J Baux

N Pébère

S Delaunay

J Tireau

M Roy

D You

J.-L Bretelle

Film-forming amines for corrosion protection of carbon steels in PWR secondary circuit conditions

During layup periods of pressurized water reactors (PWRs), corrosion phenomena can affect metallic components of the secondary circuit, mainly carbon steel. The use of corrosion inhibitors such as film-forming amines (FFAs) is an interesting way to attenuate or stop these phenomena and extend the lifetime of the materials. Film-forming amines (FFA) have both a hydrophilic head which can adsorb on metal or oxides surface, and a hydrophobic tail which can organizes rather perpendicularly to the surface and create, by accumulation, a barrier between the metal and the corrosive environment.

The aim of the present work is to investigate the behavior and the efficiency of FFA for the carbon steels protection under various physical and chemical conditions, representative of the PWR secondary circuits. Carbon steel plates was submitted to treatment at temperatures between 120 °C and 275 °C in an autoclave with or without FFA: here the octadecylamine (ODA). The adsorption of the FFA and the corrosion inhibition efficiency of the films formed on the steel surface were characterized by contact angle measurements, scanning electron microscopy (SEM) and electrochemical impedance spectroscopy (EIS).

Introduction

During layup periods of PWRs for maintenance operation, corrosion phenomena can occur on carbon steel materials due to oxygen entrance. Use of alternative corrosion inhibitors such as film-forming amines (FFA) is a promising approach to reduce or to stop the corrosion of carbon steel, which constitute the major part of the PWR secondary circuit [START_REF] Betova | Film-forming amines in steam / water cycles -structure, properties, and influence on corrosion[END_REF][START_REF] Turner | Fouling of nuclear steam generators : fundamental studies, operating experience and remedial measures using chemical additives[END_REF]. A key benefit of film-forming amines is to improve the corrosion protection during startups, shutdowns and layups of the PWR and, as a consequence, startup will be quicker, leading to significant economic repercussions [START_REF] Betova | Film-forming amines in steam / water cycles -structure, properties, and influence on corrosion[END_REF]. Even though FFA have been used successfully for decades to treat boiler feed water in industrial power plants [START_REF]Technical guidance document: Application of film-forming amines in fossil, combined cycle, and biomass power plants[END_REF], their behavior and the physical characteristics of the film formed on the carbon steel surface are not well described. The FFA tested in this paper was octadecylamine (ODA), CH3(CH2)17NH2, a well-known corrosion inhibitor for steel in nuclear power plants and fossil fuel power plants for years [START_REF] Betova | Film-forming amines in steam / water cycles -structure, properties, and influence on corrosion[END_REF][START_REF]Technical guidance document: Application of film-forming amines in fossil, combined cycle, and biomass power plants[END_REF][START_REF] Liao | Characterisation of surface film on iron samples treated with octadecylamine[END_REF][START_REF] Ge | A study of anti-corrosion behavior of octadecylamine-treated iron samples[END_REF][START_REF] Wagner | Preservation of boilers and turbines with the surface active substance octadecylamine (ODA)[END_REF][START_REF] Chernyshev | Increasing the corrosion resistance of equipment due to the use of film-forming amines[END_REF][START_REF] Kurshakov | Heat transfer enhancement in steam-turbine condensers with the use of surface-active substances[END_REF][START_REF] Hater | The chemistry and properties of organic boiler feedwater additives based on film-forming amines and their use in steam generators[END_REF][START_REF] Sipilä | Effect of octadecylamine on carbon steel corrosion under PWR secondary side conditions[END_REF].

First, this paper introduces a key characteristic of ODA which is the water/steam distribution coefficient. Then, the ODA deposition on carbon steel samples in representative thermochemical secondary circuit conditions was investigated. Surface morphology observations and contact angle measurements were supplemented by electrochemical analysis in order to assess the anti-corrosion efficiency of the film formed. Experimental devices and tests conditions were described. The obtained results were discussed in order to highlight the remaining gaps on FFA behavior understanding. The contact angle test, commonly used to state upon the formation of an FFA film on a metallic surface, was also discussed.

Chemical Composition and Action Mode of ODA

The ODA molecule is constituted by one aliphatic alkyl group (hydrophobic part) of 18 carbon atoms with one amine function (hydrophilic head). ODA has the simplest FFA structure. FFA molecules can adsorb on metal or oxide surfaces to form an insulating hydrophobic film, thus creating a barrier which limits the access of water and corrosive species (oxygen, carbon dioxide, chlorides…) toward the metal or metal oxide surface. The strong affinity of FFA for metal surfaces is due to the lone pair of electrons on nitrogen [START_REF] Jäppinen | Corrosion Behavior of Carbon Steel Coated with Octadecylamine in the Secondary Circuit of a Pressurized Water Reactor[END_REF]. The mechanism of ODA adsorption could be physisorption or chemisorption, depending on physico-chemical parameters (temperature, pH…). The latter would lead to stronger bonds to the surface.

Experimental Part

Materials

The P275 low carbon steel was chosen as representative material of PWR. Its composition is reported in Table 1. The steel sample was machined to obtain small coupons (dimensions were 30x20x3 mm). Before the deposition step, the surface was wet ground with silicon carbides (SiC) papers down to grade 800, cleaned ultrasonically with ethanol, rinsed with deionized water and dried.

ODA Deposition

The ODA deposition was performed in an autoclave (Figure 1) in representative conditions of the secondary circuit in static conditions. The autoclave was made of titanium. It contained a sampling hose, a temperature probe and a rotating axis to ensure a good homogeneity of the liquid phase. The rotation speed was fixed at 100 rotations per minute (rpm). Three carbon steel plates were placed in the mesh and two at the bottom of the autoclave to evaluate the reproducibility.

The solution was chemically conditioned a pH25 °C between 9.6 and 9.8 with ethanolamine (ETA) and ammonia (NH3). The media was deaerated with nitrogen bubbling during 10 min before each experiment. Hydrazine was added to ensure a complete deaeration (reducing conditions). For security reasons, the autoclave was not completely filled with the solution. A steam phase was present above the liquid solutions.

The effect of ODA was studied at 120 °C, 220 °C and 275 °C. To be consistent with site conditions, the samples treated at 120 °C and 220 °C were immersed in the liquid phase. And those treated at 275 °C were placed in the steam side. The ODA concentration in the liquid phase was controlled to be close to 2 mg/kg (maximum aimed concentration for ODA based conservation).

Surface Characterizations

Optical observations were performed with a binocular loupe and an Olympus PMG3 optical microscope. The contact angles were measured using a Krüss Mobile Surface Analyze. The protocol consisted in depositing a deionized water drop of an accurate volume of 1 µL on the sample surface (bare metallic surface or treated surface with ODA) and then measuring the static contact angle (θ) 60 s after the deposition. The contact angle value was obtained by calculating the slope of the tangent to the drop at the liquid-solid interface thanks to the Advance software. To assess the homogeneity of the surface properties, 20 measurements were performed on different locations on the samples surface and the average contact angle was calculated with the standard deviation. All the experiments were performed at room temperature.

Electrochemical Measurements

The electrochemical cell used was a Bio-Logic ® three electrodes cell with a saturated sulfate reference electrode and a platinum grid as counter electrode. The working electrode consisted of the carbon steel plate. Electrochemical impedance measurements were performed using a Solartron 1286 electrochemical interface connected to a Solartron 1250 frequency response analyzer. Impedance diagrams were obtained under potentiostatic regulation at the corrosion potential (Ecorr) in the frequency range of 65 kHz to 10 mHz with 8 points per decade, using a 15 mVrms sinusoidal voltage. Impedance data beyond 6.5 kHz were not taking into account due to experimental artifacts. Data modeling was carried out using the SimAd software developed at the LISE CNRS (Paris). The electrolyte solution was prepared from analytical grade Na2SO4 10 -3 M and deionized water in equilibrium with air ([O2]25 °C = 8 mg/kg). pH25 °C was adjusted at 9.6 -9.8 with NH3 and ETA. The choice of this medium was based upon its low electrical conductivity, close to that encountered in industrial water circuit, its low corrosiveness toward carbon steel and because it is an easily reproducible baseline solution.

Results and Interpretation

Water/Steam Distribution Coefficient of ODA

Preliminary experiments were performed to determine apparent water/steam distribution coefficient of ODA at 120, 220 and 275 °C. As a steam phase was always present in the autoclave during the tests, the ODA behavior according to the different tested temperatures was a key parameter to foresee its concentration both in the liquid and the steam phase. The desired ODA concentration in the liquid phase during the deposition close to 2 mg/kg.

