Thomas Rogaume
email: thomas.rogaume@univ-poitiers.fr

Thermal decomposition and Pyrolysis of solid fuels: Objectives, challenges and modelling

Keywords: Thermal decomposition, thermal degradation, solid fuels, model of pyrolysis, scaling-up, heat transfer, mass transfer, Arrhenius, thermo-physical properties

The need to describe the complexity of the situation concerning fires with more accuracy and taking all the different phases of the fire behaviour (such as the ignition, its growing, its extinction…) into consideration, has led to a great effort from the international research community in the past few years to improve the performance of the numerical simulation codes. These codes consist of different sub-models in order to simulate the different physical, thermal and chemical aspects of the fire, such as the thermal decomposition. This plays a significant role on the fire ignition and its evolution since it represents the fuel quantity supplied to the combustion process. Then, the requirement of a more detailed characterization and description is one actual and important challenge. This is the objective of this article which deals with pyrolysis of solid fuels in the conditions of fire, addressing both the experimental investigations and the modelling development aspects. After a presentation of the objectives and challenges of pyrolysis, this article details the different steps required for its description, based on an experimental and numerical scaling-up approach. Examples of application are explained. Special focus is given on different scientific bolts, issues and challenges for pyrolysis modelling.

Nomenclature

Pre-exponential factor of the reaction i [s -1 ] Solid material specific heat [kJ•kg -1 •K -1 ] dV Volume element [-] Apparent activation energy of the reaction i

[kJ•mol - ,

Initial mass fraction of solid fuel [g•g -1 ] ,

Gaseous mass fraction of combustible [g•g -1 ]

,

Oxygen mass fraction in the gas [g•g -1 ] Oxygen mass fraction [g•g -1

INTRODUCTION

The material used in the buildings, furnishings and in different applications (for example transport) have evolved considerably in the past few years, generating new challenges and requirements concerning fire safety science (complexity). In parallel, the progress of the experimental investigations, computational capacities, as well as the knowledge permit to take into account and to describe other and more complex phenomena. All around the world, fire safety engineering made a strong progress in developing the fire simulation models. For example, Fire Dynamics Simulator [START_REF]FDS -Fire Dynamics Simulator and Smokeview[END_REF] or FireFoam [START_REF] Firefoam | [END_REF]. Finally, the requirement of a better description of all the steps of fire, particularly the non-established events (such as ignition, fire growing, extinction…) is principal and conductible to several works to improve those models.

The Computational Fluid Dynamics (CFD) models are composed by several sub-models, each one has specific functions to solve: heat transfer, radiation, flows and turbulence, combustion… and pyrolysis. Specifically, the sub-model of pyrolysis permits describing the chemical evolution (reactions) of a small volume element (dV) of the material as a function of temperature and atmosphere (%O2). It expresses the relation between temperature, atmosphere and the kinetic's of thermal decomposition, which is considered as homogeneous at each time step in terms of temperature and composition. Thus, the sub-model of pyrolysis permits describing the kinetic of mass loss of a solid material and then the quantity of fuel brought to the combustion, currently nicknamed as the « source term ». It has a large influence on the ignition process (limits of flammability), growing fire, flame propagation and characteristics, gaseous emissions, extinction, etc. However, the thermal decomposition model is just a part of the pyrolysis process of a solid fuel and it must be coupled to other ones in order to describe heat transfers (temperature), mass transfers (%O2, devolatilization transfer, reactive mixture, etc.) and the boundary conditions of the volume element considered for the pyrolysis modelling.

Classically four models of pyrolysis are proposed. They are empirical, thermal, polynomial and finally comprehensive models:

-With the empirical models, the methodologies are based on analysis of small-scale experimental data for extrapolation to larger scale experiments. Classically bench-scale data are extrapolated to larger scales due to the fact that results are expressed in relation to the surface of the material (surfacic assumption): mass loss rate, heat release rate and incident radiation are commonly expressed per unit area. Moreover, it is considered that no mass loss occurs before the surface has reached a critical ignition temperature. -Thermal models increase the level of complexity. Reaction rates are not included in the thermal pyrolysis model however pyrolysis is represented by heat transfer rates. Decomposition is assumed to start when the material-dependent pyrolysis temperature is reached. Mass loss rate is mathematically determined by solving an energy balance at the thin pyrolysis front, and the chemical kinetic is not considered. -The polynomial pyrolysis models are categorized as an intermediate step between thermal and comprehensive models. The simplifications incorporated into the thermal models are also principle here, but those models permit to take into account the environment: mass loss rate is tabulated as a function of incident heat flux and oxygen concentration. Each input parameter is described by the use of a polynomial equation, in which thermal parameters and enthalpies are masked in the coefficients of the polynomial. -Comprehensive models include chemical kinetics, then the pyrolysis is described by reaction rates which are calculated based on local temperature. In many cases chemical kinetics is modelled by some form of an Arrhenius expression. The description of the pyrolysis requires properties which are determined from micro-scale apparatuses, e.g. thermo-gravimetric analysis (TGA), or differential scanning calorimetry (DSC). Due to its performance, this method has been favoured by different authors recently.

In this context, the objective of the present contribution is to describe how to model the pyrolysis of a solid fuel, using a comprehensive approach. The development of the sub-model of thermal decomposition and its characteristics will be presented, as well as its coupling with other sub-models, permitting taking into account the strong coupling between the condensed and the gaseous phases and to describe the pyrolysis of a solid. This is realized based on the scaling-up approach presented by Torero [START_REF] Torero | Scaling up fire[END_REF] and addressing both experimental and numerical aspects. Finally, the presentation of the different steps of development of a model of pyrolysis permits explaining the actual scientific challenges and issues.

In the first part, the challenges and the phenomenology associated with the pyrolysis of a solid are presented.

Next, the steps of the development of the model of pyrolysis are detailed, treatment of the reaction mechanism, the kinetic law and the function of conversion. Attention is brought on the Arrhenius law and on the determination of the kinetic parameters associated to this one. The modelling of the heat and mass transfers requires the knowledge of the values of different thermal, physical and chemical parameters. Their investigations and limits are presented. At the end, the scaling-up validation of the model is addressed.

The different aspects are illustrated by examples of applications on several materials, at different scales.

I. DEFINITION AND PHENOMENOLOGY OF THE PYROLYSIS

This first part is dedicated to a presentation of the pyrolysis phenomenon of a solid fuel.

The pyrolysis of a solid is represented by its mass loss and the quantity of combustible gases emitted which decide whether the combustion will occur or not. Thus, the coupling between the pyrolysis of the condensed phase and the combustion into the gaseous phase is very important: pyrolysis controls the fuel gases but it is dependent on the oxygen diffusion and the heat transfer (which permits the heating of the material) for its decomposition. This strong coupling is presented in Figure 1 and is summarized in Figure 2 [4].

Figure 1: Illustration of the coupling between the pyrolysis and the combustion into the gaseous phase.

