DOCTORAL DISSERTATION - Archivo Digital UPM -...

UNIVERSIDAD POLITÉCNICA DE MADRID

ESCUELA TÉCNICA SUPERIOR DE INGENIEROS INDUSTRIALES

Core physics and safety analysis of

Generation-IV Sodium Fast Reactors using

existing and newly developed

computational tools

DOCTORAL DISSERTATION

RAQUEL OCHOA VALERO

Industrial Engineer

by Universidad Politécnica de Madrid

2013

NUCLEAR ENGINEERING DEPARTAMENT

ESCUELA TÉCNICA SUPERIOR DE INGENIEROS INDUSTRIALES

UNIVERSIDAD POLITÉCNICA DE MADRID

Core physics and safety analysis of

Generation-IV Sodium Fast Reactors using

existing and newly developed

computational tools

RAQUEL OCHOA VALERO

Industrial Engineer

by Universidad Politécnica de Madrid

Thesis Director:Dña. Nuria García HerranzAssociate Professor of Nuclear EngineeringUniversidad Politécnica de Madrid

2013

Tribunal nombrado por el Magfco. y Exmo. Sr. Rector de la Universidad Politécnica de

Madrid, el día 2 de Diciembre de 2013.

Presidente: Carolina Ahnert Iglesias

Secretario: Eduardo Gallego Díaz

Vocal: Alfredo Vasile

Vocal: Francisco Martín-Fuertes Hernández-Sonseca

Vocal: Rafael Miró Herrero

Suplente: Sergio Chiva Vicent

Suplente: Federico Puente Espel

Realizado el acto de defensa y lectura de la Tesis el día 10 de Enero de 2013 en la E.T.S.

Ingenieros Industriales.

CALIFICACIÓN:

EL PRESIDENTE LOS VOCALES

EL SECRETARIO

The research leading to this doctoral dissertation has received funding from the EuropeanCommission's Seventh Framework Programme [FP7/2007-2013] under grant agreement#232658 within the Collaborative Project CP-ESFR ``European Sodium Fast Reactor'' andfrom the Cátedra Federico Goded from the Spanish Nuclear Safety Council (Consejo deSeguridad Nuclear).

Acknowledments

Many people have helped me during the development of this work and deserve to bementioned in this section. First of all, I would like to give my most sincere gratitude tomy thesis director, Dr. Nuria García Herranz. For all her support, patient and advice, notonly in the technical matters. Thanks to her, this doctorate dissertation is now a reality.A special recognition is given to Prof. José María Aragonés, who sadly is no longer amongus. He believed in me and thanks to him I enrolled myself into this adventure some yearsago. I would also like to thank Prof. Carol Ahnert and Prof. Eduardo Gallego, for all theirsupport.

I would like to thank the two external reviewers, Dr. Konstantin Mikityuk from PSI and Dr.Laurent Buiron from CEA, who so kindly agreed to review my manuscript, investing theirtime and eorts in providing quality recommendations and improvements to this thesis.They have clearly contributed to increase the quality of this manuscript.

I would also want to acknowledge all the people who have been by my side throughoutall these years. In particular I would like to dedicate this work to my family: my parentsElena and Paco, and brother Javier. They have always supported me and given me thestrength to achieve this goal. I also remember my friends and fellow PhD students at theUPM Nuclear Engineering Department (and Nuclear Fusion Institute), and also the people Imet during my two doctorate internships at CEA Cadarache (France) and KIT (Germany).They all have contributed in cheering my life up while facing the diculties that the thesisbrought. Many of them are today good friends which are worth to keep.

Ahora en español.

Quisiera dar mi más sentido agradecimiento a mi directora de tesis, Dra Nuria GarcíaHerranz. Es todo un ejemplo de profesionalidad, temple y sobre todo calidad humana. Sinsu ayuda y dedicación nada de esto hubiera sido posible. Todo lo que me ha enseñado, yno sólo en el terreno técnico, es inmesurable, y la tesis se la debo a ella.

Quisiera dedicar esta tesis al Prof. José María Aragonés, quien desgraciadamente ya no seencuentra entre nosotros y a quien le dedico esta tesis de corazón allá donde esté. Él conóen mi cuando ni yo misma sabía lo que era capaz de hacer y siempre le estaré agradecida.

Me gustaría acordarme de todas aquellas personas que han estado a mi lado durante estaetapa, aguantándome en los momentos duros y ayudándome siempre a tener una actitudpositiva ante las adversidades.

En especial, querría dedicar esta tesis a mi familia: mis padres Elena y Paco, y mi hermanoJavier. Gracias a ellos soy hoy quien soy y esta tesis es para ellos.

No me olvido de todos mis compañeros y personal del Departamento de Ingeniería Nucleary del Instituto de Fusión Nuclear, quienes han compartido tanto momentos buenos comomalos, y sobretodo han ayudado a hacer que el ir a trabajar cada día sea incluso divertido.Por nuestras conversaciones en el comedor, por las patatas de la casa y por nuestras tardesde mus, que lamentablemente no pudieron formar parte de mi rutina hasta el nal. Vuestratambién es esta tesis. En especial, me gustaría agradecer a Dr. Javier Jiménez, Dr. JuanAndrés Lozano y Dr. Jose Javier Herrero todo lo que me han ayudado durante esta tesis;

todas las horas que han dedicado en ayudarme a entender a nuestro amado COBAYA, aprogramar en fortran, a sobrevivir a los Segmentation Faults. Por todo os estaré siempreagradecida y esta tesis también es para vosotros.

Un reconocimiento quería dar a mi compañero y amigo Gonzalo Jiménez, por abrirme laspuertas a Jóvenes Nucleares en el momento en el más necesitaba tener una perspectivadiferente de mi trabajo. Sin duda, mi involucración en Jóvenes Nucleares ha supuesto elimpulso que me hacía falta para ayudarme a acabar este proyecto. Le dedico esta tesistambién a todos mis compañeros y amigos de la asociación.

A mis amigos y amigas, tanto del colegio, como de la universidad, por haberme acompañadodurante este viaje. Ellos saben realmente lo duro que ha sido y han estado ahí cuando loshe necesitado.

Many thanks / Gracias a todos.

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Resumen

El futuro de la energía nuclear de sión dependerá, entre otros factores, de la capacidadque las nuevas tecnologías demuestren para solventar los principales retos a largo plazoque se plantean. Los principales retos se pueden resumir en los siguientes aspectos: lacapacidad de proporcionar una solución nal, segura y able a los residuos radiactivos;así como dar solución a la limitación de recursos naturales necesarios para alimentar losreactores nucleares; y por último, una mejora robusta en la seguridad de las centrales que endenitiva evite cualquier daño potencial tanto en la población como en el medio ambientecomo consecuencia de cualquier escenario imaginable o más allá de lo imaginable. Siguiendoestas motivaciones, la Generación IV de reactores nucleares surge con el compromiso deproporcionar electricidad de forma sostenible, segura, económica y evitando la proliferaciónde material sible. Entre los sistemas conceptuales que se consideran para la Gen IV, losreactores rápidos destacan por su capacidad potencial de transmutar actínidos a la vez quepermiten una utilización óptima de los recursos naturales. Entre los refrigerantes que seplantean, el sodio parece una de las soluciones más prometedoras.

Como consecuencia, esta tesis surgió dentro del marco del proyecto europeo CP-ESFRcon el principal objetivo de evaluar la física de núcleo y seguridad de los reactores rápidosrefrigerados por sodio, al tiempo que se desarrollaron herramientas apropiadas para dichosanálisis.

Efectivamente, en una primera parte de la tesis, se abarca el estudio de la física del nú-cleo de un reactor rápido representativo, incluyendo el análisis detallado de la capacidad detransmutar actínidos minoritarios. Como resultado de dichos análisis, se publicó un artículoen la revista Annals of Nuclear Energy [105]. Por otra parte, a través de un análisis deun hipotético escenario nuclear español, se evalúo la disponibilidad de recursos naturalesnecesarios en el caso particular de España para alimentar una ota especíca de reacto-res rápidos, siguiendo varios escenarios de demanda, y teniendo en cuenta la capacidadde reproducción de plutonio que tienen estos sistemas. Como resultado de este trabajotambién surgió una publicación en otra revista cientíca de prestigio internacional como esEnergy Conversion and Management [108]. Con objeto de realizar esos y otros análisis, sedesarrollaron diversos modelos del núcleo del ESFR siguiendo varias conguraciones, y paradiferentes códigos.

Por otro lado, con objeto de poder analizar la seguridad en reactores rápidos, son necesariasherramientas multidimensionales de alta delidad especícas para reactores rápidos. Dichasherramientas deben integrar fenómenos relacionados con la neutrónica y con la termo-hidráulica, entre otros, mediante una aproximación multi-física. Siguiendo este objetivo,

iii

se evalúo el código de difusión neutrónica ANDES para su aplicación a reactores rápidos.ANDES es un código de resolución nodal que se encuentra implementado dentro del sistemaCOBAYA3 y está basado en el método ACMFD. Por lo tanto, el método ACMFD fuesometido a una revisión en profundidad para evaluar su aptitud para la aplicación a reactoresrápidos. Durante ese proceso, se identicaron determinadas limitaciones que se discuten alo largo de este trabajo, junto con los desarrollos que se han elaborado e implementadopara la resolución de dichas dicultades. Por otra parte, se desarrolló satisfactoriamente elacoplamiento del código neutrónico ANDES con un código termo-hidráulico de subcanalesllamado SUBCHANFLOW, desarrollado recientemente en el KIT. Como conclusión de estaparte, se presenta la evaluación y vericación de los desarrollos implementados.

En paralelo con esos desarrollos, se calcularon para el núcleo del ESFR las secciones ecacesen multigrupos homogeneizadas a nivel nodal, así como otros parámetros neutrónicos, me-diante los códigos ERANOS, primero, y SERPENT, después. Dichos parámetros se utilizaronmás adelante para realizar cálculos estacionarios con ANDES. Además, como consecuenciade la contribución de la UPM al paquete de seguridad del proyecto CP-ESFR, se calcu-laron mediante el código SERPENT los parámetros de cinética puntual que se necesitanintroducir en los típicos códigos termo-hidráulicos de planta, para estudios de seguridad. Enconcreto, dichos parámetros sirvieron para el análisis del impacto que tienen los actínidosminoritarios en el comportamiento de transitorios. Además, en la línea de los parámetroscinéticos, se efectúo un análisis colateral del impacto de las incertidumbres en las seccionesecaces sobre los coecientes de reactividad.

Concluyendo, la tesis presenta una aproximación sistemática y multidisciplinar aplicada alanálisis de seguridad y comportamiento neutrónico de los reactores rápidos de sodio de laGen-IV, usando herramientas de cálculo existentes y recién desarrolladas ad' hoc para talaplicación. Se ha empleado una cantidad importante de tiempo en identicar limitacionesde los métodos nodales analíticos en su aplicación en multigrupos a reactores rápidos, y seproponen interesantes soluciones para abordarlas.

iv

Abstract

The future of nuclear reactors will depend, among other aspects, on the capability to solvethe long-term challenges linked to this technology. These are the capability to provide adenite, safe and reliable solution to the nuclear wastes; the limitation of natural resources,needed to fuel the reactors; and last but not least, the improved safety, which would avoidany potential damage on the public and or environment as a consequence of any imaginableand beyond imaginable circumstance. Following these motivations, the IV Generation ofnuclear reactors arises, with the aim to provide sustainable, safe, economic and proliferation-resistant electricity. Among the systems considered for the Gen IV, fast reactors have arepresentative role thanks to their potential capacity to transmute actinides together withthe optimal usage of natural resources, being the sodium fast reactors the most promisingconcept.

As a consequence, this thesis was born in the framework of the CP-ESFR project with thegeneric aim of evaluating the core physics and safety of sodium fast reactors, as well as thedevelopment of the appropriated tools to perform such analyses.

Indeed, in a rst part of this thesis work, the main core physics of the representative sodiumfast reactor are assessed, including a detailed analysis of the capability to transmute minoractinides. A part of the results obtained have been published in Annals of Nuclear Energy[105]. Moreover, by means of the analysis of a hypothetical Spanish nuclear scenario,the availability of natural resources required to deploy a specic eet of fast reactor isassessed, taking into account the breeding properties of such systems. This work also ledto a publication in Energy Conversion and Management [108]. In order to perform thoseand other analyses, several models of the ESFR core were created for dierent codes.

On the other hand, in order to perform safety studies of sodium fast reactors, high delitymultidimensional analysis tools for sodium fast reactors are required. Such tools shouldintegrate neutronic and thermal-hydraulic phenomena in a multi-physics approach. Follow-ing this motivation, the neutron diusion code ANDES is assessed for sodium fast reactorapplications. ANDES is the nodal solver implemented inside the multigroup pin-by-pin dif-fusion COBAYA3 code, and is based on the analytical method ACMFD. Thus, the ACMFDwas veried for SFR applications and while doing so, some limitations were encountered,which are discussed throughout this work. In order to solve those, some new developmentsare proposed and implemented in ANDES. Moreover, the code was satisfactorily coupledwith the subchannel thermal-hydraulic code SUBCHANFLOW, recently developed at KIT.Finally, the verication of the dierent implementations performed is presented.

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In addition to those developments, the node homogenized multigroup cross sections andother neutron parameters were obtained for the ESFR core using ERANOS and SERPENTcodes, to be employed afterwards by ANDES to perform steady state calculations. More-over, as a result of the UPM contribution to the safety package of the CP-ESFR project,the point kinetic parameters required by the typical plant thermal-hydraulic codes werecomputed for the ESFR core using SERPENT, which nal aim was the assessment of theimpact of minor actinides in transient behaviour. In addition, the eect of uncertainties onthe nuclear data over the typical reactivity eects of a Sodium Fast Reactor was analysed.

All in all, the thesis provides a systematic and multi-purpose approach applied to theassessment of safety and performance parameters of Generation-IV SFR, using existingand newly developed analytical tools. An important amount of time was employed inidentifying the limitations that the analytical nodal diusion methods present when appliedto fast reactors following a multigroup approach, and interesting solutions are proposed inorder to overcome them.

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Contents

Lista de guras xix

Lista de tablas xxiii

I INTRODUCTION 1

1 Introduction 3

1.1 Thesis Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Objectives and original contributions . . . . . . . . . . . . . . . . . . . . . 4

1.3 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Sodium Fast Reactors 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Historical context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Basic Fast Reactor Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Flux Spectrum and Slowing-Down Properties . . . . . . . . . . . . . 18

2.3.2 Breeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.3 Actinide Transmutation Capability . . . . . . . . . . . . . . . . . . . 20

2.3.4 Reactivity Feedback Eects . . . . . . . . . . . . . . . . . . . . . . 22

2.3.4.1 Doppler coecient . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.4.2 Sodium void reactivity feedback . . . . . . . . . . . . . . . . 23

2.3.5 Thermal expansion reactivity coecients . . . . . . . . . . . . . . . 24

2.3.5.1 Steel cladding expansion . . . . . . . . . . . . . . . . . . . 26

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2.3.5.2 Steel wrapper expansion . . . . . . . . . . . . . . . . . . . . 27

2.3.5.3 Fuel expansion . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.5.4 Diagrid expansion eect . . . . . . . . . . . . . . . . . . . 27

2.3.5.5 Vessel expansion . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.5.6 Strongback expansion . . . . . . . . . . . . . . . . . . . . . 28

2.3.6 Sodium properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.1 Safety events in Sodium Fast Reactors . . . . . . . . . . . . . . . . . 31

II REVIEW OF SIMULATION TOOLS FOR FAST REAC-TORS 33

3 Multiphysics approach 35

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Neutronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.2 Deterministic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.2.1 Transport solution . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.2.2 Diusion approach . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Neutron kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Cross section Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5 Depletion and Fuel Cycle Analysis . . . . . . . . . . . . . . . . . . . . . . . 42

3.6 Perturbation Theory and Uncertainty Analysis . . . . . . . . . . . . . . . . 43

3.7 Thermal-hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.7.1 Core thermal-hydraulics . . . . . . . . . . . . . . . . . . . . . . . . 45

3.7.2 Plant thermal-hydraulics . . . . . . . . . . . . . . . . . . . . . . . . 45

3.8 Thermal-mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Codes for 3D Sodium Fast Reactor simulation 47

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Neutronic Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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4.2.1 Monte Carlo transport methods . . . . . . . . . . . . . . . . . . . . 47

4.2.1.1 MCNP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1.2 SERPENT . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.1.3 EVOLCODE . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.1.4 KENO-VI . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.2 Deterministic methods . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.2.1 ERANOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.2.2 PARCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2.2.3 DYN3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Thermal-hydraulics codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.1 Core thermal-hydraulics codes . . . . . . . . . . . . . . . . . . . . . 65

4.3.1.1 SUBCHANFLOW . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.2 Plant thermal-hydraulics codes . . . . . . . . . . . . . . . . . . . . . 68

4.3.2.1 SPECTRA . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

III DEVELOPMENT OF ESFR CORE MODELS 71

5 Core models 73

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 Description of core congurations . . . . . . . . . . . . . . . . . . . . . . . 73

5.2.1 Working Horses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2.1.1 Minor Actinides loading patterns . . . . . . . . . . . . . . . 75

5.2.2 Optimized core: CONF2 . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2.2.1 CONF2-HET2 core . . . . . . . . . . . . . . . . . . . . . . 76

5.2.2.2 CONF2-HOM4 core . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Code models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3.1 MCNP5 & MCNPX . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3.1.1 Minor Actinides loading patterns . . . . . . . . . . . . . . . 80

5.3.2 ERANOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3.3 SERPENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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5.3.3.1 Working Horses core model . . . . . . . . . . . . . . . . . . 83

5.3.3.2 Optimized case: CONF2 core . . . . . . . . . . . . . . . . . 84

5.3.3.3 Lower transmutation case: CONF2-HET2 . . . . . . . . . . 84

5.3.3.4 Upper transmutation case: CONF2-HOM4 . . . . . . . . . . 84

5.3.4 KENO-VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

IV ASSESSMENT OF ESFR PERFORMANCES 91

6 Assessment of Minor Actinides Transmutation 93

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2 MA loading patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3 Results for the Optimized Core . . . . . . . . . . . . . . . . . . . . . . . . 96

6.3.1 Evolution of Pu and MA . . . . . . . . . . . . . . . . . . . . . . . . 96

6.3.2 ke comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3.3 Reactivity coecients . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3.4 Linear power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.4 Comparison of SERPENT and EVOLCODE . . . . . . . . . . . . . . . . . 102

6.5 Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7 Spanish Scenario of the Deployment of ESFR-type reactors 111

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.2 Reactor Technology Description Sodium-cooled Fast Reactors . . . . . . . 113

7.2.1 SFR core design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.3 Methodology and codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.4 Spanish scenario hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.4.1 Spanish nuclear park and natural resources . . . . . . . . . . . . . . 116

7.5 Results and analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.5.1 Electricity production . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.5.1.1 Electricity demand . . . . . . . . . . . . . . . . . . . . . . . 119

7.5.1.2 Thermal eciency . . . . . . . . . . . . . . . . . . . . . . . 121

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7.5.1.3 Nuclear electricity production share . . . . . . . . . . . . . . 121

7.5.2 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.5.2.1 Uranium-238 . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.5.2.2 Plutonium-239 . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.5.3 Waste reduction by transmutation . . . . . . . . . . . . . . . . . . . 127

7.6 Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8 Generation of Point Kinetic parameters 131

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.2 Calculation of point kinetic parameters . . . . . . . . . . . . . . . . . . . . 131

8.2.1 Delayed neutron fraction, decay constants and mean generation time 132

8.2.2 Doppler constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

8.2.3 Coolant reactivity coecient . . . . . . . . . . . . . . . . . . . . . 133

8.2.4 Expansion reactivity coecients . . . . . . . . . . . . . . . . . . . . 133

8.2.4.1 Steel cladding (ODS) expansion coecient . . . . . . . . . . 134

8.2.4.2 Steel wrapper expansion . . . . . . . . . . . . . . . . . . . 135

8.2.4.3 Fuel expansion . . . . . . . . . . . . . . . . . . . . . . . . 135

8.2.4.4 Diagrid eect . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.2.5 Assessment of linearity and superposition assumptions . . . . . . . . 137

8.3 Assessment of Minor Actinides Impact on Transients . . . . . . . . . . . . . 139

8.3.1 Point kinetic parameters . . . . . . . . . . . . . . . . . . . . . . . . 140

8.3.2 Transient analysis with SPECTRA . . . . . . . . . . . . . . . . . . . 142

8.4 Assessment of the impact of uncertainties of the nuclear data on the reactivitycoecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.5 Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9 Nodal Cross sections 149

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

9.2 Cross section calculation using SERPENT . . . . . . . . . . . . . . . . . . 150

9.2.1 Choice of universes . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

9.2.1.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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Contents

9.2.2 Group constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9.3 Cross section calculation using ERANOS . . . . . . . . . . . . . . . . . . 155

9.3.1 Running ECCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

9.3.2 How to Calculate the required parameters with ERANOS . . . . . . . 157

9.4 Comparison between ERANOS and SERPENT . . . . . . . . . . . . . . . . 158

9.4.1 Main dierence between ERANOS and SERPENT . . . . . . . . . . 158

9.4.2 Comparison of cross section libraries . . . . . . . . . . . . . . . . . . 159

9.5 Feedback parametrized cross sections for transient analysis . . . . . . . . . 164

9.6 Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

V DEVELOPMENT OF A N-TH CODE FOR SFR 167

10COBAYA3-ANDES 169

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

10.2 COBAYA3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

10.2.1 Introduction to the ACMFD method . . . . . . . . . . . . . . . . . 170

10.2.2 Step 1: Transverse leakage integration of the 3D diusion equation intriangular prismatic nodes . . . . . . . . . . . . . . . . . . . . . . . 172

10.2.3 Step 2: Analytic solution of the transverse-integrated one-dimensionalequation at radial interfaces in 2D triangular nodes . . . . . . . . . . 175

10.2.4 Step 3: Calculating uxes at corners in 2D trangular nodes . . . . . 177

10.2.5 Step 4: Integration of the corners uxes in the ACMFD relation . . . 178

10.2.6 Extension to 3D (triangular-Z geometry) . . . . . . . . . . . . . . . 179

10.2.6.1The ACMFD relation for axial interfaces . . . . . . . . . . . 180

10.2.6.2Modications of the ACMFD relation for radial interfaces in3D nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

10.2.7 The ACMFD method for kinetics problems . . . . . . . . . . . . . . 181

10.3 Assessment of ACMFD method to be applied to Fast Reactors . . . . . . . 183

10.3.1 Group structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

10.3.2 Transverse leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

11Developments in COBAYA3 191

xii

Contents

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

11.2 Corrected Coarse Mesh Finite Dierence method . . . . . . . . . . . . . . . 192

11.2.1 Theoretical base . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

11.2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

11.2.3 Additional implementations for hexagonal geometry . . . . . . . . . . 195

11.3 Alternative methods to avoid diagonalization . . . . . . . . . . . . . . . . . 198

11.3.1 First approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

11.3.2 Schur decomposition method . . . . . . . . . . . . . . . . . . . . . 199

11.3.2.1Theoretical background . . . . . . . . . . . . . . . . . . . . 200

11.3.2.2Application of Schur to the diusion equation . . . . . . . . 200

11.3.2.3Analytical solution of the modal equations in Cartesian geom-etry 1D without external source . . . . . . . . . . . . . . . . 201

11.3.2.4ACMFD method for Cartesian 3D Problems with transverseintegration using the Schur decomposition . . . . . . . . . . 203

11.3.2.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . 205

11.3.2.6Verication and rst tests . . . . . . . . . . . . . . . . . . . 205

11.3.3 Future solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

11.3.3.1 Shur methodology with identical eigenvalues . . . . . . . . . 206

11.3.3.2Alternate collapsing of the group structure . . . . . . . . . . 207

12Coupling between COBAYA3/ANDES and SUBCHANFLOW 209

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

12.2 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

12.3 Description of the coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 210

12.3.1 Spatial coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

12.3.2 Implementation of new subroutines . . . . . . . . . . . . . . . . . . 211

12.3.3Description of the iterative schemes for steady state and transients . . 213

12.3.3.1 Steady State Coupling . . . . . . . . . . . . . . . . . . . . . 213

12.3.3.2Transient Coupling . . . . . . . . . . . . . . . . . . . . . . . 214

12.4 New developments for the extension to Sodium Fast Reactors . . . . . . . . 215

12.4.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

xiii

Contents

12.4.2 Coolant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

12.4.3New materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

12.4.4 Feedback parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 216

12.4.5 Cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

13Verication and application of the performed developments 219

13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

13.2 Verication of the corrected CMFD method for thermal reactors . . . . . . 219

13.2.1 Cartesian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

13.2.2Hexagonal geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 221

13.3 Verication of the corrected CMFD method for fast reactors . . . . . . . . 221

13.4 Verication of the coupling ANDES / SUBCHANFLOW for thermal reactors 226

13.4.1OECD/NEA and U.S. NRC PWR MOX/UO2 Core Transient Benchmark226

13.4.2 The OECD/NEA PWR Main Steam Line Break (MSLB) Benchmark . 227

13.4.3 V1000CT1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

13.5 Verication of the corrected CMFD with the coupling . . . . . . . . . . . . 236

13.6 Application of the coupling to SFR . . . . . . . . . . . . . . . . . . . . . . 240

13.6.1 Steady states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

13.6.2 Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

VI CONCLUSIONS 245

Bibliografía 263

xiv

List of Figures

2.1 Fission and capture cross sections of U-238 . . . . . . . . . . . . . . . . . . 12

2.2 Fission and capture cross sections for Am-241 . . . . . . . . . . . . . . . . 13

2.3 EBR-1 reactor iluminating four bulbs . . . . . . . . . . . . . . . . . . . . 13

2.4 Fast reactors list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Evolution of Nuclear Power . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Designs considered by the GIF . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 Neutron spectra[142] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.8 Neutron yield [96] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.9 Doppler broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.10 Reactivity worth as a consequence of sodium void . . . . . . . . . . . . . . 24

2.11 Sodim void eect in terms of spectrum[96] . . . . . . . . . . . . . . . . . . 25

2.12 Coolant operational ranges and safety margins [96] . . . . . . . . . . . . . 25

2.13Main elements subject of thermal expansions[96] . . . . . . . . . . . . . . . 26

2.14 Scheme of vessel expansion [96] . . . . . . . . . . . . . . . . . . . . . . . . 28

2.15 Scheme of strongback expansion [96] . . . . . . . . . . . . . . . . . . . . . 29

3.1 Diagram of the multiphysics approach . . . . . . . . . . . . . . . . . . . . 36

4.1 Coupling scheme between MCNPX and CINDER90 . . . . . . . . . . . . . 50

4.2 Depletion scheme followed by SERPENT . . . . . . . . . . . . . . . . . . . 51

4.3 Depletion scheme followed by EVOLCODE . . . . . . . . . . . . . . . . . . 53

4.4 Description of ERANOS system . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5 Geometry and Boundary Conditions and Coordinates for TPEN . . . . . . . 58

xv

List of Figures

4.6 Hexagon with the considered directions es,i and ec,i . . . . . . . . . . . . . 63

5.1 MCNP model of WH core. Radial cross section . . . . . . . . . . . . . . . 74

5.2 MCNP model of WH core. Axial cross section . . . . . . . . . . . . . . . . 74

5.3 Cross section view of the optimized ESFR core, SERPENT post-processing . 77

5.4 Vertical view of the optimized ESFR core, SERPENT post-processing . . . . 77

5.5 Axial structure of the CONF-2 conguration . . . . . . . . . . . . . . . . . 78

5.6 Radial layout of the HET4 core . . . . . . . . . . . . . . . . . . . . . . . . 82

5.7 SERPENT model of the ESFR-WH core . . . . . . . . . . . . . . . . . . . 83

5.8 SERPENT model of assembly of ESFR-WH core (radial and axial view) . . . 83

5.9 Radial section at the level of the active core and axial section of the SERPENTmodel for the CONF-2 conguration . . . . . . . . . . . . . . . . . . . . . 85

5.10 Radial section at the level of the active core and axial section of the SERPENTmodel for the CONF2-HET2 conguration . . . . . . . . . . . . . . . . . . 86

5.11 Radial section at the level of the active core and axial section of the SERPENTmodel for the CONF2-HOM4 conguration . . . . . . . . . . . . . . . . . 87

5.12 3D assembly model with KENO-VI . . . . . . . . . . . . . . . . . . . . . . 88

5.13 KENO-VI model for the 3D whole core model. Radial layout . . . . . . . . 89

5.14 KENO-VI model for the 3D whole core model. Axial layout . . . . . . . . . 90

6.1 Mass balance (kg) for Pu isotopes at EOL (EOL-BOL) . . . . . . . . . . . 99

6.2 Total Minor Actinides evolution . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3 ke evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.4 Axial power distribution in the HOM4, CONF2 and HET2 models at eachtime step from SERPENT . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.5 Am-241 Capture Branching Ratio . . . . . . . . . . . . . . . . . . . . . . . 105

6.6 ke comparison in HOM4 case, SERPENT and EVOLCODE . . . . . . . . 107

6.7 Total mass dierence between SERPENT and EVOLCODE for 239Pu and238U (HOM4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.8 Flux used for depletion in EVOLCODE and SERPENT in the rst burnupstep (0-410 EFPD). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.1 Historical Series of Electricity Production . . . . . . . . . . . . . . . . . . . 115

xvi

List of Figures

7.2 Power Demand evolution according to the postulated trends . . . . . . . . . 116

7.3 Power covered by the dierent technologies of NPP . . . . . . . . . . . . . 119

7.4 Estimated reactor eet requirements for each postulated electricity demandtrends. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.5 Estimated reactor eet requirements for each postulated thermal eciency . 121

7.6 Estimated reactor eet requirements for each postulated nuclear electricityshare. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.7 Estimated reactor eet required for each postulated scenario. . . . . . . . . 122

7.8 Estimated power produced for the A5 scenario. . . . . . . . . . . . . . . . . 126

7.9 Estimated power produced for the A3 scenario. . . . . . . . . . . . . . . . . 127

7.10Main MA isotopes' masses at BOL and EOL. . . . . . . . . . . . . . . . . 128

7.11 Comparison of the decay heat of the main MA being burned in the reactor ornot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.12MA content in the wasted fuel . . . . . . . . . . . . . . . . . . . . . . . . 129

8.1 Ke vs Na density for a ESFR fuel assembly . . . . . . . . . . . . . . . . . 137

8.2 Ke vs fuel density for a ESFR fuel assembly . . . . . . . . . . . . . . . . . 138

8.3 Ke vs ODS steel density for a ESFR fuel assembly . . . . . . . . . . . . . 138

8.4 UTOP transient. Relative Power (%), cladding maximum temperature, HotChannel exit temperature and Reactivity . . . . . . . . . . . . . . . . . . . 142

8.5 UTOP transient. Contributions to reactivity for the CONF2 (left) and CONF2-HOM4 (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.6 Sensitivity of nominal state . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8.7 Sensitivity of heated state . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8.8 Sensitivity of voided state . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9.1 ESFR core with selected universes . . . . . . . . . . . . . . . . . . . . . . 153

10.1 Triangular-z node, ACMFD . . . . . . . . . . . . . . . . . . . . . . . . . . 173

10.2 Components of the modal current in the XY plane . . . . . . . . . . . . . . 175

10.3 Total and elastic scattering cross section of Na-23 and He-4 . . . . . . . . . 186

11.1 Detail of subroutines scheme with ACMFD method . . . . . . . . . . . . . 196

xvii

List of Figures

11.2 Detail of subroutines scheme with CMFD method with two-node correction . 197

12.1Multi-physics N-TH coupling through common memory . . . . . . . . . . . 212

12.2 General iteration scheme for the COBAYA3-SUBCHANFLOW coupling cal-culations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

12.3 Temporal coupling scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 215

12.4 General algorithm for coupling calculations in COBAYA3 . . . . . . . . . . . 215

13.1 Radial Power peaking factors Case 1 . . . . . . . . . . . . . . . . . . . . . 223





13.6 Power vs. time (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

13.7 Reactivity ($) vs. time(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

13.8 Average Doppler Temperature (°C) vs. time (s) . . . . . . . . . . . . . . . 229

13.9 Average core density (kg/m3) vs. time (s) . . . . . . . . . . . . . . . . . . 229

13.10Total Power (MW) vs. time (s) . . . . . . . . . . . . . . . . . . . . . . . 230

13.11Reactivity ($) vs. time(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

13.12Average Doppler Temperature (K) vs. time (s) . . . . . . . . . . . . . . . 231

13.13Maximum Doppler Temperature (K) vs. time(s) . . . . . . . . . . . . . . . 232

13.14Average core density (kg/m3) vs. time (s) . . . . . . . . . . . . . . . . . . 232

13.15Total Power (MW) vs. time (s) . . . . . . . . . . . . . . . . . . . . . . . 233

13.16Average Doppler Temperature (K) vs. time (s) . . . . . . . . . . . . . . . 234

13.17Maximum Doppler Temperature (K) vs. time(s) . . . . . . . . . . . . . . . 234

13.18Average core density (kg/m3) vs. time (s) . . . . . . . . . . . . . . . . . . 235

13.19Power change (MW) vs. time (s) . . . . . . . . . . . . . . . . . . . . . . 237

13.20Average fuel temperature change (K) vs. time (s) . . . . . . . . . . . . . . 237

13.21Average core density change (kg/m3) vs. time (s) . . . . . . . . . . . . . . 238

13.22Power vs. time (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

13.23Reactivity ($) vs. time(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

xviii

List of Figures

13.24Average Doppler Temperature (°C) vs. time (s) . . . . . . . . . . . . . . . 239

13.25Average core density (kg/m3) vs. time (s) . . . . . . . . . . . . . . . . . 240

13.26Detail of the CAs to be withdrawn . . . . . . . . . . . . . . . . . . . . . . 241

13.27Average Doppler Temperature (ºC) . . . . . . . . . . . . . . . . . . . . . 242

13.28Average density (kg/m3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

xix

List of Figures

xx

List of Tables

2.1 Actinides Fission-to-Absorption Ratios in Thermal (PWR) and Fast (SFR)Systems [142] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.1 Depleted Uranium vector (%wt) . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Plutonium vector (%wt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3 Minor Actinide vector (%wt) . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.4 MA loading cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.5 Weight-percentage contents in the fuel for the CONF2 conguration . . . . 78

5.6 Weight-percentage contents in the fuel for the CONF2-HET2 conguration . 79

5.7 Weight-percentage contents in the fuel for the CONF2-HOM4 conguration 81

6.1 Transmutation performances . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.2 Reactivity coecients at pseudo-BOC (820 EFPD) . . . . . . . . . . . . . 101

6.3 Reactivity coecients at pseudo-EOC (1230 EFPD) . . . . . . . . . . . . . 101

6.4 Computational characteristics . . . . . . . . . . . . . . . . . . . . . . . . 104

6.5 Percentage relative errors [(SERPENT- EVOLCODE)/EVOLCODE] in theprediction of isotopes masses along irradiation, HOM4 case . . . . . . . . . 106

6.6 Percentage relative errors [(EVOLCODE(ne time step)- EVOLCODE(coarsetime step))/EVOLCODE(coarse time step)] in the prediction of isotopesmasses along irradiation, HOM4 case. . . . . . . . . . . . . . . . . . . . . 108

7.1 Spanish nuclear eet characteristics and hypothetical operational lives . . . . 117

7.2 Equivalence between Gen-II and Gen-III eet . . . . . . . . . . . . . . . . . 118

7.3 Description of considered scenarios . . . . . . . . . . . . . . . . . . . . . . 120

7.4 Total required eet for each postulated scenario . . . . . . . . . . . . . . . 123

xxi

List of Tables

7.5 U-238 (ton) available and max. eet according to each scenario (B1B6). . 124

7.6 Pu-239 (ton) available and start-up reactors. . . . . . . . . . . . . . . . . . 125

7.7 Description of postulated breeding scenarios . . . . . . . . . . . . . . . . . 125

8.1 Comparison of the errors obtained from the simulated and calculated valuefor the Best Case Scenario (BCS) and Worst Case Scenario (WCS) . . . . . 137

8.2 Reactivity coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.3 Delay neutron constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.4 Mean life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.5 Uncertainty contributors at nominal state . . . . . . . . . . . . . . . . . . . 144

8.6 Main contributors to Doppler heating reactivity uncertainty . . . . . . . . . 145

8.7 Main contributors to Na voiding reactivity uncertainty . . . . . . . . . . . . 145

9.1 Parameters required for diusion calculations . . . . . . . . . . . . . . . . . 150

9.2 Multigroup constants calculated by SERPENT . . . . . . . . . . . . . . . . 152

9.3 Qualitative comparison between capabilities of ERANOS and SERPENT . . 158

9.4 Relative error between SERPENT and ERANOS for Diusion coecient forssile nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

9.5 Relative error between SERPENT and ERANOS for Diusion coecient fornon-ssile nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

9.6 Relative error between SERPENT and ERANOS for Absorption cross sectionfor the ssile nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

9.7 Relative error between SERPENT and ERANOS for Absorption cross sectionfor non-ssile nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

9.8 Relative error between SERPENT and ERANOS for Fission cross section . . 163

9.9 Parameters required for diusion calculations . . . . . . . . . . . . . . . . . 164

10.1 Conditioning numbers of A and R . . . . . . . . . . . . . . . . . . . . . . . 185

10.2 Conditioning numbers of R in 33 and 15 groups . . . . . . . . . . . . . . . 187

13.1 Verication of the corrected CMFD method for a Cartesian PWR minicore . 220

13.2 Verication of the corrected CMFD method for a hexagonal core . . . . . . 221

13.3 Case denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

xxii

List of Tables

13.4 K-e comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

13.5 Reactivity coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

13.6 Steady State HFP results for the MOX benchmark . . . . . . . . . . . . . . 227

13.7 Steady State HFP results for the heterogeneous inlet B.C . . . . . . . . . . 230

13.8 Steady State denition for the V1000CT1 benchmark . . . . . . . . . . . . 235

13.9 Steady State HZP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

13.10Steady State HP results for the heterogeneous inlet B.C . . . . . . . . . . 236

13.11Steady state for the ESFR core . . . . . . . . . . . . . . . . . . . . . . . . 241

xxiii

Part I

INTRODUCTION

1

Chapter 1

Introduction

1.1 Thesis Origin

Given the present situation of the public opinion about Nuclear Energy, the need to solvethe three main issues regarding this technology, i.e. waste management, availability ofresources and nuclear safety, is more important than ever. These aspects are particularlytaken into account for the 4th Generation of Nuclear Reactors whose systems are beingnowadays under development. In this sense, fast reactors arise as promising concepts dueto the potential capability to transmute minor actinides (Np, Am, Cm) in conjunction withtheir optimal use of natural resources.

Indeed, fast reactors are able to transmute some minor actinides, i.e. reduce the inventoryof some of the nuclides whose contribution to the long term radiotoxicity is very high.Moreover, fast reactors are able to be fueled with depleted uranium, i.e. the extremelyhigh potential of uranium in form of U238 that nowadays is stored as nuclear wastes canbe actually employed to generate energy. But of course, safety remains the maximumpriority. Therefore intense studies have to be done in order to ensure the safety in anypostulated scenario and even beyond imaginable. Following this motivation, high delitymultidimensional analysis tools for sodium fast reactors are required for both core designcharacterization and safety assessment. Such tools should integrate dierent phenomena,such as neutronics and thermal-hydraulics, in a multi-physics approach.

There are several European Collaborative Projects created under the 7th Framework Pro-gram of European Commission to study the main challenges of the 4th Generation tech-nologies. In particular, the European Sodium Fast Reactor (ESFR) was born in order toassess the previously mentioned features, as well to overcome the main challenges relatedto that technology. It is in the framework of that European project where this thesis wasdeveloped.

3

1.2. Objectives and original contributions

1.2 Objectives and original contributions

The nal goal of the thesis work is the assessment of the core physics and safety for theESFR core, exploring the dierent phenomena integrated.

Within this global objective, 3 dierent areas have been explored:

1. Core physic analyses for the ESFR core. A broad study of the behaviour of ESFR hasbeen done regarding transmutation capabilities, resources utilization and neutronicsrelated safety behaviour using Monte Carlo transport codes.

2. Development of a methodology to generate core safety parameters in the form of:

(a) Reactivity coecients and kinetic parameters for 0D transient analysis

(b) Multigroup nodal cross section libraries for diusion-corrected calculations suit-able for 3D transient analysis.

3. Development of a 3D nodal multi-group diusion solver for SFR. In order to meet thisgoal, rstly the COBAYA3/ANDES system, initially developed for thermal reactors,was assessed for its application to SFR. Important conclusions are drawn in this regard,highlighting the limitions encountered in the code. As a consequence, a signicantamount of work was devoted to the modication of the 3D nodal multi-group diusionsolver ANDES in order to overcome those diculties for fast reactors. Finally, thecode system was coupled with a SUBCHANFLOW subchannel thermalhydraulic codein order to enable the study of coupled steady-state and transient exercises.

Original contributions

• Development of a model of the ESFR core for dierent codes, and following dierentcongurations.

• Assessment of the transmutation capabilities of the ESFR core and comparison ofburnup capabilities between SERPENT and EVOLCODE.

• Assessment of a Spanish scenario of SFR deployment exploring natural resourcesavailability and some transmutation considerations.

• Methodology to compute point kinetics parameters using SERPENT, which would al-low to study transient behaviour using plant thermal-hydraulic codes, such as SPEC-TRA code.

• Methodology to compute nodal homogenized cross sections using SERPENT for theESFR core.

• Model of the ESFR assembly and core with KENO-VI, which allow to study theimpact of uncertainties in nuclear data on fast reactor reactivity coecients, usingthe TSUNAMI module of the SCALE system.

4

1.3. Thesis structure

• Assessment of the nodal solver ANDES within COBAYA3 for its applicability to SFRs.Diagnosis of limitations.

• Development and implementation of an alternative method to the ACMFD method,implemented in ANDES: the corrected CMFD method.

• Assessment of the Schur methodology to avoid the diagonalization step in the ACMFDmethod.

• Development of the neutronic/thermal-hydraulic coupling between ANDES and SUB-CHANFLOW.

• Implementation in COBAYA3 of the required developments to allow the introductionof the new cross sections feedback eects for fast reactors.

• Verication of the performed developments in ANDES and application to the ESFRcore.

1.3 Thesis structure

The thesis is structured in such a way that would allow the interested reader to follow themotivations which justify the oncoming developments.

First, serving as introduction to fast reactors, Chapter 2 will describe some aspects relatedto this technology. In particular, the main advantages, history and main basic physicsof Fast Reactors are briey described, including the breeding and actinide transmutationcapabilities, main dierences with thermal spectrum, and sodium properties. In addition,some safety related aspects are commented too.

Then, the second part of the thesis will deal with the review of the state-of-the-art. First, themotivations of the multiphysics approach are explained in Chapter 3. Following, Chapter4 will present a brief description of the main codes, which have a relevant role for thedevelopment of this thesis, excluding naturally the COBAYA3 code, which is treated indetail in the last part of the thesis.

After that, and before starting with the analyses and developments performed, an additionalpart with a single chapter is introduced, which will describe the dierent core modelsdeveloped for each code involved (Chapter 5), which were employed during the oncominganalyses.

Then, the main analyses are presented. Chapter 6 will deal with the assessment of thefast reactor's capability to transmute minor actinides, including a comparison between thedepletion performances of SERPENT and EVOLCODE, which gave rise to a publicationin Annals of Nuclear Energy [105]. Next, the breeding properties of fast reactors wereassessed through the assessment of a Spanish scenario of the hypothetical deployment offast reactors. Other considerations, such as national resources availability and consequenceson the nal inventory repository were discussed. This analysis also gave rise to a publication,in Energy Conversion and Management [108]. Later, in Chapter 8, a methodology to

5

1.3. Thesis structure

compute point kinetics parameters using SERPENT is described and applied to the studyof the impact of minor actinides on transient behaviour for fast reactors. This work wasperformed within the framework of the task 3.3.3 of the SP3 of the CP-ESFR project.Moreover, a parallel study of the impact of uncertainties in the nuclear data on the reactivitycoecients was carried out, which is also presented in Chapter 8. Finally, the methodologyto generate multigroup nodal cross section using SERPENT is described in Chapter 9. Thesame task was previously assessed using ERANOS code, and a comparison between bothcontributions is also presented.

The last part of the thesis deals with the assessment and development of a neutronic-thermal hydraulic coupled system able to simulate fast reactors' steady state and transients.Chapter 10 describes the neutron diusion code COBAYA3, emphasizing the methodologyimplemented in the nodal solver ANDES, which was targeted as the neutron diusion codeto be used for the described application. As a consequence, an intensive assessment anddiagnosis of the limitations of the described methodology is presented. Once the limitationsare identied, Chapter 11 presents the main developments implemented in ANDES code inorder to overcome them. Later, in Chapter 12 the development of the coupling betweenANDES and SUBCHANFOW is described. Finally, Chapter 13 will present the vericationof the previous developments and their application to sodium fast reactors. At the end, theglobal conclusions are presented in Chapter VI.

6

1.4. Acronyms

1.4 Acronyms

Technological

ACMFD Analytic Coarse Mesh Finite Dierences

ADF Assembly Discontinuity Factor

ADS Accelerator Driven System

AFEN Analytic Function Expansion Nodal

ANM Analytic Nodal Method

APT Accelerator Production of Tritium Project

ARI All Rods In

ARO All Rods Out

ASAP Adjoint Sensitivity Analysis Procedure

Bi-CGSTAB Bi-Conjugate Gradient Stabilized

BOL Beginning of Life

BOC Beginning Of Cycle

BOS Beginning of Step

BWR Boiling Water Reactor

CDA Core Disruptive Accidents

CEM Cascade-Exciton Model

CFD Computational Fluid Dynamics

CGNE Conjugate Gradient on the Normal Equations

CMFD Coarse Mesh Finite Dierence

CPB Corner Point Leakage Balance

CP-ESFR Collaborative Project for a European Sodium Fast Reactor

CRAM Chebyshev Rational Approximation Method

CSD Control and Shutdown Device

DSD Diverse Shutdown Device

EFPD Eective Full Power Days

EFR European Fast Reactor

EOC End Of Cycle

EOL End Of Life

7

1.4. Acronyms

..

EOS End of Step

EVET Equal Velocity-Equal Temperature

FNR Fast Neutron Reactor

GIF The Generation IV International Forum

GFR Gas Fast Reactor

HFP Hot Full Power

HZP Hot Zero Power

IAPWS International Association for the Properties of Water and Steam

INPRO International Project on Innovative Nuclear Reactors and Fuel Cycles

INS Innovative Nuclear Reactor Systems

LAQGSM Los Alamos Quark-Gluon String Model

LFR Lead Fast Reactor

LWR Light Water Reactor

MC Monte Carlo

MSR Molten Salt Reactor

NEM Nodal Expansion Method

PWR Pressurized Water Reactor

RHS Right Hand Side

SCWR Supercritical-Water-Cooled Reactor

SFR Sodium Fast Reactor

SGGP SCALE Generalized Geometry Package

TFS Transient Fixed Source

TPEN Triangle-based Polynomial Expansion Nodal

TTA Truncated Taylor Approximation

UTOP Unprotected Transient Over Power

UVET Unequal Velocity-Equal Temperature

UVUT Unequal Velocity-Unequal Temperature

VHTR Very High Temperature Reactor

8

1.4. Acronyms

Institutional

AEC U.S. Atomic Energy Commission

CEA Commissariat à l'énergie atomique et aux énergies alternatives

HZDR Helmholtz Zentrum Dresden Rossendorf

IAEA International Atomic Energy Agency

INEEL Idaho National Engineering and Environmental Laboratory

INR Institute for Neutron Physics and Reactor Technology

KIT Karlsruhe Institut of Technology

LANL Los Alamos National Laboratory

MIT Massachusetts Institute of Technology

NEA Nuclear Energy Agency

NRC Nuclear Regulatory Commission

NURESIM NUclear REactor SIMulation

OECD Organisation for Economic Co-operation and Development

ORNL Oak Ridge National Laboratory

PSI Paul Scherrer Institut

PSU Pennsylvania State University

RSICC Radiation Safety Information Computational Center

UKAEA UK Atomic Energy Authority

UNED Universidad Nacional de Educación a Distancia

UPM Universidad Politécnica de Madrid

9

1.4. Acronyms

10

Chapter 2

Sodium Fast Reactors

2.1 Introduction

Fast reactors receive their name from fast-neutron-spectrum reactors. It refers to the energyof the neutron that will induce ssion reactions within the reactor core. In contrary to theconventional thermal reactors, the neutrons just born from ssion are not slowed downthrough the so-called moderation process but are kept in the fast spectrum range instead.