To determine the ODA distribution coefficient, the following methodology was used:

1 -Preparation of a concentrated ODA solution (50 mg/kg) 2 -Performing sampling and ODA analysis at the tested temperature in the liquid phase 3 -Deducing the ODA concentration in the steam phase and calculation of the apparent distribution coefficient.

The hypothesis of a negligible adsorption of the ODA on the autoclave surfaces was made based on the following calculation: assuming that one ODA molecule occupies 20 Ų [START_REF] Benítez | Structure and chemical state of octadecylamine self-assembled monolayers on mica[END_REF], 0.72 µg ODA was calculated to cover the inner surface of the autoclave (320 cm²) and considering a monolayer. This quantity is negligible compared to the injected mass of ODA (12.5 mg), even in the case of the formation of a multilayer film. We can conclude that the lost ODA is mainly due to the distribution of ODA between the liquid and the steam phase and not to adsorption on the autoclave surface.

The determined distribution coefficient of ODA for the different tested temperatures are presented in Figure 2 and compared to literature data [START_REF] Ramminger | The application of film-forming amines in secondary side chemistry treatments of NPPs[END_REF].

As it can be seen, the experimentally calculated ODA distribution coefficients are close to those theoretically determined, especially at 220 °C. The coefficients determined at 120 °C and 275 °C, even if they follow the same trend of the curve, show a sufficient difference that makes it non-suitable to directly use the curve from literature for the concentration prediction of ODA in the liquid phase.

Influence of Temperature on Surface Oxides Formed and Hydrophobicity

The SEM images of the surface morphology of the coupons treated at different temperatures with and without ODA, are shown in Figure 3. The morphology of the samples shows the formation of a typical magnetite oxide layer for all the coupons. Dark areas are visible on the samples treated at 120 °C and at 220 °C (Figure 3.b and 3.d). Their composition, qualitatively determined by EDX (Figure 4), highlights an enrichment in carbon in the dark areas. The deposited crystal size is clearly larger for the samples treated at 275 °C (Figure 3.e and 3.f) even for a short period of time (2 h). Dark, ODA-rich areas could not be seen for this sample. The contact angle measured for the ODA treated samples at 120 °C (Figure 3.a and 3.b) and at 220 °C (Figure 3.c and 3.d) show a significant change in the surface hydrophobicity attributed to the ODA adsorption. In fact, the contact angle increases by about 22° and 25° for the treated ODA carbon steel at 120 °C and 220 °C respectiveley compared to their reference samples. The contact angle for the sample treated with ODA at 275 °C in the steam phase (Figure 3.f) does not change significantly (≈ 14°) compared to the reference sample (Figure 3.e) even though a better hydrophobic character can be noticed. In this case, the surface roughness seems to play a significant role in the contact angle measurement.

Influence of the Temperature on ODA Deposition and Efficiency

Figure 5 presents the impedance diagrams obtained for the carbon steel treated with 2 mg/kg of ODA during one week at 120 °C. This deposition time was necessary for such a low ODA concentration and has been determined from previous experiments. The impedance diagram of the treated carbon steel sample was compared with a blank obtained in the same conditions without ODA. The impedance modulus (Figure 5.a) at low frequency (10 -3 Hz), was considered as a measure of the instantaneous corrosion resistance. A significant increase of the modulus after treatment with ODA (more than two decades) is observed. The phase angle (Figure 5.b) shows, at high frequency, a constant-phase element (CPE) behavior (phase angle lower than -90 degrees) which can be associated to the presence of a dielectric film (e.g. ODA film) on the working electrode surface. This time-constant is not visible on the blank sample. The origin of the CPE behavior can be attributed to an in-depth resistivity gradient in the film [START_REF] Hirschorn | Constant-Phase-Element Behavior Caused by Resistivity Distributions in Films. II: Applications[END_REF][START_REF] Hirschorn | Constant-Phase-Element Behavior Caused by Resistivity Distributions in Films. I: Theory[END_REF]. Therefore, the high frequency impedance data can be regressed by a power-law model (Eq. 1). Then, the physical parameters of the ODA film can be extracted (e.g. the thickness and the permittivity) (Figure 6), according to our previous results [START_REF] Baux | Impedance analysis of film-forming amines for the corrosion protection of a carbon steel[END_REF]. The regressed parameters are reported in table 2.

= 1 + + - (Eq. 1)
Where: • is the ODA film thickness;

• is the ODA film permittivity;

• is the permittivity of vacuum; • is the pulsation;

• and are the resistivity at the metal/film interface and film/electrolyte interface, respectively; • is the dimensionless position; • is a parameter indicating the variation of the resistivity. Figure 7 shows the impedance diagrams of the carbon steel treated samples with and without ODA at 220 °C during 168 h. In this case, the impedance magnitude difference between the treated and untreated samples is modest compared to what have been observed for treatment at 120 °C. This is in accordance with the SEM observations showing scarce dark areas at the surface of the carbon steel treated with ODA at 220 °C. Besides, the presence of organic molecules on the surface was corroborated by drop angle measurements showing a significant improvement of the hydrophobicity of the treated ODA sample's surface (Figure 3.c and 3.d). However, data fitting was not possible in this case due to the poor definition of the film related time-constant. This could mean that the ODA film wasn't dense enough leading to weak anticorrosive properties. A thermal degradation of ODA during the treatment at 220 °C could explain the poor film quality. The ODA concentration in the liquid phase at 275 °C was followed during 24 hours (Fig. 8).

The dosage of ODA highlights a rapid diminution of the ODA concentration (80%) in less than 6 hours at 275 °C. After 24 hours the ODA concentration is almost zero. The main hypothesis to explain this result was a thermal degradation of ODA in a period of time comprised between 2 and 24 hours at 275 °C Figure 8: Evolution with time of the ODA concentration in the liquid phase at 275 °C.

Figure 9 shows a comparison of Bode plots resulted for carbon steel after treatment at 275 °C after 2 hours with 2 mg/kg ODA and without ODA, and after 24 hours with ODA. The impedance magnitude shows a significant increase after 2 hours with ODA compared to the reference sample. Besides, two time-constants are visible on the phase after 2 hours of treatment with ODA, although there is not well-defined phase rotation related to the film. The impedance magnitude decreases between 2 hours and 24 hours. The phase shows only one time constant, meaning that the response of the film disappears after 24 hours at 275 °C, in accordance with the decreasing concentration of ODA.

Discussion

First, it was seen in Figure 2 that the distribution coefficient of ODA is a key parameter that must be taken into account in order to control the ODA concentration in the two phases (water or steam). At high temperature, ODA has a pronounce affinity for the steam phase that can lead to a decrease of the ODA concentration in water and an over-concentration of ODA in the steam.

The surface observations coupled with contact angle and EIS measurements clearly revealed the presence of an ODA film on the surface of the carbon steel sample at 120 °C and seem to indicate the presence of ODA on the surface of the sample treated at 220 °C. A better hydrophobicity was highlighted as well as a higher impedance modulus at low frequency. However, the corrosion protection of the treated surfaces could not always be linked to the measure of the contact angle as the surface roughness also played a role in the hydrophobic character of the surface, particularly in the case of the carbon steel treated at 275 °C with or without ODA. In that case, the contact angle did not increase significantly after the ODA deposition. Therefore, the contact angle measurement is a qualitative way to follow the variation of the hydrophobicity of a surface but it does not allow the phenomenon responsible of this hydrophobicity (organic film or surface roughness) to be discriminated. A link between hydrophobicity and anti-corrosive properties cannot clearly be established. The quantification of ODA along time and EIS measurements after 2 h and 24 h of treatment with ODA at 275 °C showed that ODA degraded almost completely in few hours in water conditioned at pH25 °C equal to 9.6-9.8. This result, raised the question of the ODA behavior in steam generators where such temperatures are usual.

Conclusions

The use of FFA is a possible alternative to improve conservation of secondary circuit materials during layup/shutdown periods of units. ODA is a well-known FFA that can be used to hinder magnetite depositions and decrease corrosion rates of carbon steel and low alloyed steel. Even though ODA based conservation appears promising, numerous questions still persist in particular its behavior in the secondary circuit conditions of PWRs. The results presented here helped to have a better understanding of the ODA behavior in specific studied conditions.

