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

Nicolas Gascoin

Philippe Gillard

Neural-network metamodelling for the prediction of the pressure drop of a fluid passing through metallic porous medium

Keywords: Simulation metamodelling, Artificial neural networks, Backpropagation, Permeation, Porous medium, SCRamjet cooling, numerical simulation

The pressure drop across metallic porous medium is a critical element in cooling aerospace engineering application. This paper presents a metamodel based on artificial neural networks (ANN) for estimating the pressure drop through metallic porous media. The ANN is developed using experimental data obtained from an experimental bench, developed at PRISME laboratory, which ensure the monitoring of temperature, pressure and mass flow rate in stationary and transient conditions. For each case the gas pressure which crosses the metallic porous material is measured as a function of inlet gas pressure, gas mass flow rate and temperature. The optimal feedforward ANN architecture with error backpropagation (BPNN) was determined by the cross validation method. The ANN architecture having 35 hidden neurons gives the best choice. Comparing the modelled values by ANN with the experimental data indicates that neural network model provide accurate results. The performance of the ANN model is compared with a metamodelling method using multilinear regression approximation.

INTRODUCTION

One of the many issues encountered in the development of future hypersonic vehicles is the problem of protecting the engines and airframe from high heat loads encountered at hypersonic speeds. At flight speeds near Mach 4, the air taken on board these vehicles will be too hot to cool the engines and airframe. Therefore, it will be necessary to study and develop other light weight and high-temperature materials. Among the materials which are intended to be used for high-speed flight are the metallic and the composite ones. The materials and cooling techniques and their interaction with aero-thermal loads must be addressed.

In the literature, we can found different studies in relationship with this need, experimentally [START_REF] Langener | Experimental investigations on transpiration cooling for scramjet applications using different coolants[END_REF][START_REF] Gascoin | Comparaison of two permeation test benches and two determination methods for Darcy's and Forchheimer's permeabilities[END_REF] or numerically [START_REF] Jaber | Three-dimensional study of permeability effect on convection in heterogeneous porous medium filled with a ternary hydrocarbon mixture[END_REF][START_REF] Shahnazari | Permeability prediction of porous media with variable porosity by investigation of Stokes flowover multi-particles[END_REF]; even mathematically [START_REF] Kim | Fully Discrete Mixed Finite Element Approximations for Non-Darcy Flows in Porous Media[END_REF][START_REF] Martin | Simulation of pyrolysis gas within a thermal protection system[END_REF]. Such studies are not only dedicated to the flow description but also to the heat transfers [START_REF] Kim | Fully Discrete Mixed Finite Element Approximations for Non-Darcy Flows in Porous Media[END_REF][START_REF] Krishna | Natural convection in a heat generating hydrodynamically and thermally anisotropic non-Darcy porous medium[END_REF][START_REF] Younis | Cross-flow heat exchanger embedded within a porous medium[END_REF]; for example for configurations close to fuel cooled structures [START_REF] Zhao | Non-Darcy effects on nonparallel thermal instability of horizontal natural convection flow[END_REF] and also for geophysics applications [START_REF] Hadim | Numerical Study of Non-Darcy Mixed Convection in a Vertical Porous Channel[END_REF]. The flows in porous media are widely studied under common operating conditions.

A lot of studies, often under high pressure (up to 2MPa) are available for ambient conditions or average temperature conditions (under 800K) (Park and Lawrence, 2003;[START_REF] Langener | Experimental investigations on transpiration cooling for scramjet applications using different coolants[END_REF] or for low pressure and high temperatures [START_REF] Kuhn | Application of transpiration cooling for hot structures, RESPACE: Key Technologies for Reusable Space Systems[END_REF]. But only few are dedicated to both high temperature and high pressure conditions. Numerous equations (derived from Brinkman's equation) which relate pressure drop (ΔP=P in -P out )

through the porous media to the gas velocity have been published [START_REF] Martin | Simulation of pyrolysis gas within a thermal protection system[END_REF][START_REF] Khan | Second law analysis for mixed convection in non-Newtonian fluids over a horizontal plate embedded in a porous medium[END_REF][START_REF] Gascoin | Comparaison of two permeation test benches and two determination methods for Darcy's and Forchheimer's permeabilities[END_REF]. They are based on coefficient, whose physical meaning is not evident. Furthermore, the dynamic viscosity, which appears in these equations, is often considered to be that of the fluid. It remains constant in most of experimental and numerical works. Nevertheless, this is questionable because this parameter varies across the porous medium in the case of reactive flow or a non-uniform temperature system.

In this paper, we have used a simulation metamodelling to better simulate the pressure drop of nitrogen passing through the porous media (Stainless Steel) by taking into account both high temperature and high pressure conditions. Simulation metamodelling is based on the substitution of the complex simulation model by an approximation of the input-output relationship. Metamodelling, first proposed by [START_REF] Blanning | The construction and implementation of metamodels[END_REF], makes the computations much faster, allowing for more cases to be studied. This wider exploration of the input variables improves the understanding of the model and it permits to carry out otherwise time consuming sensitivity analysis or solution optimization [START_REF] Broad | A metamodelling approach to water distribution system optimisation[END_REF][START_REF] Kleijnen | Constrained optimization in expensive simulation: Novel approach[END_REF], at the cost however of a lower accuracy of the outputs. Originally based on regression methods [START_REF] Cohen | Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences[END_REF]Freedman, 2008), metamodels now use various approaches such as artificial neural networks (ANN) or kriging [START_REF] Kleijnen | Kriging metamodeling in simulation: A review[END_REF][START_REF] Ankenman | Stochastic kriging for simulation metamodeling[END_REF]. ANN has the advantage over regression that the form of the model needs not to be pre-determined.

Moreover, ANN can theoretically approximate any function to any level of accuracy [START_REF] El Tabach | Use of artificial neural network simulation metamodelling to assess groundwater contamination in a road project[END_REF], which is very interesting when the governing physical mechanisms are highly non-linear like in high velocity fluid flow in porous medium. The database was built with three input parameters (inlet gas mass flow rate, inlet gas pressure and the temperature)

and the outlet gas pressure as output parameters, using the results coming from an experimental bench. An artificial neural network has been trained and tested on this database using the error backpropagation algorithm and cross validation.

THE EXPERIMENTAL BENCH AND SPECIFIC SET-UP

The COMPARER pyrolysis test bench is used to pressurize and to heat the fuel under flow conditions [START_REF] Gascoin | High temperature and pressure reactive flows through porous media[END_REF]. Its main characteristics are the following:

 Maximum operating conditions: 1800K, 8MPa, 0.0006 kg.s -1 for liquid fluid and 0.006 kg.s -1 for gas.

 Sensors: 5 mass flow rates, 5 pressure transducers, over 10 K-type and R-type thermocouples with data acquisition system (16 bits, 48 channels, 0.1 Hz).

A permeation test cell contains the porous sample (FIG. 1). This cell is inserted inside the furnace of the COMPARER bench and it is connected to the fluid supply system and to the appropriate sensors. The permeable medium bounds the cell in two high and low pressure chambers (upstream and downstream to the porous medium respectively). These chambers are noted HPC and LPC in this paper. An inlet pipe provides the fuel into the system. This cell is connected to a dynamic sampling system to get hot pressurized samples at three location points in the cell. Despite its small size (external diameter of 40mm), it enables measuring the temperature, pressure and mass flow rate on each side of the porous sample.

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The permeable medium can be changed. In the present work, an isotropic stainless steel material is preferred to composite one to avoid considering complex microstructure (fibres, layers). It is characterized by a porosity around 30 %, a grain diameter of 14.1 µm and a pore diameter of 4.1 µm [START_REF] Gascoin | Comparaison of two permeation test benches and two determination methods for Darcy's and Forchheimer's permeabilities[END_REF]. Further geometrical information can be found in [START_REF] Gascoin | Comparaison of two permeation test benches and two determination methods for Darcy's and Forchheimer's permeabilities[END_REF].

FIG. 1: Schematic of the permeation test cell with porous medium and associated measures [START_REF] Gascoin | High temperature and pressure reactive flows through porous media[END_REF].

The different test conditions and most of the possible combinations have been achieved:

 Temperature set-up: from 300K to 1200K by step of 100K.

 Absolute pressure: from 0.05MPa (depressurized system) to 6MPa.

 Two experimental protocols have been tested: one by measuring the mass flow rate and changing the upstream pressure and a second by measuring the pressure drop and changing the upstream mass flow rate.

CONSTRUCTION OF THE ANN METAMODEL

Artificial neural networks

The metamodel adopted in this paper is based upon the use of artificial neural networks (ANN). ANN operate like a "black box" model, requiring no detailed information about the physical parameters of the system. Instead, they learn the relationship between the input and output parameters as a result of training with previously recorded data. ANN can handle large and complex systems with many interrelated parameters.