Figure 2: Presentation of the coupling between the condensed and the gaseous phases during the combustion of a solid fuel (Inspired from [START_REF] Marquis | Caractérisation et modélisation multi-échelle du comportement au feu d'un composite pour son utilisation en construction navale[END_REF])

It is shown that the pyrolysis is dependent on the local oxygen concentration, local temperature as well as the structure and the properties of the studied material (solid fuel). Its description reclaims 3 parts: a sub-model of pyrolysis; the determination of physical, chemical and thermal properties; and the coupling with other submodels to describe heat and mass transfers. These 3 aspects will be addressed further.

However, due to the multitude and the complexity of the phenomenon to characterize and model, it is classical to study independently the gas phase and the condensed one. Concerning the condensed phase, the interaction with the gas one is just considered using an external heating source (classically an external heat flux) and a local diffusion of oxygen. Torero [START_REF] Torero | Flaming ignition of solids fuels[END_REF] proposes a monodimensional representation of the phenomenon occurring into the solid (Figure 3). When a solid, initially at ambient temperature is exposed to an external heat flux ( *+ " , part of this heat flux permits the heating of the material by conduction ( and a part is lost by reradiation of the surface ( *, " ). Its temperature starts to increase in depth and as a function of time. If the temperature is sufficient to break the link between the condensed molecules (energy of linking), the solid starts to decompose at a flow rate ( . The initial mass fraction of solid fuel ( , ) is consumed to produce gaseous components ( , ), while some oxygen can diffuse into the solid matrix ( -, ). Heat and mass transfers into the condensed phase are dependant of its permeability χ. From figure 3, pyrolysis is shown as a function of time [START_REF] Torero | Flaming ignition of solids fuels[END_REF] -in a simplified 1D approach:

-Temperature T (x, t).

-Local mass fraction of combustible, (x, t).

-Local mass fraction of oxygen, -(x, t).

-Mass fraction of residual solid fuel, , (x, t).

-Permeability, χ (x, t).

-Thickness of oxygen diffusion into the solid, -(x, t).

-Thickness of the reactive zone, (t). -Kinetic parameters values of each reaction, Ai the pre-exponential factor, ni the order of the reaction i, mi the mass of i, Ei the energy of activation. In real situation, this dependence is spatial and so in 3 dimensions.

The mass loss rate due to the pyrolysis is expressed locally by Eq. (1) [START_REF] Torero | Flaming ignition of solids fuels[END_REF]:

Eq. (1)
And for each volume element dV Eq. ( 2) [START_REF] Torero | Flaming ignition of solids fuels[END_REF]:

Eq. (2)
Thus, the objective is to represent the pyrolysis, which reclaims as presented in figure 4 (inspired from [START_REF]Impact de la connaissance des propriétés matériaux dans les simulations incendies -GDR Feux[END_REF]):

-a model of pyrolysis, -physical, chemical and thermal properties, -and the description of coupling, heat and mass transfers.

Figure 4: Pyrolysis of a solid fuel (Aspects represented are inspired from [START_REF]Impact de la connaissance des propriétés matériaux dans les simulations incendies -GDR Feux[END_REF]).

Each one of this 3 aspects will be addressed. In a first part, we are interested in the model of pyrolysis.

II. THE SUB-MODEL OF PYROLYSIS

The sub-model (or model) of pyrolysis presents the chemical evolution (reactions) with respect to the temperature, atmosphere and the kinetic of thermal decomposition of a volume element dV for the studied material which is considered as homogeneous in terms of temperature and composition at each time step.

Three main approaches are classically used in order to determine a model of pyrolysis [START_REF] Witkowski | Thermal decomposition of polymeric materials[END_REF]:

-Modelistic approach -Model fitting method: it consists of a list of models out of which one is selected that fits best with TGA non-isothermal experimental curves. In this method, the Arrhenius equations are referenced to the remaining mass in the sample holder. Then it requires the definition of A, E and n for each reaction and those terms are defined as properties of the reaction. The model-fitting method is expressed in terms of the degree of conversion α, which is equal to "0" at the beginning of the test and to "1" when all the mass has been decomposed. The degree of conversion that is an overall property of the transformation kinetics, is defined in 2.2. -Isoconvertionnal approach -Free model method: Kissinger [START_REF] Kissinger | Reaction Kinetics in Differential Thermal Analysis[END_REF], described a method to determine the preexponential factor, the activation energy and the reaction order of thermal decomposition of magnesite, calcite, brucite, kaolite and halloysite. The type of reactions observed with these minerals are singlestaged. In these reactions, the solid is transformed into residue and gas. Kissinger's method is based on the analysis of the conditions needed to attain the maximum reaction rate. The postulate is: if a reaction proceeds at a rate varying with temperature (i.e. possesses an activation energy), the position of the peak varies with heating rate if the others experimental conditions are kept unchanged. It permits then determining the evolution of the activation energy as a function of the degree of conversion of the reaction. E is dependent on α (the progress of the reaction) and T. It does not use a reaction mechanism (just one reaction) but is based on Arrhenius form. The evolution of E permits presenting the mass loss rate. This approach is available in the case of 1 reaction thermal decomposition, or when the steps are clearly separated (no interaction) and chronologic (not parallel). -The hybrid approach is a combination of the modelistic and the isoconversionnal one. This method requires the numerical and experimental individualization of each reaction peak, permitting to propose a mechanism of thermal decomposition. Each peak of mass loss rate is treated with a Kissinger method [START_REF] Kissinger | Reaction Kinetics in Differential Thermal Analysis[END_REF]. Recently (in the past 15 years), great efforts have been made by the international scientific community in order to develop more detailed and performant models of pyrolysis, for example by Rein & al. [START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF][START_REF] Fateh | Kinetic and mechanism of the thermal degradation of a plywood by using thermogravimetry and Fourier-transformed infrared spectroscopy analysis in nitrogen and air atmosphere[END_REF][START_REF] Fateh | Modeling of the thermal decomposition of a treated plywood from thermo-gravimetry and Fourier-Transformed Infrared Spectroscopy experimental analysis[END_REF][START_REF] Fateh | Experimental and modelling studies on the kinetics and mechanisms of thermal degradation of poly (methyl methacrylate) in nitrogen and air[END_REF], Batiot [START_REF] Batiot | Etude et modélisation de la cinétique de décomposition thermique des matériaux solides. Application à la dégradation du bois en cas d'incendie[END_REF]. Those different groups consider the model fitting method, which is mostly used in the different software. Our interest will focus on the rest of this contribution through this approach. In the modelistic approach, an imposed and a less complex model is used to represent the detail kinetic of thermal decomposition. The model of pyrolysis required:

-A reaction mechanism, Eq. ( 3):

Eq. ( 3)

-A kinetic law and a function of conversion. The rate of reaction of the solid phase is the multiplication of k(T), the law of rate of variation and f(α), the function of conversion. The rate of variation is classically represented by a modified Arrhenius function, Eq. ( 4). Thus:

Eq. ( 4)

where, T is the temperature, α is the progress of the reaction, E is the energy of activation, A is the preexponential factor and R is the perfect gas constant .

