F Blondel

R Boisard

M Milekovic

G Ferrer

C Lienard

D Teixeira

Validation and comparison of aerodynamic modelling approaches for wind turbines

The development of large capacity Floating Offshore Wind Turbines (FOWT) is an interdisciplinary challenge for the design solvers, requiring accurate modelling of both hydrodynamics, elasticity, servodynamics and aerodynamics all together. Floating platforms will induce low-frequency unsteadiness, and for large capacity turbines, the blade induced vibrations will lead to high-frequency unsteadiness. While yawed inflow conditions are still a challenge for commonly used aerodynamic methods such as the Blade Element Momentum method (BEM), the new sources of unsteadiness involved by large turbine scales and floater motions have to be tackled accurately, keeping the computational cost small enough to be compatible with design and certification purposes. In the light of this, this paper will focus on the comparison of three aerodynamic solvers based on BEM and vortex methods, on standard, yawed and unsteady inflow conditions. We will focus here on up-to-date wind tunnel experiments, such as the Unsteady Aerodynamics Experiment (UAE) database and the MexNext international project.

Introduction

Nowadays, most of the design software rely on the Blade Element Momentum models [START_REF] Robertson | Offshore Code Comparion Collaboration Continuation Within IAE Wind Task 30: Phase II Results Regarding a Floating Semisubmersible Wind System[END_REF]. Those methods are based on strong hypotheses, partially overcame through analytical corrections, that introduce empiricism in the models. The reliability of the model is then questionable, especially in the framework of large fixed and floating offshore wind turbines, which tend to become larger and larger, reaching 10M W or more [START_REF] Chaviaropoulos | AVATAR Deliverable D1.1 Reference Blade Specification[END_REF]. On the other side, more physical models, such as the vortex methods, that require less analytical corrections, are being increasingly used. The development of algorithms and new computational architectures have strongly reduced the computational time of vortex methods, which is crucial in the light of the large databases that have to be tackled for certification purposes. Vortex methods provide different solutions to model the wake, such as vortex points, filaments or panels. The aim of this paper is to compare a BEM solver and two vortex solvers designed to be coupled with the aero-servo-hydro-elastic solver DeepLines Wind TM [START_REF] Le Cunff | Fully Coupled Floating Wind Turbine Simulator Based on Nonlinear Finite Element Method: Part I-Methodology[END_REF] against up-to-date wind tunnels experiments, identify the cases on which BEM models are not accurate enough, and evaluate the potential of vortex models on those cases. The two lifting-line solvers are based on different wake models, a free-wake vortex filaments model and a free-wake vortex panel wake model. The impact of the wake model on the lifting-line results will be analysed.

Computational tools 2.1. AeroDeeP: BEM based aerodynamic library

AeroDeeP is based on a the widely used Glauert's Blade Element Momentum method [START_REF] Glauert | Airplane Propellers[END_REF], inspired from the Rankine-Froude theory, using two-dimensional airfoil characteristics to evaluate the forces at each blade element. Induction factors, which represent the momentum loss due to the presence of the rotor, are computed in both axial and tangential directions along the blade span. In order to overcome the major limitations of the classical BEM theory, corrections have been implemented to account for hub and tip losses, turbulent wake state, tower shadowing, dynamic inflow (i.e. unsteady BEM), dynamic stall, as well as skewed rotor configurations.

CASTOR:

Lifting-Line / Free-Wake aerodynamic library CASTOR (Code Aerodynamique pour la Simulation de Turbines OffshoRe) is a free-wake vortex filament lifting-line solver based on the generalized Prandtl lifting-line theory (see [START_REF] Weissinger | The Lift Distribution of Swept-Back Wings[END_REF], [START_REF] Katz | Low-Speed Aerodynamics[END_REF]). Various flow effects such as turbulent wake state, dynamic inflow, hub and tip losses, as well as skewed rotor configurations are inherently taken into account. Similarly to the BEM method, CASTOR is based on airfoil polars, so that blade surfaces do not have to be modeled. In order to reduce the computational cost, CASTOR is based on an hybrid OpenM P / GP U implementation, thereby reducing the computational times up to two orders of magnitude when compared to sequential CP U implementations. The presence of the ground can be taken into account through the mirror technique, and the Beddoes-Leishman modified Risoe stall model [START_REF] Hansen | A Beddoes-Leishman type Dynamic Stall model in State-Space and Indicial Formulations[END_REF] has been implemented. CASTOR will be referred to as Lifting-Line (Prandtl) in the following.

PUMA: Lifting-Line / Free-Wake aerodynamic library

The PUMA (Potential Unsteady Methods for Aerodynamics) code which is developed at ONERA since 2013 is based on a long lasting knowledge about free wake methods for helicopter aerodynamics. It is built on a coupling between a kinematic module and an aerodynamic module which relies on a free wake model combined to a lifting line approach. The free wake model is based on Mudry theory [START_REF] Mudry | La Théorie Générale des Nappes et Filaments Tourbillonnaires et ses Applications à l'Aérodynamique Instationnaire[END_REF] which rigorously describes the unsteady evolution of a wake modeled by a potential discontinuity surface while the lifting line method relies on 2D airfoils characteristics and can handle some 3D corrections for blade sweep and 2D unsteady aerodynamics effects through dynamics stall models. In order to reduce computational time the code has been parallelized using OpenMP and the Multilevel Fast Multipole Method has been implemented for the computation of the velocities induced by each wake panel on any element. Thanks to the versatility offered by the modular approach of PUMA, we were able to directly plug the aerodynamic module to the Deeplines Wind TM solver, which provide the needed kinematic data in place of the built-in PUMA's module. PUMA will be referred to as Lifting-Line (Mudry) in the following.

Deeplines Wind TM and coupling strategy

DeepLines Wind TM [START_REF] Le Cunff | Fully Coupled Floating Wind Turbine Simulator Based on Nonlinear Finite Element Method: Part I-Methodology[END_REF] is an aero-servo-hydro-elastic solver developed by Principia/IFPEN that simulate loads and deformation of onshore, offshore and floating wind turbine in operation. It offers the possibility to be coupled with different aerodynamic libraries. The coupling with external solvers is realised through APIs (Application Programming Interface) designed to be as generic as possible. At each time step, information related to the wind turbine geometry and motions are sent to the aerodynamic solver, that sends back the aerodynamic loads on the blades. Those information are: • Wind velocity at any positions specified by the aerodynamic solvers.

Based on this architecture, in-house (AeroDeeP, CASTOR) and external (PUMA) aerodynamic libraries have been coupled to the DeepLines Wind TM solver. For this specific study, in order to simplify the comparison between the different aerodynamics models, the libraries were coupled with a simple kinetic module, based on a rigid body hypothesis. This kinematic solver shares the same API than the DeepLines Wind TM solver, ensuring the validity of the coupling.

Calculations setup

In order to ensure the convergence of the results, azimuthal steps of 10 • per time step for 10 to 15 rotations of the wind turbine have been used. Both AeroDeeP and CASTOR use 30 cosine distributed elements per blade, thus ensuring a good refinement near the root and the tip, and PUMA uses 25 square root distributed elements, ensuring a good refinement near the tip. All solvers share the same airfoil data.

Experimental data

Experimental data from both the UAE Phase VI experiments and the MexNext project have been used for comparison. For more details about the experiments, see [START_REF] Hand | Unsteady Aerodynamics Experiment Phase VI: Wind Tunnel Test Configurations and Available Data Campaingns[END_REF] and [START_REF] Snel | The MEXICO project (Model Experiments in Controlled Conditions): The Database and First Results of Data Processing and Interpretation, The Science of Making Torque from Wind Conference[END_REF].

Unsteady Aerodynamics Experiment Phase VI

The Unsteady Aerodynamic Experiment (UAE) Phase VI have been carried out in the 24.2 × 36.6m NASA Ames wind tunnel. A 10.1m diameter two-bladed wind turbine rotor was used. In the considered cases, the rotor rotates at approximately 72rpm, the blades are pitched by an angle θ = 3 • for the sequence H considered below, and with a variable pitch for the dynamic inflow Q sequence. The twisted and tapered blades are based on S809 airfoils. Both blade coning and hub tilt angles are zeros. Normal and tangential forces distributions are obtained through the integration of the measured pressures at five radial positions along the blades. Normal force is supposed to be positive while pointing from the pressure side to the suction side of the airfoil. Tangential force is positive while pointing from the leading edge towards the trailing edge of the airfoil. Complete description of the wind turbine and available data can be found in [START_REF] Hand | Unsteady Aerodynamics Experiment Phase VI: Wind Tunnel Test Configurations and Available Data Campaingns[END_REF]. We used the S809 airfoil polar data provided in [START_REF] Hand | Unsteady Aerodynamics Experiment Phase VI: Wind Tunnel Test Configurations and Available Data Campaingns[END_REF] (without corrections for the rotational effects), together with transition airfoils near the root (between the cylinders and the S809 airfoils).

MEXICO rotor

The MEXICO and New MEXICO experiments have been performed in the 9.5 × 9.5m 2 DNW closed walls large low-speed facility test section, using a 4.5m diameter three bladed rotor. In the considered cases, the rotor rotates at approximately 424.5rpm, and the blades are pitched by an angle of θ = -2.3 • . The twisted and tapered blades are based on DU 91-W 2-250, RISO-A1-21 and N ACA64-418 airfoils from root to tip. Both blade coning and hub tilt angles are zeros. Normal and tangential forces distributions are obtained through the integration of the measured pressures at five radial positions along the blade, using the same sign convention than the UAE experiments. A complete description of the experiment can be found in [START_REF] Snel | The MEXICO project (Model Experiments in Controlled Conditions): The Database and First Results of Data Processing and Interpretation, The Science of Making Torque from Wind Conference[END_REF]. Airfoil data have not been corrected for rotational effects. Transition airfoils have been used between the various profiles. the three solvers present a very good agreement with the experimental data at 5 and 7m/s. Even at the tip, BEM results are very close to the lifting-line solvers. It is expected that the discrepancy observed near the root (below 0.25) between the Murdy's lifting line solver and the other solvers can be explained by different blade discretizations (cosine distribution versus square-root distribution). Near the tip (r/R > 0.85) the normal force is over-estimated by about 20% by the three solvers, potentially shifting the position of the maximum normal force. Due to the low resolution of the experimental data and the strong gradient of the force near the tip, the exact position of the maximum normal force is not clear, and small deviation can lead to high error percentage. It is then difficult to conclude on the models accuracy regarding the representation of the force evolution. At 5m/s, a small deviation (below 1N ) is observed on the tangential force at mid-span, but due to the low magnitude of the force (below 5N ) it is hard to conclude if the difference is significant. At 10m/s, strong discrepancies appear in the range r/R = [0 -0.6]. The few experimental points indicate a strong oscillation of the normal force which is not captured by the BEM solvers. The lifting-line solvers predict strong oscillations, the frequencies of which depending on the model, but due to the lack of experimental data it is not possible to conclude on the accuracy of the models. The oscillations are induced by the presence of a stalled flow, which seems a limitation of the three tested solvers. The wake models seems to have an influence on the stalled flow representation as the two lifting-line models predict oscillations with different frequencies.

