Repositorio Institucional de Documentos

1 "En la vida, no hay nada que temer, sólo hay que comprender" (Marie Curie) 2 Agradecimientos A mis tutores André Xhonneux y Dr. Stephan Kelm, así como a todos mis compañeros y amigos del LRST y del IEK-6 del FZ Jülich cuya ayuda ha sido vital para la realización de este trabajo, así como agradecer el haberme ofrecido la oportunidad tan fantástica de trabajar en su equipo. Al Dr. Javier Blasco Alberto, cuya inestimable ayuda y orientación fueron un gran soporte para superar los momentos de dicultades a las que hubo que enfrentarse. A todos mis amigos y compañeros de Aachen a los que mi agradecimiento no cabe en unas simples palabras. A todos los amigos de Zaragoza, del CPS y de Rebollo a los cuales siempre pude recurrir para ayudarme a afrontar con una sonrisa las dicultades a las que tenía que encarar en el día a día. A mi madre, a mi hermano, a mis tías, a mis tíos, así como a mis primos, y en especial a todos aquellos que ya no están entre nosotros por su incondicional apoyo, y a los cuales no puedo más que estar agradecido. 3 4 Índice general I MEMORIA 15 1. INTRODUCCIÓN 17 1.1. Motivación................................. 17 1.2. Objetivos ................................. 18 1.3. Organización de la memoria . . . . . . . . . . . . . . . . . . . . . . . 19 2. TAREAS DESARROLLADAS 20 3. FUNDAMENTOS TEÓRICOS 21 3.1. Teoría de lechos uidizados . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2. Modelosteóricos ............................. 22 3.2.1. Método de Elementos Discretos (D.E.M.) . . . . . . . . . . . . 23 3.2.2. Aproximación por dinámica de medios continuos . . . . . . . . 23 3.3. Estado del arte: experimentos realizados . . . . . . . . . . . . . . . . 24 3.3.1. AVR................................ 24 3.3.2. THTR-300............................. 24 3.3.3. ANABEK............................. 25 3.3.4. TeoríadeSilos .......................... 25 3.4. Desarrollo de un modelo uido . . . . . . . . . . . . . . . . . . . . . . 26 3.4.1. Descripción del sistema granular . . . . . . . . . . . . . . . . . 26 3.4.2. Validez de la aproximación al continuo . . . . . . . . . . . . . 27 3.4.3. Acoplamiento de variables . . . . . . . . . . . . . . . . . . . . 28 3.4.4. Denición de las condiciones de contorno . . . . . . . . . . . . 29 3.4.5. Propiedades de los materiales . . . . . . . . . . . . . . . . . . 30 3.4.6. Geometría y condiciones de operación . . . . . . . . . . . . . . 30 5 ÍNDICE GENERAL 6 4. RESULTADOS 31 4.1. Introducción................................ 31 4.2. Validez de las hipótesis . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.1. Validez de las simplicaciones . . . . . . . . . . . . . . . . . . 31 4.2.2. Validez de las condiciones de contorno . . . . . . . . . . . . . 32 4.3. Resultados y validación de las simulaciones . . . . . . . . . . . . . . . 33 4.3.1. Líneas de corriente y características del ujo . . . . . . . . . . 33 4.3.2. Distribución de tiempos de residencia . . . . . . . . . . . . . . 33 4.3.3. Perles de velocidad . . . . . . . . . . . . . . . . . . . . . . . 34 5. CONCLUSIONES Y TRABAJO FUTURO 35 5.1. Conclusiones................................ 35 5.2. Trabajofuturo .............................. 36 II ANEXO I: ANALYSIS OF PEBBLE BEDS IN HIGH TEMPERATURE REACTORS 37 A. INTRODUCTION 39 B. MOTIVATION AND OBJECTIVES 41 B.1. Pebble Beds in the operation of nuclear reactors . . . . . . . . . . . . 41 B.2. Objectives in the study of Pebble Beds for reactor design . . . . . . . 41 B.3. Motivation for a new pebble bed model . . . . . . . . . . . . . . . . . 43 B.4. Improving the design of the reactor by means of Pebble Bed study . . 43 C. THEORY OF PEBBLE FLOW 45 C.1.Granularuid............................... 45 C.2. State of the art: theoretical models . . . . . . . . . . . . . . . . . . . 48 C.2.1. Discrete Element Methods (D.E.M.) . . . . . . . . . . . . . . 49 C.2.2. Kinematic Models . . . . . . . . . . . . . . . . . . . . . . . . . 49 ÍNDICE GENERAL 7 C.2.3. Stochastic Kinematic Models . . . . . . . . . . . . . . . . . . 50 C.2.4. Modeling a pebble bed for discrete simulations . . . . . . . . . 50 C.2.5. Continuum dynamics approach: . . . . . . . . . . . . . . . . . 51 C.3. State of the art: Experiments . . . . . . . . . . . . . . . . . . . . . . 53 C.3.1. AVR experiments . . . . . . . . . . . . . . . . . . . . . . . . . 54 C.3.2.Australia.............................. 55 C.3.3.THTR-300 ............................ 56 C.3.4. ANABEK-Experiments . . . . . . . . . . . . . . . . . . . . . . 57 C.3.5.MIT ................................ 58 C.3.6.ChinaHTR-10 .......................... 60 C.4. Other applications in the industry: silo theory, ow zones and pressuredistribution.............................. 60 C.5. Conclusions of experimental results . . . . . . . . . . . . . . . . . . . 62 C.5.1. Problems found in the access to information and data . . . . . 63 D. FLOW MODEL DEVELOPMENT 64 D.1. Description of the problem case . . . . . . . . . . . . . . . . . . . . . 64 D.2. Validity and theory of continuum approach . . . . . . . . . . . . . . . 66 D.3.Couplingvariables ............................ 68 D.3.1. Physical variables . . . . . . . . . . . . . . . . . . . . . . . . . 69 D.3.2. Material Properties . . . . . . . . . . . . . . . . . . . . . . . . 71 D.3.3. Geometry and operation characteristics . . . . . . . . . . . . 73 D.4. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 D.4.1.Inlet/Outlet ........................... 74 D.4.2. Wall boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . 75 ÍNDICE GENERAL 8 E. SOLUTION TO THE FLOW PROBLEM 77 E.1. Modeling ANABEK experiment . . . . . . . . . . . . . . . . . . . . . 77 E.1.1. Modeling procedure . . . . . . . . . . . . . . . . . . . . . . . . 77 E.1.1.1. Obtaining a shear stress prole from ANABEK experiments........................ 79 E.2. Computational solution . . . . . . . . . . . . . . . . . . . . . . . . . . 81 E.2.1. Particle Tracking as a technical solution . . . . . . . . . . . . 81 E.3. Geometry and material characteristics . . . . . . . . . . . . . . . . . 84 E.3.1. Resume of ANABEK ow experiment . . . . . . . . . . . . . . 84 E.3.2. Resume of HTR-300 . . . . . . . . . . . . . . . . . . . . . . . 85 E.3.3. Material properties . . . . . . . . . . . . . . . . . . . . . . . . 86 E.4.Miscellaneous ............................... 87 E.4.1. Residence time distribution calculation method . . . . . . . . 87 F. RESULTS 90 F.1.Introduction................................ 90 F.2. Validity of the assumptions and parameters chosen . . . . . . . . . . 90 F.2.1. Saturation of the viscosity . . . . . . . . . . . . . . . . . . . . 90 F.2.2. Validity of the boundaries . . . . . . . . . . . . . . . . . . . . 92 F.2.2.1. Inlet/Outlet boundary condition . . . . . . . . . . . 92 F.2.2.2. Wall boundary condition . . . . . . . . . . . . . . . . 93 F.3. Results of the ow simulations . . . . . . . . . . . . . . . . . . . . . . 98 F.3.1. Streamlines and ow characteristics . . . . . . . . . . . . . . . 98 F.3.2. Residence time distribution . . . . . . . . . . . . . . . . . . . 100 F.3.2.1. ANABEK simulations at hopper . . . . . . . . . . . 101 F.3.2.2. RTD at core wall . . . . . . . . . . . . . . . . . . . . 102 F.3.2.3. RTD at top pebble bed surface . . . . . . . . . . . . 104 F.4. Validation of the results . . . . . . . . . . . . . . . . . . . . . . . . . 105 F.4.1. Residence Time Distribution . . . . . . . . . . . . . . . . . . . 106 F.4.2. Velocity proles and streamlines . . . . . . . . . . . . . . . . . 109 Parte I MEMORIA 15 16 Capítulo 1 INTRODUCCIÓN 1.1. Motivación Desde los inicios del siglo XX la gran disponibilidad de recursos energéticos ha permitido un gran desarrollo económico y social. Sin embargo, esta situación ha cambiado. Por un lado las reservas de combustibles fósiles empiezan a mostrar signos de agotamiento siendo cada vez más difíciles de explotar. Por otro lado, hay una necesidad de reducción de las emisiones de CO2 a la atmósfera. Estos hechos, sumados a una demanda energética creciente, requieren de una apuesta por políticas a nivel global que lleven a un menor consumo energético, a mejoras de los sistemas productivos y de distribución, así como de una sustitución de las fuentes basadas en combustibles fósiles. Ante esta nueva situación, la energía nuclear juega un papel importante teniendo en cuenta las grandes cantidades de energía que es capaz de producir sin emisiones a la atmósfera. El desarrollo de las tecnologías nucleares no está exento sin embargo de riesgos. Los últimos acontecimientos acaecidos en Fukushima nos recuerdan la indudable necesidad de incrementar constantemente los esfuerzos en seguridad. Para ello la investigación y la inversión son las maneras más ables de producir energía limpia y sin repercusiones negativas para la sociedad. En este sentido avanzan los nuevos modelos de reactores, los de generación III+ que están siendo instalados en varios países, así como los futuros de IV generación cuyos modelos aún están en desarrollo. Los reactores de IV generación tienen una diferencia fundamental con sus predecesores y es su capacidad de evacuar de modo pasivo el calor generado en el núcleo ante un fallo del sistema de refrigeración, llevándolo de este modo a una parada fría. Dentro de las diferentes tecnologías, los reactores de alta temperatura refrigerados por gas, con sus siglas HTR en inglés, son una de las apuestas más decididas para los futuros modelos de reactor. Estos reactores tienen una serie de diferencias respecto a los modelos más comunes refrigerados por agua, como los instalados en España, y es que en vez de ser refrigerados por agua, es un gas, generalmente helio, el encargado de evacuar el calor producido por la sión de los núcleos de uranio o torio. Por otro lado, la moderación de los neutrones se lleva a cabo por medio de grato y no por agua por lo que tenemos una ventaja al poder tratar por separado 17 CAPÍTULO 1. INTRODUCCIÓN 18 estas variables vitales para el funcionamiento del reactor nuclear. El uso de helio, a diferencia del agua, no supone un problema en cuanto a la contaminación radiactiva ya que el helio no puede ser transmutado en isótopos radiactivos, permitiendo además alcanzar temperaturas muy elevadas, en torno a los 950 º C, gracias al uso de materiales refractarios. Estas altas temperaturas ofrecen rendimientos muy altos a la hora de producir electricidad, y son así mismo una fuente valiosa como calor de proceso. Dentro de la tecnología HTR los mayores esfuerzos se están realizando en el desarrollo de los reactores nucleares de lecho uido, Pebble Bed Reactors en terminología inglesa, donde una vasija presurizada y con forma de silo cilíndrico es llenada con cientos de miles de esferas. Las esferas caen hacia abajo movidas únicamente por medio de la gravedad y la tasa de recirculación es controlado por un mecanismo de extracción. Estas esferas pueden ser de dos tipos. O bien esferas de grato puro cuya misión es la de actuar como moderador de neutrones en las reacciones nucleares así como de mejorar la eciencia térmica del reactor, o bien, las esferas de combustible. Dichas esferas de combustible están compuestas por pequeñas partículas de uranio o torio las cuales están recubiertas de diferentes compuestos que evitan la difusión de los átomos radiactivos. Estas pequeñas estructuras se encuentran embebidas en una matriz de grato de 6 cm de diámetro. Las reacciones nucleares tienen lugar en el interior de las estructuras anteriormente mencionadas, por lo que el movimiento de las esferas de grato repercute en el desarrollo de las reacciones nucleares. 1.2. Objetivos El movimiento de las esferas dentro de la vasija no es uniforme. Como consecuencia, el tiempo que tarda una esfera en recorrer el núcleo es variable. Por esta razón, es necesario adaptar la concentración de los materiales sibles de cada esfera según la región del núcleo en el que son situadas. Así pues, es necesario ser capaces de predecir la velocidad a la que se mueven las esferas y para ello hay que comprender las fuerzas que gobiernan el movimiento del sistema. Los objetivos de este proyecto se dividen en dos grupos: En primer lugar, se requiere la compresión de los fenómenos físicos que rigen el movimiento en el seno de un lecho uido. Con ello se debe construir un modelo en base a la experiencia recogida hasta el momento presente, dicho modelo ha de ser capaz de predecir las siguientes características del ujo. Líneas de corriente Perles de velocidad Distribución de los tiempos de residencia CAPÍTULO 1. INTRODUCCIÓN 19 Aparte del conocimiento de la física del problema, es necesario también cumplir unos requerimientos técnicos, ya que el presente proyecto esta ideado para una fase de pre-diseño, lo cual exige que sea posible la variación de numerosos parámetros y que el tiempo y capacidad computacional requeridos para realizar los cálculos sea razonable y no sea mayor de 1 día para un solo procesador. Los datos obtenidos, serán comparados y en caso de ser posible, validados a través de los experimentos realizados hasta la fecha. Sin embargo, la conabilidad de algunas de estas pruebas fue puesta en duda por la falta de documentación o por no coincidir los resultados obtenidos con lo reejado en la documentación. Por este motivo, algunos de estos estudios y tras la consulta con diversos investigadores tuvieron que ser rechazados. 