Determination of the reactional mechanism

Four main methods are used to determine the reactional mechanism of thermal decomposition of a solid. In order to understand well, we can take the example of the work of Di Blasi [START_REF] Di Blasi | Modeling chemical and physical processes of wood and biomass pyrolysis[END_REF] concerning the thermal decomposition of wood (see Figure 5). The lumped parameter approach (LPA) considers that the material is homogeneous. The thermal decomposition of this material is a multiple steps mechanism proposed by Di Blasi [START_REF] Di Blasi | Modeling chemical and physical processes of wood and biomass pyrolysis[END_REF]: Wood → β Wood + gas β Wood → char + gas char → residue + gas Figure 6: Example of multiple steps mechanism of thermal decomposition of wood using the lumped parameter approach (LPA) [START_REF] Di Blasi | Modeling chemical and physical processes of wood and biomass pyrolysis[END_REF] The constituent approach considers that a material is composed of several constituents and that its thermal decomposition is the sum of the one of each component. Considering the wood, it is composed of cellulose, hemicellulose and lignin. Then, its mechanism of thermal decomposition is the one of the cellulose, the one of the lignin and the one of the hemicellulose. This approach is quite simple, but it neglects the facts that can cause some interactions between the constituents (as presented on figure 7 for the wood).

In the functional approach, the thermal decomposition is described as functional groups, due to the wellknown fragmentation of the polymers. This last one is very precise but time consuming and it requires a good knowledge of the polymer decomposition steps [START_REF] Witkowski | Thermal decomposition of polymeric materials[END_REF].

The other one consists of describing all the elementary reactions of thermal decomposition, as it is classically done for the reactions into the gas phase. One example is presented in [START_REF] Gascoin | Kinetic modelling of high polyethylene pyrolysis : Part1. Comparison of existing methods[END_REF]. However due to the complexity of the solid structures and of the thermal decomposition it is inapplicable for safety science engineering and multicomponent materials for example.

The last one that is presented by Batiot [START_REF] Batiot | Etude et modélisation de la cinétique de décomposition thermique des matériaux solides. Application à la dégradation du bois en cas d'incendie[END_REF] is an hybrid one. Associating the other approaches, notably permitting to develop a global mechanism, considering the constituent and the known functional properties. Concerning the wood, its thermal decomposition is described as [START_REF] Batiot | Etude et modélisation de la cinétique de décomposition thermique des matériaux solides. Application à la dégradation du bois en cas d'incendie[END_REF]:

) ( exp ) ( ) ( α α α f RT E A f T k dt d       - = =
Figure 7: Mechanism of thermal decomposition of wood using the lumped parameter approach (LPA) [START_REF] Batiot | Etude et modélisation de la cinétique de décomposition thermique des matériaux solides. Application à la dégradation du bois en cas d'incendie[END_REF] Then, it currently appears that the LPA approach is the more acceptable and it has been used by several authors [START_REF] Marquis | Caractérisation et modélisation multi-échelle du comportement au feu d'un composite pour son utilisation en construction navale[END_REF][START_REF] Rein | Application of genetic algorithms and thermogravimetry to determine the kinetics of polyurethane foam in smoldering combustion[END_REF][START_REF] Lautenberger | The application of a genetic algorithm to estimate material properties for fire modeling from Bench-Scale Fire Test Data[END_REF][START_REF] Lautenberger | Gpyro -A generalized pyrolysis model for combustible solids[END_REF][START_REF] Lautenberger | Generalized Pyrolysis Model for Combustible Solids[END_REF][START_REF] Stoliarov | Prediction of the burning rates of charring polymers[END_REF][START_REF] Stoliarov | Two-dimensional model of burning for pyrolyzable solids[END_REF][START_REF] Stoliarov | Parameterization and Validation of Pyrolysis Models for Polymeric Materials[END_REF][START_REF] Matala | Estimation of solid phase reaction parameters for fire simulation[END_REF][START_REF] Valencia | Experimental and numerical investigation of the thermal decomposition of materials at three scales: application to polyether polyurethane foam used in upholstered furniture[END_REF][START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF][START_REF] Fateh | Kinetic and mechanism of the thermal degradation of a plywood by using thermogravimetry and Fourier-transformed infrared spectroscopy analysis in nitrogen and air atmosphere[END_REF][START_REF] Fateh | Modeling of the thermal decomposition of a treated plywood from thermo-gravimetry and Fourier-Transformed Infrared Spectroscopy experimental analysis[END_REF][START_REF] Fateh | Experimental and modelling studies on the kinetics and mechanisms of thermal degradation of poly (methyl methacrylate) in nitrogen and air[END_REF]. It relies on the use of results of thermal decomposition of the solid studied from thermogravimetric analysis, which can be coupled to detailed gas analysis, like a Fourier transformed infra-red spectrometer, gas chromatography or mass spectrometer. During those experimental investigations, very small samples of solid (few mm and mg) were heated up under a controlled atmosphere (inert, oxygen %) from the ambient temperature to several hundred degrees (1000°C) at a controlled heating rate, classically from 5 to 20°C/min. The mass loss is measured as a function of the time and (knowing the heating rate) the temperature, as well as the gaseous emissions (if possible). The mass loss rate is secondly deducted. The evolutions of the mass loss and the mass loss rate obtained for different experimental conditions permit the determination of the reactional mechanism.

For example, the thermal decomposition of a polyurethane foam is studied [START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF]. Figure 8 presents respectively the evolution of the mass loss and of the mass loss rate, for two atmospheres and four heating rates [START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF]. It can be seen from the Figure 8 that the thermal decomposition of the PU foam is occuring in 2 global reactions of pyrolysis (under inert atmosphere) and in 3 reactions under air (oxidation). These reactions are presented in the table 1 [START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF].

Table 1: Reactions of thermal decomposition of a PU foam [START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF] The main challenge consists of determining a more realistic mechanism, and classically several can be proposed [START_REF] Valencia | Experimental and numerical investigation of the thermal decomposition of materials at three scales: application to polyether polyurethane foam used in upholstered furniture[END_REF]. The evolutions of the gaseous emissions are thus of most importance. The accuracy of modelistic approaches does not depend on the resolution technique, but rather on the description of the apparent chemical reaction pathway. It is then interesting to focus on the complexity level of the reaction mechanism, which has been done by Marquis & al. [START_REF] Marquis | Influence of reaction mechanism accuracy on the chemical reactivity prediction of complex charring material in fire condition[END_REF]. Several complex mechanisms with various levels of sophistication (number of linked reactions) are studied using a new statistical technique. The analysis shows the possibility to simplify the initial reaction mechanism's complexity, without influencing the accuracy of forecasts.

From the reactions analysis, the reactional mechanism of the PU foam proposed by Kissinger [START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF] is as follows (Figure 9):

Determination of the kinetic law

The rate of reaction of the solid phase is the multiplication of k(T), the law of rate of variation and f(α), the function of conversion. The rate of variation is classically represented by a modified Arrhenius function (see Eq. 5).

Eq. ( 5)

where, T is the temperature, α is the progress of the reaction, E is the energy of activation, A is the preexponential factor and R is the perfect gas constant.

The model-fitting method is expressed in terms of the degree of conversion: equal to 0 at the beginning of the test and to 1 when all the mass has been decomposed. The degree of conversion is defined as, Eq. ( 6):

∞ - - = m m t m m i 0 0 ) ( α Eq. (6)
where, m0 is the mass at the beginning, mi(t) is the mass at an arbitrary time, is the mass at the end of the process.

