monte carlo simulation of bowing effects in nuclear fuel...

UPTEC F 18062

Examensarbete 30 hpJanuari 2019

Monte Carlo Simulations of Bowing Effects Using Realistic Fuel Data in Nuclear Fuel Assemblies

Marcus Westlund

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Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Monte Carlo Simulations of Bowing Effects UsingRealistic Fuel Data in Nuclear Fuel Assemblies

Marcus Westlund

Deformations of nuclear fuel assemblies have been observed in nuclear powerplants since the mid-90s. Such deformations are generally called bowing effects.Fuel assemblies under high irradiation undergo growth and creep induced by highloading forces and low skeleton stiffness of the assemblies which gives permanentdeformations and modifies moderation regions. Hence, giving an unpredictedneutron flux spectrum, power distribution, and isotopic concentrations in the burntfuel. The aim of this thesis is to study the effects of local fuel bowing in terms ofpower distribution and isotopic composition changes through simulations of thereactor core.

The reactor is simulated with realistic bowing maps and previous deterministicallysimulated realistic fuel data from a present reactor by deploying the Monte Carlomethod using the nuclear reactor code Serpent 2. Two subparts of a full reactorcore with fuel from separate fuel cycles are investigated in 2D using burnup. Toquantify the impact of the bowing, the change in power distribution and the inducedisotopic composition change are calculated by a relative difference between anominal case and a simulation with perturbed fuel assemblies. The results arepresented in colormaps, for visualization. The isotopic composition for U235, U238,Pu239, Nd148, and Cm244 are investigated.

Also, statistical uncertainty estimations in the composition of the depleted fuel aredone by multiple calculations of the same geometry while changing the seed ofrandom variables in the Monte Carlo calculation. The mean value and the standarddeviation in the mass density of U235 and Pu239 are calculated for two pinstogether with histograms with a normal fit for each case to clarify the mathematicaldistribution of the calculations.

The simulations performed in this thesis have detected clear impacts of the reactorbehavior in terms of power distribution and isotopic composition in the burnt fuelintroduced by the bowing. Assembly perturbations of about 10 mm may locallyintroduce a 10 % relative difference in power density and U235 content between thenominal and the bowed case at 15 MWd/kgU burnup. The power and the isotopiccomposition changes agree with expectations from the bowing maps.

ISSN: 1401-5757, UPTEC F 18062Examinator: Dr. Tomas NybergÄmnesgranskare: Dr. Henrik SjöstrandHandledare: Dr. Dimitri Rochman

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Sammanfattning

Deformationer av bränslestavarna i en kärnreaktor är ett problem som varit käntunder de senaste årtiondena. Dessa deformationer refereras generellt som böjning,med olika tillägg för om det gäller en tryck- eller kokvattenreaktor. Första gångendet officiellt observerades var i Ringhals 4 i södra Sverige. Anledningen till dettavar bland annat att bränslestavarnas fästpunkter och materiella uppbyggnad intevar tillräckligt styv för att motstå de axiella krafter som uppstår under reaktornsgång. Åtgärder vidtogs som till viss grad förbättrade dessa fenomen men proble-men kvarstår fortfarande och beror delvis på att bränslet utnyttjas till högre grad påsenare år. Deformationer och förändringar i bränslets placering påverkar och förän-drar neutronflödet i reaktorn. Det är fissionen, eller som det kallas kärnklyvningen,som skapar energin i en kärnreaktor. Detta sker genom att en neutron absorberasav en uran-atom, som blir ostabil och tillslut delar sig och samtidigt avger 200 MeVenergi. Detta måste ske på ett kontrollerat sätt för att säkerställa en säker drift avreaktorn. När neutronflödet förändras påverkas i sin tur sannorlikheten för fissionvilket leder till att den lokala effektfördelningen i reaktorn ändras. Förutom vissasäkerhetsrisker, ger detta upphov till en förändrad isotop-sammansättning av detanvända bränslet. I slutändan leder detta till att det förbrukade bränslet kommerha en annan isotop-koncentration än vad reaktorn är designad för. På grund av ra-dioaktiviteten är detta svårt att kontrollera i efterhand och är inte möjligt att görasför allt bränsle. Osäkerheterna i det förbrukade bränslets sammansättning betyderatt längden för slutförvar av bränsle kommer justeras. I detta examensarbete hardeformationer av en schweizisk tryckvattenreaktor (PWR - Pressure Water Reac-tor) studerats med böjningsdata vid två olika cykler av reaktorn. Neutronflödet harsimulerats med den så kallade Monte Carlo metoden genom att använda Serpent 2,en kod för reaktorfysik. Genom att skriva inputfiler till Serpent kan geometrier ochinställningar för den aktuella reaktorn bli definierad. Speciell vikt har lagts vid attstudera relativa förändringar i effekt- och isotopkoncentration-fördelning genom attberäkna den relativa förändringen av en nominell (utan böjning) och en beräkningdär böjning introducerats. Två olika delar av reaktorn som uppvisar skilda böjn-ingsfenomen studeras.

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AcknowledgementsThere are a few people I would to thank a little extra. Fist of all, my girlfriend for lettingme go to Switzerland for 3 months. Dimitri, my sepervisor for all the help and to always beready to show some scripting tricks and all other colleagues that I met at PSI. Henrik, mysubject reviewer that introduced me for the opportunity to do the Master thesis at PSI and toalways be up for a quick meeting. . .