Axial Inflow

A second comparison for the axial inflow test cases is shown figure 2. Each solver is compared to the New MEXICO measurements (using the rough NACA dataset) at three wind velocities: 10, 15 and 24m/s. A very good agreement between the experimental data and the solvers is observed at 10 and 15m/s for the normal force. In the turbulent wake state regime (lower wind velocity, 10m/s), a strong discrepancy can be observed between the lifting results and the experimental tangential force while the BEM predictions are very close to the experimental results. BEM results are driven by the Glauert correction for high induction in this case, which leads to very good results. The discrepancy between lifting-line solvers and experiments is still under investigation. At high wind velocity (figure 2, right), both normal and tangential forces appear to be slightly over-predicted near the tip of the blade, and under-predicted near the root by all three methods. At this tip speed ratio, the flow is separated from the root to the midspan of the blade, and some oscillations are observed on the tangential forces. Globally, apart in the stalled flow regions, a very good agreement is observed between the three solvers and the experimental data. For the UAE case, BEM and lifting-line results are almost super-imposed. However, on the New MEXICO case, the BEM model predicted slightly lower values for the normal and tangential component of the forces. At the lower wind speed (in the turbulent wake state regime), BEM results are closer to experiments than the lifting-line results. On those steady cases, the two lifting-line formulations and wake models appear to be strictly equivalent, apart in the stalled flow regions. The stalled flow regime is clearly a limitation of the three approach and would necessitate more investigations.

Yawed Inflow

The second test cases concern the yawed inflow. To facilitate the results interpretation, both tower-shadow and dynamic stall models have been deactivated: even so dynamic stall is supposed to occur on those test cases (especially near the root of the blade), the difference in the stall models used in the three solver could complicate the solver comparison. Normal and tangential forces as a function of azimuth angle are compared to experimental data at three radial positions: r/R = 0.30, r/R = 0.63, r/R = 0.95 for the UAE experiments, r/R = 0.25, r/R = 0.60, r/R = 0.92 for the MEXICO experiments. As predicted by the axial inflow results, an overestimation of the normal force amplitude (around 20%) is expected with the three solvers at position r/R = 0.95 for the UAE experiments and at both r/R = 0.60 and r/R = 0.92 on the normal and tangential forces for the MEXICO experiments. The force will then be compared in term of variations over the azimuth rather than in term of absolute values. More specifically, we will focus on the ability of the solvers to capture the phase lag on the normal loads between the tip and the root of the blade. As explained in [START_REF] Schepers | Annexlyse: Validation of Yaw Models, on Basis of Detailed Aerodynamic Measurements on Wind Turbine Blades[END_REF], near the tip, the maximum of the normal force is supposed to be located in the upwind part (azimuth range [180 • -360 • ], the blade is located in an upwind position with respect to the non-yawed configuration), whereas near the root, the maximum is located in the downwind part (azimuth range [0 • -180 • ]). This phase shift, which is due to the presence of the tip and root vortices, leads to a destabilizing yawing moment at the root and a stabilizing yawing moment at the tip [START_REF] Schepers | Annexlyse: Validation of Yaw Models, on Basis of Detailed Aerodynamic Measurements on Wind Turbine Blades[END_REF]. Sequence H of the UAE experiments is used again. Based on the axial inflow results, we use a wind velocity of 7m/s, at which the best agreement between calculations and experiments was observed (figure 1). Experimental data presented in the following figures have been filtered using Legendre polynomials. The first case corresponds to a yaw angle Ψ = 20 • (figure 3). Near the blade root (r/R = 0.30), the shape of the tangential forces fluctuations is quite well reproduced by the three solvers with a slight underestimation of the variation amplitude. For the normal forces, a strong discrepancy can be observed between the simulations and the experimental results, with an over-estimation of the maximal amplitude of the fluctuations around azimuth 90 and 270 • . All three solvers predict a strong decrease of the normal force around azimuth 0 and 360 • which is not observed experimentally. Further investigations have shown than the dynamic stall phenomena plays an important role here. A strong discrepancy between the solvers is also observed. At mid-span (r/R = 0.63), both the amplitude of the fluctuations and the mean value of the normal and tangential forces are close to the experiments for the three solvers.

The BEM solver fails to predict the phase shift, thus over-estimating the normal forces in the downwind part and underestimating them in the upwind part. Near the tip, the normal force is over-estimated in average, as expected (figure 1). The lifting-line solvers predict the shape of the fluctuations and the phase shift accurately, while the BEM solver has a symmetric behaviour, predicting the maximal normal force around azimuth 180 • rather than 250 • . Discrepancies are observed between the Prandtl and the Mudry lifting-line solvers, for both the normal and the tangential forces: the phase shift and the amplitude of the fluctuations are better predicted with the Prandtl solver. The disabling of the tower shadow model could explain the disregard of the small kink observed around azimuth Φ = 180 • at mid-span and near the tip. This point deserves further investigations. For the second case, an higher yaw angle (Ψ = 60 • , figure 4) has been chosen. Near the root, the BEM solver largely over-predicts the tangential force fluctuations, whereas the liftingline solvers tend to under-predict them. A phase-lag of 50 • can also be noticed between the BEM prediction of maximal tangential force and the experimental results, whereas the lifting line solvers predicted it accurately. The lifting-line solvers predictions of the normal force are very good, while similarly to the Ψ = 20 • case, the BEM model largely over-estimates the fluctuations magnitudes. At mid-span, the lifting-line solvers do predict the normal loads evolution accurately, despite a small under-prediction of the load amplitude. The BEM solver increases this error. Near the tip of the blade, normal forces fluctuations are rather small in amplitude, and thus difficult to interpret. However, the Mudry's lifting-line model does not predicts the augmentation of the normal force in the upwind part accurately, contrary to the Prandtl solver. On the tangential loads, despite a very good agreement, a small phase lag can be observed between the BEM predictions and the experimental results. Globally, the Prandtl lifting-line solver is able to reproduce the phase shift in the normal forces accurately. The Mudry's lifting-line solver behaves pretty well in the downwind part, but some discrepancies have been observed in the upwind part. The results of the BEM model are less accurate than the lifting-line ones, but do follow the main experimental trends. This is due to the complex flow field in yaw: while rotating, each blade experiences an advancing and retreating motion, inducing sinusoidal variations of relative velocities, which are well caught by BEM and liftingline solvers. However, the BEM theory does not model the tip and root vortices, and thus cannot predict the phase shift phenomena. Those observation are confirmed by the results obtained with the MEXICO dataset, one which the phase shift phenomena is also clearly visible (figure 5). A comparison has been made between experimental results and simulations for a 30 • yaw angle and a wind velocity of 15m/s. Near the root of the blade, the maximum of the normal force is located near the azimuth 90 • , in the downwind part. This is well predicted by the lifting-line solvers, but they also predict a smaller second pic around 270 • , which, following the previous analysis, should be attributed to an over-estimation of the strength of the tip vortex in the downwind part. Due to the absence of tip and root vortices, the BEM model has an unrealistic symmetric behaviour. Compared to the Murdy approach, the Prandtl solver underestimate the normal load amplitude by 50N . This is one of the few configuration where a strong difference in amplitude can be observed between the two lifting-line solvers. At this spanwise location, the tangential force variations are underestimated by all three solvers. Around mid-span, the maximum of the normal force is located in the upwind part of the blade (azimuth Φ = 210 • ). The two lifting line solvers predict the amplitude of the fluctuations and the phase shift accurately, despite an over-prediction (around 20%) of the mean value, which was expected (figure 2). In contrast, the BEM underpredicts the amplitude of the fluctuations of the tangential loads and over-predicts those of the normal loads, but the mean values of the forces are more accurately predicted. Similar observations can be made near the tip. As a conclusion for the yawed inflow cases, it can be said that lifting-line solvers do reproduce the measured load fluctuations very well, predicting the phase shift accurately, which was expected (see [START_REF] Schepers | Annexlyse: Validation of Yaw Models, on Basis of Detailed Aerodynamic Measurements on Wind Turbine Blades[END_REF], [START_REF] Grasso | Development and validation of Generalized Lifting Line code for Wind Turbine Aerodynamics[END_REF]). Differences have been observed in the upwind part between the two lifting-line solvers, but only on the UAE test cases where the Prandtl lifting-line solver appears to be more accurate. The BEM model provides a good representation of the load fluctuations, due to the valid representation of the advancing and retreating motion of the blades, but neglects the phase shift, which is due to the interactions between the blade and the tip/root vortices, thus missing a large part of the ongoing physics in yawed cases. This phase shift induces a varying yawing moment that may increase the blade fatigue, and is thus of major importance for the design. For the yawed cases, lifting-line solvers appear more reliable than the BEM solver. Further investigations including dynamic stall models are ongoing.

Dynamic Inflow

The last comparison concerns the dynamic inflow phenomena. The UAE Phase VI sequence Q data are used here. The simulations have been performed using the BEM solver, with the Øye dynamic inflow correction activated, and with the Prandtl lifting-line model. Due to numerical robustness issues, we have not been able to reproduce this case with the solver based on Mudry's theory. Both rotational velocity and pitch angle time series issued from measurements have been imposed as an input condition to the solvers. Two wind velocities are considered (5 and 10m/s). Similar results have been observed for a wind velocity of 8m/s. At 5m/s, the blade tip pitch measurements and calculations. Despite the small offset on the total rotor thrust, the overshoot on the thrust for both the upward (figure 6, center) and downward (figure 6, right) facing steps is pretty well caught. This test case highlights the capability of the corrected BEM and lifting-line solvers to handle unsteady flows, even if the time scales encountered in such kind of tests are by order of magnitude larger than the one due to standard blade pitch control or wind turbine floater motions.

Conclusion

Three different solvers have been compared and validated: a first solver based on the Blade Element Momentum theory, a second based on the Prandtl lifting-line theory, and a third one based on the more recent Mudry's theory. Those solvers are intended to be coupled with the aero-servo-hydro-elastic solver DeepLines Wind TM , in order to simulate fully coupled floating offshore wind turbine systems. In the present framework, we used a rigid body hypothesis in order to get more insight into aerodynamic phenomena. Two sets of experimental data have been used, the UAE experiments and the MEXICO / New MEXICO experiments. For the axial flow cases, all three solvers compare very well against the experiments, and very similar results were obtained. In yawed conditions, lifting-line solvers behave very well and are able to catch some relatively fine phenomenon dealing with the tip and root vortices and leading to a dephasing in the location of the maximum normal force between the tip and the root of the blade. The BEM solver, which was globally in good accordance with the experimental data, is not able to catch this phenomenon, leading to a possible inacurate prediction of the global yawing moment of the wind turbine. This could be disqualifying when dealing with full floating wind turbine systems. Last, the BEM and Prandtl lifting-line solvers results have been compared to experimental data on unsteady cases, on which turbine blades were pitched sharply. Both the BEM and the Prandtl lifting-line solver behave very well, which underlines the ability of the unsteady BEM theory to handle the various types of unsteadiness involved in FOWT simulations, even so such test cases are not necessarily representative of the time scales of the unsteady phenomena present in FOWT standard operations.