Construction of the database

The database was built using experimental data which are obtained from the developed experimental bench with input parameters: gas inlet pressure (P in ), inlet gas mass flow rate (q in ) and temperature (T) varying in a range of representative values: 35 values between 0.1MPa and 1.3MPa for P in ; 35 values between 0.00008 kg/s and 0.0011 kg/s for q in and 10 values between 290K and 1174K for T. In total; the database contains an appreciable size of 350 experimental tests. Note also that each input or output parameter has been normalized relative to its minimum and maximum values observed in the data, allowing for faster training by preventing larger numbers from overriding smaller ones. Normalization of an arbitrary parameter, X, can be carried out using Eq. 1.

    min max min norm X X X X X    (1)
where X norm is the normalized value, and X max and X min are the maximum and minimum values of X.

Construction of artificial neural network models

A brief description of the model and program used in this study will be given in this section. For a detailed description, evaluation, and discussion on the overall performance of the model, readers are referred to [START_REF] Najjar | Computational neural networks for predictive microbiology: i. Methodology[END_REF].

ANN are composed of a set of elements of calculation (nodes) connected to each other.

The most popular type of network is the multilayer backpropagation neural network (BPNN) which is used in the present study. The architecture of a typical 3-layer backpropagation neural network is shown in Fig. 2. Mathematically, a 3-layer ANN with n, m, and p the number of input, hidden and output nodes respectively, is based on the following equation:

                    m j n i i ij jk k X W S W S O 1 1 (2) 8/20
where O k are the output values and X i the input values of the network; W ij , the connection weights between the input layer and the hidden layer; W jk , the connection weights between the hidden layer and the output layer; S is a transfer function. The sigmoidal function (Eq. 3)

was used in the present study. The most common convergence criterion is the average squared error (ASE) defined as:

  2 s 1 q p 1 k qk qk O t s 1 p 1 ASE         (4)
where O qk ant t qk are respectively the predicted and target value of the output node k for the pattern q, s is the number of patterns, and p is the number of output nodes. It should be noted that any level of agreement between predicted and target vectors can be achieved by providing a sufficient number of training cycles to be carried out. Such an overtraining is however detrimental to the capacity of the network to generalize from unseen data (a network that can accurately predict the output of the testing patterns is said to have generalized).

Optimum network architecture

The overall performance of an ANN is very dependent on the number of hidden layers and of hidden nodes in each hidden layer. In the usual case of a 3-layer BPNN (one hidden layer), the optimum number of hidden nodes can be assessed by cross-validation in the same way as the optimum number of training cycles (Fig. 3). Starting with a few hidden nodes, more nodes are added up as the ASE computed on the testing patterns decreases. The number of hidden nodes beyond which the error starts to increase again is taken as the optimum.

In the present study, a neural network relating inputs {X 1 , X 2 , …, X n } to outputs {O 1 , O 2 , …, O p } and containing one hidden layer with m hidden nodes will be noted:

{O 1 , O 2 , …, O p } = ANN nmp {X 1 , X 2 , …, X n } (5)
In our case, the outlet gas pressure (P out ) is sought as a function of P in , q in and T. So, it is possible to compute P out by using a BPNN model with one node in the output layer (Eq. 6). It could be noticed that usually the pressure drop through the porous medium is investigated as In this present study, the choice of the outlet pressure is preferred to limit the impact of experimental uncertainties due to the two pressure transducers (which are generally multiplied if compared to a single transducer). In addition, since the upstream pressure is an inlet parameter, looking at the pressure drop or at the pressure outlet is equivalent when focusing on the behaviour of the ANN model. As can be observed in FIG.

4 the optimal value of ASE was calculated while using 35 nodes in the hidden layer for our model.

{P out } = ANN 3-35-1 {P in , q in , T} (6)
The ASE values for the training, testing and validation phases for the optimal artificial neural network model (ANN 3-35-1 ) are respectively unseen data (validation phase). On the same graphs the best fit line through the origin is also plotted and the coefficient of determination R 2 for this line is computed according to Eq. 7:

                  N i i i N i i i Y Y N Y Y N R 1 2 1 2 2 1 1 1 1 (7)
where N is the number of data, The evolution of the predicted outlet pressure (P out ) using BPNN as a function of temperature inside the porous medium corresponds to a decrease when the temperature increases (FIG. 7). The density decreases due to the thermal rise and since the mass flow rate inside the medium is kept constant, the mean fluid velocity increases. Thus, the outlet pressure decreases; which means that the pressure drop increases (for a given curve corresponding to a fixed inlet pressure). This is clearly understandable when paying attention to the Brinkman equation. It is thus very important to note that the ANN approach is able to reproduce physical variations. FIG. 7: Model 1 predicted P out variations with the temperature for q in = 1g/s.

In addition, it can be seen that the same pressure drop (ΔP=P in -P out ) is found for several couples of temperature/inlet pressure (FIG. 8). For example a pressure drop of 0.2 MPa is reached for (P in =0.4 MPa and T=1173.15K) and for (P in =0.6 MPa and T=573.15 K). This means that the pressure is divided by a factor 2 at high temperature while for about half of this temperature, the pressure losses only 33%. To divide the inlet pressure of 0.6 MPa by a factor 2, the temperature should be about 873.15 K. As a consequence, it can be concluded physically, that the temperature has a higher effect than the pressure itself. The thermal effect much increases the fluid velocity within the porous media (more than what the pressure does). This is the thermodynamic consequence of pressure and temperature parameters on the fluid density. This could be a way to get information of phenomena within the porous media where no direct measure seems to be possible for the fluid properties. FIG. 8: Model 1 predicted ΔP variations with P in for q in = 1g/s.

CONCLUSIONS

A metamodel based on artificial neural networks has been used to predict the nitrogen pressure which crossed a metallic porous media (Stainless steel). Based on experimental data,

  each node in the previous layer (X i ) is multiplied by an adjustable connection weight (W ij ). At each node, the weighted input signals are summed and an adjustable threshold value (W j ) is added. This combined input A j is then passed through the non-linear transfer function S to produce the output of the node (O j ). The output of one node contributes to the input to the nodes in the next layer. This process is illustrated in Fig. 2. In the present study, all connection weights (W ij ) and the thresholds (W j ) are initialized to random values in the range of [+1, -1] (Najjar et al., 1997). The learning process of BPNN is based on a series of connection weight adjustments in order to minimize a global error between predicted outputs and target values. It relies on a search technique (e. g. gradient descent) of the connecting weights yielding a minimum error. Inputs are first propagated forward through each layer of the network. Errors between outputs and target values are then propagated backwards and the connection weights are modified according to a specific learning algorithm (delta rule) to reduce the overall error. This forward-backward process is carried out for each epoch (set of training patterns used to compute the global error), and is repeated until predicted outputs and target answers coincide within a given tolerance.

  FIG. 2: Architecture of a typical multilayer backpropagation artificial neural network

  the through flow rate.

  FIG. 4: Absolute squared error (ASE) variations with the number of hidden nodes for the

Y

  FIG. 5: Comparison between target and predicted values for P out using BPNN (model 1) and

  The present database was subdivided in three subsets. A first subset (174 experimental tests) is used to train the networks. A second one (88 experimental tests) is used to test the ANN models to determine when to stop the training stage. The third subset (88 experimental tests) is used to validate the performance of the selected model on unseen cases. It should be noted that the training subset should contain the widest variety of patterns, since ANN are more reliable if they are used as an interpolation tool for generalization on new cases.

Table 1 :

 1 R 2 values between target and predicted outputs for all models

	R 2	Neural network (model 1)	Multiple linear regression (model 2)
	Training phase	0.999	0.682
	Testing phase	0.997	0.639
	Validation phase	0.996	0.592

ACKNOWLEGEMENTS

The authors would like to sincerely thank Fau G. from PRISME and Blanc D. from IUT Bourges for their technical and scientific support involving the computations and the experiments.

given in Table 1. The multiple linear regression model fails to predict the variations of P out .

The poor performance of the linear model is not surprising: the physical phenomena captured in the database are quite complex and highly non linear. In artificial neural networks nonlinearity is accounted for by the use of transfer functions (Eq. 3), while complexity can be controlled by varying the number of hidden nodes. Artificial neural networks appears as a powerful prediction tool in the present case.

the optimum architecture of artificial neural network was trained and validated, in order to generalise the prediction of gas pressure which crossed the studied porous media to cases not included in the database. The validation showed excellent performance of this metamodel for the prediction of gas transfer in the porous media.

This study is a contribution to the growing evidence of the benefits of simulation metamodels in Aerospace engineering. This important result may be applied to automate P out estimations which utilized in space flight applications without prior knowledge of correct material parameters.