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Contents

Acknowledgements v

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Reactor physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Nuclear cross sections . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Neutron transport equation . . . . . . . . . . . . . . . . . . . . . 31.3.3 Bateman depletion equation . . . . . . . . . . . . . . . . . . . . . 41.3.4 Nuclear data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Monte Carlo codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4.1 Serpent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Bowing effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5.2 Bowing effects on a PWR - mechanisms and consequences . . . 7

1.6 General methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Bowing effects on 5 x 5 fuel assemblies 112.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

SERPENT input and fuel data preparation . . . . . . . . . . . . 112.1.1 SERPENT settings . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Data libraries and materials . . . . . . . . . . . . . . . . . . . . . 142.1.3 Plotting of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Subpart 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.2 Subpart 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Discussion and conclusions 333.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

A The CRAM method 37A.1 CRAM method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

B Serpent input files 39B.1 Pin set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

C Scripting and written codes 45C.1 Preprocessing of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45C.2 Post-processing of results . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Bibliography 53

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List of Figures

1.1 An arbitrary volume V with surface area S used to illustrate the neu-tron transport (James J. Duderstadt, Louis J. Hamilton, 1976). . . . . . 3

1.2 An illustrative PWR fuel assembly from Westinghouse with 17x17pins (Dahlheimer et al., 1984). . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 A sketch of different bowing shapes (Li, 2016). . . . . . . . . . . . . . . 8

2.1 Left: A geometry output of the nominal 5x5 assemblies for one sub-part studied. Right: A geometry output with the perturbed 5x5 as-semblies from one of the bowing maps. . . . . . . . . . . . . . . . . . . 12

2.2 Left: The bowing map for subpart 1 of the reactor core. Right: Thebowing map for subpart 2 of the reactor core. . . . . . . . . . . . . . . . 13

2.3 The yellow arrows show the direction and amplitude of the bowing,the scale is noted in the lower left corner. Left: The relative powerdifference between the nominal and the bowed assemblies for bur-nup at 0.1 MWd/kgU of subpart 1. Right: The relative power differ-ence between the nominal and the bowed assemblies for burnup at 15MWd/kgU of subpart 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 The yellow arrows show the direction and amplitude of the bowing,the scale is noted in the lower left corner. Left: The relative isotopicmass density difference of U235 between the nominal and the bowedassemblies for burnup at 0.1 MWd/kgU of subpart 1. Right: The rela-tive isotopic mass density difference of U235 between the nominal andthe bowed assemblies for burnup at 15 MWd/kgU of subpart 1. . . . . 18

2.5 The relative difference in isotopic mass density between the nominaland the bowed assemblies with burnup. The results are for 4 pins inthe central assembly and the most affected region. . . . . . . . . . . . . 19

2.6 The relative uncertainty in power change for figure 2.3 at 15 MWd/kgUburnup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 A Serpent output plot of the power distribution within subpart 1 at15 MWd/kgU. The power distribution is normalized, darker regionscorresponds to less power and less fission and brighter to regions tohigher power and more fission events. . . . . . . . . . . . . . . . . . . . 21

2.8 The relative uncertainty in power change (from 2.3) with burnup for4 pins in a region of high/low absolute power. . . . . . . . . . . . . . . 22

2.9 The relative difference in mass density as a function of the pin posi-tion. Pin 1 is at the assembly edge and pin 7 is at the center. . . . . . . . 23

2.10 The relative difference in mass density for U238, Pu239, Nd148, andCm244 for burnup step 10 at 15 MWd/kgU. . . . . . . . . . . . . . . . . 24

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2.11 The yellow arrows show the direction and amplitude of the bowing,the scale is noted in the lower left corner. Left: The relative powerdifference between the nominal and the bowed assemblies for bur-nup at 0.1 MWd/kgU of subpart 2. Right: The relative power differ-ence between the nominal and the bowed assemblies for burnup at 15MWd/kgU of subpart 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.12 The yellow arrows show the direction and amplitude of the bowing,the scale is noted in the lower left corner. Left: The relative isotopicmass density difference between the nominal and the bowed assem-blies for burnup at 0.1 MWd/kgU of subpart 2. Right: The relativeisotopic mass density difference between the nominal and the bowedassemblies for burnup at 15 MWd/kgU of subpart 2. . . . . . . . . . . 26

2.13 The relative difference in isotopic mass density for the nominal andthe bowed assemblies with burnup for 4 pins in the central subpartand the most affected region. . . . . . . . . . . . . . . . . . . . . . . . . 26

2.14 The relative uncertainty in power change for subpart 2 in figure 2.11at 15 MWd/kgU burnup. . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.15 The relative uncertainty of power change for 4 pins in a lower and ahigher power region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.16 The Serpent output for the normalized absolute power distributionwithin subpart 2. Darker regions corresponds to less power densityand less fission, the opposite applies for brighter regions. Left: 0.1MWd/kgU burnup. Right: 15 MWd/kgU burnup. . . . . . . . . . . . . 28

2.17 The relative difference of mass density as a function of the pin posi-tion. Pin 1 at the assembly edge and pin 7 in the center. . . . . . . . . . 29

2.18 The relative difference of mass density for U238, Pu239, Nd148, andCm244 at 15 MWd/kgU burnup of subpart 2. . . . . . . . . . . . . . . . 30

2.19 Four histograms of 50 independent simulations for the mass densitiesof U235 and Pu239 in two separate pins. (A) for U235 in the lower, (B)for U235 in the top, (C) for Pu239 in the lower and (D) for Pu239 in thetop part of the central assembly. The mean value and the standarddeviation are calculated for each case. . . . . . . . . . . . . . . . . . . . 31

B.1 The assembly-core definition for one assembly. . . . . . . . . . . . . . . 39B.2 The pin geometrics and the material definitions (A) and the division

setting for a few pins (B). . . . . . . . . . . . . . . . . . . . . . . . . . . . 39B.3 (A) the fuel material card definition with normalized isotopic densi-

ties for one pin and (B) the material definitions for a few pins. . . . . . 40

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List of Abbreviations

MC Monte CarloFA Fuel assemblyJEFF Joint Evaluated Fission and FusionENDF Evaluated Nuclear Data FileCRAM Chebyshev Rational Approximation MethodIAEA International Atomic Energy AgencyNDS Nuclear Data ServicesNEA Nuclear Energy AgencyTENDL TALYS-based evaluated nuclear data library

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Chapter 1

Introduction

This chapter introduces a general background, the purpose and aim for this thesis, abrief history of the field together with bowing observations.

The chapter will also cover general theory related to reactor physics and specifictheory of the simulation code used.

1.1 Background

Deformation of fuel assemblies in nuclear power plants has been observed in oper-ating reactors around the world since the mid-90s. Ringhals 4 was the first wherethis phenomenon was publicly reported in 1994 (Inozemtsev, 2010). One control rodwas not fully inserted during a reactor trip and four Rod Cluster Control Assemblies(RCCAs) were stuck in the dashpoint region during a drop down test (Inozemtsev,2010).