1.3. Organización de la memoria El presente proyecto fue redactado originalmente en inglés y está incluido en el segundo volumen como anexo I. Sin embargo, al ser necesaria la traducción al castellano se ha organizado de la siguiente manera: la parte I corresponde a la memoria y se estructura con una breve introducción que explica los fundamentos de la tecnología de reactores nucleares HTR junto a las razones que hacen necesario el desarrollo de un modelo para lechos uidos, así como de los objetivos a alcanzar; en el siguiente capítulo se hace un repaso a las tareas desarrolladas por el autor hasta la consecución de las metas propuestas, prosiguiendo con un apartado que resume los aspectos teóricos y el modelo desarrollado así como el estado del arte; tras este capítulo se muestran algunos de los resultados más signicativos para nalizar con las conclusiones alcanzadas y una serie de propuestas para el futuro. Esta memoria trata de resumir de un modo más simplicado todos los aspectos trabajados a lo largo del proyecto quedando a disposición del lector el trabajo completo en su versión inglesa en el anexo. Dada la necesidad de resumir todo el trabajo en un espacio pequeño se ha decidido referenciar imágenes y grácas a dicho anexo. Toda la bibliografía consultada se encuentra también en su versión original ya que se entendió que de este modo queda una estructura de documento más consistente. Capítulo 2 TAREAS DESARROLLADAS El presente proyecto n de carrera fue desarrollado dentro del Instituto de Seguridad y Tecnología Nuclear de la RWTH Uuniversity de Aachen y en colaboración con el Instituto de Investigaciones Climáticas y Energéticas del centro de Investigación de Jülich, ambos en Alemania. Fue realizado como parte de un proyecto global en el que se están desarrollando los reactores nucleares de lecho uido para su uso comercial. Las tareas encomendadas al autor del presente proyecto son resumidas en: Investigación y búsqueda de literatura e información acerca de las investigaciones desarrolladas en el pasado. Para ello fue necesaria una búsqueda intensiva en las bibliotecas disponibles, así como el uso de bases de datos y las peticiones a diversas empresas. Algunas de las cuales, ya no existían como tales lo cual complicó y alargó esta fase del proyecto. La búsqueda de nueva documentación se prolongó hasta las últimas fases de redacción de la memoria. Preparación y asimilación de los diversos conceptos teóricos. Aprendizaje de las diversas herramientas informáticas. Dentro de los cuales se destaca CFX Ansys. Proposición de diferentes soluciones teóricas para la consecución de un modelo de cálculo que pudiera cumplir con los objetivos propuestos. Para ello se partió de los modelos existentes. Desarrollo de los métodos y búsqueda de herramientas, así como estudio de los parámetros más adecuados. Análisis y estudio de los datos obtenidos. Validación de los resultados comparándolos con los datos disponibles. Documentación del trabajo realizado y organización de los datos y resultados obtenidos. Realización de presentaciones e informes periódicos a los tutores, al catedrático del departamento y al resto de compañeros. 20 Capítulo 3 FUNDAMENTOS TEÓRICOS 3.1. Teoría de lechos uidizados El movimiento de materiales granulares en la vasija del reactor es similar a la descarga de un silo o al movimiento de materiales granulares en un conducto. A través del estudio de la teoría, la cual se encuentra más desarrollada en el capítulo 3 del anexo I, se destacan las principales características que desde el punto de vista teórico pueden ser resumidas como: Las fuerzas gravitatorias son las que originan el movimiento. En el tubo de descarga se encuentra un mecanismo que regula el número de esferas que abandonan el núcleo por unidad de tiempo. Por esta razón, la velocidad media de las esferas está sujeta a la restricción que le impone este dispositivo. Las fuerzas de fricción esfera-esfera están originadas por el contacto puntual entre ellas. La magnitud de dichas fuerzas es una incógnita dada la dicultad que existe para su medida. Sin embargo, parecen tener un rol secundario según la bibliografía consultada. Las fuerzas de fricción esfera-pared son las que tienen una mayor importancia en el desarrollo del lecho uido. Las paredes del núcleo están formadas por materiales refractarios, generalmente grato. A pesar de los bajos coecientes de fricción del grato, las presiones originadas por el propio peso de las esferas generan unas tensiones muy grandes en las paredes como demuestran los estudios realizados. Además, estas fuerzas son las causantes del perl de tensiones que puede ser observado en la gura C.3. Fuerzas de larga distancia son aquellas cuyo origen y efecto se perciben en lugares distintos. Las fuerzas se transmiten mediante la interacción de esferas en el seno del ujo. En el caso de los reactores de lecho uido, en las diversas referencias bibliográcas se determina que tienen un alcance de entre 5 y 6 diámetros de esfera. Se diferencian de su equivalente para uidos viscosos en el alcance, siendo signicativamente mayor para el caso de uidos viscosos debido a la elasticidad de las colisiones entre moléculas. 21 CAPÍTULO 3. FUNDAMENTOS TEÓRICOS 22 El uido refrigerante se encarga de recoger el calor generado por las reacciones nucleares. Se trata de helio a alta presión que se hace circular desde la parte superior a la inferior. El efecto sobre el lecho de esferas se maniesta en un incremento del peso del lecho debido a la perdida de presión del uido al circular entre las esferas según ponen de maniesto los experimentos realizados. Los lechos uidos son sistemas discretos en los cuales, las propiedades de los elementos que lo forman pueden variar signicativamente aun estando en localizaciones cercanas. Este hecho supone que en caso de un reactor pequeño compuesto por unas 100.000 esferas, nos enfrentemos ante un problema de análisis muy complejo siendo necesario analizar si es posible hacer una aproximación al continuo. La densidad del lecho uido depende directamente del nivel de compactación de las esferas en el núcleo del reactor. Esta compactación es el resultado de dividir el volumen que ocupan las esferas realmente y el volumen total disponible. Los valores habituales están en torno a 0.62 frente a un máximo teórico de 0.74. Sin embargo, los valores no son constantes y junto a las paredes de la vasija presurizada, se registran oscilaciones muy fuertes. Estas variaciones inuyen en la permeabilidad del lecho al paso del uido refrigerante, pero también supone que la interacción entre las esferas será más violenta, lo cual genera más colisiones inelásticas y por ende mayor pérdida energética. La velocidad media de las esferas es muy baja, en rangos de 10 -4 y 10 -5 m/s dependiendo del modelo. Esta baja velocidad complica la realización de experimentos y genera un estado cuasi-estático en el cual es difícil saber con certeza si los coecientes de fricción de las esferas son los estáticos o los dinámicos. Como clara ventaja, podemos aseverar tras los diversos estudios y observaciones el régimen laminar del ujo. Este aspecto permite simplicar el análisis y abre la puerta al estudio como si de un sistema continuo se tratase. La geometría es una variable que no es propia del sistema granular, Ahora bien, sus efectos sobre el mismo son tan importantes, que es imprescindible destacar su inuencia. Todos los experimentos realizados muestran claramente el efecto que tiene una variación en la geometría. De hecho, la bibliografía consultada recomienda no escalar las dimensiones del reactor nuclear ya que no queda clara esta posibilidad. 3.2. Modelos teóricos La teoría para el cálculo de lechos uidizados se encuentra todavía sin un completo desarrollo. A modo de comparación, sería equivalente al estado de la mecánica de uidos antes del descubrimiento de las ecuaciones de Navier-Stokes. Este hecho supone la necesidad de usar modelos basados en resultados empíricos y en el uso de modelos de cálculo. Existen diversos métodos de cálculo que en su mayoría pertenecen a modelos discretos, los cuales tienen ventajas en cuanto a precisión de los resultados, aunque CAPÍTULO 3. FUNDAMENTOS TEÓRICOS 23 requieren de una gran capacidad de computación. Por otro lado se puede realizar una aproximación utilizando un sistema continuo. Expondremos los métodos más destacados, los cuales se encuentran ampliados en el capítulo C del anexo I. 3.2.1. Método de Elementos Discretos (D.E.M.) Método desarrollado pro Cundall y Stack en 1979 y que es la base del estudio de dinámica molecular. Este método se basa en tomar uno a uno los elementos y en realizar los cálculos sobre todas las interacciones que tienen lugar en el seno del ujo de partículas. Al calcular para cada esfera y en cada paso de tiempo todas las ecuaciones de equilibrio de fuerzas se alcanza unos niveles de precisión muy altos. Por contra, en los cálculos realizados con el código PFC 3D para un reactor pequeño requirieron más de un mes de computación. La mayoría de esfuerzos actuales van dedicados al desarrollo de métodos computacionales que permitan simplicar y acortar signicativamente los cálculos. Sin embargo, los resultados obtenidos hasta la fecha muestran una gran precisión aunque para casos concretos. Además se enfrentan a otras dicultades como son la construcción del lecho de esferas y el modelado de las irregularidades que es posible encontrar en su seno. 3.2.2. Aproximación por dinámica de medios continuos El movimiento de lechos uidizados movidos por gravedad tiene un comportamiento macroscópico similar al de un uido viscoso. Para ello se toman los valores medios de las variables de trabajo y se simplican o anulan aquellas características propias de un sistema granular que no pueden ser modeladas por medio de un sistema continuo. Esta aproximación aplicada a reactores nucleares se inició en el Centro de Investigación de Jülich (Alemania) para el desarrollo del reactor THTR-300, que será usada como base para el presente proyecto. Dicha aproximación permite mejorar la eciencia de los métodos de cálculo ya que es posible utilizar el método de elementos nitos, disminuyendo signicativamente el tiempo de cálculo a pesar de una perdida en el nivel de precisión. Se pierde también la posibilidad de estudio de las diferentes irregularidades que se pueden encontrar en el lecho de esferas, tales como la formación de cavidades, el fenómeno de puenteado (bridging en terminología inglesa) y efectos de cristalización. A pesar de ello, como hemos indicado en los objetivos, esta es la aproximación indicada para una fase de pre-diseño en la cual es más importante conocer el comportamiento general del lecho de esferas que las peculiaridades que podemos encontrar en localizaciones muy concretas. CAPÍTULO 3. FUNDAMENTOS TEÓRICOS 24 3.3. Estado del arte: experimentos realizados En esta sección se explicarán aquellos experimentos que tienen una mayor relevancia para el modelado del lecho de esferas y para su posterior comparación con los resultados de las simulaciones. En la sección C.3 del anexo I se puede encontrar un amplio resumen de los experimentos que fueron utilizados en el desarrollo de este proyecto. 3.3.1. AVR El AVR (Arbeitsgemeinschaft Versuchreaktor por sus siglas en alemán) es el primer reactor experimental basado en el concepto de reactor nuclear de lecho de esferas refrigerado por gas. Fue construido en el Centro de Investigación de Jülich en la década de los sesenta. Diversos experimentos acerca del movimiento de las esferas se llevaron a cabo durante el tiempo que estuvo operativo. Para ello, se dividió el núcleo en dos regiones: externa e interna; y se introdujeron esferas con una carga de combustible nuclear mayor que las que estaban ya en el interior. De este modo, podían ser detectadas al salir por el tubo de descarga. Sin embargo, algunas de las esferas no fueron correctamente detectadas o lo fueron en el momento en el que se desmanteló el reactor ya que habían tardado más tiempo del previsto en recorrer el núcleo. 3.3.2. THTR-300 El reactor THTR-300 (Thorio High Temperature Reactor en sus siglas inglesas y de 300 MWe) construido en la ciudad de Hamm (Alemania) fue el primer prototipo comercial instalado en el mundo. En su interior no se realizaron experimentos, pese a lo cual, para su diseño se llevaron a cabo multitud de pruebas, entre las cuales hay que destacar los realizados en una maqueta a escala hecha de metacrilato en la cual se introdujeron esferas opacas y translúcidas en un medio líquido. Estas pruebas permitieron demostrar el régimen laminar del lecho uidizado. Además se realizó un estudio acerca de la inuencia que tiene el ángulo de la tolva sobre la evolución del ujo. Una segunda serie de experimentos acerca del comportamiento del ujo de esferas consistieron en el estudio con maquetas a diversas escalas del THTR-300, introduciendo esferas marcadas y dejándolas recircular a un ritmo controlado por un mecanismo de expulsión. Lamentablemente, la escasa documentación encontrada sobre estos experimentos no permitió el uso de estos datos ya que su abilidad no pudo ser contrastada. También se desarrolló el primer modelo computacional destinado al estudio del movimiento de esferas basado en un sistema continuo y resuelto con un método de elementos nitos. Como se indica anteriormente, este modelo es la base del presente proyecto. Capítulo 4 RESULTADOS 4.1. Introducción Tras la implementación del modelo uido en el software ANSYS CFX, se realizaron múltiples simulaciones para comprobar la validez de las simplicaciones realizadas, por otro lado, fue necesario el desarrollo de técnicas para la observación y el análisis de los ujos granulares las cuales se explican detalladamente en el capítulo E del anexo I y que suponen una ventaja para el diseñador. Los resultados obtenidos son comparados con los datos experimentales para su validación. En esta memoria se incluye un resumen de los datos más relevante quedando disponibles en el capítulo F del anexo I el resto de resultados. Se simularon diversas geometrías en función de los datos disponibles. 