The analytical form of the functions of conversion, which associates α to f(α), permits to represent the large diversity of the reactions of thermal decomposition. Thus, different forms of law are proposed in order to represent the different kinds of reactions that can be met during the thermal decomposition process of solid fuel. Different functions of conversion are presented in the table 2 [START_REF] Witkowski | Thermal decomposition of polymeric materials[END_REF]:

) ( exp ) ( ) ( α α α f RT E A f T k dt d       - = =
Table 2: Different functions of conversion into the model of pyrolysis, from [START_REF] Witkowski | Thermal decomposition of polymeric materials[END_REF] The choice of the law has to be done considering the global reactions, because the microscopic ones are not known. Classically the reactional order function f(α) = (1α) n is used, since it is adequate to represent only the kinetic aspect, with n which may be linked to a degree of complexity of the studied solid, for example the degree of polymerization. However, sometime, physical phenomena are indistinguishable from the kinetic, and it is necessary to use different conversion functions. It can be the case considering the oxygen diffusion into the solid, which impacts the kinetics of decomposition (cf Jander), eq. ( 7):

= > ? @1 B 1 B > C D E
Eq. ( 7)

Concerning the rate of the reactions, it is described by a modified (for solid application) law of Arrhenius. Then, the rate of each reaction is: ' ? F . = > :

' ? H I J $ K L Eq. ( 8) where, A is the pre-exponential factor, Ea is the energy of activation, R is the perfect gas constant, T is the temperature, m is the mass, n is the order of the reaction.

It is important to note that the law of Arrhenius has been developed for gas application and its use for the solid thermal kinetic representation can be questionable. Past works have shown that its application gives good results, nevertheless the theoretical justification of this one has not been yet done. Then, the actual works notably based on the transition states objective to clarify this application. The work of Batiot [START_REF] Batiot | Etude et modélisation de la cinétique de décomposition thermique des matériaux solides. Application à la dégradation du bois en cas d'incendie[END_REF] have then permitted establishing firmly its application and its form, using a detailed upscaling approach, from the atoms to the solid at matter scale. Moreover, Batiot & al. [START_REF] Batiot | Sensitivity and uncertainty analysis of Arrhenius parameters in order to describe the kinetic of solid thermal degradation during fire phenomena[END_REF] have conducted a sensitivity analysis on the different parameters (A, E, n) of this law to study response on the mass loss and the mass loss rate. They show not only the compensation effect between A and E but also the existence and the uniqueness of the solutions obtained for those parameters, in accordance with the works of Marquis & al. [START_REF] Marquis | Existence and uniqueness of solutions of a differential equation system modelling the thermal decomposition of polymer materials[END_REF]. Because these parameters are micro-scale properties, they cannot be determined without using inverse optimisation methods, described below.

Each reaction I of the reactional mechanism has a rate ω N defined by an Arrhenius law of the form, Eq. ( 9):

δ ω 2 O n o i RT E i i Y m m e A i i         = - •
Eq. ( 9)

From those expressions, the total mass loss rate of the solid (MLRi or OP Q OR ) is expressed by the sum of the mass loss rate due to the i reactions of the reactional mechanism, Eq. ( 10):

STU ? V V ? W X Y ' Y YZ[ B W ' Z \Z]
Eq. [START_REF] Lautenberger | The application of a genetic algorithm to estimate material properties for fire modeling from Bench-Scale Fire Test Data[END_REF] where, is the mass of the intermediate compounds, and i is the number of intermediate species, J is the number of reactions of thermal decomposition of the mass, I and K are the number of reactions permitting the gain of mass i. If m R is the total mass, so

V _ V ? W V V ` aC
Eq. ( 11)

Eq. 11 presented the calculation of the mass-loss rate of a given solid species. A similar calculation is presented in Eq. ( 12), but referenced to the mass of gas produced by a reaction: The yield of gases produced by a given reaction (eq. ( 3)) is calculated as the product of the stoichiometric coefficient, & and the reaction rate ' (see table 1).

? & . ' Eq. ( 12)

The rate of the reaction revealed four unknown parameters for each reaction: A the pre-exponential factor, E the energy of activation, n the order of the reaction and ʋ the stoichiometric coefficient. The determination of those parameters cannot be done experimentally [START_REF] Batiot | Etude et modélisation de la cinétique de décomposition thermique des matériaux solides. Application à la dégradation du bois en cas d'incendie[END_REF], and can be realised from different methods:

-The direct methods permit to determine the unknown kinetic parameters directly (mathematical resolution) from the experimental data, for example the mass loss rate evolutions. There are not time consuming, however there are only applicable for simple problems, here, for a mechanism of thermal decomposition with only one step. -The metaheuristic methods, using inverse methods of optimization. In the literature, different methods are proposed and compared [START_REF] Lautenberger | Optimization Algorithms for Material Pyrolysis Property Estimation[END_REF][START_REF] Siarry | Métaheuristiques pour l'optimisation difficile[END_REF]. The algorithms of evolutionary type or behavioral type appear to be particularly well adapted to the difficult problems of optimization on strongly non-linear and very unstable models Let us quote by example: the genetic algorithms (GA) [START_REF] Houck | A genetic algorithm for function optimization: a Matlab implementation[END_REF] (the Genetic Algorithms Simulated Annealing), the Stochastic Hill Climber, the Shuffled Complex Evolution [START_REF] Duan | Shuffled Complex Evolution Approach for Effective and Efficient Global Minimization[END_REF] or the particle swarm [START_REF] Eberhart | A new optimizer using Particle Swarm Theory[END_REF].

They are applied to the mass loss and (not systematically) to the mass loss rate evolutions. The differences between the different methods permit to resolve the problem of convergence, of not exploration of the whole space of the possible solutions. The global scheme of functioning of those algorithms is as follows (Figure 10 The form of these PDFs are determined by a calibration process in which Bayes' theorem is used to update prior knowledge with experimental data. One advantage of this approach is that it permits a complete treatment of uncertainty.

In most of the cases, multi-step mechanisms of thermal decomposition are proposed, then metaheuristic methods are used. It uses an evaluation function that compares experimental and calculated curves. The accuracy between curves is called fitness ∅, Eq. ( 13). The improvement of fitness, means a best fitting between experiments and calculations. Optimum parameters are found with the maximisation of ∅. Thus, based on the comparison of experimental and calculated curves, the evaluation function has the main role in the modelling while it indicates to the code if the result is adequate or not. Different evaluation functions φ are proposed as a function of the authors [START_REF] Rein | Application of genetic algorithms and thermogravimetry to determine the kinetics of polyurethane foam in smoldering combustion[END_REF][START_REF] Valencia | Experimental and numerical investigation of the thermal decomposition of materials at three scales: application to polyether polyurethane foam used in upholstered furniture[END_REF][START_REF] Boulet | Characterization of the radiative exchanges when using a cone calorimeter for the study of the plywood pyrolysis[END_REF] and it is shown that this one has an influence on the solutions obtained [START_REF] Valencia | Experimental and numerical investigation of the thermal decomposition of materials at three scales: application to polyether polyurethane foam used in upholstered furniture[END_REF][START_REF] Batiot | Etude et modélisation de la cinétique de décomposition thermique des matériaux solides. Application à la dégradation du bois en cas d'incendie[END_REF]. The following evaluation function is the one used by Rein & al. [START_REF] Rein | Application of genetic algorithms and thermogravimetry to determine the kinetics of polyurethane foam in smoldering combustion[END_REF].

