flow were examined, as was the effect of temperature. The results of this work show that in the case of a reactive fluid, a layer of coke is formed on the material thereby leading to a temporal variation in the mean permeability. Based on these findings, the aim of this article is to study the spatial and temporal distribution of the local permeability of a material as it is being coked (becoming blocked by a layer of coke). The present article is organised as follows. The first section is devoted to the presentation of the experimental data of the mean permeability. The experimental device, the type of material, the reactive fluid, and also the operational conditions are described. This section also explains the theoretical approaches that enable the local permeability to be determined as a function of space and time. The second section reviews the findings and outlines the interpretations. The article will conclude with the prospects for future studies.

II. Equipment and methods

A. Presentation of experimental data of mean permeability

The purpose of the test bench is to study within a laboratory the phenomena that can be expected on the engine by imitating the flight conditions as closely as possible (Fig. 1). Hydrocarbon pyrolysis is possible in pressure conditions of between 10 to 60 bars, from room temperature to 900°C, and at fluid flow rates of 0.05g.s -1 to 0.5g.s -1 . Certain conditions can be met so as to study hydrocarbon pyrolysis in its supercritical phase. The device enables the constant monitoring of these flow rates, pressures and temperatures [START_REF] Gascoin | Characterisation of coking activity during supercritical hydrocarbon pyrolysis[END_REF] . It is important to note that the test bench is adapted to thin samples (less than 3 mm) to avoid the uncontrolled lateral permeation on thicker heterogeneous samples [START_REF] Gascoin | Comparison of Two Permeation Test Benches and of Two Determination Methods for Darcy's and Forchheimer's Permeabilities[END_REF] . During the trials, the combustible liquid is injected by a pump (80 bars, 0.5 g.s -1 ) inside a chemical reactor positioned in a furnace. This reactor is formed of a preheated stainless steel tubular reactor (inside diameter = 3.45mm, length = 550mm) and a cell (inside ∅ =30 mm, length=100mm), into which the porous medium is inserted (filtration diameter only 16 mm). Upon exiting the cell, the tube is cooled and transports the fluid to a phase separator. The liquid phase and the gaseous phase are thus analysed separately and transitionally. The fluid used in the experiments was n-dodecane (VWR, Rectapur 99% of purity) as this is known to be very representative of kerosene. It has been extensively studied by the authors in numerous pyrolysis configurations [START_REF] Gascoin | High temperature and pressure reactive flows through porous media[END_REF] . The porous support used was a sample of stainless steel (∅ = 30 mm, filtration diameter = 16 mm thickness of 3 mm) supplied by Federal Mogul (Poral class). This was chosen because it has a more homogeneous morphological structure and is thus easier to study than composite materials. It has an overall porosity of 30%, a pore diameter is of 4.1µm and a grain size of 14.1µm 1 . The test conditions were as follows. For a given mass flow (65mg.s -1 ), the furnace temperature is raised from room temperature to 750K, then by increments of 50K, after thermal stabilisation (approximately 45 minutes for each plateau). The pressure within the process was set at 35 bars using a pressure regulator. Three furnace temperature settings were studied (750K, 800K and 850K), which correspond respectively to maximal fluid temperatures of 725K, 765K and 820K.

Knowing the flow rates and pressure drops at all instants and based on the Brinkman equation [START_REF] Valdes-Parada | On the effective viscosity for the Darcy-Brinkman equation[END_REF] , we were able to estimate the evolution of the average Darcy permeability versus time. Figure 2 shows the evolution versus time of the average Darcy permeability K Davg and the temperature of the n-dodecane. During the initial heating of the system from room temperature to 750K, before the n-dodecane has started to pyrolyse, a constant value of 1.96×10 -13 m 2 was found for the permeability. This was in full agreement with the value found in the ambient state with a nitrogen flow rate (nonreactive fluid). The vertical lines (dotted) separate the three coking cycles of the porous material. The boundaries between these three cycles (at times 7750 and 10700s) correspond to fluid debits upstream of the porous material at each temperature step. These debits artificially created an inversion in the flow direction for a few instants resulting in the porous material being more or less cleaned. On the contrary, the reductions in Darcy permeabilities were essentially due to the build-up of carbon particles inside the pores of the stainless steel sample. In fact the permeability of the coke deposit (of a thickness of 0.3 mm at the end of the experimentation) on the surface upstream of the material was determined [START_REF] Fau | Fuel pyrolysis through porous media: Coke formation and coupled effect on permeability[END_REF] prior to the experiments at approximately 10 -8 m 2 . This high value (with respect to the initial permeability of the medium: 1.96×10 -13 m 2 ) should not affect the average Darcy permeability measurement K Davg . This conclusion was reinforced by determining the mass of the carbon deposit detected on the surface (8.66mg) with respect to that inside the sample (460mg) at the end of the experiment. The quantity of coke was 50 times greater in the medium than on the surface. However, on the opposite side of the material there was an absence of carbon deposits. To finish, after each fluid debit (cleaning) a variable time can be observed before there is a reduction in permeability. This is probably due to the chemistry. Because of the differences in temperature, the pyrolysis rate varies considerably (influence of the catalytic effect, presence of certain species previously produced, etc).

