P.-A Warth

R Glaude

R Privat

B Fournet

Sirjean

M D Le

Z El Sayah

R Benrabah

V Warth

P.-A Glaude

R Fournet

Baptiste Sirjean
email: baptiste.sirjean@univ-lorraine.fr

An Experimental and Detailed Kinetic Modeling of the Thermal Oxidation Stability

Introduction

The rapid growth of worldwide aviation requires intense research efforts to optimize aircraft engines to achieve better energy efficiency and emission reductions [START_REF] Gössling | The global scale, distribution and growth of aviation: Implications for climate change[END_REF]. Aviation liquid fuels, also called jet fuels, are used as the main energy carrier in aircraft engines but also have a role in the cooling system, which is specific to them [START_REF] Jia | A comprehensive review of the thermal oxidation stability of jet fuels[END_REF]. Indeed, if their primary role is to provide energy through the combustion in jet engines, their second role is to act as a coolant, which ensures the normal operation of several parts including, the hydraulic and the lubrication systems, the combustor and the airframe and environmental control system [START_REF] Jia | A comprehensive review of the thermal oxidation stability of jet fuels[END_REF]. Due to their use as a heat sink, jet fuels undergo thermal stresses [START_REF] Zabarnick | Chemical kinetic modeling of jet fuel autoxidation and antioxidant chemistry[END_REF]. For example, in commercial aircrafts, it can be exposed to temperature of 393 to 433 K for several seconds under cruise conditions [START_REF] Kabana | Oxidative Stability of Middle Distillate Fuels. Part 1: Exploring the Soluble Macromolecular Oxidatively Reactive Species (SMORS) Mechanism with Jet Fuels[END_REF][START_REF] Harris | On the threshold -The outlook for supersonic and hypersonic aircraft[END_REF][START_REF] Hazlett | Thermal Oxidation Stability of Aviation Turbine Fuels[END_REF]. During their logistic operations, from their storage to the jet engine injectors, jet fuels are in contact with air under various pressure and temperature conditions. Dissolved oxygen from air in the fuel promotes the liquid phase oxidation of the fuel, also called autoxidation. This phenomenon is the source of major issues in aircrafts as it forms undesired products including gums and deposits.

Nowadays, the formulation of jet fuels, including additives, allows them to remain stable throughout their use as coolant in commercial aircrafts, but to temperatures up to 433 K [START_REF] Kabana | Oxidative Stability of Middle Distillate Fuels. Part 1: Exploring the Soluble Macromolecular Oxidatively Reactive Species (SMORS) Mechanism with Jet Fuels[END_REF]. In order to increase the efficiency of future turbine engines and design supersonic aircrafts operating at higher speed it is necessary to improve the thermal oxidation stability of jet fuel so that it can be stable up to temperatures as high as ≈573 K [START_REF] Zabarnick | Studies of the Impact of Fuel Deoxygenation on the Formation of Autoxidative Deposits[END_REF]. Comprehensive studies on jet fuel autoxidation are therefore essential to understand, predict and optimize their thermal oxidation stability.

Jet fuels are blends of hundreds of hydrocarbons. To facilitate jet fuel studies, including numerical and experimental works, researchers proposed several jet fuel surrogates [START_REF] Zhang | Recent development in studies of alternative jet fuel combustion: Progress, challenges, and opportunities[END_REF]. A jet fuel surrogate is a mixture of pure compounds combined to mimic physical and ignition characteristics of real jet fuels. n-decane is commonly used as a mono-component surrogate or included in multi-component surrogate for jet fuels [START_REF] Dortz | Evaluation of the surrogates capacity to reproduce the laminar burning velocities and the sensitivity to stretching of a commercial kerosene under constant volume combustion conditions[END_REF][START_REF] Naik | Detailed chemical kinetic mechanism for surrogates of alternative jet fuels[END_REF][START_REF] Mzé-Ahmed | Kinetics of Oxidation of a Synthetic Jet Fuel in a Jet-Stirred Reactor: Experimental and Modeling Study[END_REF][START_REF] El-Sayah | Comparison of the Effects of Different Biofuels on the Oxidation Stability of a Hydrocarbon Fuel[END_REF]. Thus, the development of a kinetic model of autoxidation of n-decane, validated against experiments, is a first step towards a jet fuel surrogate kinetic model. Kinetic modelling of n-decane autoxidation is also of great interest to understand lubricant degradation [START_REF] Pfaendtner | Mechanistic Modeling of Lubricant Degradation. 1. Structure-Reactivity Relationships for Free-Radical Oxidation[END_REF]. This remains a challenge in the literature as liquid phase detailed kinetic models are at their early stages of developments compared to gas-phase models [START_REF] Jalan | An Extensible Framework for Capturing Solvent Effects in Computer Generated Kinetic Models[END_REF][START_REF] Van De Vijver | Automatic Mechanism and Kinetic Model Generation for Gas-and Solution-Phase Processes: A Perspective on Best Practices, Recent Advances, and Future Challenges[END_REF]. It is commonly accepted in the community that both gas-phase oxidation and liquid-phase oxidation of hydrocarbons undergo similar radical mechanisms [START_REF] Van De Vijver | Automatic Mechanism and Kinetic Model Generation for Gas-and Solution-Phase Processes: A Perspective on Best Practices, Recent Advances, and Future Challenges[END_REF]. While the gas-phase oxidation mechanism of large alkanes such as n-decane is well understood [START_REF] Curran | Developing detailed chemical kinetic mechanisms for fuel combustion[END_REF], its liquidphase oxidation mechanism remains to be fully understood [START_REF] Jia | A comprehensive review of the thermal oxidation stability of jet fuels[END_REF].

Several studies on n-decane autoxidation have been performed in the literature. Cullis et al. [START_REF] Cullis | The oxidation of decane in the liquid and gaseous phases[END_REF] performed n-decane liquid and gas phases oxidation experiments in the same temperature region. They found some notable differences in the mechanisms in the two phases: absence of C 10 O-heterocycles and dihydroperoxides in the liquid phase and small amounts of lower mass oxygenated compounds, compared to gas phase experiments. Experiments on the slow combustion of n-decane were performed in a closed reactor between 445 and 473 K and liquid phase oxidation was conducted in a semi-open reactor in which a flux of oxygen was passed into an isothermal batch reactor at 423 K containing ndecane. Principal products of n-decane autoxidation were determined as decane monohydroperoxides, C 10 carbonyls and alcohols.

Goosen and Morgan [START_REF] Goosen | Autoxidation of nonane and decane: a product study[END_REF] studied the products obtained during the autoxidation of n-decane and n-nonane, at significant conversion. Oxygen was passed into boiling n-decane, in a reflux, at about 451 K. In the investigated conditions, these authors described the autoxidation of n-decane as proceeding in three stages: the first stage involves the formation of alkyl hydroperoxides, the second stage occurs when the hydroperoxide decomposes into alcohols and ketones, and the last stage corresponds to the oxidation of the latter products in competition with the alkanes, yielding γ-lactones and carboxylic compounds.

Pfaendtner and Broadbelt [START_REF] Pfaendtner | Mechanistic Modeling of Lubricant Degradation. 1. Structure-Reactivity Relationships for Free-Radical Oxidation[END_REF][START_REF] Pfaendtner | Mechanistic Modeling of Lubricant Degradation. 2. The Autoxidation of Decane and Octane[END_REF] used automated mechanism generation to study the autoxidation of n-decane to understand lubricant oil degradation. They proposed condensed phase adapted rate rules and kinetic parameters, based on structure reactivity correlations, and were able to simulate the autoxidation of n-decane and n-octane and validated the results against experimental data of the literature. This pioneering work was the first to use detailed chemical kinetic modeling to simulate the liquid phase oxidation of alkanes. However, it is entirely based on gas-phase thermochemical data and most of the kinetic data adopted are based on gas-phase estimations.

Recently, Chatelain et al. [START_REF] Chatelain | Wide Range Experimental and Kinetic Modeling Study of Chain Length Impact on n-Alkanes Autoxidation[END_REF] investigated the autoxidation kinetic of several n-alkanes including n-decane by conducting Induction Periods (IP) measurements at three different temperatures (393, 413, and 433 K) in a PetroOxy device. Liquid-phase mechanisms of these alkanes were generated thanks to an automated mechanism generator, RMG [START_REF] Gao | Reaction Mechanism Generator: Automatic construction of chemical kinetic mechanisms[END_REF], and were capable of predicting IP within a factor of 3 at high temperature (T > 413 K). Note that only the autoxidation kinetic models of n-octane and ndodecane are available from this study.

Auzani et al. [START_REF] Auzani | Assessment of ethanol autoxidation as a drop-in kerosene and surrogates blend with a new modelling approach[END_REF] assessed the effect of ethanol on the IPs of a real kerosene and three surrogates. They proposed a Python code to simulate the pressure drop in the PetroOxy measurements that solves the ordinary differential equations of a Perfectly Stirred Reactor for each species and an equation for the consumption of the number of moles of O 2 in the gas phase. RMG was used to generate detailed reaction mechanisms for n-decane, ethanol and n-dodecane and simulate some of their experimental results. Upon significant changes in the kinetic data of the R• + O 2 reactions, they were able to qualitatively simulate their experimental results. The same group also proposed a skeletal kinetic mechanism for liquid hydrocarbons autoxidation to simulate experiments on a fuel composed of five normal alkanes in a near-isothermal reactor and in a PetroOxy device [START_REF] Alborzi | Construction of a reduced chemical kinetic mechanism for autoxidation of n-paraffinic solvent -A model for aviation fuel[END_REF].