M Sahli

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

F Faubert

C Bariki

N Gascoin

N Bellel

Experimental investigation on the concentration and voltage effects on the characteristics of deposited magnesium-lanthanum powder

Keywords: Chemical synthesis, Electrodeposition, Mg-La powders, Voltage effect, Concentration effect, Design of experiments

Experimental investigation on the concentration and voltage effects on the characteristics of deposited magnesium-lanthanum powder

Introduction

Since the last decades, magnesium (Mg) is attracting renewed interest. It is a potential candidate for hydrogen storage, as evidenced by numerous studies and publications on this topic [START_REF] Zaldívar-Cadena | Dispersion of niquel on the microstructure in magnesium based alloys for hydrogen storage[END_REF][START_REF] Skripnyuk | Improving hydrogen storage properties of magnesium based alloys by equal channel angular pressing[END_REF][START_REF] Cui | Synthesis and characterization of nanocrystalline magnesiumbased hydrogen storage alloy electrode materials[END_REF][START_REF] Makridis | Polymer-stable magnesium nanocomposites prepared by laser ablation for efficient hydrogen storage[END_REF][START_REF] Crivello | Improvement of Mg-Al alloys for hydrogen storage applications[END_REF]. Indeed, it is characterized by several advantages. Being a light element allows Mg to have the highest weight capacity 7.6% [START_REF] Pohlmann | Magnesium alloy-graphite composites with tailored heat conduction properties for hydrogen storage applications[END_REF]. In addition, it is characterized by various properties for instance a low density, a low cost [START_REF] Zhao | Reversibility of electrochemical magnesium deposition from tetrahydrofuran solutions containing pyrrolidinyl magnesium halide[END_REF], a very good malleability and ductility, a high strength/weight ratio and an environmentally friendly nature [START_REF] Mordike | Magnesium: Propertiesapplicationspotential[END_REF]. The use of magnesium is found also limited due to its high chemical reactivity and electronegative potential of -2.34

V [START_REF] Rosalbino | Electrochemical behaviour assessment of novel Mg-rich Mg-Al-RE alloys (RE = Ce, Er)[END_REF]. Given its poor mechanical properties and low corrosion resistance, magnesium cannot be used in pure form as structural material [START_REF] Bornapour | Magnesium implant alloy with low levels of strontium and calcium: The third element effect and phase selection improve bio-corrosion resistance and mechanical performance[END_REF]. Hence the need to combine it with other elements in order to optimize these two fundamental characteristics. In fact, adding rare earth (RE) elements as lanthanum, element retained in this work, improve the corrosion resistance, the creep and shear strengths as well as the hydrogen storage capacity [START_REF] Grillo | Cellular response to rare earth mixtures (La and Gd) as components of degradable Mg alloys for medical applications[END_REF][START_REF] Liu | Effect of rare earth element Ce and La on corrosion behavior of AM60 magnesium alloy[END_REF][START_REF] Chu | Structure, morphology and hydrogen storage properties of composites prepared by ball milling Ti 0.9 Zr 0.2 Mn 1.5 Cr 0.3 V 0.3 with La-Mg-based alloy[END_REF]. Magnesium alloys have been therefore widely used for structural compounds for several decades, particularly in aircraft and automobile industries [START_REF] Yang | Electrodeposition of magnesium-yttrium alloys by molten salt electrolysis[END_REF][START_REF] Hamid | Synthesis and protection of AM50 magnesium alloy and its composites using environmentally pretreatment electrolyte[END_REF]. At present, different technologies are utilized to prepare Mg based alloy, including thermal spraying [START_REF] Parco | Investigation of HVOF spraying on magnesium alloys[END_REF], diffusion [START_REF] Liu | Improvement of corrosion resistance of pure magnesium via vacuum pack treatment[END_REF], laser cladding [START_REF] Ignat | Magnesium alloys laser (Nd:YAG) cladding and alloying with side injection of aluminium powder[END_REF], CVD (Chemical Vapor Deposition) [START_REF] Ch | Deposition of aluminium on magnesium by a CVD process[END_REF], etc. Besides these methods, metal alloys electrodeposition in aqueous ionic liquids are receiving increased interest [START_REF] Nuli | Electrodeposition of magnesium film from BMIMBF 4 ionic liquid[END_REF][START_REF] Liu | Electrodeposition of Al on AZ31 magnesium alloy in TMPAC-AlCl 3 ionic liquids[END_REF][START_REF] Wang | Electrodeposition mechanism and characterization of Ni-Cu alloy coatings from a eutectic-based ionic liquid[END_REF][START_REF] Bakkar | Electrodeposition onto magnesium in air and water stable ionic liquids: From corrosion to successful plating[END_REF]. The thickness and quality of the deposited layer can be controlled or optimized by processing parameters, such as electroplating time, voltage of electrodeposition or electrolyte composition, etc. At the same time, it is the most economical method for preparing Mg-RE alloys and could be easily implemented on a large scale as well as it could be employed over a large temperature range.

The aim of the work presented in this paper is to study the characteristics of the magnesium-lanthanum layers dedicated to the hydrogen storage applications. These layers are elaborated using electrodeposition technique by varying the voltage and the amount of lanthanum in the initial chemical solution. The effects of these two processing parameters on the products formation are investigated through a design of experiments. This paper is organized as follows: section 2 is devoted to the description of samples preparation and characterization devices as well as design of experiments; section 3 examines the results obtained monitoring with discussions based to the ones found in literature. In the end, conclusions are drawn with some indicators to future analyzes.

Materials and methods

Chemicals and specimens preparation

Different masses of MgCl The concentration effect was performed by varying the concentration of La from 20 to 60 wt.% with a step of 20 wt.%. The different samples were prepared under three different voltage values: 6, 8 and 10 V. The deposition experiments were performed at ambient temperature. The obtained deposits were scraped from the copper cathode plate, dried for 1 h at 110 °C, milled manually in a marble mortar and subjected to further analyses. Figure 1 shows a photograph and a schematic overview of the described experimental setup.

Figure 1 -Picture and schematic overview of electrodeposition experimental setup.

Characterization devices

The morphology and chemical composition of the obtained deposits were investigated by a scanning electron microscope (SEM, LEO 1455 VP, acceleration voltage of 20 kV) at a working distance of 15 mm equipped with an X-ray energy dispersive spectrometer (EDS, Inca X-sight, Oxford Instrument). The phase identification of the alloys were determined by means of X-ray diffractometer (XRD, Philips X' PERT) using Cu Kα radiation source (λ=1.54 Å, 40 kV, 30 mA) at room temperature. The scanning rate of 0.02 °/s within the 2θ range going from 10 ° to 80 ° was used in order to increase counting statistics and to improve the signal/noise ratio. Fourier transform infrared (FTIR) spectroscopy of the powder samples was carried out with a Nicolet 380 FT-IR spectrometer using the ATR technique. All spectra were collected with the resolution of 4 cm -1 in the range of 4000-500 cm -1 .

Design of experiments

Design of experiments, referred to as DOE, is a systematic approach to understanding how parameters affect response variables. DOE is considered one of the most important methodologies for researchers handling experiments in practical applications [START_REF] Gascoin | Characterisation of oxidised aluminium powder: Validation of a new anodic oxidation bench[END_REF][START_REF] Khan | Ultrasound-assisted extraction for total sulphur measurement in mine tailings[END_REF][START_REF] Beltrán-Heredia | Optimization of the synthesis of a new coagulant from a tannin extract[END_REF] because it provides an organized approach, with which it is possible to address both simple and tricky experimental problems. Thus, by means of DOE, one obtains more useful and more precise information about the studied system, because the joint influence of factors is assessed, which will allow to decide what the next experimental step ought to be.

Results and discussion

EDS analyses of the as-deposited layers

The EDS results presented in Table 2 are calculated as the average of three measurements on three different areas, performed using spot mode, in order to obtain the elemental atomic compositions of the different powders. These results clearly show the presence of oxygen, magnesium and lanthanum originating from the used electrolyte. As we can see from this A SEM investigation is carried out in order to study the morphological behavior of the Mg-La deposited powders. Figure 2 shows the SEM pictures of the samples for different voltages as function of the La concentration. Two kind of parts could be clearly distinguished; the dark and the bright ones. The dark aggregates are attributed to magnesium phase [START_REF] Tang | Fabrication of Mg-Pr and Mg-Li-Pr alloys by electrochemicalco-reduction from their molten chlorides[END_REF][START_REF] Wei | Electrochemical codeposition of Mg-Li-Gd alloys from LiCl-KCl-MgCl 2 -Gd 2 O 3 melts[END_REF] and the bright ones are assigned to lanthanum [START_REF] Gröbner | Phase analysis of Mg-La-Nd and Mg-La-Ce alloys[END_REF]. The distribution of the Mg and La elements is not uniform due to the heterogeneous chemical structures of the powders. Figure 2 shows also when the voltage increases, the growth of the formed seeds going to increase and their number tends to reduce for each La weight percentage. Indeed, the increase of current density (i.e. decrease of voltage in this study) leads to a remoteness from the equilibrium voltage according to Tafel's law. This leads an increase in the polarization and therefore an increasing in the number of seeds. For each voltage value, the increase of La weight percentage increases the concentration of the electroactive ions leading to a decrease in the germination speed.