Based on the experimental data, it is possible for the thickness of the coked material e 1 to be evaluated using the following relation:

   i i void coke e V V e 1 ( 1
)
where V coke , V vide are respectively the volume of coked material and the volume of the empty spaces whereas 1 e is the thickness of the layer i. The two volumes may be calculated using the following relations:

coke coke coke m V   (2)      i i vide e d V   2 4
(3)

where m coke et ρ coke are respectively the mass and the volumic mass of the coke, d is the diameter of the sample and ε is the total porosity of the material. By introducing the two volumes of Eq. ( 2) and Eq. (3) into Eq. ( 1), the thickness of the coked material may be expressed by:

       2 1 4 d m e coke coke (4)

B. Modelling the variability in permeability

In this part, the variability of the permeability within a porous material made of stainless steel will be modelled as it is coking. To perform this work, the database containing the transitional measurements of the average Darcy permeability K Davg described previously is used. On the basis of these transitional measurements of the average permeability, the models developed enable the local permeabilities to be determined as a function of a single space variable K D (x). The choice of monodimensional models allows the problem to be simplified with regard to the database. Furthermore, models proposed in 1D seem to better reflect reality. Indeed, despite there being strong spatial variability of the permeability in all spatial directions, it seems coherent to assume an even coke face since the fluid tends to continually pass through preferred paths (the fluid drives coke particles which block the former preferred paths). Models of two categories were studied, and namely discontinuous mesh models and continuous analytical models.

Discontinuous mesh models

Within the category of discontinuous mesh models, the coking is considered to affect only one part of the thickness (x=3mm) of the porous material. A fist monodimensional model (1D) was developed in which the material is considered to be divided into two meshes of variable lengths e 1 and e 2 . The permeabilities K 1 and K 2 are considered to be different and homogeneous in their respective meshes. Then, a second model was developed in which a third mesh of a length e 0 and permeability K 0 was added. The aim was to create a model as close to reality as possible which also integrates the build-up of coke on the surface of the porous material (Fig. 3). As mentioned in the introduction, there are different models linking the permeability of each strata (mesh) to the average permeability of the porous material. Although the geometric and arithmetic composition law was tested, the decision was made to retain the harmonic mean (adapted to the problem) to determine the permeability of the mesh influenced by the formation of coke. The relations giving the harmonic mean of the permeability may be expressed as follows [START_REF] Tarek | Reservoir Engineering Handbook-Ch. 4: Fundamentals of rock properties[END_REF] :

1 1 n j j D avg n j j j e K e K      ( 5
)
where n is the number of meshes of the porous medium. The principle of the 2 mesh harmonic model is based on the fact that the thickness of the porous material is firstly defined and then the permeability K 2 of the second mesh, which remains constant and has a value of K 2 = 1.96x10 -13 m 2 (which is the average permeability of the porous material when not subjected to coking). By using the thickness of the first mesh calculated by the Eq. ( 1), the permeabilities K 1 in the three coking cycles are deducted from the Eq. ( 5) by using the value of K Davg at the end of each coking cycle. The permeabilities K 1 are thereafter assumed to be constant and the thickness of the first strata is calculated for the three cycles by using the following relation:

    1 1 1 2 2 Davg K K e t e e K t K   (6)
where e = e 1 +e 2 is the thickness of the porous sample. With respect to the 3 mesh model, the principle of the harmonic model is based on the addition of a third layer to assimilate the build-up of coke on the surface of the porous medium. In the same way as previously and using the experimental value of 0 K and assuming a linear evolution of the thickness of the layer of coke 0 e versus time, the thickness 1 e may be calculated as a function of time by:

      1 1 1 1 2 0 2 Davg K K K e t e a t b e K t K K      (7)
where parameters a and b may be deduced from the instants at the beginning and end of coking and the final thicknesses of the layer of coke.