First, the water/steam distribution behavior of ODA as a function of the temperature was investigated. Then, the deposition of ODA on carbon steel in particular laboratory conditions was performed by mean of an autoclave at 120 °C, 220 °C and 275 °C. The presence of ODA on the carbon steel surface treated at 120 °C and at 220 °C was confirmed by surface observations and EIS measurements. In the case of the carbon steel sample treated at 275 °C with and without ODA, the contact angle measurement did not show a significant difference to confirm the presence of an organic film. The surface morphology was similar for both samples with and without ODA. The EIS measurements show a slight difference in the electrochemical response for the ODA treated sample at 275 °C during 2 h that could not be attributed to a covering film. In parallel, a study of the kinetics of ODA degradation at 275 °C revealed a degradation of more than 80 % of this product in less than 6 hours. All of these results provide a range of elements in favor of the absence of ODA in SGs, only ODA by-products could remain. More investigations are necessary to determine ODA behavior in all the secondary circuit conditions.

The formation and the characterization of an ODA film formed at low temperature has been recently studied [START_REF] Baux | Impedance analysis of film-forming amines for the corrosion protection of a carbon steel[END_REF]. This study and the present paper provide important elements for the understanding of ODA behavior versus temperature. However, several other key parameters will be determinant on ODA behavior in the secondary circuit conditions: the concentration, the fluid velocity, pH and the nature and morphology of the substrate. In addition, it will be necessary to determine the behavior of ODA degradation products. To go ahead in predicting the conditions of an effective inhibition of the corrosion by ODA in the secondary circuit, the next step will be to determine the impact of each parameters on the characteristics and the effectiveness of the film.

Figure 1 :

 1 Figure 1: Schematic representation of the autoclave used to simulate the secondary circuit thermo-chemical conditions (left). Schematic representation of the mesh with the three carbon steel plates (right).

Figure 2 :

 2 Figure 2: Water/steam ODA distribution coefficient as a function of the temperature (in red the present study, in blue Ramminger et al. [13]). The uncertainty of the calculated coefficient was due to the dosage protocol of ODA by spectrophotometry UV/Vis.

Figure 3 :

 3 Figure 3: Surface morphologies of the carbon steel samples observed by SEM after treatments in the autoclave at a) 120 °C during 168 h without ODA, b) 120 °C during 168 h with 2 mg/kg ODA. c) 168 h at 220 °C without ODA, d) 168 h at 220 °C with 2 mg/kg ODA. e) 2 h at 275 °C in the steam without ODA, f) 2 h at 275 °C in the steam with 2 mg/kg ODA (concentration in the liquid phase).

Figure 4 :

 4 Figure 4: EDX spectra (normalized from the Fe peak) corresponding to different areas on the treated ODA carbon steel at 120 °C during 168 h (figure 3.b).

Figure 5 :

 5 Figure 5: Impedance diagrams in Bode coordinates of the carbon steel treated samples at 120 °C during 168 h with 2 mg/kg ODA (red) and without ODA (black). (a) Impedance modulus and (b) phase angle.

Figure 6 :

 6 Figure 6: Impedance diagram of the carbon steel ODA treated sample at 120 °C during 168 h with 2 mg/kg ODA fitted with the power-law model in the frequency range corresponding to the electrochemical response of the film.

Figure 7 :

 7 Figure 7: Impedance diagrams in Bode coordinates of the carbon steel treated samples at 220 °C during 168 h with 2 mg/kg ODA (red) and without ODA (black). (a) Impedance modulus and (b) phase angle.

Figure 9 :

 9 Figure 9: Impedance diagrams in Bode coordinates of the carbon steel treated samples at 275 °C in the steam phase during 2 h and 24 h with 2 mg/kg ODA (red) and during 2 h without ODA (black). (a) Impedance modulus and (b) phase angle.

Table 1 :

 1 Chemical composition of the P275 carbon steel (EN10028-3).

	Element (wt%)	C	Si	Mn	Cr	Mo	Ni	Cu	Fe
	P275 NL1	0.16 0.4	0.5-1.5 0.3 0.08 0.5 0.3	Bal.

Table 2 :

 2 Values of the regressed parameters with the power-law model. Comparison with literature data and graphical analysis.

	Power-law parameters	Regressed parameters
	(nm)	18 ± 2	From graphical analysis: 18 ± 5
		20 ± 1	From graphical analysis: 20
		2.8 ± 0.1	From literature [17]: 2.7


S Lahaye

J Luo

P Bellier

T D Huynh

A Tsilanizara

UNCERTAINTY QUANTIFICATION OF ISOTOPIC DENSITIES IN DEPLETED FUEL

Uncertainty quantification of interest outputs based on the nuclear data is an important issue for nuclear safety, from operation of nuclear facilities to long term deposit management. To be more accurate, the Verification and Validation process must take into account the uncertainty quantification due to various sources: input data, design, models... This work is part of the VVUQ process for CEA depletion codes on a PWR fuel rod application. We compare the results of DARWIN and MENDEL and experimental data. Experimental data used in this work consist of isotopic concentrations for four UO 2 fuel samples irradiated up to 64.7 GWd/t in a pressurized water reactor. Uncertainties used for nuclear constant data are JEFF-3.1.1 ones, when cross sections uncertainties data are from the CEA COMAC data base.

Concerning the choice of uncertainty quantification methods, DARWIN deterministic approach is based on sensitivity profiles computation by direct forward perturbation, thanks to INCERD module. MENDEL probabilistic method is based on Monte Carlo correlated sampling. Those samplings are achieved through the CEA uncertainty platform URANIE.

Nuclear data uncertainties are propagated and experimental data errors are taken into account. Modelisation biases or algorithm accuracy are supposed negligible. Output quantities of interest are the isotopic concentrations at given cooling times.

We show that uncertainty propagation results by both methods give similar results, compatible with experimental data. This work validates both the methods and code systems on this application of high burnup PWR fuel irradiation. We focus on the elaboration of a measure for sensitivity coefficients applicable to the stochastic approach, where thousands of eventually correlated uncertain input parameters are taken into account.

INTRODUCTION

Code system VVUQ (Verification, Validation and Uncertainty Quantification) process consists of a three step proof of the validity of the codes:

• Verification : code elementary tests and code to code comparisons;

• Validation : code to experiment confrontation;

• Uncertainty Quantification : propagating input data uncertainties to outputs of interest. This work is part of the VVUQ process for MENDEL v1.2 [START_REF] Tsilanizara | Probabilistic approach for decay heat uncertainty estimation under uranie platform by using mendel depletion code[END_REF], CEA new generation depletion code. It focuses on a high burnup pressurized water reactor (PWR) fuel rod application. Experiment data [START_REF] Sasahara | Isotopic analysis of actinides and fission products in LWR high-burnup UO 2 spent fuels and its comparison with nuclide composition calculated using JENDL, ENDF/B, JEF and JEFF[END_REF] give isotopic concentrations for four UO 2 fuel samples irradiated up to 64.7 GWd/t in a PWR. We use two fuel cycle codes developed in CEA/DEN: DARWIN/PEPIN2.4.2 [START_REF] Tsilanizara | DARWIN: an Evolution Code System for a Large Range of Applications[END_REF] and MENDEL v1.2. Nuclear data nominal values and uncertainties on decay data are taken from JEFF-3.1.1 [4], while uncertainties on cross section data are taken from the CEA COMAC data base [START_REF] De Saint-Jean | Estimation of Multi-Group Cross-Sections Covariances for 235[END_REF]. From now on, DARWIN/PEPIN2.4.2 is referred as DARWIN, and MENDEL v1.2 as MENDEL. Both code systems use a different approach for uncertainty quantification. DARWIN propagates uncertainties through its INCERD module [START_REF] Huynh | IncerD : Outil intégrant les fonctionnalités de propagation des incertitudes du code CYRUS dans le code DARWIN/PEPIN2.4.2[END_REF][START_REF] Lahaye | Comparison of deterministic and stochastic approaches for isotopic concentration and decay heat uncertainty quantification on elementary fission pulse[END_REF] thanks to a deterministic approach: forward perturbation, one at a time method. MENDEL propagation is based on a stochastic approach, using Monte Carlo correlated sampling [START_REF] Mckaya | Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code[END_REF][START_REF] Lahaye | Comparisons between a priori Uncertainty Quantification and Calculation/Measurement Discrepancies Applied to the MERCI UO2 Fuel Rod Decay Heat Experiment[END_REF]. Samplings are generated by the CEA uncertainty platform URANIE [START_REF] Gaudier | URANIE : The CEA/DEN Uncertainty and Sensitivity platform[END_REF]. Both codes solve the depletion equations (Bateman equation) and there is a simple chaining with the flux solvers, with no retroaction of isotopic concentration to neutronic flux in the presented application. Both methods are classic methods largely used in fuel cycle applications [START_REF] Rochman | Nuclear data uncertainties for typical lwr fuel assemblies and a simple reactor core[END_REF][START_REF] Ionescu-Bujor | A comparative review of sensitivity and uncertainty analysis of large scale systems -i: Deterministic methods[END_REF][START_REF] Cacuci | A comparative review of sensitivity and uncertainty analysis of large scale systems -ii: Statistical methods[END_REF][START_REF] Rochman | Nuclear data uncertainty propagation: Perturbation vs. monte carlo[END_REF]. This article is an extension of a previously published work [START_REF] Lahaye | First Verification and Validation Steps of MENDEL Release V1.0 Cycle Code System[END_REF] where validation process had not taken into account uncertainty quantification. Uncertainty is propagated to isotopic concentrations at several cooling times. For this study, only experimental measurement errors and nuclear input data uncertainties are taken into account. Indeed, algorithm accuracy can be considered as marginal in regards to input data uncertainties, as we use reference solvers for the Bateman equation resolution. For modeling biases, we use the official complete decay chain available in MENDEL (resp. DARWIN), which is the best effort chain, generally used for reference calculations. We show that both uncertainty propagation results give similar results, compatible with experimental data and validating both the methods and the code systems. We focus on the elaboration of a measure for sensitivity coefficients applicable to the stochastic approach, where thousands of eventually correlated uncertain input parameters are taken into account. Similar work for elementary fission pulse have already been achieved and published, for both nominal computations [START_REF] Lahaye | First Verification and Validation Steps of MENDEL Release V1.0 Cycle Code System[END_REF] and uncertainty quantification [START_REF] Vallet | Deterministic Approach of the Decay Heat Uncertainty due to JEFF-3.1.1 Nuclear Data Uncertainties with the CYRUS Tool and the DARWIN2.3 Depletion Code[END_REF][START_REF] Tsilanizara | Probabilistic approach for decay heat uncertainty estimation under uranie platform by using mendel depletion code[END_REF].

