A generalization of the CIRCE method for quantifying input model uncertainty in presence of several groups of experiments

The semi-empirical nature of best-estimate models closing the balance equations of thermal-hydraulic (TH) system codes is well-known as a significant source of uncertainty for accuracy of output predictions. This uncertainty, called model uncertainty, is usually represented by multiplicative (log-)Gaussian variables whose estimation requires solving an inverse problem based on a set of adequately chosen real experiments. One method from the TH field, called CIRCE, addresses it. We present in the paper a generalization of this method to several groups of experiments each having their own properties, including different ranges for input conditions and different geometries. An individual (log-)Gaussian distribution is therefore estimated for each group in order to investigate whether the model uncertainty is homogeneous between the groups, or should depend on the group. To this end, a multi-group CIRCE is proposed where a variance parameter is estimated for each group jointly to a mean parameter common to all the groups to preserve the uniqueness of the best-estimate model. The ECME algorithm for Maximum Likelihood Estimation is adapted to the latter context, then applied to relevant demonstration cases. Finally, it is tested on a practical case to assess the uncertainty of critical mass flow assuming two groups due to the difference of geometry between the experimental setups.Comment: 26 pages, 7 figure

Analysis of fluid-structure interaction mechanism of a Na-FBR core while the evacuation of a gas pocket

Cette thèse vise à améliorer la compréhension du comportement d’un cœur de réacteur à neutrons rapides refroidi au sodium (RNR-Na) lors de vibrations par l’analyse des phénomènes d’interaction fluide-structure. Notamment, nous étudions le gerbage du cœur de Phénix lors des AURNs et les oscillations séismiques pour le projet ASTRID. Pour ce faire, trois approches ont été suivies : élaboration de solutions analytiques, développement de modèles numériques et réalisation d’expériences. Nous avons réalisé une carte d’écoulement qui identifie les régimes présents dans l’interassemblage pendant les échelles de temps courtes (AURNs) et longues (séismes). Elle a servi à identifier le système d’équations (Navier-Stokes, Euler ou Euler linéarisées) le plus adapté à représenter le comportement du fluide pour les différents cas. Pour la première fois, à notre connaissance, une solution analytique du champ de pression et de vitesse a été obtenue pour le cas d’oscillations libres et de forts confinements. Nous avons conçu et dimensionné deux maquettes, composées respectivement par 2 couronnes d’assemblages (PISE-2c) et 1 assemblage hexagonal (PISE-1a). Chaque assemblage vibre avec un mouvement de type gerbage en translation (mouvement 2D) à une fréquence d’oscillation en eau du même ordre que la fréquence en sodium liquide d’un assemblage de Phénix. Des essais d’oscillations libres en air et en eau ont été réalisés pour étudier les caractéristiques dynamiques de l’assemblage. Bien que l’oscillation soit censée être 2D, un écoulement 3D du type « jambage » se produit dans l’inter-assemblage. Ceci conduit à une baisse de la fréquence de vibration par rapport à la théorie bidimensionnelle. Les essais ont été modélisés avec un modèle numérique bidimensionnel à l’aide de Cast3M pour le couplage fluide-structure. Le modèle résout les équations de Navier-Stokes couplées avec l’équation de la dynamique de corps rigide. Le « modèle upφ », composé par les équations d’Euler linéarisées couplées avec l’équation de la dynamique, a également été utilisé pour représenter en 3D la maquette PISE-1a. Afin de se rapprocher des essais, il faut imposer dans les modèles bidimensionnels une force fluide inférieure, qui prenne en compte les effets de l’écoulement 3D.The purpose of this study is to improve the knowledge about the core behavior of a sodium fast breeder reactor (Na-FBR) during vibrations through the fluid-structure interaction analysis. Namely, we investigate the flowering of the Phénix core during the SCRAM for negative reactivity (AURN) and the seismic behavior of the core of Astrid project. Three approaches are followed : experimental campaign, performing of analytical solution and development of numerical model. We create a flow regime map to identify the flow regimes in the fluid gap for very short times scales (as AURN) as well as longer time scales (as seismic oscillations). The most suitable equation system (Navier-Stokes, Euler or linearized Euler) is chosen to model the fluid flow in the numerical code. To our knowledge, for the first time, an analytical solution for free vibration and very narrow gaps is proposed. We designed two experimental apparatus (PISE-1a and PISE-2c) composed respectively by 1 and 19 hexagonal assemblies (two crowns) of Poly-methyl methacrylate (PMMA). Every PMMA assembly is fixed to a stainless steel twin-blades support allowing only orthogonal oscillations with respect to generating line of assembly. The twin-blades supports are designed to give the same range frequency of Phénix assembly in liquid sodium. The experimental equipment PISE-1a is used to determine the dynamic characteristics of PISE-2c assembly, to calibrate instrumentation and for validating our numerical model. Free vibration tests in air are performed to evaluate the dynamic characteristics of the body. Free vibration experiments in water allow to assess the added mass and added damping effect on the frequency. Even though the fluid flow during vibration should be completely bidimensional, the fluid flow is affected by a 3D effect - named "jambage" - at the top and the basis of the assembly. This effect produces a lower frequency than the theoretical value. Tests are modeled with a bidimensional numerical model through the finite-elements method with the Cast3M code. The fluid is viscous and incompressible, whereas the structure is considered as a mass-damped-spring system with a 1 degree of freedom. Our model is solved by the Navier-Stokes equations coupled by the dynamic equation of structures. Also the « upφ model » is used to have a 3D representation of PISE-1a. Because of the 3D fluid flow presence, to reproduce the oscillation of a test, we have to impose a lower fluid force in the 2D numerical model to reproduce tests