The motivations of our study are: [START_REF] Gössling | The global scale, distribution and growth of aviation: Implications for climate change[END_REF] to provide additional experimental data for the liquid-phase reactivity of n-decane; [START_REF] Jia | A comprehensive review of the thermal oxidation stability of jet fuels[END_REF] to define a rigorous framework to model IP data measured in PetroOxy devices; [START_REF] Zabarnick | Chemical kinetic modeling of jet fuel autoxidation and antioxidant chemistry[END_REF] to develop a validated kinetic model of n-decane autoxidation and (4) to refine the understanding of n-decane autoxidation. To reach these goals, IPs of n-decane were measured in a PetroOxy and were coupled with an experimental quantification of the total hydroperoxides (ROOH) content. These data allow us to define a rigorous criterion for the simulation of IPs which is based on the total amount of ROOH in the aged fuel. The kinetic model of n-decane was developed using our reaction mechanism generator (EXGAS [START_REF] Buda | Progress toward a unified detailed kinetic model for the autoignition of alkanes from C4 to C10 between 600 and 1200 K[END_REF][START_REF] Warth | Computer-Aided Derivation of Gas-Phase Oxidation Mechanisms: Application to the Modeling of the Oxidation of n-Butane[END_REF]) which has been adapted to automatically capture solvent effects on the thermochemical data based on corrections computed with an Equation of State (EoS) [START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF][START_REF] Moine | Can we safely predict solvation Gibbs energies of pure and mixed solutes with a cubic equation of state?[END_REF]. This approach has been validated in the case of the liquid-phase oxidation of n-butane [START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF]. The main reaction pathways of n-decane liquid-phase oxidation are presented and discussed in this paper. This work will contribute to improve the experimental kinetic studies performed in PetroOxy on jet fuels because a method to determine the liquid-vapor equilibrium in the test cell is proposed for the first time.

A new approach to determine the IP from homogeneous liquid phase simulations is also proposed, based on ROOH quantification. Finally, a new detailed kinetic model is proposed, based on the calculation of hundreds of new thermochemical data adapted to n-decane solvent and new reaction rate rules. This model is systematically validated against all available literature data. This model will constitute a building block for the construction of detailed multi-component jet fuel surrogate models which are necessary to understand the oxidation stability of jet fuels and to allow the optimization and formulation of future aviation energy carriers.

Experimental methods

PetroOxy description

Experiments of n-decane autoxidation were conducted in a PetroOxy, a device that has been widely used for fuel stability studies [START_REF] Bacha | Oxidation Stability of Diesel/Biodiesel Fuels Measured by a PetroOxy Device and Characterization of Oxidation Products[END_REF][START_REF] Webster | Investigation of the Thermal Oxidation of Conventional and Alternate Aviation Fuels with Comprehensive Two-Dimensional Gas Chromatography Accurate Mass Quadrupole Time-of-Flight Mass Spectrometry[END_REF][START_REF] Aminane | Experimental study of the kinetics of degradation of n-dodecane under thermo-oxidative stress at low temperature and mechanism inferred[END_REF][START_REF] Duong | New Approach for Understanding the Oxidation Stability of Neopolyol Ester Lubricants Using a Small-Scale Oxidation Test Method[END_REF] and employed as a reference instrument in ASTM D7525, ASTM D7545 and EN 16091 tests [START_REF] Nadkarni | Guide to ASTM test methods for the analysis of petroleum products and lubricants[END_REF]. A PetroOxy apparatus operates similarly as a closed batch reactor: it contains a gold-plated test cell, whose volume is 25 mL. The experiment begins with the introduction of 5 ml of fuel into the cell, which is then pressurized under 7 bar of pure oxygen. High-purity n-decane (⩾99%) from Sigma-Aldrich were employed in this work. The reactor is then heated quickly (about 3 minutes) from the ambient to the desired temperature and kept constant until the end of the test. The temperature in these tests ranged from 403 to 418 K. The gas-phase pressure is followed during each experiment and, as the autoxidation of n-decane consumes oxygen, a decrease of pressure is observed. The induction period (IP) is defined as the time corresponding to a 10% pressure drop from the maximum pressure reached at the beginning of the test (as illustrated in Figure 1). The IP characterizes the thermal oxidation stability of a fuel and decreases with increased oxidation rates of the fuels. The maximum uncertainty observed for our experimental induction periods is 10% of IP at the lowest temperature (403 K). This uncertainty was ascribed to all measured IPs (see Table S1).

Hydroperoxide quantification: iodometric method

Since the simulations of the autoxidation process are performed in a homogeneous liquid-phase reactor with a constant oxygen content [START_REF] Jalan | An Extensible Framework for Capturing Solvent Effects in Computer Generated Kinetic Models[END_REF][START_REF] Chatelain | Wide Range Experimental and Kinetic Modeling Study of Chain Length Impact on n-Alkanes Autoxidation[END_REF][START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF], it is difficult to directly simulate the, gas-phase, pressure decrease measured in the PetroOxy. In the literature [START_REF] Chatelain | Wide Range Experimental and Kinetic Modeling Study of Chain Length Impact on n-Alkanes Autoxidation[END_REF][START_REF] Amara | Toward Predictive Modeling of Petroleum and Biobased Fuel Stability: Kinetics of Methyl Oleate/n-Dodecane Autoxidation[END_REF], the definition of the IP in the simulations was assumed to correspond to a conversion of 5% of the fuels oxidized in the PetroOxy. However, the extent of fuel conversion at IP is fuel dependent definition is not universal and Mielczarek et al. [START_REF] Mielczarek | Toward the Accurate Prediction of Liquid Phase Oxidation of Aromatics: A Detailed Kinetic Mechanism for Toluene Autoxidation[END_REF] indicated that the conversion of toluene at corresponding IP was around 40%. In this work, we propose to link the IP with another experimental measure: the total hydroperoxides content at the IP. This creates a universal criterion to define the IP in the simulations, where the simulated IP is reached when the simulated total content of ROOH matches the experimental one. To quantify the total hydroperoxides concentration, two methods based on iodometric titration were used in this study.

Colorimetry

The iodometric titration method developed by Roohi and Rajabi [START_REF] Roohi | Iodometric Determination of Hydroperoxides in Hydrocarbon Autoxidation Reactions Using Triphenylphosphine Solution as a Titrant: A New Protocol[END_REF] was used to quantify the total hydroperoxides content in aged fuels at the end of each PetroOxy test. The main advantage of this method, compared to the classic titration that uses thiosulfate in aqueous solution as titrant [START_REF] Morris | AN IMPROVED METHOD FOR MEASURING HYDROPEROXIDES IN FUEL USING POTENTIOMETRY AND ALTERNATIVES TO CHLOROFLUOROCARBON SOLVENTS[END_REF][START_REF] West | Determination of Hydroperoxides in Jet Fuel via Reaction with Triphenylphosphine[END_REF],

resides in the use of triphenylphosphine diluted in n-decane as titrant. Solubility issues of n-decane in water are therefore avoided in the titrant. The titration was performed in two steps:

1) the first step involves the oxidation of iodide anion I -to iodine I 2 by hydroperoxides at a temperature of 68 °C. A volume of 1 ml of aged n-decane, sampled at the end of the IP in the PetroOxy, was added to a solution of potassium iodide (KI, anhydrous 99.9%, VWR Chemicals) and acetic acid (≥ 99.8 %, Sigma Aldrich) in absolute ethanol (≥99.8%, VWR Chemicals). The reaction is presented in equation [START_REF] Gössling | The global scale, distribution and growth of aviation: Implications for climate change[END_REF]. The resulting solution has a yellow-reddish color of iodine.

ROOH + 2H + + 2I

-I 2 + ROH + H 2 O (1)

(Colorless) (Yellow-reddish)

2) The second step involves the titration of the iodine with triphenylphosphine (TPP, 99 %, Sigma Aldrich), which produce an uncolored complex [TPPI] + [I] -, as presented in equation ( 2). The color change at the endpoint is therefore provided by the complexation of iodine. TPP was prepared in a solution of pure n-decane (99+%, ACROS Chemicals) at a concentration of 0.08 mol/L. The titration endpoint point was determined as a sharp color change was observed.

I 2 + TPP [TPPI] + [I] - (2)
(yellow-reddish) (colorless)

The uncertainty of the concentration of ROOH with the colorimetric titration depends mainly on the uncertainty of the volume of the titration endpoint, which comes from different sources of errors: the reading of the burette, the uncertainty of the burette (given by the constructor), and the manipulations.

The detection limit of the equivalence point depends on the size of the drop of the burette which is 0.05 mL. The uncertainty calculations are presented in details in the supplementary material. The uncertainty in equivalence volumes determined via colorimetry is ±0.14 mL, with a confidence interval of 95%.

