The internationally circulated Pool Critical Assembly (PCA) Pressure Vessel Benchmark was analyzed using the PENTRAN Parallel SN code system for the geometry, material, and source specifications as described in the PCA Benchmark documentation. Improvements to the benchmark are proposed through the application of more representative flux and volume weighted homogenized cross sections for the PCA reactor core, which were obtained from a rigorous heterogeneous modeling of all fuel assembly types in the core. A new source term definition is also proposed based on calculated relative power in each core fuel assembly with a spectrum based on the Uranium-235 fission spectra. This research focused on utilizing the BUGLE-96 cross section library and accompanying reaction rates, while examining both adaptive differencing on a coarse mesh basis, as well as the sole use of Directional Theta-Weighted (DTW) SN differencing scheme in order to compare the calculated PENTRAN results to measured data. The results show good comparison with the measured data, which suggests PENTRAN is a viable and reliable code system for calculation of light water reactor neutron shielding and dosimetry calculations. Furthermore, the improvements to the benchmark methodology resulting from this work provide a 6 percent increase in accuracy of the calculation (based on the average of all calculation points), when compared with experimentally measured results at the same spatial location in the PCA pressure vessel simulator.

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Neste trabalho o transporte de nêutrons é simulado em materiais presentes no núcleo de reatores. O espectro de nêutrons é decomposto como uma soma de três distribuições de probabilidade. Duas das distribuições preservam sua forma com o tempo, mas não necessariamente sua integral. Uma das duas distribuições é devido ao espectro de fissão, isto é, altas energias de nêutrons, a outra é uma distribuição de Maxwell-Boltzmann para nêutrons de baixas energias (térmicos). A terceira distribuição tem uma forma a priori desconhecida e que pode variar com o tempo, sendo determinada a partir de uma simulação Monte Carlo com acompanhamento dos nêutrons e suas interações com dependência contínua de energia. Isto é obtido pela parametrização das seções de choque dos materiais do reator com funções contínuas, incluindo as regiões de ressonâncias resolvidas e não resolvidas. O objetivo deste trabalho é implementar efeitos de up-scattering através do tratamento estat ístico da população de nêutrons na distribuição térmica. O programa de simulação calcula apenas down-scattering, pois o cálculo do up-scattering microscópico aumenta signi_cativamente tempo de processamento computacional. Além de contornar esse problema, pode-se reconhecer que up-scattering é dominante na região de energia mais baixa do espectro, onde assume-se que as condições de equilíbrio térmico para nêutrons imersos em seu ambiente são válidas. A otimização pode, assim, ser atingida pela manutenção do espectro de Maxwell- Boltzmann, isto é, up-scattering é simulado por um tratamento estatístico da população de nêutrons. Esta simulação é realizada utilizando-se dependência energética contínua, e, como um primeiro caso a ser estudado assume-se um regime recorrente. As três distribuições calculadas são então utilizadas no código Monte Carlo para calcular os passos Monte Carlo subsequentes. / In this work the neutron transport is simulated in reactor core materials. The neutron spectrum is decomposed as a sum of three probability distributions. Two of the distributions preserve shape with time but not necessarily the integral. One of the two distributions is due to prompt ssion, i.e. high neutron energies and the second a Maxwell-Boltzmann distribution for low (thermal) neutron energies. The third distribution has an a priori unknown and possibly variable shape with time and is determined from a Monte Carlo simulation with tracking and interaction with continuous energy dependence. This is done by the parametrization of the material cross sections with continuous functions, including the resolved and unresolved resonances region. The objective of this work is to implement up-scattering e ects through the treatment of the neutron population in the thermal distribution. The simulation program only computes down-scattering, for the calculation of microscopic upscattering increases signi cantly computational processing time. In order to circumvent this problem, one may recognize that up-scattering is dominant towards the lower energy end of the spectrum, where we assume that thermal equilibrium conditions for neutrons immersed in their environment holds. The optimization may thus be achieved by the maintenance of the Maxwell-Boltzmann spectrum, i.e. up-scattering is simulated by a statistical treatment of the neutron population. This simulation is performed using continuous energy dependence, and as a rst case to be studied we assume a recurrent regime. The three calculated distributions are then used in the Monte Carlo code to compute the Monte Carlo steps with subsequent updates.