( )

β β ψ φ ∑ ∫ ∫ = - -         - +         - = c Exp Calc Exp Calc dT m m dT dt dm dt dm 1 1 1
Eq. ( 13)

Where, c is the number of heating rates that are compared (four in this research). This function is formed by the addition of two terms: the first term of the right hand side accounts for the absolute value of the difference between the experimental and calculated curves of MLR. The second term of the right hand side is the absolute value of the difference of the experimental and calculated curves of change in mass as a function of temperature. Due to several shortcomings of this function, other evaluation functions have been developed. In the following equation, the fitness is based on the classical method to calculate errors: the square of the difference between the two curves. It is in agreement with the evaluation function of Esperanza & al. [START_REF] Esperanza | Pyrolysis of varnish based on a polyurethane[END_REF]:

∑∑ = = -         - = c k j Exp j Calc j dt dm dt dm 1 1 2 β β φ
Eq. ( 14)

Where, k is the dimension of the vectors that are compared (only comparison of vectors of the same size are allowed). This equation is quite restrictive because of the second power of the error.

Another evaluation function has been proposed during the work of Bustamante Valencia [START_REF] Valencia | Experimental and numerical investigation of the thermal decomposition of materials at three scales: application to polyether polyurethane foam used in upholstered furniture[END_REF][START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF], eq. ( 15). The definition of φ takes into account the scalar product y x r r , and the Euclidean norm of vectors x r .

( ) [ ]

β β φ ∑ = -               - ⋅ ∠ = c x y x y x
Considering the example of application presented in Figures 7,8 and the table 1, the total mass loss rate of the polyurethane foam [START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF] is defined as (see Eq. 16):

V V ? W STU ? V ccd V e aC + V cf`gfV + V !hi V + V jk V
Eq. ( 16)

So: V V ? X C B 1 ' C + X B 1 ' + X D B 1 ' D + X e B 1 ' e + X l B 1 ' l Eq. ( 17
)
with each reaction having a rate: ' ? Hmn oB U p q r s - t

Eq. ( 18)

During the previous study [START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF], the 4 × w (with i the number of reactions) so 20 unknown parameters have been determined by the Genetic Algorithms method [START_REF] Houck | A genetic algorithm for function optimization: a Matlab implementation[END_REF]. In Eq. ( 18), is equal to 0 when the reaction takes place without oxygen, and 1 for the thermos-oxidative reactions. The values obtained are presented in table 3. The figure 11 presents the comparison between the experimental and the numerical mass loss rate evolutions [START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF]. The results show a very good agreement, the pyrolysis model (and then the reactional mechanism and the kinetic parameters) is validated at this particular scale, the matter one. However, just the kinetic aspect of the thermal decomposition is represented and it is now necessary to be able to represent the thermal behaviour of a real fuel, considering the thermal and mass transfers. A scaling-up approach is used which, requires, notably, the determination of several thermo-physical properties of the studied fuel.

III. DETERMINATION OF THE THERMO-PHYSICAL PROPERTIES

In order to be able to simulate the heat and mass transfers during the pyrolysis of a solid, it is necessary to determine different thermo-physical properties such as:

-The thermal conductivity k can be determined by the flash method, or using guarded hot plate.

-The density ρ.

-The specific heat Cp, using a differential scanning calorimetry.

-The enthalpies of the reactions ΔH, using a differential scanning calorimetry.

-The radiative properties (emissivity, absorptivity), using and infra-red and a thermal camera approach, as done in [START_REF] Boulet | Characterization of the radiative exchanges when using a cone calorimeter for the study of the plywood pyrolysis[END_REF][START_REF] Acem | Surface temperature on a carbon composite sample during thermal degradation[END_REF]. -The heat of combustion, using a bomb calorimeter. However, it is very important to note that those properties must be determined for all the intermediate compounds formed throughout the pyrolysis process. Taking the example of the PU foam, which means that k, ρ, Cp, ΔH, ɛ… must be determined for the initial PU foam (PPUF), the "Polyol", the Char and the Residue. The determination of the certainty of those properties is one of the main problems faced by the international community, due to several reasons:

-The experimental limitations: indeed, for several parameters, the experimental investigations are not adequate and show major limitations. For example: o It is very difficult to measure the conductivity k on porous matrix. o It is not possible to measure k and Cp after a certain temperature (thermal decomposition). Indeed, the gaseous emissions generated by the thermal decomposition will affect irreversibly the set-up used. Some authors used a TGA/DSC apparatus for the determination of Cp and ΔH, however it is shown that the precision of the measure is quite low and the Cp value has to be corrected by the mass loss.

-The possibility to isolate each intermediate compounds in order to be able to realize the measurements of those properties. In fact, it is very difficult to isolate and to extract the condensed materials formed during the pyrolysis process, with notions of superposition of the reactions and so of the compound formed in the same temperature range. A solution for this difficulty would be to decompose the solid until the temperature of formation of each intermediate compound, if possible, and to stop the test. This may permit isolating each compound. Nevertheless, when cooling, the structure of the material will be modified, and that will make an important impact on the measurements.

New experimental approaches are developed by the team of S. Stoliarov in the University of Maryland, which uses the micro-calorimeter set-up to determine the heat of combustion or pyrolysis [START_REF] Stoliarov | Measurement of the global kinetics of combustion for gaseous pyrolyzates of polymeric solids containing flame retardants[END_REF][START_REF] Ding | Determination of kinetics and thermodynamics of thermal decomposition for polymers containing reactive flame retardants: Application to poly(lactic acid) blended with melamine and ammonium polyphosphate[END_REF], or the Controlled Atmosphere Pyrolysis Apparatus (CAPA) to deduce the conductivity [START_REF] Swann | Characterization of pyrolysis and combustion of rigid poly(vinyl chloride) using two-dimensional modeling[END_REF][START_REF] Ding | Pyrolysis model development for a polymeric material containing multiple flame retardants: Relationship between heat release rate and material composition[END_REF]. However, if the first method gives encouraging results, it does not permit isolating and determining the property values for each intermediate compound. Then it can be applicable in case of one mechanism of thermal decomposition with only one reaction. Recent works on CAPA show how it can be used to parameterize the thermal transport properties for 6 reaction and 11 reaction mechanisms. It can be used for different materials, and the experiments performed under different atmospheres (pure nitrogen to air).

It appears to be very difficult to experimentally determine certain of those parameters, such as k, ρ, Cp, ΔH. Two alternative solutions are actually used:

-The parameters are determined by inverse optimization methods, for example using Gpyro [START_REF] Lautenberger | Gpyro -A generalized pyrolysis model for combustible solids[END_REF]. Nevertheless, this solution cannot be considered satisfactory, because this is like the addition of some degrees of liberty, some mathematical fitting methods with more variables, the thermochemical properties. Moreover, a compensation phenomenon can be observed due to the large unknown parameters (the kinetics and the thermochemical ones). Finally, the values obtained can have no physical sense. -Classically « weighting (average) law » is used to determine the value of the thermos-physical properties, between the properties of the initial material and the final one: o Or, equivalent properties are taken for all the intermediate compounds, as for just one equivalent material. o Or, a linear evolution of the properties between the initial and the final materials is considered. An example is presented in [START_REF] Swann | Characterization of pyrolysis and combustion of rigid poly(vinyl chloride) using two-dimensional modeling[END_REF].