Potentiometry

As colorimetry, potentiometry is used to quantify I 2 after the iodometric reaction. The biggest difference between two methods is the considered indicator which is a sharp color change in iodometric titration and a sharp electric potential change in potentiometric titration. Potentiometry has been shown to improve the method for measuring hydroperoxides in fuels, thanks to a better determination of the endpoint compared to the visual detection of the color change in standard titration [START_REF] Morris | AN IMPROVED METHOD FOR MEASURING HYDROPEROXIDES IN FUEL USING POTENTIOMETRY AND ALTERNATIVES TO CHLOROFLUOROCARBON SOLVENTS[END_REF]. The electric potential during the titration of I 2 was followed by a Mettler Toledo Compact Titrator G10S.

As for the colorimetry, the uncertainty on the final ROOH concentration obtained by potentiometry method mainly depends on the uncertainty of the titration endpoint volume. The use of a potentiometer allows a better detection of end points because of lowest size of the drops of the burette. Despite the uncertainty on the measured potential (given by the constructor) the global uncertainty in the titration endpoint volume remains well below the uncertainty of the colorimetric titration with a value of ±0.04 mL, with a confidence interval of 95% (see Table S2 for more details).

Kinetic modelling

Detailed kinetic model development

In a recent work [START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF], we presented an approach to develop a detailed kinetic model for fuel autoxidation based on the corresponding gas-phase kinetic model. In order to capture solvent effects, it is essential to include thermodynamic, diffusion and kinetic corrections. The details of the method can be accessed in the work on n-butane autoxidation kinetics [START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF], and only the most important features are recalled here.

The first step in the development of the detailed chemical kinetic model for the autooxidation of ndecane was the generation of the low-temperature oxidation (gas-phase) kinetic mechanism, using the EXGAS software [START_REF] Buda | Progress toward a unified detailed kinetic model for the autoignition of alkanes from C4 to C10 between 600 and 1200 K[END_REF][START_REF] Warth | Computer-Aided Derivation of Gas-Phase Oxidation Mechanisms: Application to the Modeling of the Oxidation of n-Butane[END_REF]. This software has been extensively used and validated for the computer aided generation of combustion kinetics models of hydrocarbons and oxygenated fuels [START_REF] Buda | Progress toward a unified detailed kinetic model for the autoignition of alkanes from C4 to C10 between 600 and 1200 K[END_REF][START_REF] Giarracca | Experimental and kinetic modeling of the ignition delays of cyclohexane, cyclohexene, and cyclohexadienes: Effect of unsaturation[END_REF][START_REF] Schönborn | Auto-ignition control using an additive with adaptable chemical structure. Part I: Development of a kinetic model for 1,3cyclohexadiene and 1,3,5-hexatriene combustion[END_REF][START_REF] Schönborn | Autoignition Control Using an Additive with Adaptable Chemical Structure. Part 2. Development of a PRF Kinetic Model Including 1,3-Cyclohexadiene Mechanism and Simulations of Ignition Control[END_REF][START_REF] Bounaceur | Prediction of Auto-Ignition Temperatures and Delays for Gas Turbine Applications[END_REF][START_REF] Sirjean | Pericyclic reactions in ether biofuels[END_REF]. The generated mechanism is composed of three parts: a primary and a secondary mechanism and a C 0 -C 2 reaction basis. In the primary mechanism, all the possible elementary reactions of the reactants, and free radicals produced, are written and leads to a large number of primary products. If the primary molecular products are larger than C 2 , their decomposition reactions are generated and are lumped to yield C 0 -C 2 products in order to avoid an excessive growth of the size of the mechanism. For products lower than C 2 , the combustion reactions are included in the exhaustive C 0 -C 2 reaction base. The details of the adaptation of the n-decane gas-phase model to the liquid-phase is described hereafter.

Thermochemical data

The gas-phase thermochemical data for all the species that do not belong to the C 0 -C 2 reaction base were generated using the THERGAS code [START_REF] Muller | THERGAS: a computer program for the evaluation of thermochemical data of molecules and free radicals in the gas phase[END_REF] based on Benson group additivity method [START_REF] Benson | Additivity rules for the estimation of thermochemical properties[END_REF]. C 0 -C 2 species thermochemistry is taken form literature databases. The gas-phase data are then corrected by calculated at infinite dilution, using the equation of state (EoS) universal mixing rule Peng-Robinson UNIFAC (UMR-PRU) [START_REF] Voutsas | Thermodynamic property calculations with the universal mixing rule for EoS/GE models: Results with the Peng-Robinson EoS and a UNIFAC model[END_REF] to obtain the liquid-phase thermochemical data. Further details of calculation can be accessed elsewhere [START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF][START_REF] Moine | Can we safely predict solvation Gibbs energies of pure and mixed solutes with a cubic equation of state?[END_REF][START_REF] Moine | Estimation of solvation quantities from experimental thermodynamic data: Development of the comprehensive CompSol databank for pure and mixed solutes[END_REF].

Liquid-phase kinetic data

The typical reaction pathways occurring during the oxidation of an alkane in the liquid phase, at low conversion, can be schematized as in Figure 2 [47]. autoxidation [START_REF] Cullis | The oxidation of decane in the liquid and gaseous phases[END_REF][START_REF] Emanuel | OF THE CHAIN REACTIONS TAKING PLACE IN THE LIQUID-PHASE OXIDATION OF HYDROCARBONS[END_REF] and that this reaction class can be ignored in the reaction mechanism [START_REF] Pfaendtner | Mechanistic Modeling of Lubricant Degradation. 1. Structure-Reactivity Relationships for Free-Radical Oxidation[END_REF][START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF]. A set of new reaction rate rules, adapted to the liquid-phase oxidation of n-decane, was established for the self-termination reactions of peroxy radicals, and the H-atom abstractions by peroxy and alkoxy radicals based on the corresponding reactions in the autoxidation model of n-butane [START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF]. For the other reactions, the EXGAS reaction rate rules were adopted as the simulations demonstrated that they are not sensitive for n-decane conversion and for the products formation. The most important features of the developed mechanism are presented below. for a secondary H-atom (R2-R5) : k = 2.8 x10 13 x exp(-50650 cal mol -1 /RT) cm 3 mol -1 s -1 ). The influence of diffusive rate constants on these kinetic parameters were verified and were found to be negligible. This point is further discussed in the 3. (k 6,7,8,10,11,12,13 ) = 2.0 x10 13 x exp(-28700/RT)) and R• → alkene + •CH 3 (k 9 = 2.0 x10 13 x exp(-31000/RT)) in cm 3 , mol, cal, K, s units.

Initiations and beta-scissions

Bimolecular

Additions on oxygen and oxidations

The five decyl radicals (C 10 H 21 ) reacts with O 2 to give decyl peroxy radicals, C 10 H 21 OO• (R14-R18). EXGAS rate rules were adopted for these reactions with the rate constant k s = 1.8 x10 19 x T -2.5 for the addition of secondary radical onto O 2 (R15-R18) and k p = 9.0 x10 18 x T -2.5 in the case of the primary decyl radical (R14), in cm 3 , mol, cal, K, s units. Diffusivities were calculated for these reactions (see 3.1.3) and were found to be negligible. alkoxy radicals based on theoretical calculations with an implicit continuum solvation model.

Disproportionations

Disproportionations are known as chain-terminating processes. In the model, these reactions are included for all important peroxy radicals. For example, decyl peroxy react with HO 2 as ROO• + HO 2 ROOH + O 2 . Kinetic parameters of these reactions follow EXGAS rate rules.

Another type of disproportionation considered in the model is the self-terminations of decyl peroxy radicals. This reaction class is known crucial processes in the autoxidation of alkanes [START_REF] Pfaendtner | Mechanistic Modeling of Lubricant Degradation. 1. Structure-Reactivity Relationships for Free-Radical Oxidation[END_REF][START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF][START_REF] Emanuel | OF THE CHAIN REACTIONS TAKING PLACE IN THE LIQUID-PHASE OXIDATION OF HYDROCARBONS[END_REF]. The were proposed as k 1 = 1.0 x 10 11 x exp(2700 cal mol -1 /RT) cm 3 mol -1 s -1 , k 2 = 2.0 x 10 11 x exp(2700 cal mol - 1 /RT) cm 3 mol -1 s -1 . The ratio α = k 1 / k 2 is 0.5. It has been shown in n-butane liquid-phase oxidation [START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF] that the ratio α could be between 0.5 and 1 to get reasonable simulation results. It is interesting to note that a ratio α=1 was used in the n-butane kinetic model to obtain the best agreement between simulations and experiments. In the case of n-decane, the ROO• + ROO• → RO• + RO• + O 2 channel is favored over the ROO• + ROO• → ROH + R 1 COR 2 + O 2 by a factor of 2. It is difficult to find an explanation for this difference, it can be either a meaningful physical effect or a consequence of the lumping of the C 10 alcohols and C 10 carbonyl compounds. Indeed, in the n-butane autoxidation model, no lumping of alcohols and ketones was used.