The neutronic behaviour diers radically from thermal to fast reactors, mainly due to theenergy dependence of the cross sections. In general, those decrease with the incipientneutron energy, reducing consequently the probability of occurrence of a nuclear reaction.On the other hand, in the fast-spectrum the ssion to capture rate increases for most ofthe nuclides. As a consequence, the two main advantages of fast reactors are derived: theoptimized use of fuel resources and the actinide transmutation capability.

The natural uranium content is 99.285% of U-238 and 0.711% in U-235. In order to beused by thermal reactors, it needs to be enriched until around 4% of U-235. This impliesthat there is a great proportion of U-238 coming from the enrichment process which cannotbe used by the conventional reactors. In addition, the content of U-238 in the irradiatedfuel from the conventional reactors is around 94%, which usually is stored as waste withoutfurther treatments.

Moreover, one of the main concerns of the world energy scenario is the potential shortage ofnatural resources. In the frame of ssion nuclear energy it is translated into the limitationof uranium resources. This fact in conjunction to the under-use of this mineral, induces areconsideration of the usage and optimal exploitation of the natural resources.

Fertile isotopes, i.e., those which have a capture cross section dominant over the ssion oneat the thermal range, can be used as fuel in fast reactors. This is a result of the fact thatin the fast spectrum the ssion to capture rate is inverted for those isotopes. In particular,for the U-238, ssion reaction becomes the dominant one at high energies (>1MeV), asshown in Fig. 2.1. As a consequence, a new perspective is open in terms of resourcesoptimization, which would mean an almost innite uranium resource.

11

2.2. Historical context

Figure 2.1: Fission and capture cross sections of U-238

On the other hand, the other potential challenge of ssion energy is the one related to theproduction and management of radioactive wastes. Many solutions have been proposed tothis problem, being the transmutation the most promising one, since it is the most intuitive.However, there are challenges remaining related to this technology which will require furtherresearch on the oncoming years. Nevertheless, it is absolutely clear that the fast spectrumis the only one capable of achieving signicant transmutation rates for the main isotopesthat contributes to the long term radiotoxicity. Indeed, the ssion to capture ratio increasesfor those isotopes, as it is showed in Fig. 2.2 for the Am-241.

Both key features of fast reactors, i.e. transmutation and resources optimization will beassessed in detail in Chapters 6 and 7, respectively.

2.2 Historical context

The history of fast reactors dates back to the beginning of nuclear energy. Actually, the rsttime that nuclear energy was employed to produce electricity was in 1951 in a fast reactorcalled EBR-I at Idaho National Engineering and Environmental Laboratory (INEEL), USA[139] . It was a historical achievement which set the foundations of the civil use of nuclearenergy ( Fig. 2.3).

From that moment on, many countries with nuclear power launched sodium-cooled fastreactor programs at a very early stage owing to the breeding capacity of these reactors.The approach was the same everywhere, progressing in stages: small experimental reactors,

12


Figure 2.2: Fission and capture cross sections for Am-241

Figure 2.3: EBR-1 reactor iluminating four bulbs

13


Figure 2.4: Fast reactors list

demonstration reactors and industrial prototypes. However, due to several circumstancesthe expectation of fast reactor deployment was stopped. Among those circumstances, thesuccess of the Light Water technologies in conjunction with the low price of uranium justiesthat decision.

Fig. 2.4 enumerates the fast reactors that have operated and are currently operating indierent countries. In total, about 20 Fast Neutron Reactors (FNR) have already beenoperating, some since the 1950s, and some supplying electricity commercially. About 400reactor-years of operating experience have been accumulated to the end of 2010. Focusingin the European countries, France has operated three SFRs, one in each of the previouslymentioned stages.

The rst French prototype, Rapsodie, was cooled by sodium, used oxide fuel and belongedto the loop type. Because the intention was to focus on the core and the fuel performance,there were not steam generators. It operated remarkably with two upgrades of power (rstfrom 20 to 24 MW then to 40 MW in 1970). It allowed key breakthroughs in fuel studies(record burn-up of 26 at%) and was nally shutdown in October 1983 due to a tiny leakin the reactor vessel.

Phénix is unquestionably one of the most characteristic example of successfully operatedSFR. It was a demonstrator which operated remarkably well in the 70s and 80s. It provided

14


abundant feedback in terms of fuel, materials, closed fuel cycle, technology (SG, IE). Forexample, the possibility of inspection & repairs operations was demonstrated and the rstexperiments on transmutation were run. Moreover, some innovative fuels such as the onesconsidered for the Generation IV systems were tested. Also, the closed fuel cycle wasdemonstrated in the 90s.

An example of a existing large fast reactor is Superphénix, which was denitively stoppedin 1997, after 4,5 years of variable operation. It includes 2 years of repair after technicalproblems and 4.5 years of shutdown for administrative procedures.

After Superphénix, the European Fast Reactor (EFR) was developed as a project withthe aim of dening the features for a new generation Sodium Fast Reactor in Europe,although it was never built. Then, in 2009 the European Collaborative Project CP-ESFRstarted within the Seventh Framework Programme (European Commission)[54].The mainobjective of CP-ESFR was the assessment of the main challenges related to this promisingtechnology. In the framework of that project the present doctoral dissertation has beendeveloped.

The Gen-IV reactors

The evolution of Nuclear Power dates back to the early prototypes in the 50s till the futureGeneration IV reactors through the currently deployed Generation II of commercial powerreactors and the advanced designs of the Generation III (Fig. 2.5). It is actually in thefuture Generation IV where the fast reactors will have the most representative role. Indeed,it is expected that by the time the Gen IV technology is matured enough to be deployed,the technology advances will provide a solution for most of the challenges related to fastreactors.

The Generation IV International Forum (GIF) was founded in 2000 with the aim of identi-fying a selection of innovative core designs that will overcome the past and present designparameters in terms of sustainability, industrial competitiveness, safety and proliferation re-sistance. Additionally to the power generation, these new designs will be used to generateindustrial process heat or for hydrogen production.

As well as the GIF, another program with similar aims is coordinated by the IAEA. This isthe International Project on Innovative Nuclear Reactors and Fuel Cycles (INPRO). It waslaunched in 2001 and has 22 members including Russia. The main objective is "to supportthe safe, sustainable, economic and proliferation-resistant use of nuclear technology to meetthe global energy needs of the 21st century." It does this by examining issues related to thedevelopment and deployment of Innovative Nuclear Energy Systems (INS) for sustainableenergy supply.

In summary, the four basic goals of Gen-IV technology are:

1. Sustainable development: Optimizing the use of the fuel till the 96% of the energyutilization, using natural resources like U-238 (or Th-232) to reproduce new fuelmaterials like Pu-239 (or U-233) in fast reactors and, what is more, using the ra-

15


Figure 2.5: Evolution of Nuclear Power

dioactive materials such as actinides for power generation. Any of these technologiesare carbon-emission free, helping to reduce the overall emission for power generation.

2. Industrial Competitiveness: Like the Generation III/III+, the Generation IV reactorsare designed for a operating life of minimum 60 years, improving the capacity factor to92%. Like the previous Generations, the main economic cost is the initial investmentin building the plant while the fuel cost is only a 10% of the overall cost, which is aplus compare to gas or coal plants. Additionally, some of these designs are operatingon high temperatures that allow to link the power generation cycle to a hydrogenproduction or an industrial heat cycle.

3. Safety and reliability: The defense-in-depth concept of nuclear power is enhancedat the same time that safety design concepts like passive systems are imposed tominimize the risk of environmental and public radioactive contamination.

4. Proliferation Resistance and security: As a consequence of the Non-ProliferationTreaty, the nuclear power must guarantee the pacic use of this technology. That isone of the main goals of Gen IV reactors. There is a large quantity of military Puthat can be used as fuel in these reactors, especially in fast reactors and ADS. Theoutt of the fast reactors is that the processed Pu from the spent fuel is a mixture ofdierent isotopes of Pu, which makes it almost impossible to use it for nuclear bombfabrication, where pure Pu239 is needed.

In 2003 the Generation IV International Forum (GIF), represented by ten countries, an-nounced the selection of six reactor technologies which they believe represent the futureshape of nuclear energy (Fig. 2.6):

16


Figure 2.6: Designs considered by the GIF

a) Gas-Cooled Fast Reactor System (GFR)

b) Lead-Cooled Fast Reactor System (LFR)

c) Sodium-Cooled Fast Reactor System (SFR)

d) Very-High-Temperature Reactor System (VHTR)

e) Molten Salt Reactor System (MSR)

f) Supercritical-Water-Cooled Reactor System (SCWR)

The Sodium Fast Reactor clearly meets the Gen-IV goals. As a result, it was the designchosen for the present thesis work, which has been developed within the framework of thecollaborative project European Sodium Fast Reactor (CP-ESFR), as stated before.

17

2.3. Basic Fast Reactor Physics

Figure 2.7: Neutron spectra[142]

2.3 Basic Fast Reactor Physics

2.3.1 Flux Spectrum and Slowing-Down Properties

The adjective fast in Fast Reactors comes from fast neutron ux spectrum, i.e., thession born neutrons are not slowed down, as it happens in thermal reactors, but arekept at the fast energy instead. Therefore, the average neutron ux spectrum is harder,concentrated in the keV and MeV range.

Fig. 2.7 compares a typical neutron spectra for SFR and LWR. It can be observed thatthere are almost no neutrons slowing down to energies lower than 1 keV. As a result,the energy range where neutrons are born from ssion and where neutrons induce ssionare overlapped, while in thermal reactors are clearly distinguished. Moreover, due to thescattering resonances of intermediate atomic weight nuclides (sodium and oxygen in case ofoxide fuel) and structural material (iron,nickel, etc.), the SFR spectrum present a stronglyjagged structure. The typical 1/E (or constant Eφ(E)) spectrum of thermal reactors, basicfor the calculation of resonance absorption, is missing in fast reactors, but Eφ(E) stronglydecreases with decreasing energy.

Another consequence of operating in the fast spectrum is the low total cross sections atthis range. They are typically 100 times lower than those from the thermal spectrum.Actually, this is the main reason which justies the moderation process in thermal reactors.In order to compensate the lower ssion cross sections characteristic of FR, a higher ssilenuclide density is required together with higher ux densities. In addition, due to the lower

18


absorption cross section of all materials in fast reactor systems, a much longer neutronmean free path is achieved (∼10 cm as compared to ∼1 cm in PWR), which increases theneutron leakage from the core (> 20% in typical designs). Naturally, the larger neutronmean free path impacts on reactor design. For example, instead of control rods insertedbetween the fuel rods within a fuel assembly, FR typical designs include control assemblieswhich are inserted in between the fuel assemblies.

In addition, some threshold reactions usually negligible in thermal reactors such as (n,2n)reactions have to be considered in fast reactors. Moreover, inelastic scattering plays a keyrole in fast reactors, being the major cause of energy loss at high energies.

In a typical fast reactor the slowing-down power is about 1% of that observed for a typicalPWR. Thus neutrons are either absorbed or leaked out from the core before they can reachthermal energies.

In order to obtain a hard spectrum, moderation should be minimized by avoiding the useof low atomic mass nuclides that cause large neutron energy losses for elastic scattering.As a consequence, alternative coolants are needed, which should accomplish the followingtasks:

• Extract heat from the core: high specic heat and thermal conductivity ensure goodextraction.

• Transfer heat to an energy conversion system (steam generator or exchanger + tur-bine), or to a system which directly uses the heat: heavy oil extraction (oil shales),thermochemical production of hydrogen, desalination of sea water, etc.

• Assure safety by providing the system with a degree of thermal inertia.

On the other hand, a coolant for fast reactors must not:

• Signicantly slow neutrons.

• Activate under neutron ux, producing compounds which create unacceptable dosime-try.

• Change the behaviour of structural materials.

• Induce unacceptable safety conditions.

• Induce insurmountable operating problems.

• Lead to wastes which cannot be processed during operation or dismantling.

Following these requirements, liquid metals and gas coolants have been suggested as po-tential coolants for fast reactors, being the liquid sodium the most popular one. Moreinformation regarding sodium is described in Section 2.3.6.

19


2.3.2 Breeding

The fast reactor was originally conceived to breed nuclear fuel by converting fertile materialsinto ssile ones. There are two known breeding chains, one starting from U-238 and theother from Th-232, as shown in the following reaction equations:

U238(n, γ)U239β−(23.5m)−−−−−→Np

239β−(2.35d)−−−−→ Pu239

Th232(n, γ)Th233β−(22.2m)−−−−−→Pa

233β−(27.4d)−−−−→ U233

The breeding process is quantied in terms of the conversion ratio C, which is dened as theratio between the average number of ssile nuclides produced per ssile nuclide consumedin a nuclear reactor. When C>1 C is called breeding ratio, and the system is considered asa breeder.

There are several ways to compute the breeding ratio. One formulation, proposed by Yang[142] is:

C = (η − 1) + ε− (lA + lL + lD)

where η is the number of neutrons produced by ssile isotope ssions per neutron absorptionin ssile isotopes, ε is the number of excess neutrons produced by ssions in fertile isotopes,lAis the number of neutrons absorbed in the non-fuel materials, lL is the number of neutronslost by leakage, and lDis the decay loss of ssile isotopes during irradiation. A condition fora system to be breeder is that η must be substantially greater than 2, because one neutronis required to induce a new ssion, and at least another one is employed to be absorbed infertile material to produce the new ssile isotope (see Fig. 2.8).

Furthermore, in the fast spectrum, the ε is higher, because the ssion of fertile isotopes ispossible above a energy threshold. Actually the ssion-to-capture ratio is higher than 1 formost of the fertile and ssile nuclides in the fast range. As a consequence, the breedingration increases with increasing incident neutron energy.

Another important parameter is the doubling time, which is dened as the time intervalduring which the amount of ssile material in a reactor doubles. It seems obvious thatan increasing breeding ratio combined with a minimum critical ssile inventory results in aminimum doubling time. In order to decrease the critical ssile inventory, the fuel volumefraction is increased, and as a result, tight triangular lattice is normally used.

2.3.3 Actinide Transmutation Capability

One of the main interests of fast reactors is the potential capability to transmute actinides.As described in the previous section, the ssion-to-capture ratio increases rapidly in fastneutron region.

20


Figure 2.8: Neutron yield [96]

Table 2.1: Actinides Fission-to-Absorption Ratios in Thermal (PWR) and Fast (SFR) Sys-tems [142]

21


For major actinides, the ssion-to-absorption ratio is presented in Table 2.1. Fertile isotopeshave much higher (up to ∼60%) probability to be ssioned in the fast spectrum than in caseof thermal spectrum. In addition, in the thermal range, the fertile isotopes, by capturingneutrons, contribute to the production of higher actinides rather than destroying them.Therefore, the fast spectrum is optimal for consuming transuranic elements in an ecientway with less generation of higher actinides.

Particularly of interest are the minor actinides (Am, Cm, Np), for being the main con-tributors to the high long-term activity of the nuclear wastes, although its content in thedischarged fuel is almost negligible (∼1.5%).

Transmutation capability of these isotopes is assessed in Section 6.

2.3.4 Reactivity Feedback Eects

The reactivity feedback is the response of the core multiplication factor (k-e) to externalperturbations. In light water reactors (LWR) there are typically three parameters responsibleof reactivity feedbacks: fuel (Doppler) temperature, and coolant/moderator density andtemperature. However, for SFRs the situation is very dierent mainly due to the highpower density compared to the one of LWR and the very short core height.

The Doppler and the coolant density remain as feedback parameters. In addition, newphenomena have proved to impact strongly on the reactivity and feedback response, andconsequently should be included in the simulations. This is the case of the axial expansionof the fuel pins, the axial and radial expansion of the cladding and the structural materialexpansions (lower grid plate where the subassemblies are xed, etc.). The importance ofleakages and the greater margin to boiling in conjunction to the higher working temperaturesexplain the relevance of thermal expansions for fast reactors. The most relevant eects willbe described in Section 2.3.5.

The role of specic reactivity coecients varies markedly depending on the transient. Butundoubtedly, the most relevant reactivity coecient is the void worth eect, which meansthat in case of coolant voiding from the core, the reactivity increases sharply. This eect isparticularly important in sodium, which makes it one of the most challenging issues relatedto sodium fast reactors.

2.3.4.1 Doppler coecient

The Doppler eect is the resonance broadening eect as a consequence of a fuel temperatureincrease (Fig. 2.9). Indeed, when the fuel temperature is increased, the resonances arebroadened, which in case of a thermal reactor would mean an increase of captures in U-238 and a consequent decrease of reactivity. However, given the increase of the ssion tocapture rate in fast spectrum, the broadening of ssion resonances should also be takeninto account for these reactors. While the broadening of captures resonances is positive interms of safety, if the ssion resonances are broadened, the eect is negative for the safety,giving rise to an increase of reactivity. As a consequence, the enrichment of ssile isotopes

22


Figure 2.9: Doppler broadening

should be limited in Fast Reactors.

2.3.4.2 Sodium void reactivity feedback

The sodium void eect is a major challenge in sodium fast reactor designs due to its impacton safety.

Should the sodium be partially removed from the core, there are several phenomena to beconsidered:

1. When the coolant is removed, the absorptions in the fuel will increase, with a conse-quent increase on reactivity.

2. The mean free path increases because there will be fewer particles in the neutrontrajectory, with which the neutron could interact. Therefore, the neutrons will travelfurther, which will favour the leakages, resulting in a negative eect on the reactivity.

3. Even though the sodium produces a small moderation, it still moderates something,and in case of its removal, the spectrum will be hardened, which will entail thefollowing consequences:

(a) The ssion to capture ratio will increase for most of the isotopes, including theU-238, producing an insertion of reactivity

(b) As the energy of the incident neutrons increases, the number of neutrons bornfrom ssion (ν) will increase. As a result, more ssions will be induced, andthus the reactivity will increase.

Among the described eects (see Fig. 2.10), the dominant ones are the second and thirdones. In the center of the reactor, the predominant eect would be the one driven by thedecrease of moderation, while in the periphery of the core, the leakage would dominate.

23


Figure 2.10: Reactivity worth as a consequence of sodium void

For this reason, the voiding eect is more relevant for large sodium fast reactors ratherthan for small ones.

As mentioned before, the driven eect in the center of the core is the spectrum hardening,which induces a positive insertion of reactivity. In the nominal state, the neutron ux is verysmall close to 3 keV, where the Na has a resonance in its inelastic scattering cross section.At higher energies, the ux is higher. Moreover, this latter region where the ux is highercorresponds to the resonance of U-238 (capture) and Pu-239 (ssion). If the Na disappearsfrom the core, the spectrum is hardened and the unbalance between the absorptions inU-238 and ssions in Pu-239 is translated into a positive insertion of reactivity. Fig. 2.11illustrates this eect.

2.3.5 Thermal expansion reactivity coecients

In fast reactors, the nominal conditions for the coolants are suciently far from the boilingpoint, as shown in Fig. 2.12, where the dierent margins to boiling and freezing areindicated for the main coolants. It might be seen that the margin to boiling of sodium isquite large compared to water. Moreover, that margin for lead is even larger, although itsmargin to freezing is quite short.

Material thermal expansions aect the geometry and the density of the materials and thusthe reactivity of the system. This eect is especially relevant in liquid metal fast reactors.

The thermal expansion eects should account for the geometrical changes at the same time

24


Figure 2.11: Sodim void eect in terms of spectrum[96]

Figure 2.12: Coolant operational ranges and safety margins [96]

25


Figure 2.13: Main elements subject of thermal expansions[96]

as for the decrease of the materials densities. It can be considered to be the sum of axialand radial eects into the dierent media of the system. The materials target of thoseexpansions are mainly fuel, cladding, wrapper, diagrid, control rods driveline, strongbackand vessel, as illustrated in Fig. 2.13.

2.3.5.1 Steel cladding expansion

The cladding expansion eect on the core reactivity is positive, due to the reduction ofabsorptions in the cladding material (i.e. ODS steel in case of ESFR core), as a consequenceof its increase on height (and decrease on density). Given that the cladding radius increasestoo, there would be less passing area for the sodium, which produces an eect similar tothat of the sodium removal.

The steel cladding can undergo an axial expansion and a radial expansion. Due to anincrease of the cladding temperature, there is a reduction of the cladding density and achange of the geometry, which should be taken into account in the calculations.

26


2.3.5.2 Steel wrapper expansion

The wrapper expansion eect on reactivity is very similar to that just described for thecladding. The wrapper material (typically steel EM10 for ESFR core) is expanded axiallyand radially, and the temperature increase would result in a reduction of the steel density,to be considered in the simulations.

2.3.5.3 Fuel expansion

For the fuel, the dominant eect is the axial expansion, while the expansion in radialdirection is usually neglected. When the fuel heats up, it is axially expanded, causing anincrease of height, for once, and also a reduction of density. As a result, more parasiteabsorptions happen in the cladding, more elastic scattering reactions in the coolant andmore leakages in the radial direction. As a consequence, the global eect is a negativereactivity eect.

It should be noted that the only signicant eect of radial expansion is pushing out thesodium, but this is already taken into account by cladding radial expansion eect.

2.3.5.4 Diagrid expansion eect

The diagonal grid or diagrid is a structural design of steel able to provide a better stressdistribution on the core structure. In the CP-ESFR core, it is supported by a diagrid orby a cylindrical structure of stainless steel (SS316) with a diameter of 7.4 m. It containsmany vertical tubes where the fuel assemblies are inserted. Those wrapper tubes supportthe fuel assemblies and allow the sodium to ow from the bottom part upwards throughthe holes. The diagrid requires high stability (thermal and mechanical) in order to minimizethe geometrical changes that might cause reactivity excursions.

The diagrid can expand radially as a consequence of the change of inlet coolant tempera-ture. When it is expanded, the core radius increases, while preserving all material masses(fuel, cladding, wrapper, and control rods). However, the sodium mass present in the gapinterassembly is increased as a consequence of the volume increase of the sodium betweenassemblies, while its density remains the same. This eect causes a decrease of reactivity,opposite to the sodium void.

2.3.5.5 Vessel expansion

Eventhough it has not been considered for the purpose of the present work, in reality thevessel can expand during a transient, as a consequence of the increase of inlet coolanttemperature. An increase of the vessel axial height means that the core could move slightlydownwards with respect to the nominal state (Fig. 2.14). As a consequence, the controlrods are slightly withdrawn from the core causing a positive reactivity eect (typicallyaround 4 pcm/ºC).

27


Figure 2.14: Scheme of vessel expansion [96]

2.3.5.6 Strongback expansion

The strongback is the structure that attaches the core to the vessel. The same way asbefore, the strongback can be expanded too during a transient as a consequence of anincrease of inlet coolant temperature. If the height of the strongback is axially increased,the core would move slightly upwards, and therefore the control rods would be inserted alittle in the core (Fig. 2.15). This would mean a negative eect on the reactivity (typicallyaround -2 pcm/ºC).

2.3.6 Sodium properties

Among the coolants suitable fo fast reactors, sodium stands out for being the coolant withmore operating experience and the technology chosen for this thesis. Along this chapterthe main properties of sodium are presented.

Sodium is an alkalin metal which has wonderful characteristics as coolant:

• Excellent thermal conductivity (70 WmK

at 400°C).

• High thermal capacity (Specic heat of Na at 400°C: 1.29 kJ/kg).

• Liquid from 97.85°C to 882.85°C (at atmospheric pressure).

• Neutron compatibility: little slowing eect on neutrons produced by ssion, it doesnot change fast spectrum properties, low capturing power (small cross section), lowlevel of activation (but must be of nuclear quality).

28


Figure 2.15: Scheme of strongback expansion [96]

• Lower density than water (850 kg/cm3 at 400ºC).

• Natural convection capability.

• Viscosity comparable to that of water (Na: 2.77e-4 Pa-s at 400°C; H2O : 2.80e-4Pa-s at 100°C).

• Compatibility with metallic materials fairly satisfactory.

• No toxicity.

• Broad operating experience.

• Low cost.

In addition, the conductive properties of sodium could be used in instrumentation, ow ratemeasurements, electromagnetic pumps, Na leak detection, etc.

However, it has some disadvantages:

• Not too much margin to freezing. As a consequence, it requires a preheating systemfor the structures before lling up and should be kept over 400ºC during shortage andreloadings. In addition, rapid cooling should be avoided during transient conditions,which should lead to sodium blockages in the decay heat systems.

• Small thermal capacity (1/4 of that of water).

• Some activation after irradiation. In particular, Na-22 and Na-24 are gamma emitterswith a half life of 2.6 years and 15h respectively. In fact, Na-24 is the major responsibleof the activity of the primary circuit.

29


• Due to the sodium opacity, there are diculties for In-Service Inspection and Repairoperations.

And some serious disadvantages:

• It reacts strongly with water generating important amounts of hydrogen and heat.

Na+H2O → NaOH +1

2H2 +4H

• It has chemical anity with oxygen.

2Na+1

2O2 → Na2O ; 2Na+O2 → Na2O2

• Need of purication system to control the content of hydrides and oxides which mightcorrode the structures.

• Need of sodium leakage detection systems to avoid sodium res and sophisticatedinstrumentation systems to detect leakage in steam generators.

• It might have a positive reactivity void eect.

Nevertheless, sodium is a very promising candidate, particularly as a primary coolant, due toits very attractive physical, nuclear, and even some of its chemical properties. In addition,sodium has been always considered as the reference coolant for intermediate loops. It hasbeen world-wide developed since the 1950's.

In order to reduce the constraints due to potential sodium-water interactions on design,investment (safety instrumentation,. . . ), reactor availability, and dedicated operational pro-cedures, several alternative options are investigated:

1. To develop an alternative Energy conversion system, such as the Brayton cycle (usinggas i.e. He-N2, SC-CO2 ), or

2. To keep the Rankine option (water steam) associated with:

• Implementation of intermediate loops tted with improved safety monitoring systemNa-H2O interaction, and improved associated procedures,

• Implementation of double-wall Steam Generator Units,

• Design of an intermediate loop, using an alternative coolant, avoiding potential detri-mental eects due to coolant-water interaction.

30

2.4. Safety

2.4 Safety

The main safety issues of Sodium Fast reactors are linked to the sodium technology, asmentioned before. The contact between sodium and water and air should be preventedin any case. In addition, there is a huge concern with the so called Core Disruptive Acci-dents (CDA). This event is linked to the recriticality produced as a consequence of a corecompaction, which could result in mechanical stresses able to produce the complete coredisruption.

The recriticality issue in CDA has been one of the major safety issues of Sodium-cooledFast Reactor (SFR) from the beginning of its development history. The conventional safetyapproach has been:

1. to minimize the occurrence probability of the CDA by utilizing, for example, twoindependent reactor shutdown systems, and

2. to assess the mechanical energy release due to recriticality events assumed in a hypo-thetical CDA, and to conrm the in-vessel capability against the estimated mechanicalenergy. The resultant loading in the containment due to sodium burning that could bespilled out from the peripherals of the plugs in the reactor vessel. International eortshave been made to develop and improve the assessment method of the mechanicalenergy release. In addition, a re-criticality-free core concept was sought aiming atpractical elimination of re-criticality events during CDAs, because the mechanicalenergy released after exceeding the super-prompt excursion increases in larger ratedpower cores of commercial reactors. Furthermore, the re-criticality event issue needto be solved prior to the commercialization of SFR.

Another issue related to fast reactors is its lower delayed neutron fraction, Indeed, this factis a direct consequence of the reduction of the amount of U-238 to allow the charge of Puisotopes, with lower ÿ. In case of considering the loading of Minor Actinides, this fractionis even lower. This fact has a negative eect in safety, since it implies a dominant inuenceof prompt neutrons over the delayed ones, leading to faster transients, which could be moredicult to control.

2.4.1 Safety events in Sodium Fast Reactors

Several safety-related events have occurred in SFRs from the beginning of its developmenthistory[124]. Almost all the operating SFR have experienced sodium leaks. In particular,32 were detected in Phenix and 5 in Superphenix. Except for two relatively big leaks (KN-IIand BN600) of around 102-103 kg, respectively, there were very small. The bigger onesled to sodium res, which had to be extinguished with Marcaline powder. The overridingcauses were mainly design (thermal-mechanical lading) and manufacturing (welds).

With respect to Sodium Fires, there were one event in Monju (Japan) as a consequence ofthe late drian of a damaged pipe. The equipment in the room was deteriorated and sodiumaerosols were diused by the ventilation. As a resut, over 14 years of delay.

31

2.4. Safety

As a consequence of sodium leaks in the steam generator tubes, the sodium-water reactionmight occur. Indeed, a total of 5 reaction events were detected in Phenix, about 12 inBN600 and about 40 in PFR. Among those, three violent sodium-water reactions occurredin BN350 (1973 and 1975) and PFR (1987), respectively.

Additional events were detected. Cracking on the external storage drum in Superphenix(1987) and some more on hot pipes of Phenix were observed. An argon leak in the reactorblock of SPX (1994) was detected when an intermediate heat exchanger has been replaced.Also in SPX (1990), a primary sodium pollution caused 8 months of reactor unavailability.Moreover, in BN 600 (1987) a block of sodium impurities was detected, which led toneutron and thermal-hydraulic perturbations, plus partially obstructed sub-assemblies (cladfailures). In 1991 an event produced by oil entering in the primary circuit of PFR caused 1year and a half of cleaning before restarting.

In addition, damages of dierent fuel sub-assemblies were detected in FBTR (1987) andJoyo (2007). Blockage of control rod mechanisms is another event that occured manytimes (PFR, KNK II, Phenix). An isolated event in Phenix (1989,1990), giving rise tonegative reactivity transients occurred, whose causes have not been fully demonstrated yet.Moreover, a partial plugging of a Fuel Sub-Assembly was detected in Superphenix (1986),with no further consequences thanks to the early detection by the core instrumentation.

Finally, a partial meltdown of two subassemblies occurred in Fermi 1 (1966), which led to4 years of unavailability. A metal plate became detached from its support and obstructedcirculation of sodium in several sub-assemblies. As a consequence, two sub-assemblies werepartially melted. This one is considered to be the most important accident until now in aSFR.

32

Part II

REVIEW OF SIMULATIONTOOLS FOR FAST REACTORS

33

Chapter 3

Multiphysics approach

3.1 Introduction

In order to assess the performance of a nuclear reactor, several disciplines should be takeninto account. Neutronics plays a key role as it predicts the reactivity and thermal powerdistribution of the system; core thermal-hydraulics is also essential in terms of reactivityfeedbacks and coolant temperature and ow-rate proles; plant thermal-hydraulics makespossible to integrate the dierent components of the primary and secondary circuit, aswell as the containment, in the system model. In addition, some other disciplines such asthermal-mechanics are particularly relevant for fast systems due to the large inuence ofgeometry changes on the reactivity, together with the necessity of predicting fuel behaviorunder irradiation. Adequate depletion modules are also required for burn-up calculations.Last but not least, uncertainty analysis is gaining more relevance nowadays since there isan increasing need of quantifying the uncertainties linked to simulations. Therefore, fastreactor physics requires a multiphysics approach (Fig. 3.1).

As discussed in Chapter 2, fast reactors have special characteristics that dier from those ofthermal reactors. As a consequence, specic physics analysis tools are required for sodiumfast reactor simulations.

Before going further, it should be mentioned that the aim of this Chapter is not to performan exhaustive review of all the methods for fast reactor simulations, but to provide anoverview of the main methods and codes linked to the dierent phenomena involved. Theneed of the multiphysic approach is highlighted, which would justify the dierent develop-ments performed during the present thesis.

First, the dierent methodologies for neutronic analysis will be described, including transportand diusion approaches and neutron kinetics. Following, depletion and neutron parametergeneration will be discussed. The role of uncertainties assessment will be described next.Finally, the dierent thermal hydraulic methods will be described and the importance ofthermal mechanics tools will be highlighted.

35

3.2. Neutronics

Figure 3.1: Diagram of the multiphysics approach

3.2 Neutronics

For whole-core calculations two main methodologies have been typically followed: deter-ministic and stochastic/Monte Carlo methods. While deterministic approaches adopt somesimplications to the model, and provides the exact solution to the simplied problem,Monte Carlo approaches do just the opposite. In MC codes, the exact problem is simu-lated, no matter how complex it is, but the solution achieved is an estimation subjected tostatistical errors.

3.2.1 Monte Carlo

The main drawback of Monte Carlo approaches is the high computational time required toachieve an acceptable level of accuracy. Since it is based on statistics, the more accuracy issought, the more neutron histories and cycles are required, which impacts naturally on thecomputational time. Moreover, because the thermal-hydraulic conditions are to be givenas xed input parameters in Monte Carlo codes, their use is not recommendable for thosesimulations with strong feedback eects. This kind of applications includes planication ofoperational manoeuvres, analysis of fuel reloading, etc.

On the other hand, given the stochastic nature of the problem, the codes based on MonteCarlo methods are in principle suitable for any type of nuclear system. These codes al-low accurate representation of nuclear data details, treatment of heterogeneity eects andcomplex geometries. They also allow work in continuous energy which is of key importancefor fast reactors simulations. Monte Carlo codes can be used for lattice and whole-corecalculations.

36

3.2. Neutronics

Among the typical Monte Carlo neutron transport codes, special remark should be givento MCNP5/MCNPX [25], developed at LANL(USA); SERPENT [86], recently developedat VTT Research Center (Finland), KENO-VI within the SCALE system [12], developed atOak Ridge National Laboratory (USA); and TRIPOLI [115], developed at CEA (France).

3.2.2 Deterministic

As previously discussed, deterministic methods aim to solve the transport equation (3.1) ina precise way, but resorting to some simplications over it.

1v∂∂tφ(r, Ω, E, t) + Ω · ∇φ+ σφ =

=´ ´

σ(r, E ′) · f(r, Ω′, E ′ → Ω, E) · φ(r, Ω′, E ′, t) · dΩ′dE ′ + q(r, Ω, E, t)

(3.1)

3.2.2.1 Transport solution

Among the deterministic codes, the accurate solution is provided by the transport theoryapplied to 3D whole-core problems. The equation 3.1 might be written in dierent waysthat will lead to the dierent deterministic methods to solve the transport equation. Thedierent transport methods can be classied according to the approach followed to treat theangular variable. In the methods PN, this angular dimension is developed as an expansionin spherical harmonics. While in the method of the discrete ordinates or SN the angularvariable is discretized along a discrete number of directions using a numerical quadraturethat minimizes the integral error over the solid angle.

The method of collision probabilities Pijdo not include the angular variable for it is alreadyintegrated in the equation to be solved, while the method of characteristics MOC arecombined usually with the method of discrete ordinates SN.

Each of the previously mentioned methods have advantages and disadvantages, in termsof calculation time, memory required and level of accuracy obtained by the model. Whilethe method of collision probabilities used to be the method most widely employed for crosssection generation in the standard approach, in the past years it has been progresively re-placed by the method of the discrete ordinates, due to its higher accuracy and the increaseof the computational capacity. These calculations are also known as lattice physics calcu-lations, and as a consequence the codes that performe them are known as lattice codes.The methods of spherical harmonic and discrete ordinates have been used not only for crosssection generation, in lattice codes, but also for whole core calculations.

One of the most extended transport deterministic codes for fast reactor analysis is ERANOS[122], which include the VARIANT code [2]. The VARIANT code is a typical productioncode based on the second-order formulation. It solves multigroup transport problems in two-and three-dimensional Cartesian and hexagonal geometries. It is based upon a variational

37

3.2. Neutronics

nodal method that guarantees nodal balance and permits renement using hierarchicalcomplete polynomial trial functions in space and spherical harmonics or simplied sphericalharmonics in angle.

These codes might be used for steady-state calculations. However, when transient calcu-lations are needed, the constraints embedded in transport solutions justify the adoption offurther simplications, which give rise to the simplied spherical harmonics method (SPN)and the diusion approach. The rst one is based on the derivation of the neutron transportequation starting from the variational principle with a set of known trial or basis functionsfor the angular variable. The most extended one is the SP3. The second, on the otherhand, is a simplication of the neutron transport equation obtained from the assumptionthat the neutron ux is linearly anisotropic. Both methods are equally employed nowadaysfor whole core calculations.

3.2.2.2 Diusion approach

The most common approach is the diusion theory, which main objective is the eliminationof the angular dependence of the ux, i.e., to reduce the directional ux to a scalar ux.The diusion approach is obtained from the P1 approximation of the transport equation,which consists in a relevant elimination of the angular ux anisotropy. It is assumed thatthe angular ux is linearly anisotropic, according to 3.2.

ϕ(r, E, Ω) =1

4πφ(r, E) +

3

4π~Ω · ~J(r, E) (3.2)

From the second equation of the P1 approximation, it is obtained:

~J(r, E) = −D(r, E)∇φ(r, E) (3.3)

, which is usually known as the Fick's law. Resulting in:

−∇ ·D(r, E)∇φ(r, E) + Σt(r, E)φ(r, E) =

=´∞

0Σs0(r, E ′ → E)Ju(r, E

′) · dE ′ + χ(E)´∞

0νΣf (r, E

′)φ(r, E ′) · dE ′ +Qex(r, E)

(3.4)

In order to assess the applicability of diusion theory to fast reactors, some considerationsshould be taken into account:

• The moderation is limited in a fast reactor, which means that the spectrum averageenergy is around a few hundred keV. The spectrum shape makes it compulsory tohave a ner energy discretization that that for thermal reactors to describe as preciseas possible the resonance range [3].

38

3.2. Neutronics

• The neutron migration area is exceptionally long, which masks the intranodal hetero-geneity details.

• The diusion theory provides better results in those applications where the ux ispredominantly isotropic. For a fast reactor, the absorption cross sections tend to besmall, and the neutron transport is dominated by scattering reactions in those regionsof the core with materials of high mass number. However, in the low density regionsthe diusion theory presents more limitations. This is the case of sodium channels,control assemblies with the rods withdrawn or even in case of sodium void. Theneutron streaming eects lead to important errors, which would justify the use of 3Dtransport codes in order to treat accurately those transport eects. However, sincethose transport calculations have been prohibitively expensive in the past, the solu-tion adopted traditionally was the modication of the diusion coecient, deningan axial corrected one and another radial one to treat those eects in diusion codes.Nowadays, the solution of 3D whole core problems using transport methods is aord-able for steady state cases. However, transient simulations, with thermal-hydraulicfeedbacks remain as a major computational challenge. As a consequence, for tran-sient calculations, the most extended approach involves kinetics diusion codes. Thisis the case of FAST [98] which integrates PARCS and uses the diusion approach fortransient calculations.

There are a number of well-established deterministic tools for diusion calculations. Most ofthe current whole-core diusion theory codes are based on the advanced nodal methods thatwere developed in early 1980s to replace the expensive pin-by-pin nite dierence method.Among a number of dierent nodal methods, those based on the transverse integrationprocedure are most widely used to reduce the 3D equation into a set of one-dimensionalequations. Two methodologies are considered for advanced nodal methods: the nodalexpansion method (NEM) and the analytical nodal method (ANM).

Moreover, whole core fast reactor applications require the use of nodal diusion codes ableto treat hexagonal geometry. Some of them, such as NEM [75], use a traverse integrationprocedure over the hexagonal nodes, while most of them, such as ANC-H [125], PANTHER[135] and MGRAC [17] resort to the conformal mapping to transform the hexagonal 3Dproblem into a Cartesian 3D problem before applying traverse integration. Others employux expansion, like the AFEN method [18], the HEXNEM2 method implemented in theDYN3D [65] or the TPEN method implemented in PARCS [22]. ANDES [88], on the otherhand, resorts to the Analytical Coarse Mesh Finite Dierence (ACMFD) method, as it isdiscussed in Chapter 10.

These advanced nodal methods usually employ discontinuity factors to reduce the errorsassociated to assembly homogenization. However, given that the neutron free main pathof fast reactors is greater than in thermal reactors, the heterogeneity eect is smaller, andthus the discontinuity eects are not so demandable.

Some of the most relevant tools to be used for fast reactor whole-core calculations aredescribed in the Chapter 4.

39

3.3. Neutron kinetics

3.3 Neutron kinetics

In order to be able to analyze transient behaviour of nuclear reactors, the neutron kineticcapabilities should be included in the simulation codes. Neutron kinetic problems involveseveral time-scales, which are really dierent due to the coexistence of several physical pro-cesses. In neutron kinetics, the two processes involved are the generation of instantaneousand delayed neutrons. The rst ones follow the reactivity changes with a time constantwhich is of the order of the neutron mean generation time. This parameter varies from10−7 seconds in fast reactors to10−4 − 10−3 s in thermal reactors moderated by graphiteor heavy water. In light water thermal reactors this value is typically around 10−5 s.

On the other hand, the characteristic time associated to the delayed neutron generationdepends on the precursors decay constants (λk), and usually achieve values of the orderof seconds. This heterogeneity in the time constants associated to both phenomena is theresponsible of the typical behaviour of the multiplicative systems when after a reactivity in-sertion bellow the prompt criticality, a delay reponse in two phases is encountered. First, thepower evolves rapidly, and second the increase (or decrease) is produced much slower. Therst phase corresponds to the readjustment of the instantaneous neutron population, i.e.the prompt jump; while in the second one, the evolution of the reactor neutron populationis controlled by the delayed neutron source.

The two main driven equations for the kinetics are the neutron balance equation (3.5) andthe delayed neutron precursors equation (3.6).

1

vg

∂φg∂t

= −∇·Jg−Σrgφg+∑g′ 6=g

Σsg′→gφg′+(1− β)χgp

G∑g′=1

νΣfg′

keffφg′+

Gd∑k=1

χgdkλkCk+Sg,ext

(3.5)

∂Ck∂t

= fkβ

G∑g=1

νΣfg

keffφg − λkCk k = 1, . . . , Gd (3.6)

In terms of the spatial treatment adopted, there are three main neutron kinetic models,which include zero, one and three dimensions, respectively. The rst and simplest one isusually known as point kinetic model.

In order to consider the spatial distribution of the neutron ux (or power), the spatial-dependent transport equation should be solved, being the diusion models the most ex-tended approach for being the rst order approximation of the neutron transport method.

On the other hand, attending to the spatial dependence, two models can be derived: theone-dimensional and three-dimensional ones. The one-dimensional models are quite accu-rate in situations where the spatial dependence is dominant in the axial direction, as occursfor instance in the reactor scram. The three-dimensional models, on the other hand, aremore generic and they do not include any hypothesis neither approximation over the spatialeects. On contrary to the point kinetic models, the 3D ones require many more eorts in

40

3.4. Cross section Generation

dening the core initial state, including the steady-state calculations.

Moreover, those equations present an additional complexity which is the energy dependenceof the nuclear reactions. Usually that dependence is addressed through a discretization onthe energy range, whether in two groups approach or in multi-groups.

Most of the plant thermal-hydraulic codes include a point kinetic module in order to simulatetransients. Such is the case of TRACE in the FAST system.

In addition, many diusion and transport solvers include kinetic capabilities. For example,VARIANT-K is included in ERANOS, or the kinetic modules implemented in COBAYA3,PARCS and DYN3D, .respectively

3.4 Cross section Generation

The rst step when using nodal diusion codes for reactor core analysis is the processing ofevaluated nuclear data les for pertinent nuclides into suitable forms for use in applicationcodes. This is the consequence of the use of the standard approach. The calculation isdivided into two steps. The rst one is the calculation of the multigroup nodal homogenizedparameters with transport calculations at fuel pin and/or assembly level. Following, in thesecond step the nodal parameters are provided to a diusion code in order to perform thewhole core calculation.

The generation of multigroup cross sections from evaluated data les remains one of thefundamental problems in reactor physics due to the need for accurate treatment of resonanceeects. This aspect is especially relevant for fast reactors due to the staggered shape of theux and the fact that the average energy of the spectrum is close to the resonance rangefor most of the isotopes involved.

Particularly, the hard neutron spectrum makes that reactions which are typical for thisrange, such as anisotropic scattering, inelastic scattering, (n,2n) reaction, and resonanceself-shielding should be considered. The scattering resonances of intermediate atomic massnuclides and the lack of 1/E spectrum for the calculation of heavy isotope resonance ab-sorption require very detailed modeling for slowing-down calculations.

As a consequence, if a multigroup approach is employed, the energy intervals should becarefully assessed to make sure that all the information required is caught. This impliesthe use of big energy group structures, typically 33 groups, which might be a challenge foranalytical methods.

The neutronic nodal parameters are whether provided by some benchmark specications,or calculated by a transport code. In the latter case, both deterministic and Monte Carlotransport codes can be employed. Among the deterministic tools employed for cross sectiongeneration for fast reactors, the ECCO [121] module within ERANOS [122] system deservesspecial consideration, for being the reference code for this kind of application. ECCO hasa multi-dimensional lattice analysis capability in conjunction with sub-group methods [39].This spatial modeling capability explicitly treats the heterogeneity eect, and the sub-group

41

3.5. Depletion and Fuel Cycle Analysis

method allows some re-capture of the resonance energy details. Regarding Monte Carlotools, MCNP [25] might be used with some modications in the source code [27]. Inaddition, the recently developed SERPENT [86] has proven to be a reliable tool for multi-group constants generation [56].

3.5 Depletion and Fuel Cycle Analysis

A reliable prediction of the isotopic inventory in any nuclear system is needed in order toassess aspects related to operation, safety and waste management. In other words, it isof key importance to be able to predict the isotopic inventory and its consequences (decayheat, radiactivity, radiotoxicity, neutron emission, etc.) at dierent steps of the fuel cycle.

Among the most popular inventory/point depletion codes it is worth to highlight ORIGEN[104], developed at Oak Ridge National Laboratory; FISPACT [55], developed by UKAEA;CINDER90 [50], developed by LANL, which has been integrated in the MCNPX packagefrom version 2.6.0 on; and ACAB [123], developed by UNED-UPM.

When the inventory equations are established, it is deduced that the total ux will berequired for the estimation of the reaction rates, and the ux spectrum will also be requiredfor the calculation of the activation cross sections condensed in the target group structure.Moreover, it is not reliable to assume a constant spectrum along the burnup for thosecases where it changes signicantly. Therefore, it is recommendable to couple those pointdepletion codes with a neutronic transport code. Recently there has been more reliance onMonte Carlo tools for depletion calculations. The method is particularly useful for analysis ofspecied designs but not suciently ecient for use in parametric and trade studies requiredfor developing an optimized design. Nevertheless, the Monte Carlo technique is attractivebecause of the ability to represent accurately nuclear data details and to treat heterogeneityeects and complex geometries. Since the 90s, dierent depletion computational systemshave been developed around the world coupling a Monte Carlo neutron transport code witha depletion code. This is the case among others of MCNPX 2.6.0 [51], MONTEBURNS[136], MOCUP [44], MCODE [141], EVOLCODE [94] and SERPENT MC transport code[86].

Propagation of the Monte Carlo statistical uncertainty during depletion calculations has notbeen addressed in these tools and future work to quantify this problem is required. Withoutthis, results from such methods would be questioned due to un-quantied uncertainty.

Particularly for fast-reactor fuel cycle calculations, depletion calculations are performedtogether with the whole-core calculation, as a consequence of the global coupling describedin Chapter 2. On the other hand, for thermal reactors, the depletion is usually incorporatedwith the lattice calculation, performing depletion calculation at the assembly scale andinterpolating the homogenized assembly cross sections for the corresponding burnup states.

42

3.6. Perturbation Theory and Uncertainty Analysis

3.6 Perturbation Theory and Uncertainty Analysis

Sensitivity and uncertainty analyses are becoming a relevant tool to comprehend the relia-bility of the neutronic design of a given nuclear system. Engineering of new reactor modelsrequires computational tools capable of producing results with the adequate level of accu-racy. One source of uncertainty in the modeling arises from the employed nuclear data. Inparticular, sensitivity analysis of the quantities important for safety and design to the inputparameters, and posterior uncertainty propagation from the parameters to the results is amain tool to point out where nuclear data should be improved, with a consequent beneton design accuracy. Moreover, it is critical to evaluate the uncertainty in the reactivitycoecients due to their inuence in the safety assessment.