The determination of the thermo-physical properties for the total range of temperature of the pyrolysis, and for all the intermediate compounds into the kinetic mechanism is essential for the modelling of the pyrolysis. However, it is one of the main bottlenecks actually, requiring huge efforts from the scientific community.

IV. MODELLING OF THE PYROLYSIS -SCALING-UP

As presented before, notably in Figures 1 and2, the challenge remains the modelling of the pyrolysis at real scale, then considering the interactions between both the condensed and the gaseous phases. This requires the description of the heat and mass transfers between these two. The scaling-up approach presented by [START_REF] Torero | Scaling up fire[END_REF][START_REF] Torero | Flaming ignition of solids fuels[END_REF] and adopted by [START_REF] Marquis | Caractérisation et modélisation multi-échelle du comportement au feu d'un composite pour son utilisation en construction navale[END_REF][START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF][START_REF] Camillo | Application and Limitations of a Method Based on Pyrolysis Models to Simulate Railway Rolling Stock Fire Scenarios[END_REF] is used: it is based on experimental investigations to determine the initial and boundary conditions for the simulation, and to furnish results permitting a comparison with the numerical ones, for the validation (or not) of the model.

Several bench scales are classically used with the principle consisting of the addition of complexity in increasing scale:

-Firstly, at small scale, using the cone calorimeter [42], the Fire Propagation Apparatus [43] or the Controlled Atmosphere Pyrolysis Apparatus [START_REF] Swann | Characterization of pyrolysis and combustion of rigid poly(vinyl chloride) using two-dimensional modeling[END_REF][START_REF] Ding | Pyrolysis model development for a polymeric material containing multiple flame retardants: Relationship between heat release rate and material composition[END_REF]. This permits to work in one-dimension, with heat and mass transfers but without a flame propagation. -At product scale, using for example the radiant panel, the IMO-LIFT [44], the Medium Burning Item

[45], the Single Burning Item [46], permitting some investigation in two and three dimensions, with a flame propagation. -At real scale, like realised for example by [START_REF]The Dalmarnock Fire Tests: experiments and modelling[END_REF][START_REF] Didieux | Real-scale Fire Tests of One Bedroom Apartments with Regard to Tenability Assessment[END_REF].

The modelling approaches use dedicated codes, like Gpyro [START_REF] Lautenberger | Gpyro -A generalized pyrolysis model for combustible solids[END_REF][START_REF] Lautenberger | Generalized Pyrolysis Model for Combustible Solids[END_REF] or Thermakin [START_REF] Stoliarov | Prediction of the burning rates of charring polymers[END_REF][START_REF] Stoliarov | Two-dimensional model of burning for pyrolyzable solids[END_REF][START_REF] Stoliarov | Parameterization and Validation of Pyrolysis Models for Polymeric Materials[END_REF], or consists of the implementation of the model of pyrolysis and the thermo-physical properties previously presented in a generalized CFD code, like FDS [START_REF]FDS -Fire Dynamics Simulator and Smokeview[END_REF] or Firefoam [START_REF] Firefoam | [END_REF] and to simulate the experimental configurations studied.

Several studies consider those different approaches and just some examples are given. Considering the PU Foam previously studied at matter scale, the works of [START_REF] Valencia | Analysis of principal gas products during combustion of polyether polyurethane foam at different irradiance levels[END_REF] have permitted its characterization using the cone calorimeter coupled with several gas analysers. Some results are presented on figure 12 for the mass loss rate (MLR) and the Heat Release Rate (HRR) [START_REF] Valencia | Analysis of principal gas products during combustion of polyether polyurethane foam at different irradiance levels[END_REF].

Figure 12: MLR and HRR evolution during the pyrolysis of a PU Foam using the cone calorimeter [START_REF] Valencia | Analysis of principal gas products during combustion of polyether polyurethane foam at different irradiance levels[END_REF] Dealing with the PU foam in their study of the behaviour of a railway seat, Guillaume & al. [START_REF] Camillo | Application and Limitations of a Method Based on Pyrolysis Models to Simulate Railway Rolling Stock Fire Scenarios[END_REF] have simulated, the cone calorimeter experiments at the specific condition of 50kW/m 2 . In this study, a simplified model of pyrolysis and the thermo-physical properties have been introduced into the FDS code. The results obtained concerning the Heat Release Rate are presented on figure 13 [START_REF] Camillo | Application and Limitations of a Method Based on Pyrolysis Models to Simulate Railway Rolling Stock Fire Scenarios[END_REF]. The results show a good agreement permitting to validate the model at this scale.

Figure 13: numerical and experimental evolutions of the HRR evolution during the pyrolysis of a railway seat using the cone calorimeter [START_REF] Camillo | Application and Limitations of a Method Based on Pyrolysis Models to Simulate Railway Rolling Stock Fire Scenarios[END_REF] If classically the mass loss rate and the heat release rate are considered for the validation of the model, the work of Fateh & al. [START_REF] Fateh | Modeling of the pyrolysis of plywood exposed to heat fluxes under cone calorimeter[END_REF] shows that it is necessary to consider other parameters. Indeed, in their study of the pyrolysis of plywood [START_REF] Fateh | Modeling of the pyrolysis of plywood exposed to heat fluxes under cone calorimeter[END_REF], the authors compared two different mechanisms of thermal decomposition: one at 5 reactions (developed in [START_REF] Fateh | Kinetic and mechanism of the thermal degradation of a plywood by using thermogravimetry and Fourier-transformed infrared spectroscopy analysis in nitrogen and air atmosphere[END_REF], see Figure 14), and a simpler one at 3 reactions (see Figure 15) using Gpyro: [START_REF] Fateh | Kinetic and mechanism of the thermal degradation of a plywood by using thermogravimetry and Fourier-transformed infrared spectroscopy analysis in nitrogen and air atmosphere[END_REF].

Plywood Air Plywood Air Plywood N Plywood N Plywood N Plywood λ θ σ β α   →    →    →    →    →  2 2 2
In order to estimate the ability of this 3 steps model to represent the thermal decomposition of plywood using Gpyro, Fateh & al [START_REF] Fateh | Modeling of the pyrolysis of plywood exposed to heat fluxes under cone calorimeter[END_REF] show that it is necessary to consider the temperature evolutions in the depth of the material and the thickness evolution of the intermediate compounds. Indeed, the authors show that the 3 steps model permits to represent very well the mass loss, the mass loss rate (Figure 16) as well as the temperatures at the front and back surfaces of the solid during its decomposition in the cone calorimeter at 30 kW.m -2 (Figure 17). during the pyrolysis of plywood with the cone calorimeter -3 steps mechanism [START_REF] Fateh | Modeling of the pyrolysis of plywood exposed to heat fluxes under cone calorimeter[END_REF]. Nevertheless, even if the 3 step pyrolysis model predicts quite well the mass loss rate of the plywood, its capacity to reproduce the pyrolysis of the solid in depth is questionable. This 3 step pyrolysis model give us some non-physical results. Indeed, the figure 18 [START_REF] Fateh | Modeling of the pyrolysis of plywood exposed to heat fluxes under cone calorimeter[END_REF] represents the evolution of the mass fractions of the heterogeneous species and the evolution of the temperature along the thickness of the sample at 1000s for 2 types of the model. For the 3 step mechanism, virgin plywood is present all along the thickness of the sample and represents 40 %wt of the sample weight at the back surface of the solid. However at this time no more virgin material remains for the corresponding experiment.