  A first comparison for the axial inflow test cases is shown figure1. The BEM solver and both lifting-line solvers are compared to the Sequence H of the UAE measurements. Forces normal and tangential to the chord line are presented in function of the non-dimensional spanwise location for three wind velocities: U ∞ = 5, 7 and 10m/s. The spanwise location is non-dimensionalised by the turbine radius (starting from the hub center). Mean experimental values of the normal and tangential forces are used. With the exception of some discrepancies at the blade extremities,

Figure 1 .

 1 Figure 1. UAE Phase VI, Sequence H: wind velocity 5m/s (left), 7m/s (middle) and 10m/s (right). The green circle represent the UAE experiments, the blue line the BEM simulations, the red dashed line the Lifting-Line (Prandtl) simulations/ and the black dashed line the Lifting-Line (Mudry) simulations.

Figure 2 .

 2 Figure 2. New MEXICO: Normal (top) and tangential (bottom) forces along the blade, wind velocity 10m/s (left), 15m/s (middle) and 24m/s (right). : UAE experiments / : BEM / : Lifting-Line (Prandtl) / : Lifting-Line (Mudry)

Figure 3 .

 3 Figure 3. UAE Sequence H: Normal (top) and tangential (bottom) blade loads evolution with azimuth at radial fraction 0.30, 0.63, 0.95, wind velocity 7m/s, yaw angle 20 • . : UAE experiments / : BEM / : Lifting-Line (Prandtl) / : Lifting-Line (Mudry)

Figure 4 .

 4 Figure 4. UAE Sequence H: Normal (top) and tangential (bottom) blade loads evolution with azimuth at radial fraction 0.30, 0.63, 0.95, wind velocity 7m/s, yaw angle 60 • . : UAE experiments / : BEM / : Lifting-Line (Prandtl) / : Lifting-Line (Mudry)

Figure 5 .

 5 Figure 5. MEXICO: Normal (top) and tangential (bottom) blade loads evolution with azimuth at radial fraction 0.25, 0.60, 0.92, wind velocity 15m/s, yaw angle 30 • . : MEXICO experiments / : BEM / : Lifting-Line (Prandtl) / : Lifting-Line (Mudry)

Figure 6 .

 6 Figure 6. UAE Phase VI: Rotor thrust evolution during pitch steps (left) and close-ups (center and right), wind velocity = 5m/s (top) and = 10m/s (bottom). : UAE experiments / : BEM / : Lifting-Line (Prandtl)

Acknowledgments

The authors are grateful to the US Department of Energy funding support and the NREL's National Wind Technology Center for providing the data regarding the UAE Phase VI experiments. MEXICO data used have been supplied by the consortium which carried out the EU FP5 project MEXICO: "Model rotor EXperiments In COntrolled conditions" to which 9 European partners contribute.


Aicha Mikdam

Xavier Colin
email: xavier.colin@ensam.eu

Gaelle Minard

Noelle Billon

Romain Maurin

A kinetic model for predicting the oxidative degradation of additive free polyethylene in bleach desinfected water

Keywords:

The chemical interactions between additive free PE and bleach were investigated by FTIR spectrophotometry and viscosimetry in molten state after immersion (for a maximum duration of one hundred days) in bleach solutions maintained at a temperature of 60 °C, a free chlorine concentration of 100 ppm, and a pH = 4, 5 or 7. It was found that the polymer undergoes a severe oxidation from the earliest days of exposure in a superficial layer of about 50-100 μm thick, almost independent of the pH value. In this layer, oxidation leads to the formation and accumulation of various carbonyl products (mostly ketones and carboxylic acids) but also, after about 2-3 weeks of exposure, to a dramatic decrease in the average molar mass due to the large predominance of chain scissions over crosslinking. It was also found that the oxidation rate is maximum at pH = 5, and of the same order of magnitude at pH = 4 and 7. Based on the equilibrium diagram giving access to the relative predominance of the three main chemical species as a function of the pH value of the bleach solution, it was assumed that oxidation is initiated by radical species coming firstly from hypochlorous acid (ClOH) and secondarily from chlorine (Cl 2 ), given that hypochlorite ions (ClO -) are totally insoluble into the PE matrix. In addition, for explaining the surprisingly large value of the oxidized layer thickness despite the high reactivity of the involved radicals, it was assumed that ClOH and Cl 2 do not decompose into radicals in the water phase, but migrate deeply into the PE matrix prior to dissociating into Cl% and HO% radicals and then, initiating a radical chain oxidation. The validity of the kinetic model derived from this scenario was successfully checked by comparing the numerical simulations with all the experimental data collected in this study. This model predicts the general trends of the oxidation kinetics and its dependence on the pH value, but also gives access to the transport properties of the chlorinated disinfectants and their radical species, and the rate constants of the radical attack.

Introduction

Steel pipes (with or without polymeric coating) are traditionally used for tertiary cooling circuits in nuclear power plants. Water is commonly disinfected by chlorinated reagents in order to control the microbiological growth and limit the deposit and development of shellfish and seaweed that might clog the circuit input. However, steel is sensitive to corrosion and water disinfectants are relatively strong oxidizers susceptible to accelerate corrosion. Therefore, in recent years, the nuclear industry has launched heavy testing campaigns for qualifying polymeric materials for this type of application, in particular polyethylene (PE), in view of using them in replacement of steel.

PE pipes are currently used for conveying drinking water under a pressure of a few bars since the early 1970s. There is an abundant literature on results of isobaric and isothermal ageing tests in pure distilled water at temperatures higher than typically 40 °C (see for instance references [START_REF] Ifwarson | Life-time of polyethylene pipes under pressure and exposure to high temperatures[END_REF][START_REF] Lu | The ductile-brittle transition in a polyethylene copolymer[END_REF][START_REF] Kausch | Polymer Fracture[END_REF]). The construction of a single master curve by using adequate Arrhenius shift factors [START_REF] Lu | The ductile-brittle transition in a polyethylene copolymer[END_REF] allows extrapolating these results up to ambient temperature and thus, demonstrating that pipes perish always by brittle fracture with lifetimes exceeding typically 50 years in the domain of water pressures of practical interest.

In the past half century, considerable research efforts were accomplished for optimizing the polymer structure, at both macromolecular and morphological scales [START_REF] De Gennes | Reptation of a polymer chain in the presence of fixed obstacles[END_REF][START_REF] Lustiger | Importance of tie molecules in preventing polyethylene fracture under long-term loading conditions[END_REF][START_REF] Huang | Dependence of slow crack growth in polyethylene on butyl branch density: morphology and theory[END_REF][START_REF] Brown | Slow crack growth in polyethylene -a review[END_REF][START_REF] Trankner | Molecular and lamellar structure of an extrusion-grade medium-density polyethylene for gas distribution[END_REF][START_REF] Hedenqvist | Morphology and Morphology Sensitive Properties of Polyethylene: Fracture Behavior and Diffusivity[END_REF][START_REF] Gedde | Molecular structure, crystallization behavior, and morphology of fractions obtained from an extrusion grade high-density polyethylene[END_REF], but also the processing conditions [START_REF] Peacock | Handbook of Polyethylene. Structures, Properties and Applications[END_REF] in view of improving the pipe durability. In the last two decades however, it was discovered that the water disinfectants do not only destroy the organic substances in water by oxidizing processes (among which radical processes play a key role), but also consume the stabilizing function of phenolic antioxidants and initiate a radical chain oxidation of the polymer matrix in the inner wall of pipes, thus leading to a significant reduction the pipe lifetime. These deleterious effects are especially pronounced when chlorine dioxide is used as a disinfectant [START_REF] Colin | Aging of polyethylene pipes transporting drinking water disinfected by chlorine dioxide. I-Chemical aspects[END_REF][START_REF] Colin | Aging of polyethylene pipes transporting drinking water disinfected by chlorine dioxide. I-Lifetime prediction[END_REF][START_REF] Yu | Deterioration of polyethylene pipes exposed to water containing chlorine dioxide[END_REF], but measurable effects were also evidenced in the case of chlorine and bleach [START_REF] Bradley | The effect of chlorine on the long-term durability of crosslinked polyethylene pipe[END_REF][START_REF] Ifwarson | Results and experiences from tests on polyolefin pipes exposed to chlorinated water[END_REF][START_REF] Gill | Long-term durability of crosslinked polyethylene tubing used in chlorinated hot water systems[END_REF][START_REF] Dear | The effects of chlorine depletion of antioxidants in polyethylene[END_REF][START_REF] Hassinen | Deterioration of polyethylene pipes exposed to chlorinated water[END_REF][START_REF] Dear | Effect of chlorine on polyethylene pipes in water distribution networks[END_REF][START_REF] Colin | Stabiliser thickness profiles in polyethylene pipes transporting drinking water disinfected by bleach[END_REF].

The chemistry of water disinfection is relatively well known. Both radical and ionic species (for instance, ClO -in the case of chlorine and bleach) have a strong oxidizing power. However, PE acts as a selective absorber because highly polar species such as ions are totally insoluble into this non-polar matrix. This characteristic was first discovered by Ravens [START_REF] Ravens | The Chemical reactivity of poly(ethyelene terephthalate). Heterogeneous hydrolysis by hydrochloric acid[END_REF] in the case of the poly(ethylene terephthalate) (PET) hydrolysis. PE is part of the less polar polymers: its dipolar moment is zero and its dielectric permittivity is 2.3 [START_REF] Van Krevelen | Properties of Polymers. Their Estimation and Correlation with Chemical Structure[END_REF]. In contrast, radicals are more or less soluble into PE depending on their solubility parameter. As a result, three distinct scenarios can be considered for explaining the chemical attack of both phenolic antioxidants and PE matrix:

-S1) The disinfectant is itself a free radical in ground state which can migrate into PE. This is the case of chlorine dioxide (ClO 2 ) [START_REF] Colin | Aging of polyethylene pipes transporting drinking water disinfected by chlorine dioxide. I-Chemical aspects[END_REF][START_REF] Colin | Aging of polyethylene pipes transporting drinking water disinfected by chlorine dioxide. I-Lifetime prediction[END_REF]; -S2) The disinfectant generates radicals in water, then these latter migrate into PE; -S3) The disinfectant itself, or a non-dissociated molecule formed from this disinfectant in water, migrates deeply into PE where it dissociates into radicals.