Continuous analytical models

Within the category of continuous analytical models, the phenomenon of coking is considered to affect the whole of the porous material. The first model chosen is a simple model that stipulates a linear evolution of the Darcy permeability (K D ) as a function of the space variable x (thickness of the porous material) of the form:

      , D K x t a t x b t   ( 8
)
where t is the time, a(t) and b(t) are parameters given by the following expression:

            2 , 2 , D D a v g Davg D a t K t e K t e b t K t K t e              (9) where   13 2 , 1.96 10 m D K t e   
is the permeability of the material without coking and K Davg the average permeability of the porous material given by the expression:

    0 1 , e Davg D K t K x t dx e   (10)
In the second continuous analytical model, a spatiotemporal evolution of the local permeability K D (x,t) is proposed in the form of an exponential equation:

    2 , , 1 exp 1 D D K t e K t x x t e                 ( 11
)
where τ is a constant determined by a minimisation of the root-mean square deviation between the theoretical average permeability given by the Eq. ( 10) and that measured experimentally by using an optimisation algorithm such as a string matching algorithm [START_REF] Chetehouna | On the experimental determination of flame front positions and of propagation parameters for a fire[END_REF] . The values of this constant as well as the model/experiment comparisons of the average permeabilities are presented in the following section.

III. Results and discussion

In this section, the different models described previously are compared. Firstly, the evolutions in thickness of the first strata versus time, calculated by Eq. ( 6) and Eq. ( 7), will be compared. Thereafter, the local permeability values of the two models, linear and exponential, will also be compared. Afterwards, the coking level in the layer with thickness e 1 of the porous material will be evaluated versus time. Lastly, the mass of coke deposits, which is directly linked to this coking level, will be linked to the average Darcy permeability.

A. Temporal evolution in the thickness of the coked material

Figure 4 shows a comparison of the evolution in the coke thickness as a function of time for the three coking cycles. Note that in this study, the final thickness of the coked material is evaluated on the basis of equation ( 4) and is value is of 1.1 mm. In this Figure, the curves obtained by the 3 mesh harmonic model can be observed to be close to the 2 mesh model. This seems to be linked to the small thickness of the layer of coke e 0 on the surface of the porous material (approximately 0.3mm), which has a high permeability value (K 0 =10 -8 m 2 ) with little impact on the evolution of the thickness of the first layer versus time. The relative errors between these two models are respectively of 2.95%, 3.87% and 4.72% for coking cycles 1, 2 and 3. As can also be observed, the thicknesses of the coke layers e 1 for the three coking cycles increase first gradually and then more quickly. The material can also be observed to become more and more clogged as the coking cycles progress. This observation may be explained by two main factors. The first is the increase in the pyrolysis temperature at each coking cycle, which results in the generation of more reaction products (i.e. coke particles) in the material. The second is the build-up of fine particles which are not all expelled during the cleaning cycles, thereby leading to a quicker blocking of the porous material.

B. Comparison of two models: linear and exponential

As mentioned previously, the exponential model giving the spatio-temporal evolution of the local permeability requires the constant τ to be determined by the minimization of the root-mean square deviation. Figure 5 shows the evolutions versus time of the experimental and theoretical average permeability (calculated on the basis of Eq. ( 10) and Eq. ( 11)) of the porous material for the three coking cycles. This Figure shows that the average permeabilities reduced as a function of time. The drop in permeability is rapid at the beginning of coking and then gradually slows down and plateaus at the end of coking. It can be noted that, for a given time, the theoretical permeabilities of cycle 3 are lower than those of cycles 1 and 2. The values of these cycles, however, are close to each other. In other words, the material seems to be clogging up more and more quickly (for the same reasons as those explained previously). It can also be noted that the theoretical curves and the experimental curves of cycle 1 better coincide for values of between 0 and 300 s and between 600 and 1100s. With respect to cycle 2, the theoretical average permeability values are in good agreement with the experimental data for the lapse of time 150-280s, as well as around 600s. Lastly, the average experimental permeabilities of cycle 3 reinforce the theoretical values of the exponential model for the time lapses 100-450s and 550-650s. The different values of the constant τ as well as the relative gaps between the theoretical average permeability values and those measured, are shown in Table 1 for the three coking cycles. This Table shows that the parameter τ of coking cycle 3 is greatly inferior to that of cycles 1 and 2. On the contrary, that of cycle 2 is only approximately 1.2 times greater than that of coking cycle 1. Despite the fact that the average theoretical permeabilities are fairly close to the experimental data for most of the time, it does not seem at all obvious to explain the physico-chemical phenomena that are produced by merely using a relatively simple one-parameter model. The two continuous analytical models (linear and exponential) will now be compared. According to Table 1, coking cycle 1 for the exponential model shows the lowest relative error between the experimental and theoretical data for the average permeability. Consequently, it was decided to focus the comparison between the two analytical models on the results of this cycle. Figure 6 shows the spatiotemporal evolution of the permeabilities obtained by the linear and exponential models. This Figure shows that the initial permeabilities as well as those calculated at x=3mm, t  are identical for both models. Furthermore, the permeabilities obtained with the linear model are greater than those of the exponential model in the two spatiotemporal regions: Based on these results, it seems clear that the exponential model has greater pertinence in estimating the spatiotemporal evolution of the permeability than the linear model.