At a higher scale, Guillaume & al. [START_REF] Camillo | Application and Limitations of a Method Based on Pyrolysis Models to Simulate Railway Rolling Stock Fire Scenarios[END_REF], in their scaling-up validation of the model -to represent the pyrolysis of a railway seat, have modelled some tests using a single burning item. The figure 19 [START_REF] Camillo | Application and Limitations of a Method Based on Pyrolysis Models to Simulate Railway Rolling Stock Fire Scenarios[END_REF] presents the comparison of the numerical and the experimental HRR evolutions at the Single Burning Item scale. A good agreement is obtained.

Figure 19: Numerical and experimental evolutions of the HRR evolution during the pyrolysis of a railway seat using a single burning item [START_REF] Camillo | Application and Limitations of a Method Based on Pyrolysis Models to Simulate Railway Rolling Stock Fire Scenarios[END_REF] V.

DISCUSSION

From several years, many efforts are realized by the scientific community in order to improve the experimental and numerical investigations of the thermal decomposition and the pyrolysis of solid fuels. If progress has been done, different scientific obstacles persist, requiring specific focus and attention in the future: development, calibration, verification, and validation of predictive models of thermal decomposition and pyrolysis are the challenge.

This then requires experimental investigations in order to understand the phenomena occurring and to initiate, to develop and to validate the models. Thus, the efforts to support in the future concerns both the experimental investigations and the models development.

It is shown previously that in order to describe numerically the thermal decomposition and the pyrolysis of a solid fuel, it is necessary to develop a model of pyrolysis, to determine the physical, chemical and thermophysical properties, to describe heat and mass transfers and finally to represent the coupling between the condensed and the gas phases. Current scientific shortcomings concern both each one of those aspects and are discussed below.

Those shortcomings are dependent on the complexity of the studied solid fuel. Indeed, the phenomenon occurring during the pyrolysis of non-charring fuel are different than the ones involved for charring materials.

Other complexities are also generated when studying multicomponent solids as well as for the intumescent ones.

The first aspect concerns the experimental investigations of the thermal decomposition in order to propose a model of pyrolysis. The challenge is to obtain "good data" in controlled and known conditions and then to develop and to promote good practises when using the thermogravimetric apparatus. Each reaction of the thermal decomposition has a kinetic defined by an Arrhenius law for which parameters are unknown. Those parameters are classically determined using different inverse methods of optimization, for which problems of convergence exist, not allowing then the exploration of the entire domain of the possible solutions. Thus, the development of more robust methods is necessary. Another important aspect concern the determination of the thermal, chemical and physical properties for the modelling of the pyrolysis process. Different experimental techniques exist and are used (thermo-differential analysis, differential scanning calorimetry, micro-calorimeter, etc.). It is necessary to establish the conditions of implementation to guaranty the acquisition of "good results". Moreover, another important challenge is the possibility of determination of those properties for each equivalent condensed phase of the kinetic mechanism of thermal decomposition, as a function of time. Complexity in those determination can be generated by the charring of the material, its intumescent properties or the mechanical transformations induced by its degradation and heating. At this scale but also in the scaling-up approach, past studies have demonstrated the intake of the measurement of the gaseous emissions induced by the pyrolysis. Nevertheless, in most studies, only a qualification of the species emitted is realised. Efforts have to be made in order to quantify the gaseous emissions emitted by the pyrolysis and the thermal decomposition, both with intrusive or non-intrusive techniques.

The pyrolysis model is developed at a very small scale (matter scale), for some conditions which are not representative of real fires. A scaling-up approach is then used by different teams. However, for increasing the scale, two aspects are important: the control and the knowledge of the initial and boundary conditions for the model input and initialization; a good realization of the tests for obtaining "good data" for the model validation (comparison of numerical and experimental results). Developments are required in order to propose specific bench scales, to established good methodologies and experiment conditions, etc. at matter scale but also during the scaling-up process, it is beneficial for the model development and validation, to establish some balance sheets of the species, of energy and of mass conservation. This requires precise quantification of what happen in the condensed phase as well as in the gas phase: chemical composition of the solid phase, gaseous emissions, heat transfers, porosity, pressure evolution, species diffusion, flowrates, etc. have to be quantified. Dedicated experimental set-up and analysis have to be developed in this sense, taking into account the complexity of the materials studied: charring, intumescence, structural evolutions (cracking, delamination…).

In longer period, the study of coupling between the condensed and the gaseous phases for a better description of the ignition-extinction processes and the flame propagation are required to perform. To do this, representative experimental tests must be developed and performed, and the model must be improved and developed. Application of those different approaches and methodologies to multicomponent solids also remain a very important challenge in the future.

Due to all these factors the "IAFSS International Working Group on Measurement and Computation of Fire Phenomena" has been created and subgroup the "Condensed Phase Phenomena" within the IAFSS. Its objective is to better coordinate measurement and modeling efforts carried out in this field by various research groups. The purpose of the Condensed Phase Phenomena subgroup is to facilitate data sharing among researchers in order to improve predictions of thermal decomposition and pyrolysis in fire.

CONCLUSION

The description of the pyrolysis of the solid fuels is of the best interest and importance. It has a large influence on the ignition process, the kinetic of the fire development, its extinction and on the characteristics of the flame and its propagation. The pyrolysis is controlled by the phenomena taking place into the condensed phase but is dependent on the heat and the mass transfers associated to the gaseous phase.

The models of pyrolysis can be divided into four main categories: the empirical, the thermal, the polynomial and finally the comprehensive ones, depending on how pyrolysis is modelled. Empirical models are not pertinent to applications other than the geometry and materials which they have been validated for. Thermal models are feasible for materials with high activation energies and where little pyrolysis occurs before ignition. Polynomial models take the environment into account but the validity of scaling is yet to be established. Finally, the comprehensive methodologies are the most developed since it has the potential to be used for any application with quite a good accuracy. However it is more complex to develop and it requires many parameters. The present paper is making a state of the art of how to develop a comprehensive model of thermal decomposition of the solid fuels.

In the context of comprehensive models, the pyrolysis description requires three main parts: the development of a sub-model of pyrolysis which simulate the kinetic of decomposition of the solid phase, the determination of the thermo-physical properties of the condensed compounds, and, finally, the description of the heat and mass transfers which determine the conditions of the thermal degradation.

Those different steps required to model the pyrolysis of a solid, are presented in the present contribution.