Consequently, the growth of the formed seeds tends to increase allowing the aggregation of large seeds for higher content of La.

Figure 2 -SEM images of the as-deposited powders for different voltages and lanthanum concentrations.

X-ray diffraction analyses of the as-prepared powders

XRD diffractograms of the as-deposited layers were measured and are given in Figure 3.

According to the standard data JCPDS, the obtained deposits can be readily indexed to two phases, magnesium hydroxide (Mg(OH) 2 ) and lanthanum hydroxide (La(OH) 3

FTIR spectroscopy analyses of the as-deposited layers

In addition to the X-ray diffraction analyses, the FTIR spectroscopy study is primary for the structural confirmation of the as-prepared powders. Figure 4 shows the FTIR transition spectra of the as-deposited Mg-La powders in the range of 4000-500 cm -1 for the different samples. One can see from this Figure a sharp and strong peak at 3696 cm -1 which corresponds to deformation of OH bonds in Mg(OH) 2 as reported by Jianjun et al. [START_REF] Jianjun | The gemmological properties and infrared spectra of brucite, an imitation of nephrite and Shoushan stone[END_REF] and Gray-Munro and Strong [START_REF] Gray-Munro | A study on the interfacial chemistry of magnesium hydroxide surfaces in aqueous phosphate solutions: Influence of Ca 2+ , Cland protein[END_REF]. The band centred on 3606 cm -1 is assigned for stretching mode of OH -in lanthanum hydroxide [START_REF] Aghazadeh | A novel lanthanum hydroxide nanostructure prepared by cathodic electrodeposition[END_REF][START_REF] Salavati-Niasari | Synthesis of lanthanum hydroxide and lanthanum oxide nanoparticles by sonochemical method[END_REF]. The bands at 3455 cm -1 and 1640 cm -1 are attributed to the hydroxyl groups in water [START_REF] Aghazadeh | A novel lanthanum hydroxide nanostructure prepared by cathodic electrodeposition[END_REF][START_REF] Khosrow-Pour | Large scale and uniform La(OH) 3 nanorods prepared by template-free pulsed electrodeposition method[END_REF]. The two others distinct bands at about 1434 cm -1 and 1048 cm -1 can be associated to asymmetric stretching vibrations of the carbonate group, which originate from the reaction of the as-deposited powders with CO 2 from air during the FTIR spectroscopy analysis [START_REF] Jianjun | The gemmological properties and infrared spectra of brucite, an imitation of nephrite and Shoushan stone[END_REF][START_REF] Aghazadeh | A novel lanthanum hydroxide nanostructure prepared by cathodic electrodeposition[END_REF][START_REF] Khosrow-Pour | Large scale and uniform La(OH) 3 nanorods prepared by template-free pulsed electrodeposition method[END_REF]. Regarding the concentration effect at 6 V on the different detected bands, one should note that there is an increase in transmittance values with increasing La weight percentage for the bands of Mg(OH) )

Figure 4 -FTIR transition spectra of samples for different voltages and lanthanum concentrations.

Quantification of processing parameters effects

The data of EDS analyses have been then post-processed to evaluate the Mg(OH) 2 content according to the design of experiments. The two processing parameters (La weight percentage and voltage) are considered with their three levels noted "-1" for low level, "0" for center level and "+1" for high level (with the values given by Table 1). Table 3 summarize the number of experiment, the values taken by the La weight percentage (A), the voltage (B), their squares (A 2 , B 2 ) and their interaction AB as well as the Mg(OH) 2 amount. The values taken by AB are the scalar products results between those of A and B.

Computer generated non-linear quadratic models are given as: The goodness-of-fit of the relation ( 1) is estimated with the coefficient of determination, and its value is 2 0.8981 R .

It appears that the voltage is the most influencing parameter with an effect of 0.0214.

Indeed, the effect of voltage is about 18 times more important than the one of the La weight percentage. Moreover, it is only about 1.25 times more significant than the interaction parameter AB leading to an effect higher than the one of the La weight percentage. The square of factor A has a positive impact, while B 2 has a negative impact on the response. Let us notice also that the quadratic terms have the same order of magnitude of the parameters B and AB. Similar tendency concerning parameter influences ranking has been observed by

Conclusion

The present study deals with the synthesis of magnesium-lanthanum powders thanks to an electrodeposition process at ambient temperature. The effects of voltage and La amount on the elemental composition, the morphological characteristics and the identified phases as well as the transmittance bands have been investigated. The deposited powders are composed of Mg(OH) 2 and La(OH) 3 phases. A design of experiments was made in order to study the effects of the two processing parameters. In a future stage of this work, other parameters such as electrolyte temperature, pH and cathode nature will be treated to determine the most influential ones by using a complete design of experiments. The effect of a pressurized environment on the preparation of samples as well as a quantitative determination of the content of each powder will be also investigated. As a final point, this study is a preliminary

Figure 3 -

 3 Figure 3 -X-ray diffraction patterns of samples for different voltages and lanthanum

  c B 8V are the real values of the independent variables at the center point and A 20% and B 2 V are respectively the step changes of variables A and B.

Table 1 -

 1 Table 1 represents a threelevel design of experiments proposed to determine the influence of the two parameters (La concentration and electrodepositing voltage) on the products formation. Levels and parameters of the design of experiments.

	Level of parameter	Concentration of La [wt.%]	Voltage [V]
	+1	60	10
	0	40	8
	-1	20	6

Table ,

 , all specimens are characterized by high contents of oxygen (51 to 74.2 at.%). Except for La weight percentage of 20% at 6V, the atomic percentages of Mg are always higher than those of La. These atomic percentages vary from 12 to 30.6 % for magnesium and from 5.4 to 24 % for lanthanum. For most of EDS analyses, it can be seen that the atomic percentages of O, Mg and La have a non-monotonic behavior with the increase of voltage and La weight percentage. This can be explained by the interaction between the voltage (which presents the driving force of the reaction) and the electrolyte composition effects in the electrolysis process.

			Atomic percentage [%]
	Element
		La 20 wt.%	La 40 wt.%	La 60 wt.%
	Voltage of 6 V
	Oxygen	69.2 ± 4.2	74.2 ± 4.6	51.0 ± 5.6
	Magnesium	12.0 ± 1.3	15.3 ± 1.4	25.0 ± 4.4
	Lanthanum	18.8 ± 3.9	10.5 ± 2.1	24.0 ± 3.7
	Voltage of 8 V
	Oxygen	69.5 ± 1.7	62.6 ± 4.2	69.4 ± 3.1
	Magnesium	25.1 ± 4.1	22.9 ± 3.2	25.1 ± 3.4
	Lanthanum	5.4 ± 1.3	14.5 ± 1.6	5.5 ± 1.4
	Voltage of 10 V
	Oxygen	57.7 ± 3.0	62.6 ± 1.4	67.6 ± 2.4
	Magnesium	30.6 ± 5.6	22.9 ± 6.1	22.4 ± 1.6
	Lanthanum	11.7 ± 4.3	14.5 ± 1.6	10.0 ± 1.5

Table 2 -

 2 Chemical composition of the as-deposited powders for different voltages and lanthanum concentrations.

  and 54.9 ° are attributed to La(OH)[START_REF] Cui | Synthesis and characterization of nanocrystalline magnesiumbased hydrogen storage alloy electrode materials[END_REF] . For La percentages of 20 and 40 wt.%, it can be observed that all peak intensities of La(OH) 3 phase decrease with increasing voltage to reach a minimum at 8 V and then increase when the voltage increases to 10 V. Similar tendency is observed for almost all peak intensities of Mg(OH) 2 for La percentage of 40 wt.%. We can also see that, for La weight percentages of 20 and 60%, almost peak intensities of Mg(OH) 2

	). As we can
	see from this Figure, peaks at diffraction angle 2θ of 18.6 °, 35.8 °, 37.9 °, 50.8 ° 58.7 ° and
	62.1° are associated to Mg(OH) 2 whereas those at 2θ of 15.6 °, 27.9 °, 39.3 °, 46.8 °, 48.5 °

phase increase by increasing the electrodeposition voltage. The same behavior is exhibited concerning the La(OH) 3 phase for a La weight percentage of 60%.