VVUQ PROCESS

Bateman Equation

In this work, we focus on irradiation process and cooling down of a homogenized radioactive material, where the atom density N i (t) of nucleus i in a fissile material region is the solution of generalized Bateman [START_REF] Bateman | The solution of a system of Differential Equations Occuring in the Theory of Radioactive Transformations[END_REF] equation [START_REF] Tsilanizara | Probabilistic approach for decay heat uncertainty estimation under uranie platform by using mendel depletion code[END_REF].

dN i (t) dt = -(λ i + τ i,i )N i (t) + ∑ j =i (b j,i λ j + τ r j,i )N j (t) + ∑ k γ k,i τ f k N k (t) (1)
where b j,i λ j is the partial decay rate of nuclide j to nuclide i; τ i,i is the total disappearance rate of nuclide i by its neutronic transmutations; τ r j,i is the transmutation rate of nuclide j to nuclide i by neutronic reaction r; γ k,i is the fission yield of fissile nuclide k creating the fission product i;

τ f k
is the fission rate of fissile nuclide k.

Multiple methods of resolution exist to solve the generalized Bateman equation [START_REF] Moler | Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later[END_REF]. The ones implemented in MENDEL have been validating in other works [START_REF] Lahaye | Implementation of a CRAM solver in MENDEL Depletion Code System[END_REF]. In the work presented here, both DARWIN and MENDEL uses a Runge-Kutta 4th order method, even if CRAM method [START_REF] Pusa | Rational approxiamtions to the matrix exponential in burnup calculations[END_REF] have also been implemented in MENDEL [START_REF] Lahaye | Implementation of a CRAM solver in MENDEL Depletion Code System[END_REF]. It will be released with MENDEL v2.0.

Correlations Between Uncertain Data

For the Uncertainty Quantification process, assumptions have been done on the choice of correlation between nuclear data. Uncertain data are of five types. No correlation is taken into consideration between data of different types.

• independent fission yields: default correlation assumes that the sum of yields for a given fissile system is constant (id est, no uncertainty on the mean number of fission products);

• radioactive decay branching ratio: correlation assumes that the sum of the decay branching ratios from a father isotope is equal to 1;

• radioactive decay energies: by defaults, considered independent;

• radioactive decay periods: considered independent;

• multigroup microscopic cross sections: correlations are given by COMAC [START_REF] De Saint-Jean | Estimation of Multi-Group Cross-Sections Covariances for 235[END_REF] data base.

In the present work, DARWIN and MENDEL use uncertainty data from JEFF-3.1.1 [4] for independent fission yields, radioactive decay periods, radioactive decay branching ratios and radioactive decay energies.

Uncertainty Quantification Methods

We describe briefly in this section the propagation methods and hypotheses.

Deterministic Method Used in DARWIN

DARWIN propagates uncertainties from nuclear data to decay heat or isotopic concentrations using a direct forward first order perturbation method (one at a time method). With X the uncertain input variables (nuclear data) and Y the uncertain outputs (isotopic density), we can use the following formula:

cov(Y ) = S Y /X cov(X)S T Y /X (2)
where cov(X) (resp. cov(Y )) stands for the variance-covariance matrix for variable X (resp. Y ) and S Y /X stands for the sensitivity matrix of Y regarding X. The use of a direct forward first order perturbation method implies the hypothesis of linearity of the outputs (as a function of uncertain nuclear data). For small perturbations of the uncertain input parameters (limited to one standard deviation), this approximation is proved to be valid when comparing with propagation approaches not taking this linearity hypothesis, such as stochastic uncertainty propagation approach as the one used in MENDEL. The uncertainty propagation in DARWIN needs (n + 1) depletion calculations where n is the number of uncertain parameters. Calculations are performed in parallel mode by DARWIN through its INCERD module, which establish the sensitivity matrix before computing the interest output covariance through equation [START_REF] Sasahara | Isotopic analysis of actinides and fission products in LWR high-burnup UO 2 spent fuels and its comparison with nuclide composition calculated using JENDL, ENDF/B, JEF and JEFF[END_REF]. Some parameters are associated to a zero value of uncertainty in the JEFF-3.1.1 library, due to the lack of knowledge. If we propagate a default uncertainty level for those parameters, the one at a time method will need to launch some 47039 calculations for decay data only (cross sections from COMAC will be added to this number): 7709 for decay energies, 3626 for decay periods, 2952 for decay branching ratios and 32752 for independent fission yields related to thermal and fast fissile systems. This level of accuracy is needed to obtain good precision on all isotopic concentrations and decay heat. Those calculations being highly scalable on perturbations, it is totally realistic in fuel cycle code systems, where flux are taken from a previous calculation. Calculation time for one perturbation lasts several seconds.

Stochastic Method Used in MENDEL

MENDEL propagates uncertainty using a correlated sample method. The creation of the samples is performed by CEA/DEN URANIE platform [START_REF] Gaudier | URANIE : The CEA/DEN Uncertainty and Sensitivity platform[END_REF]. Samples are generated by the use of a LHS method [START_REF] Mckaya | Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code[END_REF] based on the Inverse Density Function methodology. For correlated sampling, we use a correlation by rank algorithm on the samples. For this study, 2000 realizations were propagated. Most random variables are chosen Gaussian if the relative standard deviation is lower than 50% and Log-Normal in the other cases. Fission yields are always considered log-normal. This choice for nuclear data probability density functions have been explicited in [START_REF] Lahaye | Choice of positive distribution law for nuclear data[END_REF].