Neste trabalho o transporte de nêutrons é simulado em materiais presentes no núcleo de reatores. O espectro de nêutrons é decomposto como uma soma de três distribuições de probabilidade. Duas das distribuições preservam sua forma com o tempo, mas não necessariamente sua integral. Uma das duas distribuições é devido ao espectro de fissão, isto é, altas energias de nêutrons, a outra é uma distribuição de Maxwell-Boltzmann para nêutrons de baixas energias (térmicos). A terceira distribuição tem uma forma a priori desconhecida e que pode variar com o tempo, sendo determinada a partir de uma simulação Monte Carlo com acompanhamento dos nêutrons e suas interações com dependência contínua de energia. Isto é obtido pela parametrização das seções de choque dos materiais do reator com funções contínuas, incluindo as regiões de ressonâncias resolvidas e não resolvidas. O objetivo deste trabalho é implementar efeitos de up-scattering através do tratamento estat ístico da população de nêutrons na distribuição térmica. O programa de simulação calcula apenas down-scattering, pois o cálculo do up-scattering microscópico aumenta signi_cativamente tempo de processamento computacional. Além de contornar esse problema, pode-se reconhecer que up-scattering é dominante na região de energia mais baixa do espectro, onde assume-se que as condições de equilíbrio térmico para nêutrons imersos em seu ambiente são válidas. A otimização pode, assim, ser atingida pela manutenção do espectro de Maxwell- Boltzmann, isto é, up-scattering é simulado por um tratamento estatístico da população de nêutrons. Esta simulação é realizada utilizando-se dependência energética contínua, e, como um primeiro caso a ser estudado assume-se um regime recorrente. As três distribuições calculadas são então utilizadas no código Monte Carlo para calcular os passos Monte Carlo subsequentes. / In this work the neutron transport is simulated in reactor core materials. The neutron spectrum is decomposed as a sum of three probability distributions. Two of the distributions preserve shape with time but not necessarily the integral. One of the two distributions is due to prompt ssion, i.e. high neutron energies and the second a Maxwell-Boltzmann distribution for low (thermal) neutron energies. The third distribution has an a priori unknown and possibly variable shape with time and is determined from a Monte Carlo simulation with tracking and interaction with continuous energy dependence. This is done by the parametrization of the material cross sections with continuous functions, including the resolved and unresolved resonances region. The objective of this work is to implement up-scattering e ects through the treatment of the neutron population in the thermal distribution. The simulation program only computes down-scattering, for the calculation of microscopic upscattering increases signi cantly computational processing time. In order to circumvent this problem, one may recognize that up-scattering is dominant towards the lower energy end of the spectrum, where we assume that thermal equilibrium conditions for neutrons immersed in their environment holds. The optimization may thus be achieved by the maintenance of the Maxwell-Boltzmann spectrum, i.e. up-scattering is simulated by a statistical treatment of the neutron population. This simulation is performed using continuous energy dependence, and as a rst case to be studied we assume a recurrent regime. The three calculated distributions are then used in the Monte Carlo code to compute the Monte Carlo steps with subsequent updates.