The perturbation theory methods are usually employed to compute kinetics parameters andreactivity coecient distributions.

The reactivity change (i.e., change in the eigenvalue of the neutron transport equation)due to some perturbations introduced in the system can be expressed by a conventionalperturbation formula that involves the neutron and adjoint ux distributions along with thesystem perturbation. The rst-order perturbation theory formula is obtained using the uxand the adjoint ux for the unperturbed reference conguration. The exact perturbationtheory formula is obtained if the solution to the perturbed conguration is used for either theux or the adjoint ux. The perturbation theory formula readily provides the contributionof a local perturbation to the reactivity change, and the reactivity coecient distributioncorresponding to a perturbed input parameter is easily obtained.

When the response parameter is extended to other than the eigenvalue of the neutrontransport equation (e.g., reaction rate ratios), the corresponding perturbation theory iscalled the generalized perturbation theory (GPT) [59]. A class of GPT to describe thetime-dependent behavior of the coupled neutron/nuclide eld in the reactor is called thedepletion perturbation theory (DPT) [140]. The generalized perturbation theory methodsare used to calculate the sensitivity coecients of a response parameter with respect toinput parameters (e.g., isotopic cross sections). The cross section sensitivity coecients areused to estimate the uncertainty of response parameter due to uncertain cross sections orto reduce the response parameter uncertainty by the use of integral experiment data [113].

Among the codes able to perform uncertainty propagation calculations using perturbationtheory, some of them are worth to highlight: VARI3D [4], GMADJ [117] and TSUNAMImodule integrated in the SCALE6.1 system [12].

3.7 Thermal-hydraulics

The thermal-hydraulic is the study of the hydraulic ow in thermal systems. Regard-ing nuclear, thermal-hydraulic is the science that account for the movement of uids thatexchange heat (among them and with the solid structures) in Nuclear Power Plants. There-fore, thermal-hydraulic gathers two branches of Physics and Engineering: Fluid Mechanicsand Heat Transfer. The thermal-hydraulic codes aim to assess the primary and secondary

43

3.7. Thermal-hydraulics

circuit in order to predict the coolant conditions at any operation moment, and thus toevaluate the safety margins.

The main diculty to predict the uid behaviour lies on the presence of internal interfacesconstantly moving within the ow (for instance, interfaces of vapour and liquid phases). Theconservation equations of mass, momentum and energy can dene the instantaneous micro-scopic behaviour of the uid under study. However, for practical purposes the microscopicbehaviour can be very dicult to determine and then simplications must be introduced bymeans of integrations and averages in order to be able to predict the macroscopic behaviourof the uid, and thus enabling large ow systems to be analyzed.

As happened with every eld of physics, the numeric methods employed to solve the thermal-hydraulic equations are continuously improving, incorporating new capacities as the technol-ogy advances. In the 70s the rst thermal-hydraulic codes appeared including very simplemodels of 3 equations assuming thermal equilibrium between the same phases and velocities(HEM, Homogeneous Equilibrium Model). This is the case of RELAP4, for example. Sincethat moment, the models have been improved continuously.

In the 80s models with 4-5 equations appeared, which included the sliding between surfaces(drift-ux) allowing dierent velocities for them. To this generation belongs COBRAIIc/MIT-2 [77], which is a code with 3 equations for the liquid plus an additional one for the vapour.In the 90s models with 6 equations with 2 uids (3 equations per phase) began to be em-ployed. This is the case of RELAP5 [132], TRAC and CATHARE [11]. To this generationalso belongs COBRA-TF [40, 30], which is a code of 6 equations, 2 uids and 3 elds(vapour, liquid and entrainment). From beginning of XXI Century, commercial CFD codes(Computational Fluid Dynamics), such as ANSYS-CFX, FLUENT, TRIO-U and STAR-CD,started to be used. Those codes allow very detailed studies, using a ne mesh renement,including turbulence models and solving a total of 7 equations in partial derivatives for eachphase.

The thermal-hydraulic codes might be classied according to the type of estimation adoptedas optimal estimation codes and license codes (conservative estimations). They can treatthree dierent scales: Macroscopic scale (system), mesoscale (components) and micro-scale (turbulence). In this regard, there are two main families of codes for reactor thermal-hydraulic analysis. One is the family of the plant codes (i.e., RELAP5-3D [131], CATHARE,TRACE, ...), which employ a coarse modelization of the reactor components; and the otherone is the family of the core codes (i.e, COBRA, FLICA4 [35], SUBCHANFLOW [72]), etc.,which perform a detailed modelization of the fuel assemblies.

In order to be able of use a thermal-hydraulic code for fast reactors applications, theuid under study should be included and properly treated by the code. In particular, theproperties of some liquid metals, such as sodium and lead; and other uids such as gas(He, CO2) need to be implemented.

44

3.7. Thermal-hydraulics

3.7.1 Core thermal-hydraulics

There are three main families of thermal-hydraulic codes that dier on the number ofequations needed to model the uid behaviour:

• The single-phase flow model, which solves three conservation equations (mo-mentum, energy and mass). This model is also known as EVET (Equal Velocity-EqualTemperature).

• the mixture model, which solves three conservation equations (momentum andenergy for the mixture; and mass for the liquid and vapour mixture) and assumesthe thermal but not dynamic equilibrium between the liquid and vapour phase. Thismodel is also called UVET (Unequal Velocity-Equal Temperature).

• the two-phase flow model, which solves more than three conservation equa-tions. This model is also known as UVUT (Unequal Velocity-Unequal Temperature).

The single-phase ow is not used for conventional reactor applications, but might be used forIV Generation reactors, where no phase transition is foreseen. As for the mixture approach,it has been widely used for conventional reactor applications due to its simplicity. However,the new codes include two-phase ow models with the aim of providing more exibility tothe calculations.

When applying thermal-hydraulics to 3D nuclear reactor problems, the number of momen-tum equations is highly increased. As a simplication, the subchannel approach arose,which has been widely used by many authors such as Todreas and Kazimi [133, 134]. Thisapproach assume the predominant movement of the uid in the axial direction and thereforereduce the three momentum equations into two equations: one for the axial direction, andanother very simple one for the transverse direction. This approach has been so widely em-ployed that nowadays all the core thermal-hydraulic codes are named generally subchannelcodes, even if they are actually 3D codes which do not use the mentioned approach.

There are numerous codes belonging to the category of core codes, but only a few areable to work with metal liquids. Such is the case, for example, of SUBCHANFLOW andCOBRA-IV [31].

3.7.2 Plant thermal-hydraulics

The thermal-hydraulic system codes are used in support of plant operation. They employtwo-phase ow models and the most popular ones are the European codes: ATHLET andCATHARE; and the North-American ones, developed by the Nuclear Regulatory Commission(US-NRC), being TRACE the most important reference among them. In addition, theUS-NRC has developed and maintained a number of codes related to the Nuclear SafetyAnalysis, such as:

• RELAP5 for small LOCAs and operational transients

45

3.8. Thermal-mechanics

• TRAC-PF1 for big LOCAs

• TRAC-BF1 for safety analysis of BWRs

• RAMONA for safety analysis of BWR including 3D kinetics

Among the previous codes, for Fast Reactor applications, CATHARE, TRACE and SPEC-TRA are the most employed in Europe.

3.8 Thermal-mechanics

For Fast Reactors simulations, the possibility of having a thermal-mechanic code coupled tothe simulation system is of high interest. Indeed, due to the excessive inuence of geometrychanges on reactivity, in order to catch properly the corresponding feedback eects, itwould be the most convenient to couple the neutronic/thermal-hydraulic system with athermal mechanic code, able to predict the fuel and structure mechanical behaviour underirradiation. In particular, the temperatures, stresses and strains in the fuel and structures(i.e., fuel pin, core diagrid, heat exchanger tubing, reactor vessel, etc.) should be calculatedfor each time step. By doing that, both the reactivity feedback eects due to core thermalexpansion and the fuel failure probability can be estimated. This is the case, for instance,of the FAST system [98], which integrates the FRED thermal-mechanics code [97] and theTRAC/AAA structure model. Both models can be used depending upon the nature of thestructure and the level of detail required.

46

Chapter 4

Codes for 3D Sodium Fast Reactorsimulation

4.1 Introduction

Throughout the development of this thesis, several codes have been employed with dierentpurposes. All of them are suitable for fast reactor applications. This Chapter aims todescribe briey the most representative characteristics of each of them.

A signicant part of the thesis covers neutronic behaviour of fast reactors. As a conse-quence, dierent neutronic codes have been used with the purposes of neutronic analy-sis, neutronic parameter generation, burn-up calculations, benchmarking, etc. Moreover,thermal-hydraulics also play a key role for the thesis, since one of the most importantachievements is the development of the coupling between a neutron diusion code and athermal-hydraulic code, which is presented in Chapter 12.

4.2 Neutronic Codes

4.2.1 Monte Carlo transport methods

Monte Carlo codes are useful for 3D whole core simulations of fast reactors mainly due tothe continuous treatment of the neutron energy and the possibility to model the detailedgeometry. In fact, since these codes are based on statistics, and therefore, do not rely onsimplications, they can be used for almost any application. Indeed, they are suitable forgeneration of homogenized cross sections in any chosen group structure. In addition, asdiscussed in Chapter 3, they are starting to be considered for burnup calculations whencoupled to a suitable inventory code. Following these lines, the most relevant Monte Carlocodes for the purposes of this thesis work are described.

47

4.2. Neutronic Codes

4.2.1.1 MCNP

MCNP5 states for general-purpose Monte Carlo NParticle code. It brings together twomain codes MCNP and MCNPX, which are merged in the last release of the code, MCNP6.During the rst phase of the thesis, MCNP5 was employed as a tool to get familiar with theneutronic behaviour of fast reactors. Some time afterwards, the code MCNPX was usedfor the ESFR core characterization and assessment of MA transmutation, within the SP2.1WP4 of the CP-ESFR project.

MCNP5

MCNP can be used for neutron, photon, electron, or coupled neutron/photon/electrontransport, including the capability to calculate eigenvalues for critical systems. The codetreats an arbitrary three-dimensional conguration of materials in geometric cells boundedby rst- and second-degree surfaces and fourth-degree elliptical tori.

Pointwise cross-section data are used. For neutrons, all reactions given in a particular cross-section evaluation (such as ENDF/B-VI) are accounted for. Thermal neutrons are describedby both the free gas and S(α,β) models. For photons, the code accounts for incoherent andcoherent scattering, the possibility of uorescent emission after photoelectric absorption,and absorption in electron/positron pair production. Electron/positron transport processesaccount for angular deection through multiple Coulomb scattering, collisional energy losswith optional straggling, and the production of secondary particles including K x-rays,knock-on and Auger electrons, bremsstrahlung, and annihilation gamma rays from positronannihilation at rest. Electron transport does not include the eects of external or self-induced electromagnetic elds. Photonuclear physics is available for a limited number ofisotopes.

Important standard features that make MCNP very versatile and easy to use include apowerful general source, criticality source, and surface source; both geometry and outputtally plotters; a rich collection of variance reduction techniques; a exible tally structure;and an extensive collection of cross-section data.

MCNPX

MCNPXTM is a general purpose Monte Carlo radiation transport code designed to trackmany particle types over broad ranges of energies. It is the next generation in the series ofMonte Carlo transport codes that began at Los Alamos National Laboratory nearly sixtyyears ago. MCNPX 2.6.0 is the latest Radiation Safety Information Computational Center(RSICC) release of the code. MCNPX 2.6.0 includes many new capabilities, particularlyin the areas of transmutation, burnup [FEN06a, FEN06b, FEN08], and delayed particleproduction. Many new tally source and variance-reduction options have been developed.Physics improvements include a new version of the Cascade-Exciton Model (CEM), theaddition of the Los Alamos Quark-Gluon String Model (LAQGSM) option, and a substantialupgrade to muon physics.

48


The MCNPX program began in 1994 as an extension of MCNP4B and LAHET 2.8 insupport of the Accelerator Production of Tritium Project (APT). The work envisioned aformal extension of MCNP to all particles and all energies; improvement of physics simu-lation models; extension of neutron, proton, and photonuclear libraries to 150 MeV; andthe formulation of new variance-reduction and data-analysis techniques. The program alsoincluded cross-section measurements, benchmark experiments, deterministic code develop-ment, and improvements in transmutation code and library tools through the CINDER90project.

The integration of CINDER90 into the MCNPX Monte Carlo transport code provided a self-contained MonteCarlo-linked depletion capability [51] . The continuous-energy reactionrates determined during the MCNPX steady-state calculation are used to calculate one-group collision rates to be used by CINDER90. The reaction rates not treated by MCNPXare obtained by collapsing the 63-group cross sections inherent in the CINDER90 librarywith the MCNPX steady-state 63-group ux. The accuracy of this procedure is dictated bythe adequacy of the inherent CINDER90 library to the spectrum of the specic problem.To solve the set of coupled depletion equations, CINDER90 reduces it to a set of lineardierential equations using the Markov method, instead of using the matrix exponentialmethod such as ORIGEN or ACAB.

MCNPX provides to CINDER90:

• Average k eective, ssion ν, and ssion Q

• 63-group ux in each material to be burned

• Isotopic atom densities and material volumes

• Absorption and ssion reaction rates for each nuclide containing transport cross sec-tions

• Power level and burn time

CINDER90 provides to MCNPX:

• Updated isotopic atom densities

• Burnup quantities

Every depletion calculation using MCNPX is made according to the scheme in gure 4.1.

Some useful information about MCNPX-CINDER:

• MCNPX calculates continuous energy-integrated reaction rates for all isotopes con-taining transport cross-section information for the following reactions: (n, gamma),(n,f), (n,2n), (n,3n), (n, alpha) & (n, proton).

• CINDER90 takes into account the rest of the interactions, as well as the reactionrates for isotopes not containing transport cross-section information, using a intrinsiccross-section library in 63-energy groups (cinder.dat).

49


Figure 4.1: Coupling scheme between MCNPX and CINDER90

• For this reason, MCNPX calculates a 63-groups ux that is sent to CINDER90 andmatched with the 63-group cross-section set inherent in CINDER in order to determinethe complete set of reactions.

• However, CINDER inherent cross section library was collapsed over a generic spectrumthat may or may not be representative of the system to be analyzed and thus maylead to discrepancies in the isotope inventory.

• To solve the set of coupled depletion equations, CINDER90 reduces it to a set oflinear dierential equations using the Markov method, instead of using the matrixexponential method such as ORIGEN or ACAB.

• In the MCNPX-CINDER coupling, no capability to run depletion steps without trans-port simulation is available.

4.2.1.2 SERPENT

SERPENT has been used with dierent purposes along this thesis. It was rst employed fornodal cross section generation (Section 9) and afterwards for assessment of minor actinidetransmutation (Section 6).

SERPENT [86] is a three-dimensional continuous-energy Monte Carlo reactor physics bur-nup calculation code. It was initiated as the program "Probalistic Scattering Game" startingfrom 2004 in VTT Technical Research Centre in Finland. Since then it has been improvingand incorporating the new outstanding methodologies. The code is specialized in lattice

50


Figure 4.2: Depletion scheme followed by SERPENT

physics calculations. The neutron transport is based on a combination of conventionalsurface-to-surface ray-tracing and the Woodcock delta-tracking method. Similar to theMCNP and MCB Monte Carlo codes, a universe based geometry model is applied to de-scribe the conguration of fuel pins and sub-assemblies.

Burnup depletion equations are solved using the matrix exponential method CRAM [119,118], providing a robust and accurate solution with a very short computation time. A com-parison between CRAM, ORIGEN solver and other TTA (Truncated Taylor Approximation)methods proved the advantages of the CRAM method in terms of accuracy and runningtime, thanks to the mathematical approach behind CRAM [73].

SERPENT uses a predictor-corrector method, as shown in Fig. 4.2. In this case, after aprovisional irradiation step (predictor step), a neutron transport problem is solved at endof step (EOS), and the average uxes and cross sections between the BOS (beginning ofstep) and EOS are employed to recalculate the EOS compositions [74]. Regarding branchingratios in metastable isotopes, SERPENT takes by default constant values, not dependenton energy. SERPENT computes directly the eective delayed neutron parameters using thesame Monte Carlo method developed at the NRG [120].

Comparing to existing most-used Monte Carlo codes, e.g. MCNP, the SERPENT MonteCarlo code has two merits. First, Monte Carlo calculation is accelerated by 5 to 15 timesfaster thanks to:

1. Evenly subdividing the energy grid in cross section spectrums, which reduces the timeconsumption for iteration process when generating cross section data;

2. Combining conventional ray-tracing and delta-tracking methods enhances neutrontransport calculation eciency.

Second, homogenized few-group constants, e.g. diusion rates, eective delayed neutron

51


fractions, and statistical error estimation can be obtained directly by including some newlydeveloped scripts. The SERPENT Monte Carlo code has been preliminarily validated withrespect to MCNP and the deviation between these two codes is reported to be less than2% [85]. However, a burn-up calculation using SERPENT code requires a lot of memorystorage capacity, which is one of the main drawbacks of the code. As a consequence, newimprovements have been implemented in the new version of the code, SERPENT2 [87],currently in a beta-testing phase.

4.2.1.3 EVOLCODE

EVOLCODE was used during this thesis as a reference code to support the burn-up calcu-lations performed by SERPENT. Indeed, Chapter 6 presents a comparison between EVOL-CODE and SERPENT in terms of burnup capabilities, applied to the assessment of minoractinide transmutation in the ESFR core was performed. The results of this work werepublished in Annals of Nuclear Energy [105].

EVOLCODE 2.0 [94] is a CIEMAT development to automatically couple MCNPX [25]and ORIGEN [104] or ACAB codes [123], both of them based on the matrix exponentialmethod. Fuel material homogeneously evolves along several burn-up steps for cells thatcan be selected by the user. The rst stage solver is the MCNPX code, which allowsan important degree of the heterogeneity description in the reactor core model. In asecond computational stage, EVOLCODE obtains a single-group cross section for materialsin every core cell under MCNPX predicted spectra conditions and feeds ORIGEN code(alternatively ACAB) to provide the inventory evolution under irradiation for a user-inputburnup step. After this stage, EVOLCODE2 automatically generates a second MCNPXinput with updated material cards to estimate new spectra and relative uxes for the nextburn-up step.

EVOLCODE uses a predictor-corrector method based on constant neutron ux value alongthe burnup step. A rst ux value is tentatively assumed to obtain a preliminary estimationof the irradiation results. This value is based on the normalization of MCNPX results(accounting for all reactor cells) before irradiation. A provisional irradiation is performedand the new composition generally leads to end of step power values relatively far fromspecied. Therefore, the ux is corrected to match the required power for the denitivestep. The assumed macroscopic cross sections to start irradiation are always taken at thebeginning of step [92].

A specic issue is the branching ratio in metastable isotopes, for which EVOLCODE con-siders energy-dependent branching ratios taken from the JEFF 3.1.1 data library.

CIEMAT activities concerning EVOLCODE 2.0 are completed with systematic comparisonof predictions to real irradiations results [93], as well as cross-checking to other codes.

The βe is calculated using a modied MCNPX version developed at CIEMAT. It is basedon a generalization of the Meulekamp method [120]: when a ssion takes place it is furtherchecked if a delayed neutron produced that ssion (all neutrons were labeled prompt ordelayed at birth). Then, the total number of ssions originated by delayed neutrons is

52


Figure 4.3: Depletion scheme followed by EVOLCODE

divided by total ssions generated by all neutrons, which directly leads to the βe estimation.Among other internal exercises, the modied MCNPX version to compute βe has beencrosschecked to other codes in the frame of the CP-ESFR project [54] with good agreements.

4.2.1.4 KENO-VI

The SCALE code system dates back to 1969, when the current Nuclear Science and Tech-nology Division at Oak Ridge National Laboratory (ORNL) began providing computationalsupport in the use of the new KENO code to the U.S Atomic Energy Commission (AEC)sta. From 1969 to 1979 the AEC certication sta relied on the ORNL sta to assist themin the correct use of codes and data for criticality, shielding, and heat transfer analyses oftransportation packages. Shortly after creation of the U.S. Nuclear Regulatory Commission(NRC) from the AEC, the NRC sta proposed the development of an easy-to-use analysissystem that would provide the technical capabilities of the individual modules with whichthey were familiar. With this proposal, the concept of the SCALE code system was born.

The initial version of SCALE was released by the Radiation Safety Information Computa-tional Center (RSICC) at ORNL in 1980. Since that time the system has been considerablyenhanced. SCALE runs on Unix, Windows, Linux and Intel Mac computer platforms.

The concept of SCALE was to provide standardized sequences. Input for the controlmodules has been designed to be free-form with extensive use of keywords and engineering-type input requirements. The most important feature of SCALE system is the capability tosimplify the user knowledge and eort required to prepare material mixtures and to performadequate problem-dependent cross-section processing.

SCALE includes two versions of KENO, known as KENO-V.a and KENO-VI. Both versionsare 3-D multigroup Monte Carlo codes employed to determine eective multiplication factors(k-e) for multidimensional systems. KENO-V.a has been part of SCALE for more than20 years. Its simplied geometry makes it very easy to use and much more computational

53


ecient than other Monte Carlo criticality safety codes. KENO-VI was introduced in 1995in SCALE 4.3 to provide SCALE users with a more general geometry system known as theSCALE Generalized Geometry Package (SGGP) for constructing more-complex geometrymodels. KENO-VI has a much larger assortment of bodies. It constructs and processesgeometry data as sets of quadratic equations. A lenghty set of simple, easy-to-use geometricfunctions, similar to those provided in KENO-V.a, and the ability to build more complexgeometric shapes are the heart of the geometry package in KENO-VI. The code's exibilityis increased by allowing the following features: intersecting geometry regions; hexagonal aswell as cuboidal arrays; regions, HOLEs, arrays, and units rotated to any angle and truncatedto any position; and the use of an array boundary that intersects the array. KENO-VImaintains all the exibility and options of KENO-V.a plus a variety of new options. InKENO-VI, units can be constructed using both the simple geometric shapes provided andthe tailored geometric shapes constructed using quadratic equations.

KENO-VI was employed to model the 3-D ESFR core, in order to assess the uncertaintiesof the nuclear data on the reactivity coecients using the TSUNAMI module.

4.2.2 Deterministic methods

4.2.2.1 ERANOS

ERANOS was employed for nodal cross section generation at the rst stage of this thesis.That application is described in more detail in Chapter 9 .

ERANOS is the reference deterministic code for fast reactors. It is a deterministic codesystem, developed at CEA, that performs core, shielding, as well as fuel cycle calculations,and includes the most recent developments in calculational methods, such as the collisionprobability method in many groups and a 3D nodal transport theory variational methodwith perturbation theory and kinetics options.

ERANOS calculations require ECCO results; ECCO is a cell code based on the sub-groupmethod combined with a ne group transport calculation to compute homogenized crosssections considering the isotopic composition of each cell. ECCO can solve either homo-geneous descriptions of assemblies and pins, or heterogeneous ones including the detailedgeometry.

These macroscopic cross sections can then be used in the ERANOS code to model the entirecore and to solve the diusion or the transport equation in RZ or 3D geometries. With theneutron ux already calculated, the user can do a depletion calculation solving Batemanequations. Once the new isotopic concentrations have been calculated, macroscopic crosssections can be directly updated to do a new core calculation, or a new ECCO cell calculationto update the macroscopic cross sections can be performed.

The rst option is usually adopted as it is a good approximation for fast reactors to assumethat microscopic cross sections calculated for every group of energy will not change much,reducing the calculation time. This core calculation employing ERANOS/ECCO system isdescribed in Fig.4.4.

54


Figure 4.4: Description of ERANOS system

For RZ models of the core, both transport and diusion equations can be solved employingBISTRO solver. For a 3D model (Cartesian or Hexagonal-Z), the diusion equation will besolved with H3D solver and the transport equation with VARIANT solver. The advantageof employing BISTRO solver is that it will provide a high capacity of space discretizationand perturbation studies.

Depending on both the type of calculation required and the library used, the ECCO com-putation can be done with 33, 172, 175 or 1968 energy groups. Most of the time, it issucient to do a 1968 group calculation for the self shielding calculation condensed to33 groups. The ERANOS ux calculation can only be done with 33 groups in transportbecause of computer memory shortages, while the diusion equation could be solved witha 172 energy group discretization.

4.2.2.2 PARCS

PARCS has been taken as reference for the developments of COBAYA3-ANDES to beapplied for fast reactors.

The PARCS Kinetics Core Simulator Module [22] was originally developed by the PurdueUniversity (USA) in 2002 [48]. Numerous computational methods have been developed andimplemented in PARCS to model each phenomenon encompassed by the spatial kineticsproblem solution. It is based on a Coarse Mesh Finite Dierence (CMFD) method to solve

55


the eigenvalue problem and the time-dependent neutron transport equations. Two-nodeproblems are solved iteratively to correct for the discretization error in the nodal interfacecurrent resulting from the nite dierence approximation in a coarse mesh structure. InPARCS, the nodal expansion method (NEM) [52] and the analytic nodal method (ANM)[126] can both be used to obtain the two-node solution.

PARCS can be used to simulate reactors with hexagonal geometries, such as fast reactors.The Triangle-based Polynomial Expansion Nodal (TPEN) method [36] is used by the hexag-onal nodal method implemented in PARCS. This section describes the theoretical basis forthis method, together with the CMFD acceleration scheme which utilizes an innovativedynamic condensation method.

One interesting characteristic of the TPEN method is that it applies the transverse integra-tion over the entire hexagon, i.e. it treats the radial x- and y- directions together, solvingan unique radial transverse-integrated neutron diusion equation. In total, two equationsare solved for the hexoctahedron node; one for the radial direction and one for the axialdirection. The radial problem is actually solved by splitting the hexagon into six triangles,for each of which a polynomial expansion of ux is employed.

There is a particular issue of this method when the multi-group problem is solved. Thesolutions of the two separate transverse-integrated equations provide two sets of nodeaverage group uxes rather than one, given the incoming current boundary conditions.Since only the outgoing currents need to be taken from the one-node solution, it is not anissue for two-group problems. However, in the multigroup problem, a unique set of groupuxes is longed, because these uxes determine the nodal spectra required in the groupconstant generation for the two-group CMFD problem, as it will be discussed later. Inaddition, in order to avoid the direct inversion of the group block, the ssion source termsare moved to the right hand side of the equations.

Following, some particularities of the multi-group TPEN method formulation are described,taken from the PARCS manual [22].

Muti-group TPEN Method

As just discussed, the transverse-integrated neutron diusion equations dened for thehex-octahedron domain (Fig. 4.5) are divided into six radial and one axial equations.Considering index m denoting the m-th triangle, the balance equations can be formulatedas:

56


−Dg

(∂2

∂x2 + ∂2

∂y2

)φRng (x, y) + Σrgφ

Rng (x, y)−

∑g′<g

Σsg′gφRng′ (x, y)

= QRng (x, y)− LZng (x, y), m = 1...6

−Dg∂2

∂z2φZg (z) + Σrgφ

Zg (z)−

∑g′<g

Σsg′gφZg′(z) = QZ

g (z)− LRg (z)

QUg (u) = χg

keff

∑g′νΣfg′φ

R,oldg′ (u) + STFSg (u), u = (x, y) or z

LZng (x, y) = 1hZ

(JZmgT (x, y)− JZmgB (x, y)

),

LRg (z) = 2√

39hR

∑u=1,3

(Jxg(u+3)(z)− Jxgu(z)

)

(4.1)

The Q source term includes the ssion and upscattering sources and, in case of transientcalculations, also the transient xed source (TFS). Regarding the spatial dependence ofthe transverse-leakage sources in Equation 4.1, there is a dierent between the original andthe actual TPEN formulation. While in the original derivation, all transverse leakages wereconsidered known, and the two balance equations could be solved independently, in thisderivaton, the transverse-leakage of the node of interest is treated as unknown, and thusthe radial and axial equations are now coupled through the transverse leakage terms.

The radial equation in Eq. 4.1 above is solved by representing the intranodal ux withinthe triangle by a nine-term, two-dimensional polynomial of the following form:

φRng (x, y)B = cng0 + angxx+ angyy + bngxx2 + bnguu

2 + bngpp2 + cngxx

3 + cnguu3 + cngpp

3 (4.2)

where

u = −1

2x−√

3

2y and p = −1

2x+

√3

2y

These nine coecients are related with nine unknowns:

• 3 surface average uxes

• 3 corner point uxes

• node average ux

• rst order x- and y- momentum of the ux

In order to determine the nine unknowns, nine constraint conditions are imposed:

57


Figure 4.5: Geometry and Boundary Conditions and Coordinates for TPEN

• 3 surface average current continuity conditions

• 3 corner point leakage balance (CPB) conditions,

• nodal balance condition

• rst-order x-moment and y-moment balance conditions

On the other hand, the axial equation is solved by the nodal expansion method (NEM)that involves ve unknowns: rst- and second-order z-moments as well as the two surfaceaverage uxes and node average ux.

Solution of the TPEN Equations

In order to solve the equations just decribed, a linear system is built up (Eq. 4.3). In total,for the hex-octahedron shown in Figure 4.5, there are 36 unknowns per group:

• 6 hexagon surface outgoing partial currents,

• 6 inner surface average uxes,

• 6 triangle average uxes,

• 6 x-moments,

• 6 y-moments,

58


• 1 center point ux,

• 2 axial partial currents (top and bottom),

• 1 rst-order z-moment,

• 1 second-order z-moment, and

• 1 hexagon average ux

To determine these unknowns, 36 constraint conditions are required:

• 6 triangle nodal balance conditions,

• 6 triangle x-moment equations,

• 6 triangle y-moment equations,

• 6 inner surface current continuity conditions,

• 6 radial incoming current boundary conditions,

• 1 center point CPB condition,

• 2 axial incoming current boundary conditions,

• 1 rst order z-moment equation,

• 1 second-order z-moment equation, and

• 1 hexagon average ux constraint which states the average of the 6 triangle averageuxes is the same as the hexagon average ux.

In Eq. 4.3, for a given group A, X, Y, S and CR are 6x6 matrices while CZ is a 2x2 matrix.Other matrices are all 1x1. The dimensions of the unknown vectors are dened accordingly.

A1 0 0 A2 A3 A4 A5 0 0 0

0 X1 0 X1 X2 X3 X4 0 0 0

0 0 Y1 Y2 0 0 0 0 0 0

S1 S2 S3 S4 0 S5 0 0 0 0

CR1 CR

2 0 0 CR3 CR

4 0 0 0 0

0 P1 0 0 P2 P3 0 0 0 0

0 0 0 0 0 0 CP2 CP

3 CP4 CP

5

0 0 0 0 ZF1 0 ZF

2 ZF3 0 0

0 0 0 0 ZS2 0 ZS

3 0 ZS4 ZS

5

I6 0 0 0 0 0 0 0 0 −6

ϕ

ϕx

ϕy

ϕs

JRo

ϕp

JZo

ϕFz

ϕSz

ϕH

=

S

Sx

Sy

Ss

SRo

Sp

SZo

SFz

SSz

0

(4.3)

59


These equations provide the solution of the multigroup one-node problem, for which themultigroup node-average uxes and interface current are useful for the next step. Probablyone of the most interesting particularities of this method is actually the dynamic conden-sation of the multigroup cross sections, which uses the nodal spectrum obtained from themultigroup node-average uxes to weighten them as follows:

ΣαG =∑g∈G

ϕgΣαg, G=1 ó 2 (4.4)

where ϕg = φg

φGand φG =

∑g∈G φg.

On the other hand, the multigroup interface current, also taken from the solution of theone-node problem, is used to determine the two-group corrective nodal coupling coecientby the following equation:

DG = −DG

(φlG − φrG

)+ JG

φlG + φrG(4.5)

where DGis the nite dierence based nodal coupling coecient and JG =∑

g∈G Jg.

In fact, this corrective coecient forces the interface current of the CMFD method to beidentical to the one obtained from the one-node problem using the TPEN method.

Once the cross sections are condensed to two groups and the corrective coupling coecientis determined, the two-group CMFD linear system can be formulated, previously deningthe interface current as a linear combination of the two-group node average uxes of thetwo nodes of the interface (Eq. 4.6).

JCMFDG = −DG

(φr,CMFDG − φl,CMFD

G

)− DG

(φr,CMFDG + φl,CMFD

G

)(4.6)

The solution of the CMFD linear system yields a two-group ux distribution. At convergenceof the iteration between the multigroup and two group solutions, the two-group nodalreaction rates preserve the corresponding multigroup reaction rates.

Once a two-group CMFD solution is obtained, it need to be prolongated into the multigroupone, by using appropriate boundary conditions, that need to be established. The informationobtained from the previous one-node calculation is used for the iterative process.

Further details about the TPEN implementation in PARCS, as well as the results of vali-dation for a series of benchmark problems is provided in Reference [36] and in the PARCSmanual [22].

60


4.2.2.3 DYN3D

As well as PARCS, the methodology implemented in DYN3D has been explored as a refer-ence for the applicabilty of ANDES to SFRs.

DYN3D is a DYNamical 3-Dimensional code initially developed for thermal reactor cores,and recently extended to fast reactor cores. The three-dimensional neutronic model utilizenodal expansion methods (NEM) to solve the multigroup neutron diusion equation foCartesian and hexagonal-z geometry. The steady state with xed source, the adjoint uxdistribution and the transient behaviour can be calculated.

Assembly discontinuity factors (ADF) can be taken into account for correction of homo-geneization errors. A ux reconstruction can be performed for chosen assemblies to calculatepowers at given positions in the assembly. It can be used for hot channel calculation. Crosssection libraries generated by dierent cell codes as CASMO, HELIOS or NESSEL are linkedto DYN3D. Recently, SERPENT has been started to be used for fast reactors.

The kinetic model is based on the solution of the three-dimensional multigroup diusionequation by nodal expansion methods. It is assumed that the macroscopic cross sectionsare spatially constant in a node. Concerning Cartesian geometry three one-dimensionaldiusion equations are solved for the traverse integrated uxes of the nodes in the threedirections x,y,z. In the case of hexagonal-z geometry, a two-dimensional diusion equationin the hexagonal plane and a one-dimensional equation of the z-direction are solved for thetraverse integrated uxes. In each energy group, the one-dimensional equations are solvedwith the help of ux expansions in polynomials up to 2nd order and exponential functionsbeing the solutions of the homogeneous equations. The ssion source in the fast group andthe scattering source in the thermal group as well as the leakage terms are approximated bythe polynomials. Considering the 2-dimensional equation in the hexagonal plane, the sideaverage values (HEXNEM1) or the side averaged and the corner point values (HEXNEM2)of ux and current are used for the coupling of nodes in the radial direction. In the steadystate, the homogeneous eigenvalue problem or the heterogeneous problem with given sourceis solved. An inner and outer iteration strategy is applied. The outer iteration (ssion sourceiteration) is accelerated by Chebychev extrapolation.

A thermal-hydraulic model was implemented in DYN3D, whose extension to sodium fastreactors is nowadays being ongoing.

Detail for the nodal expansion for the hexagonal-z geometry

As commented before, the traverse integration is used for splitting the three-dimensionalproblem into a two-dimensional problem in the hexagonal plane and a one-dimensionalproblem over the axial direction.

First, the current is connected with the ux gradient by Fick's law:

Jg(r, t) = −Dg∇Φg(r, t) (4.7)

61


Integrating the diusion equation over the axial direction, a two-dimensional equation foreach group g is obtained:

−Dg

(∂

∂x2+

∂

∂y2

)Φg(x, y) + ΣgΦg(x, y) = Qg(x, y) (4.8)

with

Qg(x, y) = Sg(x, y)− Lg(x, y) (4.9)

Lg(x, y) = −Dg

az

z1ˆ

z0

d2

dz2Φg(r)dz (4.10)

The ux in the hexagonal plane is expanded by using polynomials up to 2nd order andexponential functions being the solutions of the homogeneous equations. The HEXNEM2ux expansion for each group is expressed as:

Φ(x′, y′) ≈5∑i=0

ci.hi(x′, y′) +

6∑k=1

as,k.exp[B′(es,k.r

′x,y)]

+6∑

k=1

ac,k.exp[B′(ec,k.r

′x,y)]

(4.11)

with the vector r′x,y = (x′, y′). The unit vectors es,k point from the hexagonal center to themidpoints of the sides and ec,k from the center to the corners of the hexagon, respectively.The buckling B′ is given by:

B′ =

√Σ′

D′(4.12)

The polynomials hi(x′, y′) are given by

h0 =1

N0

; h1 =x′

N1

; h2 =y′

N2

; h3 =1

N3

(x′2 + y′2 − 5

9

); h4 =

x′2 − y′2

N4

; h5 =x′y′

N5

;

(4.13)

The source term is approximated by the polynomials

Q′(x′, y′) ≈5∑l=0

ql.hl(x′, y′) (4.14)

Inserting the ux approximation (4.11) and the source term (4.14) in the diusion equation(4.8) with (4.9), the coecients of the polynomial expansion are obtained.

62


Figure 4.6: Hexagon with the considered directions es,i and ec,i

The coecients to the exponential functions in (4.11) are obtained from the inter-nodalinterface conditions or the boundary conditions for nodes located at the outer boundary ofthe system. In the frame of diusion approximation, the current of neutrons, which aregoing in or coming from a half space with the direction e are represented by the outgoingJ+ and the incoming J−partial currents. The side-averaged or corner point values of thepartial currents are given by

J± =1

2

(1

2Φ± (eJ)

)(4.15)

with the side-averaged or corner point uxes Φ and the side-averaged or corner point netcurrents J . The directionse can be seen in Fig. 4.6 . Using the ux expansion (4.11) and(4.8) the partial currents for all sides and corners can be represented by the equations

J±s = P±s C +Q±ssAs +Q±scAc (4.16)

J±c = P±c C +Q±csAs +Q±ccAc (4.17)

with the vectors C, As, Ac of the coecients ci, as,k, ac,kand of the expansion 4.11,respectively. The side-averaged partial currents J±s,k form the vector J±s and the partialcurrents at the corners J±c,k the vector J±c

63


Transverse leakage terms

The transversal leakage terms are approximated by applying the polynomials in the hexag-onal plane and the axial direction. Focusing in the transversal leakage of the hexagonalplane, the polynomials (4.13) are used:

L′(x′, y′) ≈ a

5∑i=0

li.hi(x′, y′) (4.18)

The coecients li of the transversal leakage in the axial direction have to be calculatedfrom the averaged values of the transversal leakage of the node and the neighbouring nodes.The averaged value of a considered node n is obtained from equation 4.10 with the help ofthe partial currents

L0 =1

az

(J+z1− J−z1 + J+

z0− J−z0

)(4.19)

Within the nodes, a linear change of the transversal leakage is assumed to approximate the6 averaged valuesLk of the transversal leakage at the sides of the hexagon from the averageof the considered node L0 and the average values L0,k of the 6 neighbouring nodes (k =1,2,...,6).

Lk =D (L0 + L0,k)

(D +Dk)for k = 1, 2, ..., 6 (4.20)

D, Dkbeing the representative diusion coecients. This approximation is obtained fromthe assumption that the second derivative of the ux in the direction of z at a radialinterface between two assemblies fullls the same conditions as the ux itself. Consideringthe radial interface k of the node n, which is the interface k' of the node n + 1, the followingconditions are obtained for the leakage at the interface

1

DnLnk =

1

Dn+1Ln+1K (4.21)

∂Lnk∂x

=∂Ln+1

K

∂x(4.22)

The coecients li of polynomial expansion4.18 are obtained as:

64

4.3. Thermal-hydraulics codes

l0 = N0L0

l1 = N1

6(2L1 + L2 − L3 − 2L4 − L5 + L6)

l2 = N2

2√

3(L2 + L3 − L5 − L6)

l3 = 3N3

10(L1 + L2 + L3 + L4 + L5 + L6 − 6L0)

l4 = 3N4

16(2L1 − L2 − L3 + 2L4 − L5 − L6)

l5 = 3√

3N5

16(L2 − L3 + L5 − L6)

4.3 Thermal-hydraulics codes

4.3.1 Core thermal-hydraulics codes

4.3.1.1 SUBCHANFLOW

SUBCHANFOW has been chosen as the core thermal-hydraulic code to be coupled withCOBAYA3-ANDES. SUBCHANFLOW [69, 70] is a subchannel thermal-hydraulic codebased on the COBRA family of computer programs [20, 31] but with improved capabil-ities. It has been developed at the Institute for Neutron Physics and Reactor Technology(INR), Karlsruhe Institute of Technology (KIT). In opposite to the COBRA family of sub-channel codes, SUBCHANFLOW uses rigorously SI units internally in all modules. It solvesthe steady state and transient mixture balance equations coupled with a fuel rod heat trans-fer model, considering cross-ow at sub-channel level. Square and hexagonal fuel assemblygeometries of Light Water Reactors (LWR) or innovative reactors, such as the SodiumFast Reactor, can be modeled. Coolant properties and state functions are implementedfor water using the IAPWS-97 formulation (The International Association for the Proper-ties of Water and Steam). In addition, property functions for liquid metals (sodium, leadand lead-bismuth) and gases (helium) are available, too [72]. For all these characteristics,SUBCHANFLOW was the code chosen for the coupling with COBAYA3.

A detailed description of the code and the specic models used in it can be found in [63],or [10], for the purpose of this thesis only a brief description of the main physical modelsimplemented in SUBCHANFLOW is given.

65


Heat conduction model

A fully implicit nite dierence method is employed by SUBCHANFLOW to calculate heatconduction in the fuel pellet and within the cladding material.

The heat balance equation to be solved is given by:

ρ(r, T )Cp(r, T )δ

δtT (r, T ) = ∇ · k(r, T )∇T (r, T ) + qm(r, T ) (4.23)

where r is the position vector, T (r, T ) is the temperature as a function of time and space,k(r, T ) is the thermal conductivity of the considered material as a function of positionand temperature, q′′′(r, T ) is the heat generation rate per unit volume (volumetric heatsource), Cp(r, T ) is the specic heat of the considered material as a function of position andtemperature, and ρ(r, T ) is the density of the material function of position and temperature.

In order to solve the equation some simplications are usually done. For example, if itis assumed that the heat is transferred mainly in the radial direction (axial heat transferneglected) and the temperature distribution is isotropic in the θdirection, (δT/δθ = 0), theheat balance equation in cylindrical coordinates might be written as:

ρ(r, T )Cp(r, T )δ

δtT (r, T ) =

1

r

δ

δrrk(r, T )

δ

δrT (r, T ) + q′′′(r, T ) (4.24)

In addition, temperature dependent material properties for dierent fuels (UO2 and UO2PuO2)and cladding (Zircaloy and stainless steel) are implemented by default in SUBCHANFLOW.Fixed values can also be chosen if needed. For particular applications, such as SFR, newmaterials can be described and their associated properties correlations (density, emissiv-ity,thermal conduction, specic heat, etc.) can be easily implemented in the source code.

Heat transfer

The heat ux transferred from the external clad wall to the uid q is ruled by:

q′′ = hc(Tw − Tb) (4.25)

where Tw is the wall temperature and Tb is the temperature of the uid.

The heat transfer coecient hc is determined by using empirical correlations depending onthe heat transfer mode. The heat transfer mode along the boiling curve is described by acombination of dierent heat transfer models. Therefore, for example, for the single-phaseliquid force convection, the Dittus Boelter or the Gnielinski correlation can be used [71].The subcooled nucleate boiling region relies on the Thom model [63] or Schrock- Grossmancorrelation[10], etc. A full description of the models used in SUBCHANFLOW can be foundin the references mentioned in this paragraph.

66


Regarding heat transfer through the gap, from version 2.1 on there are three input modelsavailable in SUBCHANFLOW:

• a simple model that considers constant gap conductivity (hgap). It is usually employedfor the NEA benchmarks

• a second model that computes hgap as a function of the gap temperature, pressureand burn-up

• a complex model called URGAP [83], that takes into account the cladding swellingand dimensional variations of the fuel pin

Fluid Dynamics model

The two-phase ow model used in SUBCHANFLOW considers the uid as a mixture. Basedon three mixture conservation equations, a four-equations formulation is established: oneequation for conservation of mass, one for conservation of energy, and two for conservationof momentum (one in axial direction and the other for lateral direction in order to take intoaccount the cross ow).

If it is assumed that the velocities and the pressures of the two phases are the same (denotedas v and p respectively) and that there exists thermodynamic equilibrium.

In case of sodium, recent developments have been performed in order to extend the two-phase ow formulation to sodium ow. Therefore, SUBCHANFLOW is able to treat twophase ow mixture for both water and sodium ( 3 equations with slip)

The three conservation equations can be written as follows:

Mass conservation:

δρ

δt+∇ · (ρv) = 0 (4.26)

Energy conservation:

∂ (ρh)

∂t+∇ · (ρhv) = −∇ ·

(q′′ + q′′T

)+∂p

∂t+ q′′′ − (q′′′wv + q′′′wl) (4.27)

Momentum conservation:

∂ (ρv)

∂t+∇ · (ρvv) = −∇p+ ρg +∇ ·

(τ + τT

)+ τ ′′′ (4.28)

where ρ and h are the density and enthalpy of the mixture respectively, q′′and q′′T are themolecular and turbulent heat uxes, and q′′′ is the energy generated in the uid. Finally τand τT are the molecular and turbulent stress tensors and τ ′′′ is the interfacial wall dragtensor.

67


In case of sodium ow, the correlations implemented in SUBCHANFLOW are based on[62].

Iterative procedure

The iterative procedure implemented in SUBCHANFLOW for the solution of the uidequations is briey described as follows:

1. The mixture enthalpy h is obtained by means of the energy conservation equation4.27.

2. With the enthalpy h , the quality x can be obtained in terms of the liquid andevaporation enthalpy: x = h−hl

hev

3. The vapour volume fractionαv (x, slip) is obtained as a function of the quality andthe slip velocity model.

4. The eective mixture density ρ is calculated.

5. The momentum and mass conservation equations are solved for the pressure dropand momentum.

6. Repeat from 1 with actualized parameters.

The energy balance matrices for enthalpy are solved using a Successive Over-Relaxation(SOR) method, whereas a Gauss reduction procedure is used for the solution of the mo-mentum balance matrices for the pressure drop and momentum. Details about the solutionprocedure can be found in [10, 63].

4.3.2 Plant thermal-hydraulics codes

4.3.2.1 SPECTRA

The SPECTRA (Sophisticated Plant Evaluation Code for Thermal-hydraulic Response As-sessment) is a thermal-hydraulic system code for transient analyses in Light Water Reactors,High Temperature Reactors and Liquid Metal Fast Reactors. SPECTRA is a code designedfor accident analysis, developed at NRG (Nuclear Research and Consultancy Group, Hol-land). It can model a power plant (nuclear or conventional) with all the components,including reactor vessel, primary, reactor control and safety systems, containment and re-actor building.

SPECTRA allows the user to simulate long-term reactor operations and their eects on theaccumulation of radioactive substances in the system. It has a number of unique features.For example, it can reveal the eects of a pipeline fracture, calculating what substances arelikely to be released and how much radiation emitted in various accident scenarios.

68


One the main features of SPECTRA is the use of thermal-hydraulic models that have beencompiled after an extensive research through the thermal-hydraulic- related literature andfrom the models available in other similar codes (CONTAIN, MAAP, MELCOR, RELAP,TRAC-BF1). The best models for each case have been selected and implemented in thesource code. Therefore, SPECTRA is not only a simulation tool, but also a library ofphysical models, that have been veried, documented and easy to use.

In opposite to other codes such as RELAP or TRAC whose models are focused on thewater-vapour uid with non-condensable gases, SPECTRA allows the simulation in detailof mixtures between multiple gases, such as hydrogen, helium, nitrogen, oxygen, and carbondioxide.

The point kinetic models are an approximation to the 3D neutron diusion equation tocompute the temporal evolution of the neutron ux. The mistake made when point kineticequation are employed instead of the 3D diusion equation is not too big in subcriticalsystem (around 5%) although this mistake increases as the system approach criticality,especially when local perturbations are present.

In order to assess the impact of minor actinides on transient behaviour, SPECTRA codewas used within the framework of the SP3.1.1 of the CP-ESFR project (see Section 8.3).

69


70

Part III

DEVELOPMENT OF ESFR COREMODELS

71

Chapter 5

Core models

5.1 Introduction

For the purpose of the thesis, dierent core models of the CP-ESFR core [54] were createdfor dierent code inputs and dierent purposes. Two main core congurations were denedwithin the project: the Working Horses core, which is the initial reference core; and theoptimized CONF2 core, developed during the project [28]. In addition to those referencecongurations, further cores are dened following dierent minor actinides (MA) loadingpatterns.