4.2. Validez de las hipótesis 4.2.1. Validez de las simplicaciones La saturación de la viscosidad, a partir de cierto nivel de tensión, impide el acoplamiento de esta variable con las tensiones tangenciales cuyo origen es la fricción en el lecho de esferas ya que nunca podrá ejercer una tensión equiparable a la real. Para comprobar esta propiedad se llevaron a cabo diversas simulaciones en las que se muestran sendos perles de velocidad calculados para el modelo THTR-300, ver sección F.2 del anexo I. Se eligen diferentes viscosidades y se simula el comportamiento del ujo bajo condiciones de contorno en las paredes de no deslizamiento. Para viscosidades a partir de 1 Pa s no existe apenas diferencia en la forma del perl de velocidades. Por esta razón, el uso de una viscosidad virtual resulta aceptable. También se estudia para diferentes viscosidades el valor de las tensiones verticales sobre las paredes del reactor con la geometría de ANABEK, ver sección E.1 del anexo 31 CAPÍTULO 4. RESULTADOS 32 I. Dado que la condición de contorno sobre las paredes es de no deslizamiento, obtendremos el nivel de tensión máximo para esa conguración. Para valores pequeños de la viscosidad, la curva de tensiones cambia de pendiente, mientras que a partir de aproximadamente 1 Pa s, esta crece linealmente. Si analizamos estos resultados y los analizamos frente a la ecuación τ=µ∂vx ∂y se conrma la hipótesis mencionada en la sección 3.4.3. Por tanto, y dado que los perles de presiones son conocidos, sólo es necesario elegir un valor de viscosidad para el cual sea posible introducir directamente estos datos de presiones. Como ejemplo, se elige una viscosidad de 1x10 8 Pa s. Como comparación este valor es varios órdenes de magnitud que un uido muy denso como el aceite pesado de 10 Pa s. 4.2.2. Validez de las condiciones de contorno Se simularon diferentes condiciones de contorno de entrada/salida según las posibilidades que la literatura explica. Una condición de presión relativa en la entrada y una de ujo másico en la salida dan como resultado un comportamiento muy similar de las líneas de corriente al que se observó en los estudios visuales. Se realizan estas simulaciones con dos valores de viscosidad muy diferentes y los resultados muestran una variación relativamente pequeña para cambios de viscosidad muy grande, lo cual conrma las hipótesis anteriormente realizadas acerca de la viscosidad. Las condiciones de contorno sobre las paredes juegan el papel más importante dentro de la denición del modelo. Esta hipótesis se conrma en todos los resultados de las simulaciones realizados y deja claro que una correcta denición de la misma es fundamental en el modelado del ujo granular. Se realizan cuatro simulaciones diferentes con la geometría y la conguración de ANABEK, en ellas y bajo las mismas condiciones se estudian los perles de velocidad en diferentes regiones del núcleo para valores de tensión diferentes. Para ello se caracterizan 4 perles de tensión diferentes que pueden ser visualizados en el anexo I. El primer perl es simplemente una condición de deslizamiento sobre las paredes del reactor. Esto supone un deslizamiento total del ujo y una tensión nula. Se dene también un perl obtenido por la expresión de Janssen con unos coecientes de fricción razonables para las esferas de ANABEK, el siguiente es el denominado Step Prole que es una interpolación directa de la presión horizontal obtenida en los experimentos. Como ya se ha indicado los valores son una primera aproximación a lo esperable para las esferas de plástico que no fueron estudiadas en los experimentos de presión. Por último un perl continuo de 4000 Pa que sería similar a una condición de no deslizamiento. En las guras F.7 y F.8 del anexo I se muestran los perles de velocidad para distintos radios del reactor. Podemos comprobar cómo va evolucionando el ujo conforme va descendiendo y el efecto que tiene la tolva sobre el mismo. Al llegar la zona de la tolva se aprecia una convergencia del ujo. En ambas guras se comprueba como en la parte superior el ujo se desarrolla con una alta dependencia de las condiciones de pared, sin embargo, al llegar la tolva, esta dependencia desciende y convergen hacia los mismos CAPÍTULO 4. RESULTADOS 33 valores de velocidad. Estos resultados dan una idea de la inuencia geométrica que efectúa la tolva sobre el ujo, disminuyendo en esta región los esfuerzos de fricción. Como conclusión es necesario remarcar la importancia que tienen las condiciones de contorno sobre el ujo. 4.3. Resultados y validación de las simulaciones 4.3.1. Líneas de corriente y características del ujo En la gura F.10 del anexo I se comparan los resultados de dichas pruebas visuales con los obtenidos en las simulaciones, la concordancia de los resultados con las pruebas visuales es buena. Se conrma el régimen laminar y se pueden comprobar las líneas de corriente rectas en la zona superior que se van curvando según se acercan a la tolva. Es también reseñable la baja difusión azimutal de las esferas. Este hecho es importante para la simulación por medios continuos ya que como se explicó anteriormente, el diámetro del reactor es menor de lo que se supone necesario para cumplir con las condiciones de adaptación al continuo. En el análisis de las diferentes regiones de ujo comprobamos en la gura F.10 del anexo I la existencia de zonas de muy baja velocidad vertical. No se detectan zonas de ujo estancado, lo cual no sería posible en un uido, sin embargo es necesario tener este factor en cuenta y optimizar el diseño para disminuir en la medida de lo posible este factor. 4.3.2. Distribución de tiempos de residencia En la realización de experimentos para lechos de esferas el método más utilizado para el análisis de los ujos consiste en la inyección de esferas marcadas, llamadas TK, en las regiones objetivo del estudio y su posterior contabilización. Para obtener la distribución se dene un intervalo i compuesto por p esferas y se cuentan las esferas marcadas extraídas en ese intervalo ( 4TK ). Cada paquete de esferas en el periodo i requiere de un tiempo 4Ti sobre el total PT , que es el tiempo teórico en el que se recircularía totalmente el núcleo, hasta que han salido las esferas de ese paquete. Tras eso se representa grácamente el porcentaje de esferas marcadas sobre el total ( 4T Ki PT K ) en el periodo i . Separando cada periodo i en porciones temporales del 10%, o en otra fracción para más resolución, sobre el total podemos representar el porcentaje de esferas marcadas que han salido dentro de un periodo i que corresponde a una décima parte del tiempo total de recirculación. Como método más importante para la evaluación y calibración del modelo se utiliza éste análisis en comparación a los resultados de los experimentos de ANABEK, los cuales son explicados en la sección 3.3.3. . En los resultados, se incluyen los datos obtenidos por un método de elementos en diferencias [Niessen 2009], el cual se supone signicativamente más preciso como hemos explicado anteriormente. CAPÍTULO 4. RESULTADOS 34 Dado que en el capítulo F del anexo I pueden encontrase todas las pruebas realizadas, se exponen aquí los mejores resultados obtenidos que corresponden al perl de tensiones Step prole 2000 Pa . En la tolva, gura F.13 de anexo I, podemos comprobar que las simulaciones son capaces de aproximar con cierta precisión los datos experimentales, en comparación a otros ensayos realizados con el modelo de cálculo. Se observa que a pesar de grandes cambios en el perl de tensiones sobre las paredes, los resultados cambian sólo para una pequeña región del núcleo. Siendo por tanto menor la inuencia de las fuerzas de fricción como se suponía previamente. Comparando los resultados con el método D.E.M. vemos que éste tampoco es capaz de lograr una gran precisión. Este hecho pone de maniesto que para regiones críticas, el uso de un método discreto ofrece ventajas sobre la aproximación al continuo. La siguiente región a estudio es la más próxima a las paredes del reactor. Los resultados muestran esta vez menos capacidad de predicción. Se realizaron numerosas pruebas con otros perles de tensión, con valores mucho mayores, las cuales eran capaces de predecir con bastante precisión los resultados sobre las paredes. Sin embargo cuando con esos mismos perles se analizaba todo el núcleo, los resultados se mostraban mucho peores. Si observamos la gura F.16 del anexo I en la que se analiza todo el núcleo del reactor, vemos que en este caso los resultados concuerdan muy bien con los resultados de las simulaciones. Por tanto el modelo es capaz de predecir el momento de salida para aproximadamente el 95% de las esferas marcadas introducidas en la vasija. Esta discordancia, teniendo en cuenta los supuestos teóricos previos, esta probablemente motivada por las diferencias propias de ambos sistemas físicos, y particularmente del alcance de las fuerzas a distancia. Es decir, podemos suponer que un lecho de esferas, la fricción producida por las esferas en contacto con la pared se transmite a menor distancia dado a la inelasticidad de los choques. Sin embargo, para el caso uido esto no ocurre del mismo modo y su alcance es mayor. 4.3.3. Perles de velocidad Dado que no existen datos con los cuales comparar las simulaciones realizadas, hemos de tomar los perles de las guras F.7 y F.8 del anexo I como estimaciones que podemos esperar concuerden dado que tanto líneas de corriente, como la distribución de los tiempos de residencia son variables relacionadas unas con otras y ofrecen una buena aproximación a los datos experimentales. Capítulo 5 CONCLUSIONES Y TRABAJO FUTURO 5.1. Conclusiones El estudio de un lecho de esferas a través de una aproximación por medios continuos es una tarea compleja. Las diferencias físicas entre ambos sistemas son importantes y la creación de un modelo empírico está siempre condicionada por la información experimental disponible y la validez de la misma. Las conclusiones más importantes extraídas en el proyecto son las siguientes. El gran número de variables existente en el marco discreto ha de ser reducido. Algunas de las diferencias propias de ambos sistemas serán una fuente de pérdida de precisión. Sin embargo, esto no implica una pérdida del sentido físico del problema, lo cual es imprescindible para denir un modelo robusto. Las condiciones de contorno del problema tienen la mayor importancia a la hora de desarrollar el modelo, destacando la condición de pared. Para un correcto acoplamiento de las variables es imprescindible caracterizar correctamente el efecto que tienen las fuerzas de fricción en el lecho de esferas y extrapolarlo al ujo viscoso. La inuencia de la viscosidad es mucho menor de lo esperado en un primer momento, pero concuerda con los supuestos teóricos. Lo cual explica, junto al origen tan diferente de las fuerzas, la imposibilidad de utilizar la viscosidad como variable equivalente a las fuerzas de fricción. Este hecho sumado a la conrmación de la saturación viscosa observada en las simulaciones permite el uso de una viscosidad virtual. El uso de esta variable es un avance respecto a las simulaciones anteriores ya que permite la implementación de perles de tensiones obtenidos por medio de experimentos o de simulaciones. Las fuerzas a distancia han mostrado tener una inuencia importante en el modelado. Este fenómeno permite explicar las diferencias en el comportamiento de ambos ujos. Condiciones de pared diferentes entre sí dan lugar a ujos que se aproximan muy bien. Por ello se supuso que la utilización de un uido no Newtoniano podría mejorar 35 CAPÍTULO 5. CONCLUSIONES Y TRABAJO FUTURO 36 esta situación. Sin embargo, no se encontraron evidencias en las simulaciones que demostraran esta suposición. En la región superior de la vasija se puede observar un patrón de ujo másico, mientras que según se acerca a la zona de transición y la tolva se evoluciona hacia un ujo de embudo. Este hecho es consecuencia de una gran pérdida de energía en las paredes del reactor, lo cual puede derivar en zonas de estancamiento. Este hecho no puede ser detectado con un uido viscoso, sin embargo, se encontraron regiones de muy baja velocidad que deben ser estudiadas en detalle. El modelo sólo es capaz de predecir los valores medios en condiciones estáticas y homogéneas. Para el estudio de regiones concretas del núcleo del reactor se pierde precisión respecto a otros modelos de cálculo. Se recomienda el análisis tridimensional para facilitar el estudio de regiones especí- cas aunque los resultados en modelos bidimensionales muestran un buen nivel de precisión. El modelo fue utilizado con éxito en diversas conguraciones de reactor y de hecho está siendo utilizado actualmente como método de cálculo para el diseño de una variante del reactor HTR-M 200. 