VALIDATION OF MENDEL ON A PWR APPLICATION CASE

Description of the Experiment

The isotopic concentrations are calculated by MENDEL and DARWIN for a 3.8% enriched UOX PWR fuel cell, using JEFF-3.1.1 nuclear library data. Results are compared to the experimental data in [START_REF] Sasahara | Isotopic analysis of actinides and fission products in LWR high-burnup UO 2 spent fuels and its comparison with nuclide composition calculated using JENDL, ENDF/B, JEF and JEFF[END_REF]. This study consists of a chemical analysis of a high-burnup PWR-UO 2 fuel, whose four fuel samples (A to D) are analyzed. Burnup varies from 52.8 to 64.7 GWd/tHM. This burnup is obtained after 5 cycles, and the chemical isotopic concentrations are calculated after a cooling time from 3 years to 5 years. Four samples are extracted at four different axial positions of the highburnup PWR-UO 2 fuel, with different moderator temperature which is supposed to be constant during the irradiation. [START_REF] Huynh | Notice d'identification de la bibliothèque CEA V5.1.2 pour DARWIN/PEPIN2[END_REF] (based on JEFF-3.1.1) for all the isotopes on which the chemical analysis had been carried out. Reference chains with more than 2500 isotopes are used in both codes. Calculation of isotopic concentrations and decay heat in MENDEL and DARWIN uses neutronic data (reaction rates and neutronic flux) issued from a Saphyb file created by APOLLO2 [START_REF] Sanchez | Apollo2 year 2010[END_REF]. Cooling period between the cycles is not taken into account during the modeling process in APOLLO2, but added during the DARWIN (resp. MENDEL) calculations. During the simulation in MENDEL and DARWIN, cooling periods of 7 weeks (49 days) between two cycles are taken into consideration. A fourth order Runge Kutta solver with parabolic interpolation of neutronic reaction rates during all the five cycles is used in MENDEL. In DARWIN, the fourth order Runge-Kutta method is implemented on constant by steps neutronic reaction rates.

In the first place, comparison will be done without uncertainty quantification. Discrepancy between calculated isotopic concentrations (by both MENDEL and DARWIN), as well as between calculated isotopic concentrations and the experimental values are analysed.

In a second step, uncertainty propagation is applied in order to study the influence of nuclear data values on isotopic concentration calculation and accuracy. In this part, computed uncertainties in MENDEL and DARWIN are taken into consideration for the sake of giving the confidential interval of calculation.

Isotopic concentrations without uncertainty propagation

Comparison between MENDEL and DARWIN

In order to compare isotopic concentrations from MENDEL and DARWIN calculations, we use the same depletion chain, the same numerical method (Runge Kutta 4th order), the same assumptions of a constant by steps reaction rates. The discrepancy of calculated isotopic concentrations between MENDEL and DARWIN is shown in the figure 1. Discrepancy is calculated by

N MDL -N DAR N MDL ,
where N MDL (resp. N DAR ) represents the isotopic concentrations in MENDEL (resp.in DARWIN). Isotopic concentration discrepancies are all in the -0.7% to 0.4% interval (where there are only 3 isotopes with discrepancies larger than 0.5%, while 44 isotopes whose discrepancies less than 0.2%). Hence, MENDEL and DARWIN give nearly the same result.

Comparisons between experimental data and MENDEL results

MENDEL and DARWIN results were compared to experimental data and relative discrepancy for each nuclide. Reader can refer to mean values of figure 2 which gives the mean relative discrepancy between experimental data and numerical results over the available fuel samples (A to D) data.Those results are similar to reference paper [START_REF] Sasahara | Isotopic analysis of actinides and fission products in LWR high-burnup UO 2 spent fuels and its comparison with nuclide composition calculated using JENDL, ENDF/B, JEF and JEFF[END_REF] and to previous work [START_REF] Lahaye | First Verification and Validation Steps of MENDEL Release V1.0 Cycle Code System[END_REF]. From now on, both MENDEL and DARWIN use their own reference depletion chain.

Referring to heavy nuclides, 234 U, 235 U and 239-242 Pu are overestimated by MENDEL results compared to experimental data, and this overestimation has been confirmed when comparing with CBZ code [START_REF] Kawamato | Validation of CBZ code system for post irradiation examination analysis and sensitivity analysis (n,γ) branching ratio[END_REF] results. Detailed power history is unknown. We made an hypothesis of constant power history for each cycle, which causes the imprecise neutron flux as well as discrepancy of the concentration for 235 U. Concerning to the concentration discrepancy of 234 U, it is possibly due to the inaccuracy initial concentration.

Discrepancies for several nuclides -like plutonium, americium and curium -are outside the experimental error bars. To investigate more precisely, uncertainty quantification is decided.

Uncertainty propagation in MENDEL and DARWIN

Comparison of experimental and computed concentration with uncertainty propagation

For one isotope, let's consider we have M nz experimental data (M nz ∈ [START_REF] Tsilanizara | Probabilistic approach for decay heat uncertainty estimation under uranie platform by using mendel depletion code[END_REF]4]), corresponding to the available data for fuel samples A to D. Global discrepancy is computed as the mean discrepancy over available data. We note X the isotopic density. We show that the mean standard deviation σ ABCD (X) over N statistical samples reads:

σ 2 ABCD (X) = 1 M nz * N -1 M nz ∑ k=1 N ∑ i=1 (X k,i -X k ) 2 = 1 M nz * N -1 M nz ∑ k=1 (N -1)σ 2 k (X) (3)
where X k,i is the isotopic density calculated for sample i in case k, X k is the mean value for isotopic densities in case k and σ k (X) is the standard deviation for case k. Hence, we obtain

σ 2 ABCD = N -1 M nz * N -1 M nz ∑ k=1 σ 2 k ∼ = 1 M nz M nz ∑ k=1 σ 2 k (X) (4)
BEPU2018-290

Figure 2 shows isotopic concentrations from MENDEL and DARWIN calculations. Ordinates correspond to one standard deviation when propagating all nuclear data uncertainties (fission yields, decay periods, decay branching ratios and microscopic cross sections).

- It can be observed that for several isotopes, uncertainty propagation bars and experimental error bars at 1σ do not overlap. The seven isotopes for which it is still the case for 2σ are listed in table 2.

Isotope 86 Sr 90 Sr 156 Gd 234 U 235 U 239 Pu 241 Am Table 2

-List of isotopes out of experimental error bars with 2σ

As shown in figure 3, uncertainty on isotopic concentration is mainly due to fission yields for fission products and to microscopic cross section for actinides. A comprehensive study is needed for the 7 isotopes listed in table 2 (3 fission products and 4 actinides) will be studied particularly. For fission products, cross section effects can not be omitted, particularly for Europium, Gadolinium, Samarium, 142 Nd, 147 Pm, 133 Cs, 134 Cs and , 135 Cs.

Discrepancy between computed uncertainty for 158 Gd in MENDEL and DARWIN

Figure 2c shows a considerable different of uncertainties for 158 Gd calculated by both codes. As shown by table 3 the main contributor to 158 Gd concentration is the capture cross section of 156 Eu, for which the equivalent one group standard deviation would be 135 %. On some energy groups, 3 -Cross sections sensitivity to 158 Gd concentration uncertainty, URANIE sampler is not able to correctly sample this level of uncertainty with only 2000 samples, and needs millions of samples to do it. The same result have been shown with python native Gaussian and log normal distributions. Samplers specifically written for high level of uncertainties should be use, which is not yet available in URANIE. Thanks to this study, the high level of uncertainty in COMAC data base has been reported to COMAC team, for eventual corrections.

Sensitivity analysis

A sensitivity analysis has been carried out in DARWIN for all isotopes listed in table 2. All main contributors to their concentrations, with related sensitivities, are given in table 4.

We analyze in this paper the production of 90 Sr and 86 Sr.

• Production of 90 Sr and 86 Sr From figure 2b, discrepancy between experiment and computation is 25.5 % for 90 Sr concentration, while propagated uncertainty is 5.93 %. It should be around 12 % to match the experimental value for a 2σ error bar. The main production chain for 90 Sr is from 90 Kr as shown in table (5a). JEFF-3.1.1 uncertainty data do not include any branching ratio uncertainty for the 90 Kr decay to Rubedium. To know the influence of this branching ratio uncertainties on isotopic concentration uncertainty, a sensitivity analysis was achieved, giving 1 % to 10 % uncertainty to this parameter (cf. table (5b)). From DARWIN results, it is proven that uncertainty on this branching ratio can 4) that its concentration is sensitive to fission yield related to the γ decay of 85 Kr→ 85 Rb as well as that associated to the fast fission of 235 U to 85 Sr, where sensitivities are respectively 0.922 and 0.496. Reeavaluation of those data or a more precise distribution power in time would be a plus to fully validate concentration on this PWR case.

Sensitivity analysis in MENDEL

In this section, we introduce a way to compute sensitivity coefficients from MENDEL output distributions as shown in equation ( 5). The derivation is considered as the mean value of all linear derivations.