Étude des mécanismes d'interaction fluide-structure d'un cœur RNR-Na lors de l'évacuation d'une poche de gaz

The purpose of this study is to improve the knowledge about the core behavior of a sodium fast breeder reactor (Na-FBR) during vibrations through the fluid-structure interaction analysis. Namely, we investigate the flowering of the Phénix core during the SCRAM for negative reactivity (AURN) and the seismic behavior of the core of Astrid project. Three approaches are followed : experimental campaign, performing of analytical solution and development of numerical model. We create a flow regime map to identify the flow regimes in the fluid gap for very short times scales (as AURN) as well as longer time scales (as seismic oscillations). The most suitable equation system (Navier-Stokes, Euler or linearized Euler) is chosen to model the fluid flow in the numerical code. To our knowledge, for the first time, an analytical solution for free vibration and very narrow gaps is proposed. We designed two experimental apparatus (PISE-1a and PISE-2c) composed respectively by 1 and 19 hexagonal assemblies (two crowns) of Poly-methyl methacrylate (PMMA). Every PMMA assembly is fixed to a stainless steel twin-blades support allowing only orthogonal oscillations with respect to generating line of assembly. The twin-blades supports are designed to give the same range frequency of Phénix assembly in liquid sodium. The experimental equipment PISE-1a is used to determine the dynamic characteristics of PISE-2c assembly, to calibrate instrumentation and for validating our numerical model. Free vibration tests in air are performed to evaluate the dynamic characteristics of the body. Free vibration experiments in water allow to assess the added mass and added damping effect on the frequency. Even though the fluid flow during vibration should be completely bidimensional, the fluid flow is affected by a 3D effect - named "jambage" - at the top and the basis of the assembly. This effect produces a lower frequency than the theoretical value. Tests are modeled with a bidimensional numerical model through the finite-elements method with the Cast3M code. The fluid is viscous and incompressible, whereas the structure is considered as a mass-damped-spring system with a 1 degree of freedom. Our model is solved by the Navier-Stokes equations coupled by the dynamic equation of structures. Also the « upφ model » is used to have a 3D representation of PISE-1a. Because of the 3D fluid flow presence, to reproduce the oscillation of a test, we have to impose a lower fluid force in the 2D numerical model to reproduce tests.Cette thèse vise à améliorer la compréhension du comportement d’un cœur de réacteur à neutrons rapides refroidi au sodium (RNR-Na) lors de vibrations par l’analyse des phénomènes d’interaction fluide-structure. Notamment, nous étudions le gerbage du cœur de Phénix lors des AURNs et les oscillations séismiques pour le projet ASTRID. Pour ce faire, trois approches ont été suivies : élaboration de solutions analytiques, développement de modèles numériques et réalisation d’expériences. Nous avons réalisé une carte d’écoulement qui identifie les régimes présents dans l’interassemblage pendant les échelles de temps courtes (AURNs) et longues (séismes). Elle a servi à identifier le système d’équations (Navier-Stokes, Euler ou Euler linéarisées) le plus adapté à représenter le comportement du fluide pour les différents cas. Pour la première fois, à notre connaissance, une solution analytique du champ de pression et de vitesse a été obtenue pour le cas d’oscillations libres et de forts confinements. Nous avons conçu et dimensionné deux maquettes, composées respectivement par 2 couronnes d’assemblages (PISE-2c) et 1 assemblage hexagonal (PISE-1a). Chaque assemblage vibre avec un mouvement de type gerbage en translation (mouvement 2D) à une fréquence d’oscillation en eau du même ordre que la fréquence en sodium liquide d’un assemblage de Phénix. Des essais d’oscillations libres en air et en eau ont été réalisés pour étudier les caractéristiques dynamiques de l’assemblage. Bien que l’oscillation soit censée être 2D, un écoulement 3D du type « jambage » se produit dans l’inter-assemblage. Ceci conduit à une baisse de la fréquence de vibration par rapport à la théorie bidimensionnelle. Les essais ont été modélisés avec un modèle numérique bidimensionnel à l’aide de Cast3M pour le couplage fluide-structure. Le modèle résout les équations de Navier-Stokes couplées avec l’équation de la dynamique de corps rigide. Le « modèle upφ », composé par les équations d’Euler linéarisées couplées avec l’équation de la dynamique, a également été utilisé pour représenter en 3D la maquette PISE-1a. Afin de se rapprocher des essais, il faut imposer dans les modèles bidimensionnels une force fluide inférieure, qui prenne en compte les effets de l’écoulement 3D