For pressurized water reactors (PWR) these deformations are called assembly bow-ing and for boiling water reactors (BWR) channel or box bowing (Inozemtsev, 2010).Deformations of the fuel assemblies (FAs) might lead to contact between subassem-blies, introducing sub-channel moderation regions. As a result, this modifies theneutron flux spectrum around the fuel assemblies giving an unexpected power dis-tributions of the reactor core. Moreover, power anomalies impact the isotopic con-centrations in the burnt fuel giving unpredicted contents in the spent fuel which inthe long run may affect storage limits in final depositories.

Fuel bowing effects are well studied and these effects have previously been simu-lated using fresh fuel data, one example is (Li, 2016). As a continuation, bowingeffects are now studied using realistic fuel data from a present Swiss nuclear reactor.Spent fuel in Sweden and Switzerland are used to determine the number of assem-blies that can be stored in final depositories. Uncertainties in the spent fuel contentare translated into storage limits. This study is meant to be part of this research byinvestigating and developing a method for simulating the impacts of the fuel bow-ing with realistic fuel data and geometrics from a present reactor.

The thesis was proposed by Paul Sherrer Institute (PSI) in Villigen, Switzerland as acollaboration with Uppsala University in Sweden. The main work has been done atPSI facility at the Reactor Physics and Systems Behaviour Laboratory.

2 Chapter 1. Introduction

1.2 Goal

The aim of this thesis is to achieve a better understanding of the change in neutronflux and power density and predict their consequences due to local fuel bowing ef-fects using the Monte Carlo method with realistic fuel and bowing data. The goal isto predict the change in power distribution and the effect on the isotopic composi-tion of the burnt fuel.

Generally, there is a need to predict the fuel content (actinides, and important fis-sion products) at the end of the assembly life, after the last irradiation.

1.3 Reactor physics

The basic concept of nuclear reactor design is to create an environment where a con-trolled and sustained nuclear chain reaction can take place (Bahman, 2017). In general,slow (thermal energy) neutrons can initiate fission by neutron capture of a U235 nu-cleus. The fission process produces fission products, fast neutrons and∼ 200 MeV ofadditional energy. The newly released neutrons are essential for the chain reactionto take place but the probability for fission induced by fast neutrons are very low. Inorder to slow down the neutrons, one adds a moderator material inside the reactor,which may consist of water. When the neutrons interact with the molecules of themoderation medium, they lose energy, and in the end neutrons of a certain energyrange will initiate fission once again.

Moreover, the reactor design need to consider the production rate of neutrons andbalance it by the absorption and leakage of neutrons out of the reactor system (JamesJ. Duderstadt, Louis J. Hamilton, 1976).

Some basic concepts (relevant for this thesis) and necessities for a self-sustainednuclear chain reaction is covered in this section, although some are omitted (mul-tiplication factor, criticality etc.) due to less relevance for this thesis.

1.3.1 Nuclear cross sections

In nuclear physics, the concept of cross sections describes the fundamental probabil-ity of a certain process (Wikipedia, 2017). Nuclear cross sections are divided intomicroscopic and macroscopic cross sections. Microscopic cross sections describe theprobability for particular events of a single nucleus, like fission, elastic and inelasticscattering or neutron capture. It measures in the unit of barns, where one barn equals10−24cm2 and it is denoted as σ.

Macroscopic cross sections describe the probability of a particular reaction character-ized by the target (and the macroscopic chunk of material) (James J. Duderstadt,Louis J. Hamilton, 1976), the unit is cm−1 and is denoted Σ. The macroscopic crosssection is calculated via (Bahman, 2017),

Σ = Nσ, (1.1)

where N is the atomic density of a particular nuclide.

1.3. Reactor physics 3

Large libraries containing nuclear cross sections of different nuclides over a wideenergy rage are used in nuclear physics and are important in reactor design. Suchlibraries and the continues development of them are further described in 1.3.4.

1.3.2 Neutron transport equation

The nuclear reactions and fission events inside a nuclear reactor and the rates atwhich these reactions take place depend on the neutron distribution inside the reac-tor core (James J. Duderstadt, Louis J. Hamilton, 1976). The neutron distribution isdetermined by studying the neutron transport with loss and gain mechanisms of neu-trons. By representing the neutron population as a dilute gas and with kinetic theoryof statistical mechanics, the distribution can be calculated using the Boltzmann equa-tion. However, the Boltzmann equation is non-linear and therefore hard to solve. Inreactor physics one often uses its linear counterpart, the neutron transport equation, todetermine the neutron distribution (James J. Duderstadt, Louis J. Hamilton, 1976).

FIGURE 1.1: An arbitrary volume V with surface area S used to illus-trate the neutron transport (James J. Duderstadt, Louis J. Hamilton,

1976).

Generally the population of neutrons inside a reactor are described by gain andloss mechanisms, meaning neutrons that entering or leaving the reactor system (JamesJ. Duderstadt, Louis J. Hamilton, 1976). Imagine an arbitrary volume V with surfacearea S and neutrons travelling in direction Ω, seen in 1.1. The gain and loss mecha-nisms for a neutron of a specific energy E entering or leaving V are then translatedinto:

Gain mechanisms:

i. Neutron sources in V (fission)

ii. Neutrons entering the system

iii. Neutron scattering that changes the neutron energy into the point of interest


Loss mechanisms:

v. Neutron leakage out of V

vi. Neutron scattering changing the energy from the desired energy E to E′ orneutron absorption.

A balance relation is created by balancing the different mechanisms by whichneutrons are gained or lost within the system (James J. Duderstadt, Louis J. Hamil-ton, 1976), the relation is later translated into the Boltzmann equation (or the neutrontransport equation in this case),

the rate of change of neutrons inside V = i + ii + iii− iv− v (1.2)

For an arbitrary volume V in 1.1 and using 1.2 one finds for the neutron transport(James J. Duderstadt, Louis J. Hamilton, 1976),

∫V

d3r

[∂n∂t

+ vΩ·∇n + v ∑t

n(r, E, Ω, t))

−∫ ∞

0dE′

∫4π

dΩ′v′∑S(E′ → E, Ω′ → Ω)n(r, E′, Ω′, t)− s(r, E, Ω, t))

]dEdΩ = 0,

(1.3)

Since the volume V is arbitrary chosen one can make the integral vanish byputting its integrand to zero,∫

anyVd3r f (r = 0)⇒ f (r ≡ 0). (1.4)

The balance relation of the neutron transport equation is written,

∂n∂t

+ vΩ·∇n + v ∑t

n(r, E, Ω, t) =∫4π

dΩ′∫ ∞

0dE′v′∑

s(E′ → E, Ω→ Ω)n(r, E′, Ω′, t) + s(r, E, Ω, t).