S Conc/λ = ∂Conc ∂ λ E(λ ) E(Conc) = 1 N(N-1) 2 N ∑ i=1 N ∑ j=i+1 Conc i -Conc j λ i -λ j 1 N ∑ N i=1 λ i 1 N ∑ N i=1 Conc i S Conc/λ = 1 N(N-1) 2 ∑ N n=1 λ n ∑ N m=1 Conc m N ∑ i=1 N ∑ j=i+1 Conc i -Conc j λ i -λ j (5)
N = 2000 decay constant samples are used in MENDEL. Uncertainty propagation was done on all parameters (yields, decay parameters and cross sections). The number of computed sensitivity is N s = N(N-1)

2

= 19990000. Table [START_REF] Huynh | IncerD : Outil intégrant les fonctionnalités de propagation des incertitudes du code CYRUS dans le code DARWIN/PEPIN2.4.2[END_REF] shows MENDEL and DARWIN sensitivities associated to 242 Cm caused by uncertain decay constants. For high enough sensitivities, computation of an accurate sensibility through a 2000 samples Monte Carlo approach is possible. Nevertheless, discrepancies occur when sensitivity is lower. The best criteria found during this study to validate the sensitivity computed from samples is the use of the relative standard deviation σ sens µ sens of all sensitivity coefficients

Conc i -Conc j λ i -λ j E(λ ) E(Conc)
.

We observe that the concordance is good when µ sens σ sens is less than 1.1*10 4 , which also correspond to biggest sensitivities in DARWIN. Figures 4a and4b show the sensitivity to λ 242 Am and λ 241 Pu.

CONCLUSION

On a PWR depletion and cooling case, MENDEL has given results fully compatible with already validated DARWIN code system. Furthermore, comparison to experiment data on 54 isotopes have shown validated concentrations with a 2σ uncertainty propagation for 47 isotopes. Origin of the main uncertainty has been established, showing a preponderant effect of fission yields for fission products, and cross sections for heavy nuclides. Sensitivity analysis have been carried out to stress out the main contributors to discrepancy for the 7 isotopes not matching the 2σ uncertainty bars. Sensitivity coefficients established from MENDEL coefficients for a relatively small number of realisations can be taken into account with regard of a criteria descibed in section 3.4. In conclusion, MENDEL version can be considered as validated for PWR calculation with high burnups. Work in undergoing to improve the computation of a statistical sensitivity coefficient, and deterministic way to propagate uncertainties in MENDEL is under development.

DiscrepancyFigure 1 -

 1 Figure 1 -Comparison of isotopic concentrations between MENDEL and DARWIN

Figure 2 -

 2 Figure 2 -C/E ratio of MENDEL and DARWIN with uncertainty propagation within 1 σ

Figure 3 -

 3 Figure 3 -Contributions of Y λ Br and σ on isotopic concentrations

Figure 4 -

 4 Figure 4 -Decay constant λ -242 Cm concentration correlation

Table 1

 1

	BEPU2018-290

indicates the elapsed time from discharge to chemical isotopic analysis for each sample.

Table 1 -

 1 Cooling time from discharge to chemical isotopic analysis for each sample and isotope MENDEL and DARWN/PEPIN2.4 both compute isotopic densities with CEA-V512 nuclear data library

Table 4 -

 4 Main sensibilities to table 2 isotope concentrations

	Isotope	Principle reaction 86 Rb β -	∆X X % Sensitivity Uncertainty % 0.107 5.55E-04 3.57E-05	∆Cc ∆Cc total
		85m Kr→ 85 Rb	0.509	9.22E-01	0.465
	86 Sr	85 Br→ 85m Kr	2.069 -8.64E-06	1.79E-05	72.0 %
		235 U (n,f thermal ) Yi	85 Sr	9.884	4.96E-01	5.160
		235 U (n,f f ast ) Yi	85 Sr	12.514 1.21E-02	0.145
		239 Pu (n,f f ast ) Yi	36.694 2.31E-05	8.46E-04
		239 Pu (n,f thermal ) Yi	36.500 4.96E-03	0.181
		235 U (n,f f ast ) Yi	36.927 3.84E-05	1.42E-03
	90 Sr	235 U (n,f thermal ) Yi	37.677 3.93E-03	0.148	74.9 %
		90m Rb β -			0.411	2.48E-01	0.102
		90 Rb β -			3.160 -3.77E-06	1.19E-05
		235 U (n,f thermal ) Yi	90 Kr	8.130	5.65E-01	4.592
		235 U (n,f f ast ) Yi	90 Kr	10.200 1.16E-02	0.118
		155 Gd (n,γ)		3.836 4.66E-004	1.79E-3
		154 Eu (n,γ) 155 Eu	11.897 1.57E-01	1.862
	156 Gd	151 Pr 17.584 4.50E-02 151 Pr 239 Pu (n,f thermal ) Yi 239 Pu (n,f f ast ) Yi 18.160 5.15E-04	0.791 9.35E-03	47.0 %
		155 Eu (n,γ) 156 Eu	19.319 4.92E-02	0.951
		152 Sm (n,γ) 153 Sm	2.830	2.58E-01	0.732
		153 Eu (n,γ) 154 Eu	4.860	4.13E-01	2.010
	239 Pu	239 Np β -238 U (n,γ) σ 239 U	0.169 -1.32E-02 0.856 9.60E-01	2.24E-03 0.821	58.6 %
		239 Pu (n,γ) 240 Pu	2.295	6.71E-01	1.540
	241 Am	240 Pu (n,γ) 241 Pu 238 U (n,γ) 239 U	1.931 0.856	2.21E-01 9.90E-01	0.427 0.847	79.8 %
		241 Pu β -			0.028	8.82E-01	0.025
	245 Cm	244 Cm (n,γ)		14.691 8.75E-01	1.285	77.2 %
	246 Cm	245 Cm (n,γ)		16.242 9.07E-01	1.473	64.8 %

Table 5 -

 5 Sensitivity analysis of90 Sr % to90 Sr concentration (with the hypothesis of a 10% uncertainty on the branching ratio). In conclusion, completion of the library would be a plus, but would not alone explain the discrepancy on this isotope. From table(4), sensitivities of β decay of nuclide 90m Rb and thermal fission of 235 U to90 Kr are Sr =0.248 as well as S γ 90 Kr, 235 U , 90 Sr =0.565. Therefore, the amount of 90 Sr depends on that of 90m Rb and the reevaluation of 235 U (n,f f ast ) Yi 90 Kr fission yield is necessary.Concerning to 86 Sr, it is shown in the table (

		(a) Production of 90 Sr		(b) Branching ratio sensitivity analysis
	90 Kr	90m Rb 90 Rb	90 Sr	∆Br Br (%) Sensitivity Uncertainty % 1 % 0.108 0.108
				10 %	0.108	1.08
	only contribute to up to 1

Table 6 -

 6 Comparison of sensitivities of 242 Cm between MENDEL and DARWINIn conclusion, sensitivities computed after a Monte Carlo propagation method can be used when σ sens µ sens < 10 5 %.

	Isotope	λ ( 242 Cm)	λ ( 241 Pu)	λ ( 242m Am)	λ ( 242 Am)	λ ( 239 Np)
	Sens_DAR -5.95283E+00 9.75182E-01 1.59265E-02 3.44186E-03 -2.35120E-03
	Sens_MDL -5.97778E+00 9.64403E-01 1.99953E-02 -1.10771E-01 9.212803E-02
	σ sens µ sens (%)	-5933.6	4736.6	101647.8	-115733.0	142980.6

ACKNOWLEDGEMENT

The authors would like to thank EDF and AREVA for their financial support to APOLLO2 code developments and EDF for its financial support to DARWIN and MENDEL developments.


Maxime Soulard
email: maxime.soulard@cea.fr

Guillaume Ritter

Cindy Le Loirec

Romain Eschbach

Olivier Serot

DEM 2018 -Dismantling Challenges: Industrial Reality

Keywords: Dose-rate, TRIPOLI-4®, DARWIN, Uncertainty propagation, Activation

This paper presents a part of the validation process for performing radiological characterization of reactors in decommissioning phase. This study focuses on three uncertainties sources: stochastic neutron uncertainty, nuclear data uncertainty and gamma rays intensities uncertainty. As the propagation of those sources of uncertainties is rarely done nowadays, we propose here to quantify their impacts. We establish criteria for converging neutron flux as well as a list of important isotopes for which a revisited evaluation could help in the perspective of activation prediction, especially

Introduction

Dismantling and decommissioning of nuclear facilities is nowadays one of the most important issues one needs to tackle. Initial radiological characterization is essential to ensure the decommissioning activities to be achieved in safe and rational manners. Both simulation and experiments are conducted to perform it in an iterative way, the experiments allowing a punctual validation of the simulation step and the computation allowing a complete characterization of the facility. The current computational power of computers and their intensive use makes it possible to perform complex and complete simulations. Appropriate codes and nuclear data should be used to obtain a reliable isotopic inventory prediction and dose-rate estimation.