5.2. Trabajo futuro Para el desarrollo del presente trabajo se utilizaron diversos experimentos como fuente de datos. Sin embargo, el acceso a los mismos resultó dicultoso y no estuvo exento de problemas. Por esta razón, sería conveniente la realización de una serie completa de experimentos que contemple tanto el análisis de las características mecánicas del lecho de esferas como el de su comportamiento uido. Parte de estos estudios podrían ser conrmados con simulaciones para caracterizar mejor el comportamiento de las esferas y las interacciones con las paredes. La inuencia de las diferentes regiones y de la geometría no está clara y por ello, sería conveniente realizar un estudio acerca de estas características. No existe una formulación completa sobre la teoría de lechos uidizados, existen modelos concretos como el del presente estudio pero que abarcan casos particulares y no siempre extrapolables a otras situaciones. Por ello, cualquier avance en este aspecto debería ser tenido en cuenta para futuros estudios. Se realizaron simulaciones con diferentes modelos de uidos como pseudoplásticos y uidos de Bingham, sin embargo la falta de información experimental hizo rechazar esta posibilidad. Diversos campos en la mecánica de suelo han realizado estudios acerca de este tipo de comportamientos, sin embargo ninguno pudo ser aplicado. Se realizó una hipótesis acerca de un posible modelado de la viscosidad virtual a las variables del lecho de esferas. Esta posibilidad debe ser conrmada. El uso de métodos combinados D.E.M. - F.E.M. está siendo desarrollado actualmente, es de esperar que este tipo de métodos permitan nuevas perspectivas de estudio, sin embargo, el análisis con uidos viscosos se ha demostrado ecaz y eciente. Parte II ANEXO I: ANALYSIS OF PEBBLE BEDS IN HIGH TEMPERATURE REACTORS 37 38 Appendix A INTRODUCTION The future of energy production is at present one of the most important topics that concern the international community. The increase of oil prices is not only a temporary situation caused by a punctual event in the production. The already known reserves are probably at maximum exploitation capacity, also the political instability that aect countries where are placed more than a half of the world oil reserves requires the diversication of energy sources to ensure the future energy supplies. The use of other traditional fossil fuels such as natural gas and coal is also a problem due to the clear necessity of a global reduction in the CO2 emissions. Nuclear technologies are present in this future outlook, but the associated risks that entails every technology and recent events that took place in Fukushima remind us the necessity of continuous research in safety features and improving the facilities that are running at present, replacing old plants for modern models and always looking after the safety as the most important priority, for that reason, the development of new reactor designs which are safer and more sustainable is required. In line with this is the current deployment of III+ Generation reactors and the development of future IV Generation designs. In this context High Temperature Reactors, also known as HTR, have an important role to play. These reactors, which are cooled by gas instead of water, have several advantages in comparison with other reactor concepts. They are designed to resist coolant ow accidents by the use of heat resistant materials and passive heat removal technologies. An other advantage is the possibility of achieving high temperatures of the coolant ow, which allow better eciencies in electricity generation and also oer the possibilities of heat process that have multiple utilities in industry. An important part of HTR technologies involve the Pebble Bed reactor concept. HTR pebble bed nuclear reactors are conceptually simple, a pressured vessel lled up with spherical elements where the nuclear reaction take place and that are recirculated. Dierent pebble bed reactors have been operated in the past. Two of these reactors are the AVR located at Jülich Research Center and the THTR-300 in Hamm both in Germany. Those facilities have given a lot of worthy experience but have also 39 APPENDIX C. THEORY OF PEBBLE FLOW 46 C.1. The pressure magnitudes are comparable or even bigger that those originated by gravity. Figure C.1: Horizontal pressure distribution over the core reector. Comparison of dierent pressure models [Bedenig et al. 1968] Those eects are explained by the silo theory in section C.4. It is also necessary to explain that all the core reector models have been designed with concavities all along the wall to avoid crystallization eects. The potential eect of this design on the overall friction forces is not clear, and supposed to have minor consequences [Tingate 1974]. Forces between pebbles are originated by the slip of pebbles within the ow. The value and characterization of this ow is not clear due to the impossibility of obtaining reliable experimental data. [Bazant et al. 2002] explain the importance of the force between pebbles is undened and probably lower than gravity eect. Studies done in the past model the ow of grains with Couette ow models [Patton et al. 1987] but they do not clarify the explicit inuence of pebble-pebble friction over the ow behavior and no conclusive information could be found. In case of pebble bed reactors appears an other problematic situation referred to the extremely low ow velocity, a fast circulation rate would be around 1 pebble min −1 , because in that case and taking into account the intermittent circulation, is not clear which parameter origin the forces, static or dynamic friction coecients. APPENDIX C. THEORY OF PEBBLE FLOW 47 Figure C.2: Bridging causes a ow blockage in a conical hopper [NN 4] Short-range force s involved in the ow behavior, these forces are produced by the physical contact between spheres, as an eect of the normal component and the tangential one on punctual contact. In case of pebble-pebble interaction there is an energy loss due to inelasticity. Long-range forces are forces originated several diameters away from the location where the eect is detected. The transmission of this forces is related to the contact and collision of pebbles and the energy transmission related to them. [Bazant et al. 2002] consider that long range forces can be transmitted until 5 sphere diameters, but they are much more important for wall-pebble friction forces. Those facts were observed also in the results of ow experiments [Babilas 1992] and in the results of the present project in section F.3.2. Coolant ow eects come from the loss of pressure in the ow of helium. In modern designs, the coolant gas is introduced from the top core and ows downwards to the bottom. The friction with the thousands of pebbles has an eect of decreasing the specic density of the pebble bed. It can be conclude, that no eect over the pebble ow can be appreciated [Gougar et al. 2004]. Discrete framework is a very important characteristic of pebble bed reactors. In fact it is not only that the ow is intermittently controlled through the exhaust mechanism, but also that as it was said, does not exist a continuous uid or even a possible description on this way. Nevertheless, it can be observed that despite being a discrete system, the pebble motion behaves in some aspects like a continuous uid. An important property was observed by [Bedenig 1971] in the rst visual experiments done in Germany. It was possible to determine that the pebbles do not move randomly within the core. In fact they follow very clear ow lines with very low dispersion from the initial position. This statement made possible the development of pebble bed reactors and open the possibility of studying a complex discrete system with a continuum uid formulation. Pebble Packing Factor and granular structures have an undened eect on the ow, but they are very important to the study of pebble bed in HTR reactors. APPENDIX C. THEORY OF PEBBLE FLOW 48 This measurement can be characterized by the packing factor which is the quotient between the volume occupied by the pebbles and the total volume; the maximum value is 0.74 given for hexagonal close packing (HCP) structure in gure C.3 whereas the average packing factor in a reactor it is about 0.64. Figure C.3: Hexagonal Close Packing of spherical particles. Arise maximum compacting possible In a whole core can be observed that the packing density varies a lot depending on the position that we study. The packing density determines the permeability of the bed and thus, the ow of the coolant gas helium which is the nal responsible of the heat transported, and therefore of the output power. Test made by [Tingate 1973] show irregularities, also known as cavities, in the pebble structure that can be explain by combined eect of friction forces and the initial position when loading the reactor vessel. Bridging is a peculiar situation that happens in the outlet when it is too small, see gure C.2. Outlet diameters bigger that 5 pebble diameters avoid this situation [Bazant et al. 2004]. Low intermittent velocity has to be adapted to make possible the simulation of the ow by means of computational systems. As long as it is not possible to dene an instant velocity in every position and moment, it is necessary to work with an average eld of velocities. The low velocity has to be adapted in experiments and computational simulation, otherwise, the necessary time step in the computation process would make not feasible obtaining results. Geometry is one of the most determinant variables that are involved in the pebble motion. Event it is not a particular characteristic of granular uids, It is necessary to remark its importance and the consequences that has for the behavior of the system. It is fully explained in section D.3.3. C.2. State of the art: theoretical models The development of a general granular ow is the objective for many scientists at present. The application of these uids is very common in engineering, medicine or chemistry; and the possibilities of a better comprehension of the phenomena can APPENDIX C. THEORY OF PEBBLE FLOW 49 open many new applications. In the case of pebble bed reactors, the behavior of pebbles has a particular and important property. Despite pebbles are macroscopic elements which interact between them with complex and non-equilibrium relationships, gravity driven granular uids having a common behavior. This important property allows us to model the behavior of a pebble bed without the necessity of a pebble by pebble study which is extremely complex and not necessary to fulll the objectives of this thesis. C.2.1. Discrete Element Methods (D.E.M.) This numerical method was developed in 1979 by Cundall and Strack and it is the basis for Molecular dynamics and more recently, the behavior of granular ows in silos or uidized beds. This method calculates the interaction of all the particles that are conned in a volume, as a dierence with Finite Element Methods (FEM) where the volume is a continuum medium. For every single pebble and for every time-step all the force equilibrium equations have to be computed. This allows to study accurately the friction forces between pebbles and the interaction with all the surfaces. A study of [Niessen 2009] shows the high precision that can be achieved with this method. Nevertheless, and as a high inconvenient, appears the necessity of huge CPU time. In recent studies [Rycroft 2006] reports that for a 1E6 steps simulation in 60 modern processors more than 13h of computing time were required. At present, one of the research lines, consists on the improvement of the numerical method making it faster and more exible. In order to do it, assumptions and simplications of the model have to be made, but the problem is that in granular ow, do not appear general situations which can be simply neglected. Some of the methods are here remarked. Figure C.4: Forces between pebbles [NN 1] C.2.2. Kinematic Models Kinematic models are based on the energy conservation law. Simple kinematic models are used to simulate granular uids moved by gravity. Despite they are conceptually too simple compared with granular uid physics; they have shown certainly good accuracy in the simulation results [Rycroft 2006]. However they do not explain the interactions between pebbles either with reector walls. APPENDIX C. THEORY OF PEBBLE FLOW 50 C.2.3. Stochastic Kinematic Models These models are an evolution of simple kinematic models and are based on statistical principles to explain the movement of pebbles. One of the methods is the Void methods [Mullins 1972] which involves the introduction of voids in the outlet of the core and ow up in the core. Then by means of statistics, the voids are dispersed in the core creating a distribution. This theory predicts properly the streamlines of the ow and also the key parameters if they are studied far enough from the walls of the core. Anyway, the theory does not explain the eect of the shear stresses over the reector and their inuence in the ow behavior. An evolution of this method is the Spot Model. It was developed at the MIT (USA) . In this model the principles of the void movement are changed and it assumes a dierent way of void diusion which explains better the movement of pebbles and the way how the occupy new spaces in the pebble bed. Their studies [Bazant et al. 2004] show that exist a longer diusion length that it was expected. [Bazant et al. 2002] determine that the particles make an inuence up to 10 pebble diameters. At present the most recent studies are focus on dierent discrete models; they have better precision and go directly to the comprehension of the physical problem. In this thesis, the objectives do not cover all the granular uids and it is necessary to simulate the pebble