(1.5)

Even with the addition of adequate initial and boundary conditions the neutrontransport equation is often impossible to solve analytically for most reactor typestructures due to its seven independent variables x, y, z, θ, φ, E, t (James J. Duder-stadt, Louis J. Hamilton, 1976). Normally, it is approximated by deterministic meth-ods using the neutron defuse model (James J. Duderstadt, Louis J. Hamilton, 1976) ormodelled based on stochastic Monte Carlo methods (nuclear-power, 2018) . In thisthesis the neutron transport is solved with a Monte Carlo code, described in 1.4.1.

1.3.3 Bateman depletion equation

The isotopic composition inside a nuclear reactor is constantly changing due to fis-sion, decay and production of new isotopes. A model for the decay chain was pos-tulated by E. Rutherford in 1904-1905 (Rutherford, 1904). The equation was latersolved by H. Bateman in 1910 (Bateman, 1910).

To ensure a safe operation of a nuclear reactor it is important to predict its behaviour

1.4. Monte Carlo codes 5

by estimating nuclei concentrations. This is done via the Bateman depletion equation(Bateman, 1910) ,

dNj

dt= ∑

i 6=j(Si−>j − λjNj − φσjNj), (1.6)

Si−>j is the source term and the production rate of nuclide j, λjNj is the radioac-tive decay and φσjNj is the rate of transmutation of nuclide j, including fission. Dif-ferent methods of solving this equation are implemented in nuclear reactor codes.One method that Serpent uses is further described in Appendix A.

1.3.4 Nuclear data

The IAEA (International Atomic Energy Agency) serves as an international coordi-nator for scientific collaborations and activities within the nuclear field under theUnited Nations regime (World Nuclear, 2018). There are many different nuclearorganizations around the world, some are responsible for nuclear research and co-operations and others based on regulatory purposes. A few of them are internationalbased, whereas others are country specific.

One example of such an activity is the continuous development of reliable and im-proved nuclear data libraries. Reliable nuclear data is essential for nuclear energyproduction, research and waste management but are also used in medical dosime-try and diagnostics, laser and accelerator applications, environmental monitoring,and fusion energy research (IAEA/NDS, 2018). Thus, a continuous development ofbetter nuclear data for future libraries is still important. Two organisations responsi-ble for development of nuclear data and establishment of international networks arethe NDS (Nuclear Data Section) though the IAEA (NDS, 2018) and the NEA (Nu-clear Energy Agency) (NEA, 2018) based on the OECD countries. A few examplesof nuclear data libraries are JEFF (Joint Evaluated Fission and Fusion) (Nuclear En-ergy Agency, 2018), ENDF (Evaluated Nuclear Data File) (NNDC, 2018) and TENDL(TALYS-based evaluated nuclear data library) (Koning and Rochman, 2012).

The libraries cover different aspects depending on the purpose. Some cover nu-clear structure and decay data whereas others contain cross section for fundamentalcollision processes.

In this thesis is the U.S library ENDF/B-VII.1 (Chadwick et al., 2011) used, wherethe Cross Section Evaluation Working Group (CSEWG) in nuclear data evaluation isresponsible for the evaluation.

1.4 Monte Carlo codes

The particle reaction and transport in the reactor core is simulated with the MonteCarlo method. There are several public available Monte Carlo codes for reactor cal-culations, such as MCNP (Los Alamos National Laboratory, 1957). A relatively newone is Serpent (Leppänen et al., 2015), written at VTT Technical Research Center inFinland by Jaakko Leppänen. Serpent 1 was first distributed by the OECD/NEAData bank in 2009 and is nowadays widely used at institutions and in researchworld-wide (Leppänen et al., 2015)


Due to limitations of Serpent 1, Serpent 2 (montecarlo.vtt, 2018) was initiated andis still under continuous development. Serpent 2 is more adopted for three dimen-sional analysis and is not only suitable for reactor physics. It has a possibility to cou-ple with multi-physics calculations like thermal hydraulics, CFD (Computer FluidDynamics) and fuel performance codes. Moreover, it can be used for neutron andphoton transport simulations for a range of applications, such as medical physicsand fusion research. Serpent 2 is used in this thesis.

1.4.1 Serpent

Serpent uses a continuous energy approach and solves two- and three-dimensionalparticle transport equations with the use of external nuclear data libraries containingcontinuous neutron energy cross sections, described in 1.3.4. Separate libraries areused for moderation regions containing thermal scattering cross sections (Leppänenet al., 2015). A Doppler broadening preprocessor routine adjusts the nuclide tem-peratures (Viitanen, 2009) and a Doppler broadening Rejection Correction (Becker,Dagan, and Lohnert, 2009) accounts for the temperature dependences in the reso-nant scattering kernels. Meaning that Serpent adjusts for irregularities in the actualtemperatures and the temperature specified by the nuclear data libraries.

The geometries can be defined in multiple layer structure where subparts are de-fined individually and later imported in the main input file (Leppänen, 2010). Thisfeature is adopted in this thesis which simplifies the file structure of the model.

The particle transport equations are solved using a combination of surface track-ing and Woodcock delta tracking methods (Woodcock, E.R., et al., 1965). The Bate-man depletion equation described in 1.3.3 is solved with two options, Transmuta-tion Trajectory Analysis (TTR) (Cetnar, 2006) or Chebyshev Rational Approxima-tion Method (CRAM) (Pusa, 2011). TTR solves the particle decays based on a linearchain method and provides the analytic solution of the depletion chains. CRAM (de-scribed in Appendix A by applying CRAM to first order linear equation) applies amatrix exponential method developed for Serpent. The method handles a system ofnuclides and accounts for short lived isotopes without any approximation for steplength or numerical precision.