Decompositions of hydroperoxides

The unimolecular decompositions of hydroperoxides (ROOH) plays an important role as these processes generate free radicals (RO• and OH). The rate constant of the decomposition of n-decyl hydroperoxide is k = 1.60 x 10 11 x exp(-31700/RT) in mol, cal, K, s units. This rate constant results from a regression analysis of experimental data reported in the review of Dixon [START_REF] Dixon | Decomposition Rates of Organic Free Radical Initiators[END_REF].

Decanol sub-mechanism

Decanol is one of the major products quantified experimentally in n-decane liquid-phase oxidation [START_REF] Cullis | The oxidation of decane in the liquid and gaseous phases[END_REF][START_REF] Chatelain | Wide Range Experimental and Kinetic Modeling Study of Chain Length Impact on n-Alkanes Autoxidation[END_REF]. To limit the size of the mechanism, a simplified sub-mechanism was developed for decanol, following the validated approach used for butanol in the autoxidation model of n-butane [START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF]. In the model, the lumped alcohol C 10 H 21 OH is consumed by H-abstraction by decyl peroxy (ROO•) and OH radicals to produce an α-hydroxyl alkyl (•ROH) radical noted as RC10H21OL. In this approach, the Habstraction is assumed to solely occur on the α-site of the hydroxyl group because of a lower C-H bond dissociation energy at this position compared to other H-atoms of decanol. The •ROH radical is then oxidized by O 2 to form lumped C 10 carbonyl compounds and HO 2 . The associated rate constants are estimated using the proposed rate rules in the n-butane model [START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF].

Diffusion correction

To account for diffusive limits in the liquid phase, a diffusion rate constant (k diff ) has to be combined to the intrinsic liquid-phase constant (k int,liq ). This leads to the following effective rate constant (k eff ) for a bimolecular reaction A+B  (A-B)  Products:

k diff was calculated using the Stokes-Einstein approach for diffusion of spherical particles in the liquid phase [START_REF] Jalan | An Extensible Framework for Capturing Solvent Effects in Computer Generated Kinetic Models[END_REF][START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF]. In this approach, molecular radii of considered species are calculated based on the UNIFAC molecular size parameter included in UMR-PRU equation of state [START_REF] Moine | Can we safely predict solvation Gibbs energies of pure and mixed solutes with a cubic equation of state?[END_REF]. Temperature-dependent viscosities are read in a literature database [START_REF] Rowley | DIPPR^(! R) Data Compilation of Pure Chemical Properties[END_REF]. Diffusion calculations were conducted for all the most important bimolecular reactions included in the main reaction flux of n-decane. It is found that the values of k diff are nearly similar for the different considered reactions and increase with temperatures.

k diff varies from 4 x 10 13 at 300 K to about 2 x 10 16 cm 3 mole -1 s -1 at 500 K. As the values of k diff of the considered reactions are very high because of the relatively low viscosity of n-decane (0.29 mPa.s at 400 K), k eff = k int,liq . A similar behavior was observed for n-butane which has a much lower liquid viscosity (0.005 mPa.s at 400 K).

Calculation of liquid and gas phases initial compositions in the PetroOxy

The simulation of liquid-phase oxidation in the PetroOxy is performed using a homogeneous batch reactor model at fixed temperature (T), pressure (P) and volume (V) and with a constant dissolved O 2 concentration [START_REF] Jalan | An Extensible Framework for Capturing Solvent Effects in Computer Generated Kinetic Models[END_REF][START_REF] Chatelain | Wide Range Experimental and Kinetic Modeling Study of Chain Length Impact on n-Alkanes Autoxidation[END_REF][START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF]. In the input, T is the target temperature (e.g. 140°C) that is used to accelerate the oxidation phenomena. However, in the PetroOxy experiments, the 25 mL cell is first filled with 5 mL of n-decane and sealed under atmospheric air pressure. The cell is then purged and filled with 7 bar of O 2 at ambient temperature and heated to the target temperature. With the increasing temperature, the compositions of the liquid and gas phases evolve. It can be noted that the heating time to reach the control temperature is a few minutes.

In conjunction with the temperature increase, the pressure in the cell test increases from the initial value at 25°C (700 kPa) to a maximum value. Several competing physical phenomena affect the pressure evolution, including the thermal expansion, the evaporation of the fuel, and the solvation of oxygen in the liquid phase. An iterative calculation method has been developed to estimate molar compositions of both liquid and gas phases in a PetroOxy test, when the gas pressure reaches its maximum pressure.

These molar compositions were used as inputs in IP simulations. Figure 3 presents the simplified boundary conditions that needs to be considered in the calculations. In Figure 3, F1 is the fuel, x n and y n are the mole fractions of the compound n (1 for the liquid fuel and 2 for oxygen) in the liquid and gas phases, respectively. In the initial state (T° = 25 °C), it is assumed that the gas phase contains only oxygen (O 2 , y°2 = 1) and the initial gas pressure (P°) is equal to the initial oxygen pressure (P initial O2 = 7 bar). Similarly, the liquid phase is composed only of fuel (F1, x°1 = 1). The initial volume of the liquid phase, noted V°l iq , is 5 mL. The transient state, represented at the center of Figure 3, corresponds to the heating phase of the reactor and the final state corresponds to the thermodynamic equilibrium at the test temperature.

The system obeys the laws of conservation during the temperature increase: volume conservation (V gas + V liq = V total = 25 mL = constant) and the material balance: where n gas and n liq are the mole number of the gas and liquid phases, respectively; are the mole number of the component i in the gas and liquid phases.

The total volume of the gas phase obeys the perfect gas law, . The total volume of the liquid phase is deduced from the density of the liquid (D liq ) phase, .

In the final state, both the gas and liquid phases contain mixtures of the fuel F1 and oxygen. The molar compositions of both phases and the final pressure (P) are related and are described using the so-called thermodynamic approach in which the liquid solvent is assumed to obey Raoult's law while the liquid solute is described using Henry's law; the gas phase is considered as a perfect gas. These assumptions are justified as long as (i) the molar fraction of the solvent in the liquid phase is nearly one ( ), (ii) the molar fraction of the liquid solute approaches zero ( ) and (iii), the total pressure remains below 10 bar (ensuring that the perfect-gas assumption is acceptable).

P 1 sat corresponds to the vapor pressure of the pure fuel F1 and H O2 is the Henry's law constant of oxygen in F1 solvent. Both these properties depend on temperature only.

Remembering that and ( ), the final problem to be solved is summarized in the following system, gathering 5 equations: , , , are input variables. The 5 unknowns are , , ,

, .

In practice, the resolution of Eq. (1-5) is carried out using successive substitution methods. The proposed iterative calculation method consists of several steps and is presented in Figure 4. In the first step, a gas pressure in the final state is guessed based on the ideal gas law, assuming constant mole number and volume of the gas phase. Next, molar composition in the liquid phase is deduced thanks to the conservation of volume, material balance and relations ( 1) and ( 2). The thermodynamic properties involved in these equations, i.e., the liquid density (D liq ), vapor pressure (P sat ), Henry's law constant of oxygen (H O2 ) in solvent, are obtained from the Predictive SRK (PSRK) equation of state [START_REF] Holderbaum | PSRK: A group contribution equation of state based on UNIFAC[END_REF] implemented in Simulis software [START_REF] Baudouin | Simulis® Thermodynamics: an open framework for users and developers[END_REF]. The PSRK model, capable of guesstimating the above properties, was chosen for both its flexibility and accuracy.

At the end of a calculation process, the sum of molar composition in the liquid phase is checked whether it is equal to 1. A tolerance of 0.005 is adopted. If the checking condition is not satisfied, a new liquid density and gas pressure are estimated and the calculation loop starts again until the validation of the checking condition.

P i = P°x T/T°y 1 °, y 2 °= 1 -y 1 °x1 °, x 2 °= 1 -x 1 °Vliq °, V gaz °molecular weights M1, M2 i starts from 1 Initial estimation D liq (T, x 1 i-1 ,x 2 i-1 ) V gas i = V total -V liq i n gas i = P i x V gas i / (RT) n liq i = n total -n gas i y 2 i = 1 -y 1 i n 1,gas i = n gas i x y 1 i n 1,liq i = n 1,total -n 1,gas i x 1 i = n 1,liq i / n liq i H O2 i (T) 0.995 < x 1 i + x 2 i < 1.005 ? P ELV (T) = P i , V liq (T, n liq ) = V liq i n liq , n gas x 1 , x 2 y 1 ,y 2 Re-estimate P i -Decreasing P i if x 1 i + x 2 i > 1 -Increasing P i if x 1 i + x 2 i < 1 NO D liq i (T, x 1 i , x 2 i ) D liq (T, x 1 i-1 ,x 2 i-1 ) D liq (T, x 1 i ,x 2 i ) ? YES YES NO x 2 i = 1 -x 1 i 21
The iterative calculation presented in Figure 4 was used to determine the initial composition at the set temperature T in the PetroOxy experiments. This method enables the simulation of the gas pressure profiles as a function of temperature at the beginning of the tests, until the maximum pressure is reached (Figure 5). S3.

In general, simulations reproduce the measured pressure profiles and are able to capture the equilibrium between the liquid and vapor phases.