To ensure the validity of the code, the Nuclear Data (ND) and the coupling involved in the predictive radiological characterization, it is important to perform a Verification, Validation and Uncertainty Quantification (VVUQ) step. This process is especially recommended by the French nuclear safety authority (ASN) [START_REF] Asn | Guide de l'ASN n°28 : Qualification des outils de calcul scientifique utilisés dans la démonstration de sûreté nucléaire -première barrière[END_REF] and specifies steps one has to respect to obtain a trustful tool that can be used for any application that fits in the application domain.

The present work presents an automated coupling between both codes developed by the CEA: TRIPOLI-4® MC [START_REF] Brun | TRIPOLI-4® CEA, EDF and AREVA reference Monte Carlo code[END_REF] (code v.4.10) with the DARWIN-PEPIN2 [START_REF] Tsilanizara | DARWIN: an evolution code system for a large range of applications[END_REF] (v.2. 3) depletion and decay package. This coupling was applied for the source term determination of the CEA PHEBUS facility, a research reactor in decommissioning since 2007. Dose rate values were estimated and the contribution of several kinds of uncertainties was also evaluated by using the IncerD code developed at CEA [START_REF] Lahaye | Comparison of deterministic and stochastic approaches Figure 10 : Gamma spectrum with nuclear data uncertainties (aluminum) for isotopic concentration and decay heat uncertainty quantification on elementary fission pulse[END_REF].

Facility description

The PHEBUS reactor was an experimental reactor constructed at CEA Cadarache in 1977. With a maximum power of 40 MW, it was a crucial tool to study loss of coolant accidents (LOCA) and fusion of fuel assemblies throughout 4 Fission Product (PF) programs. The real burn-up is equal to 973 MWd/t. The decommissioning program started in 2007.

Computation scheme

Initial radiological characterization of any reactors in decommissioning phase consists in performing 4 coupled calculations steps reported in Figure 1:  Core Calculation (with fuel fission sources),

Monte Carlo modeling: Fission sources

The PHEBUS reactor is modeled twice as presented in Figure 2: in the first model called "Core model" in the following, the core is detailed whereas in the second one, called "activation model" in the following, the structures are precisely described. The first model is extremely detailed with the representation of each fuel rods in the assemblies and the main structures of the core. It consists of a twenty-six 8x8 standard fuel assemblies with four triangle assemblies around the central tube for a total of 1816 fuel rods. The middle of the core is dedicated to the experimental test-device of the FP program. Absorber rods height is such that k eff ~ 1.00 and fission sources are normalized to a thermal power of 15 MWth. This model is used to compute the neutron flux outgoing the fuel assemblies Figure 3 and Figure 4). Parameters (E,x,y,z,…) of neutrons leaving fuel assemblies are stored in text files to be reused in the second simulation.

Activation Calculation : Structures geometry and Monte-Carlo modeling

In the second simulation, neutron parameters stored in the file are reused in the dedicated activation model with detailed structures (graphite reflectors, nuts and bolts, tiny pieces) but without fuel assemblies as shown in Figure 5. In the activation model, structures of the core are modeled as precisely as possible to be able to compute activities in each part of the material. Different structures are described as reported in Figure 6. One can see graphite reflectors (turquoise), aluminum central The composition of those materials is either taken from results of destructive analysis or from literature for typical alloy. Especially, concentration of impurities is always taken equal to the maximum values given by the literature. For dismantling constraints, the method used here is conservative as it maximizes the materials activities and so the gamma emitted. The best solution should be to have precise isotopic analysis of each material. We will focus later on aluminum structures (fine composition reported below in Table 1).

The activity is extremely sensitive to the impurities concentration, as consequence a small variation in the initial composition has a major impact on the final activity of the material. For the aluminum, the Co sensitivity coefficient equals to 2.8E-3 %/%.

Atoms

Dose-rate estimation

Dose-rate estimation (Gy.h -1 ) is performed by a coupled photon-electron transport in regions of interest. We can compute the dose as well as the dose rate everywhere in the nuclear facility in order to establish a dose and dose-rate maps.

Uncertainty propagation during activation and decay

We propose here a new step forward with the addition of an uncertainties propagation method. We introduce three main sources of uncertainties, added to the previous scheme and reported Figure 7. where n is the number of particles simulated. For activation studies it is sometimes hard to reach a great convergence for all energy groups (because of the distance from the neutron source and the lack of neutrons for the tallies) .The stochastic uncertainty has an impact on the future isotopic balance calculated by the depletion solver because of the great variations of the reaction rates. In order to know the bias introduced by this statistical error, we first assume no correlation between energy groups. The statistical error given by TRIPOLI-4® is taken to be equal to the standard deviation and the flux given by TRIPOLI-4® to the mean value [START_REF] Garcia-Herranz | Propagation of statistical and nuclear data uncertainties in Monte Carlo burn-up calculations[END_REF]. This allows generating a normal distribution for each energy group and then picking a random value for each energy group. A new input file is produced for in order to compute the reaction rates. This operation is repeated up to 1000 times in a given volume to produce 1000 concentration values for each isotope of the inventory. We show that those results are perfectly fitted by a normal distribution. Eventually, for each isotope we can assess the stochastic uncertainty by : 𝜇 𝑖 𝜎 𝑖 (i is the isotope considered).

Nuclear data uncertainties

The code IncerD is based on the first order perturbation theory. The function representing the isotopic concentration is linearized at the first order thanks to a Taylor development. It is then possible to obtain several sensitivity coefficients (one for each uncertain parameter). By knowing the vector of sensitivity coefficients as well as the covariances matrices associated to those uncertain parameters (X), we can compute the standard deviation of the concentration (or the activity) for each isotope and for each uncertain parameter according to the "sandwich formula" : 𝐶𝑜𝑣(𝐶𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛) = 𝑆(𝐶𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛|𝑋)𝐶𝑜𝑣(𝑋)𝑆 𝑡 (𝐶𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛|𝑋) 𝐶𝑜𝑣(𝐶𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛) refers to the standard deviation on isotopic concentration 𝑆(𝐶𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛|𝑋) refers to the vector of sensibility of the isotopic concentration with respect to the list of uncertain parameters X.

Gamma ray intensities uncertainties:

An other source of uncertainties is relative to the gamma rays intensities. Although the energy of the rays are often well known with an extremely low uncertainty, because of high purity germanium crystals with exceptional energy resolution (1 eV of uncertainty for 1 MeV gamma ray), intensities are far less known (about few percents) and one can observe variation of few percent for each ray. For each isotope of interest it is possible to get from the JEFF3.1.1 Decay Data the uncertainty associated with each gamma and X ray intensity. As the evolution solver computed the activities for each isotope of the isotopic balance, a multiplication between activities and intensities is needed to get the amount of photons emitted by the activated matter. From there we can generate two different gamma source files for further photon transport with TRIPOLI-4®. One with the maximum of gamma emitted according to the level of uncertainties and the other with the minimum of gamma emission. It may not be rigorously physical but it gives precious margins that have never been taken into account before.

Results

The new methodology is applied to the PHEBUS reactor case with a specific focus on the aluminum structure which envelops the fuel assemblies and the graphite material. We present the results of the propagation of the three sources of uncertainties described above on both activation and dose-rate. 4 are automatically created at the end of the calculation process. The tables contain one column for the activity computed without any uncertainty propagation (i.e Total_Activity_Nom) which is the standard values always presented. Next columns contain Decay uncertainties, Branching Ratio uncertainties and Reaction Rate uncertainties obtained with IncerD. Each one of those columns represents the proportion of uncertainty of the parameter of interest (decay constant, branching ratio or reaction rates) on the total activity of the material per isotopes. We consider here only the isotopes with a contribution to the activity greater than 1 Bq.cm -3

. Co}. Another source of uncertainties comes from decay values, especially for 93m Nb (20%) and 93 Mo (20%). Some parameters such as the branching ratio of 93m Nb do not have any uncertainty value in the literature so that a default 40% of uncertainty is applied to this parameter corresponding to the quantile at 95% of uncertainties associated to all branching ratios.