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Doses absorvidas e doses efetivas podem ser calculadas utilizando códigos computacionais de transporte de radiação. A qualidade desses cálculos depende dos dados nucleares, no entanto, são raras as informações sobre as diferenças nas doses causadas por diferentes bibliotecas. O objetivo desse estudo é comparar os valores de dose (absorvida e efetiva) obtidos utilizando diferentes bibliotecas de dados nucleares devido a uma fonte externa de nêutrons na faixa de 10-11 a 20 MeV. As bibliotecas de dados nucleares são: JENDL 4.0, JEFF 3.1.1 e ENDF/B-VII.0. Cálculos de doses foram realizados utilizando o código MCNPX considerando o modelo antropomórfico da ICRP-110. As diferenças nos valores das doses absorvidas utilizando as bibliotecas JEFF 3.1.1 e a ENDF/B.VII são pequenas, em torno de 1%, porém os resultados obtidos com a JENDL 4.0 apresentam diferenças de até 85 % compara aos resultados da ENDF/B-VII.0 e JEFF 3.1.1. Diferenças nas doses efetivas são em torno de 1,5% entre ENDF/B-VII.0 e JEFF 3.1.1, e 11 % entre ENDF/B-VII.0 e JENDL 4.0. / Absorbed dose and Effective dose are usually calculated using radiation transport computer codes. The quality of the calculations of absorbed dose depends on nuclear data utilized, however, there are rare information about the differences in dose caused by the use of different libraries. The objective of this study is to compare dose values obtained using different nuclear data libraries due to external source of neutrons in the energy range from 10-11 to 20 MeV. The nuclear data libraries used are: JENDL 4.0, JEFF 3.3.1 and ENDF/B.VII. Dose calculations were carried out with the MCNPX code considering the anthropomorphic ICRP 110 model. The differences in the absorbed dose values using JEFF 3.3.1 and ENDF/B.VII libraries are small, around 1%, but the results obtained with JENDL 4.0 presented differences up to 85% compared to ENDF and JEFF results. Differences in effective dose values are around 1.5% between ENDF and JEFF and 11% between ENDF/B.VII and JENDL 4.0.

Des milieux hétérogènes et désordonnés émergent dans plusieurs applications de la science et de l'ingénierie nucléaires, en particulier en ce qui concerne la propagation des neutrons et des photons. Les exemples sont très répandus et concernent par exemple la double hétérogénéité des éléments combustibles dans les réacteurs à lit de boulets ou l'évaluation de la probabilité de re-criticité suite aux arrangements aléatoires du combusitble résultant d'accidents graves. Dans cette thèse, nous étudierons le transport linéaire de particules dans des milieux aléatoires. Dans la première partie, nous nous concentrerons sur quelques modèles mathématiques qui peuvent être utilisés pour la description de matériaux aléatoires. Une attention particulière sera accordée aux tessellations stochastiques, où un domaine est partitionné en polyèdres convexes en échantillonnant des hyperplans aléatoires selon une probabilité donnée. Les inclusions stochastiques de sphères dans une matrice seront également brièvement introduites. Un code informatique sera développé afin de construire explicitement de telles géométries par des méthodes de Monte Carlo. Dans la deuxième partie, nous évaluerons ensuite les caractéristiques générales du transport de particules dans des milieux aléatoires. Pour ce faire, nous allons considérer quelques benchmarks assez simples pour permettre une compréhension approfondie des effets des géométries aléatoires sur les trajectoires de particules tout en conservant les propriétés clés du transport linéaire. Les calculs de transport seront réalisés en utilisant le code de transport de particules Monte Carlo Tripoli4, développé au SERMA. Les cas de modèles de désordre quenched et annealed seront considérés séparément. Dans le premier, un ensemble de géométries sera généré en utilisant notre code, et le problème de transport sera résolu pour chaque configuration: des moyennes d'ensemble seront alors prises pour les observables d'intérêt. Dans le second cas, un modèle de transport efficace capable de reproduire les effets du désordre dans une seule réalisation sera étudié. Les approximations des modèles annealed seront élucidées, et des améliorations significatives seront proposées. / Heterogeneous and disordered media emerges in several applications in nuclear science and engineering, especially in relation to neutron and photon propagation. Examples are widespread and concern for instance the double-heterogeneity of the fuel elements in pebble-bed reactors, or the assessment of re-criticality probability due to the random arrangement of fuel resulting from severe accidents. In this Thesis, we will investigate linear particle transport in random media. In the first part, we will focus on some mathematical models that can be used for the description of random media. Special emphasis will be given to stochastic tessellations, where a domain is partitioned into convex polyhedra by sampling random hyperplanes according to a given probability. Stochastic inclusions of spheres into a matrix will be also briefly introduced. A computer code will be developed in order to explicitly construct such geometries by Monte Carlo methods. In the second part, we will then assess the general features of particle transport within random media. For this purpose, we will consider some benchmark problems that are simple enough so as to allow for a thorough understanding of the effects of the random geometries on particle trajectories and yet retain the key properties of linear transport. Transport calculations will be realized by using the Monte Carlo particle transport code Tripoli4, developed at SERMA. The cases of quenched and annealed disorder models will be separately considered. In the former, an ensemble of geometries will be generated by using our computer code, and the transport problem will be solved for each configuration: ensemble averages will then be taken for the observables of interest. In the latter, effective transport model capable of reproducing the effects of disorder in a single realization will be investigated. The approximations of the annealed disorder models will be elucidated, and significant ameliorations will be proposed.