The physical model predicts that at the end of the heating time of 5 minutes, the volume of liquid increases from 5.0 to 5.7 mL, going from 298 to 413 K. At 413 K, the mole fractions of n-decane and dissolved O 2 in the liquid phase are 0.98 and 0.02, respectively. For this condition, the corresponding gasphase mole fractions are 0.04 for n-decane and 0.96 for O 2 .

The mole compositions of the liquid phase corresponding to the maximum gas pressure are also determined, at different temperatures (Table 1). The calculated compositions were used as concentration inputs to simulate the induction periods of n-decane at different temperatures. It is interesting to note that the calculated concentration of O 2 dissolved in n-decane increases with temperature. A similar behavior is well known for O 2 in jet fuels [START_REF] Hazlett | Thermal Oxidation Stability of Aviation Turbine Fuels[END_REF].

All the simulations were performed using a homogeneous liquid batch reactor (constant T, P, and V) solver based on Chemkin II library [START_REF] Kee | Chemkin-II: A Fortran chemical kinetics package for the analysis of gas-phase chemical kinetics[END_REF] with constant O 2 concentration [START_REF] Jalan | An Extensible Framework for Capturing Solvent Effects in Computer Generated Kinetic Models[END_REF][START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF][START_REF] Mielczarek | Toward the Accurate Prediction of Liquid Phase Oxidation of Aromatics: A Detailed Kinetic Mechanism for Toluene Autoxidation[END_REF].

Results and Discussions

Experimental results in PetroOxy are first presented, then, the simulations and the performances of the n-decane autoxidation model are shown. Based on the validated model, the reaction pathways of ndecane autoxidation are analyzed and the key reactions are highlighted thanks to a sensitivity analysis. with a mean relative deviation of 16% over the temperature range studied. Further validations of the kinetic model were performed using oxidation products profiles.

Liquid-phase reactivity of n-decane

For one condition of oxidation stability test, at 413 K and 7 bar of oxygen, the total concentration of ROOH as a function of time was measured. To do so, PetroOxy tests were stopped every t = t(previous test) + 15 min until the IP time was reached (Figure 9). The total [ROOH] content was measured by potentiometric titration. The titration of fresh, unaged, n-decane revealed that the liquid already contained hydroperoxides at a concentration of 0.016 mol/L. This initial hydroperoxides content was also employed as inputs in all simulations of n-decane autoxidation. S6.

The experiments show that the concentration of hydroperoxides increases with time as n-decane oxidation progresses. For these conditions, and within the IP time range, no extremum is observed in the The reaction scheme of n-decane is found to be very similar to the one of n-butane [START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF].

Sensitivity analysis of the autoxidation kinetic model of n-decane

Sensitivity analyses have been performed to determine the most sensible reactions in an oxidation mechanism of fuel [START_REF] Le | Experimental and numerical investigation of the promoting effect of a cetane booster in a low-octane gasoline fuel in a rapid compression machine: A study of 2-ethylhexyl nitrate[END_REF]. These analyses were conducted on the conversion of n-decane at 413 K and 7 bar initial of oxygen. Sensitivity analyses were initiated by the modification of the A-factor of the rate constant of each reaction, which was multiplied by a factor of ten. For each modification, a simulation was performed to determine the modified conversion (X i ) of n-decane. The sensitivity coefficient (τ i ) of each reaction was then calculated as follows:

X 0 corresponds to the simulated conversion of n-decane with the original unmodified model. A positive sensitivity coefficient represents a promoting effect of the considered reaction on the conversion of n-decane. Conversely, a negative coefficient shows an inhibiting effect of the considered reaction on the reactivity of n-decane. Figure 12 summarizes the results of the sensitivity analyses.

Conclusion

Experiments and modeling works were conducted to investigate the autoxidation of n-decane. The induction periods of n-decane were measured in a PetroOxy and the total hydroperoxydes content in oxidized n-decane was systematically quantified using colorimetric and potentiometric titrations.

Modeling works were performed to examine, in depth, the liquid-phase reactivity of n-decane. For the first time, a physical model was proposed to estimate liquid-phase composition at the beginning of a Kinetic analyses indicate that n-decane autoxidation follows a classical chain mechanism applied for the liquid-phase oxidation. The main consumption pathway of n-decane is the H-abstraction by C 10 peroxy radicals leading to the formation of C 10 hydroperoxides as major products. Similar to n-butane autoxidation modeling, the self-reactions of peroxides radicals (ROO•) and its exit channels branching ratios play an important role in the reaction fluxes. The uncertainty of the kinetic parameters of this type of reactions remains high and deserves more studies. The proposed kinetic model is validated for limited conversions (up to about 15%) of n-decane. In order to expend its validity for higher conversions, the secondary mechanism needs to be further detailed.

This work contributes to the systematic understanding of the oxidation stability of n-decane and the validated kinetic model is a first step towards the development of a detailed kinetic model for the autoxidation of a more complex jet fuel surrogate.

Supplemental material

Figure 1 .

 1 Figure 1. Evolution of measured pressure during a typical PetroOxy test and definition of the induction period (IP) Fuel: n-decane, test temperature: 413 K, initial O 2 pressure: 7 bar.

Figure 2 .

 2 Figure 2. Simplified reaction pathways of an alkane in the liquid phaseThe main consumption channels of the reacting alkane (RH) are H-atom abstractions by peroxy radical

  self-termination of C 10 peroxy radicals has two exit channels: (1) producing two decoxy radicals (C 10 H 21 O•) and O 2 , and (2) producing O 2 , C 10 alcohols and C 10 carbonyl compounds. In our model, isomers of C 10 carbonyl compounds were lumped as C 10 H 20 OK. For these two reaction classes, kinetic parameters

Figure 3 .

 3 Figure 3. Simplified scheme presenting the different steps when the temperature increases in the PetroOxy.

F1

Figure 4 .

 4 Figure 4. Iterative calculation method to estimate the maximum pressure and molar composition in PetroOxy experiments

Figure 5 .

 5 Figure 5. Simulations of pressure profiles as a function of time at the beginning of PetroOxy tests (heating phase up to the set temperature T) of n-decane at different T: (a) 403 K, (b) 408 K, (c) 413 K, (d) 418 K. Symbols are experimental data and solid lines are simulated pressure profiles. Experimental pressure profiles of heating phases are given in TableS3.

Figure 6

 6 Figure 6 presents the measured induction periods of n-decane as a function of the inverse of the

Figure 6 .

 6 Figure 6. Induction periods of n-decane measured in a PetroOxy at 7 bar initial of oxygen and comparison with data reported byChatelain et al.[START_REF] Chatelain | Wide Range Experimental and Kinetic Modeling Study of Chain Length Impact on n-Alkanes Autoxidation[END_REF].Total hydroperoxides content in aged n-decane was quantified by iodometry, using colorimetric and

Figure 8 .

 8 Figure 8. Comparison between simulated and experimental induction periods of n-decane in a PetroOxy device, at 7 bar of oxygen. Criteria for simulated IP: simulated [ROOH] total = [ROOH] total measured at IP. Under the investigated conditions, the measured IP of n-decane presented in logarithmic scale varies

Figure 9 .

 9 Figure 9. Simulated and measured concentrations of total hydroperoxides formed during a PetroOxy test of n-decane at 413 K and 7 bar initial of oxygen. Experimental data are given in TableS6.

[

  ROOH] curve as a function of time. The simulations agree with the ROOH profile measured in the PetroOxy rig. Simulations show that the formed ROOH are solely C 10 H 21 OOH. Under these conditions, the conversion of n-decane at the end of the test (190 minutes) reaches 3.6% and the product branching ratios are 0.76, 0.18 and 0.06 for C 10 H 21 OOH, C 10 H 21 OH and C 10 H 20 (C=O), respectively (C 10 H 20 (C=O) are lumped decanones and decanals and C 10 H 21 OH are lumped decanols). Figure S1 presents these product profiles as a function of time. The flux analysis shows that almost all C 10 alcohols come from Habstractions from n-decane by C 10 alkoxy radicals, which are produced in the unimolecular decomposition of the C 10 H 21 OOH. Our new kinetic model of autoxidation of n-decane was also compared with experimental data of the literature. Cullis et al. [17] investigated n-decane autoxidation in a semi-open reactor in which a flux of oxygen was passed through the liquid fuel, at a constant temperature of 423 K. Major products of ndecane autoxidation includes C 10 -hydroperoxides, C 10 -alcohols, and C 10 carbonyls that were quantified as a function of time. Figure 10.a presents the formation of C 10 hydroperoxides per initial mole of n-decane and figure 10.b illustrates the formation of sum of C 10 -decanols and C 10 -decanones/decanal per initial mole of n-decane.

Figure 10 .

 10 Figure 10. Simulation of products formed during the autoxidation of n-decane and comparison with experiments from Cullis et al. [17] at 423 K, P ini, O2 = 700 kPa, V ini, liquid = 5 ml. Symbols: experiments, lines: simulations. (a): formation of C 10hydroperoxides, (b): formation of C 10 -ketones/decanal and C 10 alcohols.

4. 2 . Kinetic model analysis 4 . 2 . 1 .

 2421 Reaction pathways of n-decane autoxidationRate of production (ROP) analyses were carried at 413 K, P ini,O2 = 700 kPa, V ini,liq = 5 mL for a conversion of n-decane of about 4%. Figure11presents the main reaction pathways of n-decane liquid-phase oxidation at the examined condition.