Theoretical and computational analysis of transient heat transfer in plate-type core nuclear reactors

Aim of this work is to investigate transient heat transfer phenomena of interest for the thermal-hydraulic analysis of nuclear reactor core during the course of a postulated reactivity insertion accident (RIA). An instantaneous extraction of control rods may cause the so called prompt jump, if the reactivity insertion is larger than the reactor β, usually referred to as reactor dollar. Associated to the prompt jump of the neutron flux, an exponential increase of the thermal power produced within the fuel rods is observed. Consequently, the heat flux from the fuel to the coolant also increases, making the coolant to heat up and eventually boiling. The thermal-hydraulic feedbacks caused by the heating and, most of all, the formation of voids, cause an insertion of negative reactivity that tends to stabilize the reactor core power. It may happen however, that before the thermal-hydraulic feedbacks produce a stabi- lizing effect on the neutron flux, the energy stored within the fuel is large enough to cause the fuel rod to melt down. Then, the fuel melted and released in the bulk of the coolant may lead to the undesired phenomenon of steam explosion. Whether the thermal power is stabilized to a safe level before fuel melting could occur or not depends on the growth rate of the thermal power (and therefore the reactivity inserted) and the time delay between the production of the thermal energy within the fuel and the transfer of this energy to the coolant. Indeed, due to finite heat capacities and thermal conductivities within the different structures (fuel and claddings), the heat transferred to the coolant can be significantly lower than the heat released in the fuel at the same moment, leading to an accumulation of energy within the fuel. In order to contribute to the understanding and the prediction of power excursion transient, a fundamental analysis of transient heat transfer phenomena has been performed. Aim of this analysis is to provide a clear quantification of the rate of energy transfer between the fuel and the coolant during the transient, as a function of the subchannel geometry, material properties and the characteristic time scale of the exponential power excursion (the so called period). A multi-slices configuration is considered, representative of plate fuel type subchannels, typical of experimental assemblies, like those of the SPERT and the BORAX reactors described in chapter 2. A two-step analysis is proposed. In a first instance, the analysis of purely conductive systems is carried out. Then, the effects of coolant convection are investigated. In the purely conductive case, both the numerical (CFD) and the analytical solutions of the heat transfer equation have been obtained. The analytic solution has provided the exact formulation for the spatial temperature profile all along the transient. Basing on the temperature profile, other relevant quantities have been resumed, i.e. the disequilibrium between the energy produced within the fuel end the energy transferred to the coolant (defined as R) and the heat transfer coefficient between the solid structures and the coolant. Last but not least, the mutual verification of the models has been achieved. The CFD model, verified against the analytic solutions for the purely conductive systems, has been thus applied to investigate convection effects associated to the coolant motion. The boundary conditions of the SPERT-IV experimental campaign have been thus addressed in order to define the phenomenology of heat transfer associated to the different tests. This aims at drawing guidelines for the application of correlations for the thermal disequilibrium and the heat transfer coefficient in system codes and subchannel codes