s t x

C. Relation between coking and average permeability

This section focuses on the determination of the temporal evolution of the coking rate for the three cycles as well as the average Darcy permeability as a function of the mass of deposited coke. The coking rate is evaluated using a relation proposed in a previous work [START_REF] Fau | Fuel pyrolysis through porous media: Coke formation and coupled effect on permeability[END_REF] :

        t V t S t m m t R f g f coke f coking       (12)
      t e d t m coke coke 1 2 1 4           (13)       t e d d t S g g 1 2 1 2 3         (14)     t e d t V f 1 2 1 4       (15)
By substituting Eq. ( 13) to Eq. ( 15) in Eq. ( 12), the temporal evolution of the coking rate may be given by:

      t e d d m t R g f f coke coking 1 2 1 1 1 3 2                         (16)
where ε 1 is the porosity of the coked material, which may be linked to its permeability K 1 and to the steel grain size d g by the relation [START_REF] Gascoin | High temperature and pressure reactive flows through porous media[END_REF][START_REF] Yamamoto | Combustion simulation with Lattice Boltzmann method in a threedimensional porous structure[END_REF] :

1 36 1 1 1     K h d k g  (17)
Figure 7 shows the coking rates in the thickness e 1 of the porous stainless steel, evaluated by Eq. ( 16) for the three coking cycles. This Figure demonstrates that the initial coking rate of cycle 1 is greater than that of cycle 2, which is itself greater than that of cycle 3. The coking rate of cycle 1 is quasi-constant up to 180 s then decreases very rapidly from this instant to 220s. After this time it slowly varies until reaching the end of the cycle with a quasi-linear trend.

As for the other cycles, the coking rate decreases as the time increases at an exponential rate. It can be observed that for cycle 3, the coking rate reaches an asymptotic value of approximately A similar tendency was observed [START_REF] Gandarillas | Influence of the Reactor Material Composition on Coke Formation during Ethane Steam Cracking[END_REF] for the coking of an aluminium alloy during the decomposition of naphtha at 1098K. The decline in the coking rate as a function of time for the three cycles may be explained by the fact that the contact surface (and thus the catalytic surface) between the material and the n-dodecane reduces as the coke builds up in the thickness e 1 of the porous material. In practice, the mass of the coke deposit is generally one of the accessible parameters in industrial installations and/or the porous materials within the scope of hydrocarbon cracking. The coking rate is a value that depends on the nature of the fluid, the operating conditions and the type of reactor. The mass of coke deposits is directly linked to this coking rate and it may be usefully linked to the average Darcy permeability. Thus, the transitional models intended to predict the thermal exchange between solid and liquid in the case of structures that are cooled by regeneration will be able to take into account more accurately those changes due to the chemical effect (strongly coupled with multiphase flows with heat and mass transfer). Indeed, Fig. 8 illustrates the evolution (on log-log scales) of the average permeability as a function of the mass of coke deposited in the material for three coking cycles. This Figure shows that the average Darcy permeability decreases quasi-linearly when the mass of coke increases. The values of this permeability for the first coking cycle are lower than for those of the other cycles. Furthermore, the latter two coking cycles have very close permeability values which tends to show the asymptotic behaviour of surface effect on coking compared to homogeneous phase coking rate.

IV. Conclusion

The purpose of the study was to produce information on the circulation of a reactive fluid (n-dodecane) in a stainless steel porous material, so as to improve understanding of the physical processes involved in active transpiration cooling in aeronautics. Most of this work consisted in developing different models enabling the spatiotemporal variability of the permeability of the stainless steel porous material to be modelled during the coking process. Two categories of models were studies, termed "discontinuous models" and "continuous models". In the discontinuous models, it was assumed that the porous medium was firstly stratified in two and then three layers of different permeabilities. The study revealed that the latter two layers were almost equivalent in evaluating the evolution of the thickness of the coked material versus time. This is no doubt linked to the small thickness of the coke layer at the surface of the porous material and to its high permeability. In the second category, two models (linear and exponential) were proposed and the coke was considered to affect the whole of the thickness of the porous material. Compared to the linear model, the exponential model more adequately describes the spatiotemporal evolution of the permeability. The coking rate in the porous material for the three cycles decreases as a function of time and this decrease may be linked to the reduction in the contact surface between the latter and the fluid. Lastly, the permeability values of the different cycles declines quasi-linearly as a function of the mass of coke deposited inside the porous material.

In future works, the study will be widened to include other materials and fluids. In fact, each material has its own permeability, which depends on the properties, such as the porosity and grain size, of its internal structure. As for the fluid, its capacity to be more or less reactive (to pyrolyse) will also impact the permeability. We are also planning to study the influence of micro-cracking which may appear and increase the permeability of a material over time (given the high pressure and temperature exerted).