Among the purposes for which the models were developed, some might be highlighted suchas the core characterization analyses, assessment of MA transmutation, and the generationof nodal neutron parameters.

Along this Chapter, rst the main core congurations are described. Following, furtherdetails are explained regarding the particular core models developed for each code.

5.2 Description of core congurations

5.2.1 Working Horses

The so-called Working Horses is a 3600 MWth Sodium-cooled Fast Reactor that consistsof two separated driver fuel regions of 225 inner fuel assemblies (FA) (red, blue and yellowin Fig. 5.1) and 228 outer fuel assemblies (green in Fig. 5.1) with 271 fuel pins/FA. Theassembly pitch is 21.08 cm while the fuel pellet diameter is 0.943 cm. The FAs are loadedwith MOX fuel. Three radial rings of reector assemblies surround the active core (brownin Fig. 5.1). The control rod system is composed of 9 DSD (Diverse Shutdown Device,90%w 10B), located in the second ring of control assemblies, and 24 CSD (Control andShutdown Device, ~19.9%w 10B).

The axial layout of the Working Horses core presents, just above the active core, a small

73

5.2. Description of core congurations

Figure 5.1: MCNP model of WH core. Radial cross section

Figure 5.2: MCNP model of WH core. Axial cross section

gas plenum (Upper Gas Plenum), in purple in Fig. 5.2. On top of it there is a steelreector (Upper Axial Reector), dark brown in Fig. 5.2. Just below the active core, thereis another reector (Lower Axial Reector), gray in Fig.5.2, and the big gas plenum (LowerGas Plenum), purple as well.

Fuel specications

A mixture of Uranium and Plutonium oxide core (MOX) was considered, with a dierentplutonium content in the inner and outer fuel assemblies (14.43 %w and 16.78 %w re-spectively). The plutonium vector (Table 5.2) is estimated as the one coming from ~4500MWd/THM burnup, after 20 years cooldown. As a consequence, a small fraction of Am-241coming from Pu-241 decay is considered.

74


U-235 U-238

0.25% 99.75%

Table 5.1: Depleted Uranium vector (%wt)

Pu-238 Pu-239 Pu-240 Pu-241 Pu-242

3.6% 47.76% 28.89% 8.29% 10.46%

Table 5.2: Plutonium vector (%wt)

5.2.1.1 Minor Actinides loading patterns

In addition to the basic Working Horses core, some derived cores are proposed in orderto assess the transmutation capabilities and core performances when minor actinides areloaded in the core. All of them follow the same minor actinide vector (Table 5.3).

Three loading patterns were considered: two with a minor actinide content of 4 and 10%wt(MA/HM), respectively, homogeneously distributed in the fuel (HOM4 and HOM10); andone with an external radial blanket loaded with 15%wt (MA/HM) of minor actinides (HET4,bright blue in Fig.5.6).

Further information regarding the fuel composition of each core is described in Table 5.4.For the Pu and U vectors, the same tables as in the basic WH core are considered.

5.2.2 Optimized core: CONF2

The so-called CONF2 core [28] is basically an evolution from the Working Horses concept.It is also a 3600 MWth oxide fuelled Sodium-cooled Fast Reactor with two separated driverfuel regions of 225 inner fuel assemblies (FA) and 228 outer fuel assemblies with 271 fuelpins/FA. The assembly pitch and fuel pellet diameter are exactly the same: 21.08 cm and0.943 cm, respectively. The FAs are loaded too with MOX fuel with a small fraction ofAm-241 coming from Pu-241 decay during fresh fuel storage. The inner and outer regionspresent dierent Pu contents, which in this case dier from the previous core (14.76 wt%and 17.15 wt% respectively), aim to provide low peaks of the neutron ux and local power.The considered isotopic vectors, taken from [28], dier also a little bit from the onesspecied for the WH core:

Np-237 Am-241 Am-242m Am-243 Cm-242 Cm-243 Cm-244 Cm-245 Cm-246

10.86% 60.62% 0.24% 15.7% 0.02% 0.07% 5.14% 1.26% 0.09%

Table 5.3: Minor Actinide vector (%wt)

75


UO2 PuO2 (MA)O2

HOM4Inner fuel 81.57% 14.43% 4%

Outer fuel 79.22% 16.78 % 4%

HOM10Inner fuel 75.16% 15.04% 9.80%

Outer fuel 75.16% 15.04% 9.80%

HET4

Inner fuel 85.94% 14.06% 0%

Outer fuel 83.64% 14.36% 0%

Additional ring 68.88% 16.42% 14.7%

Table 5.4: MA loading cases

- Uranium isotopic composition: U-235 / U-238 = 0.25 / 99.75 (wt%). - Plutoniumisotopic composition: Pu-238 / Pu-239 / Pu-240 / Pu-241 / Pu-242 / Am-241 = 3.57 /47.39 / 29.66 / 8.23 / 10.37 / 0.78 (wt%).

Three radial rings of reector assemblies surround the active core, identicaly as in theWH. However, while the radial layout of the core remains the same when compared to theprevious core, the axial layout presents some dierences. Just above the active core, thereis a large sodium plenum. Further above in the axial direction there is a layer of neutronabsorbent material (boron carbide) and another layer of steel reector. In the lower part,just below the active core, there is a fertile region in order to provide neutron absorptions.

Just as in the WH core, the control rod system is composed of 9 DSD (Diverse ShutdownDevice, 90%w B-10), located in the second ring of control assemblies, and 24 CSD (Controland Shutdown Device, ~19.9%w B-10). Cross section and vertical views of the reactor coreare shown in Figs.5.3 and 5.4 respectively. In the vertical view, the following parts can bedistinguished from the top to the bottom: the upper shielding structure, absorber layer,sodium plenum, upper plug, upper gas plenum, active core, fertile blanket, lower gas plenumand lower plug. In addition, Fig. 5.5 shows the sub-assembly axial structure.

Fuel composition

Fuel composition diers from the previous core, as discussed before. It is given for the innerfuel, outer fuel and fertile blanket, respectively in Table 5.5.

5.2.2.1 CONF2-HET2 core

CONF2-HET2 consists of the same conguration as CONF2, with the same isotopics ininner fuel, outer fuel and lower fertile blanket, but with an additional ring of 84 radialblanket sub-assemblies. Fig. 5.10 shows the axial and radial layout of the model.

76


Figure 5.3: Cross section view of the optimized ESFR core, SERPENT post-processing

Figure 5.4: Vertical view of the optimized ESFR core, SERPENT post-processing

77


Figure 5.5: Axial structure of the CONF-2 conguration

Inner fuel Outer fuel Lower fertile blanket

U 85.24% 82.85% 100%

Pu 14.76% 17.15% 0%

MA 0% 0% 0%


O-16 11.825% 11.821% 11.858%

U-235 0.188% 0.182% 0.223%

U-238 74.870% 72.756% 87.919%

Pu-238 0.468% 0.544%

Pu-239 6.216% 7.223%

Pu-240 3.891% 4.521%

Pu-241 1.080% 1.254%

Pu-242 1.360% 1.580%

Am-241 0.102% 0.119%

Table 5.5: Weight-percentage contents in the fuel for the CONF2 conguration

78


MA ring fuel

U 80.00%

Pu 0.00%

MA 20.00%

MA ring fuel

O-16 11.723%

U-235 0.179%

U-238 74.443%

Np-237 2.977%

Am-241 10.703%

Am-242m 0.042%

Am-243 2.772%

Cm-242 0.004%

Cm-243 0.012%

Cm-244 0.907%

Cm-245 0.222%

Cm-246 0.016%

Table 5.6: Weight-percentage contents in the fuel for the CONF2-HET2 conguration

Fuel composition

The fuel composition of the driven fuel is identical to the reference CONF2 core. However,the fuel composition of the additional ring is dierent, given in Table 5.6. Note that boththe active region and lower blanket parts of the additional ring have the same isotopics.

5.2.2.2 CONF2-HOM4 core

The geometry layout of the CONF2-HOM4 core is exactly the same as the reference CONF2core. The only dierences lay on the fuel composition, due to the introduction of 4%wtMA in the core.

Fuel composition

Fuel composition for the inner fuel, outer fuel and fertile blanket is given in Table 5.7. Theweight-percentage contents of Pu are the same than in CONF2, reducing the U contents

79

5.3. Code models

to include the 4% of MA. Note that MA are included also in the lower fertile blanket.

5.3 Code models

In this section additional details of the models, particularly developed for each code, arepresented.

5.3.1 MCNP5 & MCNPX

The world-wide known MCNP code is a reference for neutron transport Monte Carlo codes.MCNP5 was rstly used during this thesis to allow a rst approach to the neutronic be-haviour of fast reactors, using MCNP5. Later on, MCNPX was used to study the isotopicevolution of the main nuclides along the burn-up. Following those motivations, a modelof the reference code was developed with dierent level of detail. In the very rst step, ahomogeneous denition was followed, to be compared afterwards with the heterogeneousone.

In order to assess the inuence of the heterogeneity on the reactivity two models weredeveloped. A rst one, were the dierent media were considered as homogenized materials;and a second one where the active region of the fuel assemblies were modeled in detail.For all of them, no axial discretization was considered, i.e., just one axial node per activeregion.

A single batch cycle was considered with 2050 EFPD lenght. The operating conditionsconsidered for the simulations are:

• Hot Full Power (3600 MWth), All rods (CSD, DSD) withdrawn

• Average coolant temperature: 470ºC

• Average structure temperature: 470ºC

• Average fuel temperature: 1227ºC

5.3.1.1 Minor Actinides loading patterns

As contribution from UPM to the WP4 SP2.1 of the CP-ESFR project, further modelswere developed following dierent minor actinides loading patterns. The motivation behindthis task was the assessment of the transmutation capabilities of the reference core. Theisotopic content of the minor actinides vector is shown in Table 5.3.

The dierent models developed with MCNP are described as follows

1. Homogeneous distribution of 4%wt (MA/HM) in the fuel (HOM4)

80

5.3. Code models


U 81.24% 78.85% 96.00%

Pu 14.76% 17.15% 0.00%

MA 4.00% 4.00% 4.00%


O-16 11.731% 11.730% 11.743%

U-235 0.182% 0.176% 0.215%

U-238 71.528% 69.425% 84.512%

Np-237 0.595% 0.595% 0.595%

Pu-238 0.469% 0.545%

Pu-239 6.222% 7.230%

Pu-240 3.894% 4.525%

Pu-241 1.080% 1.255%

Pu-242 1.363% 1.583%

Am-241 2.140% 2.140% 2.140%

Am-242m 0.008% 0.008% 0.008%

Am-243 0.554% 0.554% 0.554%

Cm-242 0.001% 0.001% 0.001%

Cm-243 0.002% 0.002% 0.002%

Cm-244 0.181% 0.181% 0.181%

Cm-245 0.044% 0.044% 0.044%

Cm-246 0.003% 0.003% 0.003%

Table 5.7: Weight-percentage contents in the fuel for the CONF2-HOM4 conguration

81

5.3. Code models

Figure 5.6: Radial layout of the HET4 core

2. Homogeneous distribution of 10%wt (MA/HM) in the fuel (HOM10)

3. Heterogeneous loading of 15%wt in a radial blanket, bright blue in Fig. 5.6 (HET4)

Finally, these and other loading patterns were assessed in [106].

5.3.2 ERANOS

As previously stated, ERANOS code was employed to have a rst view at the methodologyrequired to generate cross sections and neutronic parameters, to be used for a diusioncode. Since ERANOS is a deterministic code, the model created was not that detailedas in the Monte Carlo codes. In fact, homogeneous cells dene most of the regions withthe exception of the active region of the fuel assemblies, which are described in detail.The model created follows the Working Horses denition, as described in the CP-ESFRspecications.

In addition to the model of the whole core, a few mini-cores were dened with the aim toserve as trials for miscelaneous studies.

5.3.3 SERPENT

SERPENT code was used in the present thesis following mainly two purposes: cross sectiongeneration and assessment of minor actinides transmutation. First, SERPENT capabilitiesfor cross section generation were assessed for the preliminary model of the CP-ESFR core,the so-called Working Horses core, as explained in Chapter 9. Later on, the burnupcapabilities of SERPENT were evaluated in the framework of the Work Package WP5 ofoptimization in the SP2.1 CP-ESFR project . As aconsequence, the new proposed optimized

82

5.3. Code models

Figure 5.7: SERPENT model of the ESFR-WH core

Figure 5.8: SERPENT model of assembly of ESFR-WH core (radial and axial view)

core was modeled with SERPENT, following dierent congurations with regard to minoractinides loading (Chapter 6).

5.3.3.1 Working Horses core model

The same specications employed for the model of the ESFR-WH core with MCNP ( 5.3.1)were followed for the model of SERPENT. Figs. 5.7 and 5.8 show the core and fuel assembly,respectively of the ESFR-WH core. Given that in this case the objective was to generatehomogenized cross sections, the fuel assemblies' active length was simulated in detail, asshown in Fig. 5.8. However, the non-ssile regions remain as homogenized media.

Furthermore, a mini-core constituted from the assemblies of the whole core was taken toofor some specic analysis.

83

5.3. Code models

5.3.3.2 Optimized case: CONF2 core

As a consequence of the UPM contribution to the optimization task of the SP2.1 CP-ESFRproject [28], a new model was developed using SERPENT.

Geometry layout of the SERPENT model

Fig. 5.9 shows an axial and radial section of the SERPENT model.

• The control assemblies were considered withdrawn for the purpose of the analysis.

• Just a part of the axial structure is modeled: doted box underlines in Fig. 5.5, frombelow the upper shielding layer up to the lower plug (head and foot of the SAs areneglected).

• The radial reector was modeled according to the specications provided in the frameof the CP-ESFR project, maintaining the same geometry used in the Upper ShieldingSection (UPS), i.e., the wrapper tube with 19 pins consisting of a EM10 steel claddingand F17 steel pellets.

• Dimensions correspond to cold conditions (20ºC). However, the temperature of thecoolant and structure is 470ºC (743 K), the temperature of the ssile fuel is 1227 ºC(1500 K) and the temperature of the blanket is 667 ºC (900 K).

In addition, a more detailed model was developed for its comparison with the EVOLCODEmodel from CIEMAT (Figures 5.3 and 5.4).

5.3.3.3 Lower transmutation case: CONF2-HET2

The CONF2-HET2 conguration is based on the CONF2 core with HET2 loading (20% byweight of MA heterogeneously included within the core) dened in [34].

Geometry layout

CONF2-HET2 consists of the same conguration as CONF2, with the same isotopics ininner fuel, outer fuel and lower fertile blanket, but with an additional ring of 84 radial blanketsub-assemblies. Fig. 5.10 shows an axial and radial cut of the corresponding Serpent model.

5.3.3.4 Upper transmutation case: CONF2-HOM4

The CONF2-HOM4 conguration is based on the CONF2 core with HOM4 loading (4% byweight of MA homogeneously included within the core) dened in [34].

84

5.3. Code models

Figure 5.9: Radial section at the level of the active core and axial section of the SERPENTmodel for the CONF-2 conguration

85

5.3. Code models

Figure 5.10: Radial section at the level of the active core and axial section of the SERPENTmodel for the CONF2-HET2 conguration

86

5.3. Code models

Figure 5.11: Radial section at the level of the active core and axial section of the SERPENTmodel for the CONF2-HOM4 conguration

Geometry layout

Fig. 5.11 shows an axial and radial cut of the Serpent model for this conguration.

5.3.4 KENO-VI

At the Nuclear Engineering Department (UPM), a preliminary assessment of the uncertain-ties in nuclear data (XS, nu-bar, chi, . . . ) on the reactivity coecients for a fast reactorwas intended in order to conrm their reliability. As a consequence, a model of KENO-VIwas developed for a 3D fuel assembly, rst, and for a 3D whole core, after, following thespecications of the CP-ESFR project. KENO-VI is the Monte Carlo module within theSCALE system and it was used in conjunction with TSUNAMI and TSAR modules for thispurpose.

The 3D assembly model, shown in Fig. 5.12 is taken from the specications of the CP-ESFRproject, corresponding to the CONF2 optimized core .

87

5.3. Code models

Figure 5.12: 3D assembly model with KENO-VI

88

5.3. Code models

Figure 5.13: KENO-VI model for the 3D whole core model. Radial layout

Finally, the 3D whole core was modeled for KENO-VI, as shown in Figures 5.13 and 5.14following the specications of the optimized core CONF2 [28] from the CP-ESFR project(see section 5.2.2), without minor actinides loading.

89

5.3. Code models

Figure 5.14: KENO-VI model for the 3D whole core model. Axial layout

90

Part IV

ASSESSMENT OF ESFRPERFORMANCES

91

Chapter 6

Assessment of Minor ActinidesTransmutation

6.1 Introduction

Minor Actinides (MA) transmutation is a main design objective of advanced nuclear systemssuch as Generation IV Sodium Fast Reactors. In advanced fuel cycles, MA contents in nalhigh level waste packages are main contributors to short term heat production as well asto long-term radiotoxicity.

In addition to the traditional roles in a medium or large Sodium-cooled Fast Reactors (SFR)for electricity production and plutonium breeding, a new tentative strategy is nowadayspursued concerning MA transmutation. This strategy would allow minimization of alreadyaccumulated nuclear wastes coming from LWR open fuel cycles, which in turn may have apositive impact on nal repository requirements. However, it is well known that MA loadingin a reactor has large impact on aspects such as safety parameters (i.e. Doppler coecient,βe, void eect), helium production in the fuel, mechanical fuel behaviour under irradiation,and batch cycle considerations that need a careful assessment.

Dierent alternatives for MA loading have been studied in the past (see Section 6.2). Theperformed studies show that the introduction of MA in the reactor fuel deteriorates thecore reactivity coecients, where the magnitude of impact depends on the MA contentand the type of recycling approach standard homogeneous or heterogeneous managementor intermediate solutions. But, as pointed among others in [101, 114], there is no xedlimit of MA loading, but for each particular system conguration a limit should be denedaccording to dierent aspects, such as safety limits for reactivity coecients. Therefore,optimized core designs around basic designs could limit the potential negative impact ofthe MA content in the reactivity eects, and then the limits should be carefully assessedfor each particular conguration.

This study was performed within the framework of the CP-ESFR Project [54]. One of theobjectives in the project was to explore dierent designs and operation strategies of Sodium-cooled Fast Reactors in order to enhance the nominal core performances as well as the core

93

6.1. Introduction

safety mostly in the frame of unprotected transients. Optimization studies were carriedout in the project, and an optimized oxide core with a reduced sodium void reactivity wasproposed (see Section 5.2.2), for which an assessment of the MA transmutation possibilitiesis of major interest.

From the computational point of view, most of the performed studies in CP-ESFR and otherprojects used deterministic calculation approaches like ERANOS [122] to assess the eectsof MA recycling on the core performances [15]. Deterministic methods for transport cal-culations utilize spatial, angular and energy discretization, as well as the prior developmentof appropriate multi-group constants. Nevertheless, due to the acceptable computationaltime, they are the most common approach nowadays for fast reactors neutron analyses.

However, the increased relevance of Monte Carlo transport codes, mainly due to the im-provement of computing performances, makes that such codes are starting to be an al-ternative for design calculations [142]. Its integration with depletion modules providestools able to model very detailed and complex three-dimensional core geometries usingcontinuous-energy cross-section data. This approach eliminates the concern of generationof multigroup cross sections, which remains one of the fundamental problems in fast reactorphysics because of the need for accurate self-shielding treatment [142].

Since the 90s, dierent depletion computational systems have been developed around theworld coupling a Monte Carlo neutron transport code with a depletion code. This is the caseamong others of MCNPX 2.6.0 [51], MONTEBURNS [136], MOCUP [44] and MCODE[141]. For the purpose of this work, two codes were selected: i) EVOLCODE [94], whichautomatically links the MCNPX Monte Carlo transport code with the depletion code ORI-GEN; ii) SERPENT MC transport code [86], which incorporated the depletion capability bythe coupling with a burnup algorithm. EVOLCODE is being developed at CIEMAT sincesome years ago, and related activities include a number of successful validation exercises[92, 93]. SERPENT is recently being used at UPM and it is expected to obtain very similarresults as MCNPX for criticality calculations, with faster execution. The code is releasedwith an open source.

Along this Chapter the promising Monte Carlo transport-coupled depletion code SERPENTis used to examine the impact of MA burning strategies in a SFR core. The core conceptproposal for MA loading in two congurations is the result of an optimization eort upon apreliminary reference design to reduce the reactivity insertion as a consequence of sodiumvoiding, one of the main concerns of this technology. The objective pursued with thiswork is double. Firstly, eciencies of the two core congurations for MA transmutationare addressed and evaluated in terms of actinides mass changes and reactivity coecients.Results are compared with those without MA loading. Secondly, as a benchmark , the resultsfrom SERPENT are compared with the ones from EVOLCODE, provided by CIEMAT. Thediscrepancies in the results are quantied and discussed.

This work is the result of a collaboration between CIEMAT and UPM within the CP-ESFRproject, which led to a publication in Annals of Nuclear Energy, [105].

94

6.2. MA loading patterns

6.2 MA loading patterns

A number of core congurations with MA loading has been proposed and assessed byresearchers around the world in the last years. A short general classication is the following:

• Homogeneous mixture of MA distributed throughout the whole core fuel, i.e., a mix-ture of ssile (Pu), fertile (U) and MA. The Pu content is tuned, per reactor zones,according to reactivity swing, breeding ratio and power peak requirements, while theMA content is typically low, below 4 or 5% weight of Heavy Metal (HM)[112, 143] .There are also studies with MA contents higher than 5% in the fresh fuel [114, 28].

• Heterogeneous recycling approach, where MA loading is concentrated in radial fuelassemblies at high contents, 10 to 30% wt, mixed with U fertile material, and no Pucontent; the Pu is only loaded in the core driver in this conguration [14]. A variantof this case is to dedicate a lower or upper axial blanket to MA. In that congurationthe purpose is to take benet from the neutrons leaking from the reactor for MAtransmutation and Pu breeding. As the neutronic importance is low in the reactorperiphery, the impact on neutronic parameters due to MA is small.

• Intermediate radial or axial blankets with high MA content, similar to the previousfully heterogeneous conguration, but where some Pu content is loaded also in theblanket together with the uranium and MA [15]. The purpose is to avoid largedecrease of the peaking factors in those positions close to the ones developing power.This kind of solutions can be considered as homogeneous since MA are mixed withthe driver fuel, and heterogeneous since only some limited core regions are involved.

• Special fuel subassembly targets with high contents of MA and no fertile material.The MA mixture is contained in a neutronically inert matrix, metallic or ceramicmaterial. These targets are distributed throughout the reactor in a certain number offuel assembly positions in the core. On the other hand, some rods of the target fuelassembly may be occupied by moderating rods (e,g., ZrHx), as transmutation ratecan increase when the local spectrum is softened [8].

Behind such congurations there are several possibilities of spent fuel reprocessing andseparation. They are in general advanced processes at laboratory stage which could reachtechnological readiness levels after some decades [109, 110]. For instance, all transuranicscan be separated together in a single stream, or all MA can be extracted together in astream while Pu is separated in a dierent one, or Pu and Np are separated together or Cmis extracted from the MA stream and put aside for specic management as a consequenceof originating high irradiation levels [111].

Among all above possibilities, the rst two recycling models are explored in this chapter forthe CP-ESFR optimized CONF2 core:

• The homogenous distribution of MA with other fuel components, with 4 wt% ofMA content, is HOM4 case. The lower axial blanket presented in Section 5.3.3.4is assumed also 4 wt% MA loaded (and no Pu content). This case is intended to

95

6.3. Results for the Optimized Core

provide insights in line with moderate transmutation values in homogeneous loadingcompatible with aordable deterioration of safety parameters.

• The heterogeneous core model, with a radial blanket of MA located surroundingthe last row of outer fuel, in the place of the rst reector row, as described inSection 5.3.3.3) and depleted uranium, with 20 wt% of MA content is HET2 case.Higher contents of MA can be considered too much optimistic as the curium contentcomplicates fuel management.

In both cases oxide fuel form is assumed in driver and blanket regions.

6.3 Results for the Optimized Core

Three-dimensional SERPENT geometry models were developed (CONF2, CONF2-HOM4,CONF2-HET2), from the lower plug up to the upper shielding layer in the reactor, ne-glecting the head and foot of the sub-assemblies. Helical wire wrap spacers between fuelpins were considered as merged in the cladding (see Section 5.2.2). Global transmutationperformances as well as neutronics safety parameters for the mentioned cases are providedin this section.

A single batch approach for the whole irradiation period is assumed. In this approach,a cycle length of 2050 equivalent full power days (EFPD) is computed starting from thefresh core. The 2050 EFPD cycle is divided into 5 periods of 410 EFPD. Though no batchreloading is assumed, real core reloading would be envisaged each 410 EFPD. As a rstapproach, 820 and 1230 EFPD can be used as a rough estimate of core conditions forbeginning and end of equilibrium cycles (BOEC, EOEC), as the cumulated ssion productpopulation is representative.

A total number of 150 000 neutron histories and 200 active cycles were employed. Theaverage statistical uncertainties obtained in ke are 10 pcm. JEFF3.1.1 cross section librarywas used for neutron reactions under the assumed material working temperatures. Fissionyields were also taken from JEFF-3.1.1 library.

For isotope depletion, the burnable region was divided in a number of computational cells,intersection of 12 radial rings (8 for the inner core, 4 for the outer core). For the axialdiscretization, only 2 axial levels were considered, one for the active core, 100 cm high,and one axial level for the lower blanket, 30 cm high. A more detailed axial renement wasinitially considered but due to the the high memory requirements for a burn-up calculationusing SERPENT, it could not t in the memory of the machine employed.

6.3.1 Evolution of Pu and MA

Table 6.1 shows the core mass balances for the dierent cases. The transmutation rate(TR) presented in the table is dened as the relative mass produced from beginning of life(BOL, 0 EFPD) to end of life (EOL, 2050 EFPD), eq. 6.1.

96


TR(%) =M(EOL)−M(BOL)

M(BOL).100 (6.1)

Assuming a 40% thermodynamic eciency, the minor actinide balance per electricity pro-duction is also provided.

It can be observed in Table 6.1 that all cases under study are breeders concerning the totalplutonium mass balance (with Pu Breeding Ratio values of the order of 1.11 for CONF2,1.15 for HOM4 and 1.17 for HET2, when averaged in the 820-1230 EFPD period). Thetotal plutonium breeding is more important when MA are loaded, producing HET2 thehighest amount of Pu due to the extra radial ring.

Fig. 6.1 shows the mass balance of Pu isotopes from BOL to EOL in the dierent coreregions. Pu-239 is the main responsible of the increase of Pu mass in all congurationsand accumulates in all regions, being its maximum production in the axial fertile blanket,as expected (it comes from captures in U-238). The Pu-238 mass increases in the coresloaded with MA, since it originates in chains starting in Am-241 and Np-237.

Concerning the change of MA masses, there is a global accumulation in CONF2, a moderatedestruction in HOM4 (with a net Am and Np destruction competing with Cm accumulation)and a slight but net MA destruction in HET2. The MA behaviour is also depicted in Fig.6.2.

6.3.2 ke comparison

The ke results along irradiation in the single batch approach are depicted in Fig.6.3. It canbe seen in the gure that addition of MA in a fresh core has a negative eect in reactivity atBOL, as the ke is lower in HOM4 and HET2 congurations. This eect is a consequenceof the replacement of initial uranium isotopes by minor actinide isotopes (see Chapter 5).

Also, as a result of much more Pu-238 breeding capability in the HOM4 driver core, thepositive reactivity swing is higher, of the order of 600 pcm for the period 820-1230 EFPD. Infact the increase of reactivity in the HOM4 is explained by the production of more reactiveisotopes from some MA isotopes. Such is the case of Np-237 which gives rise to Pu-238 andAm-241 which is transmuted into Am-242m. Both products have a ssion-to-capture ratiomuch larger than their parents, and thus their impact on reactivity is positive. However, asthese and the other ssile isotopes are burned in the reactor, the eect on reactivity beginsto be negative and the k-e decreases.

6.3.3 Reactivity coecients

Important information is required for Safety Analysis concerning reactivity coecients,which have been calculated in the representative equilibrium sub-cycle and are shown inTables 6.2 and 6.3, respectively.

The Doppler constant is the fastest compensating safety parameter against positive reac-tivity insertions. Here, the Doppler constant estimation comes from equation 6.2.

97


CONF2 CONF2-HOM4 CONF2-HET2

Charged mass (kg)

U 87190.4 80676.1 101514.2

Np 0.0 650.1 603.9

Pu 11855.9 11856.8 11855.9

Am 93.2 2951.8 2835.5

Cm 0.0 253.8 235.7

MA 93.2 3855.7 3675.1

Discharged mass (kg)

U 78017.4 72632.4 92160.2

Np 44.1 406.0 583.8

Pu 13120.8 13417.4 13711.6

Am 338.8 1900.4 2701.7

Cm 66.1 472.0 333.6

MA 449.0 2778.4 3619.1

Transmutation Rate (%) at EOL

U -10.5 -10.0 -9.2

Np - -37.5 -3.3

Pu 10.7 13.2 15.7

Am 263.5 -35.6 -4.7

Cm - 86.0 41.5

MA 381.7 -27.9 -1.5

Mass balance (kg/TWhe) at EOL

U -129.5 -113.5 -132.0

Np 0.6 -3.4 -0.3

Pu 17.9 22.0 26.2

Am 3.5 -14.8 -1.9

Cm 0.9 3.1 1.4

MA 5.0 -15.2 -0.8

Table 6.1: Transmutation performances

98


Figure 6.1: Mass balance (kg) for Pu isotopes at EOL (EOL-BOL)

99


Figure 6.2: Total Minor Actinides evolution

Figure 6.3: ke evolution

100


CONF2 CONF2-HOM4 HET2

Doppler, pcm -891 -562 -762

βeff 373 350 372

Core void worth, pcm 1476 (3.96 $) 1714 (4.9 $) 1527 (4.11 $)

Extended void worth, pcm 719 (1.93 $) 1036 (2.96 $) 781 (2.1 $)

Table 6.2: Reactivity coecients at pseudo-BOC (820 EFPD)

CONF2 CONF2-HOM4 HET2

Doppler, pcm -727 -570 -723

βeff 367 345 365

Core void worth, pcm 1636 (4.5 $) 1778 (5.2 $) 1622 (4.5 $)

Extended void worth, pcm 896 (2.4 $) 1145 (3.3 $) 907 (2.5 $)

Table 6.3: Reactivity coecients at pseudo-EOC (1230 EFPD)

Kd =ρ(2500)− ρ(1500)

ln(2500)− ln(1500)(6.2)

where ρ(2500) is the reactivity when the fuel temperature has been increased by 1000 Kand ρ(1500) is the nominal reactivity value (the whole core fuel is assumed at 1500 K).Thisapproach was agreed among the CP-ESFR Project partners and is justied by the lineardependence of Doppler eect with logarithmic temperature increments. A more rigorousmethod for Doppler constant estimation should account for more temperature intervalsbelow and above the 1500 K nominal one and data interpolation; however the procedurewould require much more time.

In the HOM4 case the Doppler constant is lower than in CONF2 conguration due to the4 wt% of U content replaced by MA, being the broadening of 238U capture cross sectionone of the main contributions to Doppler eect, while broadening of the substitute 241Amcapture cross section takes place at much lower neutron energies, below the fast spectrum.

The eective delayed neutron fraction, βe, has an important role in the power responseto a transient. It has been calculated with the method developed by [120] as described inSection 4.2.1.2. βe in the HOM4 conguration is only 6% worse than in the referenceoptimized conguration. In the HET2 case, its value is similar than in the reference casesince the MA content is located only in a radial blanket around the active core. It has beenestimated that βe is reduced by 25 pcm during 2050 EFPD burnup.

Coolant density eect is a challenging parameter in large fast reactor design due to its largepositive value. Coolant voiding in the core has three eects in reactivity: a positive contri-bution because of spectral hardening (increase of the Pu-239 ssion to U-238 capture rate

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6.4. Comparison of SERPENT and EVOLCODE

at higher energies); a negative contribution as a result of neutron leakage increase depend-ing on the reactor size and on the height to radius rate; and nally a positive contributionas a result of decrease in coolant captures. In the case of commercial reactors like theESFR concept the void worth has a global positive value. In the optimized congurationused in this Chapter the sodium plenum above the core increase leakages if a bubble occursin this zone, while the upper absorber layer avoids backscattering to the active region (seeSection 5.2.2).

The void worth is the reactivity dierence between the voided and the reference core. Corevoid worth represents the worst possible scenario and has been calculated by decreasing Nadensity to 10-10 g/cm3 between rods in the 1 metre active length (wrapper interstitials,control rod channels and steel reectors were not voided).. The core void worth has a valuearound 4.5 $ for CONF2 and HET2 congurations and around 5 $ for the HOM4 option.Core void worth slightly increases from BOC to EOC.

In addition, an extended reactor void worth has been calculated, where not only the coolantin the core is voided, but also the upper part until the sodium plenum, that is, active length,upper ssion gas plenum, upper plug and upper sodium plenum. In the HET2 congurationthe radial blanket is also emptied. The extended void worth has a value around 2.5 $, clearlymore easy to compensate with the negative reactivity coecients such as the Doppler andthermo-mechanical expansion of the core.

6.3.4 Linear power

Complementary information obtained with SERPENT is provided in this section concerningthe linear power, a major core design parameter. Fig.6.4 shows the comparison of the powerfrom the three designs considered, and their evolution along the burnup. At beginning oflife the higher power peak is in the outer core due to the higher Pu content in this zone.Due to the higher ux in the outer core the Pu is burned faster and at EOL the power hasa more attened shape.

6.4 Comparison of SERPENT and EVOLCODE

A detailed comparison between EVOLCODE and SERPENT codes is now explained in thissection. The benchmark is carried out for the HOM4 case using the same axial nodalizationin both codes: one for the active core and one for the axial blanket, while the same 12 radialrings are preserved. Isotope mass dierences are provided in Table 6.5, second to seventhcolumn. It can be seen that the agreement is very good, being the dierences lower than3% wt. in all cases. An important contribution to discrepancies takes place already in therst irradiation step, 0-410 EFPD.

Around 1100 independent isotopes are tracked with EVOLCODE and around 500 by SER-PENT for the particular models under study (the isotopes not considered in this set areassumed reasonably neglected in terms of masses and macroscopic cross section values).A total number of 150 000 neutron histories and 200 active cycles were employed in both

102


Figure 6.4: Axial power distribution in the HOM4, CONF2 and HET2 models at each timestep from SERPENT

103


SERPENT EVOLCODE

(using 1 core) (Parallel 128 cores)

Calculation time of a ~8 h proc 3GHz ~2h15m proc 3GHz

single burnup step 24GB RAM (2GB/core)

Calculation time of the~48h ~13h

whole calculation

Table 6.4: Computational characteristics

codes. The average statistical uncertainties obtained in ke are 10 and 8 pcm respectivelyfor SERPENT and EVOLCODE. In both codes JEFF3.1.1 cross section library was used forneutron reactions under the same assumed material working temperatures. Fission yieldswere also taken from JEFF-3.1.1 library.

For isotope depletion, the burnable region was divided in a number of computational cells,intersection of 12 radial rings (8 for the inner core, 4 for the outer core) and dierent axiallevels depending on the model:

• In the analysis with SERPENT, only 2 axial levels were considered, one for the activecore, 100 cm high, and one axial level for the lower blanket, 30 cm high. Thisis a consequence of the high memory requirements for a burn-up calculation usingSERPENT, which makes a ner renement not possible in the machine employed(Table 6.4 ).

• In the analysis with EVOLCODE, 11 axial levels were considered as a best estimatecase (10 axial levels in the active core, 10 cm high each and one axial level at thebottom, for the lower blanket, 30 cm high). This proposal of meshing is a result of theCiemat team experience acquired in past exercises and it tries to capture the spatialinuence on the spectra. Results are provided along this Section. In addition, a casewith 2 axial levels, exactly the same than SERPENT, will be provided to ease a directcode to code comparison. This case is also a sensitivity analysis for EVOLCODE.

The computational characteristics of the machines employed to run SERPENT and EVOL-CODE are showed in table 6.4.

As explained in Chapter 4, SERPENT and EVOLCODE present dierent treatment of themetastable branching ratios. While SERPENT takes by default constant values, EVOL-CODE considers energy-dependent branching ratios taken from the JEFF 3.1.1 data library.As a consequence of the capture reaction in Am-241, isotopes Am-242 and Am-242m ap-pear with a fraction given by the reaction branching ratio. Am-242 then decays dominantlyinto Cm-242 with a short half-life (16 hours) while Am-242m is a ssile isotope with along life (141 years). After examining the dependence upon energy of the Am-241 capturebranching ratio (Fig. 6.5), it is realized that it varies noticeable with energy above 0.1 eV.

As a consequence, when the results of SERPENT and EVOLCODE were rst compared,

104


Figure 6.5: Am-241 Capture Branching Ratio

i.e. when the SERPENT default branching ratio was used, some relevant discrepancies wereobserved (see last column in Table 6.5). In this case, EOL dierences concerning some Amand Cm isotopes (as well as Pu-238) were much higher. However, when the SERPENTdefault value was modied to employ a spectrum-averaged value, similar to the EVOLCODEone (fraction to the isomeric state equal to 0.133), Am and Cm isotopes exhibit comparabledierences in both codes as shown second to seventh column of Table 6.5.

It should be mentioned now that the results from SERPENT presented in the previoussections were calculated using the spectrum-averaged Am-241 capture branching ratio,since it proved to be more accurate for the application under study.

Fig.6.6 shows the ke during the total irradiation period. The dierence between codesincreases in each burn-up step with a maximum value of 630 pcm at 2050 EFPD. It isinteresting to observe that EVOLCODE always predicts a higher reactivity than SERPENT,as the Pu content estimation at the end of each step is higher with EVOLCODE. Fig. 6.7shows total mass dierences between codes in Pu-239 and U-238 isotopes. In addition, inFig. 6.6 ke results for the 10 axial levels case are also plotted. In this case the sensitivityis some 300 pcm at EOL between both EVOLCODE cases, although the agreement is verygood until 1230 EFPD.

Looking for potential explanations of discrepancies between SERPENT and EVOLCODE,the burnup methodology arises as the main candidate. Firstly, it has been checked thatconcerning the normalization issue, both codes dier very little in their ssion Q-valueassumptions. On the other hand, although both codes use approaches that belong to thegroup of predictor-corrector methods, the algorithms employed are not exactly the same(as explained in Sections 4.2.1.2 and 4.2.1.3).

The eect is illustrated in Fig. 6.8 which shows total ux values employed for both codesin the rst burnup step, where core composition signicantly changes starting from fresh

105


EPFD 0 410 820 1230 1640 2050 2050

(BOL) (BOC) (EOC) (EOL) (default branching ratio)

U-235 0.01 -0.27 -0.60 -0.90 -1.12 -1.29 -1.35

U-236 1.34 1.32 1.16 0.91 0.66

U-237 0.96 1.60 1.90 1.78 1.77

U-238 0.02 0.03 0.03 0.04 0.05 0.05 0.04

Np-237 0.03 -0.16 -0.37 -0.56 -0.71 -0.83 -0.88

Np-238 1.59 1.67 1.29 0.73 0.52 0.60

Np-239 -0.78 -0.52 -0.42 -0.45 -0.48 -0.39

Pu-238 0.02 0.52 0.92 1.05 1.00 0.86 2.02

Pu-239 0.02 -0.26 -0.44 -0.60 -0.77 -0.92 -0.91

Pu-240 0.02 0.01 -0.01 -0.02 -0.03 -0.06 -0.06

Pu-241 0.03 -0.04 -0.01 -0.12 -0.09 -0.07 -0.09

Pu-242 0.03 -0.05 -0.12 -0.20 -0.30 -0.39 -0.16

Pu-243 0.69 0.35 0.11 -0.09 -0.35 0.83

Am-241 0.02 -0.19 -0.43 -0.64 -0.80 -0.91 -0.97

Am-242 1.66 1.56 1.31 0.51 0.26 2.39

Am-242m 0.34 1.20 1.15 0.84 0.39 -0.01 -12.96

Am-243 0.02 -0.10 -0.21 -0.32 -0.42 -0.50 -0.65

Cm-242 -0.90 2.19 2.19 1.84 1.27 0.86 3.03

Cm-243 1.18 1.62 2.23 2.55 2.53 2.14 4.13

Cm-244 0.04 0.34 0.49 0.54 0.51 0.44 0.42

Cm-245 0.06 -0.05 -0.04 0.03 0.10 0.13 0.14

Cm-246 0.72 0.85 0.94 1.02 1.08 1.17 1.20

U 0.02 0.04 0.04 0.04 0.04 0.05 0.05

Np 0.03 -0.17 -0.38 -0.56 -0.70 -0.82 -0.86

Pu 0.02 -0.11 -0.21 -0.28 -0.38 -0.48 -0.39

Am 0.03 -0.15 -0.34 -0.52 -0.66 -0.77 -1.36

Cm 0.06 0.72 0.81 0.75 0.62 0.50 0.81

Table 6.5: Percentage relative errors [(SERPENT- EVOLCODE)/EVOLCODE] in the pre-diction of isotopes masses along irradiation, HOM4 case

106


Figure 6.6: ke comparison in HOM4 case, SERPENT and EVOLCODE

Figure 6.7: Total mass dierence between SERPENT and EVOLCODE for 239Pu and 238U(HOM4)

107


EPFD 0 410 820 1230 1640 2050

(BOL) (BOC) (EOC) (EOL)

U-235 0.00 -0.06 -0.11 -0.14 -0.10 -0.03

U-236 0.23 0.21 0.15 0.02 -0.09

U-237 0.15 0.18 -0.21 0.14 -0.21

U-238 0.00 -0.01 -0.02 -0.03 -0.03 -0.03

Np-237 0.00 -0.02 -0.04 -0.05 -0.01 0.03

Np-238 -0.25 -0.30 -0.32 -0.55 -0.34

Np-239 0.10 0.09 0.16 0.08 0.17

Pu-238 0.00 0.08 0.13 0.13 0.05 -0.04

Pu-239 0.00 0.15 0.25 0.32 0.34 0.34

Pu-240 0.00 0.00 0.01 0.03 0.06 0.08

Pu-241 0.00 0.00 0.01 0.04 0.07 0.11

Pu-242 0.00 0.01 0.03 0.04 0.04 0.05

Pu-243 -1.00 -1.04 -1.07 -0.93 -0.74

Am-241 0.00 -0.04 -0.07 -0.07 -0.02 0.05

Am-242 -0.12 -0.23 -0.35 -0.66 -0.46

Am-242m 0.00 0.19 0.14 0.02 -0.16 -0.29

Am-243 0.00 -0.04 -0.07 -0.10 -0.12 -0.13

Cm-242 0.00 0.32 0.23 0.08 -0.26 -0.39

Cm-243 0.00 -0.14 0.05 0.23 0.11 -0.19

Cm-244 0.00 0.04 0.03 0.00 -0.06 -0.13

Cm-245 0.00 -0.02 -0.02 0.00 0.00 -0.03

Cm-246 0.00 0.08 0.10 0.13 0.14 0.16

Table 6.6: Percentage relative errors [(EVOLCODE(ne time step)- EVOLCODE(coarsetime step))/EVOLCODE(coarse time step)] in the prediction of isotopes masses alongirradiation, HOM4 case.

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6.5. Outcomes

Figure 6.8: Flux used for depletion in EVOLCODE and SERPENT in the rst burnup step(0-410 EFPD).

fuel. While SERPENT averages out the uxes from the predictor and corrector method(SERPENT MOS in the gure) to obtain the denite value, the EVOLCODE denitivevalue (EVOLCODE BOC in the gure) is very similar to the SERPENT predictor value,i.e., previous to correction. As a consequence, dierences in the denitive ux estimationswill lead to dierences in compositions.

Finally, the burnup time step size has been also investigated with EVOLCODE to demon-strate that the inuence on isotope masses and ke is very small. An additional case wascalculated with a smaller size, 51.25 EFPD (ne time step) to compare results with theoriginal 410 EFPD (coarse time step) case. The found dierences in terms of actinidemasses are shown in Table 6.6. When compared to coarse time step results, dierences arelower than 0.8% at EOL (the higher value is for Pu-243, having a short half-life of 4.96 h).Concerning ke, dierences were 100 pcm at EOL. Therefore, it might be concluded thatthe use of a ner time mesh has a minimal impact on the isotopics and k-eective.

6.5 Outcomes

A detailed analysis of the optimized CP-ESFR CONF2 core concept has been done with twolast generation fuel depletion Monte Carlo codes for two dierent MA loading strategies.Conclusions are provided under two points of view: (i) core performances and (ii) computercodes comparison.

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6.5. Outcomes

(i) All cases under analysis lead to total plutonium production as a result of the lower axialblanket (CONF2), the axial and radial blankets (HET2) or the lower axial blanket and MAloading in the driver region (HOM4). Total plutonium breeding ratios range from 1.1 to1.17.

Concerning MA behaviour, the reference CONF2 core is a net MA producer (some 4.5 timesmore than the small initial and unintended loading).

The HOM4 core conguration leads to the net elimination of an important amount of MA,specically americium and neptunium. Concerning curium, there is a net accumulation inspite of the important amount of Cm ssioning. As the management of curium entailsdiculties and costs, a careful analysis of possibilities taking into account Cm-244 (18.11years half-life) during storage is required. Compared to a similar core with no MA loading,the HOM4 core void deteriorates some 12% and the extended void some 20%.

On the other hand, HET2 model presents similar safety characteristics than the CONF2and a nearly neutral behaviour regarding MA transmutation (in fact, a small destruction isestimated).

Even though detailed transient analysis is still needed in order to conclude the licensing ofsuch cores, the estimated safety coecients are encouraging. Even for HOM4 core, thecore void is some 5.2 $, while extended void is some 3.3 $.

(ii) In what concerns comparison of codes, a great special care was taken to develop BOLsimilar models: same core geometries, initial fuel and material compositions and masses,neutron libraries and working temperatures. Then, results predicted by EVOLCODE andSERPENT codes are very close in ke estimations before activation of burnup models.After a typical long irradiation period of 2050 EFPD, when a similar reasonable assumptionis taken concerning the branching ratio, discrepancies are quantied in the order of 3%concerning actinide isotope masses, 600 pcm as much in ke, and 10% concerning reactivityparameters. The dierent corrector-predictor approaches in the fuel depletion model mostlikely explain the observed dierences.

110

Chapter 7

Spanish Scenario of theDeployment of ESFR-type reactors

7.1 Introduction

The most famous denition of sustainability was included in the Brundtland report in 1987[1]: ``Sustainable development is development that meets the needs of the present withoutcompromising the ability of future generations to meet their own needs''. Another technicalpoint of view about sustainability is the fact of no decreasing neither the environmentalquality nor the individual one [53]. The human activities linked to sustainability are aectedby three kinds of limitations: environmental, economic and social ones [19], [33] and [129].

Many authors have classied the dierent energies in terms of sustainability [42] accordingto dierent parameters. They can be set into three: resources consumption, environmentalimpact and economical and technical availability. In case of nuclear energy, parameterssuch as no proliferation and nuclear safety should be also considered.

Regarding nuclear ssion, the use of the current technology (Generation I to III+) hasdisadvantages in terms of resources availability for long term. An interesting option forFuel Cycle is what is called ``closed cycle'' as it is the most sustainable option possible.Closed cycle means that the irradiated fuel, after being burnt and then cooled in the powerplant spent fuel pool, is reprocessed as it still has around 0.6% Pu-239 and 0.7% U-235,which are the most common ssile materials. This Pu-239 is used together with U-238 tomake new fresh fuel, called MOX (Mixed Oxide) fuel, to be used again in power plants.Almost 94% of the irradiated fuel is U-238, which might be reused for fresh fuel too. Therest are ssion products (FP) and minor actinides (MA), which are the most radioactivecomponents of the spent fuel. Those materials (FP and MA) are normally vitried tomanage them in a much more stable solid form than their natural one. Those wasteshave been considered to be stored in a geological deep repository, to avoid human andenvironmental contact for thousands of years.