  2 , La(OH) 3 , H 2 O and CO 2 . Similar tendencies have been also remarked for bands of the water and the CO 3 2-group at 10 V. Non-monotonic behaviors at 8 V, characterized by a decreasing phase in the La concentration range of 20-40 wt.% and an increasing one in 40-60 wt.%, have been observed for the all detected bands. For the Mg(OH) 2 and La(OH) 3 phases, it can be seen also that the transmittance values increase in the La amount range of 20-40 wt.% and decrease in 40-60 wt.% at an electrodepositing voltage of 10 V.

				Voltage of 6 V						Voltage of 8 V
				La 20 wt.%							La 20 wt.%
	Transmittance (a.u.)	3606		La 40 wt.% La 60 wt.%	1434	1048	Transmittance (a.u.)		3606		La 40 wt.% La 60 wt.%	1434	1048
	3696					1640			3696				1640
		3500	3000	2500	2000	1500	1000	500	4000	3500	3000	2500	2000	1500	1000	500
				Wavenumber (cm	-1 )						Wavenumver (cm	-1 )
								Voltage of 10 V
								La 20 wt.%
				Transmittance (a.u.)		3606		La 40 wt.% La 60 wt.%	1434	1048
					3696				1640
					4000	3500	3000	2500	2000	1500	1000	500
								Wavenumber (cm	-1

Table 3 -

 3 Design of experiments matrix with Mg(OH) 2 content results.

	Trindade et al. [35].

work to figure out the usability of the obtained samples as potential materials for hydrogen storage.


G Fau

N Gascoin

Numerical Investigations of Chemical Kinetics for Methane Catalysis and Pyrolysis

In the framework of the active and regenerative cooling of hypersonic and space structures, a kinetic mechanism should be proposed to investigate numerically the fuel pyrolysis and the coke formation with a possible surface effect (catalytic reactions). This chemical mechanism should be suitable for Computational Fluid Dynamics approach (limited number of species and reaction). For this purpose, this work presents some homogeneous and heterogeneous calculations of methane chemistry with metallic catalysts in order to test different kinetic schemes. A parametric study is achieved and several reactors natures are tested.

Introduction

Hypersonic applications involve the management of a large thermal load applied to the whole structure and more specifically, the engine. The temperature overpasses 1600 K (Mach 6), which imposes an efficient method to cool the materials. In regards with this objective, the circulation at counterflow of an endothermic fuel is explored as a regenerative cooling system [START_REF] Gordon | Pyrolysis of methane flowing through a porcelain tube in the region 1000°[END_REF][START_REF] Billaud | Thermal coupling of methane in a tubular flow reactor: parametric study[END_REF]. The thermal degradation of the fuel would act as a heat sink before entering in the combustion chamber. The choice of the fuel is critical and numerous studies have been conducted

in order to explore the potential of several hydrocarbons [START_REF] Billaud | Thermal decomposition of pure methane at 1263 K. Experiments and mechanistic modelling[END_REF][START_REF] Davis | The use of methane as a fuel for hypersonic propulsion[END_REF][START_REF] Rice | The Thermal Decomposition of Organic Compounds from the Standpoint of Free Radicals. I. Saturated Hydrocarbons[END_REF][START_REF] Kossiakoff | Thermal decomposition of hydrocarbons, resonance stabilization and isomerization of free radicals[END_REF][START_REF] Doue | Mechanism of pyrolysis of some normal and branched C6-C9 alkanes. Composition of their pyrolysis products[END_REF]. Based on this bibliography, methane has been selected as a reference fuel for this work. Generally, the pyrolysis of the different fuel is realized in a reactor manufactured in stainless steel, alumina or other metallic materials. Sometimes, it is made with quartz which is an inert material and so prevent from potential reaction between the pyrolyzed fuel and the surface of the reactor. Indeed, it can be assumed that the molecule which compose the inner wall may react with the species present inside the reactor (moreover with the increase of temperature). It would be of great interest to review the studies which deal with that surface catalytic effect and, in particularly, the interaction between pyrolyzed methane and a composite inner wall. Unfortunately, there are a limited number of available works and no study corresponding to the specific conditions mentioned above was found. Consequently, the more appropriate works will be presented below, then several catalytic thermal decomposition studies will be discussed.

The objective of this study is to obtain a mechanism which permits to compute methane pyrolysis considering the surface catalytic effect. This one should be represented by the addition of about a few dozen of reactions.

Studies on the catalytic effect of the inner wall of a reactor are relatively uncommon and were essentially conducted in the seventies. The works of Crynes and/or Albright (et al.) are of prime importance [START_REF] Crynes | Pyrolysis of propane in tubular flow reactors. Kinetics and surface effects[END_REF][START_REF] Ghaly | Reactor Surface Effects During Propylene Pyrolysis[END_REF][START_REF] Dunkleman | Surface Effects During Pyrolysis of Ethane in Tubular Flow Reactors[END_REF][START_REF] Dunkleman | Pyrolysis of propane in tubular flow reactors constructed of different materials[END_REF] even if they deal with simple hydrocarbons (ethane, propane, propylene). Materials of the construction or surface-volume ratio (S/V) were considered as an influent factor on the conversion of hydrocarbon and product distribution.

In [START_REF] Crynes | Pyrolysis of propane in tubular flow reactors. Kinetics and surface effects[END_REF], they studied the pyrolysis of propane in some tubular flow reactors of different natures: stainless steel, low-carbon steel or nickel, treated or not by oxygen, hydrogen sulfide (after O 2 treatment) or steam. The experimental conditions were: temperature range of 873 K -1023 K, residence time of 2.8 s and atmospheric pressure. For untreated stainless steel reactors, methane and ethylene are the major products. No influence was demonstrated of the ratio variation S/V and a kinetic model was proposed. Nickel reactors tests lead to similar results. The wall effects have been highlighted for the treated inner surface.

Crynes, in association with Ghaly also explored the wall activity during the thermal cracking of propylene in a flow system [START_REF] Ghaly | Reactor Surface Effects During Propylene Pyrolysis[END_REF]. Seven tubular flow reactors (two stainless steel AISI 304, one low-carbon steel AISI 1015, one nickel, one inconel 600, two incoloy 801) were compared under the following conditions: 973 K-1123 K, atmospheric pressure and residence time of 0.1 s -3 s. For a reference reactor (stainless steel), the major products were methane and ethylene for a mid-conversion rate (20 %). The carbon yield was also evaluated and Crynes et al.

assumed that coke was directly formed by the interaction between propylene and the metallic surface. Indeed, they observed that the carbon yield at low conversion was higher than for inert surface. The authors also calculated the reaction rates and compare them with some precedent data. Even if some differences were noticed, the values were quietly similar. Considering the low-carbon steel reactor, wall activity increased during the first stage of pyrolysis (coke and hydrogen yield increased during the first 20 min) and then dropped and reached a steady-state after one hour. Reaction rates increased systematically in a range of 25-35 % higher. A quite similar profile was found for nickel reactor. The highest activity was observed at the beginning of the run (slightly higher conversion than reference) before decreasing progressively according to the increase of carbon deposition. At steady-state, propylene conversion was comparable to the reference one but the product distribution remained modified. Inconel and incoloy reactors presented a lower activity. For the first mentioned, a catalytic effect was observed for about 30 minutes with a maximum during the first two minutes. Only product distribution differed during this period and no modifications were noticed for larger time. The same results were found with a reactor made in incoloy with a slight higher activity period (40 minutes). To the authors, wall activity is essentially due to the presence of iron which is a catalyst for carbon and hydrogen production. However, alloys of iron generally contain a significant ratio of chromium which inhibits the activity of iron. So, surface containing active carbon layer such as low-carbon steel may present a higher activity by carrying iron atoms in this layer. Concerning nickel, it is assumed that the active sites have a lower affinity with hydrocarbons than iron. At last, alloys containing both iron and nickel may be less effective than pure metal so that explains the results obtained with inconel and incoloy.

The surface effect was also explored for the pyrolysis of ethane in several flow reactors (stainless steel, incoloy 800 and Vycor glass) [START_REF] Dunkleman | Surface Effects During Pyrolysis of Ethane in Tubular Flow Reactors[END_REF]. The experimental conditions were closed to the previous studies and surface different treatments have been done for several runs. Hydrogen and ethylene were the dominant decomposed species and coke was also widely represented.

Comparison between reactors showed that the ethylene yields is higher while hydrogen amount and coke formation are lower for Vycor glass than for metal. Furthermore, the composition of produced gases changed with the time excepted for the non-metallic reactors. Another observation, the treatment of the metal reactor had an impact on the product distribution as it was demonstrated in the precedent works. For example, a reduced inner surface leads to a product composition similar at the Vycor glass one. It was also noticed that the ethylene yields varied to the opposite of the hydrogen and coke ones.