Figure 1 .

 1 Figure 1. The permeation test cell.

Figure 2 .

 2 Figure 2. Evolution of the average Darcy permeability and the fluid temperature versus time.

Figure 3 .

 3 Figure 3. Discretisation model of the porous medium: (a) Model with 2 meshes and (b) Model with 3 meshes.

Figure 4 .

 4 Figure 4. First layer thickness evolution versus time: (filled symbols) model with 2 meshes and (non-filled symbols) model with 3 meshes.

Figure 5 .

 5 Figure 5. Average permeability evolution as a function of time for 3 coking cycles: symbols represent experimental data and the lines represent the model.

Figure 6 .

 6 Figure 6. Spatio temporal evolution of permeability obtained by linear and exponential models for 1 st coking cycle.

  mass flow rate of the n-dodecane and its volumic mass. The coke mass m coke (t), the total surface area of the steel grains S g (t) and the volume reserved for the fluid in the coked material V f (t) are given by the following expressions:

Figure 7 .

 7 Figure 7. Evolution of the coking rate versus time for the 3 coking cycles.

Figure 8 .

 8 Figure 8. Evolution of the average Darcy permeability versus the mass of coke deposit for the 3 coking cycles.

Table 1 . Values of τ and the relative gap between experimental and theoretical data for different 3 coking cycles.

 1

		1 st Coking cycle	2 nd Coking cycle	3 rd Coking cycle
	τ(s)	91.17	105.97	34.78
	Relative gap (%)	4.07	14.17	7.87


N Grange

K Chetehouna

N Gascoin

S Senave

Numerical Study of the Thermal Behaviour of a Thermo-Structural Aeronautical Composite under Fire Stress

The use of composite materials for aeronautical applications has been growing since several years because of the opportunity to produce lightweight structures reducing the fuel bills and emissions. The need for fireproof certification imposes costly and time consuming experiments that might be replaced or complemented in the years to come by numerical calculations. The present work creates a CFD numerical model of a fireproof test. As an example, a composite part (plenum) located in an aircraft APU (auxiliary power unit) which provides power to the aircraft is investigated. A numerical calibration of the flame is conducted according to the fireproof standards. The results of fireproof tests demonstrate a good evaluation of the plenum temperature (discrepancies lower than 19%). The influence of an internal air jet within the studied part is also evaluated observed to evaluate how this could lower the requirements of certification rules. A thermal decrease as high as 38 % is found for a velocity of 10 m/s.

Introduction

The use of composite materials for aeronautical applications has been growing since several years because of the opportunity to produce lightweight structures reducing the fuel bills and emissions. The growing use of these materials leads to technical and design challenges to comply with safety standards and certifications, especially when fire safety requirements are concerned. Aircraft parts dedicated to firewall applications or located in a designated fire zone, should meet a fireproof requirement. Therefore the composite parts have to pass fire tests according to ISO 2685 [START_REF]Aircraft-Environmental Test Procedure for Airborne Equipment-Resistance to Fire in Designated Fire Zones[END_REF] or FAA -AC20-135 (FAR-25) [START_REF]Powerplant installation and propulsion system component fire protection test methods, standards and criteria[END_REF] standards. Both standards use an oil burner to heat the part with a minimum temperature of 1100°C for 15 minutes. In this work, a 3D numerical model of a fireproof test using a CFD code is created to investigate the predictivity of a numerical fireproof test. This numerical step is expected to replace experimentation during the development phases of the composite part before the certification test to reduce development cost. This numerical tool would help designers to choose between different composite materials and designs options to avoid critical temperature increases at certain areas and perforation in this composite part during fireproof tests. The second section is dedicated to the presentation of the experimental setup and the third one will present the physical and numerical modelling approaches. In the fourth section the computed temperatures are compared to the experimental ones to validate the presented numerical approach and the results are discussed. The influence of an internal air jet within the studied part is also evaluated The feasibility of replacing a thermal protection by an internal air jet is also presented in this paper as a first design variable case.