In terms of eciency, the total primary energy (the natural uranium at the mine) estimatedto be transformed into heat in the ssion process in a once-through cycle is 0.5%. The

111

7.1. Introduction

85% of the potential energy is still stored in the natural uranium unused in the processof fuel fabrication and the other 14.5% is stored in the spent fuel [80]. In terms of mass,in light water reactors with oncethrough cycles, only 3 kg of U-235 from each 1000 kg ofnatural uranium (99.3% U-238 and 0.7% U-235) are used for generating heat. In a closedcycle with reprocessing, the used resource is doubled to 1%. Therefore, there is a need ofimprovement in the fuel cycle as the major part of the resource is treated as a sub-product(depleted uranium) or as waste (spent fuel).

The most advanced fuel cycle strategy nowadays is not only based on spent fuel reprocessingto make new MOX fuel, but also on separation and transmutation of minor actinides andfuel breeding. The MA transmutation is meant to reduce the radioactive inventory of thespent fuel, while fuel breeding consists on generating ssile material (Pu-239) from non-ssile U-238. Actually fast reactors can produce more fuel than what they use in a cycle.The advanced fuel cycle strategy needs fast reactors as the sodium cooled one, which is thechosen system for the present analysis. The use of these reactors can wide the availabilityof resources for much more than 300 years, as they use very eciently the fuel [137].

It is also important to underline the case of Thorium (Th-232) as a fuel for fast reactors.It can be transmuted into U-233, a ssile element that might be used in the actual thermalreactors. Thorium is a huge resource to be used, which is estimated to be 4.4 millions oftons among the known resources [102]. This resource can play an important role to makesure a long term development of nuclear energy, especially in countries like India [64].

Nuclear ssion produces low CO2 emissions, even taking into account all fuel cycle process(65 g CO2/kW h), compared with other energies like thermal plants (6001200 g CO2/kWh), solar panels (90 g CO2/kW h), and higher than wind or hydro power (3065 g CO2/kWh) [37, 41, 47, 58, 84].

In addition, a sustainable strategy for nuclear energy has to take into account the noproliferation and safety aspects [116, 81]. In case of the no proliferation aspects, the fuelreprocessing process makes very dicult the Pu-239 isolation, as it is mixed with otherisotopes such as Pu-240 and Pu-241. One advantage of a closed fuel cycle with MOXfuel is the possibility of recycling the military Pu for energy production. Regarding safety,sodium fast reactors have inherent safety characteristics due to the physical properties ofthe materials and the type of system (for example, pool type avoids coolant losses). Thereis a long experience of commercial operation with these reactors and some of them haveoperated for more than 35 years like Phenix, in France. However, they face some challengesin the next years for the massive implementation of this technology. As a consequence ofUSA president Carter decision in 1977 [5], the United States stopped reprocessing in anattempt to limit the proliferation of nuclear weapons material. Since then spent fuel isstored intact in a repository, what is called once-through cycle.

Spain, as other countries like Canada, Finland or Sweden, followed later on that decision[26]. Before that moment Spain was reprocessing its spent fuel in other countries like Franceor UK. After that strategic change, the spent fuel pools of the Spanish nuclear plants slowlybecame full, as they were not designed to handle with the spent fuel of their operationallife. Several actions were planned by the power plant owners such as re-racking the spentfuel pool or building an Individual Temporary Storage to store each plant's own wastes in

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7.2. Reactor Technology Description Sodium-cooled Fast Reactors

shielded casks. All these expensive actions could have been avoided with a CentralizedTemporary Storage (CTS). However the decision was delayed in the last years for politicalreasons till nally it was approved in December 2011. This storage will have enough roomto cope with all the Spanish spent fuel and with the waste from reprocessing waiting inFrance to come back to Spain. Nevertheless, as its own name indicates, the CTS facility isa temporal solution for the high-level radioactive wastes. Alternative technologies should bepursued to provide a denite solution to those wastes, and the reduction of the radioactiveload of the wastes in fast reactors could be an interesting option to be investigated. Indeed,one feasible option is the mentioned advanced closed cycle, since the existing reprocessingplants have demonstrated the correct operability of the irradiated waste treatment and theoccidental countries respect the non-proliferation treaties. These two aspects are the mainreason to support the advanced fuel cycle in the study.

In this Chapter, a hypothetical Spanish energy scenario with sodium fast reactors is analyzed,testing if a eet of nuclear fast reactors could solve any of the challenges that the nuclearindustry is facing, such as nuclear waste management, security of supply and sustainability.

In the rst part, the Generation IV reactors characteristics are briey summarized, as theyrepresent a step up to the nowadays Generation II and III/III+ reactors. Later on, the Span-ish energy scenario is presented, introducing some sensitivity studies. Finally the detailedresults and conclusions of the study are presented to show the inuence in terms of energyproduction, resources availability and nuclear waste reduction.

This work was published in Energy Conversion and Management [108] in 2013.

7.2 Reactor Technology Description Sodium-cooled

Fast Reactors

The new nuclear power plant designs considered in Generation IV initiative pretend toovercome past and present design parameters in terms of sustainability, industrial compet-itiveness, safety and proliferation resistance. Additionally to power generation, these newdesigns will be used to generate industrial process heat and hydrogen production.

Sodium fast reactors meets these goals clearly and have many years of operational experi-ence. Actually, they are in operation nowadays in countries like Japan, China or Russia andin a very short term India [45, 144, 49].

The two main advantages of SFR in terms of sustainability were explained in the intro-duction: the possibility of minor actinides transmutation and the capacity of breeding fuel.The main disadvantages are related with operational safety. Sodium could have a positivereactivity feedback in case of coolant voiding. This fact has a strong inuence on thecore design. Therefore many analyses have been and are being done to get an optimalconguration design that avoids power excursions [54, 28] On the other hand, sodium isvery reactive with water and air (oxygen), so the interfaces between those componentsmust be reliable enough to avoid any contact in any case. Related with fuel fabrication,if transmutation of minor actinides is intended, diculties in manufacturing (U,Pu,MA)O2

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7.3. Methodology and codes

should be overcome, since fuel loaded with MA is highly radioactive.

7.2.1 SFR core design

The SFR core design used in this analysis corresponds to the new optimized core of theCP-ESFR European Project, from the European Community's Seventh Framework Program[54]. The basic core, called CONF2, with 3600 MWth contains oxide fuel in two concen-trically regions, distributed in an inner and outer region. The outer region presents slightlyhigher plutonium content in order to atten the radial power shape at the end of the cycle.Further details concerning the core description are presented in Section 5.2.2.

Some other congurations derived from the CONF2 have been used for dierent purposesalong this work. CONF2-HET2 (see Subsection 5.3.3.3) consists of the same congurationas CONF2, i.e. with the same isotopics in inner fuel, outer fuel and lower fertile blanket,but with an additional ring of SA's located in the place of the rst reector ring. Thatring is lled with 20% MA in the (U,Pu,MA)O2 matrix. This conguration is employed inSubsection 7.5.2.2 as a consequence of its better breeding properties. The CONF2-HOM4conguration (see Subsection 5.3.3.4), on the other hand, consists in the same geometricalconguration as CONF2, but with 4% MA homogeneously loaded in the driven fuel andfertile blanket. It was used in Subsection 7.5.3 to analyse the transmutation performances.

A single batch approach for the whole irradiation period was assumed. In this approach,a cycle length of 2050 equivalent full power days (EFPD), divided into 5 cycles of 410EFPD is considered, starting from fresh fuel, with no batch reloading. However, in reality areload is envisaged each 410 EFPD. For this reason, the Pu production is estimated for eachreloading cycle taking into account that 1/5 of the assemblies of the core are substitutedby fresh fuel assemblies after each 410 days.

7.3 Methodology and codes

In this section the computational methodology used to obtain the isotope inventories at thedierent stages of the SFR commercial operation is presented. For the cores described inthe previous section, burn-up and decay calculations have been performed using the MonteCarlo transport-coupled depletion code SERPENT [86] (see Subsection 4.2.1.2 for moreinformation). In addition, some pre- and post-processing tools were developed in order totreat the data and obtain the results.

7.4 Spanish scenario hypotheses

2011 Spanish electricity consumption was 252.848 GW h, according to the Spanish nuclearindustry forum [21]. Although the installed electrical power was 103.625MW (this wouldcorrespond to 907.755 GW h working 365 days 24 h, or 28% average utilization factor).

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7.4. Spanish scenario hypotheses

Figure 7.1: Historical Series of Electricity Production

With the current energy mix, nuclear electricity power represents 7.5% of the total installedpower and 19.5% of the total production. This last value is much lower than in the nineties,when around 30% of the power plants were nuclear.

The historical electricity generation series is shown in Fig. 7.1 [21]. The average annualgrowth was about 5.5% during the period 19952007. Last 4 years were aected by theeconomic contraction and therefore tendencies present a dierent slope and even a decreaseduring recession periods.

For the study of the future energy scenario in Spain we link the evolution of electricityconsumption to national economic activity. We assume that electricity demand is a kind ofmacroeconomic variable with pro-cyclical behavior. This means that it increases during ex-pansions and decreases during contractions. Regarding timing, it is not exactly a coincidentvariable, but just a slightly leading variable that reaches a peak or trough before the turningpoints of the business cycle. The period of study in this work is so extensive (20502110)that the trend of both variables, electricity demand and Gross Domestic Product (GDP)can be taken as coincident.

The relation between electricity consumption and economical activities is easy to understandin an industrial occidental economy, where mostly all economical businesses require elec-tricity to work (industrial activities, public and private services, tourism, transport, leisuretime, communication and information technologies). According to dierent internationaleconomical organisms such as the International Monetary Fund (IMF) and the EuropeanCentral Bank (ECB) the growth rate for next decades will get back to stable positive values.

Since it is really complicated to predict the evolution of the national economy in a periodof 50 years, three dierent scenarios are proposed in this study. For the period 20112020

115


Figure 7.2: Power Demand evolution according to the postulated trends

contraction and expansions seem to be the trend, so we consider neither increase nordecrease of GDP in this period. Therefore the electricity demand remains constant. For20202050 time period a positive 1.5% annual growth rate is considered. And for thefollowing 60 years (20502110), we propose three possible tendencies, shown in Fig. 7.2:

1. Moderate constant development (+0.5% annual growth rate). This option is verymoderate and represents the case where important energy saving policy as well aselectricity distribution network improvement are applied.

2. High development by periods (combined cycles of 2.5% annual growth rate for 10years plus zero growth for next 2 years and repeatedly). This option seems to bethe most realistic one. According to economic cycle theories in dierent countries,in every decade there is a small contraction of the economy. Sometimes it can bedeeper and sometimes lighter, but in average every decade undergoes a trend change.

3. Medium constant development (+1.5% annual growth rate). This option representsthe case of a stable expansion economy and this will be the upper limit of mostprobable electricity demand.

7.4.1 Spanish nuclear park and natural resources

The characteristics of the Spanish nuclear eet are shown in Table 7.1 [21]. Currently thereare 7 nuclear reactors in commercial operation. They have a total power of 7753 MW, andin 2011 their contribution was 7.5% of the total power installed in Spain.

116


Electrical Power (MWe) Operational Starting Estimated end of operation

José Cabrera 160 1968 2006*

Garoña 466 1970 2012

Almaraz I 980 1980 2020

Almaraz II 984 1983 2023

Asco I 1032 1982 2022

Asco II 1027 1985 2025

Cofrentes 1097 1984 2024

Vandellos I 480 1972 1989*

Vandellos II 1087 1987 2027

Trillo 1066 1987 2027

*Real End of operation

Table 7.1: Spanish nuclear eet characteristics and hypothetical operational lives

Initially the lifetime of the reactors was 40 years, but the license for operating is renewedevery 10 years. According to last Spanish energy policy, further extension beyond 40 yearsare being studied for the future, following the procedures of other countries such as USAand Sweden where they extend the lifetime up to 60 years.

According to ENRESA PGRRS (General Radioactive Waste Plan) [26] data, the total nu-clear waste accumulated from the Spanish nuclear power plants is calculated to be 6675 tonin 2030, if 40 years of operating life is assumed. Based on the possible lifetime extension,we consider that all current nuclear reactors will operate up to 60 years. That assumptionis important, not only for the necessity of meeting the demand, but also for knowing theamount of irradiated fuel and specically the plutonium content that could be reused inthe future as fuel for fast spectrum reactors.

In addition to the lifetime extension and in order to give continuity in the nuclear park, anintermediate deployment of Gen-III eet has been considered. According to this hypothesis,the current nuclear reactors would be substituted by their equivalent Gen-III technologyright after the shut down of the previous generation, as explained in Table 7.2.

In Spain there are uranium mines, closed for exploitation nowadays due to high economicalcosts compared to uranium price in international markets, but still natural resources thatmight be restarted someday. According to the Red Book Uranium 2009 [103], there isin Spain a total of 12,300 tons of natural uranium located in dierent regions such asSalamanca or Andalucia.

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7.5. Results and analyses

Gen-II Gen-III

Electrical Power (MWe) Technology Electrical Power (MWe) Technology

Almaraz I 980 PWR-W 1154 AP1000

Almaraz II 984 PWR-W 1154 AP1000

Asco I 1032 PWR-W 1154 AP1000

Asco II 1027 PWR-W 1154 AP1000

Cofrentes 1097 BWR 1400 ABWR

Vandellos II 1087 PWR-W 1154 AP1000

Trillo 1066 PWR-KWU 1650 EPR

Table 7.2: Equivalence between Gen-II and Gen-III eet

7.5 Results and analyses

Along this section, the main results of the performed analysis are presented. They havebeen divided into three groups. The rst part will deal with the electricity production issues,and some sensitivity analysis will be described. The second part will analyse the capacity ofSpain to deploy a nuclear eet of Gen-IV SFRs in terms of availability of national resources.Finally, some considerations regarding the transmutation capability of the proposed eetwill be addressed.

7.5.1 Electricity production

The rst study focuses on the estimation of ESFR-like reactor eet needed to supply elec-tricity in a hypothetical reference case. As described in Section 7.4, a progressive evolutionfrom Gen-II to Gen-IV has been considered through the deployment of Gen-III reactors, asshown in Fig. 7.3.

Fig. 7.3 shows the target Gen-IV electricity production, which would cover circa 20% of theelectricity share. The selected reference scenario for the Gen-IV deployment assumes: 1.5%increasing rate for the electricity demand, 38% thermodynamic eciency of the nuclearplant and 20% nuclear electricity share (this will be called case A5).

The analysed period goes from 2050 (when the technology is assumed to be mature enoughto start to be implemented) to 2110 (when the electricity demand is assumed to start tobe constant and just the replacement would be considered). An operational life of 60 yearshas been considered for the SFR reactors. Under those assumptions, a total of 17 reactorswould be required to supply the nuclear energy demand until 2110, after which the rstreactors should be replaced by new ones. The availability of resources will be taken intoaccount in the following section. The considered scenarios are described in Table 7.3 , as

118


Figure 7.3: Power covered by the dierent technologies of NPP

a combination of the most representative variables: electricity demand, thermal eciencyand nuclear share of production.

Moreover, some sensitivity analyses have been performed around the target variables, whichare presented in the following sections.

7.5.1.1 Electricity demand

Based on the reference case assumptions (38% eciency and 20% nuclear share), threeelectricity demand scenarios were studied, according to Section 7.4, with dierent demandincreasing trends:

• Moderate constant development (+0.5% annual growth rate) (A1).

• High development by periods (combined cycles of 2.5% annual growth rate for 10years plus zero growth for next 2 years and repeatedly) (A3).

• Medium constant development (+1.5% annual growth rate) (A5).

The estimated reactor eet for each of them is presented in Fig. 7.4.

A total amount of 9, 24 and 17 reactors, respectively, are estimated to be required for eachof the postulated scenarios until 2110. After that moment, the rst reactors would need tobe replaced by new ones. Concluding, it can be inferred that the proposed electricity demandevolution has a strong impact on the number of reactors required, as it was expected.

119


NameIncreasing Demand rate Thermal eciency Nuclear Share

0.5 2.5* 1.5 38% 42% 20% 25%

A1 x x x

A2 x x x

A3 x x x

A4 x x x

A5 x x x

A6 x x x

A7 x x x

A8 x x x

A9 x x x

A10 x x x

A11 x x x

A12 x x x*Combined cycles of 2.5% annual growth rate for 10 years plus zero growth for next 2

years and repeatedly

Table 7.3: Description of considered scenarios

Figure 7.4: Estimated reactor eet requirements for each postulated electricity demandtrends.

120


Figure 7.5: Estimated reactor eet requirements for each postulated thermal eciency

7.5.1.2 Thermal eciency

Based on the reference case (A5), dierent eciencies for the conversion of thermal energyinto electricity have been postulated: 35%, 38% (A5) and 42% (A11).

The total estimated eet for each of the postulated eciencies presented is 19, 17 and15, respectively (Fig. 7.5). It can be drawn that the eciency impacts but not as muchas the electricity demand scenario. Due to the improvement of thermodynamic eciency,settled as one of the primary objectives of the Gen-IV reactors, an eciency of 35%, as aconservative option, or even 42%, as an optimistic one, could be achievable.

7.5.1.3 Nuclear electricity production share

Following the assumptions of the reference case, the sensitivity of the share has also beenstudied. Three dierent shares have been postulated: 20% (A5), 25% (A6) and 30%. Foreach of them, the required eet has been calculated as 17, 21 and 26, respectively, as shownin Fig. 7.6.

Giving the importance of renewal energies in Spain, the energy mix is pursued to be asdiverse as possible, and therefore, a nuclear share larger than 25% is not foreseen. Thusthe two rst values have been selected as the most representative ones.

In order to summarize the previous analysis, the most representative scenarios have beengathered in Table 7.4, and the estimated reactor eet for each of them is presented in Fig.7.7.

121


Figure 7.6: Estimated reactor eet requirements for each postulated nuclear electricityshare.

Figure 7.7: Estimated reactor eet required for each postulated scenario.

122


A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12

9 12 24 30 17 21 8 10 22 27 15 19

Table 7.4: Total required eet for each postulated scenario

7.5.2 Resources

The second approach consists on the study of the availability of the resources for eachconsidered case. This study aims to analyze the capability of Spain to supply the eet ofSFRs calculated in the previous section with national resources.

The limiting resources in this case, to be considered for the analysis, would be the uraniumand plutonium resources. In particular, regarding uranium the study focuses on U-238 beingthe main contributor to the fuel and the responsible of the production (breeding) of ssilematter (Pu-239) through the following reaction:

U23892 + n −→ U239

92

β−

−→ Np23993

β−

−→ Pu23994 (7.1)

As for plutonium, the Pu-239 isotope is the relevant one for being the ssile materialresponsible of the criticality at Beginning of Life (BOL).

7.5.2.1 Uranium-238

The aim of this section is to verify the availability of Spanish national resources to supplythe postulated eet according to the reference electricity scenario (A5). U-238 could beobtained from three main channels: mining, reprocessing and from waste of fuel enrichmentprocess (depleted uranium). Thus, dierent combinations of these three sources werepostulated:

• B1. U-238 obtained only from mining.

• B2. U-238 obtained from both mining and reprocessing.

• B3. U-238 obtained from both mining and enrichment wastes.

• B4. U-238 obtained from mining, reprocessing and enrichment wastes.

In addition, the extension of the Spanish nuclear power plants operational lives from 40 to60 years has been considered, giving rise to two further scenarios:

• B5. U-238 obtained from both mining and reprocessing (after 60 years).

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B1 B2 B3 B4 B5 B6

U-238 available (ton) 12213.9 64380.4 18484.6 70651.1 21619.9 80057.2

Max. eet 18 97 28 107 32 121

Table 7.5: U-238 (ton) available and max. eet according to each scenario (B1B6).

• B6. U-238 obtained from mining, reprocessing (after 60 years) and enrichmentwastes.

No U-238 reprocessing for Gen-III eet has been considered. However, it remains as anadditional source in case of needed.

First, the amount of U-238 required for a whole life of a nuclear ESFR-type reactor wasestimated as circa 659 ton. No U-238 recycling from the SFR operating eet was envis-aged. Second, the total availability of U-238 following the dierent paths (B1B6) wascalculated (Table 7.5), and consequently, the maximum number of reactors able to be fedwas estimated for each scenario.

Therefore, comparing Tables 7.4 and 7.5, it is concluded that the Uranium-238 is not alimitation for most of the postulated combinations. In particular, for the reference scenario(A5), where 17 reactors were required, no limitation is expected until 2110. In fact, thereonly exist a limitation when the U-238 can just be achievable from the mining (B1) incombination to the most demanding scenarios (A3, A4, A6, A9, A10, and A12), or in caseof considering just mining and reprocessing without lifetime extension (B3) in combinationwith the electricity scenario A4.

7.5.2.2 Plutonium-239

This section deals with the study of the availability of the ssile isotope employed in FastReactors, Pu-239. This isotope cannot be obtained from the nature, as uranium is, but onlyfrom spent fuel reprocessing. Thus, along these lines the amount of Pu-239 extractablefrom the spent fuel disposal coming from the Spanish nuclear power plants will be estimated.

Three possibilities have been considered for the estimation of the Pu-239 requirements:

• No lifetime extension of current eet (Gen-II).

• Lifetime extension to 60 years of current eet (Gen-II).

• Lifetime extension to 60 years of Gen-II and substitution by Gen-III (until 2050).

Then, the amount of Pu-239 required for a whole life of a ESFR-like reactor is estimated asaround 61 tons. Comparing it with the total amount of Pu-239 available, it can be veriedthat there would be enough to feed maximum a single reactor whole life, in case of lifetimeextension, as shown in Table 7.6.

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40y 60y 60y + GIII (2050)

Pu-239 available (ton) 49.29 73.94 82.03

Whole life reactors 0.81 1.21 1.34

Start up reactors 4 6 7

Table 7.6: Pu-239 (ton) available and start-up reactors.

NameCore Operational life Electricity scenario

HET2 CONF2 40y 60y+ GenIII A3 A5

C1 x x x

C2 x x x

C3 x x x

C4 x x x

Table 7.7: Description of postulated breeding scenarios

However, there is enough plutonium to feed a number of startup cores, which can startproducing plutonium, thanks to the breeding properties of this kind of reactors. Therefore,the number of start-up cores able to start operating with autonomy of 2050 EFPD wasestimated. A constant reprocessing strategy is considered in order to obtain the requiredresources from the operating reactors. In this regard, the plutonium availability taking intoaccount the production and consumption is updated every year. The produced plutonium isassumed to come from both Gen-IV and Gen-III operating reactors, depending on the case,while the plutonium is assumed to be consumed only by the Gen-IV operating reactors. Forthe reprocessing cycle an eciency of 99.9% was considered, and the assumption of 5 yearsof spent fuel cooling followed by 1 year of reprocessing was employed.

The nal objective of the present analysis is to verify if there would be enough plutonium togenerate the power demanded by the reference scenario (A5), by means of reprocessing theplutonium obtained by the operating reactors. Following this purpose, a burn-up calculationwas performed using SERPENT for the CONF2 and CONF2-HET models. The reloadingwas envisaged each 410 days, but for the rst cycle, no reprocessing is expected until theequilibrium cycle (2050 EFPD) is achieved. Both designs include a fertile blanket aimedto produce plutonium. In addition, the CONF2-HET2 conguration was employed for itsbetter breeding properties, thanks to the production of Pu-239 in the additional externalring.

A total of four scenarios have been postulated including a combination of the followingspecications as described in Table 7.7:

• Core: HET2 core or CONF2.

• Operational life: 40 years (Gen-II) or 60 years + GEN-III, where the available pluto-

125


Figure 7.8: Estimated power produced for the A5 scenario.

nium is updated after each Gen-III operational year.

• Electricity scenario: Reference (A5) or a more demanding electricity scenario (A3).

The prior objective is to verify the Pu-239 availability in order to achieve the electricityscenario demanded by Subsection 7.5.1. In particular, to verify if the reference scenario(A5) is achievable with the hypotheses presented.

Following this motivation, Figs. 7.8 and 7.9 show the estimated power production with theavailable Pu-239 according to the postulated scenarios for the reference electricity scenario(A5) and for the more demanding one (A3), respectively.

It is inferred from Fig. 7.8 that there would be enough Pu-239 to produce circa 20% of theelectricity demand according to the reference scenario A5, when the Pu-239 is reprocessedfrom the spent fuel of both Gen-II and Gen-III (C3). The HET2 core has proved to breedmore plutonium, as expected. As the Gen-III NPPs start to shut down, the required eetof Gen-IV increases to cover their share. As a consequence, for the cases C1 and C2, from2102 to 2120 the nuclear share is lower than 20%.

In case of the more demanding electricity scenario (A3), assuming reprocessing from bothGen-II (with lifetime extended to 60 years) and Gen-III, there would be enough plutoniumto generate 20% of the electricity share.

As a conclusion, there would be enough Pu-239 to contribute circa the 20% to the electricityproduction in the reference scenario as long as reprocessing both Gen-II and Gen-III wastesis considered.

126


Figure 7.9: Estimated power produced for the A3 scenario.

7.5.3 Waste reduction by transmutation

For MA transmutation assessment the HOM4 model was employed where all fuel pellets inthe ESFR reactor core contain 4% of MA homogeneously distributed (Section 7.2.1) Thetransmutation performances of the three designs described in Section 7.2.1 were previouslycompared in Chapter 6 and [105]. Moreover, for the purpose of the present analysis, adierent approach was considered. Two dierent assumptions were compared: in one casethe HOM4 fuel was burned during 2050EFPD, while in the second one the same fuel fromthe HOM4 design is left to decay in the temporal storage centre instead.

Depletion calculations performed with SERPENT code estimate the mass of MA that willbe eliminated during the 2050 days of permanence in the reactor. Fig. 7.10 shows thecomparison between the MA isotopes at Beginning of Life (BOL) and End of Life (EOL).

Complementary, the decay heat of the two postulated cases were compared, when the samefuel is burned in the reactor and when it is left to decay instead. Fig. 7.11 shows theevolution of decay heat of those materials after around 60 years after disposal. The netreduction in terms of decay heat might be drawn from Fig. 7.11 as well. The decay heat ofthe most characteristic elements is also represented, comparing the impact of being burnedin the reactor.

Fig. 7.11 shows that the contribution of Am-241 and Am-243 to the decay heat is signi-cantly reduced in the reactor, as well as the one from Np-237. However, the contributionof Cm isotopes increases.

The total decay heat from the MA in case of being burned in the reactor is lower than

127


Figure 7.10: Main MA isotopes' masses at BOL and EOL.

Figure 7.11: Comparison of the decay heat of the main MA being burned in the reactor ornot.

128

7.6. Outcomes

rate

Figure 7.12: MA content in the wasted fuel

the corresponding to the temporal storage after 60 years of discharge, even though it ishigher during the rst period. The main minor actinides elements are also represented incase of going through the reactor (IN) or not (OUT). The main responsible of reducingthe total MA decay heat is the Am, which is eliminated signicantly in the reactor. On theother hand, although the Cm isotopes mass increases in the reactor, its activity is quicklyreduced.

Furthermore, the total nuclear waste accumulated from the Spanish nuclear power plantsis calculated to be 6675 tons by 2030, as mentioned in Section 7.4. It is assumed that the0.1% are minor actinides, which means a 6675 kg of minor actinides (Am, Np, and Cm)following the usual proportion in spent fuel, Fig. 7.12.

Using the previous data, it has been estimated that the 39% and 43% of the initial Am andNp, respectively, will be consumed in each cycle. However, a small quantity of Cm will becreated. Only one SFR is needed to operate with MA in the rst cycle, later on some morecycles would be needed to transmute the generated MA.

7.6 Outcomes

An analysis of a Spanish energy scenario using one type of Gen- IV reactor has beenperformed in terms of electricity production, availability of resources and wastes reduction.

Three sensitivity studies to the required eet of ESFR-like reactors have been done: dierentelectricity demand scenarios, thermal to electrical conversion eciencies and shares coveredby nuclear electricity. It was conrmed that the most relevant factor is the electricitydemand.

To study deeply some parameters, a reference scenario has been xed with the hypothesisof 1.5% of annual growth of the demand, 38% of thermal eciency and 20% of nuclearpower. For this reference scenario, the availability of the most representative resources, i.e.uranium (U-238) and plutonium (Pu-239), has been assessed.

129

7.6. Outcomes

For uranium, it can be concluded that there is no limitation for the exploitation of theproposed eet for the reference case scenario assuming dierent alternatives in the waysthe uranium might be obtained. When the study is extended to further energy scenarios,a limitation appears in the most restrictive scenarios. In case of using only resources fromnational uranium mines, there would not be enough uranium to feed the required eet forthe most demanding energy scenarios. Nevertheless, if reprocessing is possible and theoperational life is extended to 60 years, there would not be any limitation to feed therequired eet by any of the postulated scenarios.

Regarding plutonium, it can be concluded that as the only way to obtain Pu is fromthe reprocessing of the spent fuel, there is not enough plutonium to feed the completeeet. However, thanks to the breeding properties of the ESFR-like reactors, a strategyinvolving the continuous reprocessing of plutonium has been proposed in order to obtainenough plutonium to feed reactors' cycles until the desired production level is achieved.The real power production capacity has been assessed for the reference scenario accordingto dierent parameters. It is concluded that for the CONF2 core it would be possible toachieve the reference case scenario energy demand, as far as the on-the-y reprocessingfrom the 60-yearsof- operational-lifetime Gen-II and future Gen-III is considered.

In addition to those studies, the capability of reducing the quantity and decay heat of themost radioactive isotopes presented in the spent fuel, i.e. the minor actinides, is assessed.The study concludes that the whole decay heat of the MA is reduced after 60 years whenfuel with MA is burned in the reactor, compared with the case when it is left to decayin a temporal storage centre. The main responsible of this result is the reduction of Amisotopes when the fuel is burned.

130

Chapter 8

Generation of Point Kineticparameters

8.1 Introduction

In order to perform transient calculations for safety analyses, there are typically two ap-proaches. Either 3D kinetics is used in neutron diusion codes coupled with an appropriatethermal hydraulic code, or point kinetics is employed in a plant thermal hydraulic code.The main dierence lays on the dimensional approach. In case of 3D kinetics, parametrizedcross section libraries are required for each selected variation of the feedback parameters,while in point kinetics, reactivity coecients are required in order to catch the eect of thefeedback parameters variation on the system reactivity. For both cases, additional kineticparameters, such as delayed neutron fractions, decay constants, etc. would also be needed.

While Chapter 9 will deal with the alternative approach of nodal cross sections, this Chapterwill focus on point kinetic parameters. First, the methodology used to calculate them isdescribed. Then, as a result of the collaboration between J. M. Mendez and the authorwithin the CP-ESFR project [99], the required parameters were calculated and applied to theassessment of the impact of minor actinides on transients behaviour. Finally, an additionalstudy is presented, where the eect of uncertainties in nuclear data on reactivity coecientsis quantied. This latter study is the result of a collaboration of the author with some fellowresearchers at the Nuclear Engineering Department (UPM), which led to a presentation atthe International Nuclear Data Conference for Science and Technology (ND) held in NewYork, NY, USA in 2013 [68].

8.2 Calculation of point kinetic parameters

Along this section, the methodology employed to calculate the required parameters for pointkinetics is described. It was dened within the specications of the CP-ESFR project [46].

Among those parameters, the reactivity coecients have special relevance. The reactivity

131

8.2. Calculation of point kinetic parameters

feedback coecients represent the core reactivity change driven by any change in theoperating nominal conditions, as a consequence of a core transient.

Those coecients are calculated during the design phase in order to have a rst estimationof the safety performances of the core under design. The total reactivity is computed as thesum of the initial reactivity and the feedback reactivity, being the latter usually expressedas the product of the dierent reactivity feedback coecients considered (pcm/ºC) andtemperature changes in coolant, fuel, cladding, etc.

For the purpose of this study, reactivity coecients were calculated using the Monte Carlotransport code SERPENT. The direct method was chosen over the perturbation theory, andthe reactivity coecients are computed by direct calculations as the reactivity dierencebetween nominal state and perturbed states.

The statistical uncertainty on the k-eective in all the Monte Carlo calculations does notexceed 3 pcm and the perturbed states have been dened so that the reactivity dierenceswith respect to the nominal case are much higher.

Nominal state corresponds to nominal densities and temperatures specied in [13, 28]; thegeometrical dimensions at nominal state are supposed to be the ones specied in thosereferences at 20ºC.

8.2.1 Delayed neutron fraction, decay constants and mean gen-eration time

Delayed neutron fraction and decay constants are provided by default by SERPENT code.

The mean generation time (Λ), dened by Λ = 1νΣfv

= lkis also estimated by SERPENT

code. However, after benchmarking the results with other codes, a discrepancy of aroundone order of magnitude was noticed.

8.2.2 Doppler constant

The core reactivity progresses with the absolute fuel temperature following the rule 1/Tfuel,dening the Doppler constant as the proportionality factor:

dρ

dT=KDoppler

Tfuel(8.1)

The Doppler constant is computed from the results of two Monte Carlo simulations for thenominal and perturbed state:

• Nominal state: ssile mediums temperature is 1500 K.

• Perturbed state: same fuel isotopic composition, ssile mediums temperature is 2500K.

132


Moreover, it was conrmed through several experiments that the Doppler costants calcu-lated according to this denition provides results close to the real ones. It is important tounderline that for both states the isotopic compositions remains constant, changing onlythe cross sections.

Therefore, the Doppler constant can be calculated integrating Eq. 8.1 as [46]:

KDoppler =ρDoppler − ρNominal

ln(T2

T1

) (pcm) (8.2)

8.2.3 Coolant reactivity coecient

The reactivity change as a consequence of a sodium density change can be computed byreducing the density in 10% with respect to the nominal state. In addition, the locationof the sodium to be voided has a relevant impact on the results, as a consequence of thedierent inuence of the leakage term. In fact, the worst considered scenario would be justremoving the sodium from the active core, and keeping it for the rest of the core. Thisway, the positive eect is boosted while the leakages are not increased. Therefore, in orderto have a conservative estimation of the sodium reactivity coecient for safety analyses,only the active core of the fuel assemblies within the wrapper is voided. Moreover, if anoptimized core is considered, with axial regions particularly designed in order to reduce thiseect, the sodium density from these upper parts should also be reduced.

Here, a method to calculate the sodium reactivity coecient based on the linear expansioncoecient is described, according to [46], by comparison of the nominal and perturbedstate:

• Nominal state: Sodium with nominal density.

• Perturbed state: Sodium density is reduced only inside the wrapper by 10% in thessile core along with the Na-plenum and internal blankets.

For isotropic materials and small expansions, the linear expansion coecient is 1/3 of thevolumetric expansion coecient, assuming that 1/3 of the volumetric expansion takes placein each axis. Then, representing the variation in reactivity for each % density variation asCNa,we get the expression to calculate the void reactivity coecient:

KNa(T ) =dρ

dTNa=

∆ρ

∆ρNa(±x%)

∆ρNa(±x%)

∆TNa= −CNa(T ).3.αNa.100

(pcmK

)(8.3)

8.2.4 Expansion reactivity coecients

Among the dierent material susceptible to thermal expansion in SFR described in Chapter2, only the fuel, cladding, wrapper and diagrid are considered in this analysis.

133


Computing the expansion eects of the dierent media with a Monte Carlo code (by com-parison of nominal and perturbed states) requires important changes in the geometric modelof the core, as well as in the material densities. A script in Python was developed to carriedout this task in an automatic way.

8.2.4.1 Steel cladding (ODS) expansion coecient

Axial expansion

The cladding axial expansion is proposed to be modeled as a reduction of the ODS steeldensity, which would entail a decrease on absorptions and consequently a positive reactivityeect. The perturbed state propose would consist in a reduction of 10% of the claddingdensity with respect to the nominal value.

• Nominal state: nominal conditions

• Perturbed state: the cladding density is changed by -10% with respect to the nominalvalue in the ssile core and internal blanket

Taking CClad as the change of reactivity relative to the cladding density change of 1%, wecould obtain [46]:

KClad,ax =dρ

dTClad=

∆ρ

∆ρClad(±x%)

∆ρClad(±x%)

∆TClad= −CClad.αClad.100

(pcmºC

)(8.4)

Radial expansion

The radial expansion will be modeled according to [46] as the increase on volume of thecladding, with the consequent removal of the same volume of sodium. This eect is thesame as that of the sodium void eect, which was modeled as a reduction of sodium density.Being XVOLClad and XVOLNa the cladding and sodium volume fraction in the consideredmedium:

KClad,rad =dρ

dTClad=

∆ρ

∆ρNa

∆ρNa∆TClad

= −CNa.2.αClad.100XV OLCladXV OLNa

(pcmºC

)(8.5)

As deduced from 8.5, there is no need to perform new calculations, but the results alreadygot from the simulation to obtain the sodium void eect are taken.

Therefore, the total Cladding expansion coecient would result in:

KClad = KClad,ax +KClad,rad

(pcmºC

)(8.6)

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8.2.4.2 Steel wrapper expansion

Axial expansion

The wrapper axial expansion is modeled as a reduction of density of EM10 steel, whichwould entail less absorptions and a positive reactivity eect. The perturbed state wouldconsist in a reduction of 10% of the steel density with respect to the nominal value in thessile core and internal blanket.

Assuming CWrap as the change of reactivity relative to the wrapper density change of 10%,we could obtain [46]:

KWrap,axial = dρdTWrap

= ∆ρ∆ρWrap(±x%)

∆ρWrap(±x%)

∆TWrap=

= −CWrap.αWrap.100(pcmºC

) (8.7)

Radial expansion

The radial expansion of the wrapper will be modeled according to [46] as the increase onthe wrapper volume, with the consequent removal of the same volume of sodium. Thiseect is the same as that of the sodium void eect, which was modeled as a reduction ofsodium density. If XVOLWrap and XVOLNa represent the wrapper volume fraction and Navolume fraction in the considered medium, respectively:

KWrap,rad = dρdTWrap

= ∆ρ∆ρNa(±x%)

∆ρNa(±x%)

∆TWrap=

= −CNa.2.αWrap.100XV OLWrap

XV OLNa

(pcmºC

) (8.8)

As it might be inferred from Eq. 8.8, no more calculations are needed, since the requiredinformation can be taken from the results of the sodium void eect.

Therefore, the total steel wrapper expansion coecient would be obtained as:

KWrap = KWrap,ax +KWrap,rad

(pcmºC

)(8.9)

8.2.4.3 Fuel expansion

Axial expansion

The axial expansion of the fuel has two main eects: the decrease of the fuel density(strong negative eect on reactivity) and the increase of the fuel height (positive eect onreactivity). Both eects have to be addressed.

The result would be the sum of the previous eects. Therefore, two perturbed states aredened:

135


• Perturbed state 1: Fuel density is changed by -1% in ssile and fertile zones

• Perturbed state 2: Fuel height is increased by 1% in ssile and fertile zones

Kfuel,ax =dρ

dTfuel=

∆ρ

∆ρfuel

∆ρfuel∆Tfuel

+∆ρ

∆Hfuel

∆Hfuel

∆Tfuel(8.10)

Therefore, assuming that Cfuel,ρrepresents the reactivity change for each % density change,and Cfuel,Hthe reactivity change as a consequence of a active core height change (%):

Kfuel,ax = −αfuel (Cfuel,ρ + Cfuel,H) 100 (8.11)

Radial expansion

The radial fuel expansion is normally neglected because in case of a gaseous gap (i.e.helium) this eect is zero. Therefore, the total fuel expansion coecient is:

Kfuel = Kfuel,ax (8.12)

There are two approaches: the closed and the open gap. The linear expansion coecientto be used depends on that approach. It would be the cladding one in case of closed gapand the fuel one in case of open gap.

For the calculation it was employed the open gap approach, and the linear expansion coef-cient used was the same for MOX fuel with and without MA.

8.2.4.4 Diagrid eect

If CDiag represents the reactivity change as a consequence of the diagrid area change (%):

KDiag =dρ

dTDiag=

∆ρ

∆SDiag(±x%)

∆SDiag(±x%)

∆TDiag= CDiag.2.αDiag.100 (8.13)

This eect has also been calculated by comparison of a nominal state and a perturbedstate:

• Nominal state: nominal geometries and densities

• Perturbed state: the mass and density of the system remains the same, except forthe sodium mass present in the gap interassembly which is increased as a conse-quence of the volume increase of the sodium between assemblies, while its densityremains the same. The whole region within the EM10 wrapper (wrapper included)does not change in volume, mass neither density. The sub-assembly pitch (only af-fecting the sodium) is increased by 0.8784% and the whole core radius is increasedin consequence.

136


Figure 8.1: Ke vs Na density for a ESFR fuel assembly

Case BCS error WCS error

1 3.66% 14.14%

2 2.62% 14.26%

3 -1.76% 20.22%

4 -2.44% 15.85%

5 5.35% 17.70%

Table 8.1: Comparison of the errors obtained from the simulated and calculated value forthe Best Case Scenario (BCS) and Worst Case Scenario (WCS)

8.2.5 Assessment of linearity and superposition assumptions

In order to validate the assumptions made for the previously discussed methodology, furtheranalyses were carried out. The core taken for this analysis is the optimized core of the CP-ESFR project (see Chapter 5).

First, the linearity of the main reactivity coecients was veried. Indeed, all those guresrepresent the variation of ke as a consequence of a variation in the density of dierentmaterials. Through Figs. 8.1 to 8.3 the linearity of some representative expansion reactivitycoecients can be checked. Further details can be found in [99].

Moreover, regarding the second hypothesis typically taken, i.e. superposition of eects,the following analysis was performed. Two approaches were followed to estimate the totalreactivity produced by a combination of dierent eects. First, the total reactivity wasestimated as the sum of the dierent reactivity coecients computed each by simulating

137


Figure 8.2: Ke vs fuel density for a ESFR fuel assembly

Figure 8.3: Ke vs ODS steel density for a ESFR fuel assembly

138

8.3. Assessment of Minor Actinides Impact on Transients

the isolated perturbations considered for each case. Second, the considered perturbationswere simulated together in one case with SERPENT in order to get a unique reactivityvalue consequence of the accumulated eects. Finally both approaches were compared asshown in Table 8.1. The errors were computed as the ratio of the dierence between thesimulated (all eects considered in one input) and estimated (summing up the reactivitycontributions using the superposition theory) values to the dierence between the nominalcase and the simulated values:

error(%) =Xsim−XsumXnom−Xsim

The following combinations of eects were assessed:

1. ODS density and EM10 density

2. ODS density, EM10 density and Sodium density

3. ODS density, EM10 density, Sodium density and fuel density

4. EM10 density and Sodium density

5. ODS density and Sodium density

In order to have a better estimation of the errors between the direct simulation and thesuperposed eect, the two limits of the statistical uncertainties were taken. This way aBest Case Scenario (BCS) was dened as the one that produces the lower dierence, andthe Worst Case Scenario (WCS) as the one responsible of the greater dierences. It canbe observed that in the WCS the calculated value is underestimated with respect to thesimulated one for all the combinations, while in the BCS it is underestimated only for cases1,2 and 5.

8.3 Assessment of Minor Actinides Impact on Tran-

sients

The analysis presented in this section was originated as a consequence of the participation ofUPM in the subproject SP3.3.3 of the CP-ESFR project. The UPM contribution consistedof providing to NRG the core neutronic data necessary to perform transient analysis withthe SPECTRA code. This code models the neutronic behavior with a point kinetics model,and the coupling with thermal (fuel, cladding) and thermalhydraulics (Na) is carried out bymeans of the total reactivity term, expressed as the sum of the initial reactivity and the totalreactivity feed-back (product of the reactivity coecient in pcm/ºC and the temperaturedierence with respect to the nominal state). Consequently, point kinetics parameters andreactivity coecients were provided.

The reference case without initial MA is the optimized oxide fuelled core dened in SP2.1,WP5 by Rineiski et al. [28], called CONF2 conguration. This conguration was proposed

139


Coecient CONF2 CONF2-HOM4

Fissile Doppler (KDoppler) -988.607 -599.1921 pcm

Na density effect (KNa) 0.1913 0.2806 pcm/ºC

Total Steel cladding exp. effect (KClad) 0.0403 0.0477 pcm/ºC

Axial Steel cladding exp. eect (KClad,ax) 0.0331 0.0372 pcm/ºC

Radial Steel cladding exp. eect (KClad,rad) 0.0072 0.0105 pcm/ºC

Total Steel wrapper exp. effect (Kwrap) 0.0225 0.0281 pcm/ºC

Axial wrapper exp. eect (Kwrap,ax) 0.0177 0.0211 pcm/ºC

Radial wrapper exp. eect (Kwrap,rad) 0.0047 0.0069 pcm/ºC

Total Fuel axial exp. effect (KFuel) -0.1435 -0.1297 pcm/ºC

Fuel Density contribution -0.3133 -0.2909 pcm/ºC

Fuel Height contribution 0.1698 0.1612 pcm/ºC

Diagrid exp. effect (K_diag) -0.7752 -0.8091 pcm/ºC

Table 8.2: Reactivity coecients

to reduce the total sodium void reactivity. The case considered with MA loading is calledCONF2-HOM4 and corresponds to a high level of minor actinide consumption capable oftransmuting not only the minor actinides produced within the ESFR core but also minoractinides from thermal reactors. The HOM4 conguration consists of 4% by weight of MAhomogeneously included within the core fuel [34]. The analysis is restricted to beginningof life (BOL), and in all calculations the control assemblies (CA) are supposed to be attheir upper parking positions. The whole information of the models employed is presentedin Section 5.2.2

In order to compute the neutronic data, the SERPENT Monte Carlo code was used. Itallows to give reliable and realistic estimates of reactor core features. On one hand, thecalculated data will allow assessing the eect of MA on the transient results by means of thedynamic code SPECTRA. On the other hand, neutronic data can be compared with the onescalculated by other Monte Carlo code as MCNP, or by a deterministic code as ERANOSso that the impact of the code used to compute the neutronic data can be evaluated andthen the condence in the transient results.

8.3.1 Point kinetic parameters

The coecients evaluated at BOL as indicated in the previous section are summarized inTables 8.2 and 8.4.

140


λ: Decay constants (s)Eective beta

CONF2 CONF2-HOM4

0.0124667 5.9318E-05 5.44571E-05

0.0282917 0.00062253 0.000581948

0.0425244 0.00022723 0.00020921

0.133042 0.00056522 0.000526835

0.292467 0.00123052 0.00114862

0.666488 0.00054826 0.000506965

1.63478 0.0004599 0.000430239

3.5546 0.00022915 0.000215719

Table 8.3: Delay neutron constants

Λ: MEAN LIFE(s)

CONF2 CONF2-HOM4

2.0566E-06 1.9066E-06

Table 8.4: Mean life

141


Figure 8.4: UTOP transient. Relative Power (%), cladding maximum temperature, HotChannel exit temperature and Reactivity

8.3.2 Transient analysis with SPECTRA

Once the reactivity coecients were computed for both cases, with (CONF2-HOM4) andwithout (CONF2) minor actinides, they were provided to NRG in order to perform somerepresentative transients with SPECTRA. Among the transients analyzed, one is presentedherewith in order to illustrate the impact of MA in transient behaviour.

The UTOP (Unprotected Transient Over Power) consists in the unprotected expulsion ofcertain control rod assemblies from the core. As a result, an increase of reactivity is expectedwhich would lead to an increase of power. Even though the Doppler eect and thermalexpansions of fuel and diagrid will minimize the reactivity insertion, if the increase of thecladding temperature of some hot channels exceeds 700ºC, the cladding integrity mightbe compromised. In addition, the sodium boiling temperature at working conditions is of882.8ºC (934ºC at ~1.6 bar) , which should not be exceed in any case in order to avoidthe reactivity insertion driven by the sodium void.

142

8.4. Assessment of the impact of uncertainties of the nuclear data on the reactivitycoecients

Figure 8.5: UTOP transient. Contributions to reactivity for the CONF2 (left) and CONF2-HOM4 (right)

In the case under study, the grouped control rods are withdrawn at a speed of 4mm/s in10s, corresponding to 25 pcm/s, according to the specications of one of the transientsdescribed within the SP3.3.1 of the CP-ESFR project. In terms of reactivity, it will meana total insertion of 250 pcm. Results of the evolution of some characteristic parametersare shown in Fig. 8.4. In particular, the power, cladding maximum temperature, hotchannel exit temperature and total reactivity are compared for the CONF2 and CONF2-HOM4 models. It is observed that the power peak is around 167% (CONF2) and 199%(CONF2-HOM4) at 10s which is reduced afterwards due to the Doppler. As for the claddingmaximum temperature, it achieves 669ºC in the CONF2 at 350s and 709ºC in the CONF2-HOM4 at 17s. Moreover, the hot channel exit temperature achieves a maximum value of631ºC at 390s (CONF2) and 661ºC at 15s (CONF2-HOM4).