The identification of the reactive species and the determination of their respective concentration in the PE matrix remain an open issue, especially in the case of chlorine and bleach for which the two last scenarios S2 and S3 are conceivable. It is now well known that bleach solutions contain three main chemical species whose the relative proportions depend essentially on the pH value [START_REF] Durliat | L'eau de javel: Sa chimie et son action biochimique[END_REF], as shown for instance at 25 °C in Fig. 1. These latter can be determined from the kinetic analysis of the two following chemical equilibria:

The temperature dependence of the equilibrium constants K d and K a was reported in the literature [START_REF] Connick | The hydrolysis of chlorine and its variation with temperature[END_REF][START_REF] Morris | The acid ionization constant of HOCl from 5 to 35°C[END_REF]. Typically between 0 and 45 °C, it can be written:

= - + pK
Since K d and K a are very slowly decreasing functions of temperature, the chemical composition of bleach is almost insensitive to temperature. In fact, the chemical equilibria are very slightly shifted towards lower pH values when increasing the temperature. It can be thus concluded that chlorine (Cl 2 ), hypochlorous acid (ClOH) and hypochlorite ion (ClO -) predominate respectively in highly acidic (pH < 3), weakly acidic (3 < pH < 7.5) and basic media (pH > 7.5), whatever the temperature. Moreover, the maximum yield of ClOH is reached at pH = 5.

Based on these observations, it is thus possible to select carefully pH conditions for identifying what chemical species are responsible for the polymer degradation, but also for deciding between the two possible scenarii S2 and S3.

According to Holst [START_REF] Holst | The chemistry of bleaching and oxidizing agents[END_REF], when they coexist (at pH = 7.5 ± 1.5), ClOH and ClO -would generate ClO% and HO% radicals in water:

ClOH + ClO -→ ClO • + Cl -+ HO • HO% + ClO -→ HO -+ ClO% ClO% + ClO -+ HO -→ 2Cl -+ O 2 + HO%
Thus, scenario S2 could be considered. This scenario was supported by some authors who effectively found a maximum degradation rate at pH = 7 for polyether-based polyurethane fibers [START_REF] Khatua | Chlorine degradation of polyether-based polyurethane[END_REF], or at pH = 8 for polysulfone membranes [START_REF] Rouaix | Experimental study of the effects of hypochlorite on polysulfone membrane properties[END_REF][START_REF] Causserand | Ageing of polysulfones membranes in contact with bleach solution: role of radical oxidation and of some dissolved metal ions[END_REF][START_REF] Gaudichet-Maurin | Ageing of polysulfone ultrafitration membranes in contact with bleach solutions[END_REF].

On the contrary, for other authors [START_REF] Dear | The effects of chlorine depletion of antioxidants in polyethylene[END_REF][START_REF] Dear | Effect of chlorine on polyethylene pipes in water distribution networks[END_REF][START_REF] Epstein | Kinetics of the oxidation of cotton with hypochlorite in the pH range 5-10[END_REF][START_REF] Fukatsu | Degradation of poly(ethylene oxide) by electro-generated active species in aqueous halide medium[END_REF], Cl 2 and ClOH could directly dissociate into Cl% and HO% radicals (presumably within the PE matrix) because Cl-Cl and Cl-O bonds are characterized by a very low dissociation energy (of respectively 242 and 247 kJ mol -1 [START_REF] Calvert | Photochemistry[END_REF]): Cl 2 → 2 Cl% ClOH → Cl% + HO% In this case, the degradation rate would be maximum at pH < 7 and scenario S3 could be envisaged.

Regardless the chosen scenario, it will remain to determine the relative proportion of each radical involved in the polymer degradation events. Since HO% radicals are of the smallest molecular size, they have a high diffusivity into the PE matrix, presumably very close to that of water [START_REF] Mccall | Solubility and diffusion of water in low-density polyethylene[END_REF]. However, as they are also extremely reactive with respect to hydrocarbon substrates (typically more than 10 10 times more reactive than PO 2 % radicals at 25 °C [START_REF] Hickel | Absorption spectra and kinetics of methyl and ethyl radicals in water[END_REF][START_REF] Rudakov | Low selectivity of reactions of OH radicals with alkanes in aqueous solution[END_REF][START_REF] Getoff | Advancements of radiation induced degradation of pollutants in drinking and waste water[END_REF]), they should be scavenged into a sub-micrometric superficial layer of PE pipes. In contrast, chlorinated radicals (ClO% and Cl%) should diffuse more slowly into the inner pipe wall because of their larger molecular size, but they should be also much less reactive than HO% radicals. In fact, it is expected that their reactivity ranks in the following order:

≫ ≫ HO ClO Cl • • • (6)
In a first approach, the depth L of penetration of a radical species can be estimated from a simple scaling law [START_REF] Audouin | Role of oxygen diffusion in polymer ageing: kinetic and mechanical aspects[END_REF]:

≈ ⎛ ⎝ ⎞ ⎠ L D K 1/2 ( 7
)
where D is the diffusion coefficient and K the first-order constant for the consumption of the diffusing species by the chemical reaction with the polymer.

As an example, this equation can be used to compare HO% and ClO% radicals:

⎜ ⎟ = ⎛ ⎝ ⎞ ⎠ L L D D K K HO ClO HO ClO ClO HO 1/2 • • • • • • (8)
The diffusion coefficients of both radicals into the PE matrix are not known, but by analogy with more common vapors and gases of very close molecular size, for instance H 2 O [START_REF] Mccall | Solubility and diffusion of water in low-density polyethylene[END_REF] and CO 2 [START_REF] Van Krevelen | Properties of Polymers. Their Estimation and Correlation with Chemical Structure[END_REF], it can be written:

< < 1 D D 10 HO ClO • • (9)
So that, presumably:

≈ - - - L L 10 10 HO ClO 3 2 • • (10)
Chlorinated radicals are thus expected to penetrate more deeply into the inner wall of pipes. The order of magnitude of the depth of this radical attack should also allow us to decide between scenarii S2 and S3.

The objectives of the present article are twofold. The first objective is to evidence and analyze by conventional laboratory techniques the chemical interactions between additive free PE and bleach in view of proposing a general degradation mechanism. Since ClOH is often considered as the main source of radicals in the literature [START_REF] Holst | The chemistry of bleaching and oxidizing agents[END_REF][START_REF] Khatua | Chlorine degradation of polyether-based polyurethane[END_REF][START_REF] Rouaix | Experimental study of the effects of hypochlorite on polysulfone membrane properties[END_REF][START_REF] Causserand | Ageing of polysulfones membranes in contact with bleach solution: role of radical oxidation and of some dissolved metal ions[END_REF][START_REF] Gaudichet-Maurin | Ageing of polysulfone ultrafitration membranes in contact with bleach solutions[END_REF][START_REF] Epstein | Kinetics of the oxidation of cotton with hypochlorite in the pH range 5-10[END_REF][START_REF] Fukatsu | Degradation of poly(ethylene oxide) by electro-generated active species in aqueous halide medium[END_REF], three distinct pH values, corresponding to the maximum yield (pH = 5) and lower but equivalent yields of ClOH (pH = 4 and 7), will be investigated. The two last values are necessary to see if Cl 2 may be a secondary source of radicals. If so, the degradation rate will be less reduced in acidic (pH = 4) than in neutral medium (pH = 7). In these exposure conditions, two aging indicators will be used for deciding between scenarii S2 and S3: the oxidation rate and the thickness of the oxidation layer. The second objective is to derive a kinetic model from the degradation mechanism and to check its validity by comparing the numerical simulations with all the experimental data collected in this study.

Experimental

Materials

An unstabilized and unfilled PE powder was supplied by Borealis Company for this study. PE films of thicknesses ranging between 150 and 350 μm were formed by compression molding with a Gibrite laboratory press under a pressure of 3 MPa for 2 min at 180 °C. After demolding, the films were characterized by conventional laboratory techniques.

The FTIR spectrophotometry was used in a transmission mode to check that the molding conditions had avoided a premature oxidation of the films. No trace of oxidation was detected after processing and it was necessary to expose the PE films for long periods at moderate temperatures to finally see the appearance of oxidation products. Typical values of induction period of unoxidized PE [START_REF] Khelidj | Oxidation of polyethylene under irradiation at low temperature and low dose rate. Part II-Low temperature thermal oxidation[END_REF] were found: 27 h at 120 °C and 125 h at 100 °C in air.

The main characteristics of the PE films are: melting point T m0 = 128.3 °C, crystallinity ratio X C0 = 51%, density ρ 0 = 0.934 g cm -3 , weight average molecular masses M w0 = 105 ± 5 kg mol -1 and polydispersity index: I p0 = 2.5.

Ageing conditions

PE films were immersed in bleach solutions in glass carboys whose the outer surface was painted with a metal color for avoiding undesired effects of UV radiations throughout the duration of ageing tests. The bleach solutions were maintained at a temperature of 60 ± 1 °C in airventilated ovens, and weekly controlled to be re-adjusted (if necessary) at nominal values of free chlorine concentration [free Cl] W = 100 ppm (i.e. 100 mg L -1 ) and pH = 4, 5 or 7. These solutions were prepared in advance by diluting with pure distilled water and adding a few drops of hydrochloric acid in a highly concentrated (38 g L -1 ) and very basic (pH = 11.5) commercial solution supplied by Sigma-Aldrich. The corresponding concentrations of the three main chemical species (i.e. Cl 2 , ClOH, and ClO -) in water were simply deduced as follows:

= [Cl ] [free Cl] . % 2 W W Cl2 (11) = [ClOH] [free Cl] . % W W C l O H (12) = - - [ClO ] [free Cl] . % W W C l O (13)
where % i is the relative proportion of the chlorinated species i under consideration depending essentially on the pH value (see Fig. 1).

Experimental methods

At defined immersion durations, PE films were removed from the bleach solutions and dried at room temperature in a desiccator containing silica-gel. After complete drying, they were characterized by conventional laboratory techniques in order to determine the resulting changes at both molecular and macromolecular scales.

FTIR analysis

The course of oxidation was monitored by FTIR spectrophotometry according to two different but complementary methods. On one hand, IR spectra across the entire film thickness were determined in a transmittance mode within the 400-4000 cm -1 range (with a minimal resolution of 4 cm -1 ) with a Perkin-Elmer Frontier device. The comparison between the IR spectra obtained for virgin and oxidized PE films (Figs. 234) revealed the development of two broad and intense absorption bands within the 3150-3700 cm -1 and 1680-1780 cm -1 ranges, whatever the pH value. These bands were respectively assigned to the stretching vibrations of O-H and C=O bonds of a large variety of hydroxyl and carbonyl compounds. However, since the latter is centered at 1715 cm -1 , it was concluded that carbonyls are mainly composed of ketones and carboxylic acids [START_REF] Cruz | Thermo-oxidative degradation of additive free polyethylene. Part I. Analysis of chemical modifications at molecular and macromolecular scales[END_REF]. The carbonyl concentration across the entire PE film thickness [CO] global was then calculated from the absorbance Abs at 1715 cm -1 by using the classical Beer-Lambert's law and taking a coefficient of molar extinction of ε = 300 L mol -1 .cm -1 [40]:

= [CO] Abs ep . ε ( 14
)
where ep is the sample thickness.