Experimental setup

To be labelled "fireproof" as it is requested in most of the APU (Auxiliary power unit) part specifications and according to the related standards, the concerned part (here a composite plenum) has to resist 15 minutes to a calibrated flame. Criteria to establish the test is passed include no burn through of the part structure, as well as no ignition of the emitted smokes (backside part inner surface self-ignition). This second criteria is here investigated by measuring the part material temperature increase. The Figures 1 and2 present respectively a picture and an overview of the experimental setup. The composite part is located at 100 mm from the outlet of the cone burner above a vibrating table (sinusoidal vibration of 0.4 mm amplitude and 50 Hz frequency). The oil burner (kerosene-air) operates with a kerosene flow rate of 7.58 l/h and the air flow rate is adjusted to 25 l/s to generate a diffusion flame with: (i) Flame temperature of 1100 °C measured at 100 mm and (ii) Heat flux of 120 KW/m². The flame temperature is measured at 100 mm by a six thermocouples rack. When the temperature is adjusted and stabilized around 1100 °C the heat flux is measured, thanks to a specific device where water is circulating along a copper pipe exposed to the flame. The water flow rate and temperature are 226 l/h and 300 K respectively. According to the standards, the minimum temperature increase have to be 5 K. For experimental information, two thermocouples (TC 1 and TC 2) are added and located on the plenum to monitor the internal wall temperatures (cf. Figure 3).Two cameras are also used to record the experimental fire test.

Physical and numerical modelling 3.1. Numerical modelling approach

The present study is based only on the evaluation of the thermal phenomenon, so the flame is modelled with a hot inert gas [START_REF] Remie | Analysis of the heat transfer of an impinging laminar flame jet[END_REF][START_REF] Remie | Heat-transfer distribution for an impinging laminar flame jet to a flat plate[END_REF][START_REF] Cremers | Thermochemical heat release of laminar stagnation flames of fuel and oxygen[END_REF]. The hot gas jet is defined by two parameters which are the burner outlet velocity and the flame temperature given above (1100 °C). The fluid

           (1)
Using equation [START_REF]Aircraft-Environmental Test Procedure for Airborne Equipment-Resistance to Fire in Designated Fire Zones[END_REF] and the definition of the mass stoichiometric ratio as well as the experimental values of kerosene and air flow rates, the computed value of equivalence ratio is φ=0.88. This equivalence ratio permits to estimate the burner outlet velocity, thanks to a chemical equilibrium calculation performed by the CEA software [START_REF] Mcbride | Computer program for calculation of complex chemical equilibrium compositions and applications: II. User's manual and program description[END_REF]; its value is equal to 19.5 m/s. The CFD code employed in this paper solves the three-dimensional time-dependent equations governing fluid motion, and heat transfer. This CFD code uses a finite volume method with the SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm [START_REF] Caretto | Two calculation procedures for steady, threedimensional flows with recirculation[END_REF] to solve the fluid motion equations. Concerning turbulence treatment, the k-ε realizable turbulence model is chosen in this study. This model consists of a new model dissipation rate equation and a new realizable eddy viscosity formulation [START_REF] Shih | A new k-ϵ eddy viscosity model for high Reynolds number turbulent flows[END_REF]. The k-ε realizable turbulence model has been tested in different benchmark configurations and results show that this model has better performances than the standard k-ε model [START_REF] Suryan | Comparative study of turbulence models performance for refuelling of compressed hydrogen tanks[END_REF][START_REF] Chen | Turbulence modelling for flow in a distribution manifold[END_REF][START_REF] Ziani | Numerical simulations of non-premixed turbulent combustion of CH4-H2 mixtures using the PDF approach[END_REF].

Numerical calibration of the hot jet

Before studying the temperature evolutions of the plenum exposed to the burner's flame, it is required to perform a preliminary numerical simulation to reproduce the experimental calibration procedure and respect the specifications guideline. The couple of parameters defined in the previous section is used to achieve this numerical calibration. In case the calibration would not be respected, these parameters will have to be adjusted to respect the specified values of the standards. The Figure 4 presents the computational grid of the numerical calibration process which is meshed using an unstructured grids. The outlet region of the model is an environmental pressure boundary (101325 Pa). No-slip adiabatic boundary condition is applied on the burner wall and coupled wall is applied to the copper tube surface.

Thermal properties of the Plenum material

The modelling of the plenum material thermal behaviour needs the utilisation of accurate temperature dependent properties. The A first material studied used in this work is a carbon-phenolic composite which is of interest commonly used in the aerospace industry due to its low thermal conductivity [START_REF] Trick | Mechanisms of the pyrolysis of phenolic resin in a carbon/phenolic composite[END_REF][START_REF] Srikanth | Mechanical, thermal and ablative properties of zirconia, CNT modified carbon/phenolic composites[END_REF]. To the best knowledge of the authors, there is no data in the literature on the thermal properties of the studied material. Nevertheless, Engelke et al. [START_REF] Engelke | Thermal and mechanical properties of a nondegraded and thermally degraded phenolic-carbon composite[END_REF] gives experimental values of thermal conductivity and specific heat as functions of the temperature for a carbon phenolic material very similar to the one used in this work (i.e. equivalent fibre volume ratio). According to their values, the temperature dependent properties can be given as:

Computational domain and conditions

The computational domain considered for this study reproduces the experimental bench presented in the first section. For a second configuration, to study the benefit effect of an air jet at room temperature (300 K) in the internal part of the plenum, an air injector of a 70 mm diameter located at 200 mm above the plenum is added. This distance has been chosen to avoid any perturbation of the flow in this area. Figure 4 illustrates an overview of the burner and the plenum as well as the air injector. In some defined real operating conditions, the APU is flown by air circulation that could decrease the maximum temperature found on the plenum. A numerical study is conducted to evaluate this air circulation impact. To achieve this, three air flow velocities are selected from 0 to 10 m/s. Their values as well as the related flow rates are presented in the Table 1. The size of the computational domain is a hemisphere with a diameter of 3 m which correspond to ten times the diameter of plenum. A 3,000,000 unstructured mesh is adopted to mesh the plenum and the whole domain (cf. Figure 5). Like the numerical calibration case, the outlet region of the model is the environmental pressure boundary (101325 Pa). A no-slip adiabatic boundary condition is applied on the burner wall and a coupled wall is imposed to the whole plenum geometry. The enhanced wall treatment was used for near-wall modelling. This wall treatment is called the low-Reynolds number approach; it resolves the viscous sublayer and computes the wall shear stress from the local velocity gradient normal to the wall. This wall treatment requires a very fine mesh resolution in wall normal Direction [START_REF] Lateb | Comparison of various types of k-ε models for pollutant emissions around a two-building configuration[END_REF]. The y + values obtained close to the walls are less than 1, which demonstrates the suitability of the grid used in this paper.

Results and discussion

As mentioned in the introduction, the aim of this work is to develop a numerical model of the fire safety tests as an investigation tool for aeronautical certification process. From the velocity and temperature chosen in the previous section (19.5 m/s and 1373 K), it is possible to get the temperature evolution of the water at different times of simulation when regarding the calibration phase. Figure 7 exhibits the numerical evolution of the water temperature as well as the experimental one for this preliminary step of the tests. As one can see from this Figure 7, the computed water temperature is stabilised at 305.9 K after 50 s. The relative gap between the numerical and the experimental temperature elevations do not exceed 1% showing that the criterion imposed by the norms (ISO 2685 and FAA -AC20-135) is valid for the values of velocity and temperature chosen. Now regarding the full fireproof test, the temperature evolution of the two thermocouples in the internal surface of the plenum is are compared to the measured values (Figure 8). It is clear, reading this Figure 8 that the temperatures measured by the two thermocouples on the internal wall of the plenum are very close experimentally (discrepancies lower than 6%). They are also close to the computed values of the thermocouple 2 before 30 s and of the thermocouple 1 after 40 s, not to be unnoticed that the gap between the numerical temperatures increases with the time. The relative error calculated between the experimental and numerical results for the thermocouple 1 is about 19 % whereas the one for thermocouple 2 is around 13 %. These errors could be due to three reasons. First, the approximation on the thermal properties of the carbon-phenolic material used to design the plenum. Indeed, there are no values in the literature on the thermal conductivity and specific heat of the studied material. and the only possibility For this early step of the study, it was therefore decided to use data of a similar material with an equivalent fibre volume fraction. For quantifying the effect of properties, a 50% variation of each physical property was tested. It was found for example that 10% of variation on the specific heat conduct to 10% of variation on the temperature. Second, the simplification of the flame by an inert hot jet which leads to different air temperature fields surrounding the plenum. Indeed, modelling the entire flame will probably show that combustion is not complete at the outlet of the burner and hence it continues near the plenum. Third, the unmodeled thermal degradation of the carbon-phenolic material. Indeed, over 220°C, it was found through some preliminary tests that a mass loss of 1 wt.% is measured. It is expected that over 700°C, the matrix of the composite get completely pyrolysed and/or burned. In spite of these relative errors between experimental and numerical results, the present thermal study gives a first good evaluation of the material temperature and can be used as first indicator in the design process. In some defined real operating conditions, the APU is flown by air circulation that could decrease the maximum temperature found on the plenum. Thus, it is decided to examine whether an air injection in the internal part of the plenum could have a beneficial effect and hence if certification rules should be adapted in the future. Figures 9 and 10 present the computed temperature evolutions of the two thermocouples for different air jet velocities as well as the experimental ones of the thermally protected same studied plenum but equipped with an additional thermal protection. These Figures show the influence of the air jet on the internal temperature approximately after 100 s and both profiles have very different evolutions. Numerical values of TC 1 have a rapid increase before 100 s accompanied by a stabilisation phase during the remaining computational times for different jet air speeds. Compared to numerical data, the experimental temperature of TC 1 exhibits a slower evolution. The increase of this air jet air velocity leads to a decrease of the TC 1 temperature. Before 150 s all computed temperatures of TC 1 are higher than the measured one. After this time, numerical temperatures of TC 1 for air jet velocities of 0 and 1 m/s are higher than experimental data whereas they are lower for 5 and 10 m/s (cf. Figure 8). Table 2 presents the final TC 1 temperatures reached at the stabilisation phase for different velocities and their reduction percentages calculated from the initial case (i.e. without air jet). From this Table it can be seen that an increase of the velocity from 1 to 5 m/s induces a final TC1 temperature 1.25 times lower. At the same time an increase of the velocity from 1 to 10 m/s decreases the TC1 temperature by 1.38 times.