Regarding the reactivity, it might be seen that the reactivity peak is higher in case of loadingMA in the core. Around 77 pcm is achieved at 10s in the CONF2-HOM4 while 66pcm areachieved in case of CONF2. The dierent contributions from each particular eect areshown in Fig. 8.5. It is remarkable to observe that the Doppler eect clearly compensatesthe insertion of reactivity, being its inuence more noticeable in the case of CONF2.

As a conclusion of this analysis, the MA impacts negatively in the transient performances,being susceptible to lead to a cladding failure in some rods that exceed the 700ºC.

8.4 Assessment of the impact of uncertainties of the

nuclear data on the reactivity coecients

Sensitivity and uncertainty analyses are becoming a relevant tool to comprehend the relia-bility of the neutronic design of a given nuclear system; in particular, it is critical to evaluatethe uncertainty in the reactivity coecients due to their inuence in the safety assessment.

143


Reaction Contribution %Δk/k Cumulative contribution (%)

U-238 (n,n') 1.2965E+00 ± 1.1752E-03 81.53

Pu-239 υ 5.6580E-01 ± 1.8243E-06 88.95

U-238 (n,γ) 3.3170E-01 ± 1.7394E-05 91.37

Pu-239 χ 2.9968E-01 ± 2.4770E-06 93.29

Pu-240 υ 2.2127E-01 ± 2.6682E-07 94.32

Pu-239 (n,γ) 2.2035E-01 ± 7.8671E-06 95.33

Table 8.5: Uncertainty contributors at nominal state

As a consequence, an uncertainty quantication from nuclear data (XS, nu-bar, chi, . . . )to the reactivity coecients of sodium fast reactors was performed in order to conrmtheir reliability. The results of the work were presented in the International Nuclear DataConference for Science and Technology (ND) held in New York, NY, USA in 2013 [68].The objective of the study was to identify the nuclear reaction data where an improvementwould certainly benet the design accuracy.

As a personal contribution to this work, the ESFR full core was modeled with SCALE6.1(see Chapter 5). It was then used to compute a series of steady states with KENO-VI MonteCarlo code using the available 238 energy groups cross sections library based on ENDF/B-VII.0 evaluation. In addition, an adjoint calculation was also performed to apply AdjointSensitivity Analysis Procedure (ASAP) to obtain sensitivities using SCALE tools. First itwas applied for each steady state ke, and then for the reactivity coecients between thereference and perturbed states. Propagated uncertainty data comes from the 44 energygroups evaluation included in SCALE.

Two perturbated states were dened, which would further used to calculate the reactivitycoecients:

1. A heated state, where the fuel temperature is increased in 1000K This result will leadto the calculation of the Doppler coecient.

2. A voided state, where the sodium is voided from the active core up to the sodiumblanket region. This state will lead to the calculation of the sodium density reactivitycoecient.

As a sample of the results obtained, Tables 8.5 to 8.7 show the main reaction cross sectionsthat contributes to the total uncertainty values for the nominal state, Doppler heated state,and sodium voided state, respectivelly. In general, it is dominated by Pu-239 and U-238reactions.

In general, uncertainties are higher when propagated to the reactivity coecients than forthe k-e of each state. It should be noted that for the Doppler reactivity eect and forthe voided state, not only the Pu-239 and U-238 reactions are present, but also the elasticscattering reactions with Na-23 and O-16 are relevant.

144



U-238 (n,n') 4.7772E+00 ± 3.4628E-01 82.73

O-16 elastic 1.5308E+00 ± 1.0979E-01 86.87

Na-23 elastic 1.4566E+00 ± 8.3134E-02 90.46

Pu-239 χ 9.8054E-01 ± 6.6798E-04 92.04

U-238 elastic 8.8004E-01 ± 1.2216E-03 93.29

Pu-239 (n,γ) 8.4592E-01 ± 2.5672E-03 94.43

Pu-239 υ 8.3095E-01 ± 2.9677E-04 95.52

Table 8.6: Main contributors to Doppler heating reactivity uncertainty


U-238 (n,n') 2.1909E+01 ± 3.6239E-01 70.61

Na-23 (n,n') 1.2629E+01 ± 2.1610E-02 81.50

U-238 elastic vs. U-238 (n,n') 7.4737E+00 ± 3.2129E-02 84.99

Na-23 elastic 7.4694E+00 ± 7.2949E-02 88.33

U-238 (n,γ) 7.4333E+00 ± 7.0998E-03 91.52

Pu-239 υ 7.0537E+00 ± 4.9805E-04 94.30

Pu-239 ssion 4.2774E+00 ± 4.9529E-04 95.31

Table 8.7: Main contributors to Na voiding reactivity uncertainty

145


Figure 8.6: Sensitivity of nominal state

Figure 8.7: Sensitivity of heated state

146

8.5. Outcomes

Figure 8.8: Sensitivity of voided state

In addition, Figs. 8.6 to 8.8 shows the sensitivity of the most representative cross sectionsover the reactivity and reactivity eects, respectively, as function of the energy.

As a consequence of the study it was concluded that an uncertainty improvement in thecross section evaluation is required, specially for the elastic and inelastic scattering crosssection for U-238 and Na-23.

8.5 Outcomes

Along this Chapter, the main point kinetic parameters were computed using SERPENT tobe provided to a plant thermal hydraulic code. Collateral assessments were also performedrelated to the methodology. The obtained parameters were then provided to NRG withinthe CP-ESFR in order to enable them to perform transient calculations using SPECTRA.The nal aim of this collaboration was the assessment of the impact of minor actinides intransient behaviour (SP3.3.3).

In addition to those analyses, the assessment of the impact of uncertainties of cross sectionson typical reactivity coecients of a SFR was performed, as a result of the collaborationbetween the author and some fellow researchers at DIN. As a consequence, the importanceof quantifying properly some relevant cross sections was highlighted.

147

8.5. Outcomes

148

Chapter 9

Nodal Cross sections

9.1 Introduction

Cross section generation is an essential step for nodal diusion codes, as disussed in Chapter3.

Here we aim to analyze two dierent codes able to generate these nodal parameters in orderto be provided to our home-made nodal diusion solver ANDES, within the COBAYA3system.

There are two main ways in which the cross sections can be provided to diusion codes: astabulated cross sections and as derivatives. The rst one is easier to calculate, but requires ahuge amount of memory available in the system in order to have all the needed information.As for the cross sections given as derivatives, they would required more preparation work inorder to obtain a correlation with enough accuracy. Among the tabularized cross sectionsformats, the NEMTAB is the most generalized.

Depending on the type of calculation desired, there are two kinds of cross section libraries:

• For steady states, which for most of the diusion codes would include the parametersshown in the second column of Table 9.1.

• For transient analysis, which would require additional parameters to account for thekinetics, as shown in last column of Table 9.1.

Moreover, when a coupled calculation is seeked, these libraries need to be parametrized infunction of the dierent feedback parameters.

Note that macroscopic cross sections rather than microscopic ones are required.

This Chapter will deal with the methodology to compute nodal cross sections followingtwo dierent approaches: deterministic and MC codes. Some conclusions are inferred fromthe comparison between both results. Finally, further intructions are provided regardingthe extension of the methodology to compute parametrized cross sections for transientcalculations.

149

9.2. Cross section calculation using SERPENT

Static Transient

Diusion coecient cm x x

Absorption cross section 1/cm x x

Fission neutron-production cross section 1/cm x x

Fission energy-production cross section Ws/cm x x

P0-Scattering matrix (Multi-group diusion) 1/cm x x

Fission spectrum 1 x x

Neutron inverse velocity s/cm x

Precursor decay constant (λi) 1/s x

Delayed neutron fraction for each precursor (βi) 1 x

Table 9.1: Parameters required for diusion calculations

9.2 Cross section calculation using SERPENT

Among the many capabilities available in SERPENT, the generation of homogeneized multi-goup parameters has been used. SERPENT is based on a universe-based approach, and thecode allows the user to generate group constants simultaneously to multiple universes. Foreach steady state problem, a set of cross sections homogeneized over the selected universesare provided in the output le of SERPENT. The group structure at which the constantsare homogeneized can be selected as well.

9.2.1 Choice of universes

SERPENT is a universe-based code, similar to other Monte Carlo codes, such as MCNP andKENO-VI from the SCALE system. As any Monte Carlo codes, it allows to describe in detailany geometry, no matter how complex it is. In addition, it is possible to group the dierentregion within the same universe, following the user's will. For cross section homogeneization,the user is allowed to choice any number of universes where the homogeneization is desired.The only limitation is that the universes in the list cannot be sub-universes to each other.

For the ESFR-WH core, the following universes were considered (see Chapter 5):

• Active height of the inner fuel assemblies

• Active height of the outer fuel assemblies

• Rod follower

• CSD absorber assembly

• DSD absorber assembly

150


• Lower/Upper Gas Plenum (L/U-GP)

• Upper Axial Shielding (UAB)

• Lower Axial Blanket (LAB)

9.2.1.1 Methodology

In order to develop an adequate methodoloy to generate group constants, it is importantto identify the dierent regions to be considered. Not only the material compositions arerelevant for the calculation of those homogeneized cross sections, but also the neighbouringeect, which might impact on the real ux perceived in the zone. Particularly challengingwould be the calculation of those parameters in non-ssile regions, such as the controlassemblies. In this regard, Fig. 9.1 shows the radial section of the ESFR core, identifyingthe dierent types of contol assemblies, attending to the situation on the core. The six CSDassemblies in the inner ring are considered identical, as well as the nine DSD assembliesin the second ring. For the third ring of CSD control assemblies, two additional type ofassemblies could be considered: the ones surrounded by two inner and four outer FAs, andthe ones surrounded by three inner and three outer FAs. Therefore, there would be onetype of DSD control assembly and three types of CSD control assemblies.

Here, we propose two alternative methodologies to calculate group constants in the selecteduniverses:

1. One single 3D whole core calculation, where the universe homogeneization wouldattend only to the material composition. Some alternatives to this method could alsodistinguish regions with the same material compositions but dierent neighbours,such as the case of the FAs located at the core boundaries, or the control assemblies.

2. Eleven calculations using SERPENT, which would account for the dierent mediasepparately.

(a) Two 3D Fuel assembly calculations, one for the inner fuel, and another one forthe outer fuel. The axial layers of each FA will be taken into account in theuniverse subdivision.

(b) Eight 2D minicore calculations with the control assembly surrounded by six FAs,as indicated in Fig. 9.1. For each control type, one calculation for the controlinserted and another one withdrawn.

(c) One 2D core calculation to get the average parameters for the radial reector

9.2.2 Group constants

In the output le SERPENT provides by default the parametes shown in Table 9.2 for eachuniverse dened.

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Parameter Values Description

FLUX 2G+2 Integral ux

LEAK 2G+2 Leakage rate

TOTXS 2G+2 Total cross section

FISSXS 2G+2 Fission cross section

CAPTXS 2G+2 Capture cross section

ABSXS 2G+2 Absorption cross section

RABSXS 2G+2 Reduced absorption cross section

ELAXS 2G+2 Elastic scattering cross section

INELAXS 2G+2 Inelastic scattering cross section

SCATTXS 2G+2 Total scattering cross section

SCATTPRODXS 2G+2 Total scattering production cross section

N2NXS 2G+2 (n,2n) cross section

REMXS 2G+2 Group-removal cross section

NUBAR 2G+2 Average number of emitted ssion neutrons

NSF 2G+2 Fission neutron production cross section (νΣf,g)

RECIPVEL 2G+2 Inverse mean neutron speed

FISSE 2G+2 Average ssion heating value (in MeV)

Table 9.2: Multigroup constants calculated by SERPENT

152


Figure 9.1: ESFR core with selected universes

All the parameters described in Table 9.2 present the average and associated relative sta-tistical error. Independently to the group structure chosen, the one-group constants areprovided. All cross sections are macroscopic. Capture cross section includes all (n,0n)reactions. Absorption as usual includes capture and ssion reactions. Elastic scatteringincludes thermal bound-atom reactions. The group removal cross sections includes absorp-tion and scattering out of the energy group. The reduced absorption cross section is denedas absorption minus production in (n,xn) reactions. It is useful for some diusion codesdepending on the way the neutron balance equation is formulated.

Further parameters are provided in the output le of SERPENT, such as the energy spectrumof prompt neutrons (CHIP), of delayed ssion neutrons (CHID) and the energy spectrum ofall the ssion neutrons (CHI). As for the scattering matrixes, they are named group-transfercross section matrix (GTRANSFXS) or group-production cross section matrix (GPRODXS),depending once more on the way the neutron balance equation is formulated. The dier-ence really lies on where the (n,xn) reactions are considered. For example, if the reducedabsorption cross section is considered, the group-production cross section matrix should beemployed.

Diusion coecient

Regarding diusion coecients, it is important to realize that calculating the diusioncoecient using Monte Carlo tools is a very dicult task because diusion theory does notcombine easily with the continuos-energy simulation. Therefore, SERPENT computes itthrough four dierent methods:

153


1. Based on the migration area (DIFFCOEF). The code essentially tallies the meansquare distance the neutrons travel within each energy group. Therefore, this def-inition does not really make sense unless the calculation is carried over the entiregeometry.

2. Based on the transport cross section. The denition is based on the collision es-timator, which means that it is not exactly valid if subdomains are considered:P1_DIFFCOEF

3. Based on the leakage mode. Leakage diusion coecient is dened as bucklingdivided by leakage, which can be physical or from a leakage model. Although thetheoretical basis is very questionable.

4. Based on the B1 critical spectrum calculation [56]: B1_DIFFCOEF

In order to account for the non-physical innite-lattice approximation, B1 methodology,routinely used by deterministic lattice transport codes, was considered for generation ofleakage-corrected few-group cross sections with SERPENT. This capability is available inSERPENT from version 1.1.14 on and calculates the same parameters as provided with thedefault method.

Assembly Discontinuity Factors

An additional parameter usually required for nodal diusion codes is the Surface Disconti-nuity Factors (also called Assembly Discontinuity Factors, ADFS). By default, SERPENTcomputes the surface discontinuity factors and the corner discontinuity factors (ADFC).However, these parameters only account for 2D problems and have been only tested whenthe group constant generation is extended over the entire geometry. In addition, the ADFSare calculated according to 9.1.

ADF =φhets

φhet(9.1)

The required parameters for COBAYA3 would be the Interface Discontinuity Factors denedas 9.2, since the objective is to be able to reproduce the transport heterogeneous results, andtherefore to perform nodal homogenized diusion calculations. In particular, for the nodalsolver ANDES within COBAYA3 system, the homogenized surface ux can be expressed asa relation between the surface heterogeneous current and the average heterogeneous ux(Eq. 9.3).

IDF =φhets

φhoms

(9.2)

φhoms = φANDESs = f(Jhets , φhet) (9.3)

Meaning that it would be required to have from SERPENT:

154

9.3. Cross section calculation using ERANOS

• the surface currents Jhets

• the surface integrated uxes φhets

• the average uxes φhet

In fact, they are allready available. The φhet is calculated by default in SERPENT, and thesurface currents are provided by SERPENT since version 1.1.17.

9.3 Cross section calculation using ERANOS

ERANOS is a deterministic tool especially designed for fast reactor simulations. It is com-posed by dierent modules with specic functions. It can perform diusion and transportcalculations. It can also perform perturbation analysis and burnup (see section 4.2.2.1).

Among the dierent modules, the ECCO cell code prepares self-shielded cross sections andmatrices for use in fast reactor core or shielding calculations (performed with the ERANOScode system). The use of the subgroup method to treat the resonance self-shielding isparticularly suitable for calculations involving complex heterogeneous structures. Associatedwith the slowing down treatment in many groups (1968 groups with the current library), themethod improves the standard of accuracy to a level required by current reactor designs,and deals with numerous dierent situations (various core and subcritical subassemblies)and geometries (hexagonal and rectangular subassemblies in particular).

The ECCO cell code includes the following features :

• the exact description of the usual fast reactor geometries,

• pin cells and plate cells of critical facilities,

• hexagonal subassemblies of power reactors (wrappers, sodium void), as well as de-tectors or experimental devices in critical facilities or in reactors,

• the subgroup method for self-shielding calculations with dierent algorithms for ux-weighted cross sections such as capture or ssion, and for current-weighted crosssections like transport or Legendre order 1 type cross sections and matrices. Anexponential mode of the subgroup method has been implemented specically forshielding applications which calculates cross sections and matrices for any Legendreorder,

• the ux and current P1 consistent equations for both homogeneous and heteroge-neous descriptions of cells, which enables an accurate treatment of leakage with bothanisotropy of scattering and streaming eects,

155


• the anisotropy of streaming eects in voided cells and low density channels,

• elaborated algorithms and procedures for blankets (heterogeneity of the cell and ofthe incoming source) and for shielding (the higher order of the elastic scatteringcross-sections, and of the weighting uxes).

Collision probabilities are used in ECCO for dierent purposes such as self-shielding, uxand current as well as streaming eect calculations.

Collision probabilities are obtained for various geometries and are then used to calculateuxes and eective cross sections with dierent types of approximations (resonance shieldingand ux calculations). These many options make the use of the ECCO cell code very exible.However, recommended routes and options are necessary to users in practice.

9.3.1 Running ECCO

In practice, the input of ECCO is not straight forward to comprehend. But in summary,the neccesary information is the following:

• Nominal conditions, dpa and expansion coecients

• Description of each medium composition

• Geometry data of each cell

• Boundary conditions

• Operating conditions

• Steps denition for the considered route.

For each step, some options are available:

1. GEOMETRY:Type of geometry modelling to be used in this step

(a) Original/Homogeneous

2. GROUP STRUCTURE: Group structure to be used

(a) Fine /Broad/Other

3. ELEMENTS: Denes elements for which resonance shielding is to be performed.

4. FIND_ELEMENTS_IN_LIST: To use a pre-dened list loaded in LU variable beforecalling ECCO

5. FLUX_SOLUTION: Method to be used for ux solution

156


(a) Fundamental Mode (FM)/Collision Probabilities (CP)

(b) P1/B1

(c) CONSISTENT/INCONSISTENT

(d) Order

6. BSEARCH: Search for buckling which gives the specied value of ke

(a) Buckling is to be taken from the specied step (BFROM) or to be read (BUCK-LING)

7. LEAKAGE: This section denes the method of treating leakage in CP ux

(a) Add DB2 absorption (DBBABS)/Apply non-leakage factor (CP only,NLFACT)

(b) Use cell averaged D (CELL)/Use region D(REGION)

(c) Formula employed for D calculation: BENOIST/RSTR/MURAL

(d) Type of averaging for cell D: Volume weighting (VOLWT)/Volume x ux weight-ing(FLUXWT)

(e) Mean D to be used(MEAN)/ Directiona(DIR)/Axial(DIRAX)/Radial(DIRAD)

8. SHELF-SHIELDING: This species whether or not DB2 is added to the total cross-section in the self shielding calculations

9. HOMOGENISE: Keyword indicates that a library from the previous step is to be usedin conjunction with geometry specied by GEOMETRY HOMOGENEOUS

10. DIAGNOSTICS: Indicates diagnostic printout is required.

11. PRINT DATA: Indicates various quantities are required to be printed.

It is important to remark that the neighbouring eect is not considered by ECCO when themicroscopic cross sections are calculated. For the non ssile media, a ssile ux should beprovided to ECCO.

After ECCO is run, the whole core is built and the macroscopic cross sections are thencalculated using the updated densities coming from the burnup module.

The available group structures are divided into 33, 172, 175 or 1968 groups of energy.

9.3.2 How to Calculate the required parameters with ERANOS

All the macroscopic cross sections as well as the matrix were printed using MACRO_EDITION.However, some parameters required a further study to be obtained, such as the energy re-leased per ssion (Kappa-Fission) and the kinetic parameters. The former one was nallyobtained indirectly through the MICRO_EDITION and MEDIUM_EDITION: the ssion

157

9.4. Comparison between ERANOS and SERPENT

ERANOS (ECCO) SERPENT

Code type Deterministic Monte Carlo

Energy distribution Discrete (33, 172, 175 or 1968) Continuous

Computational time Faster (~minutes) Slower (~hours)

Geometries allowed 1D, 2D (XY, hexagonal, RZ) Detailed 2D or 3D

Cross section Per mediums More exible

homogenization (compositions) (per universes)

Burnup capability YES YES (CRAM method)

Directional diusionTheorically yes No

coecient

ADFs No Yes (Surface ux)

Macroscopic cross sections Yes Yes (B1 and Kinf)

Table 9.3: Qualitative comparison between capabilities of ERANOS and SERPENT

importance of each isotope is given and together with the energy released by each isotopeit is possible to estimate the Kappa.

As for the kinetic parameters, a post-processing module to calculate the eective betawas spotted. This module requires the adjoint ux calculation, so a full calculation withERANOS had to be performed.

9.4 Comparison between ERANOS and SERPENT

For all the calculations, in both codes, a 15-energy-groups structure was taken, and thecross sections were taken from the JEFF-3.1.1 and JEFF-3.1 libraries at 20ºC

9.4.1 Main dierence between ERANOS and SERPENT

Table 9.3 summaries the main dierences between the codes under study.

Note that cross section homogenization refers to the way the homogenization is performed.ERANOS does not considerer anything but the composition of the media and the detailedradial description is just available for fuel subassemblies. The localization of the mediawithin the core is not taken into account for cross section generation. While on the otherhand, SERPENT generates the homogenized cross sections per each universe describedrequested, which provides a much more exibility as well as a better consideration of thereal spatial distribution of the media in the core.

158


DIFF-COEF

EnergyFuel1 Fuel2

Group

1 -10.3 -21.13

2 -3.5 -14.78

3 8.0 -3.6

4 0.1 -9.69

5 16.37 6.68

6 20.7 11.88

7 16.07 9.27

8 22.74 18.29

9 6.79 3.96

10 38.58 30.12

11 128.04 141.68

12 810.77 569.96

13 737.29 516.95

14 11628.96 6282.27

15 3594.42 9975.23

Table 9.4: Relative error between SERPENT and ERANOS for Diusion coecient forssile nodes

9.4.2 Comparison of cross section libraries

In this section the comparison between the cross sections generated by SERPENT andERANOS codes will be shown. In order to take into account all the materials constitutingthe core, the whole core model is employed. For all the SERPENT calculations, 100000neutrons per cycle in 1200 cycles (200 inactive ones) were used.

To analyze the dierences between both codes, the relative error is computed as

Relative error =ΣSERPENT − ΣERANOS

ΣERANOS

From Tables 9.4 and 9.5 it can be drawn that the diusion coecient presents clearly bigdiscrepancies between both codes, especially for the absorber assemblies (CSD and DSD).It can be explained taking into account that the diusion coecient calculated accordingto the migration area in SERPENT does not work when the homogenization is done overdierent universes within the whole geometry. As a consequence, it is concluded that the

159


DIFF-COEF

EnergyReector CSD DSD

RodLGP LAB UGP UAB

Group Follower

1 288.11 304.23 315.6 -64.92 327.92 305.37 -53.84 840.53

2 232.79 272.4 220.16 -62.47 204.62 243.4 -57.32 637.23

3 109.42 243.53 165.56 -63.96 91.1 107.31 -66.77 304.98

4 51.55 310.76 340.56 -47.96 63.23 26.51 -73.52 134.69

5 26.29 428.66 415.79 -57.11 9.46 0.11 -77.96 53.35

6 20.3 423.99 424.81 -58.02 -5.59 6.37 -77.95 38.88

7 29.63 273.55 342.15 -49.18 -3.67 32.5 -69.05 54.03

8 5.02 375.14 539.55 -48.17 -22.28 5.7 -77.12 12.09

9 11.47 84.87 199.43 -22.72 -11.69 32.95 -46.06 33.17

10 14.11 377.15 654.82 -69.67 -18.44 56.57 -78.23 26.16

11 12.88 716.48 1208.04 -70.56 -25.43 66.08 -86.5 -7.49

12 12.11 1326.15 3096.54 -51.85 -20.66 78.19 -79.5 -6.81

13 13.38 5054.42 27713.6 -57.04 -19.05 99.98 -76.99 -5.66

14 15.61 201634.12 2102765.5 -51.67 -10.02 111.2 -57.56 -2.77

15 7 271605.59 264826.66 -32.73 15.89 279.9 -62.15 6.73

Table 9.5: Relative error between SERPENT and ERANOS for Diusion coecient fornon-ssile nodes

160


ABSORPTION XS

EnergyFuel1 Fuel2

Group

1 -2.63 -2

2 -1.6 -1.29

3 -0.06 0.22

4 0.22 0.37

5 0.03 0.14

6 0.74 0.8

7 2.02 1.96

8 1.15 1.15

9 0.3 0.4

10 0.52 0.64

11 8.05 11.42

12 -23.24 -27.7

13 -12.15 -18.16

14 29.83 -15.7

15 -96.52 -94.28

Table 9.6: Relative error between SERPENT and ERANOS for Absorption cross section forthe ssile nodes

diusion coecient cannot be accurately calculated using SERPENT.

As for the absorption cross section, a better agreement is observed. The highest errors arelocated in the slower groups, which is a consequence of the limited number of neutronsachieving these groups. It should be mentioned too that the dierence encountered in theabsorber material might be a consequence of the dierent treatment of this regions by eachcode. ERANOS uses the homogenized region while SERPENT the detailed description ofeach absorber rod and dierent wrappers.

From table 9.8 a good agreement is also observed regarding ssion cross sections.

All in all, as a general comment, large discrepancies between the cross section providedby SERPENT and ERANOS are observed in the thermal range. This can be explained asthe poorer statistics characteristic of this region, where very few neutrons with that energyrange achieve the regions of interest. As a consequence of this eect, some authors haveproposed a energy group structure where the thermal range has been collapsed [57].

Moreover, it is also veried that SERPENT provides worst results for the materials as theyare further from the ssile regions, and as a consequence, fewer neutrons achieve their

161


ABSORPTION XS

EnergyReector CSD DSD

RodLGP LAB UGP UAB

Group Follower

1 6.25 -4.99 42.58 1.92 39.54 6.68 3.36 6.59

2 -6.54 -28.36 -15.5 0.45 -15.68 -6.21 -6.67 -7.23

3 -3.7 -29.18 -16.28 0.29 -3.08 -3.27 -1.23 -3.43

4 -1.9 -29.54 -24.18 -1.21 -3.82 -1.11 -1.95 -1.82

5 -2.58 -33.94 -31.38 -3.12 -9.36 -1.55 -5.56 -2.91

6 -3.26 -33.49 -36.41 -5.85 -10.24 -1.56 -7.91 -2.87

7 -5.23 -29.71 -43.46 -4.99 -12.66 -4.5 -9.67 -5.04

8 0.98 -36.57 -59.52 8.02 -0.55 0.74 1.84 2

9 3.96 -32 -73.98 5.25 3.34 4.26 4.07 4.33

10 -1.8 -63.12 -86.8 -13.41 -17.86 -1.88 -9.66 -2.13

11 8.64 -75.89 -92.86 29.67 36.42 5.99 26.79 13.45

12 2.39 -93.53 -98.17 11.53 15.31 1.73 11.34 0.35

13 1.81 -98.44 -98.81 -10.79 8.3 1.06 3.39 1.24

14 1.73 -80.9 -88.87 -0.53 -2.41 1.41 -13.89 1.47

15 -0.87 -99.61 -100 -89.62 -47.19 -12.41 -98.37 -14.7

Table 9.7: Relative error between SERPENT and ERANOS for Absorption cross section fornon-ssile nodes

162


FISSION XS

EnergyFuel1 Fuel2

Group

1 -0.31 -0.05

2 -0.04 0.22

3 -0.08 0.21

4 0.27 0.4

5 -0.09 0.05

6 -0.03 0.11

7 0.19 0.31

8 0.15 0.23

9 -1.31 -1.15

10 0.28 0.59

11 8.32 12.34

12 7.34 -1.68

13 -13.23 -19.26

14 29.5 -26.35

15 18.64 12.47

Table 9.8: Relative error between SERPENT and ERANOS for Fission cross section

163

9.5. Feedback parametrized cross sections for transient analysis

UnitsDependence on

feedback parameters

Diusion coecient cm yes

Absorption cross section 1/cm yes

Fission neutron-production cross section 1/cm yes

Fission energy-production cross section Ws/cm yes

P0-Scattering matrix (Multi-group diusion) 1/cm yes

Assembly discontinuity factors 1 yes

Fission spectrum 1 No

Inverse neutron velocity s/cm No

Delayed-neutron precurser decay constant (λi) 1/s No

Delayed-neutron fraction for each precursor (βi) 1 No

Delayed-neutron spectrum 1 No

Table 9.9: Parameters required for diusion calculations

location.

Finally, regarding the calculation of the diusion coecient, it has been veried that SER-PENT cannot calculate it accurately for the regions of interest.

9.5 Feedback parametrized cross sections for tran-

sient analysis

Transient analyses usually need to be simulated using the diusion approach rather thanthe transport one, as discussed in Chapter 3. There are two main ways of performingtransient analyses. One uses the point kinetics approach in conjuction with a systemthermal-hydraulic code, while the other uses a diusion code coupled with a core and/orsystem thermal-hydraulic code. In case of using the point kinetic approach, a set of reactivityfeedback coecients need to be calculated beforehand, as explained in Chapter 8. On theother hand, when the diusion approach is employed, a set of parametrized cross sectionlibrary as function of the dierent feedback parameters is required instead. Table 9.9 showsthe required parameters/cross sections for a transient calculation using a diusion code,such as COBAYA3.

As discussed in Chapter 2, the feedback reactivity coecients in fast reactors dier fromthose in thermal reactors. Therefore, new feedback parameters are to be considered. Giventhe importance of thermal expansions on reactivity, those are taken intro account as feed-back parameters to the cross sections. Usually, the cross section feedback parameters in

164

9.6. Outcomes

fast reactors are:

• Doppler temperature (TF)

• Coolant density/temperature (ρC)

• Radial expansion (R)

• Axial expansion (H)

The acurate calculation of the expansion feedback parameters is one the most challengingissue in the simulation of fast reactor transients. In case of point kinetics, the expansionreactivity coecients are treated in more detail (see Chapter 8). However, in order toprovide a rst simple approach, the average core radius (R) and the average core height(H) can be taken as representation of the core expansion, as proposed by K. Mikityuket al. in [98]. For the purpose of the FAST system, the macroscopic cross sections arerecalculated according to:

Σ(TF , ρc, R,H) = Σ0 +[∂Σ∂Z

]Z0

(Z − Z0) +[

∂Σ∂ lnTF

]TF0

(lnTF − lnTF0)+

+[∂Σ∂ρC

]ρC0

(ρC − ρC0) +[∂Σ∂R

]R0

(R−R0) +[∂Σ∂H

]H0

(H −H0)(9.4)

The control rod insertion is also considered in the FAST system as feedback parameters.However, in ANDES a special treatment of the control insertion is performed, in order tocatch accurately the rod cusping eect.

As explained before, there are two ways to recalculate the cross sections or parametersthat are subject to feedback dependance (indicated in Table 9.9) attending to the actualfeedback parameteres. One way is by mean of derivatives, as in 9.4, while the other wayis by the interpolation on a tabulated library. For both cases, the parameters should becalculated for each combination of the four feedback parameters in order to obtain thederivative or the tabulated library, respectively.

9.6 Outcomes

When 3D kinetics are pursued with diusion codes, adequate cross sections need to beprovided. Here, two dierent codes are employed for cross sections generations, and themethodology adopted is described. Results provided by the two codes are compared andsome conclusions are drawn. In general, while the statistics is good enough for SERPENT,the agreement between codes is quiet good. However, for those regions of energy groupsthat few neutrons achieve, big discrepancies are obserbed between codes. As a consequence,adequate denition of the thermal range, together with variance reduction techniques arenowadays been implemented in SERPENT in order to reduce this discrepancies.

165

9.6. Outcomes

As future developments, sensitivity analyses to the group structure are required, in orderto limit the eect of the thermal range. Moreover, the methodology needs to be extendedin order to account for changes in feedback parameters, such as fuel temperature, sodiumdensity and also geometry variations.

166

Part V

DEVELOPMENT OF A N-THCODE FOR SFR

167

Chapter 10

COBAYA3-ANDES

10.1 Introduction

As described in Chapter 2, fast reactors simulations entail some singularities that dierfrom those linked to conventional thermal reactors calculations. Thus, in order to extendthe applicability of COBAYA3 to SFRs, those singularities should be carefully addressed.

In particular, as a consequence of the typical fast spectrum, a ner energy discretizationshould be considered in order to treat correctly the unresolved resonance region. The needof more energy groups could introduce a complexity when advanced analytical methods areemployed to solve the neutron diusion equation.

Moreover, fast reactors are designed in hexagonal geometry in order to maximize the fuelvolume fraction, which decreases the critical ssile inventory, while taking into account heatremoval constraints. The hexagonal geometry entails a bonus diculty for deterministiccodes, compared to Cartesian geometry. In particular, the transverse leakage integrationrequired to transform one 3D equation into three 1D equations is one of those additionaldiculties.

Another aspect related to fast reactor design is that they usually involve large axial layersof non-ssile materials, such as ssion gas plenums, axial blankets or even neutron absorberlayers and sodium plenums, being the latter ones particularly designed to optimize coreperformances within the framework of the CP-ESFR project [54]. Those regions add apiece of complexity to the simulations when deterministic codes are to be employed.

In addition, the long mean free path, consequence of the small elastic scattering cross sec-tions characteristic of fast spectrum, implies a global coupling of the core. As a consequence,fast reactor calculations require detailed whole core calculations. The hard neutron spec-trum makes that reactions which are typical for this range, such as anisotropic scattering,inelastic scattering, (n,2n) reaction, and resonance self-shielding, should be considered. Thescattering resonances of intermediate atomic mass nuclides and the lack of 1/E spectrumfor the calculation of heavy isotope resonance absorption require very detailed modeling forslowing-down calculations.

169

10.2. COBAYA3

In this Chapter, rst the methodology implemented in ANDES, the nodal solver of COBAYA3,for the hexagonal geometry is described. Second, the assessment of this method applied toSFR is carried out and the main diculties are discussed.

10.2 COBAYA3

The reactor dynamic code COBAYA3 was developed at the Technical University of Madrid(UPM). It is a multi-scale, parallelized code that solves the steady state and time-dependentdiusion equation at nodal and pin-cell level for hexagonal and square geometries. COBAYA3can perform pin-by-pin calculations of mini-cores and whole 3D cores using domain decom-position through alternate dissections methodology [78, 67]. COBAYA3 is composed oftwo dierent solvers in charge of developing pin-wise solutions (COBAYA3k) [79] and fuelassembly (nodal) base solutions (ANDES) [90]. In addition, COBAYA3 has already beencoupled with other thermal-hydraulic codes, such as COBRA-III, COBRA-TF and FLICA[78].

The 3D nodal multi-group diusion solver ANDES [88] is based on the Analytical CoarseMesh Finite Dierence (ACMFD) method for solving the multi-group diusion equationat nodal level. ANDES is able to simulate steady state and transient problems by solvingthe time dependent diusion equation plus the delayed neutron balance equations. Sinceit was already extended to hexagonal-Z geometry , it was selected for its application toSFRs. On the other hand, COBAYA3k code solves the diusion equation through nitedierence method at pin level. Both codes include a transport correction factor to accountfor homogenization errors (Discontinuity Factors-DF).

Hereby the description of ANDES is presented, emphasizing the treatment of hexagonalgeometry, according to [88, 91].

10.2.1 Introduction to the ACMFD method

The Analytic Coarse Mesh Finite Dierence (ACMFD) method, developed by Y.A. Chao[16], was rst fully generalized to multigroup [90] and multidimensional [60] problemsand implemented in the ANDES solver for PWR core analysis. Then, following the sameguidelines, the ACMFD methodology was extended to Triangular-Z geometry to be appliedto VVER reactors [91]. Excellent performances have been proved in both rectangular andhexagonal geometries, which were considered as a promising starting point for its applicationto Gen IV advanced core designs that use hexagonal fuel assembly congurations, such asSodium Fast Reactors.

The ACMFD method is based on two steps aimed to obtain an analytical solution of themultigroup diusion equation within homogeneous nodes. The rst one is the decouplingof the multigroup diusion equation by means of diagonalization of the multigroup diusionmatrix A, in such a way that we obtain a set of uncoupled diusion equations over themodal uxes which are linearly related to the physical uxes.

170

10.2. COBAYA3

∇2 |φg (r)〉 −A |φg (r)〉 = −D−1 |Sg (r)〉 (10.1)

A|um〉 = λm|um〉 ; R−1 = [um] ; A = R−1 [λm]R (10.2)

Where λm and um are the eigenvalue and eigenvector for mode m. Thus, pre-multiplyingequation 10.1 by matrix R and considering that:

|ψm(r)〉 = R|φg(r)〉 ; |φg(r)〉 = R−1|ψm(r)〉

|sm(r)〉 = RD−1|Sg(r)〉 ; |Sg(r)〉 = DR−1|sm(r)〉

∣∣∣~jm(r)⟩

= RD−1| ~Jg(r)〉

(10.3)

The G multigroup coupled equations 10.1 are reduced to another G uncoupled modalequations which analytical solution is aordable 10.4.

∇2ψm(r)− λmψm(r) = −sm(r) ; m = 1, G (10.4)

According to the formulation of the ACMFD method, a relationship between the ux atthe interface in terms of the average ux at the adjacent node and of the current at thesame interface is established. This relation comes from the analytic solution of equation10.4, which in Cartesian 1D is expression 10.5, and where pm(x) is a particular solution ofthe heterogeneous equation with a transverse leakage source.

ψm(x) = Ame+αmx +Bme

−αmx + pm(x) ;d2

dx2pm(x)− λmpm(x) = −sm(x) (10.5)

where αm =√λm.

Determining the constants Am and Bm from the values of ux and current on interfaces:

ψm(∓H2

)− pm(∓H2

) = Cfm

[ψm − pm

]± Cj

m

H

2

[Jm(∓H

2) + p

′

m(∓H2

)

](10.6)

Where Cf and Cj are constants for every mode and depend only on the eigenvalue, the crosssections, and the nodal width over the considered direction.

Cfm =

2αmH

e+αmH − e−αmH; Cj

m =e+αmH + e−αmH − 2

e+αmH − e−αmH· 2αmH

(10.7)

171

10.2. COBAYA3

Finally, transforming from the modal uxes to the physical group uxes, using 10.3, weobtain:

|φg(∓H

2)〉 = Af |φg〉 ±

H

2AjD−1

g |Jg(∓H

2)〉 −R−1|Tm(∓H

2)〉 (10.8)

Af = R−1CfR ; Aj = R−1CjR (10.9)

The spectral technique that relies on the use of similarity transformation to diagonalize themultigroup diusion equation was rst proposed by D.L. Vogel and Z.J. Weiss [138].

The second step consists in performing transverse integration, to reduce the n-dimensionaldiusion equation into n one-dimensional equations, coupled through the transverse leakageincluded as an external source term.

While the rst step is independent of the geometry and only related to the material proper-ties of each node represented by its cross sections; the second step, i.e. the transverse leak-age integration procedure, required additional developments when extended to triangular-Zgeometry [91].

For hexagonal geometry, ANDES chose to perform a transverse integration over a trianglesince it has been broadly demonstrated its satisfactory performance [91]. Triangular nodesinstead of hexagonal ones were preferred in order to avoid the singularities that appearwhen applying transverse integration to hexagonal nodes, allowing moreover the advantageof the mesh subdivision capabilities implicit within that geometry.

In order to obtain the ACMFD coupling equation at each nodal interface of the triangularright prisms, the following steps are carried out:

Step 1. Transverse integration procedure of the 3D diusion equation, yielding to oneequation per direction, coupled to other directions through the transverse leakage, with thetransverse-integrated ux as unknown.

Step 2. Analytic solution of the one-dimensional equation and calculation of the modalaverage ux, interface uxes and interface currents. A relation among them is found,which involves the uxes at corners and the values of the particular solution.

Step 3. Calculation of uxes at corners.

Step 4. Integration of the corners uxes in the ACMFD relation

10.2.2 Step 1: Transverse leakage integration of the 3D diu-sion equation in triangular prismatic nodes

Transverse integration is employed in most of the nodal methods to convert the 3D diusionequation into 3 separate 1D equations. In Cartesian geometry it is quite straightforwardprocedure. However, it leads to non-regular terms for the transverse leakage in hexagonal-z geometry. The reason for that is the need to perform the integration over a variable

172

10.2. COBAYA3

Figure 10.1: Triangular-z node, ACMFD

transverse surface. In ANDES, the hexagon is divided into six triangular nodes, whichallows us to rene as much as wanted. Fig. 10.1 highlights the variable trasnsverse areaon the x-direction.

In general, the transverse integration method could be applied to geometrical domains withthe following characteristics: the area perpendicular to the x- direction is a rectangle denedby its two sides in directions y and z, respectively (See Fig. 10.1). The length of the sideon the z-direction should be independent of coordinate x, and the length of the side on they-direction depends on x according to the functionsf(x)and g(x):

A(x) = (x, y, z) | g(x) 6 y 6 f(x), 0 6 z 6 Hz (10.10)

Let us start from the 3D neutron diusion modal equation within our homogeneous triangular-prismatic nodal volume V shown in 10.1:

−∇ ·~jm(~r)− λmψm(~r) = −sm(~r) ; ~jm(~r) = −∇ψm(~r) ; m = 1, 2, ..., G (10.11)

where λmis the modal eigenvalue, ψm(r) the modal ux, sm(r) the modal external sourceand G the number of groups.

The transverse integration over the x direction is obtain by integrating 10.11over a slice ofwidth dx perpendicular to the x axis.

´A(x)

jxm.dydz −´A(x+dx)

jxm.dydz −´P (x)

jnm.dydz − λm´Vψm(r) · dV =

= −´Vsm(r) · dV

(10.12)

173

10.2. COBAYA3

[dÁ(x)

∂2ψm∂x2 ·dydzdx

− 2√3

´ HZ0

(jn+m − jn−m ) dz −

´ ax−ax (jz+m − jz−m ) dy

−λmÁ(x)

ψm(r) · dydz]dx =

[−Á(x)

sm(r) · dydz]dx

(10.13)

where jxm(r) = ∂ψm(r)∂x

and jnm(r) is the component of the modal current perpendicular tothe peripheral area P(x).

[dÁ(x)

∂2ψm∂x2 ·dydzdx

− 2√3

´ HZ0

(jn+m − jn−m ) dz −

´ ax−ax (jz+m − jz−m ) dy

−λmÁ(x)

ψm(r) · dydz]dx =

[−Á(x)

sm(r) · dydz]dx

(10.14)

where jn+m = jnm

(x, x√

3, z), jn−m = jnm

(x,− x√

3, z), jz+m = jzm (x, y,Hz) and jz−m =

jzm (x, y, 0).

Applying the Leibniz'rule to the rst term of 10.14 we nd that

Á(x)

∂ψm∂x.dydz =

d(Á(x) ψm.dydz)

dx

− 1√3

(´ Hz0

(ψm

(x, x√

3, z)

+ ψm

(x,− x√

3, z))

.dz)

Substituting this expression into 10.14, we nally obtain:

d2ψ1Dm (x)

dx2− λmψ1D

m (x) = −s1Dm (x) + lm(x) (10.15)

ψ1Dm (x) =

´ Hz0

[´ ax−ax ψm(x, y, z) · dy

]dz; a = 1√

3

s1Dm (x) =

´ Hz0

[´ ax−ax sm(x, y, z) · dy

]dz

(10.16)

lm(x)is a term equivalent to the transverse leakage term in Cartesian nodes, although witha higher complexity.

lm(x) = −23

´ Hz0

(jt+m (x) + jt−m (x)) .dz + 2√3

´ Hz0

(jn+m (x) + jn−m (x))

+´ ax−ax (jz+m + jz−m ) · dy

(10.17)

where jt+m and jt−m are the modal ux derivatives along the tangential direction to the radialinterfaces.

As occurs in Cartesian geometry nodes, the transverse-integrated ux is the unknown ofthe resulting 1D diusion equation obtained after integration of the original 3D equation

174

10.2. COBAYA3

Figure 10.2: Components of the modal current in the XY plane

along a radial direction. It is important to remark that radial transverse leakage does not tthe net leakage through the side faces of the prism and a tangential component of currentarises as a consequence of the non-constant transverse area. However, the name is keptsince it plays the same role as in Cartesian geometry.

10.2.3 Step 2: Analytic solution of the transverse-integratedone-dimensional equation at radial interfaces in 2D tri-angular nodes

For simplicity, we will consider rst the 2D geometry case, that is, the triangular nodeshown in Fig. 10.2.

The transverse integrated ux being solution of 10.15 can be expressed analytically as acombination of a homogeneous solution and a particular one pm(x).

ψ1Dm (x) = Ame+αmx +Bme−αmx + pm(x)

d2

dx2pm(x)− λmpm(x) = lm(x)

αm =√λm

(10.18)

Particularizing in the interface at the right side of the triangle, a relation between theaverage ux at the interface and the constants Amand Bm is obtained.

175

10.2. COBAYA3

Computing interface average modal ux:

First, let us calculate the interface average modal ux at a radial interface (being L thelength of the triangle side) Ψm :

Ψm =Ψ1Dm

(x =

√3

2L)

L

Particularizing the 1D integrated ux in x =√

32L , we realize that average ux at this

interface can be related to constants Am and Bm as follows:

Lψm(

√3

2L) = Ame

γ +Bme−γ + pm(

√3

2L); γ =

√3

2Lαm (10.19)

Computing interface average modal current:

Doing the same for the rst derivative of expression 10.18, we aim to obtain a relationbetween the interface average current and the constants Am and Bm.

dψ1Dm

dx= 1√

3(ψ1

m + ψ2m)− L · jm(

√3

2L)

dψ1Dm

dx= αm (Ame

γ −Bme−γ) + p′m(

√3

2L)

⇒

⇒ 1√3

(ψ1m + ψ2

m)− L · jm(√

32L) = αm (Ame

γ −Bme−γ) + p′m(

√3

2L)

(10.20)

From 10.20 and10.19, the constants Amand Bm are obtained:

Am = L2eγ

¯ψm + 12√

3αmeγ(ψ1

m + ψ2m)− L

2αmeγjm −

pm(√

32L)

2eγ−

p′m

(√3

2L)

2αmeγ

Bm = L2e−γ

¯ψm − 12√

3αme−γ(ψ1

m + ψ2m) + L

2αme−γjm −

pm(√

32L)

2e−γ+

p′m

(√3

2L)

2αme−γ

(10.21)

Computing nodal average ux:

Finally, substituting Amand Bm in the equation of the node-average modal ux ¯ψm (10.22),the ACMFD modal relation at every radial interface in triangular mesh is obtained (10.23).

¯ψm =1

√3

4L2

ˆ √3

2L

0

ψ1Dm (x) · dx (10.22)

176

10.2. COBAYA3

(ψm −

pm(√

32L)

L

)=

= Cfm2

(¯ψm − 2pm

L

)+ Cjm

4(ψ1

m + ψ2m)−

√3

4LCj

(jm +

p′m(√

32L)

L

) (10.23)

where L is the triangle side length and Cfmand C

jm are the analytical coecients dened as:

Cfm =

2γ

eγ − e−γ; Cj

m =eγ + e−γ − 2

eγ − e−γ· 2

γ(10.24)

Let us compare expression 10.23with the equivalent relation in Cartesian geometry 10.25,for the right side in a square node of width H:

ψm − pm(H) = Cfm

[¯ψm − pm

]− Cj

m

H

2

[jm + p

′

m(H)]

(10.25)

First, it is remarkable the dierence in the terms containing the particular solution. InCartesian geometry they are not divided by the triangle side as they come from a transverse-integrated equation already divided by the constant transverse area, so that the transverseleakage and the particular solution implicitly are also divided by H (transverse area in the2D case).

It is important to remark that the main drawback in the triangular case is the presence ofthe modal uxes at the two corners adjacent to the interface, which have to be carefullycomputed.