bed as a single unit, thus, the idea is to recover the rst approaches done in the past my means of continuum dynamic models and improve them as much as possible. C.2.4. Modeling a pebble bed for discrete simulations D.E.M. methods and its dierent variants are capable to simulate the interaction of the pebbles and their movement, but they require of an initial situation in which the core is already lled of pebbles because those codes are not able to reach this rst stage, the complexity is high as can be observed in gure C.5. This fact can appear trivial in a rst sight, but it has a great importance and has to be properly prepared with the desired structure. The later dynamic simulations will be strongly inuenced by the initial pebble bed structure. Figure C.5: Construction of an ordered pebble bed for a D.E.M analysis [Ooms 2008] APPENDIX C. THEORY OF PEBBLE FLOW 51 Studies done in the past [Tingate 1973] show the decisive inuence of the initial packing bed structure to the later evolution of the ow, a random packing of pebbles has as consequence a lower packing rate which has advantages for the coolant gas ow. As a negative point is in such a structure, the density is not homogenous and it is not proper from a heat transfer point of view. The other option is to ll the vessel ordered with a pre-dened structure. It is interesting from a theoretical point of view, but in the practice, the situation may not be stable. In a series of experiments [Babilas 1991] used in this work as main reference, the pebbles are recirculated in dierent percentages of the core coming from an initial random packing situation. With the goal of measure the dierences and how long does it take to reach a stable situation. Recent works done by the U.S. Department of Energy [Abderra et al. 2005] and in the T.U. Delft [Ooms 2008] propose dierent computation methods to ll a core with dierent possibilities Most of the actual research lines are focus on the discrete approach, as long this is not the main approach of the present work, the author encourages a full revision of the literature about those methods in case of future studies based on discrete approach. C.2.5. Continuum dynamics approach: Gravity driven pebble bed framework shows a group behavior despite the fact that it is made by macroscopic particles. This property allows an intuitive approach to the problem. The pebble bed is considered as a uid which can be done due to these long range pebble relationships. The rst step was to visualize the ow behavior of packing pebbles. For that reason, were done a series of experiments [Bedenig 1966] in the Jülich Research Center (Germany) with glass spheres and color spheres. It was proved that the ow lines were very regular and straight, in fact can be ensure that we face to a laminar behavior. This important characteristic is necessary for a continuum approach, otherwise the solution of the equations would be much more complicated, making this approach useless. APPENDIX C. THEORY OF PEBBLE FLOW 52 Figure C.6: The Five-Channel method for calculating the velocity of the ow Taking this into account and after the glass spheres experiments; the rst mathematical model developed for a pebble ow was done in Jülich Research Center [Bedenig 1967]. See gure C.6. In this model, the core from a frontal view, is divided in n canals. Then, from azimuthal view, the core is divided in the same number of rings, making then ow 3D canals. The velocity is calculated by means of mass conservation equations. This allows a simple way to make a simulation. The main problem of this model, is that needs the initial velocity at top of the core vessel and the travel time of the pebbles. That requires the experimental measurements and evidently they are time and cost intensive. Nevertheless it can calculate the shape of the ow lines, which accurate quite well to the experiment observations. This method either takes into account the interactions between pebbles and walls. The next simulations were done in the early 80's for the design of the THTR-300 reactor prototype. The model was also in Jülich Research Center developed by [Scherer 1989] and it is one of the main theoretical supports of the present thesis. The procedure of equation simplication is here explained. The starting point are the Navier-Stokes equations: ∂ρ ∂t +∂ ∂xi (ρui) = 0 (C.1) ρ∂v ∂t +ρv ·∇·v=−∇ P+∇τik +ρfm (C.2) ∂ ∂t(ρe) + ∇ · (ρev) = ∇τ0v+ρgv (C.3) APPENDIX C. THEORY OF PEBBLE FLOW 53 The model assumes incompressible ow ∇ · v= 0 and stationary ow dv dt , therefore is possible to linearized the equation. The stress tensor: τik =ξσik ∂vi ∂xi +η(∂vi ∂xk +∂vk ∂xi ) (C.4) With incompressibility condition: τik =η(∂vi ∂xk +∂vk ∂xi ) (C.5) Thus, ρ∂v ∂t +ρv ·∇·v=−∇P+∇η(∂vi ∂xk +∂vk ∂xi ) + ρfm (C.6) Introducing the stress tensor into the quantity of movement equation and if constant viscosity in a local area is supposed: ∂2vk ∂xi∂xk Can be neglected Then: ρ∂v ∂t +ρvk ∂vi ∂xk =−∂P ∂xi +∇η∂2vk ∂x2 i +ρg (C.7) Velocities in the problem are very low, therefore: ρ∂v ∂t =−∇ · P+η4v+ρg (C.8) Equation C.2.5 is the linearized conservation of momentum equation. The solution was carried out by a F.E.M., however the complexity of this approach is to dene the boundary conditions and all the domain characteristics. C.3. State of the art: Experiments Granular materials were studied from nineteenth century with the construction of the rst silos for the storage of bulk materials. Recently the eld of application for granular ow has increase exponentially. Nowadays many industrial and technical processes require a good knowledge of those materials and the laws that rule their movement in dierent scenarios. For that reason from the very beginning of the APPENDIX C. THEORY OF PEBBLE FLOW 54 research, it became necessary to make experiments. Actually, most of the assumptions and models are based on empirical information and the experience achieved is very helpful in order to develop a general granular ow model. For these reasons, in this section the Experiments related to gravity driven pebble ows done until the present will be introduced. Also they will be briey resumed to get a general idea of the methods used and nally they will be put together to extract the conclusions and assumptions that are the starting point for the development of this work. C.3.1. AVR experiments The AVR reactor (Arbeitsgemeinschaft Versuchsreaktor in German) was the rst HTR pebble bed reactor built in the 60's at Jülich Research Center. AVR was a test nuclear facility where a lot of experiments were carried out to test the feasibility of the pebble bed concept. Series of experiments were performed in the 80's introducing high-enriched uranium pebbles (HEU) in two dierent parts of the core, central and outer regions. In the moment of the pebble injection, the already placed pebbles were in an advanced stage of burn-up, therefore was possible to distinguish them in the outlet with a radiation detector [AVR 1990]. Figure C.7: Top view from of the empty AVR core reactor APPENDIX C. THEORY OF PEBBLE FLOW 55 In the rst trial, 300 HEU pebbles were introduced in the central section of the core. In the second trial, the same procedure was done in the outer core, see gure C.8. Unfortunately not all the pebbles appeared in the expected time or were not properly detected. Some of them appeared years later when the reactor was decommissioned. With the information obtained, it was tried to improve the 5-Channel model using a new version with more canals, up to seven. See gure C.6. The results were not satisfying enough and the accuracy of the method was discussed as it is explained by the authors [Pohl 2009]. Figure C.8: Geometry of the AVR reactor and the two core regions A visual inspection inside the reactor vessel which showed up an irregular distribution of the pebbles inside the core and the formation of small hills due to the location of the inlet tubes. This experience is not important from the pebble ow point of view, but it is interesting to understand that asymmetry in the pebble bed characteristics is common in pebble beds. C.3.2. Australia In 1970 a large research about the ow of pebbles took place in the laboratories of the Australian Atomic Energy Commission. The objective was to determine the inuence of about 30 parameters [Tingate 1973] in the pebble bed behavior. All the details and the data from the experiments can be found in [Tingate 1970]. The experiments [Gatt 1977] were carried out in an aluminum cylinder with a conical discharge hopper. The angle of the hopper was variable and the vessel was lled with plastic pebbles coated with dierent friction properties materials. For the ow paths, radioactive pebble tracers were introduced in the pebble bed. With the help of gamma detectors was possible to measure the position in the 3 axes, the velocity was APPENDIX C. THEORY OF PEBBLE FLOW 62 Figure C.10 show the two basic ow types. In case of pebble bed reactors, the ow type is a mixture of both mechanism. In the upper part, they type is clearly mass ow, but as long as the ow crosses half of the core, it starts to be noticeable a change on the ow pattern. C.5. Conclusions of experimental results The experimental data provided by experiments has a great importance for the development of this work; the author considers very important that all the experiments are properly studied and understood for future development, nevertheless all the assumptions necessary have to be physically realistic and have to be proved by experiments. The insucient documentation available about some of the previous explained trials and the possibility that contain measurements fails makes necessary to discard them in a validation process. Under the dierent experiments previously explained and the theory about granular ows, we can ensure that these conditions are common for pebble ow in case of pebble bed reactors. Laminar regime Low radial and tangential dispersion. Possible 2D or symmetrical simulation Limited scale variable possibilities, thus necessary most precision possible when modeling simulation Discontinuous ow of pebbles can be approximated by mean values of variables Not only direct friction forces; long range forces play important role. Other characteristics such as adhesion of particles, clump rolling friction, arching cannot be modeled Possibility of viscous models to characterize the pebble bed properties but lack of experimental data. Coolant gas eects can be simplied by an increase/decrease of the mean viscosity value. Friction forces coecient distribution for dierent parts of the core: Temperature, concentration of H2O, dust are the main distortion variables Geometry has a major role in the ow behavior. Remark the eect of the hopper and the transition cylinder-hopper. APPENDIX C. THEORY OF PEBBLE FLOW 63 C.5.1. Problems found in the access to information and data Due to the specic problem and the non-conventional elements of study; the research task was hard and took more time than expected. It was also problematic to obtain the experimental data of some test because they are under industrial right laws and others were lost. This lack of information is the cause to reject some data because it was not possible to conrm all the details that were involved in the realization of the experiments. In the validation section will be remarked those test where some information deciencies have been found. Appendix D FLOW MODEL DEVELOPMENT D.1. Description of the problem case No general governing equations for granular ows are available in the present moment, therefore the use of experiments, observations and simulations are required as starting point for the development of a model using a continuum uid approach. In the previous chapter the most important dierences between both systems have been exposed, but also it is a resume itself of the common characteristics that can be observed in pebble beds applied to nuclear reactors, and that allow to distinguish them from other granular ow types. In order to guarantee that the physical meaning of the variables coupled is correct and after an extensive literature research and an analysis of the experiments explained in the previous sections; all the variables, parameters, characteristics, properties and observations are put together to have a simple overall view of the problems to solve. The problem could be dened in simple terms as a cylindrical vessel which ends in a conical hopper and is lled with spheres which leave the core intermittently. But even in the ideal case, the description of the pebble motion through a common viscous uid is very limited. The macroscopic behavior is similar in operation conditions but they respond to forces which are very dierent in origin, thus a direct transformation of the variables, parameters and scales is not acceptable. For these reasons it is necessary to couple the variables based on similar phenomena that can be experimentally demonstrated. 