Added mass and damping of an hexagonal rod vibrating in highly confined viscous fluids

International audienceThis paper deals with fluid–structure interaction analysis of an hexagonal rod enclosed in a narrow viscous gap. A new analytical solution for a two-dimensional (2D) cylindrical case is derived and described. A numerical solution of 2D Navier–Stokes equations coupled with a harmonic structure model is applied to both cylindrical and prism geometries. The comparison between the numerical tool and the analytical solution is discussed and a method to apply the analytical solution to the hexagonal case is proposed. An original definition of the added mass and damping based on an energetic approach is provided avoiding the dependence from the geometry and the type of forcing (free or forced vibration). An experimental facility is provided accounting for an hexagonal prism vibrating within a 7 mm enclosure. Free vibration experiments in water allow assessing the added mass and added damping effect on the modal parameters. The fluid flow is affected by a three-dimensional (3D) effect—named down-strokes flow—at the top and the base of the assembly because of free surface and stocky geometry. This produces a higher frequency than the 2D theoretical value given both by the analytical solution and the numerical simulation. A geometry-based correction factor is suggested to taken into account in the 2D numerical simulation the 3D effect. Velocity measured within the gap provides further insight on this phenomenon and agrees well with the prediction of the transposed cylindrical analytical model

Sensitivity Analysis on the Critical Mass Flowrate Based on Sobol’ Indices Through Replicated LHS

International audienceIn order to quantify uncertainties of constitutive relationships in thermal-hydraulic system codes, the setup of an experimental database based on Separated Effect Tests (SET) is a critical step. Basically, series of experiments are carried out over various experimental input conditions to study some thermal-hydraulic Quantities of Interests (QoIs). Those quantities usually depend on several constitutive relationships whose individual impact should be assessed by performing a sensitivity analysis.In the paper, a sensitivity analysis based on first-order Sobol’ indices through replicated Latin Hypercube Sampling (rLHS) is proposed to identify the most influential constitutive relationships. Using rLHS allows to strongly reduce the number of simulations needed to compute the whole set of first order Sobol’ indices.A comparison between SET series will be performed. This method is applied to a database composed by two series of SETs in which the critical mass flowratesimulated with the thermal-hydraulic system code CATHARE2 is the QoI. Generally, the relevance of the constitutive relationships of the code on the QoI depends on the experimental input conditions and the geometry of the mock-up. With this study, we can shade such dependences and establish a database based on the impact of the physical models.Finally, the purposes of this analysis are to define which constitutive relationships can be judged as significant and to rank the experiments of the two SETs according to the effect of these constitutive relationships on the mass flowrate

RIVA: a new experimental facility for fast blowdown and pressurization of small containment