L'objectif de ce travail de thèse est le développement d'une approximation polynomiale axiale dans un solveur basé sur la Méthode des Caractéristiques. Le contexte, est celui de la solution stationnaire de l'équation de transport des neutrons pour des systèmes critiques, et l'implémentation pratique a été réalisée dans le solveur "Two/three Dimensional Transport" (TDT), faisant partie du projet APOLLO3®. Un solveur MOC pour des géométries en trois dimensions a été implémenté dans ce code pendant un projet de thèse antécédent, se basant sur une approximation constante par morceaux du flux et des sources des neutrons. Les développements présentés dans la suite représentent la continuation naturelle de ce travail. Les solveurs MOC en trois dimensions sont capables de produire des résultats précis pour des géométries complexes. Bien que précis, le coût computationnel associé à ce type de solveur est très important. Une représentation polynomiale en direction axiale du flux angulaire des neutrons a été utilisée pour réduire ce coût computationnel.Le travail réalisé pendant cette thèse peut être considéré comme divisé en trois parties: transport, accélération et autres. La première partie est constituée par l'implémentation de l'approximation polynomiale choisie dans les équations de transmission et de bilan typiques de la Méthode des Caractéristiques. Cette partie a aussi été caractérisée par le calcul d'une série de coefficients numériques qui se sont révélés nécessaires afin d'obtenir un algorithme stable. Pendant la deuxième partie, on a modifié et implémenté la solution des équations de la méthode d'accélération DPN. Cette méthode était déjà utilisée pour l'accélération et des itérations internes et externes dans TDT pour les solveurs deux et trois dimensionnels avec l'approximation des flux plat, quand ce travail a commencé. L'introduction d'une approximation polynomiale a demandé plusieurs développements numériques regardant la méthode d'accélération. Dans la dernière partie de ce travail on a recherché des solutions pour un mélange de différents problèmes liés aux premières deux parties. En premier lieux, on a eu à faire avec des instabilités numériques associées à une discrétisation spatiale ou angulaire pas suffisamment précise, soit pour la partie transport que pour la partie d'accélération. Ensuite, on a essayé d'utiliser différentes méthodes pour réduire l'empreinte mémoire des coefficients d'accélération. L'approche qu'on a finalement choisie se base sur une régression non-linéaire au sens des moindres carrés de la dépendance en fonction des sections efficaces typique de ces coefficients. L'approche standard consiste dans le stockage d'une série de coefficients pour chaque groupe d'énergie. La méthode de régression permet de remplacer cette information avec une série de coefficients calculés pendant la régression qui sont utilisés pour reconstruire les matrices d'accélération au cours des itérations. Cette procédure ajoute un certain coût computationnel à la méthode, mais nous pensons que la réduction de la mémoire rende ce surcoût acceptable.En conclusion, le travail réalisé a été concentré sur l'application d'une simple approximation polynomiale avec l'objectif de réduire le coût computationnel et l'empreinte mémoire associées à un solveur basée sur la Méthodes des Caractéristiques qui est utilisé pour calculer le flux neutroniques pour des géométries à trois dimensions extrudées. Même si cela ne constitue pas une amélioration radicale des performances, l'approximation d'ordre supérieur qu'on a introduit permet une réduction en termes de