64 APPENDIX D. FLOW MODEL DEVELOPMENT 65 PEBBLE FLOW CONTINUUM VISCOUS FLUID Granular Flow: Molecular dynamics models. Solid mechanics equations VISCOUS FLUID: Governed by Navier-Stokes equations NON-CONTINUUM MEDIUM CONTINUUM FRAMEWORK 1. Possible to identify a single particle of the uid 2. s pebble face distance → inelastic collisions 3. Normal and tangential forces between pebbles involved → Force and momentum 1. Not possible to identify a single uid particle 2. Assumption of the elastic collisions between molecules 3. Assumption of normal forces interaction INTERMITTENT PEBBLE FLOW: A pebbles leaves the core within discrete time interval CONTINUOUS FLOW REQUIRED: A transient analysis complicates the resolution and does not give more valuable information FRICTION FORCES have major inuence in the pebble bed behavior FRICTION plays a minor role in the ow at very low velocities. Viscous forces are small at low velocities NON LINEAR PRESSURE DISTRIBUTION: Pressure prole dependent on friction forces, cohesion and location of pebbles HYDROSTATIC PRESSURE DISTRIBUTION which is determined by P=ρ·g·H MECHANICAL PROBLEMS during the ow of pebbles: Crystallization, angle of repose, bridging and blocking which cause irregularities in the ow HOMOGENEITY in the properties and structure of the uid. Irregularities are not stable in time HETEROGENEITY OF PROPERTIES within the uid. The density of the pebble bed is variable in dierent in dierent locations COMPRESSIBLE FLUID needed. Characterize the variation in the density involve much more complexity and does not give more information HETEROGENEITY OF PROPERTIES: The friction coecients in graphite are strongly inuenced by atmosphere, temperature and material irregularities. VISCOSITY MODELS allow dierent behavior patterns of the uid INITIAL PACKING of pebbles has a noticeable inuence over the ow. Evolution to a stable situation TWO-PHASE uids are capable to experience heterogeneity in that way MEASUREMENTS are dicult to be obtained. Necessary average values EULERIAN approach does not allow to obtain information of a dened particle → Lagrangian approach NON-REPRODUCIBILITY of results → Statistical mechanics required Macroscopic reproducibility of results APPENDIX D. FLOW MODEL DEVELOPMENT 66 Also some particular characteristics of pebble bed reactors must be taken into account. CHARACTERISTICS CONSEQUENCES Pebbles of dierent diameters in some designs Necessity of a two-phase uid or approximation by a single uid Reactor not cylindrical, but n-side polygon Angle corners and sharp regions dicult a smooth ow and create irregularities Reactor walls with irregular texture to avoid crystallization Walls are not at and have roughness and cavities that aect the ow Recirculation rate can be understood as mass ow An average velocity can be an operation variable to calibrate the model Spherical pebbles No adhesion of grains, thus smoother ow but unknown viscosity model Several devices like control rods and columns are introduced during the pebble motion Eects over the pebble ow must be individually evaluated Coolant ow Changes in the specic density of the pebble ow Extraction procedure of pebbles A non careful extraction of pebbles can cause irregularities and rotational components in the ow[Gatt 1977] Straight and regular ow lines. No azimuthal and angular diusion Laminar ow regime simplies the resolution of the dierential problem D.2. Validity and theory of continuum approach Despite the several remarkable dierences between granular and continuum uids, the use of a continuum uid framework is useful to obtain simple, fast and feasible solutions to the problem, but this approach is not possible in every situation. [Ha 1983] did a research on the conditions that a granular ow has to fulll to be described properly by a viscous uid. He assets that the biggest dierence between classical uids and granular ones is the fact that in case of granular uids, the collisions between particles are essentially inelastic, which means a strong loss of energy in form of heat, meanwhile in viscous uids, the collisions take place between molecules and they can be considered as fundamentally elastic because in that situation, not only Coulomb forces between molecules have to be taken into account but also quantum mechanics. APPENDIX D. FLOW MODEL DEVELOPMENT 67 Figure D.1: Inelastic collisions between solid grains [Frank 2009] This inelasticity of the system requires the usage of the equation of mass, conservation of momentum and energy; The Navier-Stokes set of equations is the basis of Kinetic Theory [Nedderman and Tuzun 1979]. In a packing of pebbles it is related with the distance s between the faces of the neighboring pebbles. The smaller is the distance, the less energy is lost and less inelasticity can be observed, the criteria set is s<<d where d is the diameter of the sphere. The value of 0.61 for common reactors plus the low velocities of pebbles ensures that the inelasticity inherent to granular ows is small enough to consider the option of continuum approach. The second condition required is that the quotient between the core diameter and the pebble diameter is bigger than 1000. That means, that in case of 60 mm pebbles, the core must be at least 60 m of diameter, then the biggest designs are in order of 10 m diameter. That condition is not fullled for HTR-Reactors but experimental data [Bedenig et al. 1967, Tingate 1970, Bazant et al. 2004] assets that the ow regime is clearly laminar and the particles follow in an uniform ow [Patton et al. 1987], therefore we assume that the low velocity and the ordered movement of pebbles is APPENDIX D. FLOW MODEL DEVELOPMENT 68 sucient condition to use a continuum approach such as show the model developed by [Scherer 1989]. Once accepted the viability of the continuum framework approach it is necessary to simplify the Navier-Stokes set of equations. It is previously explained that mass, conservation of momentum of energy equations govern the ow in case of big energy dissipation uids. Mass conservation equation: ∂ρ ∂t +∂ ∂xi (ρui) = 0 (D.1) Conservation of momentum simplied and linearized: ρ∂v ∂t =−∇P+η∆v+ρg (D.2) And the equation of energy: ∂ ∂t(1 2ρui+1 2ρ~ v2) = −∂ ∂xk [ρuk(p ρ+1 2u2+1 2 ~ v2)−uiη(∂ui ∂xk +∂uk ∂xi )−κ∂ ∂xk 1 2ρ~ v2]+ρuig−1 (D.3) But in fact the energy equation can be simplied because the term of dissipation can be assumed as very small compared with the other variables. [Ha, Hui et al. 1984] approximate the viscous dissipation term to I=γ·ρ·~ v3 s ; where gamma is proportional to 1−e2 and e the coecient of restitution that we assume one because elastic collisions are supposed [Bazant et al. 2002]. For this reason in the present work only mass and momentum conservation equations are taken. Mass conservation is implicit to the pebble bed inside a reactor where constant density, thus packing factor too, and no leakages of mass or changes in the volume are coherently supposed. Therefore the stress tensor in the conservation of momentum equation must be dened, but in case of common uids, the inuence of frictional forces is signicantly smaller than in case of granular ow [Hanes and Inman 1985], hence it is necessary to couple the variables that govern the friction eects in case of granular uids and to nd a correlation between them and the viscous term of the Navier-Stokes equation. D.3. Coupling variables Obtaining a meaningful model requires that the most inuencing variables that govern the behavior in a pebble bed must have an equivalent variable which causes the same eect over the viscous uid and with the same relative weight respect APPENDIX D. FLOW MODEL DEVELOPMENT 69 other variables; this process is the key point in the development of the model. In order to simplify the process of comprehension, the variables are divided into 3 dierent groups: Physical variables, material properties and problem operation characteristics. D.3.1. Physical variables Friction and gravitational forces are the main motion mechanism in case of uids under gravitational inuence, but despite that fact the origin of those forces is not the same as it is previously explained in Paragraph C.1. In case of gravity the specic weight per volume unit is the variable that controls the value of the force, for viscous uids that is represented by density, which is dependent on temperature and pressure and it is dened as the quotient of a dierential mass portion and the dierential volume that it occupies. But this denition is not possible in case of a non-continuum framework like pebble beds. In that case, it is chosen an average value called packing factor or void fraction and it is dependence is more complex than in uids including temperature, pressure, shape and roughness of the spheres, elasticity of the material, adhesion and the surrounding atmosphere. Figure D.2: Void fraction in reactor core [Ooms 2008] In fact, these conditions are not constant during the reactor operation and as a consequence the value of the packing factor is neither constant. Specially in the areas closest to the reector walls, this eect is amplied and it is related to the pressure distribution and friction eects [Ooms 2008] but this situation is found in a relative small area and for this reason it is assumed an average value for the viscosity which is the product of the average packing factor in operation conditions and the density of the pebbles. Pressure in the storage of bulk materials has some dierences compared with a uid in containment. In case of a common uid, the hydrostatic pressure ensures that at the same height, pressure has a constant value; thus pressure is dependent only on the depth for an incompressible uid; and taking into account that velocities are very slow dynamic pressure can be neglected. In case of the storage of bulk materials the pressure distributions does not answer to any already known analytic expression APPENDIX D. FLOW MODEL DEVELOPMENT 70 and must be estimated with empirical expressions based on the properties of the material, the geometry, and the friction forces created within the uid and with the walls which origin forces in the opposite direction of the pressure. See gure C.1 But this is not the only reason to explain the big dierences and the irregularities of the pressure distribution in a pebble bed. Taking into account how the forces are transferred, we can imagine a very complex situation where the force created by i.e. the mass of pebble, is distributed between its neighbor pebbles, but simultaneously they transfer also the forces. That fact and the inelasticity of the collisions makes it impossible to talk about hydrostatic pressure in case of bulk materials. Figure D.3: Stress transmission in a pebble bed [NN 2] Pressure forces originated by temperature gradients are not taken into account in this study, because in case of operation condition, the changes on temperature are expected to be low. But it is necessary to remark, that the thermal stress studies performed in ANABEK [Babilas 1992], the eect over the pebble bed are stronger stress forces which can be added to those produced by friction. Viscosity is the main friction mechanism that is found in common uids but there is not any feasible equivalence with the friction between pebbles than can ensure the same behavior. In case of pebbles, the friction forces are in origin the result of Fr=µfriction ·N meanwhile in uids is the product of viscosity and shear rate deformation per time unit. This idea is used by [Scherer 1989] to think about viscosity as virtual variable and assumes that it is not possible to nd a relation between its value and the frictional parameters of the pebble bed. Following this idea and taking into account studies done in case of Couette ows applied to simulation of granular ows, when shear rate is big enough, it saturates and remains constant [Thompson and Grest 1991] having as a consequence the saturation of the shear stress. In other words, the possible tangential deformation of the uid is limited to a certain value, which will occur when the walls of the containment are rough enough to ensure that velocity on the wall surface is zero. A maximum value for the shear rate is obtained when there is a pure non-slip wall condition and a minimum value when the boundary condition is free-slip. APPENDIX D. FLOW MODEL DEVELOPMENT 71 Then in the equation of the shear stress at a wall τ=η∗du dy ; we have to x the value of the velocity term, and as it is proposed that the viscosity term is a virtual variable, hence is possible to make that the shear stress term has the same value to the friction forces over the core wall, these forces have to be obtained by experimental measurements or can be calculated if the velocity at wall surface is known, which is really dicult to determine. Then, it would be only necessary to calibrate the viscosity depending on the shear rate and we will obtain the same relation between relative velocities inside the pebble uid and he pebble bed. Then, once the virtual viscosity that corresponds to a specic case is known and with the use of shear stress prole expressions given by silo theory, it is feasible to obtain a good approximation to the velocity eld found at the core reector. The last statement requires an expression which can give the value of this virtual viscosity as a function of geometry, operation condition, material properties. Due to the lack of experimental data required. This expression is approximated in the present study. Despite the fact that bingham and pseudoplastic viscosity models are more similar to a pebble ow, nally and after some trials in section F.2 was found that for high values of viscosity and very low velocities of the ow, the inuence of viscosity is very low and it conrms in the results obtained by Scherer. Figure D.4: Viscosity models D.3.2. Material Properties The so called pebble uid is the continuum material approach to the ow of pebbles. For this reason it has to be modeled as a liquid in stable thermodynamical state where no heat transfer is allowed because