Serpent is adopted for burnup calculations and can handle a large number of iso-topes and depletion zones in an efficient way (Isotalo and Aarnio, 2011). This is onereason why Serpent was considered in this thesis.

Most of the theory in this section apply for both Serpent 1 and 2.

1.5 Bowing effects

1.5.1 Observations

Incomplete Rod Cluster Control Assembly (RCCA) insertions (IRI) was reported byseveral operators around the world in the mid 1990s during emergency shuts downor drop time tests, developed for reactor safety analysis. The root cause reportedfor the fail of such tests was extensive fuel assembly deformations causing stickingof control rods during the drop down (Inozemtsev, 2010). As stated in the Back-ground section 1.1 such deformations are refereed as Bowing effects. The Bowing

1.5. Bowing effects 7

effects were first publicly reported for Ringhals 4 1994 (Inozemtsev, 2010) in Swe-den. Investigations showed that FAs (fuel assemblies) bowed in an S-shape withdisplacement amplitudes reaching 20 mm. Because of that, several monitoring andpreventive actions were implemented by the safety regulators. Such as limitation offuel burnup in the assemblies close to the RCCA position and introduction of ad-ditional drop tests at the end of the fuel cycle. Such drop tests of RCCAs are aneffective way to find deformation issues of the nuclear fuel (Inozemtsev, 2010).

In recent years, extensive investigations and safety arrangements have made thebowing effects of FAs less critical but FA shape measurements are still performed byutilities in the reactor core and storage pools (IAEA, 2005). A higher utilization ofthe nuclear fuel nowadays (related to economy etc.) impacts the bowing, since theamplitude of the displacement is related to the fuel burnup (among other things).Implying that, it is still important to study the consequences of fuel bowing effects.

1.5.2 Bowing effects on a PWR - mechanisms and consequences

A typical pressure water reactor (PWR) fuel assembly is shown in figure 1.2 froma Westinghouse 17x17 pin assembly design and with the corresponding definitionsof its parts. Although a 15x15 pin assembly design is used in this thesis, the mainfeatures are the same.

FIGURE 1.2: An illustrative PWR fuel assembly from Westinghousewith 17x17 pins (Dahlheimer et al., 1984).


Beside handling damages on fuel and grid spacers during transport, investiga-tions showed a few root causes to the fuel deformations. One major contributorwas low skeleton stiffness of the FAs together with high hold down forces from theending springs (IAEA, 2005) (Inozemtsev, 2010). This phenomenon is illustrated infigure 1.3 and shows the basic concept causing the deformations.

FIGURE 1.3: A sketch of different bowing shapes (Li, 2016).

It is observed that the bowing shape has a strong dependence of the FA locationwithin the core at the end of a fuel cycle whereas the burnup has a milder impor-tance to the global deformations. Hence, the bowing effects evolves as a stepwisefunction of its positioning in the core throughout its lifetime. As a consequence ofthis were fuel shifting schemes invented, relaxing such effects. The bowing effectsare basically seen as something that evolves with increased deformations over time.

Actions, such as design changes and improvement of structural materials in reac-tors and fuel assemblies were recommended by utilities. Generally, materials withenhanced mechanical properties were implemented by reactor operators and fuelmanufacturers based on the recommendations (Inozemtsev, 2010). A few actionsproposed by suppliers to enhance the structural behaviours and attenuate the bow-ing effect were an advanced cladding and guide thimble material with low growthrate and high creep resistance and increasing the grid width to reduce inter assem-bly lateral gaps (Inozemtsev, 2010). A low growth recrystallized zircaloy-4 claddingmaterial was implemented, which increased the skeleton stiffness.

As a consequence of the deformations, the friction between control rods and theguide thimbles may increase. Which in the end could break or cause sticking ofRCCAs during drop-down tests or result in incomplete RCCA insertion during anemergency shut down, as mentioned in 1.5.1.

1.6. General methodology 9

Moreover, the increased/ decreased water gap due to the bowing redistributes thelocal power distribution within the reactor core in respect with nominal conditionsand thus it is critical to assure that the perturbations in power do not exceed limita-tions in the reference safety analysis (Inozemtsev, 2010) of the reactor.

1.6 General methodology

Realistic data from an operating Swiss reactor is used both regarding fuel composi-tion and bowing data. The dimensions, the fuel data and direction and amplitudesof the bowing come from the same PWR.

The fuel contents are previously simulated for the full core using two determinis-tic codes (CASMO5 and SIMULATE3). The fuel data for a specific height of thereactor core are extracted by the supervisor of this thesis and the data are providedas a starting point. However, all nuclear data generated from the deterministic sim-ulations is not essential for this thesis and is therefore not used. The isotopic massdensities are used and are extracted from the simulated fuel data (at the specificheight), formatted in a Serpent readable way and used as input for the Monte Carlosimulations. Isotopic contents less than 10 g/t fuel are not considered and assumedto have a minor effect of the reactor behaviour.

Each pin within the subparts simulated is defined with its own unique fuel com-position and all calculations in this thesis are done in 2 dimensions using Serpent 2.Since already a 2D calculation with burnup and separate depletion zones in each pinis memory demanding, only two subparts of the full reactor core are chosen for thestudy and presented in this report. The Serpent input files are constructed in such away that these regions can easily be moved throughout the reactor core, thus moreinteresting parts can be isolated, such as larger bowing regions. The bowing mapsof these subparts are presented in chapter 2. Generally, the 2D bowing maps showthe extreme displacement in one plane of each FA together with its correspondingdirection.

By using Bash scripting, the fuel data for the subparts is extracted and formattedfor each pin which makes the analysis efficient and less time-consuming. The dis-placement of the bowing map is however read and updated manually. Examples ofthe Bash scripts are found in Appendix C.

All result data are post-processed with Bash and plotted in Matlab.

11

Chapter 2

Bowing effects on 5 x 5 fuelassemblies

This chapter covers the methodology used together with the main results.

2.1 Methodology

Two subparts of the full reactor core with different bowing characteristics are in-vestigated and presented in this report. The bowing maps 2.2 are measured fromdifferent fuel cycles, one quite recent and another a bit older, but both of them comefrom the same Swiss reactor.