mémoire et de temps de calcul d'un facteur compris entre 2 et 5, selon le cas. Nous pensons que ces résultats constitueront une base fertile pour des futures améliorations. / The purpose of this PhD is the implementation of an axial polynomial approximation in a three-dimensional Method Of Characteristics (MOC) based solver. The context of the work is the solution of the steady state Neutron Transport Equation for critical systems, and the practical implementation has been realized in the Two/three Dimensional Transport (TDT) solver, as a part of the APOLLO3® project. A three-dimensional MOC solver for 3D extruded geometries has been implemented in this code during a previous PhD project, relying on a piecewise constant approximation for the neutrons fluxes and sources. The developments presented in the following represent the natural continuation of this work. Three-dimensional neutron transport MOC solvers are able to produce accurate results for complex geometries. However accurate, the computational cost associated to this kind of solvers is very important. An axial polynomial representation of the neutron angular fluxes has been used to lighten this computational burden.The work realized during this PhD can be considered divided in three major parts: transport, acceleration and others. The first part is constituted by the implementation of the chosen polynomial approximation in the transmission and balance equations typical of the Method Of Characteristics. This part was also characterized by the computation of a set of numerical coefficients which revealed to be necessary in order to obtain a stable algorithm. During the second part, we modified and implemented the solution of the equations of the DPN synthetic acceleration. This method was already used for the acceleration of both inners and outers iteration in TDT for the two and three dimensional solvers at the beginning of this work. The introduction of a polynomial approximation required several equations manipulations and associated numerical developments. In the last part of this work we have looked for the solutions of a mixture of different issues associated to the first two parts. Firstly, we had to deal with some numerical instabilities associated to a poor numerical spatial or angular discretization, both for the transport and for the acceleration methods. Secondly, we tried different methods to reduce the memory footprint of the acceleration coefficients. The approach that we have eventually chosen relies on a non-linear least square fitting of the cross sections dependence of such coefficients. The default approach consists in storing one set of coefficients per each energy group. The fit method allows replacing this information with a set of coefficients computed during the regression procedure that are used to re-construct the acceleration matrices on-the-fly. This procedure of course adds some computational cost to the method, but we believe that the reduction in terms of memory makes it worth it.In conclusion, the work realized has focused on applying a simple polynomial approximation in order to reduce the computational cost and memory footprint associated to a Method Of Characteristics solver used to compute the neutron fluxes in three dimensional extruded geometries. Even if this does not a constitute a radical improvement, the high order approximation that we have introduced allows a reduction in terms of memory and computational times of a factor between 2 and 5, depending on the case. We think that these results will constitute a fertile base for further improvements.