Parameters

Uncertainty due to Gamma ray intensities

The last part of the uncertainty propagation consists in checking whether or not the uncertainties on gamma ray intensities have an impact on the dose-rate itself. The process developed here involves getting all the intensity uncertainties from the JANIS database (xml format) on each gamma ray emitted by each isotope in the composition and perform the product between activity (without uncertainties) and intensities (with uncertainties). We basically obtain the maximum and the minimum number of gamma rays that can be emitted by the considered composition and so assess the maximum spread in which the dose value will be due to uncertainties on gamma ray intensities. The gamma spectrum is binned according to the common MERCURE energy grid (see in appendix).. The simulation is performed on different materials (graphite, aluminum, stainless steel, Zircaloy).

We present on Figure 9 the results for the aluminum. One can observe basically no difference for the aluminum material. The only subtle variation is concerning the very first bin with a variation equals to 0.8%. This result is consistent with all the other materials. The uncertainty on gamma intensities can be totally neglected for dose estimation.

As we can neglect the gamma ray intensities uncertainties, the only other source of uncertainties is coming from the other nuclear data uncertainties, mentioned in the nuclear data part above.

We can observe on Figure 10 the result of a complete propagation of the nuclear data uncertainties on a gamma spectrum of a typical aluminum material. The impact on the spectrum is variable. The maximum variation is observed for the very first bin due the large uncertainty of the 54 Fe capture cross-section. The variation is about ∓ 13% on the number of gamma emitted by the material. All the other variations are below 1%. This result is consistent with the other material we have been working with. For a dose estimation, ND uncertainties can be neglected. In spite of the important uncertainty in the first energy bin, the X rays have so little energy that they will never contribute to the dose itself.

Conclusion

We present in this paper a new methodology to take into account three sources of uncertainties neglected in the dismantling studies. It can bring attention to some particular ND that should be improved to better estimate both material activities and dose-rate. Our results highlight that 62 Ni and 54 Fe cross sections need to be improved as well as the decay value for 93 Mo. A work has also to be conducted on data without uncertainties to provide uncertainties values in order to estimate rigorously the impact on the results. A work on ND evaluation will bring confidence in the waste classification estimation during the initial radiological characterization of the installation but will not have as much impact on dose estimation. It is possible to neglect uncertainties propagation for dose rate estimation but not for waste management. This new method is fully automated and user friendly to help integrate uncertainties to all future dismantling and decommissioning studies. Work is currently ongoing to properly compare dose-rate measurements and dose-rate simulations to achieve the complete validation process of the code system.



  Neutron transport beyond fuel region,  Depletion calculation (isotopic inventory and gamma rays sources),  Gamma rays transport. The Monte-Carlo code TRIPOLI4® (v.4.10) is used with the JEFF3.1.1 nuclear data library [5] for the estimation of the fission sources and the neutron transport. Neutron fluxes are tallied in specific meshes depending on the volume considered and then collapsed in 315 energy groups. Those fluxes are then used by the package DARWIN-PEPIN2 (v.2.3) to perform activation calculation with the JEFF-3.0/A library [6] for reactions rates calculations and JEFF3.1.1/DD as Decay Data library. Resulting gamma sources are then injected in TRIPOLI4 to estimate dose rate values in volumes of interest. Gamma transport is also performed with TRIPOLI4® (v.4.10) and the JEFF3.1.1 nuclear data library.

Figure 2 :Figure 1 :

 21 Figure 2: Two steps fission sources calculation

Figure 3 :

 3 Figure 3: Radial neutron flux distribution (n.cm -2 .s -1 ) Figure 4: Axial neutron flux distribution (n.cm -2 .s -1 )

Figure 5 :

 5 Figure 5: Neutron transport from storage files (without fissile materials) (n.cm -2 .s -1 )

Figure 6 :

 6 Figure 6: Top view of the Monte-Carlo model with detailed structures and experimental device (green) Left : z=0cm (midplane), Right : z=45cm



  Neutron Transport : Stochastic uncertainties (only due to statistical convergence)  Depletion/Activation/Cooling: Nuclear data uncertainties (for the activation reactions)  Photon Transport: Gamma rays intensities uncertainties (direct impact on the dose) When estimating a multi-group flux, TRIPOLI-4®, like any other Monte-Carlo code, gives for each energy group a statistical uncertainty which varies according to the classic law 1 √𝑛

Figure 7 :

 7 Figure 7: New scheme with uncertainties propagation (purple stars)

  𝐶𝑜𝑣(𝑋) refers to the matrix of correlation of those uncertain parameters. 𝐶𝑜𝑣(𝑋) = 𝐼 𝑛 if no correlation are taken into account.Here is the list of the uncertain parameters taken into account for the uncertainty propagation:𝑋 = { 𝜆, 𝑑𝑒𝑐𝑎𝑦 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑏𝑟, 𝑏𝑟𝑎𝑛𝑐ℎ𝑖𝑛𝑔 𝑟𝑎𝑡𝑖𝑜 𝜎, 1 𝑔𝑟𝑜𝑢𝑝 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑖𝑐 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒 = 1𝑔 𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 }As the evolution package uses JEFF3.1.1 as nuclear data library, we use for the uncertainty propagation a table of uncertainties associated with JEFF3.1.1 to properly propagate the uncertainties.

Figure 8

 8 Figure 8 presents the results of the stochastic uncertainty propagation for the 60 Co in a graphite reflector. It represents the cumulative distribution function relative to 1000 different concentrations coming from 1000 random draws in a 315 multi-group energy flux. We use the cumulative distribution because it shows without misinterpretation the shape of the distribution itself. The assumption of a

Figure 8 :

 8 Figure 8: Stochastic uncertainty on the concentration of 60 Co (Graphite)

Figure 9 :

 9 Figure 9: Gamma spectrum with intensities uncertainty (aluminum)

Table 1 : Aluminum AG3 elemental concentration As

 1 the geometry has changed over the past few decades according to the experiment, we used up to 3 different geometries to properly represent the structures of the reactor (especially the thickness of graphite increased from 5 cm to 20 cm, to gain on neutron balance). Neutron fluxes are obtained in all structures of interest (graphite reflectors, nuts, etc.), reaction rates are then used by the depletion solver which solves the Bateman equations. To properly compute one group reaction rates and get the right isotopic balance, isotopic activities and gamma spectrum for each structure volume considered one should use the accurate irradiation history reported in Table2. The output is filtered as we only keep the isotopes with an activity at the last cooling time (the date of the beginning of the dismantling) greater than 1 Bq.cm

		C (%)	Atoms	C (%)	Atoms	C (%)	Atoms	C (%)
	Na	0.0008	Fe	0.4000 Rb		0.0001 Ba	0.0003
	Mg	2.6000	Co	0.0002 Sr		0.0015 Ce	0.0001
	Al	95.9766 Ni	0.0075 Zr		0.0020 Pr	0.0001
	Si	0.4000	Cu	0.1000 Mo		0.0002 Nd	0.0001
	Ca	0.0319	Zn	0.2000 Cd		0.0001 Dy	0.0001
	Ti	0.1500	Ga	0.0159 Sn		0.0051 W	0.0001
	Cr	0.0950	Ge	0.0121 Te		0.0001 Hg	0.0001
	-3 .
			Cycle	# Days	BU MWd/t
			Cycle 1	18			359
		Inter-Cycle	960			0
			Cycle 2	7			141
		Inter-Cycle	1529			0
			Cycle 3	10			207
		Inter-Cycle	1485			0
			Cycle 4	13			266

Table 2 : Irradiation History

 2

Isotopes Capture cross-section

		54 Fe,	59 Co,	150 Nd,	58 Ni,	62 Ni,	92 Mo, 153 Eu, 154 Eu
	Decay	59 Ni, 121 Sm, 93 Mo,	93 Nd,	121 mSn,	121 Sn
	BR	(	125 Sb->	125m Te), (	93 Mo-> 93 Nb),(	121m Sn-> 121 Sn)

Table 5 : Sum-up of the parameters of interest

 5

Table 5

 5 presents a list of isotopes and nuclear data that remains after severe sorting criteria consisting of retaining the isotopes with a contribution to the isotopic activity considered greater than 5%. This list

Table 3 : Nuclear data uncertainties by isotopes for aluminum material Table 4: Nuclear data uncertainties by isotopes for stainless steel material would

 3 help gain confidence in activation calculation if their evaluation was revisited and improved in the next JEFF evaluation.

Appendix

MERCURE energy grid : Energy bounds (keV)

Energy bounds (keV)