Van Der Zwet et al. studied the role of the surface area for several different catalysts (essentially aluminum oxides) on the methane pyrolysis [START_REF] Vanderzwet | Pyrolysis of methane and the role of surface area[END_REF]. The experiments were performed in a non-reactive flow reactor made in quartz, at 1398 K and at atmospheric pressure. The authors confirmed that the ratio (S/V) had a clear impact on the product distribution. However, even if the product distribution was modified, the authors noted that the conversion of methane was poorly affected (increase only by a factor 10 when the S/V ratio is increased by 4 orders). To explain such facts, Van Der Zwet assumed that the radical or other dislocations sites of the surface may promote the hydrogen abstraction from CH4. Plus due to the autocatalysis phenomenon, the surface also may have an influence on the termination (by capturing from the gas phase the species involved in the autocatalysis). These two opposite effects lead to very complex surface area dependence. Finally, they suggested that a large catalytic surface is favorable to the production of coke and hydrogen while the gaseous and liquid pyrolyzed species are preferentially produced in a reactor with a low S/V ratio. They concluded that the nature of the catalyst has a poor impact in comparison with the surface influence.

The works of Li et al. could be also mentioned [START_REF] Li | Catalytic cracking of endothermic fuels in coated tube reactor[END_REF]. Their study consisted in coating the inner wall of a stainless steel tube by a mixture of different zeolites and a ceramic-like binder.

The pyrolysis of two jet fuels (n-dodecane and aviation fuel RP-3) were performed according to supercritical conditions (873 -898 K and 4 MPa) and continuous flow. Heat sink capacity and conversion rate were both explored. It was demonstrated that those two parameters were enhanced by the zeolite coating: increase of about 25 % for the heat sink capacity and the dodecane conversion pass from 66 to 80 %. Plus it was also observed that the coke deposit was lower in the case of RP-3 fuel.

Fang and Yeh [START_REF] Fang | Catalytic pyrolysis of methane[END_REF] proposed an extensive analysis considering numerous metal oxides catalysts [START_REF] Sinaki | Development of a reaction mechanism for predicting hydrogen production from homogeneous decomposition of methane[END_REF] deposed on a silica gel. Methane pyrolysis was performed at 1400 K and 2.10-3 MPa in a quartz reactor. 4 g. of catalysts was deposed inside it. The goal of this study was to produce C2 species from methane so that the catalytic C2 selectivity of the metals was quantified with regards to CH4 conversion rate while the dehydrogenation rate of methane served as activity reference. Fang and Yeh have then proposed a criterion to evaluate the catalysts.

A more recent study further explored the interaction between CH4 and the surface of the catalysts [START_REF] Ferreira-Aparicio | Methane interaction with silica and alumina supported metal catalysts[END_REF]. It exposed the effect of different metallic catalysts (cobalt, nickel, ruthenium, rhodium, iridium and platinum) supported on alumina or silica on the dehydrogenation of methane. A temperature programmed surface reaction (TPSR) of methane from 298 to 873 K (10 K.min-1) has been performed in parallel with a temperature programmed hydrogenation (TPH) of the carbonaceous adspecies.

The works of Zadeh and Smith are also of great interest notably for the numerical aspect.

They studied the pyrolysis of methane in presence of supported cobalt catalysts (Co-Sio2, K-Co-SiO2, Co-Al2O3 and K-Co-Al2O3) [START_REF] Zadeh | Kinetics of CH4 decomposition on supported cobalt catalysts[END_REF]. A mixture of CH4 and Ar (respectively 95 and 5 %) was injected in quartz fixed-bed microreactor heated at 723 K. In those conditions, they observed different phenomenon such as the metal support interaction (MSI) which consists on the migration of the SiO2 support on the prepared cobalt. The MSI impact is the increase of methane decomposition activity and the facilitation of the migration of carbonaceous species from catalyst surface to the support. This phenomenon is notably increased by the decrease of the initial Co loading. Temperature had also an effect: the higher it is the higher the CH4 conversion and the metal coverage. As it was mentioned in the precedent paper, the support had also an effect: The conversion is greater for an alumina support than a silica one.

Numerous other works on the decomposition of methane with metallic catalyst have been conducted those past years, notably according to three main objectives: the production of CO2free hydrogen; the production of carbon nanofibers and the production of heavier hydrocarbons (e.g. [START_REF] Choudhary | Nonoxidative Activation of Methane[END_REF][START_REF] Pinilla | Parametric study of the decomposition of methane using a NiCu/Al2O3 catalyst in a fluidized bed reactor[END_REF][START_REF] Ashok | Catalytic Decomposition of Methane to Hydrogen and Carbon Nanofibers over Ni-Cu-SiO2 Catalysts[END_REF][START_REF] Snoeck | Kinetic study of the carbon filament formation by methane cracking on a nickel catalyst[END_REF]). Even if the goal of those studies is different of the present one of this paper, it can be cited the works of Hussein et al. [START_REF] Hussain | Effect of surface structure on the catalytic behavior of Ni: Cu/Al and Ni: Cu: K/Al catalysts for methane decomposition[END_REF]. They investigated the catalytic decomposition of methane and the consequent production of H2 and carbon. Nickel (13 wt %) was chosen as catalyst and titanium oxide as support. The catalysis was conducted at atmospheric pressure and for a temperature range of 823 -1173 K. Methane was mixed with argon (1:1 in mole) and then injected in a stainless steel reactor where 1 g of catalyst was placed in its center. The reactor dead volume was limited by a filling of quartz particles. No specific analysis on the product distribution was done but the hydrogen and carbon deposit rates as the CH4 were carefully monitored.

Another work of great interest is the one of Wolf et al. [START_REF] Wolf | Kinetic model of an oxygen-free methane conversion on a platinum catalyst[END_REF] which investigated and developed a model which treats exclusively the reactions that occur at the surface of a platinum catalyst. Neglecting the gas-phase reactions was justified by the studied temperature (300 -700 K): indeed, the authors showed that there were no or negligible thermal degradation reactions for such low temperature. According to this fact, they created a mechanism composed of 39 elementary surface reactions involving 14 surface species. The species are limited to 2 carbons in order to avoid the very numerous isomerizations reactions that exist for C3+ molecules. The kinetic model also permitted to take into account the thin monolayer of carbonaceous overlayer that quickly covers the surface of the catalyst. This mechanism, in addition with the three precedent ones could be a good base to develop a kinetic model for the catalytic decomposition of CH4. Some of the reactions could be implanted in a pure pyrolysis mechanism. Nevertheless, it appears that various adjustments (thermodynamic factors) should be done thanks to experimental results in order to obtain convenient results. Indeed, those described mechanisms were originally developed to predict the catalytic production of species, generally without taking into account the gas-phase reactions (low temperature).

Numerical simulations and discussion

The literature survey presented in [START_REF] Fau | Methane Pyrolysis: literature survey and comparisons of available data for use in numerical simulations[END_REF] permitted to identify several interesting surface mechanism which could be added to reduced Hamdullahpur's one [START_REF] Sinaki | Development of a reaction mechanism for predicting hydrogen production from homogeneous decomposition of methane[END_REF]. Calculations were performed for methane pyrolysis and they consisted of testing 8 existing kinetic mechanisms.

Those models have been compared against experimental data (Gordon [START_REF] Gordon | Pyrolysis of methane flowing through a porcelain tube in the region 1000°[END_REF]; Billaud et al. [START_REF] Billaud | Thermal coupling of methane in a tubular flow reactor: parametric study[END_REF], [START_REF] Billaud | Thermal decomposition of pure methane at 1263 K. Experiments and mechanistic modelling[END_REF]), numerical data (Davis et al. [START_REF] Davis | The use of methane as a fuel for hypersonic propulsion[END_REF]) and equilibrium data. Then, the most accurate mechanism was selected and a reduction work was achieved by using the dedicated software Workbench. Two methods were tested and associated to this purpose: the Detailed Reduction (DR) which consists of decreasing the number of reactions as a function of their flux; and the Direct Relation Graph (DRG) which limits the number of species and, consequently, the number of reactions. The monitoring of methane decomposition and acetylene prediction served as a first comparison criteria.

Thus, in a first approach, it was planned to integrate the reactions proposed by Wolf et al. in [START_REF] Wolf | Kinetic model of an oxygen-free methane conversion on a platinum catalyst[END_REF]. Unfortunately, it was not possible to gather all the kinetic and the thermodynamic parameters so that this model could not be tested. So, the authors focused on the mechanism proposed by Deutschmann et al. [START_REF] Deutschmann | Numerical modeling of catalytic ignition[END_REF][START_REF] Deutschmann | Natural gas conversion in monolithic catalysts: interaction of chemical reactions and transport phenomena[END_REF] 1 . His team has elaborated several surface mechanisms considering different catalysts: nickel, platinum, rhodium and palladium. These ones were not fully designed for pyrolysis since they integrate reactions with some species containing oxygen (e.g., O2, CO2, H2O…). Nevertheless, by removing these molecules, it can be assumed that the surface models were suitable for the purpose of this study.