Figure 11 .

 11 Figure 11. Reaction pathways of n-decane in the liquid phase at 413 K. ≈ 4% conversion of n-decane (190 min residence time).

Figure 12 .

 12 Figure[START_REF] El-Sayah | Comparison of the Effects of Different Biofuels on the Oxidation Stability of a Hydrocarbon Fuel[END_REF]. Sensitivity analyses on the conversion of n-decane during its autoxidation at 413 K The reactions that promote the conversion of n-decane are H-atom abstractions from n-decane (n-

  -OO => C10H21 + C10-OOH C10-OO + C10-OO => C10-O + C10-O + O2 C10-OO + C10-OO => C10-OH + C10=O + O2

  initiations involve the reactions R1-R5 of n-decane (n-C 10 H 12 ) with O 2 . Five isomers of n-decyl radicals (1-C 10 H 21 , 2-C 10 H 21 , 3-C 10 H 21 , 4-C 10 H 21 , 5-C 10 H 21 ) are produced. The kinetic parameters were taken

from EXGAS rate rules (for a primary H-atom, (R1): k = 4.2 x10

13

x exp(-53033 cal mol -1 /RT) cm 3 mol -1 s -1 ;

  In the model, the additions on O 2 of the decyl radicals compete with the oxidation reactions, which consist in the oxidation of decyl radicals by oxygen to form decene (R19-R23). All decene isomers were lumped into one species (C 10 H 20 ) to reduce the complexity of the model. H-atom abstractions are the principal consumption flux of n-decane in liquid-phase oxidation, yielding five n-decyl radicals. These reactions involve the transfer of a H-atom from n-decane to radicals that are present in significant amounts. In our mechanism, these radicals are: hydroxyl (OH), hydroperoxyl (HO 2 ), methyl (•CH 3 ), methoxy (CH 3 O•), methyl peroxy (CH 3 OO•), five isomers of decoxy radicals (C 10 H 21 O•) and five isomers of n-decyl peroxy radicals (C 10 H 21 OO•). In the model, the five isomers of n-decanol were lumped as C 10 H 21 OH and the five isomers of n-decyl hydroperoxide were lumped as C 10 H 21 OOH. In the experiments of n-decane liquid-phase oxidation[START_REF] Cullis | The oxidation of decane in the liquid and gaseous phases[END_REF][START_REF] Chatelain | Wide Range Experimental and Kinetic Modeling Study of Chain Length Impact on n-Alkanes Autoxidation[END_REF], decanol was determined as one of the major products. Most H-atom abstractions follow EXGAS rate rules except H-atom abstractions by alcoxy and peroxy radicals, which were shown to be very sensitive in n-butane autoxidation. In the model of n-

	EXGAS rate rules were used for the kinetic data.
	3.1.2.3	H-atom abstractions
		1-•C 10 H 21 + O 2 = 1-C 10 H 21 OO•	(R14)
		2-•C 10 H 21 + O 2 = 2-C 10 H 21 OO•	(R15)
		3-•C 10 H 21 + O 2 = 3-C 10 H 21 OO•	(R16)
		4-•C 10 H 21 + O 2 = 4-C 10 H 21 OO•	(R17)
		5-•C 10 H 21 + O 2 = 5-C 10 H 21 OO•	(R18)
		1-•C 10 H 21 + O 2 = C 10 H 20 + HO 2	(R19)
		2-•C 10 H 21 + O 2 = C 10 H 20 + HO 2	(R20)
		3-•C 10 H 21 + O 2 = C 10 H 20 + HO 2	(R21)
		4-•C 10 H 21 + O 2 = C 10 H 20 + HO 2	(R22)
		5-•C 10 H 21 + O 2 = C 10 H 20 + HO 2	(R23)

decane, the rate constants of H-atom abstractions by C 10 peroxy radicals are estimated using an analogy with the H-abstraction from n-butane by butylperoxy radicals

[START_REF] Le | Development of a Detailed Kinetic Model for the Oxidation of n-Butane in the Liquid Phase[END_REF]

. The kinetic parameters of H-atom abstraction by C 10 alkoxy radicals were estimated by analogy with H-atom abstractions of n-butane by C 4

Table 1 .

 1 Calculated molar composition of the liquid phase corresponding to the maximum gas pressure during PetroOxy tests of n-decane.

	Cell temperature	[n-Decane]	[Oxygen]
	(K)	(mol/L)	(mol/L)
	403	4.51	8.56 x 10 -2
	408	4.48	8.56 x 10 -2
	413	4.45	8.75 x 10 -2
	418	4.41	8.97 x 10 -2

  The simulation agrees with experimental results. The ratio [C 10 alcohol] / [C 10 carbonyls] is constant over the reaction time and is equal to 3. At a reaction time of 117 min, 3.6% of conversion is reached and the same branching ratios as those found in the simulation plotted in Figure9are found for C 10 H 21 OOH, C 10 H 21 OH and C 10 H 20 (C=O) (0.76, 0.18 and 0.06). At 250 min, n-decane conversion is 16% and the branching ratios of C 10 H 21 OOH, C 10 H 21 OH and C 10 H 20 (C=O) are 0.58, 0.31 and 0.10, respectively. It can be noted that water is also a major product of n-decane autoxidation but is not included in the latter branching ratios because it does not contain carbon atoms. The concentration of H 2 O is always similar to that of C 10 H 20 (C=O) in the simulations of PetroOxy experiments and of the data of Cullis et al. [17]. For example, simulated [H 2 O] is equal to 6.7 10 -2 mol l -1 at the end of Cullis et al. experiments vs 6.67 10 -2 mol l -1 for C 10 H 20 (C=O).

	1
	(a)
	(a)
	(b)
	(b)
	2

  The latter radicals entirely yield decyl peroxy radicals (C 10 H 21 -OO•) through addition on O 2 . The majority of decyl peroxy radicals (81%) leads to the formation of decyl hydroperoxide through H-atom abstractions from n-decane. The remaining part of C 10 H 21 -OO• reacts in the self-termination reactions to produce decanol (C 10 H 21 -OH), C 10 decanone/decanal (C 10 H 20 -OK), and C 10 alkoxy radicals (C 10 H 21 -O•). Decyl hydroperoxide decomposes into two fragments, C 10 alkoxy radicals, and OH radicals. This reaction is the most important reactions governing the liquid-phase reactivity of n-decane as it transforms a neutral molecule into two reactive radicals. Sensitivity analyses presented in Figure 12 confirm the central kinetic role of this reaction. C 10 alcohols are produced by both self-termination reactions of decyl peroxide radicals and by H-abstraction by C 10 H 21 -O• from n-decane. The major part of decanols is produced in the latter process. Once formed, C 10 alcohols are principally consumed by reactions with C 10 peroxy radicals (97.7%), through H-abstractions producing C 10 alcohols radicals (•C 10 H 20 -OH). The latter radical is then oxidized into C 10 carbonyl compounds by successive processes: addition of O 2 , and elimination of HO 2 .

The main consumption flux of n-decane is H-atom abstractions by n-decyl peroxy radicals (C 10 H 21 -OO•) (80.8%), C 10 alkoxy radicals (C 10 H 21 -O•) (12.4%), and OH (6.4%) radicals, yielding five isomers of decyl radicals.

  10 H 21 OO• + C 10 H 21 OO•  C 10 H 20 C=O + C 10 H 21 OH + O 2 C 10 H 21 OO• + C 10 H 21 OO•  C 10 H 21 O• + C 10 H 21 O• + O 2 n-C 10 H 22 + C 10 H 21 OO•  C 10 H 21 • + C 10 H 21 OOH C 10 H 21 OOH  C 10 H 21 O• + •OHpetroOxy experiment. Indeed, the knowledge of the liquid-phase composition is essential to rigorously simulate the autoxidation of fuels in PetroOxy experiments. A detailed kinetic model for n-decane oxidation in the liquid phase was developed using a state-of-the-art methodology which was developed based on solvent-adapted corrections for thermodynamic data (with an equation of state) and for the most sensitive kinetic data. Kinetic modeling shows a good agreement with experiments. The developed kinetic model is able to reproduce reliably the measured IP of n-decane, at different temperatures, with a mean relative deviation of 16%. The model was also able to satisfactorily simulate our new experimental profiles of hydroperoxides measured during PetroOxy tests. In addition, the kinetic model was validated against literature data, which reported experimental concentration profiles of C 10hydroperoxides, C 10 -ketones and C 10 -alcohols during n-decane autoxidation.

	Normalized sensitivity
	C10H21OO

C

Acknowledgments

This work was financially supported by the ANR BioACe project, grant ANR-18-CE05-002 of the French National Research Agency. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 101003318). High performance computing resources were provided by the EXPLOR center hosted by the University of Lorraine. This work was also granted access to the HPC resources of IDRIS under the allocation 2022-AD010812434R1.