10.2.4 Step 3: Calculating uxes at corners in 2D trangularnodes

In order to give consistency to the method, the modal ux at the corners is derived ana-lytically. The procedure employed is detailed as follows. Expression 10.18 is particularizedat the coordinate (x=0), which corresponds to the triangle vertex opposed to the analyzedinterface.

ψ1Dm (0) = 0 = Am +Bm + pm(0) (10.26)

dψ1Dm

dx(0) =

2√3ψVm = αm (Am −Bm) + p′m(0) (10.27)

√3

4L2 ¯ψm =

Amαm

(eγ − 1)− Bm

αm

(e−γ − 1

)+

√3

2L · pm (10.28)

177

10.2. COBAYA3

If we focus on the values of the function (10.26) and its rst derivative (10.27) at thiscoordinate and substitute them in the expression for the nodal average ux (10.28), thenthe following relation between the ux at vertex and the nodal average ux results:

(ψVm −

√3·p′m(0)

2

)= Df

m

(¯ψm − 2·pm

L

)+Dj

mpm(0)L

Dfm = γ2

eγ+e−γ−2; Dj

m =γ·(eγ−e−γ)eγ+e−γ−2

(10.29)

with L the triangle side length.

10.2.5 Step 4: Integration of the corners uxes in the ACMFDrelation

Until now, it has been demonstrated that, as occurs in Cartesian geometry, it also existsan ACMFD relation for triangular nodes (Eq.10.23). Nevertheless, a new term where theuxes particularized at the adjacent vertexes appears.

The most straightforward option to treat equation10.23 would be to take the term ofuxes at vertex from the previous iteration, including them in the non-linear iteration loop.However, this option has been reported to be unfeasible as the iterative process becomesunstable as a consequence of the excessive inuence of the transverse leakage term. Otherpossibility would be to set the vertex uxes as unknowns in the linear system, in additionto the nodal average uxes. It could be done by introducing the corner-point balanceequations. Nevertheless, in order to preserve the same eciency of the ACMFD method inCartesian geometry, where the linear system includes the nodal average uxes as the uniqueunknowns, the new relation 10.29 is applied to eliminate the corner-point uxes from thenal linear system.

Consequently, substituting the uxes at vertex in 10.23 with its analytical denition (10.29),andgrouping together the transverse leakage terms, the following expression is obtained:

ψm = Cfm

¯ψm −√

3

4LCj

mjm − T ∗m (10.30)

where T ∗m includes all terms related to the particular solution driven by the transverseleakage, i.e., including the component from the integrating direction x and from the termscoming from the relation at the vertexes (10.31).

178

10.2. COBAYA3

T ∗m = Cfm

2·p∗mL− 2·p∗m(

√3

2L)

L+√

32Cjmp′∗m(√

32L)

p∗m = 12pSm + 1

4pV 1m + 1

4pV 2m

p∗m(√

32L) = 1

2pSm(

√3

2L) + 1

4pV 1m (0) + 1

4pV 2m (0)

p′∗m(√

32L) = 1

2p′Sm(√

32L)− 1

4p′V 1m (0)− 1

4p′V 2m (0)

(10.31)

This way a very similar expression as in Cartesian geometry is obtained although the trans-verse leakage prole has a more complex physical meaning than in the previous one.

However, this approach (in which nodal average uxes are the unique unknowns) did not leadto the level of accuracy achieved with Cartesian geometry. The reason for this assertion isthat the accuracy of the analytical nodal method relies on a good estimation of the averagevalue of the transverse leakage. In Cartesian mesh this average value only depends on theinterface average currents whose errors vary from 0.1% to 1% in most of nodal solutions.Nevertheless, attending to the transverse leakage expression in triangular mesh (10.17), itis noticed that the transverse leakage not only gathers the perpendicular component ofcurrent to the surface, but also the tangential one. This makes compulsory to compute theux at the three vertexes of every node.

lm =2√3

(j+m + j−m

)+

2

3·(ψV 1m − ψV 3

m

)+(ψV 2m − ψV 3

m

)L

(10.32)

Basically the uxes at corners are computed using the corner point balance equations, in-volving the intranodal ux distributions of the six triangular nodes adjacent to the consideredvertex. Following this motivation, the analytical expression (10.29) is used for each one ofthe six nodes, obtaining a relation between ux at a given corner and the six nodal averageuxes. Thus the corner uxes are iteratively updated in transverse leakage loop, once thenodal average uxes and the interface average uxes and currents have been computed.

For both analytical relations at interfaces (10.23) and vertex (10.29) a polynomial prole hasbeen assumed for the transverse leakage lm(x). Particularly, a cubic t has been adopted,

which has four coecients that are determined guessing the values of lm, lm(0), lm

(√3

2L)

and one of the values of l′m(0), l′m

(√3

2L). To compute those values a parabolic interpola-

tion of the average currents at radial interfaces has been adopted.

10.2.6 Extension to 3D (triangular-Z geometry)

In triangular-Z geometry, the main diculties arise from the fact that the transverse areachanges in the radial direction. Once the formulation for 2D triangles is obtained, its exten-sion to the 3D case is not dicult. However, two additional aspects should be overcome.

179

10.2. COBAYA3

First, an ACMFD relation for axial top and bottom interfaces has to be derived. Second, thetransverse leakage has to be redened to include the axial component as well. It means thatthe neutron leakage through the top and bottom interfaces has to be taken into accountin the particular solution of 10.23 and the transverse leakage is modied accordingly.

10.2.6.1 The ACMFD relation for axial interfaces

In axial direction the transverse area is constant, so the procedure to obtain the transverseleakage term is the same as in Cartesian geometry. Starting from the 3D modal equations,a 1D modal equation over the 1D average ux is obtained:

d2ψ1Dm (z)dz2 − λmψ1D

m (z) = −s1Dm (z) + lm(z)

lm(z) = 4√3L·

3∑r=1

jrm(z)(10.33)

where jrm(z) is the averaged outgoing current through radial interfaces at height z.

Independently of the way the transverse leakage lm(z) is obtained, 10.33 leads to an ACMFDrelation similar to the Cartesian one.

∣∣φSg ⟩ = Afz

∣∣φg⟩− HZ2Aj

z

∣∣JSg ⟩−R−1∣∣T Sm⟩

∣∣φIg⟩ = Afz

∣∣φg⟩+ HZ2Aj

z

∣∣J Ig ⟩−R−1∣∣T Im⟩

(10.34)

10.2.6.2 Modications of the ACMFD relation for radial interfaces in 3D nodes

In 3D nodes, 10.17 can be written in the following way:

lm(x) = lRm(x) +

ˆ ax

−ax(jm,z(x, y,HZ)− jm,z(x, y, 0)) · dy = lRm(x) + lZm(x) (10.35)

where lR and lZrefer to leakage through radial and axial interfaces, respectively.

The transverse leakage prole in z-direction leads to an additional particular solution pZ .Then 10.18 is modied:

ψ1Dm (x) = Ame+αmx+Bme−αmx+pRm(x)+pZm(x) ;

d2

dx2pZm(x)−λmpZm(x) = lZm(x) (10.36)

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10.2. COBAYA3

ψm =ψ1Dm

(x=√

32L)

L·HZ

dψ1Dm

dx= HZ ·

(1√3

(ψ1m + ψ2

m)− L · jm(√

32L))

¯ψm = 1√3

4HZ ·L2

´ √32L

0ψ1Dm (x) · dx

(10.37)

With these denitions, the new ACMFD relations would be:

(ψm −

pm(√

32L)

L ·HZ

)=Cfm

2

(¯ψm −

2pmL ·HZ

)+Cjm

4

(ψ1m + ψ2

m

)−√

3

4LCj

m

(jm +

p′m(√

32L)

L ·HZ

)(10.38)

And the analytical corner relation:

(ψVm −

√3 · p′m(0)

2HZ

)= Df

m

(¯ψm −

2 · pmL ·HZ

)+Dj

m

pm(0)

L ·HZ

(10.39)

With the transverse leakage term:

T S∗m = T S∗m,2D + Cfm

2p∗ZmL·HZ

− 2·p∗Zm (√

32L)

L·HZ+√

32Cjmp′∗Zm (

√3

2L)

HZ

p∗Zm = 12pSZm + 1

4pV 1Zm + 1

4pV 2Zm

p∗Zm (√

32L) = 1

2pSZm (

√3

2L) + 1

4pV 1Zm (0) + 1

4pV 2Zm (0)

p′∗Zm (

√3

2L) = 1

2p′SZm (

√3

2L)− 1

4p′V 1Zm (0)− 1

4p′V 2Zm (0)

(10.40)

Finally, it is worth it to remark that if a polynomial t is desired for lZm(x), it should betaken into account that the integration length in 10.35 varies with x.

10.2.7 The ACMFD method for kinetics problems

In order to assess transient conditions, the 3D kinetic equations were implemented in AN-DES source. In that case, the time-dependent multigroup diusion equation (10.41) need tobe solved together with the balance equations of the six/eight neutron precursoss (10.42).Therefore, the neutron-kinetic equations can be solved by utilizing the same methodologyalready implemented for the steady-state xed source problem (FSP) and assuming that

181

10.2. COBAYA3

the temporal dependence can be treated in such a way that it yields a FSP. Moreover, giventhat the geometry has no inuence in this methodology, it might be applied to triangular-Znodes with very few modications with respect to Cartesian nodes [89].

Using the matrizial notation, the neutron-kinetic equations can be written as:

∣∣∣∣ 1

Dgvg

∂φg∂t

⟩= ∇2 |φg〉 −A |φg〉 − Fd |φg〉+

Gd∑k=1

λkCk

∣∣∣∣χgdkDg

⟩(10.41)

∂Ck∂t

= fkβ

G∑g=1

νΣfg

keffφg − λkCk k = 1, . . . , Gd (10.42)

with A being the steady state multigroup diusion matrix; χand χd refering to the promptand delayed neutron spectrum, respectively; and being the ig-jg element of the Fd matrix

dened as Fd(ig, jg) = βχig

Dig

νΣigfkeff

.

Then, in order to treat the time dependence, rst the time domain needs to be discretizedinto discrete time steps. Using as a rst approach a nite dierence implicit scheme forthe time derivatives, and focusing on the neutron precursor balance equation, a relationbetween the prompt neutron ux per energy group and every precursor concentration canbe derived:

Ck,∆t(r) =

Ck,0(r) + ∆t · fk · βG∑g=1

νΣfgkeff

φg(r)

1 + λk∆t(10.43)

Now, for the time derivative of the neutron-kinetics equation, the method of exponentialextrapolation is employed (10.44).

φg(∆t) = φg,0 · eωg∆t (10.44)

As for the time derivative of form function, an implicit linear forward dierence scheme isemployed. However, this approximation can fail for those nodes that are close to movingcontrol rods. As a result, the neutron ux time derivative can be expressed as:

[∂φg∂t

]t=∆t

=[ωg + 1

∆t

]φg(∆t, r)− eωg∆t

∆tφg,0 (10.45)

Then, using the nodal average ux, the frecuency ωg can be recursively computed for eachenergy group and node.

ωg =1

∆tln

(φg (∆t)

φg,0

)(10.46)

182

10.3. Assessment of ACMFD method to be applied to Fast Reactors

This iterative process will be stopped as soon as the ssion souce distribution is converged.

In those transient calculations where the ux evolution follows the fundamental mode,the eciency of the solver is noticeably improved by using the exponential extrapolationmethod. This conditions are achieved in the phases of the transients where there is norelevant change in neutronic properties, mainly due to thermal hydraulic conditions. As aconsequence, the transient time steps might be increased when this method is employed inthese phases.

On the other hand, althought it is known that the delayed neutron spectrum is softer thanthe prompt one, it will be assumed henceforward the same neutron spectrum for prompt(χ) and delayed neutrons (χd).

Using the described approximation, the diusion equation would result in:

∇2 |φg〉 −A |φg〉 −Aadd |φg〉 = −D−1g |Skin,g〉 (10.47)

with

Skin,g =Gd∑k=1

χgdkλkCk,01 + λk∆t

+eωg∆t

vg∆tφg,0 + Sext,g (10.48)

Aadd(ig, jg) =1

Dig

[(Gd∑k=1

∆t · fk1 + λk∆t

)χigp β

νΣjgf

keff+ δig,jg

(ωigvig

+1

vig∆t

)](10.49)

After looking at 10.47 it can be observed that it is a FSP with additional terms with respectto the steady state FSP. The multigropu matrix will include A + Aadd, and the externalsource will have also an additional term which is determined by the ux spatial distributionand the precursor cncentrations in the previous step (φg,0, Ck,ic0).

Finally, the terms to be included in the multigroup matrix (A + Aadd) and the spatialdistribution of the external source should be carefully chosen for the obtention of an accurateACMFD relation (10.50)

|φg(∓H

2)〉 = Af |φg〉 ±

H

2AjD−1

g |Jg(∓H

2)〉 −R−1

∣∣T∓m⟩−R−1∣∣T∓m,kin⟩ (10.50)

10.3 Assessment of ACMFD method to be applied

to Fast Reactors

In principle, the main equations that govern the neutron transport in a nuclear reactor coreare the same for thermal and fast reactors, so the same codes could be run in principle for

183


both of them. However, as previously discussed, some dierences were identied that couldimpact on the convergence of the solution.

The rst dierence regards the cross sections. It has been mentioned that these parametersare strongly dependent on the incident neutron energy, and the behaviour on the fast range isquite distinct. As a consequence, those parameters need to be adequately generated takinginto account the particularities of the fast spectrum. This topic is thoroughly covered inChapter 9. Moreover, fast reactor calculations require a ner energy discretization than theusually employed in thermal reactors. This fact obliges our nodal diusion solver ANDES towork with more energy groups. The reference group structure considered for fast reactorsincludes 33 energy groups. ANDES was already extended to multigroups and satisfactorilyvalidated for thermal reactors applications [88]. However, the maximum group structureemployed included 8 energy groups. As mentioned before, the ACMFD method entails astep where the diagonalization of the multigroup diusion matrices is performed. Now, ithas been proven that this step entails some convergence problems as it will be discussedlater on.

Another important dierence of fast reactors is related to the geometry. As previouslymentioned, fast reactors are usually designed in hexagonal geometry. However, that is notan unique characteristic of fast reactors, but some thermal reactors such as VVER cores arealso designed in hexagonal geometry. In fact, in order to model those reactors, the ANDESsolver was extended to triangular-Z nodes [91]. Nevertheless, this challenging geometrycould entail some diculties for deterministic methods, which would be more noticeabledepending on the employed methodology. In the case of the ACMFD method, the inuenceof geometry is present in the transverse integration step. As a consequence of the variabletransverse area present in the triangular nodes, some singularities appear in the formulation,which could be a source of instability, as it will be discussed in the last part of this chapter.

Some other geometry-related issues in fast reactors account for the presence of some specicregions or nodes without ssile materials. In particular, as a result of the longer free mainpath, the control rods are not inserted in between the fuel rods within a fuel assembly, asit happens in thermal reactors, but they are all arranged together in the control assembliesinstead, and distributed along the whole core in substitution of some fuel assemblies (forfurther explanation, visit Chapter 5) . A simple way to understand this eect could beto compare the rod scale in thermal reactors with the assembly scale in fast reactors. Infact, this design is a direct consequence of the longer neutron mean free path. The usualassembly pitch in fast reactor is of the same order as the free main path, while in thermalreactors, the pin diameter is of the order of the mean free path. Continuing with thenon-ssile regions, fast reactor designs include several axial layers with dierent purposesthat provide a prolongation of non-ssile region above and below the core. These regionsimpact signicantly in the cross sections, and induces additional convergence problems, asit will be further described in Section 10.3.1.

Paying attention now to the applicability of the diusion to Fast Reactors, there will besome consequences which should be analyzed. The errors committed for applying thespatial discretization are small in principle, due to the potential renement of the triangular-Z mesh. In addition, the errors derived from the spatial homogenization would be smallas well, taking into account the longer free path length of the neutrons. Moreover, the

184


Material Diusion matrix A Eigenvector matrix R

Rodfollower(92%Na) 2220.85 6.55E+17

Lowergas plenum(52%He) 639.4 9.45E+13

Uppergas plenum(52%He) 640.04 9.16E+13

Loweraxial blanket(steel+Na) 565.38 3.03E+09

Upperaxial blanket(steel+Na) 624.66 1.17E+09

Reector (steel+Na) 596.24 2.37E+09

Table 10.1: Conditioning numbers of A and R

assembly discontinuity factors for an isolated fuel assembly of fast reactor are close to 1.

At the end, the ANDES code should be evaluated in comparison with other codes, such asERANOS or FAST system, for steady and transient analysis.

10.3.1 Group structure

Fast reactors requires a ner energy discretization compared to thermal reactors, as a resultof the need of an adequate treatment of the resonance range, as discussed in Chapter 2.

From the very beginning, when the cross sections generated with ERANOS in 33 energygroups (see Chapter 9) were introduced in ANDES, the nodal solver of COBAYA, con-vergence could not be achieved. Therefore, an intensive study was carried out in orderto identify the source of those troubles. The results of this study were presented in theInternational Conference on Mathematics and Computational Methods Applied to NuclearScience and Engineering (M&C 2011), held in Rio de Janeiro (Brazil) in May 2011 [107].

During the study, it was realized that 2D fuel-assemblies with reective boundary conditionsdid not present any convergence trouble. However, when the same fuel-assembly wasmodeled in 3-D, adding the axial layers, the convergence could not be achieved.

Further studies revealed that the convergence problems arise only when the non-ssile ma-terials (axial blankets, axial gas plenums, rod follower, radial reectors, etc) are included inthe calculation. After several tests, it was concluded that the step where the diagonalizationof the multigroup matrixes is performed was one of the main sources of troubles, as it willbe described in the following sections.

After examining the relations in 10.7, it is clear that the ner the mesh is, the closer to 1the Cf

m and Cjmcoecients have to be, since Cf

m → 1 and Cjm → 1, when H → 0.

When applied to a sodium fast reactor mode, both coecients were veried to tend to1 as the mesh gets ner in ANDES. However, when we analyzed the matrices Af and Aj,corresponding to a node of reector material (axial, blanket, gas plenum) it happened thatthey do not tend to the identity as it should have been from 10.9, taking into account that

185


Figure 10.3: Total and elastic scattering cross section of Na-23 and He-4

186


Material R 33 groups R 15 groups

Rod follower(92%Na) 6.55E+17 1.30E+08

Lower gas plenum(52%He) 9.45E+13 1.98E+07

Upper gas plenum(52%He) 9.16E+13 2.92E+07

Lower axial blanket(steel+Na) 3.03E+09 4.71E+03

Upper axial blanket(steel+Na) 1.17E+09 2.41E+04

Reector (steel+Na) 2.37E+09 3.16E+03

Inner fuel 2238.57 26.51

Table 10.2: Conditioning numbers of R in 33 and 15 groups

both Cf and Cj are almost 1. This fact gave us a clue about the inconsistency of both theeigenvector matrix R and its inverse and led us to a more accurate study of the multi-groupdiusion matrix and the procedure to calculate their eigenvalues and eigenvectors.

As it is usual in fast reactor calculations not to consider up-scattering, the matrix A in 11.12corresponding to a non-ssile material is lower triangular, which means that the eigenvaluesmight be easily extracted from the main diagonal; it can be seen that some of them arevery close one to another. This fact deteriorates the condition number of the matrix A,so that their associated eigenvectors are not numerically well determined, and the inversecalculation solver does not work properly. In order to check this, the conditioning numbersof the matrices involved were computed as:

C(A) = ‖A‖2

∥∥A−1∥∥

2=σmaxσmin

(10.51)

Table10.1 shows the conditioning numbers computed for the multigroup matrix and itsassociated eigenvector matrix R for the non-ssile materials. It seems clear that due to thehigh condition number of the eigenvector matrixes the inverse cannot be calculated andthus the diagonalization could not be performed.

After having a look at the cross sections of the main isotopes presented in the troublingmaterials (i.e. Na-23 and He-4), it can be observed that they present very at regions (Fig.10.3), which could justify that the more energy groups selected the easier is that the crosssections of two of the groups are almost identical, which in this case, it will impact directlyon the eigenvalues.

Furthermore, the high condition number of matrices this size makes any numerical errorpropagate erratically through the code, giving rise to convergence problems. When a few-groups structure is employed, the condition number of the matrices has less impact andthus the convergence of the code might be easily achieved, as shown when a 15 groupstructure was employed (Table 10.2).

Therefore, since fast reactor calculations typically involve at least 33 energy groups, themultigroup diusion matrices generated do not present optimal numerical properties re-

187


garding their conditioning. This fact which has proved to be relevant in the non-ssilematerials, might be the source of convergence failures.

10.3.2 Transverse leakage

In order to isolate the dierent aspects identied as potential source of convergence troubles,new cases in Cartesian geometry were run using a group structure collapsed into two groups.The consequences were that even when the conditioning of the matrixes involved was notan issue, some cases still failed to convergence using the ACMFD method. After a deepexamination, it resulted that the transverse leakage loop performed in the ACMFD methodwas extremely unstable.

Particular challenges of Hexagonal geometry

The method implemented in ANDES to treat the transverse leakage in hexagonal geometryis extremely complex, as described before. In general, applying the transverse integrationover triangular nodes induces some singularities, that arise from the fact that the transversearea changes in the radial direction. The aim followed in ANDES was to keep the sameformulation used for Cartesian geometry, while preserving the eciency showed by theACMFD method. It means that the unique unknowns of the linear system are the nodalaverage uxes. However, it was demonstrated that as a consequence of the transverseintegration over the variable surface of a triangle, new terms appeared in the formulation.These new terms are the ux on the vertexes (2D) or corners (3D). The method followed inANDES implies the calculation of those uxes analytically by means of an equation derivedfrom the ACMFD formulation ( 10.29). This aims to preserve the coherency with the restof the formulation. However, those uxes at the corners cannot be explicitly introducedin the ACMFD relation, as it would be the straightforward method, due to convergenceproblems. In addition, as just mentioned, it was decided not to treat them as unknowns inthe linear system in order to keep the same structure as in Cartesian geometry. So the naldecision taken by J.A. Lozano in [88]was to substitute their analytical denition (10.29)in the ACMFD relation (10.23), gathering all the transverse leakage terms together. As aconsequence, the resulting transverse leakage term T ∗m is much more complicated than theone for the Cartesian case (10.31). Actually, the nonlinearity of this term is likely to be asource of convergence troubles when working with more complex cores.

Having a deep look at the nonlinearity of the problem, the methodology implemented inANDES implies the treatment of the uxes at vertexes in two dierent ways. On one hand,they are substituted in the ACMFD relation using the analytic relation 10.39. On the otherhand, they need to be calculated explicitly in order to calculate the transverse leakage term.

Nevertheless, this methodology works ne with thermal reactors, such as the VVER corefrom the V1000 Benchmark [29], as proven in [91, 88]. Then, why is not working with fastreactors? It should be understood that there is not an unique cause of convergence failure.And the transverse leakage complexity is nothing but another piece of the puzzle that leadsto the global system failure. Its combination with the large axial conguration composed of

188


non-ssile nodes together with the radial non-ssile nodes (control assemblies) embeddedin the core contribute to the convergence diculties.

In order to solve those challenges, a new methodology is proposed in the next Chapter.

189


190

Chapter 11

Developments in COBAYA3

11.1 Introduction

Once the main challenges of the application of the nodal solver ANDES from the COBAYA3package to sodium fast reactors were presented, the next step should be the proposal andimplementation of solutions that would overcome the discussed limitations. Indeed, thischapter will deal with the dierent developments implemented in ANDES.

As discussed in Chapter 10, there are two main limitations in ANDES when sodium fastreactors are to be studied. First, when a broad energy group structure is to be used,the ACMFD method fail to converge as a consequence of diagonalization step performedover some ill-conditioned matrixes, which correspond to some specic materials. The rstobvious solution to this problem would be to reduce the number of groups until thoseproblems disappear. However, accuracy of fast reactor calculations is strongly dependenton the correct choice of the energy group structure, as discussed in Chapter 2. Moreover,it has been broadly veried through long experience with the ERANOS code in France,that an optimal group structure is the one presented in the ERANOS package in 33 energygroups. Therefore, in order to keep working with that energy structure, alternative methodsthat substitute the diagonalization step should be assessed. The method that was assessedand implemented for this matter is the Schur decomposition, which is described in Section11.3.2.

The second main limitation is related to the performance of the transverse leakage inte-gration over hexagonal geometry. The ACMFD method entails an internal loop in aims ofupdating the transverse leakage prole after each ux solution. This approach was analyzedas it was candidate to be a potential source of convergence troubles. Indeed, due to thehigh non-linearity of this process, when some piece of additional complexity is added tothe calculation, the global convergence is dicult to achieve. Even for Cartesian geometry,some convergence problems were reported, although in case of hexagonal geometry thecomplexity is higher. Since the transverse leakage approach is broadly employed by manynodal diusion codes, some similar codes were analyzed in order to nd a solution to thisproblem. At the end, the Coarse Mesh Finite Dierence method with a nodal correction,similar to what is implemented in PARCS code was proposed. Further details regarding the

191

11.2. Corrected Coarse Mesh Finite Dierence method

description and implementation of this alternative method is discussed in Section 11.2.

In addition to the developments performed with the purpose of solving the limitationsdiscussed, further developments were accomplished. In particular, one of the main achieve-ments of the thesis is the successful implementation of the coupling between ANDES andSUBCHANFLOW. Indeed, in order to be able to perform safety studies, our neutron dif-fusion code needed to be coupled with an appropriate thermal-hydraulic code. Given therelevance of these implementations, the description, motivation and implementation of thecoupling will be presented in the next Chapter 12. Finally, linked to the development of thecoupling, new developments were required in order to allow the dierent thermal-hydraulicand geometrical feedbacks linked to SFR on the cross sections during coupled calculations.This last part will also will described in Chapter 12.

11.2 Corrected Coarse Mesh Finite Dierence method

As discussed before, in order to reduce the convergence problems linked to the transverseleakage loop, alternative methods were investigated. In particular, similar diusion codessuch as PARCS and NESTLE were assessed in order to understand the methodology imple-mented in those. Both codes coincide in not solving the whole core with advanced nodalmethods but solving it in two steps. First, the coarse mesh nite dierence (CMFD) prob-lem is dened for the whole core and then a correction coecient is calculated for eachtwo-node local problem. This process is performed iteratively until the whole convergence isachieved. The CMFD problem involves a linear system with a block of septa-diagonal ma-trix (in case of three-dimensional problems). The two-node problems are solved to correctfor the discretization error in the nodal interface current resulting from the nite dierenceapproximation in a coarse mesh structure. They can be solved using any of a number ofso-called advanced nodal diusion methods. In PARCS the nodal expansion method (NEM)and the analytical nodal method (ANM) can be employed, while NESTLE uses only theNEM method.

Therefore, the proposal herewith presented is to remove the transverse leakage loop andimplement the CMFD method in replacement of the ACMFD to be applied to the wholecore. In addition, correction coecients could be used to force the surface current of theCMFD to be the one calculated by the advance nodal method for each 2-node problem.

11.2.1 Theoretical base

The Coarse Mesh Finite Dierence Method consists in solving the nite dierence diusionequation in the coarse mesh. The balance equation for each node is coupled to that of theneighbouring nodes through the leakage terms. The nodal coupling is resolved by using thenonlinear nodal method in which the interface current between any two nodes is representedin terms of the node average uxes of the two facing nodes by the following relation:

Ji→i+1 = −Di→i+1(φi+1 − φi)− Di→i+1(φi+1 + φi) (11.1)

192


where i and i+1 are two consecutive nodes in the direction of interest. Here Di→i+1 isthe base nodal coupling coecient or Finite Nodal Coupling Coecient (FNCC), sinceit represents the rst order estimate of the nodal coupling based on the nite dierenceapproximation which is given as:

Di→i+1 =2DiDi+1

f−1i Dihi+1 + f−1

i+1Di+1hi(11.2)

On the other hand, Di→i+1 is the Corrective Nodal Coupling Coecient (CNCC), denedas:

Di→i+1 = −Ji→i+1ACMFD + Di→i+1(φi+1 − φi)

(φi+1 + φi)(11.3)

where J i→i+1ACMFD is the interface current computed by the ACMFD method for the two-node

problem.

1

V

6∑l=1

∣∣J lg⟩ · Sl + DA∣∣φg⟩ =

∣∣Sg⟩ (11.4)

Once we got the coupling coecients, the linear system can be established. The equation11.1 is inserted into 11.4, resulting in a expression that contains only node average uxesas unknowns.

M∣∣φ⟩ = |S〉 (11.5)

The resulting matrix M is a block diagonal matrix, where each block is a GxG matrix,complete in case of the main diagonal, and diagonal in case of the outer block diagonals.There are a total of 1 + nf block diagonals, where nf is the number of interfaces thatsurround the node. This way, for 2D problems there are 5 blocks of diagonal matriceswhile for 3D there would be 7 blocks. Each diagonal block represents for each node, thedependency of the nodal ux with the own node properties, while the o diagonal elementsrepresents the spatial coupling with the neighboring nodes, which in this formulation wouldbe not-null just for the main diagonal blocks.

M =

[. . .] [. . .] [. . .] [. . .] [. . .][. . .] [. . .] [. . .] [. . .][. . .] [. . .] [. . .] [. . .] [. . .][. . .] [. . .] [. . .] [. . .][. . .] [. . .] [. . .] [. . .] [. . .][. . .] [. . .] [. . .] [. . .]

(11.6)

193


∣∣φ⟩ =

φ1

...

φG

1,1,1

, · · · ,

φ1

...

φG

k,j,i

, · · · ,

φ1

...

φG

NK,NJ,NI

(11.7)

The linear system just established will be then solved following a Krylov subspace method,BiCGSTAB, preconditioned with the ILU3D preconditioner, the same way as in the ACMFDmethod.

11.2.2 Implementation

For the implementation of the new methodology, the intention to be as less intrusive aspossible with the source code was followed. As mentioned before, the ACMFD methodpresents lots of parallelisms with the CMFD method. In both of them, a linear system isachieved with the nodal uxes as only unknowns, and the way to solve that linear systemis kept. However, the elements of the matrix change from one method to the other asindicated in the previous section. Furthermore, an optional nodal update was implementedin order to correct the CMFD current with the one coming from the advanced nodal method,ACMFD.

Figs. 11.1 and 11.2 show a detail of the control logic scheme followed by the ACMFDmethod and the CMFD method with two-node correction, respectively. The major changesare the removal of the iteration loop over the traverse leakage and the substitution ofsetblocks by twonodeanm and setupcmfd. In addition, the subroutine facecuris changed by a similar one that calculates the interface currents according to the newmethod, facecur2.

The CMFD method was implemented rst to Cartesian geometry and then extended tohexagonal geometry. In order to simplify the formulation, the description of the imple-mentation will be focused on the Cartesian geometry and Section 11.2.3 will provide someadditional remarks related to the particularities of hexagonal geometry.

Control Logics

The rst step of the implementation was to dene the control scheme where the newsubroutines would work. In this regard, rst the original control scheme followed by ANDESis showed in Fig.11.1, compared to the new one in Fig. 11.2. All the variables needed forthe new formulation were dened in appropriate modules, allocated when required andinitialized.

Within the outer loop over the eigenvalue k-e, rst the cross sections from each materialare assigned to the proper node in ANDES (multidif), and the ux is initialized for therst iteration (initflux). Then the transverse leakage loop starts. Within this loop (onlyin the original method), the subroutine coefandes (vertex and coefandes-TZ, incase of hexagonal-z geometry) is called, where all the parameters required for the ACMFD

194


formulation are calculated, i.e., transverse leakage terms, and Af and Aj matrixes. Rightafter that, the linear system is established. The original method would call the subroutinesetblocks for this matter, while the new one would call setupcmfd. In case of using thenodal update, to correct to CMFD current with the one from the advanced nodal method,the subroutine twonodeanm would be called. Once the linear system is established, theinner loop to solve it begins, through the Krylov subspace using the BiCGSTAB solver.When the convergence is achieved in the inner loop, the currents are calculated accordingto the ACMFD method (facecur) or to the new CMFD method (facecur2). In caseof the original scheme, the convergence on the transverse leakage would be checked now.Then the new eigenvalue is updated and the outer loop continues until the convergence isachieved.

The new subroutines are described in detail as follows:

• Setupcmfd is the subroutine in charge of computing the Finite Dierence CouplingCoecients, Di→i+1, using equation 11.2,and establishing the linear system.

• Twonodeanm is the optional subroutine where the nodal update is performed. Itprepares the parameters from coefandes to be used by facecur and calls it tocalculate the interface currents according to the ACMFD method, eq.11.9. All theinformation related to the traverse leakage integration is included in the ACMFDcurrent denition. Once the ACMFD currents are calculated, it computes for eachtwo nodes interface the Correction Coupling Coecient, Di→i+1, using equation 11.3.

• Facecur2 is the equivalent of facecur for the CMFD method. It calculates theinterface currents following the relation 11.1.

Moreover, with the aim of providing more exibility to the code, some logical controlvariables were introduced, enablig to use whether the ACMFD method (by default), or theCMFD with or without correction.

During the rst iteration over the k-e the CNCCs are zero. Until the second sweep thenodal update is not performed. Then, the user could dene the desired periodicity wherethe correction is to be made, .i.e., the number of iterations in between two consecutivecorrections.

11.2.3 Additional implementations for hexagonal geometry

Once the limitations of the transverse leakage formulation in hexagonal geometry are under-stood, it seems clear that the rst approach would be to simplify the formulation as muchas possible. It would reduce the high non-linearity of the processes involved. Following thismotivation, several approaches were adopted according to a try-and-check methodology.

First, a very simple denition of the ux at the vertex φVg was adopted, as the averagebetween the nodal average uxes of the six neighbours to the vertex (11.8). However,this approach did not result in the convergence of the method and a damping factor wasrequired to make it work. In addition, it provided results that diered relevantly to the

195


Figure 11.1: Detail of subroutines scheme with ACMFD method

196


Figure 11.2: Detail of subroutines scheme with CMFD method with two-node correction

197

11.3. Alternative methods to avoid diagonalization

results from the ACMFD original method, and did not make the code to converge eitherwhen the ACMFD original method failed.

φVg =6∑i=1

φig (11.8)

Another approach was to consider the possibility of avoiding the substitution of the uxesat the vertexes φVg in the ACMFD coupling relation. Instead of that, those uxes wouldbe treated explicitly using the same denition as it was used for the transverse leakageterms, coming from the previous iteration. However, as it was reported in [88], this stepled to a not convergence due to the excessive inuence of the transverse leakage terms inthe analytical denition of the ux at vertex. As a consequence, the simplied denition(11.8) was adopted in combination with the explicitly treatment. After some checks, it wasdemonstrated that treating those terms explicitly did not improve the convergence but theother way around. Therefore, this path was abandoned.

In addition, a damping factor was introduced in the surface currents, aiming to help theconvergence. However, the convergence was not improved by the damping.

Finally, the idea to consider the uxes at vertexes as unknowns of the linear system wasconsidered. Actually, other codes treat the hexagonal geometry in a similar way, consideringthese new terms as unknowns and solving the TPEN method, as is the case of PARCS.

There are two ways of doing this. One implies the resolution of the ANM method in thewhole core (ACMFD) with the addition of the uxes at vertexes as unknowns. The secondone consists in solving a linear system with the ANM methodology for the two-node problemconsideringφVg as unkowns, and computing afterwards the surface currents to be used tocorrect the CMFD method. From a optimal point of view, the second approach wouldbe better, because it would reduce the memory requirements and the computational time.However, when it was implemented, it was found that given strong variations from oneiteration to the next, the method fail to converge.

At the end, it was veried that by choosing appropriately the frequency of corrections,i.e. the number of iterations between two consecutive nodal updates, the code succeed inconverge.

Nevertheless, in order to get a general-purpose solution, it is recommended to avoid theACMFD method and implement a NEM solution to correct the surfaces currents for eachtwo-node problems.

11.3 Alternative methods to avoid diagonalization

In this section, some solutions are proposed in order to overcome the limitations linked to theuse of a ne energy group structure in ANDES. First approaches are discussed beforehand,and the Schur decomposition method will be described next, including its implementation,verication and future developments.

198


11.3.1 First approaches

In the previous Chapter, the limitations encountered when a ne energy group structure isemployed were presented. It was veried that for some specic materials, i.e. the non-ssileones, the multigroup diusion matrixes are ill-conditioned and therefore, the diagonalizationstep fail to converge.

The rst solution proposed was to reduce the group structure from 33 to 15 groups, follow-ing the group structure of ERANOS usually employed for sensitivity analysis. As shown inSection 10.3.1, when the group structure is reduced, the conditioning of the matrixes im-proves and therefore the convergence could be achieved. However, as described in Chapter2, due to the staggered shape of the neutron ux, the choice of the right group structureis of maximum relevance. Therefore, the use of 15 groups cannot be a denitive solution.

As previously discussed, in order to overcome the limitations linked to the use of a neenergy group structure, the diagonalization step in the ACMFD method should be avoided.Therefore, the most direct solution would be to apply pure diusion to the core. Aspreviously discussed, the convergence problems seem to arise due to the numerical propertiesof the matrices considered in the calculation. To check the absence of these problems whenthe diagonalization of the multigroup diusion matrices is not required, a Fine Mesh FiniteDierence (FMFD) scheme has been successfully implemented in ANDES code. It canbe used for non-ssile material nodes until new numerical methods are implemented andtested. In Chapter 13 the calculations using FMFD in ANDES are discussed. These resultsare intended to conrm that the ANDES code with the ACMFD method will work as soonas the conditioning problems of the matrices are solved. When comparing the half-nodeACMFD formula 11.9 with the Fick's Law 11.10 the former coincides with the latter if thematrices Af and Aj are the identity.

∣∣J i+1/2g

⟩= (Aj

iD−1i + Aj

i+1D−1i+1)−1

Af∗i |φig〉 −Af∗

i+1|φi+1g 〉

H/2(11.9)

∣∣J i+1/2g

⟩= (D−1

i + D−1i+1)−1

|φig〉 − |φi+1g 〉

H/2(11.10)

Therefore, forcing the matrices to be the identity and removing the traverse leakage eect,we achieve the implementation of the Finite Mesh Finite Dierence diusion approach inANDES. This way, the convergence has proved to be achieved easily. However,results canstill be improved.

11.3.2 Schur decomposition method

As it was previously discussed, in order to be able to work with a representative enoughenergy group structure, the diagonalization should be avoided. After doing some researchon mathematical methods, it was decided to try the Schur decomposition [66], which intheory works well for non-invertible matrices.

199


11.3.2.1 Theoretical background

Before starting with the description of the methodology, some theory should be introduced.

The classical theory tells that any symmetric matrix can always be diagonalized by orthog-onal transformations. Moreover, Schur (1909) allows us to extend this classical theory tonon-symmetric matrices. A real matrix Q is called orthogonal if its column vectors aremutually orthogonal and of norm 1, i.e., if QTQ = I or QT = Q−1. A complex matrixQ is called unitary if Q∗Q = I or Q∗ = Q−1, where Q∗ is the adjoint matrix of Q, i.e.,transposed and complex conjugate.

Schur postulated a theorem in 1909 that reads that for each complex matrix A there existsa unitary matrix Q such that

Q∗AQ =

λ1 b1,2 b1,3 ... b1,n

0 λ2 b2,3 ... x

0 0 ... ... ...

0 ... ... λn−1 bn−1,n

0 0 ... 0 λn

(11.11)

Winter and Murnaghan postulated in 1931 an additional theorem that reads: For a realmatrix A the matrix Q can be chosen real and orthogonal, if for each pair of conjugate

eigenvalues λ, λ = α± iβ one allows the block

(λ x

λ

)to be replaced by

(x x

x x

) .

Therefore, using the foregoing theory, any non-symmetric complex or real matrix could betransform into a upper triangular form.

11.3.2.2 Application of Schur to the diusion equation

Let's start with the diusion equation (11.12):

∇2 |φg (r)〉 −A |φg (r)〉 = −D−1 |Sg (r)〉 (11.12)

Given that for our problem, A is real, the Schur decomposition would result in:

ZTAZ = T (11.13)

being T the Schur upper triangular matrix. Note that in order to simplify the oncomingexpressions, the index naming start on the last row:

200


T =

λN bN,2 bN,3 ... bN,N

0 λN−1 bN−1,3 ... bN−1,N

0 0. . . ...

...

0 .... . . λ2 b2,N

0 0 ... 0 λ1

(11.14)

And the relations between the modal and physical space are:

|ψm(r)〉 = ZT |φg(r)〉 ; |φg(r)〉 = Z|ψm(r)〉

|sm(r)〉 = ZTD−1|Sg(r)〉 ; |Sg(r)〉 = DZ|sm(r)〉

∣∣∣~jm(r)⟩

= ZTD−1| ~Jg(r)〉

(11.15)

11.3.2.3 Analytical solution of the modal equations in Cartesian geometry 1Dwithout external source

Formulating the solution of the modal equations for the simplest problem using the Schurdecomposition, the following expressions are obtained.

For the last row, the expression would be the same as in the regular ACMFD formulation:

d2ψ1(x)dx2 − λ1ψ1(x) = 0 (11.16)

ψ1(x) = A1e+α1x +B1e

−α1x ; (11.17)

However, for the other rows the expressions dier as a consequence of including the non-diagonal terms bij. Actually, for the rest of rows, instead of an homogeneous solution, aheterogeneous one should be encountered.

For the second-to-last row (m=2), assuming that λ1 6= λ2:

d2ψ2(x)dx2 − λ2ψ2(x) = b21ψ1(x) (11.18)


−α2x + Ce+α1x +De−α1x ; (11.19)

Taking into account that the particular solution must be solution of the general equation,the constants C and D can be obtained. And using the recursiveness of the expression11.33 could be rewritten as:

201



−α2x + w21ψ1(x) ; (11.20)

In general, for any m 6= 1,assuming all eigenvalues dierent from each other (λi 6= λj):

d2ψm(x)dx2 − λmψm(x) =

∑m−1j=1 [bmjψj(x)] (11.21)


−αmx +m−1∑k=1

[Cm,ke

+αkx +Dm,ke−αkx

](11.22)

with

Cm,k =

∑m−1j=1 [bm,jCj,k]

λk − λm; Dm,k =

∑m−1j=1 [bm,jDj,k]

λk − λm(11.23)

being Ck,k = Ak and Dk,k = Bk

Applying now the recursiveness of the equations:


−αmx +m−1∑k=1

[wm,kψk(x)] (11.24)

with

wm,k =

∑m−1j=1 [bm,jwj,k]

λk − λm(11.25)

It should be noted that the previous expressions are only valid when all the eigenvalues aredierent from each other.

Taking the average of the modal ux:

ψm = 1H

´ H0ψm(x) · dx =

= 1H

[Amαm

(e+αmH − 1

)− Bm

αm

(e−αmH − 1

)]+∑m−1

k=1

[wm,kψk

] (11.26)

Constants Am and Bm can now be calculated as function of the modal current and ux atthe surface, and introduced in the equation 11.34. Reordering:

ψm(∓H2

)−∑m−1

k=1

[wm,kψk(∓H

2)]

=

= CfTm

[ψm −

∑m−1k=1

[wm,kψk

]]± CjT

mH2

[Jm(∓H

2) +

∑m−1k=1

[wm,kJk(∓H

2)]] (11.27)

202


And multiplying by Z, going back to the physic space, we obtain the Chao's equation 1Dfor the physic space using Schur desomposition:

|φg(∓H

2)〉 = Af

T |φg〉 ±H

2Aj

TD−1g |Jg(∓

H

2)〉 (11.28)

with

AfT = ZS−1

T CfTm Z∗;Aj

T = ZS−1T CjT

m Z∗ (11.29)

and

ST =

1 −wN,N−1 · · · · · · −wN1

1 −wN−1,N−2 · · · .... . . −w32 −w31

1 −w21

1

;

CfT =

CfN −Cf

NwN,N−1 · · · · · · −CfNwN1

CfN−1 −Cf

N−1wN−1,N−2 · · · .... . . −Cf

3w32 −Cf3w31

Cf2 −Cf

2w21

Cf1

CjT =

CjN −Cj

NwN,N−1 · · · · · · −CjNwN1

CjN−1 −Cj

N−1wN−1,N−2 · · · .... . . −Cj

3w32 −Cj3w31

Cj2 −Cj

2w21

Cj1

(11.30)

with Cfg and Cj

g being the same expression as in the ACMFD method.

11.3.2.4 ACMFD method for Cartesian 3D Problems with transverse integra-tion using the Schur decomposition

In order to extend this methodology to 3D problems, the transverse integration is required,as it happened in the original ACMFD method. However, these additional terms make thetraverse leakage term a bit more complicated.

203


d2ψm(x)dx2 − λmψm(x) =

∑m−1j=1 [bmjψj(x)] + lm(x) (11.31)


−αmx +m−1∑k=1

[Cm,ke

+αkx +Dm,ke−αkx

]+ plm(x) (11.32)

with plm(x)being the particular solution corresponding to the traverse leakage term. Usingnow the recursive property of the equation:


−αmx +m−1∑k=1

[wm,k(ψk(x)− plk(x))] + plm(x) (11.33)

Calculating the average modal ux:

ψm = 1H

´ H0ψm(x) · dx =

= 1H

[Amαm

(e+αmH − 1

)− Bm

αm

(e−αmH − 1

)]+∑m−1

k=1

[wm,k ¯(ψk − ¯plk)

]+ ¯plm

(11.34)

Once again calculating Am and Bm as a function of the surface ux and current, andintroducing them in equation 11.34:

ψm(∓H2

)−∑m−1

k=1

[wm,kψk(∓H

2)]

=

= CfTm

[ψm −

∑m−1k=1

[wm,kψk

]]+

±CjTm

H2

[Jm(∓H

2) +

∑m−1k=1

[wm,kJk(∓H

2)]]

+

+[plm(∓H

2)−

∑m−1k=1

[wmkplk(∓H

2)]]

+

±H2CjTm

[pl′m(∓H

2)−

∑m−1k=1

[wm,kpl

′

k(∓H2

)]]

+

−CfTm

[plm −

∑m−1k=1

[wm,kplk

]]

(11.35)

|φg(∓H

2)〉 = Af

T |φg〉 ±H

2Aj

TD−1g |Jg(∓

H

2)〉 −Z|Tm(∓H

2)〉 (11.36)

204


Tm(∓H2

) = CfTm

[plm −

∑m−1k=1

[wm,kplk

]]+

−[plm(∓H

2)−

∑m−1k=1

[wmkplk(∓H

2)]]

+

∓H2CjTm

[pl′m(∓H

2)−

∑m−1k=1

[wm,kpl

′

k(∓H2

)]]

(11.37)

11.3.2.5 Implementation

The Schur formulation was implemented satisfactorily for Cartesian geometry in the sourceof ANDES. First of all, the logical variable SCHUR was created in order to indicate wherethis option is to be used. Moreover, a new subroutine was created: eigenschur ; and twomain subroutines were modied: multidif and coefandes.

• EIGENSCHUR: It is the subroutine responsible of performing the Schur decompo-sition. For this matter, it can call either the subroutine ZGEESX or the DGEESXfrom the Lapack library, depending upon the type of matrix: complex or real. Sincethe eigenvalues obtained from these subroutines are not ordered from greatest tosmallest, a variable was created to point the position of the maximum eigenvalue.

• MULTIDIF: For each node, it builds rst the multigroup diusion matrix A, and thenit computes the eigenvalues and eigenvectors needed to move from the physic to themodal mode and vice versa. While for the original version, it would call EIGEN, whenthe Schur decomposition is applied, EIGENSCHUR is called instead. Therefore, thematrices Z are computed in this subroutine for each node.

• COEFANDES: This is the subroutine in charge of computing the coecients neededfor the ACMFD method, i.e., Cf andCj, which will be transformed into Af andAj;and the transverse leakage terms. In case of using the Schur option, the coecientsCfT and Cj

T are calculated instead. In addition, the new coecients wij are computedin this subroutine to build the matrices ST , which will need to be inverted in orderto obtain the new AT

f and AjT.

In addition to the previous developments, and particularizing for the actual application, asubroutine to distinguish between upper triangular, lower triangular and complete matriceswere developed which will be useful to identify non-ssile nodes.