Concerning the TC 2 temperature, numerical data show a different behaviour compared to the TC1 temperatures. The evolution can be divided into three phases: a fast increase until 100 s, a decrease between 100 and 300 s and a stabilisation from 300 s till the end of the calculation times. Like the computed TC 1 temperature, the final temperature decrease with the increase of the jet air velocity. However, measured TC 2 temperature present a similar tendency that of TC 1 temperature (cf. Figure 10). Despite the fact that TC1 and TC2 are placed symmetrically for the plenum, a slight temperature shift of about 100 K is found between both because the burner does not impact symmetrically the plenum but it is slightly shifter near TC2. This explains why TC2 receive a higher heat flux from the flame than TC1. Table 3 present the stabilised temperature values and the relative reduction as functions of the jet air velocities. From 1 to 5 m/s, the final computed TC 2 temperature is 1.2 time lower. For an increase of the velocity from 1 to 10 m/s, the computed TC 2 temperature is decreased by 1.3 times. From the above results, it is clear that injecting a defined air flow level an air injection in the internal wall of the plenum seems would facilitate the contributes to lower the part material temperatures (backside part inner surface). This is beneficial to reduce the risk of released smoke and part backside to ignite during the test flame application time, which is a fail criteria for "fireproof" requirement as defined required in the ISO 2685 and FAA -AC20-135 standards.

Conclusion

The numerical study conducted on the thermal behaviour of a composite plenum made of carbonphenolic material presents a good evaluation of the heating process compared to the experimental measurements performed on a composite part tested according to the AC20-135 and ISO 2685 standards compared to the experimental measurements. Also, it has been shown that the air jet in the internal part of the plenum leads to the diminution of the wall temperatures showing its benefit effect. The authors anticipate performing a more reliable analysis with more accurate thermal properties of the studied carbon-phenolic material used to design the plenum as well as additional studies on other materials. To do so, an extensive testing of materials under various stress conditions is expected to get the density, specific heat, thermal conductivity, emissivity and heat release rate as a function of time and of temperature. A better modelling of the flame (with combustion description) and of the thermal degradation of the carbon-phenolic material (Arrhenius law with release of products and modification of properties) are also expected within the next two years. In future works, a new metrology will be setup on the experimental test bench with possibility of gas sampling for chemical analysis and an in-house heat flux sensor [START_REF] Chetehouna | On the experimental determination of flame front positions and of propagation parameters for a fire[END_REF] coupled to image processing from previous studies [START_REF] Rudz | Metrological Tool for the Characterization of Flame Fronts Based on the Coupling of a Heat Flux Approach with Image Processing Data[END_REF]. This metrology will allow a better understanding of the different phenomena and will enable the improvement of the numerical model.

Figure 1 :

 1 Figure 1 : Photograph of the fire test bench.

Figure 2 :

 2 Figure 2: Schematic overview of the experimental setup.

Figure 3 :

 3 Figure 3: Overview of the TC 1 and TC 2 thermocouple locations.

Figure 4 :

 4 Figure 4: Computational domain for the calibration procedure.

Figure 5 :

 5 Figure 5 : Overview of the burner, the plenum and the air injector.

Figure 6 :

 6 Figure 6: Illustration of the computational grid.

Figure 7 :

 7 Figure 7: Experimental and numerical water temperature evolutions.

Figure 8 :

 8 Figure 8: Experimental and numerical temperatures on the internal wall of the heated plenum.

Figure 9 :

 9 Figure 9: TC 1 temperature evolution in the cooling configuration Table 2: Effect of the air jet air velocity on TC 1 computed temperature.Jet velocity (m/s)

Figure 10 :

 10 Figure 10: TC 2 temperature evolution in the cooling configuration.

Table 1 :

 1 Jet velocity and flow rate.

Table 3 :

 3 Effect of the jet air velocity on TC 2 computed temperature.

	Jet velocity	Final T (K)	Final T Reduction
	(m/s)		(%)
	0	624	-
	1	448	28.2
	5	386	38.1
	10	338	45.8

Acknowledgements

This research has been done under the support of the PCT² project which is partially funded by the Conseil Général du Loir-et-Cher.