Simulations of products formation in a Petroxy test, measured IPs, pressure profiles in the PetroOxy,

[ROOH] concentrations, calculations of uncertainties in [ROOH] and the simulation of product profiles as a function of time for the autoxidation of n-decane at 413K are given in supplemental material.

presented as a function of the PetroOxy cell temperature in Figure 7. IP and [ROOH] data are given in The ROOH concentrations obtained from potentiometry are generally higher than those measured by colorimetry by a about 5 x 10 -3 mol l -1 . Our uncertainty calculations showed that potentiometric titration is more accurate than colorimetric titration since an electronic signal is used in the former method (see Table S2). Consequently, the potentiometric total ROOH concentrations were used as the criterion to determine the IP time in the simulation: the simulated IP is reached when the simulated total concentration of ROOH matches the experimental one.


domain: phys spi

text:
Fares Bougourzi
email: fares.bougourzi@isasi.cnr.it

Cosimo Distante
email: cosimo.distante@cnr.it

Fadi Dornaika
email: fadi.dornaika@ehu.eus

Abdelmalik Taleb-Ahmed
email: abdelmalik.taleb-ahmed@uphf.fr

CNR-IEMN-CD and CNR-IEMN-CSD approaches for Covid-19 Detection and Covid-19 Severity Detection From 3D CT-scans

Keywords: Covid-19, Deep Leaning, CNNs, Recognition, Severity

Since its appearance in late 2019, Covid-19 has become an active research topic for the artificial intelligence (AI) community. One of the most interesting AI topics is Covid-19 analysis from medical imaging. CT-scan imaging is the most informative tool about this disease. This work is part of the 2nd COV19D competition, where two challenges are set: Covid-19 Detection and Covid-19 Severity Detection from the CT-scans. For Covid-19 detection from CT-scans, we proposed an ensemble of 2D Convolution blocks with Densenet-161 models (CNR-IEMN-CD). Here, each 2D convolutional block with Densenet-161 architecture is trained separately and in the testing phase, the ensemble model is based on the average of their probabilities. On the other hand, we proposed an ensemble of Convolutional Layers with Inception models for Covid-19 severity detection CNR-IEMN-CSD. In addition to the Convolutional Layers, three Inception variants were used, namely Inception-v3, Inception-v4 and Inception-Resnet.

Our proposed approaches outperformed the baseline approach in the validation data of the 2nd COV19D competition by 11% and 16% for Covid-19 detection and Covid-19 severity detection, respectively. In the testing phase, our proposed approach CNR-IEMN-CD ranked fifth and improved the baseline results by 18.37%. On the other hand, our proposed approach CNR-IEMN-CSD ranked third in the test data of the 2nd COV19D competition for Covid-19 severity detection, and improved the baseline results by 6.81%.

Introduction

Since the appearance of the Covid-19 pandemic in the late of 2019, reverse transcription-polymerase chain reaction (RT-PCR) has been considered as the golden standards for Covid-19 Detection. However, the RT-PCR test has many drawbacks including inadequate supply of RT-PCR kits, time-consuming consumption, and considerable false negative results [START_REF] Jin | A rapid advice guideline for the diagnosis and treatment of 2019 novel coronavirus (2019-nCoV) infected pneumonia (standard version)[END_REF] [23] [START_REF] Kucirka | Variation in False-Negative rate of reverse transcriptase polymerase chain Reaction-Based SARS-CoV-2 Tests by time since exposure[END_REF]. To overcome these limitations, medical imaging techniques have been widely used as tools. These imaging techniques include X-rays and CT-scans [START_REF] Vantaggiato | Covid-19 recognition using ensemble-cnns in two new chest x-ray databases[END_REF] [START_REF] Bougourzi | Recognition of COVID-19 from CT Scans Using Two-Stage Deep-Learning-Based Approach: CNR-IEMN[END_REF]. In fact, CT-scans are not only used to detect Covid-19 infected cases, but they could also be used to track the patient's condition and predict the severity of the disease [START_REF] Bougourzi | Per-COVID-19: A Benchmark Dataset for COVID-19 Percentage Estimation from CT-Scans[END_REF] [START_REF] Bougourzi | CNR-IEMN: A Deep Learning Based Approach to Recognise Covid-19 from CT-Scan[END_REF].

In the last decade, Deep Learning methods have become widely used in most computer vision tasks and have achieved high performance compared to traditional methods [START_REF] Bougourzi | Fusing Transformed Deep and Shallow features (FTDS) for image-based facial expression recognition[END_REF] [START_REF] Bougourzi | Deep learning based face beauty prediction via dynamic robust losses and ensemble regression[END_REF]. However, the main drawback of Deep Learning, especially the CNN architecture, is the need for huge labeled data, which is difficult to obtain in medical fields [START_REF] Bougourzi | Per-COVID-19: A Benchmark Dataset for COVID-19 Percentage Estimation from CT-Scans[END_REF]. On the other hand, most of the proposed CNN architectures have been used for static images (single image input) [START_REF] Bougourzi | CNR-IEMN: A Deep Learning Based Approach to Recognise Covid-19 from CT-Scan[END_REF].

In this work, we used pre-trained CNN architectures to detect Covid-19 infection and Covid-19 severity from 3D CT scans in the context of the 2nd COV19D challenge. We also trained a CNN model to filter out the sections that do not show a lung region. For Covid-19 severity detection, an Att-Unet model was trained to segment the lung regions and remove unimportant features. In addition, two volume input branches were used to reduce information loss due to resizing and varying the number of slices from one CT-scan to another. The main contributions of this work can be summarized as follows:

-For Covid-19 detection from CT scans, we proposed an ensemble of Convolution 2D blocks with Densenet-161 models, which we call CNR-IEMN-CD. Each convolution 2D block with densenet-161 architecture was trained separately, and the ensemble model is based on the average of their probabilities. -To detect the severity of Covid-19 using CT scans, we proposed an ensemble of Convolutional Layer with Inception models, which we call CNR-IEMN-CSD. In addition to the Convolutional Layers, three Inception variants were used, namely Inception-v3, Inception-v4, and Inception-Resnet. The Inception variants were trained twice separately and the ensemble model is the average of the probabilities of the six models. -Our proposed approaches outperformed the baseline approach in the validation data of the 2nd COV19D competition by 11% and 16% in Covid-19 detection and Covid-19 severity detection, respectively. Moreover, our proposed approach CNR-IEMN-CD ranked fifth and improved the results of the baseline approach by 18.37%. On the other hand, our proposed approach CNR-IEMN-CSD ranked third in the test data of the 2nd COV19D competition for Covid-19 severity detection and improved the baseline results by 6.81%. -The codes used are publicly available at https://github.com/faresbougourzi/ 2nd-COV19D-Competition. (Last accessed 28 th June 2022).

This paper is organized as follows: Section 2 describes our proposed approaches for Covid-19 detection and severity detection. The experiments and results are described in Section 3. In Section 4, the obtained results of Covid-19 Detection Challenge are discussed. Finally, we conclude our paper in Section 5.

Our Approaches

In the following two sections, we will describe our proposed approaches for Covid-19 detection and severity prediction, respectively.

Covid-19 Detection

Our proposed approach to Covid-19 detection for the 2nd COV19D competition is summarized in Figure 1. Because the 3D CT scans have a different number of layers depending on the scanner and acquisition settings and contain multiple slices with no lung regions, we trained a model to remove the non-lung slices. For this purpose, we manually labelled 20 CT-scans from the training data as lung and non-lung. In addition, we used the following datasets: COVID-19 CT segmentation [START_REF]COVID-19 CT-scans segmentation datasets[END_REF], Segmentation dataset nr.2 [START_REF]COVID-19 CT-scans segmentation datasets[END_REF], and COVID-19-CT Seg dataset [START_REF] Ma | Toward data efficient learning: A benchmark for COVID 19 CT lung and infection segmentation[END_REF] to have more labelled data as lung and non-lung slices. After creating the lung and non-lung sets, we trained the ResneXt-50 model [START_REF] Xie | Aggregated residual transformations for deep neural networks[END_REF] to remove the slices that showed no lungs at all. Since the CT scans have different numbers of slices, we concatenated all lung slices and then reduced them to 224 × 224 × 64 as shown in Figure 1. To distinguish between Covid-19 and non-Covid-19 CT scans, we separately trained three Densenet-161 [START_REF] Huang | Densely connected convolutional networks[END_REF] models. Since the input size of the Densenet-161 model is an RGB image and we have a 3D volume of size 224 × 224 × 64, we added Convolution block with a 3 x 3 kernel, a stride of 1, and a padding of 1. The 3 x 3 convolution block takes 64 input channels and produces 3 output channels. It should be noted that each CNN model was trained separately. In the decision or test phase, we propose to form an ensemble of the three trained CNN architectures by averaging their probabilities.