stationary reactor operation in thermodynamical equilibrium is supposed. The density values will be xed as constant and as it was explained in the previous section, an average value will be taken. The viscosity value have not a major role in the behavior of the pebble uid behavior when shear stress saturation point is reached, and it is used as a calibration variable to x the mass ow and thus, the average velocity of the ow. APPENDIX E. SOLUTION TO THE FLOW PROBLEM 78 Figure E.1: Maximum shear stress values over the cylindrical reector core for dierent viscosity values. No-slip boundary condition xed In gure E.1 are the results of the simulation. The vertical shear stress is linear from a viscosity of 1Pa s . As a reference, water at 25 º C has a viscosity of 8.9·10−4 Pa s , and a heavy oil is around 100 −500 Pa s . Before that value is reached, from 1·10−6Pa s to 1Pa s a change of the slope can be observed, that means a change on the shear rate. In other words, in that region, uids with dierent viscosity values show dierent velocity proles. It is necessary to remark, that the values obtained in this test are the maximum possible with that conguration, because a no slip wall boundary condition was chosen. A value for the viscosity of 1·108Pa ·s is assumed for an initial approximation because the stresses found in the ANABEK experiments are around 10 kPa, the results are resumed in section E.1.1.1. Finally the expected shear stress prole and its nominal pressure value are implemented in the ow model. In order to calibrate precisely the model, a simple procedure is followed. First the slope of the simulated residence time distribution curve was tted to the data. The slope represents the stress prole, and in case of free slip walls, that means no friction with walls, the curve would be almost vertical because if no friction exist, the velocity prole remains at and all pebbles leave within a very short time dierence. On the other side, if no-slip walls are xed, the residence time curve would have a low inclination, that means a high residence time dispersion. Thus, the model must be between both situations. The next step is to adjust the nominal stress values until both curves t as accurate as desired. APPENDIX E. SOLUTION TO THE FLOW PROBLEM 79 E.1.1.1. Obtaining a shear stress prole from ANABEK experiments Obtaining a realistic stress prole is probably the most critical step in all the modeling process. The inuence of the shear prole and their nominal values over the ow are very high. However, in the ow and stress experiments of ANABEK pebbles are made of dierent materials, then there is not any experimental data available about both stresses and ow behavior in same conditions. For those reasons, it is necessary to interpolate a possible stress prole for the pebble ow. The ANABEK stress tests are carried out with ceramic and graphite spheres, while the ow experiment is done with spheres made of polyacetat, whose properties are between graphite and ceramic spheres, closer nevertheless to graphite. In both gures E.2 and E.3, the horizontal pressure given by silo expression C.9 is plotted for two sets of parameters, cover the complete range of possible values and compared with the results of the ANABEK horizontal pressure simulations. Figure E.2: Horizontal pressures over the core reector with ceramic spheres in ANABEK compared with Janssen's proles for dierent friction coecients APPENDIX E. SOLUTION TO THE FLOW PROBLEM 80 Figure E.3: Horizontal pressures over the core reector with graphite spheres in ANABEK compared with Janssen's proles for dierent friction coecients The validity of the shear stress prole expression is very limited by the characteristics of the experiments carried out. The model is 1:6 scaled and the pressure distribution due to height is evidently totally dierent in the real core. Figure E.3 show that in case of small graphite spheres the agreement is quite low, but is specially noticeable that there is not any pressure measured at all until 0.5 m below the top surface. It is intuitive to imagine that until a certain height for a light material as graphite, whose density is 1700 kg/m3 , the pressures found are too low to be measured. As a comparison, in the next series of experiments. Spheres are made of a heavy ceramic material, whose density is 2600 kg/m3 , nevertheless the same eect as in graphite test can be observed, the values close to the top surface are smaller than expected. It is not acceptable to use Janssen's prole in this situation, and for this reason, a similar stress prole is supposed. However, the agreement of the measured experimental values with Janssen's prole is very good and permits thinking about the application to every reactor design. As long as polyacetat pebbles have closer characteristics to graphite pebbles than to ceramic ones, it is dened a prole called Step_prole that has a value of 0 Pa until the height is 1250 mm, similar to the prole in gure E.3. Then, the shear stress grows linearly until a chosen nominal value. The linear prole has been used to simplify the implementation and to test the validity of such a prole. Several other proles with dierent nominal values have been tried. In gure E.4 three examples of applied proles and their nominal pressure values are shown. Although they do not t exactly with the stress proles of gures E.2 and E.3, the approximation is acceptable because rst was necessary to measure the real inuence of the prole, and then, the goal was to nd the prole that makes the ow behave more similar to the one obtained in the ow experiments. APPENDIX E. SOLUTION TO THE FLOW PROBLEM 81 The nominal pressure values, from the stress proles of the experiments are referred to horizontal pressure and no to shear stress . As a result, for the chosen value of viscosity 1·108Pa s is acceptable to x smaller values than in the experiments. As it is explained, viscosity is a virtual variable which has not any correlation known at present. Figure E.4: Dierent stress proles tried in the simulations E.2. Computational solution The solution of the ow problem, once the theoretical model is clear, will be carried out with the software CFX ANSYS which uses a Finite Element Method (FEM). This software is chosen because it is one of the most extended commercial codes for uid analysis and has already been validated for many cases. E.2.1. Particle Tracking as a technical solution One of the biggest problems found during the development of the present work was the diculties that involve obtaining information of a single uid particle or a group of them. In the experiments carried out, THTR-300 and ANABEK test were performed with spherical elements which could be easily identied. But in case of a FEM ow analysis exist two dierent approaches to solve the problem: Lagrangian and Eulerian approach. APPENDIX E. SOLUTION TO THE FLOW PROBLEM 82 Figure E.5: Lagrangian vs. Eulerian problem ow modeling [Frank 2009] The Eulerian approach does not distinguish between the uid particles and makes the analysis in a control volume without taking into account what happens to the particles inside this control volume. Then, it is possible to obtain the ow lines integrating the velocity, but this approach is not capable to calculate what happens to a single particle. For that porpoise is necessary a Lagrangian system, the problem is that this approach requires high computation time and the solving process gets more complicated. CFX ANSYS allows a new method to use a Lagrangian-Eulerian method to simulate the eect of particles in a ow, for example, to study the accumulation of sand inside a pipe, the eect of a two-phase ow with a solid phase, etc. In this work, this method has been used as a part of the uid dening a two-phase problem where the two phases are exactly the same. Figure E.6: Comparison between Lagrangian and Eulerian ow models [Ma and Srinivasa 2008] The conguration of a two-phase method is challenging for the robustness of the APPENDIX E. SOLUTION TO THE FLOW PROBLEM 83 solution and the load of computational work is strongly increased. There is also a problem that comes from the fact of adding a second medium into the main uid. The interaction between the uid and the solid particle causes changes in the ow behavior which is not desired at all. For this reason, the tracking particles are congured with one-way equation coupling, in other words, the ow will aect to the solid particle, but not the other way around. Thus, it is possible to work with this virtual particles as an inherent part of the uid but with the security that they are neutral for the uid behavior. The tracking particles are dened with the same density and the friction interaction with the Schiller-Naumann model. The interaction between particles are simplied with inelastic collisions, that means a coecient of restitution e =1. Figure E.7: Visualization of tracking particles for the study of dierent core regions This method oers good reliability on the results, but to ensure that the results of the calculations are realistic, it is necessary to implement a code that is used as a pebble chronometer. This code injects a line of marked uid through the inlet. That means, that from the beginning of the simulation, the rst layer of nite elements receives in all of them this marked uid. The solver does not stop calculating until all the marked uid particles have left the core. The use of this technique permits also a better visual comprehension of the grain motion and allows more exibility in the analysis because the complete history of a APPENDIX E. SOLUTION TO THE FLOW PROBLEM 84 single particle is available. To avoid statistical problems, the tracking particles are uniformly injected, controlling the location and their diameter. Figure E.8: Time already traveled by the marked uid while the solver is calculating E.3. Geometry and material characteristics E.3.1. Resume of ANABEK ow experiment In this section a resume of the ANABEK ow experiment [Babilas 1992] is showed. Geometry: Model scale 1:6 HTR-Modul 200 Core diameter 500 mm Core height 1600 mm Outlet diameter 40 mm Bottom angle 30 º Refueling rate 4600 pebbles h −1 . Equivalent to 1.28% Vol recirculated per hour. APPENDIX E. SOLUTION TO THE FLOW PROBLEM 85 Pebbles: Material Polyacetat Number 360000 Pebble density 1.41g/cm3 Friction pebble-wall 40 mm Friction pebble-pebble 0.31-0.4 Refueling rate 0.37-0.51 Figure E.9: Geometry of ANABEK model experiment Colored marked pebbles No cylindrical core, but 24-side polygon Cooling system not represented, no injection of cooling gas Pebble bed is randomly packed, the it is recirculated to achieve a stable structure. Refueling rate is 20 times higher than in real HTR-M 200 core project, no remarkable eects are shown. E.3.2. Resume of HTR-300 HTR-300 reactor was the pebble bed prototype reactor located in Hamm (Germany). Before its construction a series of experiments called HRB [Bedenig 1966] was carried APPENDIX E. SOLUTION TO THE FLOW PROBLEM 86 out to check the ow behavior. The information about that experiment was lost and it is only possible to use the parameters used by [Scherer 1989] in his simulations. Core diameter 5600 mm Core height 5800 mm Outlet diameter 800 mm H/D: 0.81 Bottom angle 30 º Refueling rate 2.4 pebbles min−1 Number of inlets 8 Pebbles TRISO pebble fuel Fuel pebbles and graphite pebbles mixed Figure E.10: Geometry of THTR-300 core reactor E.3.3. Material properties In the operation of pebble bed core reactor thermal resistant materials are required, the most common material in all the devices inside the reactor vessel is graphite, but the composition and properties of every single component is variable, a research done in China [Yu, Lu et al. 2010] shows the properties for dierent nuclear graphite composes. Fuel pebbles are the main object of research in this work. The pebbles are known as TRISO, tristructural-isotropic fuel, and consist on a graphite matrix where small Uranium particles are embedded. APPENDIX E. SOLUTION TO THE FLOW PROBLEM 87 Figure E.11: TRISO fuel pebble [NN 3] E.4. Miscellaneous E.4.1. Residence time distribution calculation method Experiments about pebble ow explained in section C.3 were performed using marked pebbles, also called test pebbles. They were positioned in a thin layer over the core surface, or in dened regions depending on the study goals. Then, they were recirculated and counted as long as they leave the core vessel. When a marked pebble appears must be separately counted. Was developed by [Bedenig 1967] a statistical calculation method called Residence Time Distribution, Verweilspektra-methode in German literature to measure the distribution of the time that pebbles need to travel across the core. To obtain the distribution, it is dened an interval of p pebbles, then marked pebbles in an interval ( ∆TK ) are plotted in function of the total number of pebbles that already left in the same interval. In other words, for discrete packs of pebbles, the number of marked pebbles in those packs are counted, and then represented. It is expressed by the formula: (Vuc) = 1 PTK lim4VU C →0 ∆TK(Vuc) ∆Vuc =1 PTK dTK(Vuc) dVuc (E.1) Where P TK is the sum of test pebbles, TK means Testkugel in German, and dVuc is the dierential volume from which the test pebbles are extracted. In recent studies done by [Niessen 2009] the spectra distribution is summed up and represented in a single curve. This method have some advantages when analyzing APPENDIX F. RESULTS 94 Figure