In addition, in some ageing cases (see for instance Fig. 4), it was clearly observed the development of a third broad, but much less intense, absorption band within the 1560-1680 cm -1 range. This band was attributed to the stretching vibration of the C=O bond of carboxylate ions COO -M + , presumably sodium carboxylates COO -Na + [START_REF] Al-Malaika | A comparative study of the degradability and recyclability of different classes of degradable polyethylene[END_REF][START_REF] Stuart | Infrared Spectroscopy: Fundamentals and Applications[END_REF] resulting from the partial conversion of carboxylic acids by sodium hypochlorite according to the following reaction [START_REF] Johnson | Invitation à la chimie organique[END_REF].

R C OH O + NaOH R C O -Na + O + H 2 O
However, since carboxylate ions never exceed 10 mol% of all the carbonyl compounds, their formation was neglected thereafter.

On the other hand, some PE films were cut in the thickness direction and embedded in a commercial epoxy-amine resin which was crosslinked for 12 h under primary vacuum at room temperature. Then, film cross-sections were polished with silicon carbide abrasive papers of decreasing particle size (typically from 80 to 2400 granulometry) before being examined by FTIR microscopy. IR spectra were determined every 1.56 μm throughout the film thickness in an Attenuated Total Reflectance (ATR) mode within the 750-4000 cm -1 range (with a minimal resolution of 2 cm -1 ) with a Perkin-Elmer Spotlight 300 device equipped with a germanium crystal. Oxidation profiles were then calculated from the carbonyl absorbance Abs measured at 1715 cm -1 as described above (Equation ( 14)). Each profile was validated by checking that the sum of the local concentrations along the PE film thickness [CO] gives access to an average concentration [CO] global very close to the experimental value measured by the first FTIR method.

Viscosimetry

The consequences of oxidation on the PE macromolecular structure were determined by viscosimetry in molten state (typically at 160 °C) in inert atmosphere (nitrogen) with a TA Instruments ARES device equipped with a coaxial and parallel plate geometry. The plate diameter was chosen at 25 mm and the gap between plates was fixed at 1 mm. Sweep angular frequency experiments were performed between 0.1 and 100 rad s -1 with a strain amplitude of 5% on stacks of about ten PE films. These experiments revealed that the rheological behavior of PE is Newtonian in the low frequency range, typically for ω ≤ 1 rad s -1 , before and after ageing (Fig. 5). In addition, it was found that the Newtonian viscosity η varies non-monotonically with exposure time, whatever the pH value. After an initial period of about 2 or 3 weeks, during which it remains almost constant or increases very slowly, η decreases catastrophically with exposure time, indicating that PE undergoes a largely predominant chain scission process. Although their progressive branching cannot be totally excluded, it was concluded that oxidation leads mainly to a progressive shortening of the polymer chains.

The resulting changes in the weight average molar mass M W global were determined by using the classical Bueche's equation [START_REF] Bueche | Viscosity, self-diffusion and allied effects in solid polymers[END_REF][START_REF] Bueche | The viscoelastic properties of plastics[END_REF]:

= η K . M w 3.4 (15)
where K is a constant depending only on molecular structure and temperature. K was preliminary determined from a series of linear PE calibrants of perfectly known distribution of molar mass. It was found that K = 2.46 × 10 -4 Pa mol 3.4 .kg -3.4 for PE at 160 °C.

Results and discussion

Experimental results

The changes in the average carbonyl concentration [CO] global and weight average molar mass M W global of PE films are reported in Fig. 6 for all the exposure conditions under study. It appears clearly that the polymer undergoes a severe oxidation from the earliest days of exposure, but the average oxidation rate is faster at pH = 5 than at pH = 4 and 7. In addition, the kinetics curves reported at pH = 4 and 7 superimpose perfectly (if taking into account the measurement uncertainties). Based on the equilibrium diagram of Fig. 1, it can be concluded that oxidation is initiated by radical species coming mainly from ClOH, the eventual contribution of Cl 2 being clearly insufficient to be detectable. In addition, based on the mechanisms proposed in the literature [START_REF] Dear | The effects of chlorine depletion of antioxidants in polyethylene[END_REF][START_REF] Dear | Effect of chlorine on polyethylene pipes in water distribution networks[END_REF][START_REF] Holst | The chemistry of bleaching and oxidizing agents[END_REF][START_REF] Epstein | Kinetics of the oxidation of cotton with hypochlorite in the pH range 5-10[END_REF][START_REF] Fukatsu | Degradation of poly(ethylene oxide) by electro-generated active species in aqueous halide medium[END_REF], it can be concluded that these radicals are formed according to scenario S3.

Examples of oxidation profiles determined after different immersion durations of PE films in bleach solutions are reported in Figs. 7 and8. It is found that the radical attack operates in a superficial layer of about 50-100 μm thick, almost independent of the pH value. This high thickness value also supports scenario S3. However, it appears that the general shape of the oxidation profiles depends on the pH value (Fig. 8). Indeed, these profiles are clearly in chain-shape in neutral medium, indicating that the oxidation kinetics is of the first-order as demonstrated in Ref. [START_REF] Audouin | Role of oxygen diffusion in polymer ageing: kinetic and mechanical aspects[END_REF]. In addition, they are composed of a horizontal plateau (of about 30 μm length) in the most superficial part of the oxidized layer in acidic media. This plateau presumably results from the presence of a secondary source of radicals, in this case Cl 2 , leading to a significant increase in the local oxidation rate and consequently, to a change of the local oxidation kinetics. In the most superficial part of the oxidized layer, the oxidation kinetics is obviously of the zero-order [START_REF] Audouin | Role of oxygen diffusion in polymer ageing: kinetic and mechanical aspects[END_REF].

Based on these experimental observations, the following causal chain can be proposed for explaining both the initiation of oxidation by bleach and the development of oxidation profiles in additive free PE, whatever the pH value: Formation of Cl 2 , ClOH and ClO -in the water phase → Migration of Cl 2 and ClOH into the PE sample → Dissociation of Cl 2 and ClOH into Cl % and HO% radicals → Radical attack of the PE matrix initiating its oxidation.

Kinetic modeling

This causal chain was introduced into the closed-loop mechanistic scheme developed during the two last decades in our laboratory for predicting other cases of PE oxidation at low temperature [START_REF] Colin | Aging of polyethylene pipes transporting drinking water disinfected by chlorine dioxide. I-Chemical aspects[END_REF][START_REF] Colin | Aging of polyethylene pipes transporting drinking water disinfected by chlorine dioxide. I-Lifetime prediction[END_REF][START_REF] Khelidj | Oxidation of polyethylene under irradiation at low temperature and low dose rate. Part I-the case of "pure" radiochemical initiation[END_REF][START_REF] Khelidj | Oxidation of polyethylene under irradiation at low temperature and low dose rate. Part II-Low temperature thermal oxidation[END_REF][START_REF] Colin | Lifetime prediction of polyethylene in nuclear plants[END_REF]. Then, a kinetic model was derived from this new mechanistic scheme by applying the classical rules of the chemical kinetics. The Fick's second law was directly introduced into this system of 14 differential equations in order to establish a strong coupling between the diffusion of all the molecular reagents (i.e. Cl 2 , ClOH, Cl%, HO% and O 2 ) and their consumption by the chemical reaction. All these theoretical tools are detailed in Appendix A.

The kinetic model contains 27 parameters, consisting in boundary conditions, rate constants, chemical yields and diffusion coefficients, of which many have been already determined in previous studies [START_REF] Colin | Aging of polyethylene pipes transporting drinking water disinfected by chlorine dioxide. I-Chemical aspects[END_REF][START_REF] Colin | Aging of polyethylene pipes transporting drinking water disinfected by chlorine dioxide. I-Lifetime prediction[END_REF][START_REF] Khelidj | Oxidation of polyethylene under irradiation at low temperature and low dose rate. Part I-the case of "pure" radiochemical initiation[END_REF][START_REF] Khelidj | Oxidation of polyethylene under irradiation at low temperature and low dose rate. Part II-Low temperature thermal oxidation[END_REF][START_REF] Colin | Lifetime prediction of polyethylene in nuclear plants[END_REF]. Their values at 60 °C are recalled in Table 1.

Finally, only 12 parameters are unknown, all being relative to chlorinated disinfectants (Cl 2 and ClOH) and their radical species (Cl% and HO%). The following procedure was applied for their determination: a) An indirect approach based on structure/solubility relationships was used to access the value of the equilibrium concentration of Cl 2 . This approach, which had been already applied to ClO 2 in a previous study [START_REF] Colin | Aging of polyethylene pipes transporting drinking water disinfected by chlorine dioxide. I-Chemical aspects[END_REF], is detailed in Appendix B. It leads to:

= × - ⎛ ⎝ - ⎞ ⎠ [Cl ] 4.7 10 (1 X )[Cl ] exp 19130 RT 2 S 2 C 2W (16)
where X C is the PE crystallinity and [Cl 2 ] W is the concentration of Cl 2 in water, depending mainly on the pH value (see Equation ( 11)).