SERPENT input and fuel data preparation

All fuel data for each FA used is first simulated and validated for the full core and thecorresponding fuel cycle. As stated in 1.6, these initial calculations were performedwith deterministic codes (CASMO5 and SIMULATE3) before the start of the thesis.The simulated fuel data for one assembly is stored into a separate file containing fuelcontent for each pin at a specific height of the reactor core. Fresh fuel is consideredfor the FAs that are being replaced in the present fuel cycle.

The amount of simulated nuclear- and isotopic-data is large and before one can useit in Serpent it needs prepossessing and formatting. Initially, the fuel mass densityis in the unit of g/t but it is normalized to 1 during the formatting. Serpent handles alarge number of different isotopes and their corresponding distribution better if theisotopic contents are first normalized. For simplicity, materials with low content areomitted and assumed to a have minor impact.

Each pin is separated from each other by creating individual names and depletionzones. All pins are associated with a unique file containing the fuel data. The extract-ing, the normalizing, the renaming of isotopes in Serpent readable way, and prepar-ing of the material card definition for each pin is programmed with Bash scriptingin the Linux environment. A few scripting examples are found in Appendix C anda file with fuel contents in Appendix B.

For simplicity, each fuel assembly is also separated in a two-file layer structure. Filesare constructed for material pin inputs, pin layouts and definitions and for deple-tion zones with the possibility to add several depletion zones of each pin. The pinradius and the assembly size and distances are set accordingly of the reactor underconsideration. Definitions are set for pins in a first layer. The pin definitions are col-lected and formed as assembly definitions in a second layer, forming assembly files.

12 Chapter 2. Bowing effects on 5 x 5 fuel assemblies

The assembly files are later imported into the main Serpent input file, where a 5x5assembly matrix and all specific simulation settings are defined. Examples of eachfile type and the main input file are found in Appendix B. Keeping a file structurelike this makes the structure easier to follow and possible errors easier to find.

For the chosen subparts, the 5x5 assembly matrix is first simulated with a nominaldistance of the adjacent FAs and later with a perturbed. The assembly perturbationsare in accordance with the bowing maps in figure 2.2 of each subpart. The physicalbehaviour of a bowed assembly in 2 dimensions is represented by using the nominaland the perturbed case and calculate the relative difference between in terms of thedesired quantities one like to study.

Figure 2.1 shows a geometry output plot of the 5x5 assemblies from Serpent. Thesmall circles are the fuel pins with random colors due to the specific fuel data defini-tion of each pin (a feature of Serpent). The blue background is water and the emptycircles are the channels for the control rods. To the left is the nominal case with theunperturbed assemblies and to the right the case with perturbed assemblies.

FIGURE 2.1: Left: A geometry output of the nominal 5x5 assembliesfor one subpart studied. Right: A geometry output with the per-

turbed 5x5 assemblies from one of the bowing maps.

The bowing maps for the two subparts under consideration are seen in figure2.2 with the scale of the perturbations marked in the lower left corner. As beforementioned these maps do not come from the same fuel cycle of the reactor. Meaningthat some assemblies are replaced or at least moved during their lifetime, accordingto the fuel shifting scheme of the reactor.

As seen, displacement measurements are not taken for each FA and even for the twosubparts under consideration some measurements are missing. Since these positionsare not measured, it does not mean that no displacements occurs at these positions.These assemblies are perturbed as the assembly straight above or below, wherevermeasurements exist. Additionally, for the methodology to work, the displacementin 2D cannot be greater than the corresponding distance to the adjacent assembly,since physically they cannot overlap. As confirmed in 2.2, most assemblies perform

2.1. Methodology 13

a collective displacement behaviour, although some deviations are seen. Some as-semblies at the left border in the left map of 2.2 move more than their inner neigh-bours, which is not physically allowed in 2D. To avoid collisions, these are perturbedas much as allowed by the distance of the adjacent FAs. In reality, the extreme dis-placement of the FAs in the bowing maps may occur at different heights, which arenot covered in the 2D plots. Meaning that the amplitude maxima of the s- and c-shape perturbations showed in figure 1.3 varies at different heights throughout thereactor core.

FIGURE 2.2: Left: The bowing map for subpart 1 of the reactor core.Right: The bowing map for subpart 2 of the reactor core.

2.1.1 SERPENT settings

Two general phenomena are investigated, power and isotopic composition changesinduced by FA displacements. The power calculation in Serpent is performed byplacing power detectors covering each assembly. These detectors produce an outputfile with power content and uncertainty for each pin.

The fuel is depleted with 10 burnup steps from 0.1 - 15 MWd/KgU and a set ofisotopes is defined in the Serpent input file, which is followed. Serpent generatesisotopic data with burnup in output files for the defined isotopes. Since all pins areseparately defined, the output allows a study of isotopic data with burnup for eachpin.

The nominal calculation, described in section 2.1, is first performed and later is thesame calculation performed again but with perturbed assemblies and a differentseed. The seed is changed manually to ensure that the random variables come froma different set.

A periodic boundary condition is used, periodic means that the defined geometry ofthe subpart is copied and repeated periodically at the boundary of each side of the5x5 assemblies.

Serpent performs the MC calculation in a set of cycles with the defined number ofneutrons for each set. The uncertainty of the calculation depends on the size of the


neutron population and the number of cycles. The number of neutrons and cycleshave been changed and steadily increased during the simulation process, in orderto improve the statistical uncertainties. However, due to the statistical importanceare all the simulations kept and used in the analysis. The results are weighted andaveraged according to the number of cycles and neutrons, as described in 2.1.3.

The assembly linear power density is set to 38.6−3kW/g. This value (the total av-erage power density in the system) is not substantial for a relative study but is usedas a normalization factor. Serpent provides normalized power data based on thisfactor in the result output files.

A possibility to investigate subdivisions of every pin are defined. Every pin aredivided into radial- or section-wise divisions (or both), allowing radial or azimuthalinvestigation with burnup. Results for this feature are not presented in this reportbut it may be used for pins of specific interest. This feature is preferably not used forall pins since sub-divisions in each pins are very memory demanding and all pinsare already defined with separate depletion zones.

The predictor-corrector method is used to solve the neutronics with depletion (bur-nup). A linear extrapolation on the predictor and a linear interpolation on the cor-rector in 10 steps (with the command set pcc leli 10 10) is used.