11.3.2.6 Verication and rst tests

Several checks were performed and it was demonstrated that for very simple problems,without the presence of the non-ssile material, both the ACMFD method and its variationusing the Schur decomposition provided very similar values. However, it was encounteredthat when the conictive materials were involved in the calculation, this new method failedto converge. After a deep study, it appeared that the triangular matrix ST that had to

205


be inverted presents very high conditioning number, and therefore the convergence couldnot be achieved. This is the result of the assumption of all eigenvalues dierent from eachother. Actually, it was checked and some of them diered only in 10−5. This causes theparameters wij to be too high for some specic groups, getting even worst the higher thematrix is (i.e., with more energy groups).

In order to consider the presence of very similar eigenvalues, the procedure complicates toomuch. Indeed, there are multiple combinations of solutions depending on the position atwhich the almost-equal eigenvalues appear.

11.3.3 Future solutions

As discussed previously, the Schur decomposition cannot be applied because there is not adirect way to implement it generically when there are identical eigenvalues. So, there aretwo alternatives:

1. To try to develop a methodology to account for the complete casuistry of identical andnot identical eigenvalues at dierent positions

2. To develop a methodology where the group structure is collapsed and extended afterwardsfor the calculation of the non-ssile materials.

11.3.3.1 Shur methodology with identical eigenvalues

The limitation of the Schur methodology really lays on the fact that if the considered matrixpresents almost-equal eigenvalues the methodology fails. The reason underneath is that theparameters that account for the dependence of the solution at each specic group upon thesolution of the previous groups are divided by the eigenvalue dierence (see 11.25). Thus,if the eigenvalues are too similar, this term is quite big. As a consequence the conditioningnumber of the matrix ST is too high, which makes it impossible to be inverted. Actually, itcan be seen as a translation of the ill-conditioning properties from the R matrix from thediagonalization to the matrix ST from Schur decomposition.

The problem of the identical eigenvalues has been recently addressed by Dreher [23] inits application to the Bateman depletion equations (Eq.11.38). In that paper, a derivedmethodology is proposed to treat the case.

dNi(t)

dt+ ki·Ni(t) = ki−1,i·Ni−1(t) (11.38)

As a result, that methodology was assessed for the application to our problem. However,there is an important dierence between both applications. While the Bateman dierentialequation (11.38) only depends on the immediate previous solution Ni−1(t), the diusionequation through the Schur methodology (Eq.11.21) depends on the solutions of the previ-ous modes, which makes it much more complicated. As a consequence and giving the lackof resources, this implementation stays as recommended future work.

206


11.3.3.2 Alternate collapsing of the group structure

As it has been discussed, the limitation of the energy groups only appears in the non-ssile nodes. Therefore, a new methodology consisting in the alternation of energy groupstructures from one iteration to the next has been assessed. The idea would be to run oneiteration in a broad energy group structure, such as 8 energy groups for the whole core, andexpanding it afterwards to the right group structure. Then a new iteration is performed inthe ne group structure, where the results from the non-ssile nodes are frozen. However, itwould need to be checked which alternative provides a better solution, whether the ACMFDmethod in 8 groups or the diusion approach in 33 groups.

207


208

Chapter 12

Coupling betweenCOBAYA3/ANDES andSUBCHANFLOW

12.1 Introduction

In order to perform safety studies of sodium fast reactors, high delity multidimensionalanalysis tools for sodium fast reactors are required for both core design optimization andsafety assessment. Such tools should integrate neutronic and thermal-hydraulic phenomenain a multi-physics approach.

Consequently, the aim of this work is the development of a code system able to modelSodium Fast Reactors that allows to simulate transient scenarios and consequently to per-form safety analysis. Therefore, the neutronic code developed at UPM (COBAYA3) neededto be coupled with a thermal-hydraulic code that works with sodium as coolant. The codeselected is the KIT subchannel thermal-hydraulic code SUBCHANFLOW.

12.2 Context

In the last two decades, many coupled neutronic/thermal hydraulic system codes weredeveloped and validated for the transient analysis of LWR. In this case, the core behavior ismainly described by 3D nodal diusion solvers with direct consideration of local feedbacksbetween the two domains. Recently, these codes are being extended by the introductionof multi-group diusion solutions and SP3-based time-dependent transport solvers. Onthe contrary, the transient analysis of fast reactors relies mainly on system codes usingpoint kinetic models or 3D solvers with quasi-static approximations to solve the time-dependent problem. Recent investigations in dierent places are focused on the adaptationof LWR-best estimate codes for their use in transient analysis of e.g. liquid metal fastreactors. To do so, many challenging issues needs to be solved e.g. the consideration of

209

12.3. Description of the coupling

the FR-relevant feedback eects (axial, radial elongation of the core, etc.) in the process ofgeneration of cross section sets for transient analysis with coupled codes e.g. look-up tables,parameterization of XS or development of polynomial functions, etc. In [98] for example,a cross-section feedback model for PARCS was developed which uses the cross sectionsgenerated by ERANOS [122]so that the analysis of e.g. sodium fast reactor transients isfeasible using the coupled system TRACE/PARCS which is embedded in the FAST system.Similar work is done at ENEA for RELAP5/PARCS to simulate lead cooled fast reactors[43]. Previous work was done at UPM coupling SIMTRAN and RELAP5 [6].

The reference transport neutronic code for Fast Reactors is ERANOS [122], as it is con-sidered the most validated code through many experiments performed in the CEA facilities(France). However, ERANOS is only applicable to steady state calculation and transientsare not available so far. Therefore, CEA is implementing heat transfer correlations andthermo-physical properties of sodium in the LWR system code CATHARE to be used inconnection with CRONOS core solver for fast reactor transient analysis [24, 32].

On the mentioned coupling schemes, the thermal hydraulic models are based either on 1Dor 3D coarse mesh to describe the thermal hydraulic phenomena within the core. Theseapproaches do not allow the direct prediction of local safety parameters such as maximalfuel and cladding temperature. Hence, the coupling of advanced neutronic solver at celllevel with subchannel codes is urgently needed not only for LWR but also for fast reactoranalysis.

Therefore, the development of an integrated code system able to perform detailed 3Dtransient analysis for the future Sodium Fast Reactors was settled as an objective of thethesis. In order to do that, the rst step consists of coupling the UPM neutron diusioncode COBAYA3 consisting of a nodal and a cell solver with a sub-channel thermal-hydrauliccode, and since the coolant uid for the ESFR is liquid sodium, a thermal-hydraulic codedesigned for this particular coolant is required. That is the case of the SUBCHANFLOWcode, recently developed at KIT [69, 70, 72], and able to perform detailed 3D core thermal-hydraulics. Moreover, the main part of this work was performed during a three months stayat INR department, KIT. This unique opportunity was possible due to the collaborativeagreement between UPM and KIT to share knowledge, codes and exchange students.

12.3 Description of the coupling

The motivation of the N-TH coupling lays on the fact that both physics are strongly de-pendent on each other by means of the so called feedback eects. In practical applications,the coupling can be seen as the exchange of information between codes modeling dierentphysical phenomena. In the neutronics for example, the cross sections are to be used atthe actual operating conditions of each neutronic node/cell, and, as this information is notknown in advance, it leads to an iterative problem between the solution of the thermal-hydraulic and the neutronic balance equations. The nature of this coupled problem is alsonon-linear because the variables in the coupled system of equations appear in both sides ofthe equations.

210


The coupling between COBAYA3 and SUBCHANFLOW was performed by the integrationof SUBCHANFLOW in the COBAYA3 system. The variables involved in the coupling areshared by means of the internal memory. This type of coupling is called tight coupling dueto the fact that the two independent codes are compiled in a unique executable and shareinternal memory structures. It should be noted that the coupling scheme has only beenimplemented for the nodal solution, i.e. for the coupling between ANDES (the nodal solverof COBAYA3) and SUBCHANFLOW. Therefore, from now on COBAYA3 will be synonymof ANDES.

As mentioned before, COBAYA3 has already been coupled to several thermal-hydrauliccodes [78], and as a consequence its internal structure was already modularized to allow thecoupling. The same structure has been employed for the coupling with SUBCHANFLOW,keeping the parallelism between SUBCHANFLOW subroutines and the ones from other THcodes, such as COBRA-III or COBRA-TF.

On the other hand, SUBCHANFLOW has already being integrated in the SALOME plat-form [38] and coupled with some other neutronic codes like DYN3D, and therefore, somemodularization was already performed.

12.3.1 Spatial coupling

At the nodal scale (whole core calculation with one or some mesh points per fuel assembly),the thermal-hydraulic mesh is usually assigned in a way that one thermal-hydraulic channelis related to one fuel assembly. In COBAYA3/ANDES one hexagon corresponds to at least6 triangles, due to the triangular mesh denition. Consequently, one TH hexagonal channelcorresponds to the 6 triangular nodes from ANDES and no renement has been considered.

As for the axial dimension, each TH axial node corresponds to one TH neutronic node. Thismeans that the neutron mesh is superposed to the TH mesh and vice versa. Consequentlyno interpolation is required in the properties transferred from one mesh to another, and theabsolute value is introduced in the node/channel. Moreover, no axial renement has beenconsidered so far.

12.3.2 Implementation of new subroutines

New subroutines were developed or modied in order to implement the coupling.

For the SUBCHANFLOW source, the main change consisted in splitting the programsubchanow into three subroutines: initsubchf, calcscf, and edit_subchf, whilethe new subroutine subchanflow has the unique function to call three subroutines:paso_nk2thS, calcscf and densandS

As for the COBAYA3 source, three new subroutines were created: prepasoS, paso_nk2thSand densandS, and another one was modied, paso_th2nk.

• initsubchf is in charge of reading the SUBCHANFLOW input, calling clean_input

211


Figure 12.1: Multi-physics N-TH coupling through common memory

to rewrite the input skipping comments and blank lines, and initializing all the variablesin the common of SUBCHANFLOW.

• prepasoS is in charge of reading the le cobradat.gen, which contains the deni-tion of the radial mapping to connect the nodes from SUBCHANFLOW and the onesfrom ANDES.

• paso_nk2thS is in charge of transferring the power from ANDES to SUBCHAN-FLOW. It is in this subroutine where the radial mapping read in prepasoS will betaken into account in order to assign the power from the triangular nodes of AN-DES to the hexagonal nodes of SUBCHANFLOW. The most relevant variables ofSUBCHANFLOW in this subroutine are powdis and powmap. The former one is thenormalized power distribution to be used in steady state calculations, while the latteris the time-dependent normalized power distribution but to be used during transients.

• calcscf consists basically the same instructions that are presented within the timeloop of the SUBCHANFLOW program, i.e., it is the core of the program, wherethe main calculations are performed and the convergence is achieved, with slightmodications.

• densandS is in charge of transferring all the feedback data, i.e., Doppler temper-ature, coolant density and temperature, to the neutronic code ANDES, in order toupdate the cross sections.

• paso_th2nk introduces the feedback parameters (Doppler temperature, coolanttemperature and density) into the commons of COBAYA3.

• edit_subchf is in charge of shut down SUBCHANFLOW, once the execution isterminated.

Figure 12.1 shows the data exchange through the memory modules between the neutronicand the thermal-hydraulic part with the names of the most relevant subroutines.

Additionally, some modules were developed specially for the coupling:

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• nodalmod stores the nodal variables to be exchanged between the codes (power,temperatures and densities)

• couplingmod stores secondary variables required for the coupling (i.e. logicalvariables involved).

• acopmod stores other variables required for the coupling (i.e. damping factors)

• ncanmod stores the radial mapping for the spatial coupling

12.3.3 Description of the iterative schemes for steady state andtransients

The sequence for the coupled calculations in COBAYA3 follows the following steps, asindicated in the gure 2. It should be noted that for steady state calculations, the neutroniccode starts the iterative process while for transient calculations, the thermal-hydraulic codestarts.

Reading of input les and loading of nuclear data libraries on memory. First neutronic cal-culation that provides the rst 3D power distribution. The iterative process starts betweenNK and TH until the error between iterations is converged to the desired level in both Keand tridimensional distributions. Once the coupled calculation has converged, the controlreturns to the leading subroutine in COBAYA3. If a transient calculation or a successionof steady state is requested, then the sequence goes on in the point 5. Otherwise, theexecution is terminated. In case of transient, the equilibrium concentration of the delayneutron precursors, i.e., the kinetic equations are initialized and the sequence go on in thepoint 3, with a semi-implicit coupling temporal scheme. In case of a succession of steadystates, the neutronic conditions are changed and the sequence goes to the point 3, wherea new steady state is calculated.

In order to get a quick convergence in the non-linear iteration of the thermal-hydraulic feed-back, a damping factor should be applied to the new values of the feedback parameters thatwill be used to interpolate the cross sections and the power distributions to be introducedagain in SUBCHANFLOW. The damping factor DAMP oscillates between 0 and 1, being0.7 an optimal value for most of the analyzed cases.

X = DAMP Xnew + (1−DAMP )Xold (12.1)

12.3.3.1 Steady State Coupling

The steady state coupling sequence is shown in the left part of the Figure 12.2. As men-tioned before, the neutronic calculation precedes the TH one in the iterative process. Thepursued goal is to get a neutronic and thermal-hydraulic solution which should be autoconsistent, i.e., in which all the balances are achieved, including those that depend on thecommon variables.

213


Figure 12.2: General iteration scheme for the COBAYA3-SUBCHANFLOW coupling calcu-lations

12.3.3.2 Transient Coupling

The transient coupling sequence is shown in the right part of Figure 12.2. In this case,since the previous steady state has already converged, the TH is in charge of initializing theiterative process and the goal is to advance the solution through each time step keepingthe numerical scheme within the stability margins and focusing in the accuracy obtained.

As for the transient temporal coupling scheme, it is done employing a modication of thealternate temporal scheme staggered alternate time mesh scheme, developed by Merino in1993 [95]. The most remarkable feature of the proposed algorithm is that the calculationsof the neutronic and thermal-hydraulic variables are not performed at the same time step,but they are done half a time step out of phase. This means that the neutronic calculationswill be performed at an instant of time in which the thermal-hydraulic variables have notbeen calculated yet.

Given that the characteristic time of the neutronic variables is very small compared to theone of the thermal-hydraulic variables, it was decided to extrapolate linearly the thermal-hydraulic variables half a time step in order to advance to the conditions where the neutroniccalculation is performed in COBAYA3. The extrapolation is performed from the two previouscalculations of SUBCHANFLOW, using the same damping scheme as in the steady state(12.1) but with a dierent damping factor (12.2).

DAMP = 1 +∆tn

2∆tn−1

(12.2)

214

12.4. New developments for the extension to Sodium Fast Reactors

Figure 12.3: Temporal coupling scheme

Figure 12.4: General algorithm for coupling calculations in COBAYA3

According to this scheme, the energy is conserved in a rst order, which means that incase of linear variations of the power, the energy is completely preserved, while in case ofsharpener power variations (exponential changes) this scheme is proved to be much morerobust and stable than a pure explicit one.

Figures 12.3 and 12.4 show the temporal coupling scheme and the general algorithm fol-lowed, respectively.

12.4 New developments for the extension to Sodium

Fast Reactors

In order to extend the N-TH coupling between COBAYA3 and SUBCHANFLOW to SFRs,some especial considerations should be taken into account, which might be divided into 5

215


sections: geometry, coolant, materials, feedback parameters and cross sections.

12.4.1 Geometry

Fast reactors cores are usually characterized by a hexagonal wrapper tube that surroundseach Fuel Assembly (FA) in order to avoid transversal ows. Therefore, hexagonal channelsshould be modeled in SCF and no neighborhood eect is considered under a thermal-hydraulic point of view. This might impact the convergence of the thermal-hydraulic code,and forced boundary conditions might be recommendable to accelerate the convergence.

Moreover, the control rods are placed all together in the control assemblies, instead ofbeing embedded in the FA, which means that some dummy assemblies should be denedto consider the mass ow through the control channels, with no power generation. Thisis also possible with SCF. Furthermore, according to the new optimized model especiallydesigned to avoid the reactivity insertion due to a sodium void, further axial layers belowand over the active core are dened, whose consideration should be assessed in detail.

12.4.2 Coolant

As for the use of sodium as coolant, some thermal-hydraulic considerations should bemade. First, the use of sodium requires specic uid correlations, which have been alreadyimplemented in SUBCHANFLOW. And secondly, thanks to recent developments, two phaseow equations are now implemented for the sodium in SUBCHANFLOW, which would allowto model sodium voiding scenarios. However, since SUBCHANFLOW resorts to a mixturemodel to treat two phase ow, the directional eects of the sodium voiding cannot becaught, which might induce to errors in sodium boiling scenarios.

12.4.3 New materials

For Sodium Fast Reactors, new materials have been developed. Those materials shouldbe properly modeled in the thermal-hydraulic code in order to consider their main thermalproperties (thermal conductivity, specic heat, density, porosity, expansion coecients,swelling, gap conductance, etc.) whether as a correlation function of the temperature, oras a xed value.

SUBCHANFLOW already has implemented some correlations for a typical fuel and steel ofa SFR. However, these properties should be adapted to the specic problem to be modeled.The specic materials properties can easily be implemented in SCF.

12.4.4 Feedback parameters

In LWRs there are typically four feedback parameters: fuel (Doppler) temperature, coolant/moderatordensity and temperature, and boron concentration (in case of a PWR). However, for SFRs

216


the situation is very dierent mainly due to the high power density compared to the one ofLWR and the very short core height.

The Doppler and the coolant density remain as feedback parameters, while the boronconcentration disappears, due to the absence of boron in the coolant. In addition, a setof new parameters have been proved to impact strongly on the reactivity and feedbackresponse, and consequently should be included in the simulations: the axial and radialexpansion of the core components (pins, lower grid plate where the subassemblies are xed,etc.) feedback parameters.

For the two rst parameters, the same treatment as in LWRs is kept in COBAYA, whilefor the expansion feedback parameters, new developments were implemented in COBAYAin order to take into account the eects of geometrical changes on the neutronic part.A simple model based on the linear expansion coecients was employed to update thegeometry according to the new thermal-hydraulic conditions. Indeed, for each iteration,the height and radius of the core are updated according to the average fuel and coolanttemperature, respectively, and used as feedback parameter to update the cross sections. Asfor the thermal-hydraulic part, simplied models are under development in SCF to predictthe axial and radial elongation of the fuel rods.

12.4.5 Cross sections

The required parametrized cross sections for the SFR core according to the previouslymentioned feedback parameters should be whether provided or generated, as discussed inChapter 9.

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218

Chapter 13

Verication and application of theperformed developments

13.1 Introduction

As discussed along the thesis, the ACMFD method implemented in ANDES entails somediculties when applied to SFR. As a consequence, the alternative methodology CMFDmethod with nodal correction update was implemented in ANDES, as described in Chapter11. In order to verify its implementation and behaviour, several steps were followed:

First, the methodology was veried for thermal reactors against the original ACMFDmethod, in both Cartesian and hexagonal geometries (Section 13.2). Then, the method-ology was applied to SFR, similar to the ones described in Chapter 5, and veried againstPARCS and DYN3D (Section 13.3).

Moreover, besides the improvements performed in the ANDES code in order to apply it toSFRs, it was coupled to the thermal hydraulic code SUBCHANFLOW (Chapter 12). Again,some steps were followed for the verication of its performances:

First, it was veried for thermal reactors using the original ACMFD method (Section 13.4).Then, the new methodology CMFD was veried with the coupling for a thermal reactor,benchmarking it against the ACMFD method. Finally, the coupling was applied to the SFRcore, in conjunction to the CMFD method. However, given the lack of enough informationit could not be propperly benchmarked.

13.2 Verication of the corrected CMFD method for

thermal reactors

After implementing the methodology described in Section 11.2 in ANDES and debuggingnext, some tests were run in order to check the performance of the new methodologycompared to the ACMFD method.

219

13.2. Verication of the corrected CMFD method for thermal reactors

Method NOXEL 2 NOXEL 4

ACMFD 1.17594039 1.17593653

CMFD 1.17589570 1.17570025

corrected CMFD 1.17594038 1.17593650

fuel-corrected CMFD 1.17609674 1.17590478

Table 13.1: Verication of the corrected CMFD method for a Cartesian PWR minicore

First the Cartesian geometry was tested, for being the transverse leakage term much simpler.Then, the methodology was applied to hexagonal problems. The verication for bothgeometries in thermal reactors is presented along this Section.

13.2.1 Cartesian geometry

First of all the pure CMFD method, without correction was tried and the convergencecould be easily achieved. Then the correction was applied and no convergence problemsarose either. The results of both of them were compared with the resulting ones from theACMFD method. However, when the group structure was increased, the method started tofail. This result is perfectly coherent with the limitation described in the previous Chapter11. For these cases, a third modication was tried, where only the ssile nodes were subjectof correction.

For the verication in Cartesian geometry, a very simple model was used. A minicore of 3x3fuel assemblies, based on the specications of the benchmark PWR MOXUO2 [82] in twoenergy groups, was chosen for this purpose. In order to quantify the error committed whenthe pure CMFD approach was used, two levels of renement were considered. Table 13.1shows the eigenvalue results for two dierent renement levels using the previously describedmethods. NOXEL 2 comes for a rst level of renement, where each subassembly isradially divided into 4 nodes. On the other hand, NOXEL 4 comes for the case whereeach node is radially divided into 64..

After checking the Table 13.1 several conclusions are drawn. First, the greater renementwe considered, the lower dierences are found between the pure CMFD method and theACMFD. This is something expected, given that the diusion approach improves with themesh renement. Moreover, a pretty good agreement is observed between the correctedCMFD method and the ACMFD method. This is a desired result, since the former is theproposed method to replace the ACMFD in those cases where the latter fail to converge.Finally, the case where the CMFD method is only corrected in the fuel nodes also provesto provide comparable results, getting better with the increasing renement.

Once it was found that both methods provide comparable results, some cases that used notto converge with the ACMFD method were tried. For those cases, it was proven that theconvergence was achieved satisfactorily, and what is more, the Wielandt acceleration, whichcould not be employed for those cases using the ACMFD method, succeeded in accelerating

220

13.3. Verication of the corrected CMFD method for fast reactors

Method K-e

ACMFD 1.03011430

CMFD 1.03292630

Corrected CMFD 1.03011421

Fuel-Corrected CMFD 1.03193126

Table 13.2: Verication of the corrected CMFD method for a hexagonal core

the convergence with the new method.

Concluding, in Cartesian geometry, the method proposed and implemented solved in prin-ciple the limitations related to the transverse leakage loop.

13.2.2 Hexagonal geometry

As explained in Chapter 10, the hexagonal geometry entails further diculties when thetransverse leakage integration is to be used. As a consequence, the validation for thisgeometry is even more critical than for the Cartesian one.

In order to verify the new methodology in hexagonal geometry, a VVER core from theV1000CT benchmark [29] was used. Table 13.2 presents the comparison between theAMCFD method and the CMFD method with dierent level of corrections. After examthe results, it can be conrm that the corrected CMFD method is in good agreement withthe ACMFD method. Moreover, when the radial power distribution was compared, it wasobserved that those methods provide identical results.

13.3 Verication of the corrected CMFD method for

fast reactors

In order to be able to verify the performance of the new developments in SFR applications,an adequate benchmark is required. Following this motivation, an agreement with PSI wascarried out in order to compare the results obtained with the ones provided by PARCS. Inaddition, as a consequence of a potential contribution to the FR'13 Conference in 2013,the results were to be compared with the ones from PARCS and DYN3D. Following thatmotivation, the cross sections generated with PARCS were provided to both UPM andHZDR, together with further specications.

As a rst approach, single steady states calculations were proposed, following the thermal-hydraulic conditions specied in table 13.3. For all those cases, the developments on thecross section feedback according to the specic parameters required for SFR (Section 12.4)were also tested.

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Case number Fuel temperature (ºC) Coolant density (g/cm3) Position of control rods

1 20 0.95 Not present




5 20 0.95 Critical position**

Table 13.3: Case denition

Case numberk-eective

FAST DYN3D COBAYA

1 1.03591 1.03536 1.03654

2 1.01517 1.01448 1.01570

3 1.0354 1.03453 1.03601

4 1.03657 1.03603 1.03720

5 1.01963 1.01895 1.02179

Table 13.4: K-e comparison

In order to imitate as much as possible the conditions in which the other participantsmade their calculations, the 33 group structure was employed in ANDES. Then, the cor-rection could not be performed in the non-ssile nodes, due to the bad conditioning of themultigroup matrixes for those nodes in such structure, as it was discussed in Chapter 11.Therefore, the results in Table 13.4 illustrate the case where the correction of the CMFDmethod is only applied to the fuel nodes. When compared to the other codes, the method-ology in ANDES provides similar results, although still might be improved as soon as thecorrection could be applied to the whole core, i.e., when the energy groups issue is solved.

Moreover, the reactivity coecients obtained from the comparison of the eigenvalues ofperturbed cases with the reference one (Case 1) is presented in Table 13.5. It can beinferred that the agreement is quite good for all of them, except for the control rod worthwhich present higher dierences, as expected. Indeed, it was predictable given that thediusion approach works worst in regions with strong neutron absorbers.

Finally, Figures 13.1 to 13.5 show the comparison of radial power distribution for the vecases under study, respectively.

The agreement observed in the power distribution is really good with respect to the othercodes.

222


FAST DYN3D COBAYA

Doppler constant, pcm 1343 1353 -1347.61

Coolant density coecient, pcm/(kg m3) 0.95 0.2 63 103 -96.72

Control rod worth for critical position, pcm 1541 1555 -1440.01

Table 13.5: Reactivity coecients

Figure 13.1: Radial Power peaking factors Case 1

223




224




225

13.4. Verication of the coupling ANDES / SUBCHANFLOW for thermal reactors

13.4 Verication of the coupling ANDES / SUB-

CHANFLOW for thermal reactors

Following the verication of the CMFD methodology, the neutron/thermal-hydraulic cou-pling between ANDES and SUBCHANFLOW was veried. In order to isolate eects, it wasrstly validated for thermal reactors through several international benchmarks exercises.

A total of three benchmark exercises were modeled for both steady state and transientsimulations.

OECD/NEA and U.S. NRC PWR MOX/UO2 Core Benchmark [82]: Based on the West-inghouse 4-loops PWR core partially loaded with MOX fuel. Transient: Rapid Control Rodejection departing from a Hot Zero Power (HZP) state.

OECD/NEA PWR Main Steam Line Break (MSLB) Benchmark [76]: Based on theTMI-1 PWR core, using forced boundary conditions (inlet temperature, mass ow rate andpressure). Two transient scenarios were studied, one with a return to power (RP) andanother one with no return to power (no RP), departing from an End of Cycle (EOC)conditions.

V1000CT1 Benchmark [29]: Based on the unit 6 of the Kozloduy Nuclear Power Plant(Bulgaria), VVER design with hexagonal geometry, using forced boundary conditions alongthe transient time. The departing time for the transient is Beginning of Cycle (BOC).

13.4.1 OECD/NEA and U.S. NRC PWRMOX/UO2 Core Tran-sient Benchmark

The reactivity insertion accident benchmark exemplies a local perturbation of parametersfor both NK and TH due to a rapid Control Rod (CR) ejection (done in 0.1 s) applied to acore partially loaded with Mixed Oxide (MOX) FA, departing from a Hot Zero Power (HZP)state. The design of the core is based on the Westinghouse 4-loops PWR nuclear powerplant with 193 FA. Core models for COBAYA3 and SUBCHANFLOW were developed usingthe data and boundary conditions given in the benchmark specications [82]. The core wasdivided into 193 nodes/channels and 20 uniform axial levels.

First a Hot Full Power (HFP) steady state is analyzed and afterwards the transient resultsare presented in 1s transient showing a comparison with other participants' results.

Steady state (HFP)

Results of the steady state are presented in Table 13.6. The agreement observed is quietsatisfactorily.

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Case HFPCritical Boron

Fz Fxy A.O% Tdop ºC Tmax ºCppm

COBAYA3-SUBCHANFLOW 1685.705 1.4168 1.3727 -9.884 560.38 1536.19

COBAYA3- COBRA-III 1685.4 1.415 1.3741 -9.62 562 1541

COBAYA3-COBRA-TF 1682.3 1.416 1.3782 -10.3 561.4 1538.3

Table 13.6: Steady State HFP results for the MOX benchmark

Transient

The results of COBAYA3-SUBCHANFLOW (denoted SUBCHANFLOW) are compared withthe ones of COBRA-III using the same nodalization of SUBCHANFLOW (denoted COBRA-III) and using a rened mesh (ANDES/COBRA-III). Moreover, results of PARCS, CORE-TRAN and COBAYA3-COBRA-TF (denoted ANDES/COBRA-TF) are shown.

The main results analyzed are the relative power ( Fig.13.6), reactivity (Fig. 13.7), averageDoppler temperature (Fig. 13.8) and average core density (Fig. 13.9). For all of them avery good agreement is observed.

13.4.2 The OECD/NEA PWR Main Steam Line Break (MSLB)Benchmark

The PWR core data for this exercise was taken from the TMI-1 MSLB Benchmark [76].The initial steady state is characterized by End of Cycle (EOC) conditions with a boronconcentration of 5 ppm and equilibrium Xe and Sm concentrations. In addition, externalboundary conditions (inlet temperature, mass ow rate and pressure) for each FA/channelwere established for the steady state and transient solutions in the TH side. Two transientscenarios have been studied, one with a return to power (RP) and another one with noreturn to power (no RP). For both cases, SCRAM is done at 6.65 s from the beginningof the transient, where most of the reactivity worth is stuck and stay in the fully extractedposition, leaving the core in All Rods In situation (ARI-1) for the rest of the transient.After this point, the temporal scheme is guided by the TH boundary conditions set inSUBCHANFLOW, particular for each case. The independent models of COBAYA andSUBCHANFLOW were developed using 177 radial elements, and 24 uneven axial levels.The importance of this axial discretization is crucial for the NK and TH models since thecore is highly heterogeneous along its active height.

In the next sections the results obtained are presented for the Steady State calculationand the two transients. The results of SUBCHANFLOW coupled with COBAYA3 are com-pared with the ones from the benchmark reference, COBAYA3-COBRA-TF and COBAYA3-COBRA-III.

227


Figure 13.6: Power vs. time (s)

Figure 13.7: Reactivity ($) vs. time(s)

228


Figure 13.8: Average Doppler Temperature (°C) vs. time (s)

Figure 13.9: Average core density (kg/m3) vs. time (s)

229


Case HFP Ke Fz Fxy A.O(%) Tdop ºC Tmax ºC

COBAYA3-SUBCHANFLOW 1.005246 1.0705 1.3315 -2.025 530.31 1131

COBAYA3- COBRA-TF 1.005690 1.065 1.3312 -1.46 530.2 1148.2

reference 1.003900 1.0858 1.3536 -1.58 - -

Table 13.7: Steady State HFP results for the heterogeneous inlet B.C

Figure 13.10: Total Power (MW) vs. time (s)

Steady state

Results of the Hot Full Power steady state are presented in Table 13.7, where a pretty goodagreement is observed.

Transient RP

The transient with Return to Power is characterized by a slow return to a 30% of thenominal Power. Figures 13.10 to 13.14 shows the evolution of the total power, reactivity,average Doppler temperature, maximum Doppler temperature and average core density,respectively, where a pretty good agreement is observed.

230



Figure 13.12: Average Doppler Temperature (K) vs. time (s)

231


Figure 13.13: Maximum Doppler Temperature (K) vs. time(s)


232


Figure 13.15: Total Power (MW) vs. time (s)

Transient no RP

On the other hand, the transient without return to power, as its own name points out, thepower is not recovered after the transient. Figures 13.15 to 13.18 shows the evolution ofthe total power, average Doppler temperature, maximum Doppler temperature and averagecore density, respectively, where a pretty good agreement is observed.

13.4.3 V1000CT1

The purpose of this exercise is to prove the core response when the thermal-hydraulicboundary conditions (mass ow rate distribution, inlet temperature distribution, and outletpressure) are forced along the transient time.

The reference design for the V1000CT1 benchmark was taken from the reactor geometryand operational data of the unit 6 of the Kozloduy Nuclear Power Plant (Bulgaria) of 3000MWth, which are fully described in the benchmark specications [29].

The proposed steady state scenarios are characterized for a reactor at Beginning Of Cycle(BOC), with an average burn-up of 30.7 EFFP and a boron concentration of 5.95 g/kgH2O. The experiment starts with 3 pumps (1st, 2nd and 4th) on operation and one out oforder (3th), with a power level at 27.45% of the nominal power. Those conditions describea Hot Power (HP) state. In addition, a Hot Zero Power (HZP) state is dened with apower level at 0.1% of the nominal power, uniform fuel temperature spatial distribution at

233


Figure 13.16: Average Doppler Temperature (K) vs. time (s)

Figure 13.17: Maximum Doppler Temperature (K) vs. time(s)

234



552.15K and a moderator density of 767.1 kg/m3.

Table 13.8 shows the denition of the two steady states considered.

Steady states

For the two steady states considered, the results of COBAYA3-SUBCHANFLOW are com-pared with the other participants of the benchmark. Tables 13.9 and 13.10 show thecomparison for the HZP and HP cases, respectively. A very good agreement is observed.

Case TH Conditions Control Rods Position

0 HZP

Groups 1-8 ARO

Group 9 36% inserted

Group 10 ARI

1 HPGroup 1-9 ARO

Group 10 36% inserted

Table 13.8: Steady State denition for the V1000CT1 benchmark

235

13.5. Verication of the corrected CMFD with the coupling

Case HZP Ke Fz Fxy A.O (%) Tdop ºC Tmax ºC


COBAYA3- COBRA-III 0.99977 1.517 1.4047 -16.39 279.2 279.9

Reference 1.00022 1.519 1.416 -16.44

Table 13.9: Steady State HZP

Case HP Ke Fz Fxy A.O (%) Tdop ºC Tmax ºC


COBAYA3- COBRA-III 1.00037 1.4322 1.3392 -18.287 371.2 530.4

Reference 0.99998 1.406 1.341 -16.29

Table 13.10: Steady State HP results for the heterogeneous inlet B.C

Transient

The results of COBAYA-SUBCHANFLOW are compared with the ones of COBAYA-COBRAIIIand the mean values of all the contributions to the benchmark, along with some of themost representative ones. It should be noted that at nodal scale COBAYA and ANDES arethe same code.

Figures 13.19 to 13.21 show the power change evolution, the Doppler temperature changeevolution and the density change evolution, respectively. A big dispersion is observed be-tween the participants to the benchmark, but still the COBAYA-SUBCHANFLOW solutionis presented close to some of the most representative values.

13.5 Verication of the corrected CMFD with the

coupling

Once the coupling was satisfactorily veried for thermal reactors in both Cartesian andhexagonal geometries, the next step is to verify the performance of the two main develop-ments together, i.e., the corrected CMFD methodology plus the coupling with SUBCHAN-FLOW. Following this motivation, the transient from the PWR-MOX benchmark [82] wassimulated with both the ACMFD method and the corrected CMFD method. Results arepresented in Figures 13.22 to 13.25, where a perfect agreement is observed.

236


Figure 13.19: Power change (MW) vs. time (s)

Figure 13.20: Average fuel temperature change (K) vs. time (s)

237


Figure 13.21: Average core density change (kg/m3) vs. time (s)

Figure 13.22: Power vs. time (s)

238



Figure 13.24: Average Doppler Temperature (°C) vs. time (s)

239

13.6. Application of the coupling to SFR


13.6 Application of the coupling to SFR

Finally, the performed developments are applied to a SFR core according to the specica-tions provided by PSI. However, since some information regarding thermal properties andconductivities of the materials were missing, the results could not be properly benchmarked.Therefore, the aim of this section rather than verifying the performance of the recent de-velopments for fast reactors, is to demonstrate the applicability of the methodology to fastreactors.

13.6.1 Steady states

The steady state proposed correspond to the hot state, with the control rows positiondened as follows:

• 1st and 2nd ring at the upper parking position. 3rd ring 30 cm inserted in fuel region

Results are presented in Table 13.11, for both the pure CMFD method, and the samemethod but with a nodal correction in the fuel nodes.

13.6.2 Transients

Finally, the last step will be to verify the performed developments for its application toa transient scenario of a SFR core. However, given the lack of the required benchmark

240


Ke Fz FxyTdop Tmax Tdop max

ºC ºC ºC

CMFD 0.99937421 1.246 1.37166249 1024.16 2146.33 1428.75

Fuel corrected0.99869455 1.244 1.36919506 1024.15 2146.32 1428.56

CMFD

Table 13.11: Steady state for the ESFR core

Figure 13.26: Detail of the CAs to be withdrawn

specications to properly compare and verify the results, this last step cannot be completed.

Nevertheless, as a preliminary proof that the coupling worked with the new CMFD method-ology applied to a SFR transient scenario, a runaway of some specic control rods wassimulated.

For this preliminary test, the same SFR core as for the steady state cases was used, withthe nodal cross sections provided by PSI. The transient denition consists in the withdrawalof a certain group of 6 control assemblies (CAs), marked with red circles in Fig. 13.26.The rods move 30 cm in 7.4 s, i.e., with a speed of 4 cm/s.

Since some thermal-hydraulic details were not available, the SUBCHANFLOW default cor-relations employed for the sodium and typical MOX fuel were used.

Some results of the transient, such as Doppler temperatures and average density, are shownin Figures 13.27 and 13.28, respectively. However, after looking at the results for the powerand reactivity, an issue with the rodcusping treatment was observed, which should beaddressed in future works.

Nevertheless,the objective of this exercise was not the assessment of the transient results,

241


Figure 13.27: Average Doppler Temperature (ºC)

but the verication that the system works with the new developments just implemented.Due to several circumstantial constraints, the few adjustments in the input parameters thatwould be required in order to obtain realistic results will be left as future works.

242


Figure 13.28: Average density (kg/m3)

243


244

Part VI

CONCLUSIONS

245

Conclusions

The thesis herewith presented deals with the simulation tools required for fast reactorapplications. It can be divided into three dierent parts.

• The rst one includes the assessment of dierent capabilities of Sodium Fast Reactors,i.e., assessment of minor actinide transmutation capabilities and core performance

• The second one deals with the generation of neutron core parameters required for fastreactor simulations, i.e. point kinetic parameters and multigroup nodal cross sections

• The last part includes the dierent developments performed in ANDES in order tomake it suitable for fast reactor calculations.

For all those purposes, a series of core models were specically developed for dierentcongurations of the ESFR core and for dierent codes

Assessment of capabilities of Sodium Fast Reactors

First the transmutation capabilities were assessed in order to conrm the potential elim-ination of minor actinides achievable in such systems. In order to do that, three corecongurations with dierent contents of minor actinides were studied: a reference one, aone homogeneously loaded with 4% of MA and a third one with a heterogeneous loadingof 20% of MA located in an outer ring.

It was concluded that while the core without MA is a net MA producer, the other congura-tions are able to eliminate minor actinides with dierent level of eciency. In particular thehomogeneous case is able to eliminate an important amount of MA, specically americiumand neptunium. Concerning curium, there is a net accumulation in spite of the importantamount of Cm ssioning. As the management of curium entails diculties and costs, acareful analysis of possibilities taking into account Cm-244 (18.11 years half-life) duringstorage is required.

However, loading the core with minor actinides brings many diculties. It deteriorates thecore performances and safety characteristics, as it is pointed out while comparing the corereactivity coecients. Nevertheless, it was proven that the heterogeneous loading is notthat sensitive to the deterioration driven by the introduction of minor actinides. Indeed,when the reactivity coecients are compared with the other congurations, it is observedthat those are quite similar to the ones of the reference core, without minor actinides.

247

In what concerns comparison of codes, a great special care was taken to develop BOLsimilar models: same core geometries, initial fuel and material compositions and masses,neutron libraries and working temperatures. Then, results predicted by EVOLCODE andSERPENT codes are very close in ke estimations before activation of burnup models.After a typical long irradiation period of 2050 EFPD, when a similar reasonable assumptionis taken concerning the Am-241 capture branching ratio, discrepancies are quantied inthe order of 3% concerning actinide isotope masses, 600 pcm as much in ke, and 10%concerning reactivity parameters. The dierent corrector-predictor approaches in the fueldepletion model most likely explain the observed dierences.

In addition to the transmutation assessment, an analysis of a Spanish energy scenario usingthe Gen- IV SFR reactor has been performed in terms of electricity production, availabilityof resources and wastes reduction.

Three sensitivity studies to the required eet of ESFR-like reactors have been done: dierentelectricity demand scenarios, thermal to electrical conversion eciencies and shares coveredby nuclear electricity. It was conrmed that the most relevant factor is the electricitydemand.

To study deeply some parameters, a reference scenario has been xed with the hypothesisof 1.5% of annual growth of the demand, 38% of thermal eciency and 20% of nuclearpower. For this reference scenario, the availability of the most representative resources, i.e.uranium (U-238) and plutonium (Pu-239), has been assessed.

For uranium, it can be concluded that there is no limitation for the exploitation of theproposed eet for the reference case scenario assuming dierent alternatives in the waysthe uranium might be obtained. When the study is extended to further energy scenarios,a limitation appears in the most restrictive scenarios. In case of using only resources fromnational uranium mines, there would not be enough uranium to feed the required eet forthe most demanding energy scenarios. Nevertheless, if reprocessing is possible and theoperational life is extended to 60 years, there would not be any limitation to feed therequired eet by any of the postulated scenarios.

Regarding plutonium, it can be concluded that as the only way to obtain Pu is fromthe reprocessing of the spent fuel, there is not enough plutonium to feed the completeeet. However, thanks to the breeding properties of the ESFR-like reactors, a strategyinvolving the continuous reprocessing of plutonium has been proposed in order to obtainenough plutonium to feed reactors' cycles until the desired production level is achieved.The real power production capacity has been assessed for the reference scenario accordingto dierent parameters. It is concluded that for the CONF2 core it would be possible toachieve the reference case scenario energy demand, as far as the on-the-y reprocessingfrom the 60-years of operational-lifetime Gen-II and future Gen-III is considered.

In addition to those studies, the capability of reducing the quantity and decay heat of themost radioactive isotopes presented in the spent fuel, i.e. the minor actinides, is assessed.The study concludes that the whole decay heat of the MA is reduced after 60 years whenfuel with MA is burned in the reactor, compared with the case when it is left to decayin a temporal storage centre. The main responsible of this result is the reduction of Amisotopes when the fuel is burned.

248

Generation of neutron core parameters

A methodology to compute point kinetics parameters using SERPENT has been describedand applied to the study of the impact of minor actinides on transient behaviour for fastreactors. This work was performed within the framework of the task 3.1.1 of the SP3 ofthe CP-ESFR project. It was concluded that loading the core with minor actinides inducesa worst respond to transient scenarios. In addition, further assessments regarding thevalidation of the hypotheses of superposition and linearity were performed, which conrmtheir reliability. Moreover, a parallel study of the impact of uncertainties in the nuclear dataon the reactivity coecients was carried out using the SCALE system.

On the other hand, the methodology to generate multigroup nodal cross section usingSERPENT has been described. The same task was previously assessed using ERANOScode, and a comparison between both contributions has also been presented.

Developments in ANDES

First, the ANDES solver was assessed for its application to sodium fast reactors. TheACMFD (Analytical Coarse Mesh Finite Dierence) method implemented in ANDES cor-responds to an Analytical Nodal Method (ANM) and consists in two steps: diagonalizationto decouple energy groups and transverse leakage integration to transform the 3D equationinto three 1D equations. The rst step has been identied as the main responsible of theconvergence diculties when a ne energy group structure is employed. Indeed, the ma-trixes involved in the problem have been proven to be non-diagonalizable for some non-ssilenodes when the typical 33 energy group structure is employed. In fact, after revisiting theliterature, it has been veried that for multigroup problems (>2 groups), NEM methodsare used instead.

On the other hand, the way the transverse leakage integration is treated by ANDES forhexagonal geometry leads to a very complex methodology, strongly non-linear. As a conse-quence, the system becomes very unstable and dicult to converge. This fact is enhancedwhen big and heterogeneous cores are used, such as the case of the ESFR core.

As a consequence, it is concluded that the ACMFD method is not suitable for fast reactorcalculations.

Then, in order to solve those limitations, further developments were assessed and imple-mented in ANDES.

In particular, the corrected CMFD method is proposed in order to overcome the limitationslinked to the transverse leakage integration. It consists in applying the CMFD methodto the whole core, and using a correction factor, which force the interface currents to beidentical to the ones obtained for the two-node problem, when the advance nodal methodis employed. In this case, the ACMFD like method is selected for consistency. This newmethodology was implemented in ANDES and satisfactorily veried.

As for the limitations related to the use of a ne energy group structure, an alterna-tive method to the diagonalization was assessed: the Schur decomposition. However,

249

after developing the formulation and implementing it in ANDES, it was found that theill-conditioning of the matrixes persisted. This is a consequence of not considering inthe formulation the presence of identical eigenvalues, which would allow us to reduce theconditioning number of the matrix. Should the identical eigenvalues be considered, theformulation would have been too complex, leaving it far from the scope of the thesis. As aconsequence, in order to work in the typical 33 groups, the correction could only be appliedto fuel nodes, where the ill-conditioned matrixes are not present.

Nevertheless, the results of ANDES applied to a SFR using the previously described method-ology were compared with the ones from codes such as PARCS and DYN3D, showing apretty good agreement.

In addition to those developments, the neutron/thermal-hydraulic coupling between AN-DES and SUBCHANFLOW was satisfactorily performed. It was rstly veried for thermalreactors through a collection of international benchmarks in both Cartesian and Hexagonalgeometry, showing a really good agreement for all the cases under study. Then, in orderto apply it for fast reactors, new developments were required, mainly related to the intro-duction of new cross section feedback eects. Thus, the coupling was applied to SFRs,although it could not be contrasted with benchmarks due to the lack of some of the requiredinformation.

All in all, it can be concluded that several implementations have been included in the ANDESsolver in order to adapt it for SFR applications. While the problems with the transverseleakage have been satisfactorily addressed, the problem with the 33 energy group persists aslong as the ACMFD-like methodology is used in the non-ssile nodes. Due to the importanceof choosing a representative group structure, the option of collapsing the energy structureinto a broader one is rejected. Thus, in order to use ANDES with an accurate enoughgroup structure, such as the 33 energy groups, dened for ERANOS, the correction to theCMFD method can just be applied to the fuel nodes. Nevertheless, thanks to the greateorts performed in the code, the new developments (suggested in the next section) arenow ready to be implemented, without much diculties.

Future works

As future works, concerning the assessment of minor actinides transmutation, the heteroge-neous loading would need further studies to come up with an optimal conguration whichenable high MA transmutations rates. On the other hand, regarding the cross section gen-eration, further developments could include sensitivity analyses to the group structure inorder to limit the eect of the thermal range. Moreover, the methodology proposed wouldneed to be extended in order to account for changes on feedback parameters, such as fueltemperature, sodium density and also geometry variations.

As for the developments in ANDES, in order to solve the limitations linked to the use ofthe energy group structure in 33 groups, two possible paths are proposed:

• In case of keeping the ANM method for multigroups, it is suggested to develop amethodology to deal with almost identical eigenvalues for the Schur approach.

250

• Should the ANM method be abandoned, the NEM method should be implementedinstead for the 2 node problem in conjunction with the CMFD method, using thedevelopments already performed in ANDES.

In case that the diusion approach is to be avoided, new paths are open:

• Use a transport deterministic or Monte Carlo code for the 3D steady state calculation,which will initialize the transient and then use ANDES for the transient analysis.

• Modify the resolution method in ANDES including a transport approach of a higherorder than the diusion one. For whole core calculations, the use of low order trans-port approaches such as SPN , have proven to improve signicantly the solution ac-curacy [127], with a computational consume much lower than the discrete ordinatesmethods (SN) or spherical harmonics (PN). The SPN was derived initially from anad-hoc formulae substituting the derivatives of the one-dimensional PN equation inat geometry with the three-dimensional Laplacian operators [61]. That substitutionresults in a multidimensional generalization of the one-dimensional PN equations,which avoids the complexity of the complete spherical harmonics approach. Sincethe theoretical basis of the SPN equations was rigorously established [100], its ap-plication has become more popular, and dierent codes based on this method havebeen developed so far for thermal reactor applications[130, 9, 7, 128]. Moreover,given that the transport eect is more noticeable for fast reactors than for thermalreactors, the SP3 approach could be considered as a promising tool for fast reactoranalysis.

Finally, additional developments could include:

• To explore the introduction of discontinuity factors, to overcome the dierences withthe transport approaches.

• Introduction of burnup capabilities in ANDES.

251

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