Unfortunately, it cannot be applied to ClOH due to the total ignorance of some key characteristics, in particular its vaporization enthalpy. Since this chlorinated species contains a hydroxyl group able of establishing strong molecular interactions (i.e. hydrogen bonds) with polar side groups along the polymer chain, it was considered that it behaves like H 2 O. Thus, its equilibrium concentration was deduced from water sorption experiments reported in the literature between 25 and 60 °C for ldPE [START_REF] Mccall | Solubility and diffusion of water in low-density polyethylene[END_REF]:

= × ⎛ ⎝ - ⎞ ⎠ [H O] 3.8 10 exp 37600 RT 2 S 3 (17)
The introduction of X C and [ClOH] W in this equation leads finally to:

= × - ⎛ ⎝ - ⎞ ⎠ - [ClOH] 5.1 10 (1 X )[ClOH] exp 37600 RT S 3 C W (18)
b) The order of magnitude of the rate constant of the polymer attack by HO% radicals was directly taken in the literature: k 1h = 10 9 L mol -1 .s -1 [START_REF] Hickel | Absorption spectra and kinetics of methyl and ethyl radicals in water[END_REF][START_REF] Rudakov | Low selectivity of reactions of OH radicals with alkanes in aqueous solution[END_REF][START_REF] Getoff | Advancements of radiation induced degradation of pollutants in drinking and waste water[END_REF]. In a first approach, the temperature variation of this rate constant was neglected owing to its very low activation energy (E 1h = 1-10 kJ mol -1 ) [START_REF] Greiner | Hydroxyl radical kinetics by kinetic spectroscopy. VI. Reactions with alkanes in the range 300-500[END_REF]. In addition, since HO% radicals and H 2 O are almost of the same molecular size, it was assumed that their diffusion coefficients into the PE matrix are also very close and thus, obey the following Arrhenius law [START_REF] Mccall | Solubility and diffusion of water in low-density polyethylene[END_REF]:

≈ = × ⎛ ⎝ - ⎞ ⎠ - D D 2.9 10 exp 37600 RT HO H O 4 • 2 (19)
Finally, the rate constant of addition of HO% radicals onto P% radicals (k 7h ) was determined by adjusting as closely as possible the experimental results obtained for PE films immerged in hydrogen peroxide solutions. Indeed, it has been shown in the literature that H 2 O 2 is an efficient initiator of radical chain oxidation in polyolefins, leading to a predominant chain scission process and thus, to a catastrophic decrease in the molecular mass [START_REF] Moad | Aqueous hydrogen peroxide-induced degradation of polyolefins: a greener process for controlled-rheology polypropylene[END_REF]. The kinetic model used for this identification was established on the same principles as for the bleach This model is detailed in Appendix C. For the same reasons mentioned for HO% radicals, Equations ( 18) and ( 19) could be used for proposing a first set of values for the transport properties of H 2 O 2 into the PE matrix. However, it is noteworthy that H 2 O 2 is capable of establishing two hydrogen bonds (against only one for H 2 O) with polar side groups. Thus, its equilibrium concentration is expected to be slightly higher while, according to structure/water permeability relationships [START_REF]Non-empirical kinetic modelling of non-Fickian water absorption induced by a chemical reaction in epoxy-amine networks[END_REF], its diffusion coefficient should be somewhat lower than for water. Fortunately, [H 2 O 2 ] S was reported in the literature for a ldPE sample immerged in a highly concentrated hydrogen peroxide solution at T = 35 °C [START_REF] Radl | Quantifying absorption effects during hydrogen peroxide decontamination[END_REF]: This value was selected for this study. In contrast, the coefficient of The values of all the parameters used for these simulations are given in Tables 1 and2. c) Finally, the last 7 unknown parameters, i.e. the coefficients of ClOH, Cl 2 and Cl% diffusion (D ClOH , D Cl2 , D Cl %) and the rate constants of ClOH and Cl 2 decomposition (k 0a , k 0c ), polymer attack by Cl% radicals (k 1c ) and addition of Cl% radicals onto P% radicals (k 7c ), were determined by adjusting as closely as possible the experimental results

= × - - [H O ] 1.1 10 (1 X )[H O ]
This large difference in reactivity is difficult to perceive through the rate constants (k 7h and k 7C ) of addition of these radicals onto P% radicals due to the high mobility (by valence migration) of P% radicals within the PE matrix: P% + PH → PH + P%.

In addition, it can be pointed out that the values of k 1C and k 7C are very close to those reported for ClO 2 in a previous publication [START_REF] Colin | Aging of polyethylene pipes transporting drinking water disinfected by chlorine dioxide. I-Chemical aspects[END_REF]. Thus, these two chlorinated radicals would have almost the same reactivity.

-The diffusion coefficients of the molecular reagents ranks in the right order when considering their molecular size:

≪ < Cl Cl O 2 • 2 (22)
but also their polarity:

< < H O ClOH HO 2 2 • ( 23
)
-It is thus confirmed that Cl% radicals migrate more deeply than HO% radicals into the PE sample thickness, thus contributing, in addition to scenario S3, to the large value of the superficial oxidized layer. In a first approach, their depth of penetration can be estimated from Equation [START_REF] Brown | Slow crack growth in polyethylene -a review[END_REF]:

⎜ ⎟ ⎜ ⎟ ≈ ⎛ ⎝ ⎞ ⎠ ≈ ⎛ ⎝ ⎞ ⎠ L D K D k [PH] Cl Cl Cl 1 2 Cl 1c 1 2 • • • • ( 24
)
It comes:

L Cl % ≈ 10 μm
In contrast, the application of Equation ( 7) to HO • radicals leads to L HO• values lower than 1 nm. Thus, these radicals are scavenged as soon as they are formed within the PE matrix.

Once validated at the molecular level, the kinetic model was used to predict the consequences of oxidation at the macromolecular scale. The classical Saito's equation [START_REF] Saito | On the effects of high energy radiation to polymers. I. Crosslinking and degradation[END_REF][START_REF] Saito | On the effects of high energy radiation to polymers. II. End-linking and gel fraction[END_REF] was used for simulating the changes in weight average molar mass of PE films reported in Fig. 6b:

- = - 1 M 1 M S 2 2X W W 0 (25) = + - M 2M 2 (S 4X)M W W0 W0 (26)
Here again, it can be noticed a satisfying agreement between theory and experiments. The kinetic model perfectly accounts, without making additional assumption and without adding additional parameter, that, after a few weeks, oxidation leads to a catastrophic decrease in the average molar mass due to the large predominance of chain scissions over crosslinking. It was concluded that crosslinking results only from the coupling of PO 2

• and P • radicals, while chain scissions result only from the rearrangement by β scission of alkoxy (PO • ) radicals.

Conclusion

The chemical interactions between additive free PE and bleach have been investigated in solutions maintained at a temperature of 60 °C, a free chlorine concentration of 100 ppm, and a pH = 4, 5 or 7. A scenario was proposed for tentatively explaining why oxidation reaches its maximum rate at pH = 5 and occurs in a relatively large superficial thickness (about 50-100 μm thick), almost independent of the pH value, despite the high reactivity of the involved radicals (in particular HO • ). According to this scenario, ClOH and Cl 2 would migrate deeply into the PE matrix prior to dissociate into radicals and then, initiate a radical chain oxidation.

A kinetic model has been derived from this scenario. Many of its parameters were determined in previous studies dedicated to simpler cases of oxidative ageing. The remaining 12 unknown parameters have been determined from structure/transport property and structure/reactivity relationships reported in the literature, but also from complementary ageing experiments of immersion in hydrogen peroxide solutions. This kinetic model predicts accurately all the experimental data collected in this study. It gives access to the orders of magnitude of the transport properties of chlorinated disinfectants (Cl 2 and ClOH) and their radical species (Cl • and HO • ), but also to the orders of magnitude of the rate constants of the elementary reactions in which they are involved. It was found that Cl • radicals are much less reactive than HO • radicals, but are of the same reactivity as ClO 2 . The values of these parameters will have to be refined in future studies.

The next challenges will consist in introducing the stabilizing effects of commercial blends of antioxidants into the kinetic model for tentatively predicting the degradation states (in particular, the profiles of antioxidant consumption and oxidation products) in the inner wall of industrial PE pipes in service.

Appendix A. Kinetic model for predicting PE oxidation in a bleach solution

The mechanistic scheme describing the radical chain oxidation of additive free PE in a bleach solution can be summarized as follows. It is composed of five main stages:

Decomposition: (0a) ClOH → Cl% + %OH (k 0a ) (0c) Cl 2 → 2Cl% (k 0c ) Initiation: (Iu) POOH → 2P% + γ CO P = O + (1-γ CO )P-OH + γ S S (k 1u ) (Ib) 2POOH → P% + PO 2 % + γ CO P = O + (1-γ CO )P-OH + γ S S (k 1b ) (Ic) Cl% + PH → HCl + P% (k 1c ) (Ih) HO% + PH → H 2 O + P% (k 1h ) Propagation: (II) P% + O 2 → PO 2 % (k 2 ) (III) PO 2 % + PH → POOH + P% (k 3 ) Bimolecular recombination: (IV) P% + P% → γ 4 P-P + (1-γ 4 )PH + (1-γ 4 )F + γ 4 X (k 4 ) (V) P% + PO 2 % → γ 5 POOP + (1-γ 5 )PH + (1-γ 5 )POOH + γ 5 X (k 5 ) (VIa) PO 2 % + PO 2 % → [PO% %OP] cage + O 2 (k 6a ) (VIb) [PO% %OP] cage → POOP + X (k 6b ) (VIc) [PO% %OP] cage → P=O + P-OH (k 6c ) (VId) [PO% %OP] cage → 2P% + 2γ CO P = O +2(1-γ CO )P-OH + 2γ S S ( k 6d ) Grafting: (VIIc) P% + Cl% → P-Cl (k 7c ) (VIIh) P% + HO% → P-OH (k 7h )
in which: a) ClOH and Cl 2 account respectively for hypochlorous acid and chlorine having penetrated into the PE matrix b) PH accounts for secondary C-H bonds (i.e. methylene groups) c) Cl%HO%, P%, PO 2 % and PO% account respectively for chlorine, hydroxyl, alkyl, peroxy and alkoxy radicals; d) POOH, POOP, P=O and P-OH account respectively for hydroperoxides, dialkyl peroxides, carbonyls and alcohols as macromolecular oxidation products; e) P-Cl is an attempt at representing the grafting of chlorine onto the polymer chain as organic chlorides f) F, S and X account for double bonds, chain scissions and crosslinking nodes (i.e. covalent bridges) respectively; g) γ S is the yield of β-scission of PO% radicals (leading to aldehydes) in competition with hydrogen abstraction (leading to alcohols), irrespectively of the molecularity of the initiation reaction; h) From a practical point of view, it is more convenient to use an apparent yield γ CO for the carbonyl production owing to the high uncertainty on the nature of these species and the value of their corresponding molar extinction coefficients [START_REF] Cruz | Thermo-oxidative degradation of additive free polyethylene. Part I. Analysis of chemical modifications at molecular and macromolecular scales[END_REF]; i) γ 4 and γ 5 are the respective yields for the coupling of alkyl-alkyl and alkyl-peroxy radicals in competition with disproportionation j) k i are the elementary rate constants of the different chemical reactions under consideration. Thus, two main types of initiation reaction are in competition: the decomposition of hydroperoxides (POOH) according to an unimolecular (Iu) or bimolecular mode (Ib), and the direct polymer attack by radical species (Cl% and HO%) formed by dissociation of chlorinated species (ClOH and Cl 2 ) having beforehand migrated into the PE matrix. The former type of initiation is strongly thermo-activated: the activation energy of the corresponding rate constant ranges typically between 140 (for k 1u ) and 105 kJ mol -1 (for k 1b ) [START_REF] Khelidj | Oxidation of polyethylene under irradiation at low temperature and low dose rate. Part II-Low temperature thermal oxidation[END_REF]. contrast, because of the high reactivity of HO% radicals [START_REF] Greiner | Hydroxyl radical kinetics by kinetic spectroscopy. VI. Reactions with alkanes in the range 300-500[END_REF], but presumably also of Cl% radicals, the two last initiations are almost independent of temperature. As a result, at low temperature close to ambient, the oxidation should be mainly initiated by chlorinated species.