The material volumes are crucial for the burnup calculation and these are estimatedby sampling 106 random points inside the geometry (with the command set mcvol100000000).

2.1.2 Data libraries and materials

Light-water is used as moderator. Thermal scattering data for hydrogen at 600 Kis used via lwe7.12t in the ENDF/B-VII.1 evaluation (Chadwick et al., 2011). Thenuclear decay data for the burnup calculation and the neutron induced fission yieldsare used via endfb7.dec and endfb.nfy libraries in the ENDF/B-VII.1 evaluation.Zircaloy-4 at 610 K is used as cladding material (PNNL-15870, Rev.1).

2.1.3 Plotting of data

As described in section 2.1.1, are two general phenomena investigated, a power andan isotopic composition change induced by the bowing impact.

As discussed, Serpent produces output files with power distribution and isotopiccomposition data for each pin and burnup step. Results from several separate cal-culations of the same geometry but with different seed are combined to increasethe statistics. The nominal and the bowed simulations are combined independentlywith Bash scripting and one example is found in Appendix C. All results are plottedwith MATLAB.

One power detector is defined for each of the 5x5 assemblies which calculates thepower dissipation in each pin. The output from the power detectors are straightfor-ward to plot and one output file is generated for each burnup step.

One assembly is defined by 15x15 pins and the power detectors produces power

2.1. Methodology 15

outputs for those 225 positions. Thus, the positions for control rods with zero powerare also detected. However, these positions are distinguished by plotting them asblack squares in the color maps that show the relative difference between the nomi-nal and perturbed calculation. The color maps are seen in 2.2.

The power files contain matrices with normalized power densities and the corre-sponding relative power uncertainty of each pin. Results with different neutronpopulation and number of cycles, discussed in 2.1.1, are combined and weightedaccording to their significance. The total weighting factor for the number of cyclesand the number of neutrons are calculated as,

ωtot = ∑n(wc,nwp,n), (2.1)

for all simulations. Where ωtot is the total weighting factor of all simulationscombined, wc,n is the number of cycles and wp, n the size of the neutron populationof simulation n, respectively.

The average power density of all simulations is calculated as,

Pj,m =1

ωtot∑n

wc,nwp,naj,m,n, (2.2)

aj,m,n is the power density in pin j, burnup m and simulation n. The pin index jis increased by moving to the next pin. The power density is calculated for the nom-inal and bowed simulation case (described in section 2.1) and a relative differencebetween them are plotted using equation 2.4 (see result section 2.2).

It is not straight forward to use the result files containing isotopic composition datafrom Serpent. The output files contain more than 3 million rows for the 80 differentisotopes that are followed, in this case. The files consist of matrices with compositiondata for each burnup step and every fuel pin. The rows are for different isotopes andthe columns are for burnup steps. Naturally, no data is generated in places wherecontrol rods are, since these cells do not contain any fuel. In this sense, some modifi-cation of the output files simplifies the plotting while plotting the relative change ofthe nominal and perturbed simulation in color maps, similar as for the power results.Matrices with zeroes of the same size are added at the same position as the emptycells with a corresponding name. Thus, can control rods easily be distinguished inthe color maps.

The output files with composition results contains various data, but the isotopicmass density is only studied here. The mass densities are extracted from the restby creating one new file for each assembly that contains the isotopic mass densityfor the different burnups and pins.

As for the power density, results form several runs are combined with a weightedaverage for each burnup step,

Mdens =1

ωtot∑n

wc,nwp,nbj,m,n, (2.3)

where bj,m,n is the mass density in pin j, burnup step m and simulation n.


The statistical uncertainty of the calculation improves by combining results fromseveral independent simulations using equation (2.2) and (2.3).

The bowing effects are illustrated in MATLAB by using equation (2.2) and (2.3) andby calculating the relative change in power and isotopic mass density between thenominal and the perturbed case with Bash. Hence,

∆i =CN

aj/bj− CB

aj/bj

CBaj/bj

, (2.4)

where i applies for the mass density of isotope i, CNaj/bj

and CBaj/bj

are the case forthe nominal and the perturbed power or mass densities of the fuel assemblies, re-spectively. The bowing maps used are considered constant as a function of burnup.

The Serpent input file is prepared in such a way that the power and the relativeuncertainty in power are calculated for each pin. The combined mean absolute un-certainty in power is calculated for the nominal and the perturbed case of severalindependent simulations,

εabs =1

ωtot∑n

√(wc,nwp,naj,m,nεrelj,n)

2, (2.5)

where εrelj,n is the relative uncertainty for one pin and one simulation. The com-bined absolute uncertainty is used to propagate and calculate the relative uncer-tainty of the relative change in power between the nominal and the bowed calcula-tion, described in equation 2.4,

εrel =

√√√√ (PBεNabs)

2 + (PNεBabs)

2

(PB)4, (2.6)

PN and PB are the nominal and bowed mean power distributions from (2.2)whereas εN

abs and εBabs are the nominal and bowed mean absolute uncertainty from

(2.5).

The relative uncertainty of the relative change between the nominal and the bowedcalculation is illustrated with a colormap for each subpart of the reactor core (seeresult section 2.2).

Serpent does not provide any uncertainty data for the isotopic composition. Insteadan uncertainty estimation is done for subpart 2 (in section 2.2.2) by running 50 in-dependent simulations, calculating the standard deviation in mass density betweenthem and plotting mass densities in a histogram. One can consider the statistics tobe enough, if the standard deviation is low and the distribution of the histogram isGaussian.

2.2 Simulation results

The results of the two 5x5 assembly subparts from figure 2.2 of the reactor core, bothwith different characteristics, are presented separately in this section. The two partsare denoted as subpart 1 and 2, respectively. The bowing direction and amplitude

2.2. Simulation results 17

of each subpart are denoted by yellow arrows in some of the result plots that showthe global distribution of a relative power or a mass density change induced by thebowing effects. The arrows allow for a better understanding of the impact from thedisplacement of the fuel assemblies.

2.2.1 Subpart 1

The corresponding bowing map of subpart 1 is to the left in 2.2.