Standardmäßig erfolgt die Datenaufbereitung der Neutronenwirkungsquerschnitte für Reaktorkernrechnungen mit 2D-Zellcodes. Ziel dieser Arbeit war es, einen 3D-Zellcode zu entwickeln, mit diesem Code 3D-Effekte zu untersuchen und die Notwendigkeit einer 3D-Datenaufbereitung der Neutronenwirkungsquerschnitte zu bewerten. Zur Berechnung des Neutronentransports wurde die Methode der Erststoßwahrscheinlichkeiten, die mit der Ray-Tracing-Methode berechnet werden, gewählt. Die mathematischen Algorithmen wurden in den 2D/3D-Zellcode TransRay umgesetzt. Für den Geometrieteil des Programms wurde das Geometriemodul eines Monte-Carlo-Codes genutzt. Das Ray-Tracing in 3D wurde auf Grund der hohen Rechenzeiten parallelisiert. Das Programm TransRay wurde an 2D-Testaufgaben verifiziert. Für einen Druckwasser-Referenzreaktor wurden folgende 3D-Probleme untersucht: Ein teilweise eingetauchter Regelstab und Void (Vakuum oder Dampf) um einen Brennstab als Modell einer Dampfblase. Alle Probleme wurden zum Vergleich auch mit den Programmen HELIOS (2D) und MCNP (3D) nachgerechnet. Die Abhängigkeit des Multiplikationsfaktors und der gemittelten Zweigruppenquerschnitte von der Eintauchtiefe des Regelstabes bzw. von der Höhe der Dampfblase wurden untersucht. Die 3D berechneten Zweigruppenquerschnitte wurden mit drei üblichen Näherungen verglichen: Lineare Interpolation, Interpolation mit Flusswichtung und Homogenisierung. Am 3D-Problem des Regelstabes zeigte sich, dass die Interpolation mit Flusswichtung eine gute Näherung ist. Demnach ist hier eine 3D-Datenaufbereitung nicht notwendig. Beim Testfall des einzelnen Brennstabs, der von Void umgeben ist, erwiesen sich die drei Näherungen für die Zweigruppenquerschnitte als unzureichend. Demnach ist eine 3D-Datenaufbereitung notwendig. Die einzelne Brennstabzelle mit Void kann als der Grenzfall eines Reaktors angesehen werden, in dem sich eine Phasengrenzfläche herausgebildet hat.

Standardmäßig erfolgt die Datenaufbereitung der Neutronenwirkungsquerschnitte für Reaktorkernrechnungen mit 2D-Zellcodes. Ziel dieser Arbeit war es, einen 3D-Zellcode zu entwickeln, mit diesem Code 3D-Effekte zu untersuchen und die Notwendigkeit einer 3D-Datenaufbereitung der Neutronenwirkungsquerschnitte zu bewerten. Zur Berechnung des Neutronentransports wurde die Methode der Erststoßwahrscheinlichkeiten, die mit der Ray-Tracing-Methode berechnet werden, gewählt. Die mathematischen Algorithmen wurden in den 2D/3D-Zellcode TransRay umgesetzt. Für den Geometrieteil des Programms wurde das Geometriemodul eines Monte-Carlo-Codes genutzt. Das Ray-Tracing wurde auf Grund der hohen Rechenzeiten parallelisiert. Das Programm TransRay wurde an 2D-Testaufgaben verifiziert. Für einen Druckwasser-Referenzreaktor wurden folgende 3D-Probleme untersucht: Ein teilweise eingetauchter Regelstab und Void (bzw. Moderator mit geringerer Dichte) um einen Brennstab als Modell einer Dampfblase. Alle Probleme wurden zum Vergleich auch mit den Programmen HELIOS (2D) und MCNP (3D) nachgerechnet. Die Abhängigkeit des Multiplikationsfaktors und der gemittelten Zweigruppenquerschnitte von der Eintauchtiefe des Regelstabes bzw. von der Höhe der Dampfblase wurden untersucht. Die 3D berechneten Zweigruppenquerschnitte wurden mit drei üblichen Näherungen verglichen: linearer Interpolation, Interpolation mit Flusswichtung und Homogenisierung. Am 3D-Problem des Regelstabes zeigte sich, dass die Interpolation mit Flusswichtung eine gute Näherung ist. Demnach ist hier eine 3D-Datenaufbereitung nicht notwendig. Beim Testfall des einzelnen Brennstabs, der von Void (bzw. Moderator geringerer Dichte) umgeben ist, erwiesen sich die drei Näherungen für die Zweigruppenquerschnitte als unzureichend. Demnach ist eine 3D-Datenaufbereitung notwendig. Die einzelne Brennstabzelle mit Void kann als der Grenzfall eines Reaktors angesehen werden, in dem sich eine Phasengrenzfläche herausgebildet hat.