Different methods are used to develop a model of pyrolysis: the modelistic approach, the isoconversionnal and the hybrid (coupling the two last) ones. However, the modelistic one is used the most, then the attention is focused on it. It requires a mechanism of thermal decomposition and a kinetic law, which is composed by a function of conversion and by a law of variation of the rate. Different approaches are used in order to propose a mechanism of thermal decomposition, where one can have different levels of complexity: one single reaction to several sequential or parallel reactions. In many studies, the lumped parameter approach is used, or the hybrid one proposed by Batiot [START_REF] Batiot | Etude et modélisation de la cinétique de décomposition thermique des matériaux solides. Application à la dégradation du bois en cas d'incendie[END_REF]. Different functions of conversion are also proposed, it is a choice to be dependent on the characteristics of the reactions of thermal decomposition. Thus, classically a reactional order function is chosen. Concerning the rate of the reactions, it is described by a modified (for solid application) law of Arrhenius. Four unknown parameters appear for each reaction of thermal decomposition and cannot be determined experimentally. Inverse metaheuristic methods of optimization are then used by the international community in order to define the best value of the kinetic parameters. To do this, evaluation functions are used in order to compare the experimental and the numerical results. The development and the validation of the model of pyrolysis is realized at matter scale, using thermogravimetric analysis. A stake is then to evaluate the ability of the model to represent the pyrolysis at upper scale.

The determination of the values of the thermo-physical properties is necessary for the modelling of the heat and mass transfers into the condensed phase but it remains an important challenge.

The modelling of the pyrolysis is realised by a scaling up approach, permitting furnishing experimentally the initial and boundary conditions as well as results for the model application and development. Generally complexity is added step by step. First considering the modelling of experimental tests with cone calorimeter or the fire propagation apparatus. Tests with a flame propagation are subsequently studied until full scale situations.

Despite the progress of the research, different scientific obstacles persist, requiring specific focus and attention in the future. They concern notably: the experimental investigations whatever the scale in order to obtain "good" data as well as the control of the initial and boundary for the development and the validation of the models; the determination of the thermo-physical properties; etc. and in a longer period, the coupling between the condensed and the gaseous phases for a better description of the ignition-extinction processes and the flame propagation.

Figure 3 :

 3 Figure 3: Simplified (1D.) representation of the phenomenon occurring during the pyrolysis of a solid[START_REF] Torero | Flaming ignition of solids fuels[END_REF]

  [9], Lautenberger & al. [10-12], Stoliarov & al. [13-15], Matala & al. [16], Bustamante Valencia [17], Rogaume and Fateh & al.

Figure 5 :

 5 Figure 5: Comparison of experimental and numerical curves of thermal decomposition of the woodand its constituents at à 5°C.min -1[START_REF] Di Blasi | Modeling chemical and physical processes of wood and biomass pyrolysis[END_REF]

Figure 8 :

 8 Figure 8: Evolution of the mass loss and the mass loss rate during the thermal decomposition of a PU foam at TGA scale [18].

Figure 9 :

 9 Figure9: Mechanism of thermal decomposition of a PU foam[START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF]

  Figure 10: global scheme of functioning of the metaheuristic methods of optimization of the unknown parameters

  arrays of experimental and calculated mass-loss rates as a function of temperature. These vectors are of the same dimension, i.e. the data points are taken at the same temperature.

1 EFigure 11 :

 111 Figure11: comparison of the experimental and the numerical mass loss rate evolutions[START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF]

Figure 14 :

 14 Figure14: 5 steps mechanism of thermal decomposition of a plywood[START_REF] Fateh | Kinetic and mechanism of the thermal degradation of a plywood by using thermogravimetry and Fourier-transformed infrared spectroscopy analysis in nitrogen and air atmosphere[END_REF]

Figure 15 :

 15 Figure15: 3 steps mechanism of thermal decomposition of wood[START_REF] Fateh | Kinetic and mechanism of the thermal degradation of a plywood by using thermogravimetry and Fourier-transformed infrared spectroscopy analysis in nitrogen and air atmosphere[END_REF].

Figure 16 :

 16 Figure 16: Experimental and numerical mass and mass loss rate evolutions as function of time at 30 kW.m -2during the pyrolysis of plywood with the cone calorimeter -3 steps mechanism[START_REF] Fateh | Modeling of the pyrolysis of plywood exposed to heat fluxes under cone calorimeter[END_REF].

Figure 17 :

 17 Figure 17: Experimental and numerical temperatures evolutions as a function of time (30 kW.m -2 ) at the front and the back surface of the plywood sample during cone calorimeter study -3 steps mechanism [50]

Figure 18 :

 18 Figure 18: Evolutions of mass fractions of heterogeneous species as a function of time (30 kW.m -2 ) at the back surface of the solid at 1000 s during cone calorimeter experiments, for the two mechanisms [50]

Table 3 :

 3 kinetic parameter values of the different reactions of thermal decomposition of a PU foam[START_REF] Rogaume | Development of the thermal decomposition mechanism of polyether polyurethane foam using both condensed and gas phase release data[END_REF]

	Reaction	Parameter	Value	High	Range	Low	Units
	PPUF pyrolysis			169.9	190		150	kJ•mol -1
		1 A	6.09 x 10 13	1 x 10 22		1 x 10 7	s -1
		n 1 υ 1 1 τ	0.91 0.69 _	1 0.9 9 x 10 9		0.1 0.1 1.5 x 10 9	_ kg•kg -1 _
	Polyol pyrolysis	E	2	243.9	260		100	kJ•mol -1
		A	2	4.42 x 10 17	1 x 10 19		1 x 10 7	s -1
		n 2 υ 2 2 τ	1.26 0.10 4.9 x 10 9	1.5 0.81 9 x 10 9		0.1 0.1 1.5 x 10 9	_ kg•kg -1 _
	PPUF oxidization	3 E	214.1	240		161	kJ•mol -1
		3 A	3.07 x 10 18	1 x 10 20		1 x 10 7	s -1
		3 n	0.48	3		0.2	_
		3 υ	0.44	0.7		0.1	kg•kg -1
		3 τ	8.9 x 10 4	1.5 x 10 5		3 x 10 4	_
	Polyol oxidization	E	4	213.6	240		161	kJ•mol -1
		A	4	1.26 x 10 18	1 x 10 22		1 x 10 7	s -1
		n 4 υ 4 4 τ	0.95 0.56 8 x 10 5	3 0.7 2.2 x 10 6		0.3 0.1 2 x 10 4	_ kg•kg -1 _
	Char oxidization	5 E	160.8	240		160	kJ•mol -1
		5 A	4.30 x 10 12	3 x 10 15		1 x 10 11	s -1
		5 n	1.64	3		0.5	_
		5 υ	0.25	0.8		0.1	kg•kg -1
		5 τ	3.4 x 10 6	9 x 10 6		1.7 x 10 5	_

ACKNOWLEDGMENTS

The author wishes to thank: the "Institut des Risques Industriels Assurantiels et Financiers" for their technical and human support into the HESTIA platform ; his colleagues, particularly Benjamin Batiot, Jocelyn Luche, Marc Poisson and Franck Richard ; his past and present PhD students ; Hussain Najmi for the English corrections. This work was partially funded by the French Government program "Investissements d'Avenir" (LABEX INTERACTIFS, reference ANR-11-LABX-0017-01).