Covid-19 Severity Detection

Our proposed approach to predict Covid-19 severity for the 2nd COV19D competition is summarized in Figure 2. Similar to the Covid-19 detection in section 2.1, we used the trained ResneXt-50 model to remove non-lung slices. To segment the lungs, we used the Att-Unet [START_REF] Oktay | Attention U-Net: Learning Where to Look for the Pancreas[END_REF] [6] architecture trained with the following datasets COVID-19 CT segmentation [START_REF]COVID-19 CT-scans segmentation datasets[END_REF], Segmentation dataset nr.2 [START_REF]COVID-19 CT-scans segmentation datasets[END_REF], and COVID-19-CT-Seg [START_REF] Ma | Toward data efficient learning: A benchmark for COVID 19 CT lung and infection segmentation[END_REF]. After removing the non-lung slices and segmenting the lung regions to remove unnecessary features, we divided generated for each scan, from the obtained slices, two volumes 299 × 299 × 32 and 299 × 299 × 16. The goal of the two volumes is to reduce the information lost due to resizing and to obtain two views of each CT scan. The two input volumes were fed into 3 Experiments and Results

The COV19-CT-DB Database

The COVID19-CT-Database (COV19-CT-DB) [ COVID19-CT-DB contains 3-D CT scans of the chest, which have been examined for the presence of COVID-19 . Each of the 3-D scans contains a different number of slices, ranging from 50 to 700. The database was divided into a training set, a validation set, and a test set. The training set contains a total of 1992 3-D CT scans. The validation set consists of 494 3-D CT scans. The number of COVID-19 and non COVID-19 cases in each set is given in Table 1.

Further subdivision of the COVID-19 cases was based on the severity of COVID-19 , which was given by four medical experts in a range of 1 to 4, with 4 denoting the critical state. The training set contains a total of 258 3-D scans CT. The validation set consists of 61 3-D CT scans. The number of scans in each severity class in these sets is shown in Table 2.

Experimental Setup

For Deep Learning training and testing, we used the Pytorch [START_REF] Paszke | Pytorch: An imperative style, highperformance deep learning library[END_REF] Library with NVIDIA GPU Device GeForce TITAN RTX 24 GB. The batch size used consists of 16 CT-scan volumes for both tasks. We trained the networks for 40 epochs. The initial learning rate is 0.0001, which decreases by 0.1 after 15 epochs, followed by another 0.1 decrease after 30 epochs.

Covid-19 Recognition

Table 3 summarizes the Covid-19 detection results (F1 score) obtained with the validation data. As described in section 2.1, we trained three Convolutional Block + Densenet-161 architectures. For simplicity, we refer to Convolutional Block + Densenet-161 with Densenet-161 and the training iteration. This table shows the results of the individual Densenet-161 models and the ensemble of 2 and three models. The ensemble of three models is denoted by (CNR-IEMN-CD). From these results, the ensemble of two models achieves better results than the individual models. The ensemble of three models, on the other hand, achieves the best performance. Compared to the baseline results, our proposal CNR-IEMN-CD improved the results by 13%. Since five submissions were allowed in the 2nd COV19D competition, our submissions are 1, 2, 4, 5 and 6. The obtained results are summarized in Table 4. From these results, we notice that the ensemble of the three Densenet-161 models achieved the best performance than using single models and the ensemble of two models as well. Furthermore, our approach achieved higher performance than the baseline method by 18.37%. On the other hand, our approach ranked the fifth in the 2nd COV19D competition for Covid-19 Detection.

Covid-19 Severity Detection

Table 5 summarizes the results (F1 score) of detecting the severity of Covid-19 using the validation data. As described in section 2.2, we designed three models consiting of convolutional layers and CNN backbones. The CNN backbones used are Inception-V3, Inception-V4, and Inception-Resnet, and each model was trained twice to obtain a greater variety of predictors. In total, we have six models. For simplicity, we refer to ConvLayer + CNN backbone with the backbone architecture name. Table 5 shows the obtained results of the individual models and the ensemble of two models of the same architecture and the ensemble of the six models (CNR-IEMN-CSD). The obtained results show that the ensemble of two models improves the performance compared to the single models. Moreover, the ensemble of the six models achieved the best performance and was 17% better than the baseline. Since five submissions were allowed in the 2nd COV19D competition, our submissions are 1, 7, 8, 9 and 10. The obtained results on the testing data of the 2nd COV19D Severity Detection competition are summarized in Table 6. From these results, we notice our proposed CNR-IEMN-CSD approach achieved the best performance than using single models and the ensemble of two models as well. Furthermore, our approach achieved higher performance than the baseline method by 6.8%. On the other hand, our approach ranked the fifth in the 2nd COV19D competition.

Discusssion

To understand the behavior of our proposed approach for Covid-19 detection challenge, we visualize some slices from predicted CT-scans as TP, TN, FP, FN as shown in Figure 3. As stated in [START_REF] Sun | A systematic review of chest imaging findings in covid-19[END_REF], the COVID-19 lung imaging manifestations The first row shows that many Non-Covid cases are classifies as Covid-19, these examples shows the appearance of infection manifestations, however, the signs are not related to Covid-19 infection. The second row shows similar infection signs and the CT-scans were correctly classified as Covid-19. From the third row shows that our approach can misclassify when the Covid-19 infection appears in small region of the lungs. The fourth row shows that our approach is able to classify Non-Covid CT-scans despite the appearance of the infection signs of the other diseases. Especially, with using the ensemble of multiple models. From these examples, we argue that separating other diseases as separate class can help the trained models to distinguish between the Covid-19 and Non-Covid cases.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Our proposed CNR-IEMN-CD Approach for Covid-19 Detection.

Fig. 3 :

 3 Fig. 3: Visualization illustration of some example slices from the predicted CTscans. The first to the fourth rows show examples of wrongly predicted CT-scans as Covid-19, correctly predicted CT-scans as Covid-19, wrongly predicted CTscans as Non-Covid-19, and CT-scans as Non-Covid-19, respectively.

Table 1 :

 1 Data samples in each Set.

	10] [11] [12] [14] [13] [1] consists

Table 2 :

 2 Data samples in each Severity Class

	Severity Class	Training	Validation
	1	85	22
	2	62	10
	3	85	22
	4	26	5

Table 3 :

 3 Covid-19 Detection Results on the 2nd COV19D competition validation data.

	Model Architecture	F1-Score
	-	Baseline	77
	1	Densenet-161 Model1	88.34
	2	Densenet-161 Model2	88.83
	3	Densenet-161 Model3	87.81
	4	Ensemble (1&2)	89.61
	5	Ensemble (2&3)	89.85
	6	Ensemble (1&2&3)	90.07

Table 4 :

 4 Covid-19 Detection Results on the 2nd COV19D competition testing data.

	Model Architecture	F1-Score
	-	Baseline	66.00
	1	Densenet-161 Model1	81.91
	2	Densenet-161 Model2	77.03
	4	Ensemble (1&2)	83.12
	5	Ensemble (2&3)	84.12
	6	Ensemble (1&2&3)	84.37

Table 5 :

 5 Covid-19 Severity Detection Results on the 2nd COV19D Severity Detection competition validation data.

	Model Architecture	F1-Score
	-	Baseline	63.00
	1	Inception-v3 Model1	74.86
	2	Inception-v3 Model2	76.25
	3	Inception-v4 Model1	75.60
	4	Inception-v4 Model2	73.48
	5	Inception-Res Model1	73.37
	6	Inception-Res Model2	72.01
	7	Ensemble Models(1-2)	79.10
	8	Ensemble Models(3-4)	79.54
	9	Ensemble Models(5-6)	79.10
	10	Ensemble Models(1-6)	80.08

Table 6 :

 6 Covid-19 Severity Detection Results on the 2nd COV19D Severity Detection competition Testing data. with those of the viral pneumonia and this makes the diagnosis of Covid-19 challenging even for the radiologists. It can be noticed that in the 2nd COV19D competition for Covid-19 Detection, only two classes were defined, Covid-19 and Non-Covid. Thus, that Non-Covid class may presents No infection at all or other lung diseases.

	Model Architecture	F1-Score Mild Moderate Severe Critical
	-	Baseline	40.30	61.10	39.47	37.07	23.54
	2	Inception-v3 Model2	39.89	34.86	35.44	58.00	31.25
	7	Ensemble Models(1-2)	43.04	53.09	18.52	63.52	37.04
	8	Ensemble Models(3-4)	45.49	54.66	26.79	63.48	37.04
	9	Ensemble Models(5-6)	44.37	53.42	26.67	60.36	37.04
	10	Ensemble Models(1-6)	47.11	55.67	37.88	55.46	39.46
	are highly overlapped

Conclusion

In this work, we proposed two ensemble CNN-based approaches for the 2nd COV19D competition. For Covid-19 detection from CT scans, we proposed an ensemble of 2D convolution blocks with Densenet-161 models. Here, the individual convolution 2D blocks with Densenet-161 architecture are trained separately, and the ensemble model is based on the average of their probabilities at testing phase. On the other hand, we proposed an ensemble of Convolutional Layers with Inception models for Covid-19 severity detection. In addition to the Convolutional Layers, three Inception variants were used, namely Inception-v3, Inception-v4, and Inception-Resnet.

In the testing phase, our proposed approaches ranked fifth and third for Covid-19 detection and Covid-19 Severity detection, respectively. Furthermore, our approaches outperformed the baseline results by 18.37% and 6.81% for Covid-19 detection and Covid-19 Severity detection, respectively. In future work, we will evaluate our approaches in other datasets and use the attention mechanism to select the most representative slices for Covid-19 detection.

Acknowledgments

The authors would like to thank Arturo Argentieri from CNR-ISASI Italy for his support on the multi-GPU computing facilities.