F.4: Vertical velocity for dierent shear stress proles at core height 150 cm In the upper part of the core, the eect that shear stress has over the ow can be observed. In the three overlapped curves in gure F.4 the shear stress value is 0 Pa. Meanwhile the prole dened by a constant value of 4000 Pa shows an already developed velocity prole. In the middle core region the dierent velocity proles are caused by the development of the ow under dierent shear stress patterns. No friction at walls, like with freeslip boundary, causes a completely at velocity prole, meanwhile higher pressures cause more developed proles. APPENDIX F. RESULTS 95 Figure F.5: Vertical velocity for dierent shear stress proles at height 50 cm However, those eects are reduced at transition core-hopper zone. In gure F.6 the velocity proles are very similar independently of the shear stress value. Nevertheless, in the hopper the friction forces are supposed to be lower than in the cylindrical part [Ravenet 1992], but despite that fact, this similar behavior must be produced by the geometry eects of the core. It seems clear that hopper geometry has great importance and conrms the experiments done about the inuence of core geometry [Tingate 1974]. APPENDIX F. RESULTS 96 Figure F.6: Vertical velocity for dierent shear stress proles at height 15 cm Figure F.7: Vertical velocity for dierent shear stress proles at Radius 10 cm APPENDIX F. RESULTS 97 In both gures F.7 and F.8 can be better appreciated the inuence of the shear stress prole and the hopper geometry eects. Free-slip and 4000 Pa stress proles are both constant and origin also very regular velocity proles. Meanwhile variable stress proles create variable velocity proles. This fact conrms the supposition that it is possible to model every shear rate with the proper value of shear stress. As a second consequence, the inuence of the hopper is again very clear. When the pebbles arrive at the transition region cylinder-hopper, the velocity eld converges to the same values. Figure F.8: Vertical velocity for dierent shear stress proles at Radius 24cm Velocity proles in gures F.7 and F.8 show big dierences. In the upper part of the core, the velocities are almost the same, or very similar in case of shear stress proles whose values are low in that part. The 4000 Pa prole shows clearly the dierence between the inuence of friction forces at wall. However, in the case that velocity is measured close to the core center, R=10 cm, the velocity increases when the ow arrives to the transition region, but in the simulations done close to the side reector, the velocity decrease quickly. After the simulation results, it can be concluded that wall boundary condition have the largest inuence over the ow behavior. This fact is helpful because reduces the number of variables that have to be coupled, but also requires of accuracy experimental data in order to obtain a robust solution. APPENDIX F. RESULTS 98 F.3. Results of the ow simulations The ow behavior model must be able to predict the evolution of the streamlines, the residence time distribution and the velocity proles. F.3.1. Streamlines and ow characteristics In case of streamlines, the only valuable experimental data found comes from the experiments done by Bedenig and their visual results [Bedenig 1966]. The boundary conditions chose after the initial simulations done are a relative pressure condition at the inlet and a mass ow condition at the outlet. The other possible conditions and the reason the reject them are in section D.4.1. The material is dened with two dierent values of the viscosity to check the real inuence of that value over the ow. Figure F.9: Streamlines comparison for dierent material viscosities. Relative pressure boundary condition at inlet APPENDIX F. RESULTS 99 Figure F.10: Comparison between THTR-300 model experiment and simulation results for dierent viscosities [Bedenig 1966] In gure F.10 the streamlines obtained from the simulation in gure F.9 are compared directly with the results of the visual experiments performed for the THTR- 300 design in the 1:6 model. The agreement with the experimental results is good. The trajectory of the streamlines obtained in the simulation t properly with the ow of colored pebbles in the experiments. The experimental procedure carried out is explained in section C.3.3. Finally the contour-plot of the vertical velocity is also shown in gure F.11. The zones close to the wall and specially in the transition cylinder-hopper have very low velocities, which can cause the stagnant of the ow. Despite that fact, the velocity never decreases to zero but it is necessary to analyze carefully that region of the core in order to optimize the geometry. APPENDIX F. RESULTS 100 Figure F.11: Vertical velocity contour-plot. Danger of stagnant ow in low velocity regions F.3.2. Residence time distribution Residence time distribution, from now RTD, is the chosen method to study the ow behavior and the velocity elds, it is explained in section E.4.1. The validity of the present model requires that results obtained in the simulation are comparable to the data derived form ANABEK ow experiments [Babilas 1992] which are explained in detail in section C.3.4 . The results are also compared with the the simulations performed with a D.E.M. method [Niessen 2009]. The simulations have been performed for the hopper and the whole core. A geometry and a mesh was built for each one of the parts because it was not possible the injection of tracking particles apart from the inlet. The hopper is individually studied because its inuence over the ow requires a careful verication. Then, the whole core is studied for dierent radial positions. APPENDIX F. RESULTS 101 The goal is to test the inuence of the dierent core regions over the ow, but specially the wall eects over the whole ow. The tracking particle injection positions are represented in gure F.12. Figure F.12: Injection of tracking particles in ANABEK core model F.3.2.1. ANABEK simulations at hopper The simulation of the hopper is carried out injecting a layer of pebbles over the whole surface of the hopper geometry model showed in gure F.12. Meanwhile in the ANABEK experiment, the marked pebbles were located on rings. This dierence in the simulation method could derive in statistical error due to the dierences in the number of particles analyzed for each region. APPENDIX F. RESULTS 102 Figure F.13: RTD in the hopper for dierent wall shear stress proles in the boundary conditions The results obtained in the hopper, gure F.13, show an intermediate level of agreement. The most interesting fact is that all the RTD from the simulations follow the same pattern until the 80% of the spheres have already been extracted. Then, they separate depending on the xed shear stress. Friction forces in that region as previously said, are lower than geometrical inuence, or at least comparable, otherwise the dierence should be signicantly bigger. The change of the ow direction was assessed as the probable geometrical cause for that dierence. The results obtained by Niessen do not match with higher precision. However, they predict better the tendency of the curve. In that region, is probable that dierences between discrete and continuum approach are big from a theoretical point of view, because the discrete method can study the interactions with the wall and other pebbles for each pebble separately, but in case of continuum framework there is an average calculation of the interactions with a loss of accuracy. In all cases, linear shear stress proles are chosen. Nominal pressure value at hopper beginning is 500 Pa and decrease linear until 0 Pa in the outlet. This prole is a coarse approximation to the expected values at hopper. This approximation is acceptable compared to the results of ANABEK stress prole patterns in section E.1.1. F.3.2.2. RTD at core wall At core wall, friction between pebbles and side reector govern the pebble behavior. Figure F.14 shows the eects over the ow for dierent shear proles. The use APPENDIX F. RESULTS 103 of dierent proles was essential to understand the relative weight of this section compared to the rest of the core. Despite the fact that stress proles have similar nominal pressure values, the ow has a completely dierent behavior, therefore an accurate description of the friction forces at the side reector is absolutely necessary. This variable in relative terms, has the highest inuence over the ow. Figure F.14: RTD for dierent shear stress proles For that simulation a 3 cm ring of pebbles in the outer ring is located on the top of the whole core. In case of the experimental procedure, it is supposed that pebbles are in contact with the core reector but there is not more information available. See locations in gure F.12. The simulation results do not match well with the results of the experiments. This fact was a source of problems because in case of choosing a more similar prole, for example in gure F.14 4000 Pa stress prole ts better, the results obtained in the other regions of the core did not match, in gure F.21 it is shown. That fact is explained by the dierent eect of long range forces in continuum and granular ows as it was explained in section D.2. Therefore, long range forces produced by the transmission of energy between pebbles have to been taken into account. Their eect reach about 5 pebble diameters. Appendix G CONCLUSIONS AND OUTLOOK G.1. Conclusions of the pebble bed study The study of a granular ow with a continuum framework requires the construction of a model that takes into account the most number of variables possible. During the development of the present work many assumptions had to be made due to the complexity of the system. However, after testing the most important simplications made, it can be assessed that it does not suppose a large loss of accuracy for the nal development of the model and that it is compensated by a more exible, more simple and faster solution. As a conclusion, the most important facts conrmed during the research are here resumed. The great number of variables which have to be reduced and the inherent dierences between both systems entail a loss of accuracy but do not imply important changes of the ow behavior. The boundary conditions, specially in case of the wall boundary conditions, play the most important role for the proper coupling of both granular and uid systems. In case of the wall boundary conditions, the inuence of friction forces with the core reector dene the main characteristics of the ow behavior and for this reason, the knowledge and comprehension of their physical basis is essential. The inuence of the viscosity was much lower than expected. The inuence over the ow is low compared with the forces over the reector wall and for this reason, a virtual viscosity can be accepted. The saturation of viscosity and therefore the existence of a maximum shear rate has been proved in all the simulations performed. Long range forces reach longer distances in case of viscous uids than in pebble beds and is a negative consequence of the continuum approach. A no Newtonian viscous uid was supposed to solve this problem but no evidences of that were found. In the upper region mass ow was observed, but close to the transition region, the ow evolved to a funnel ow pattern. This fact implies large energy losses of the pebbles due to friction. No stagnant zones were detected, but very low velocity regions appeared in the transition region between core and hopper. Thus, the geometry of that region requires intensive research. 110 APPENDIX G. CONCLUSIONS AND OUTLOOK 111 In case of the physical results it is conrmed that the study of ow lines and the ow regions match with good precision with the experiments performed in the past. The residence time distribution shows high precision for the rst 95% of the pebbles in the simulations performed. Nevertheless, the results for the regions close to the reector wall and hopper are not satisfactory, probably due to the irregularity of the pebble ow and all the diculties that requires the comprehension of complex ows. The velocity proles could not be compared with experimental data, despite of that fact and compared with the ow patterns, a good agreement can be supposed. The possibilities of use in dierent reactor models are successful. Dierent calculations of dierent models were carried out. Nevertheless the lack of experimental data or empirical expressions leave uncertainty in the validity of the solution. There is no a general formulation that can explain all the characteristics and the behavior of a granular ow, and therefore the development of continuum models is necessary in a middle future. The comprehension of the friction forces with the walls and the transmission mechanism of forces between pebbles requires still a lot of eorts. G.2. Outlook For the development of the present work and in order to achieve a better comprehension of pebble beds a whole experiment, in which all the ow characteristics, stress studies and material properties are studied, is required. The inuence of the dierent reactor core regions is still not clear and especially an empirical analysis of the hopper would be also needed for an optimized vessel design. According to the silo theory a precise expression for the horizontal and vertical pressures can be obtained by means of D.E.M. methods. However, experimental procedures should also be performed. 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