Hence several comparisons have been made according to the following criteria: the catalyst, i.e., platinum, nickel and rhodium; the site density, which can be estimated considering as one face of a unit cell of a crystal structure. By dividing the number of sites on this face by the area, a first value of the site density is obtained. Typically, the value goes from 1015 to 1020 sites/cm2. Then, this value should be balanced by each species which could react. By simplification, it was decided to not consider this last element and to fix site density values which would efficiently frame the real ones. Hence, computations were made considering site density going from 10 -10 mole.cm -2 to 0.1 mole.cm -2 . Therefore, it must be noticed that a value of 10 -4 mole.cm -2 should be a theoretical maximum. The reactor design (i.e. closed homogeneous batch reactor or perfectly stirred reactor) was also considered.

a. Platinum catalyst

Deutschmann's platinum surface mechanism was initially used for the catalytic combustion of methane on platinum (13 species, 24 reactions) 2 . The pure combustion reactions have been deleted and only pyrolysis ones were kept (7 species, 7 reactions). These reactions are then injected in the reduced mechanism of Hamdullahpur The first computations were performed considering a closed homogeneous batch reactor considering a temperature of 1600 K, a pressure of 1.013 MPa and a residence time of 100 s. The internal surface of the reactor was fixed at 1.436 cm 2 , the volume at 1.6.10 5 cm 3 (which correspond to the reference volume and internal area proposed by the software) and the density of sites to 2.72.10 -9 cm -2 (value fixed by Deutschmann). Firstly, the catalytic results were compared with reference ones, i.e. without surface reactions (Figure 1). Two kinds of species were selected as point of comparison: predominant light species (hydrogen, methane, acetylene and ethylene -Figure 1a) and heavy coke precursors (benzene, styrene, naphthalene, anthracene and pyrene -Figure 1b). It can be noted that no differences are found for the lights while very minor variations are observed for naphthalene and pyrene. Plus, these minor variations do not affect final composition so that they could be neglected. The site density could be increased up to 10 -5 mole.cm -2 (beyond that value, the software could not solve the problem even by increasing the absolute and relative tolerances). In this case, the production is practically not affected. Thus, it can be noticed a very slight decrease of methane consumption before 1 ms of computations but the fractions do not change at the end of the simulation (Figure 2a). Concerning solid PAH, a slight decrease of pyrene and anthracene productions are predicted but as the precedent case, those differences are too minor to be considered. Regarding the repartition of surface species, the coke (C(S)) becomes rapidly predominant and the only surface species whatever the site density (Figure 3). The rise of the site density has a visible effect, i.e., it critically accelerates the production rate of coke (more than 4 orders) with a faster consumption of platinum sites. The surface is quickly deactivated by the coke deposit so that the catalytic effect is suppressed by this wall of solid carbon. No particular effect of the platinum site density is found up to 10 -5 mole.cm -2 on light and heavy coke precursors. Higher site density may conduct to an increase of its effect (as it will be seen for other catalysts).

Nevertheless, as it is mentioned above, higher values of site density should be not realistic so that it can be assumed that platinum has no impact on methane pyrolysis for such conditions (considering Deutschmann mechanism). Thus, the experimental results presented in section III should not be affected by the presence of platinum coil and can be considered representative of pure methane thermal degradation.

b. Nickel catalyst

Deutschmann developed a specific mechanism to predict the reforming and oxidation of methane on nickel (42 reactions involving 13 species) 3 . By removing the oxidized molecules, the number of species fell to 7 species and the reactions one to 11.

This mechanism was injected in the Hamdullahpur's one and tested according to the same conditions described above. A first comparison was made according to the increase of site density (Figure 4). It can be noted that as for platinum catalyst, the site density has no impact up to 10 - 5 mole.cm -2 for light species (Figure 4a). Nevertheless, for heavy coke precursors, some differences have been found (Figure 4b). Indeed, whereas no notable variation is found from C 6 to C 14 , a high difference is predicted for the initial production of pyrene (C 16 H 10 ) which the formation is clearly delayed. This phenomenon is not fully understood considering the fact pyrene is issued from lighter species (notably acetylene) which productions are not affected.

Finally, such deviation may be caused by a delayed production of intermediary species which intervene in the reaction pathway of pyrene.

Regarding surface molecule profiles (Figure 5), strong variations can be noticed. A higher delay is observed for the decrease of Ni(s) and the increase of H(s) ratios while the C(S) production is delayed and clearly lower. These phenomena, notably the lower consumption of nickel sites, appeared to be understandable because its absolute amount is higher considering a higher site density, so that both the "saturation" and the surface coke formation required a longer delay. Thus, it can be assumed that the different dynamic of anthracene production is linked to these surface variations. Considering this surface mechanism, the software can solve a problem with a higher value of site density (up to 0.1 mole.cm -2 ) and reveals strong differences in the gas and surface specie productions. For the lights, the critical difference is the total consumption of methane (Figure 4a).

As a consequence, the production of other lights, if it could be higher for the first milliseconds of computation (C 2 H 2 and C 2 H 4 ), it becomes null after 0.1 s for hydrogen and ethylene and after 1 s for acetylene. In parallel the production of heavy coke precursors is also affected (Figure 4b).

Excluding pyrene ratio, the production of the other species is earlier and more important but felt to 0 after 1 s of computation. The pyrene production is delayed of about one order and reached a higher value (about 3.6 % against 2.1 %). After 100 s of computations, its formation is not still stabilized. At last, the dynamic of surface species appears to be strongly slower: the Ni site ratio remaining higher than 90 % while the H(S) and C(S) ones are limited (respectively < 10 % and < 0.2 %). This is probably due to the quick extinction of methane and the high density of sites. Then due to the fact that all those gas species are totally consumed since 0.1 s, this should be balanced by the production of other species. The computations reveal that C 6 H 3 becomes the main gas species after 1 ms and C 6 H 2 the second one (Figure 6). C 6 H 2 is a precursor of C 6 H 3 which is notably involved in the production of acetylene. The accumulation of such precursors is questionable since they are highly hydrogenated compared to methane. This may be related to the validity of the mechanism and further analysis of the catalytic effect should be provided with help of specialists. All these observations suggest that nickel catalyst enhances the methane production (up to its total consumption) and favors the coke formation. It is also noted that the higher the site density the faster the decomposition and the more important the amount of coke (illustrated by the presence of anthracene as only heavy coke precursors).

Finally, the well-known catalytic effect of Nickel is qualitatively verified notably with an increase of coking effect.

c. Rhodium catalyst

This mechanism was developed by Deutschmann's team in order to simulate the partial oxidation of methane on rhodium 4 . This kinetic model is more complex than the precedent ones, involving 12 species in 38 reactions (which represents 7 species for 12 reactions).

As the precedent case, the impact of the site density was investigated. As the nickel case, a higher value of site density could be treated (up to 100 mole.cm -2 ). Strong differences are found considering this catalyst. Thus, for light species, a moderate site density (10 -5 mole.cm -2 ) has a very relative effect on the kinetics dynamic: the consumption of methane as the production of hydrogen are faster for the first milliseconds (Figure 7a). Regarding the heavy species, it can be considered that only the pyrene production is slightly affected by the presence of rhodium (< 1 % -Figure 7b). For surface species, the consumption of rhodium is delayed (same profile as nickel catalyst) and coke becomes the dominant species after a few ms of computation (Figure 8).

For a higher value of site density (0.1 mole.cm -2 ), the profile is totally modified. Indeed, a complete consumption of methane occurs before 1 ms. In parallel, acetylene and ethylene are produced transiently while H 2 ratio rapidly reached 100 % (Figure 7a). As a consequence, the carbon atoms initially present in methane are adsorbed at the surface. Since the surface species also take the Rhodium into account, which is largely predominant, only 1 % of C is found (Figure 8). Nevertheless, this 1 % represents the totality of initial carbon. Furthermore, it is difficult to provide a mass balance of the atoms in the heterogeneous calculations because of the two phases (solid and gas/liquid). The species are quantified independently in each phase.

Finally, using rhodium as catalyst lead to a faster and complete decomposition of methane associated with a huge production of hydrogen (which became the only produced species) while solid carbon is deposited on the surface (Figure 8). This signs the critical impact of a catalyst which may involve, in case of large excess, the total decomposition of methane at the reactor surface into gaseous hydrogen and solid carbon (equilibrium state). A site density of 10 -5 mol.cm -² clearly shows a site saturation around 0.01 s, which shows that the catalytic effect get negligible for test duration over 10 ms. However, for very high site density (0.1 mol.cm -²), the Rhodium sites are still around 100%, which clearly reveals that no site saturation is found. The catalytic effect can continue, up to 100 s here.