International audienceIn case of Main Steam Line Break scenarios, several thermal hydraulics phenomena, such as two-phase critical flow, wall condensation, and stratification of the atmosphere of the containment are involved. The RIVA facility is devoted to the study of such phenomena by means of a fast blowdown of a vessel discharging in a containment. A valve near the inlet of the containment vessel with a fast opening actuator is used to simulate the sudden break of the secondary loop of a nuclear reactor. The entire device is highly instrumented, especially the containment with more than 200 thermocouples. Moreover, two instrumentation sections on the connection pipe allows us to measure local pressure, pressure drop and local temperature before the inlet of the vessel. At first, some experiments are realized by replacing steam by nitrogen, to understand the behavior of the fluid during a very fast blowdown, changing the outlet section of the line. These experiments indicate that a chocked flow is present and allow us to study the influence of temperature, mass flow rate and pressure at the injection section on the jet plume in the vessel. Subsequently, experimental tests with steam and then with some structures in the containment vessel have been realized. They have allowed a better understanding of the wall condensation phenomenon and the volume occupation rate effects. The analyse of the results are not presented in the document but will in coming articles.In the same way, further experiments with a steam blowdown have been performed to take into account the wall condensation and will be the subject of a future article. Experimental results will be provided for validation of system as much as CFD codes

SAPIUM - Follow-up activities, perspectives for future development

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A comprehensive Bayesian framework for the development, validation and uncertainty quantification of thermalhydraulic models

International audienceThe development, validation and uncertainty quantification of closure laws used into thermal-hydraulic system codes is a key issue before applying the BEPU (Best Estimate Plus Uncertainty) methodology for safety studies and licensing of nuclear reactors. The assessement of those physical models requires tuning some parameters against available experimental data. This paper presents a methodology called Bayesian calibration, which allows a more robust and reliable assessment, selection and uncertainty quantification of physical models. In this work, the experimental and predicted values are linked by means of a multiplicative random variable which represents the model uncertainty. This hypothesis is suitable for models that scale many orders of magnitude as it is the case in nuclear thermal-hydraulics. Several empirical models, which describe the same physic with different formalisations, are calibrated. Then, the one which is the best-suited according to different statistical indicators is chosen. A statistical validation is performed with a Leave One Out (LOO) technique which allows using the same database for both assessment and validation of the physical model. Then, the uncertainty on the selected best-fitting model is estimated neglecting the experimental errors. This framework is applied to condensation heat transfer correlations for safety injection based on the COSI (COndensation at Safety Injection) experiments

MODEL ASSESSMENT FOR DIRECT CONTACT CONDENSATION INDUCED BY A SUB-COOLED WATER JET IN A CIRCULAR PIPE

International audienceIn this paper, the condensation induced by a sub-cooled water injection in a circular horizontal pipe with a two-phase stratified flow is investigated. The focus of the work is to review the physical models or cor- relations predicting the condensation heat transfer coefficient and assess them against an experimental database.Three experiments, namely COSI, TOPFLOW-PTS and UPTF, are consolidated in a substantial database. They have different configurations and complexity, covering a wide range of injection mass flowrate, tem- perature and pressure. A thermal-hydraulic analysis is performed, resulting in reliable and coherent ex- perimental data.The condensation models found in the literature are based on the modelling of the Nusselt number through several dimensionless numbers. The assessment of these correlations against the experimental database provides poor results. Thus, a new approach is proposed.The cold jet is modelled as a heat exchanger, which is described by a condensation potential. The analytical formula of the potential is found starting from an energy balance at the injection, showing that the condensation depends on the jet geometrical shape and a parameter η.A new correlation for the parameter η is calibrated against the COSI and TOPFLOW-PTS experiments, significantly reducing the average standard deviation between evaluations and experimental data. The new correlation is then applied to an independent database, i.e. the UPTF experiments. The results show good agreement between the calculated and experimental values, proving the capability of the new model to accurately predict the condensation at the injection