A kinetic model was derived from this mechanistic scheme by applying the classical rules of the chemical kinetics. It is composed of 14 non-linear differential equations:

= - + ∂ ∂ d[ClOH] dt k [ClOH] D [ClOH] z 0a ClOH 2 2 (A.1) = - + ∂ ∂ d[Cl ] dt k [Cl ] D [Cl ] z 2 0c 2 Cl 2 2 2 2 (A.2) = + - - + ∂ ∂ d[Cl ] dt k [ClOH] 2k [Cl ] k [Cl ][PH] k [Cl ][P ] D [Cl ] z • 0a 0c 2 1c • 7c • • Cl 2 • 2 • (A.3) = - - + ∂ ∂ d[HO ] dt k [ClOH] k [HO ][PH] k [HO ][P ] D [HO ] z • 0a 1h • 7h • • HO 2 • 2 • (A.4) = - - + + - d[POOH] dt k [POOH] 2k [POOH] k [PO ][PH] (1 γ )k [P ][PO ] 1u 1b 2 3 2 • 5 5 • 2 • (A.5) = + + + d[P ] dt 2k f [POOH] k f [POOH] k [Cl ][PH] k [HO ][PH] • 1u PH 1b PH 2 1c • 1h • (A.6) - + - - k [O ][P ] k [PO ][PH] 2k [P ] k f [P ][PO ] 2 2 • 3 2 • 4 • 2 5 PH • 2 • + - - 2k f [PO OP] k [Cl ][P ] k [HO ][P ] 6d PH • • cage 7c • • 7h • • = + - - d[PO ] dt k f [POOH] k [O ][P ] k [PO ][PH] k f [P ][PO ] 2 • 1b PH 2 2 2 • 3 2 • 5 PH • 2 • (A.7) -2k [PO ] 6a 2 • 2 = - + + d[PO OP] dt k [PO ] (k k k f )[PO OP] • • cage 6a 2 • 2 6b 6c 6d PH • • cage (A.8) = -+ -+ - d[PH] dt (2 γ )k f [POOH] (1 γ )k f [POOH] k [Cl ][PH] S 1u PH S 1b PH 2 1c • (A.9) - - -- k [HO ][PH] k [PO ][PH] (1 γ )k f [P ][PO ] 1h • 3 2 • 5 5 PH • 2 • - + - - 2(1 γ )k f [PO OP] k [Cl ][P ] k [HO ][P ] S 6d PH • • cage 7c • • 7h • • = - + + ∂ ∂ d[O ] dt k [O ][P ] k [PO ] D [O ] z 2 2 2 • 6a 2 • 2 O 2 2 2 2 (A.10) = = + + d[P O] dt γ k f [POOH] γ k f [POOH] k [PO OP] CO 1u PH CO 1b PH 2 6c • • cage (A.11) + 2γ k f [PO OP] CO 6d PH • • cage - = d[P Cl] dt k [Cl][P ] 7c • • (A.12) = + + dS dt γ k f [POOH] γ k f [POOH] 2γ k f [PO OP] S 1u PH S 1b PH 2 S 6d PH • • cage (A.13) = + + dX dt γ k [P ] γ k f [P ][PO ] k [PO OP] 4 4 • 2 5 5 PH • 2 • 6b • • cage (A.14)
where z is the depth beneath the sample surface and D ClOH , D Cl2 , D Cl %, D HO % and D O2 are the respective diffusion coefficients of ClOH, Cl 2 , Cl%, HO% and O 2 in the sample thickness.

PH is a mathematical function introduced in the system of differential equations for avoiding that the substrate concentration becomes negative at high conversion ratios of oxidation process. In a first approach, a hyperbolic form has been chosen to describe the changes of f PH versus [PH]: [START_REF] Bradley | The effect of chlorine on the long-term durability of crosslinked polyethylene pipe[END_REF] with typically ε = 10 -2 ≪ 1. This function does not induce significant changes of the oxidation kinetics below a conversion value of about 99%.

= + f [PH] [PH] ε PH (A.
The boundary conditions are: The numerical method for solving such a system of differential equations has been already described in a previous publication [START_REF] Colin | Determination of thermal oxidation rate constants by an inverse method. Application to polyethylene[END_REF]. Very briefly, it is based on implicit or semi-implicit algorithms recommended for solving stiff problems of chemical kinetics. Indeed, these algorithms give a satisfying approximate solution for a reasonable cost of calculation. The results reported in this study were obtained by applying the ODE23s solver of Matlab commercial software based on the semi-implicit Rosenbrock's algorithm [START_REF] Hairer | Solving Ordinary Differential Equations II. Stiff and Differential-algebraic Problems[END_REF].

Eqs A-1 to A-14 were integrated numerically for accessing the concentration profiles (in the sample thickness, along the z abscissa) of all the chemical species, and their changes against exposure time. The corresponding average concentrations were deduced by integrating these profiles over the entire sample thickness with the trapezoid method.

Appendix B. Equilibrium concentration of ClO 2 and Cl 2

The transport properties of chlorinated species into PE have never been studied to our knowledge. But, they would be close to those of much better known vapors or gases which can be thus used for estimating some key characteristics of chlorinated species, in particular their Lennard-Jones temperature T LJ :

= T ε k LJ (B.1)
where ε is the pre-factor of the Lennard-Jones potential and k the Boltzmann constant. In Table B-1, it can be seen that SO 2 and C 5 H 12 exhibit close molecular size (assimilated to molar mass M) and cohesive properties (represented by the ratio ΔH V /R T b ) than ClO 2 and Cl 2 respectively. Since their T LJ is known, it can be assumed that:

= T (X) T (Y) ΔH (X) ΔH (Y) LJ LJ V V (B.2)
where X and Y designate a couple of molecules, and ΔH V is their vaporization enthalpy. The numerical application of Equ. B-2 gives:

T LJ = 369 K for ClO 2 T LJ = 353 K for Cl 2
The solubility of chlorinated species in PE at 298 K can be calculated from an empirical equation first established by Van Amerongen [START_REF] Van Amerongen | Diffusion in elastomers[END_REF] for little polar elastomers, then used by Michaels and Bixler [START_REF] Michaels | Solubility of gases in polyethylene[END_REF][START_REF] Michaels | Flow of gases through polyethylene[END_REF] for the amorphous phase of PE:

= - + Log S (298 K) 7,0 0,010 T a LJ (B.3)
where S a is expressed in cm 3 STP.cm -3 .Pa -1 . The numerical application of Equ. B-3 gives: S a (298 K) = 4.9 × 10 -4 cm 3 STP.cm -3 .Pa -1 i.e. S a (298 K) = 2.2 × 10 -5 mol L -1 Pa -1 for ClO 2 S a (298 K) = 3.4 × 10 -4 cm 3 STP.cm -3 .Pa -1 i.e. S a (298 K) = 1.5 × 10 -5 mol L -1 Pa -1 for Cl 2

For a PE sample of crystallinity ratio X C , one can write:

= - S S (1 X ) a C (B.4)
The same authors gave the expression for the heat of dissolution H S into the PE matrix:

= - ± H R (1000 10 T ) 500 K S LJ (B.5)
The numerical application of Equ. B-5 gives:

= - ± H R 2690 500 K for ClO S 2 = - ± H R 2530 500 K for Cl S 2
Thus, the dissolution of chlorinated species into the PE matrix is an exothermic process. In other words, S is a decreasing function of temperature: For determining the concentration of chlorinated species in PE, it remains to know the partial pressure of chlorinated species p in the atmosphere, in equilibrium with the water solution. According to the equilibrium diagrams of water/chlorinated species systems reported in French technical publications [START_REF] Masschelein | Le dioxyde de chlore pour la maîtrise de la qualité des eaux[END_REF][START_REF] Irsn | Dioxyde de chlore[END_REF][START_REF] Gmbh | Chlorine, Physical Data Sheet SD[END_REF][START_REF] Irsn | [END_REF], it can be written: and E p = 40 190 J mol -1 for Cl 2 where [ClO 2 ] W and [Cl 2 ] W are the corresponding mass fractions of the chlorinated species in water, expressed in ppm. Let us remember that [Cl 2 ] W is given by Eq. ( 11), if knowing the pH of the bleach solution.

= ⎛ ⎝ - ⎞ ⎠ S S
Finally, the equilibrium concentration of chlorinated species in PE is given by the usual Henry's law: The mechanistic scheme describing the radical chain oxidation of additive free PE in a hydrogen peroxide solution can be summarized as follows. The numerical methods for solving such a system of differential equations, then deducing the average concentrations of all chemical species over the entire sample thickness, are indicated in Appendix A.

= × = ⎛ ⎝ - ⎞ ⎠ [ClO ] S p [ClO ] exp H RT 2 S 2 S0 C (B.8) with = × - - - [ClO ] 1.8 10 (1 X ) [ClO ] mol. L 2 S0 4 C 2W 1 and H C = 4450 J mol -1 = × = ⎛ ⎝ - ⎞ ⎠ [Cl ] S p [Cl ] exp H RT

Cl 2 (

 2 aq) + 2H 2 O ⇆ ClOH + Cl -+ H 3 O + (Kd) ClOH + 2H 2 O ⇆ ClO -+ H 3

Fig. 1 .

 1 Fig. 1. Chemical composition of a bleach solution in function of the pH value at 25 °C.

Fig. 2 .

 2 Fig. 2. Modification of the carbonyl (a) and hydroxyl regions (b) of the IR spectrum during immersion of additive free PE films in a bleach solution regulated at [free Cl] W = 100 ppm, T = 60 °C and pH = 4.

Fig. 3 .

 3 Fig. 3. Modification of the carbonyl (a) and hydroxyl regions (b) of the IR spectrum during immersion of additive free PE films in a bleach solution regulated at [free Cl] W = 100 ppm, T = 60 °C and pH = 5.

Fig. 4 .

 4 Fig. 4. Modification of the carbonyl (a) and hydroxyl regions (b) of the IR spectrum during immersion of additive free PE films in a bleach solution regulated at [free Cl] W = 100 ppm, T = 60 °C and pH = 7.

Fig. 5 .

 5 Fig. 5. Melt viscosity at 160 °C versus angular frequency after immersion of additive free PE films in a bleach solution regulated at [free Cl] W = 100 ppm, T = 60 °C and pH = 7.

Fig. 6 .

 6 Fig. 6. Kinetic curves of average carbonyl concentration (a) and weight average molar mass (b) during immersion of additive free PE films in bleach solutions regulated at [free Cl] W = 100 ppm, T = 60 °C and pH = 4, 5 or 7. Comparison between experimental data (symbols) and kinetic modeling (solid lines).