,00E-03 1,00E-02 1,00E-01 1,00E+00 1,00E+01 1) a) 0,00% 0,50% 1,00% 1,50% 2,00% 2,50% 1

d. Comparison of catalytic mechanisms

A comparison was made according to site density and catalyst nature. The results are given, for the light species in Figure 9, for heavy coke precursors in Figure 10 and for surface species in Figure 11.

For lights, two values were retained 10 -5 mole.cm -2 (Figure 9a) and 0.1 mole.cm -2 (Figure 9b). Considering the first value, the differences are very limited between the three catalysts (i.e.

faster but limited production of hydrogen linked with methane decomposition in presence of platinum). On the contrary, strong variations are found for a higher amount of site density.

Methane is totally consumed after 1 ms of computation (faster consumption considering rhodium). Nevertheless, whereas for nickel, only transient precursors (C 6 H 3 and C 6 H 2 -Figure 6)

and pyrene remain in the gas phase after 1 ms, H 2 is the only remaining gas specie considering rhodium catalyst for the same computation delay. For heavy coke precursors, the differences are negligible for a site density of 10 -5 mole.cm -2 (Figure 10). For a higher value, the production is practically null with rhodium catalyst while anthracene (4 %) is predominant considering nickel.

0,00% 0,50% The consumption of site density is complete whatever the catalyst (Figure 11). This consumption is clearly faster for platinum (1 ns) while for the two others, the profiles are similar

(

e. Impact of the reactor configuration on catalytic computations

The precedent computations were realized considering a closed homogeneous batch reactor (CHBR). Nevertheless, it was shown in the precedent study [START_REF] Fau | Fuel Pyrolysis through Porous Media : Coke Formation and Coupled effect on Permeability[END_REF] that the product distribution obtained during the pyrolysis of reactive fuel through a porous media was quite similar with the one obtained in perfectly stirred reactor (PSR). The CHEMKIN software permits to make the computations regarding PSR. Thus, first simulations were made to compare CHBR and PSR with the same software. No differences are found using only Hamdullahpur's mechanism (Figure 12). Then, a comparison was done with nickel catalyst and a site density of 10 -5 mole.cm -2 but as previously, no significant variations are found (Figure 13). The site density has to be increased to a value of 0.1 mole.cm -2 to notice relative discrepancies. The CHBR presents a residence time equal to the computation time (closed system) while the PSR (open system) has a residence time around 1000 s (linked to the mass flow rate 0.5 g.s -1 and methane density at 625 K -3.0812 kg.m -3 ). For both reactors, the surface/volume ratio is constant to have a similar catalytic effect. The predictions for lights are closed excluding C 6 H 3 ratio which the fall is faster and stronger in the PSR configuration (Figure 14a). This drop is notably compensated by the increase of pyrene ratio which rises after 1 s of computations (Figure 14b). Finally regarding nickel catalyst, the ratio predictions are closed whatever the reactor nature so that the reactor configuration cannot be considered as a significant criterion for the study computations. This fact is also confirmed for platinum catalyst.

Therefore, for rhodium mechanism, the predictions are more affected by the reactor type especially for a high value if site density (0.1 mole.cm -2 -Figure 15b). The consumption of methane is delayed of about one order (which delayed the production of other species). The ethylene ratio is multiplied by 2 while the acetylene one by 3. Finally the transient production varies according to the reactor nature but after a few seconds of computations, the final composition remains unchanged with a H 2 ratio of 100 % (which is high -equilibrium state-thus questionable). Finally, the choice between PSR and CHBR is numerically of secondary importance regarding the final composition and the conditions imposed by the authors.

Conclusions and Perspectives

Regenerative and active cooling represents a large subject gathering several domains as material withstanding or reactive fuel thermal decomposition. Methane was chosen as a reference fuel. The goal was to select an available mechanism which could predict its pyrolysis for a large set of conditions. The size of the kinetic model should be restricted in order to easily implement it in a CFD code. In parallel, a work has been done on the integration of catalytic reactions into the kinetic model. Several catalysts has been considered (because a number of materials are found in cooling systems where methane is used as a coolant). Rhodium, platinum and nickel mechanisms were studied and injected in the reduced Hamdullahpur's mechanism. The effect of these catalysts on the methane conversion has been observed. The chemical mechanisms were extensively tested to estimate the site density effect and the effect of the material nature.

Experiments are expected in order to obtain data for nonmetallic materials for which no heterogeneous kinetic scheme has been found.

Figure 1 .

 1 Figure 1. Comparison between Hamdullahpur's reduced mechanism with and without surface reactions (platinum catalyst, density, 2.72.10 -9 mol.cm -2 ). a) Light species. b) Heavy coke precursors.

Figure 2 .

 2 Figure 2. Comparison between Hamdullahpur's reduced mechanism with and without surface reactions (platinum catalyst, density 10 -5 mol.cm -2 ). a) Light species (mol.%). b) Heavy coke precursors (mol.%).

Figure 3 .

 3 Figure 3. Surface species distribution in mol.% (platinum catalyst).

Figure 4 .

 4 Figure 4. Comparison between Hamdullahpur's reduced mechanism with and without surface reactions (nickel catalyst). a) Light species (mol.%). b) Heavy coke precursors (mol.%).

Figure 5 .

 5 Figure 5. Comparison between the surface specie predictions (mol.%) of Hamdullahpur's reduced mechanism with catalytic reactions according to different site density (2.66.10 -9 mole.cm -2 , 10 -5 mole. cm -2 and 0.1 mole. cm -2 ).

Figure 7 .

 7 Figure 7. Comparison between Hamdullahpur's reduced mechanism with and without surface reactions (rhodium catalyst). a) Light species (mol.%). b) Heavy coke precursors (mol.%).

Figure 8 .

 8 Figure 8. Comparison between the surface specie predictions (mol.%) of Hamdullahpur's reduced mechanism with catalytic reactions according to different site density (2.72.10 -9 mole.cm -2 , 10 -5 mole. cm -2 and 0.1 mole. cm -2 ).

Figure 9 .

 9 Figure 9. Effect of catalyst nature on light specie production (mol.%). a) Site density = 10 -5 mole.cm -2 . b) Site density = 10 -1 mole.cm -2 .

Figure 10 .

 10 Figure 10. Effect of catalyst nature on heavy coke precursor ratio (mol.%)for a site density of 10 -5 mole.cm -2 .

Figure 11 .

 11 Figure 11. Consumption of catalyst sites (mol.%) for an initial density of site = 10 -5 mole.cm -2 .

Figure 12 .

 12 Figure 12. No prediction differences considering PSR or CHBR for the reduced mechanism. a) Light species (mol.%). b) Heavy coke precursors (mol.%).

Figure 13 . 5 mole.cm- 2 )

 1352 Figure 13. No prediction differences considering PSR or CHBR with nickel catalyst (site density = 10-5 mole.cm-2). a) Light species (mol.%). b) Heavy coke precursors (mol.%).

Figure 14 .

 14 Figure 14. Comparisons between PSR and CHBR predictions with nickel catalyst (site density = 0.1 mole.cm-2). a) Light species (mol.%). b) Heavy coke precursors (mol.%).

Figure 15 .

 15 Figure 15. Comparisons between PSR and CHBR predictions (mol.%) with rhodium catalyst. a) Site density = 10-5 mole.cm-2. b) Site density = 0.1 mole.cm-2.

Predominant species (mol.%) in gas phase after the total consumption of methane.

		100,00%
		90,00%	C6H3 C6H2
		80,00%
		70,00%
	Molar fraction	40,00% 50,00% 60,00%
		30,00%
		20,00%
		10,00%
		0,00%
		1,00E-03	1,00E-02	1,00E-01	1,00E+00	1,00E+01	1,00E+02
				Time (s)
	Figure 6.

  1 ms for Rh and above 10 ms for Ni). Coke is mainly and quickly produced with platinum (and with a certain delay for rhodium) while H(S) is predominant considering nickel catalyst. This could explain why a strong amount of asphaltic coke formation is expected in presence of nickel while a catalytic one is predicted with other catalysts.

	Molar ratio	0,00% 10,00% 20,00% 30,00% 40,00% 1,E-15 1,E-13 1,E-11 1,E-09 1,E-07 1,E-05 1,E-03 1,E-01 1,E+01 Time (s) 50,00% 60,00% 70,00% 80,00% 90,00% 100,00% Pt (S), 1E-5 Rh(S), 1E-5 Ni(S), 1E-5

Available mechanisms at: http://www.detchem.com/mechanisms.html

Mechanism available at: http://www.detchem.com/mechanisms/sm_Pt_CH4_O2_1.2_CHEMKIN

Available at : http://www.detchem.com/mechanisms/sm_Rh_CPO_CH4_1.0_CHEMKIN

Acknowledgments

The authors would like to gratefully acknowledge J. Steelant from ESA-ESTEC, M. Bouchez and B. Le Naour from MBDA-France, M. Kuhn and W. Roatermel from DLR.