Fig. 7 .

 7 Fig. 7. Development of oxidation profiles in additive free PE films during their immersion (in days) in a bleach solution regulated at [free Cl] W = 100 ppm, T = 60 °C and pH = 7. Comparison between experimental data (symbols) and kinetic modeling (solid lines).

  X C = 50% and [H 2 O 2 ] W = 10 ppm, it comes: [H 2 O 2 ] S = 5.5 × 10 -8 mol l -1

Fig. 8 .

 8 Fig. 8. Effect of pH on oxidation profiles in additive free PE films after 21 days of immersion in bleach solutions regulated at [free Cl] W = 100 ppm, T = 60 °C and pH = 5 (a) and pH = 7 (b). Comparison between experimental data (symbols) and kinetic modeling (solid lines).

Fig. 9 .

 9 Fig. 9. Kinetic curve of average carbonyl concentration during the immersion of additive free PE films in a hydrogen peroxide solution regulated at [H 2 O 2 ] W = 10 ppm and T = 60 °C. Comparison between experimental data (symbols) and kinetic modeling (solid lines).

Fig. 10 .

 10 Fig. 10. Development of oxidation profiles in additive free PE films during their immersion (in days) in a hydrogen peroxide solution regulated at [H 2 O 2 ] W = 10 ppm and T = 60 °C. Comparison between experimental data (symbols) and kinetic modeling (solid lines).

  a) When t = 0, [ClOH] = [ClOH] S and [Cl 2 ] = [Cl 2 ] S (equilibrium concentrations of chlorinated species) at z = 0 and z = L (i.e. at the sample surfaces, L being the sample thickness) and [ClOH] = [Cl 2 ] = 0 everywhere else. b) When t = 0, [Cl%] = [HO%] = [P%] = [PO 2 %] = [PO% %OP] cage = [P=O] = [P-Cl] = S = X = 0, [POOH] = [POOH] 0 = 10 -2 mol l -1 , [PH] = [PH] 0 = 60 mol l -1 (initial concentration in methylene groups), and [O 2 ] = [O 2 ] S = 3.8 10 -4 mol l -1 (equilibrium concentration of oxygen) at every z. c) When t > 0, [O 2 ] = [O 2 ] S = 3.8 10 -4 mol l -1 , [ClOH] = [ClOH] S and [Cl 2 ] = [Cl 2 ] S at z = 0 and z = L.

  mol -1 for Cl 2

- 1 4 [ 5 [

 145 For a PE sample of X C = 50% immerged in a water solution regulated at [ClO 2 ] W = 100 ppm, it comes [ClO 2 ] S = 1.5 × 10 -3 mol l -1 at 25 °C [ClO 2 ] S = 1.8 × 10 -3 mol l -1 at 60 °C Thus, in the temperature interval of practical interest, [ClO 2 ] S would keep the same order of magnitude. The same conclusion can be made for [Cl 2 ] S . It appears clearly that Cl 2 is considerably more soluble in PE than ClO 2 , but its equilibrium concentration depends mainly on the pH value. As an example, for a PE sample of X C = 50% immersed in a bleach solution regulated at [Cl free] W = 100 ppm and T = 60 °C, it comes: [Cl 2 ] S = 2.8 mol l -1 at pH = Cl 2 ] S = 3.2 × 10 -1 mol l -1 at pH = Cl 2 ] S = 2.2 × 10 -3 mol l -1 at pH = 7 Appendix C. Kinetic model for predicting PE oxidation in a hydrogen peroxide solution

  ][PH] 2k [P ] k f [P ][PO ] 2k f [PO OP] ][PH] (1 γ )k f [P ][PO ] 2(1 γ )k f [PO OP]

+

  O] dt γ k f [POOH] γ k f [POOH] k [PO OP] 2γ k f [PO OP] CO 6d PH • • cage where z is the depth beneath the sample surface and D H2O2 , D HO• and D O2 are the respective diffusion coefficients of H 2 O 2 , HO • and O 2 in the sample thickness. The mathematical expression of f PH is given in Appendix A. The boundary conditions are: a) When t = 0, [H 2 O 2 ] = [H 2 O 2 ] S (equilibrium concentration of hydrogen peroxide) at z = 0 and z = L (i.e at the sample surfaces, L being the sample thickness) and [H 2 O 2 ] = 0 everywhere else. b) When t = 0, [HO%] = [P%] = [PO 2 %] = [PO% %OP] cage = [P=O] = 0, [POOH] = [POOH] 0 = 10 -2 mol l -1 , [PH] = [PH] 0 = 60 mol l -1 (initial concentration in methylene groups), and [O 2 ] = [O 2 ] S = 3.8 10 -4 mol l -1 (equilibrium concentration of oxygen) at every z. c) When t > 0, [O 2 ] = [O 2 ] S = 3.8 10 -4 mol l -1 , [H 2 O 2 ] = [H 2 O 2 ] S at z = 0 and z = L.

Table 1

 1 Values at 60 °C of parameters used for modeling the thermal oxidation of PE.

	Kinetic parameter	Value at 60 °C	Unity
	k 1u	9.1 × 10 -10	s -1
	k 1b	9.7 × 10 -8	L.mol -1 .s -1
	k 2	10 8	L.mol -1 .s -1
	k 3	5.4 × 10 -2	L.mol -1 .s -1
	k 4	8.0 × 10 11	L.mol -1 .s -1
	k 5	1.8 × 10 11	L.mol -1 .s -1
	k 6a	1.4 × 10 7	L.mol -1 .s -1
	k 6b	2.0 × 10 6	s -1
	k 6c	2.0 × 10 5	s -1
	k 6d	9.0 × 10 6	s -1
	γ CO	100	%
	γ S	41	%
	γ 4	50	%
	γ 5	1.1	%
	D O2	1.4 × 10 -10	m 2 .s -1

0h ) and addition of HO • radicals onto P • radicals (k 7h ) were determined by using the kinetic model as an inverse resolution method. As an example, the simulations of the kinetic curve of average carbonyl concentration and the development of oxidation profiles in PE films, during their immersion in a hydrogen peroxide solution regulated at [H 2 O 2 ] W = 10 ppm and T = 60 °C, are reported in Figs. 9 and 10. It can be noticed a satisfying agreement between theory and experiments.

Table 2

 2 Values at 60 °C of the additional parameters used for modeling the PE oxidation in hydrogen peroxide solutions.

	Kinetic parameter	Value at 60 °C	Unity
	k 0h	8.0 × 10 -1	s -1
	k 1h	10 9	L.mol -1 .s -1
	k 7k	10 10	L.mol -1 .s -1
	D H2O2	8.0 × 10 -11	m 2 .s -1
	D HO•	3.7 × 10 -10	m 2 .s -1

Table 3

 3 Values at 60 °C of the additional parameters used for modeling the PE oxidation in bleach solutions.obtained for PE films immersed in bleach solutions (Figs.6a, 7 and 8). It can be noticed a satisfying agreement between theory and experiments. The values of all the parameters used for these simulations are given in Tables1-3. They call for the following comments:-ClOH is much more unstable than Cl 2 within the PE matrix. It is thus the main source of radicals. It is firstly responsible for the polymer degradation in bleach solutions according to scenario S3. -As expected, HO% radicals are much more reactive than Cl% radicals:

	Kinetic parameter	Value at 60 °C	Unity
	k 0a	1.8	s -1
	k 0c	3.0 × 10 -5	s -1
	k 1c	1.5 × 10 -3	L.mol -1 .s -1
	k 7c	3.0 × 10 9	L.mol -1 .s -1
	D ClOH	10 -10	m 2 .s -1
	D Cl2	10 -14	m 2 .s -1
	D Cl %	8.0 × 10 -12	m 2 .s -1

Table B1

 B1 Molar mass (M), boiling point (T b ), critical temperature (T cr ), vaporization enthalpy (ΔH V ), ratio ΔH V / R T b and Lennard-Jones temperature (T LJ ). All the characteristics, except T LJ of ClO 2 and Cl 2 , were taken from Handbooks, for instance from Ref.[START_REF] Durliat | L'eau de javel: Sa chimie et son action biochimique[END_REF].

	Molecule	M	T b	T cr	ΔH V	ΔH V /RT b	T LJ
		(g.mol -1 )	(K)	(K)	(kJ.mol -1 )		(K)
	SO 2	64	263	431	24.8	11.3	335
	ClO 2	67.5	284	426	27.3	11.5	369
	Cl 2	71	239	417	20.4	10.3	353
	C 5 H 12	72.1	309	470	26.4	10.3	457

  It is composed of five main stages: 2P% + γ CO P = O + (1-γ CO )P-OH + γ S S (k 1u )(Ib) 2POOH → P% + PO 2 % + γ CO P = O + (1-γ CO )P-OH + γ S S (k 1b ) (Ih) HO% + PH → H 2 O + P% (k 1h ) Propagation: (II) P% + O 2 → PO 2 % (k 2 ) (III) PO 2 % + PH → POOH + P% (k 3 ) Bimolecular recombination: (IV) P% + P% → γ 4 P-P + (1-γ 4 )PH + (1-γ 4 )F + γ 4 X (k 4 ) (V) P% + PO 2 % → γ 5 POOP + (1-γ 5 )PH + (1-γ 5 )POOH + γ 5 X (k 5 ) (VIa) PO 2 % + PO 2 % → [PO% %OP] cage + O 2 (k 6a ) (VIb) [PO% %OP] cage → POOP + X (k 6b ) (VIc) [PO% %OP] cage → P=O + P-OH (k 6c ) (VId) [PO% %OP] cage → 2P% + 2γ CO P = O +2(1-γ CO )P-OH + 2γ S S ( k 6d ) Grafting: (VIIh) P% + HO% → P-OH (k 7h )in which H 2 O 2 accounts for hydrogen peroxide having penetrated into the PE matrix.A kinetic model was derived from this mechanistic scheme by applying the classical rules of the chemical kinetics. It is composed of 9 non-linear differential equations:

	Decomposition:
	(0h) H 2 O 2 → 2HO%				(k 0h )
	Initiation:
	(Iu) POOH → = -d[H O ] dt 2 2 k [H O ] D + 0h 2 2	H O 2 2	∂	2	∂ [H O ] 2 2 z 2


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  Downscaling Dynamic Time Warping (IDDTW) : Estimation de la dissimilarité des séries temporelles de précipitations-Analyse et interprétation Djallel Dilmi, Laurent Barthès, Cécile Mallet To cite this version: Djallel Dilmi, Laurent Barthès, Cécile Mallet. Iterative Downscaling Dynamic Time Warping (ID-DTW) : Estimation de la dissimilarité des séries temporelles de précipitations-Analyse et interprétation. Journée Scientifique SIRTA 2017, Jun 2017, Palaiseau, France. insu-01663710