The relative power difference between the nominal and the perturbed assembliesis calculated using equation 2.4 and the result is plotted in figure 2.3. The left col-ormap corresponds to the first burnup step at 0.1 MWd/KgU and the right figure tothe last burnup step at 15 MWd/KgU. Each simulation is performed with burnupin 10 steps, but only the two extremes are shown here. The red regions refer to anincrease of power dissipation and blue regions to a power decrease, introduced bythe displacement of the FAs in respect with the nominal non-displaced case. Theyellow arrows show the 2D direction and amplitude of the perturbations from thecorresponding bowing map, with the scale noted in the lower left corner.

The relative difference in power is directly comparable with the expected behaviourby the displacement of the FAs. An increased water gap in-between adjacent as-semblies results in an increased moderation region, meaning that the neutrons aremore efficiently thermalized, which increases the probability of fission and thus in-creases the power dissipation in that region. The opposite applies to regions witha decreased moderation region. The relative difference is greatest for the first bur-nup step. However, the central part of the FAs at the first burnup are whiter thanat higher burnup, meaning that the power dissipation of the central parts is not af-fected, as much, at low burnup. The central parts of the FAs are affected for thehighest burnup, where one sees a blue-shift (decrease) of the FAs in the center of thesubpart whereas the assemblies at the top and bottom row are red-shifted (increase).The cause for this might be related to the periodic boundary condition used (and theinput settings), but also a result of the FA perturbations. Possible reasons are furtherdiscussed in the discussion3.1.

Rel

ativ

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in p

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[%]

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10 mm

FIGURE 2.3: The yellow arrows show the direction and amplitude ofthe bowing, the scale is noted in the lower left corner. Left: The rela-tive power difference between the nominal and the bowed assembliesfor burnup at 0.1 MWd/kgU of subpart 1. Right: The relative powerdifference between the nominal and the bowed assemblies for burnup

at 15 MWd/kgU of subpart 1.


The isotopic concentration and depletion of U235 are closely related to the powerdistribution inside the reactor, regions of high power dissipation corresponds to highdepletion of U235. Figure 2.4 shows the relative mass density difference of U235 forthe first and last burnup step. Note that the scales are not the same in this cases,the first burnup step has a very low relative difference, but still notable. The relativedifference is at least 100 times higher for the last burnup step.

Regions of an increased relative power corresponds to a decrease of U235 mass den-sity, as expected, by comparing the relative power difference in figure 2.3 with therelative change of U235 mass density in figure 2.4. The color settings are adopted forshowcasing this phenomenon, where the red regions for increased power becomesblue for the relative decrease in U235 content. Generally is the power map directlytranslated into the mass density map of U235, except for a few places. In the cen-tral FA of the second row form the bottom are all pins in the bottom row dark blue.That should, with the discussed logics, correspond to a red region in figure 2.3 ofincreased power, which is not the case.

Rel

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5 [%

]

-0.1

-0.08

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0

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0.1

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5 [%

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10 mm

FIGURE 2.4: The yellow arrows show the direction and amplitudeof the bowing, the scale is noted in the lower left corner. Left: Therelative isotopic mass density difference of U235 between the nominaland the bowed assemblies for burnup at 0.1 MWd/kgU of subpart 1.Right: The relative isotopic mass density difference of U235 betweenthe nominal and the bowed assemblies for burnup at 15 MWd/kgU

of subpart 1.

Figure 2.5 shows the relative mass density difference with burnup for U235, U238,Pu239, Cm244, and Nd148. The results are for the 4 most affected pins in the centralassembly. Two pins are at the border of the FA and the other two are adjacent (inner)to them. The ones at the border exhibit the greatest difference and are most affectedby the bowing. Naturally, the relative mass density of U235 and Pu239 decreases withdepletion whereas Cm244 and Nd148 increases since more isotopes are produced withdepletion. The mass density of U238 stays roughly the same depending on its lowfission cross section in the present energy region.


0 5 10 15Burn up [MWd/kgU]

-15

-10

-5

0

5

10

"i m

ass

dens

ity [%

]

U235

U238

Pu239

Cm244

Nd148

Burnup

FIGURE 2.5: The relative difference in isotopic mass density betweenthe nominal and the bowed assemblies with burnup. The results are

for 4 pins in the central assembly and the most affected region.

The relative uncertainty for the relative power change is calculated and propa-gated for each pin of the combined simulations results, as mentioned in section 2.1.3.The global distribution of uncertainty is plotted in a colormap to easily display thedifferences. Figure 2.6 shows the relative uncertainty of figure 2.3 at 15 MWd/kgU.The uncertainty is generally low (∼ 0.2%) for all FAs, although some differencesare seen. The assemblies in the second column from the left together with the twoassemblies, one in the end of the first row and the other in the end of the last row,originates from fresh fuel at the start of the simulation and these show the lowestrelative uncertainty. The power dissipation is highest in these fresh assemblies (com-paring with others from older fuel cycles), meaning that there might be a absolutepower relation to the relative uncertainty of the power change. The absolute powerdistribution is shown in figure 2.7. However, it could also be related to the numberof different isotopes considered in the beginning of the simulation, comparing thefresh fuel with previously used fuel.


"0 re

l

[%

]

#10 -3

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2-1

FIGURE 2.6: The relative uncertainty in power change for figure 2.3at 15 MWd/kgU burnup.

Since the FAs have separate fuel composition and some originates from differ-ent fuel cycles, i.e consisting of fuel of different age, which implies that the powerdistribution inside the reactor is not uniform. Unfortunately, this behaviour is notrealised in figure 2.3, since only the relative difference is calculated and the differ-ences introduced by the bowing are only covered.

The absolute power distribution within subpart 1 (for one simulation run) is shownby the Serpent generated geometry plot in figure 2.7, for the perturbed assembliesat 15 MWd/kgU. Brighter regions corresponds to higher absolute power and theopposite for the darker regions. Basically, one sees a similar pattern for the relativeuncertainty and the absolute power distribution comparing figure 2.6 and figure 2.7.


FIGURE 2.7: A Serpent output plot of the power distribution withinsubpart 1 at 15 MWd/kgU. The power distribution is normalized,darker regions corresponds to less power and less fission and brighter

to regions to higher power and more fission events.

The previous reasoning of a power dependency for the uncertainty is confirmedby figure 2.8. The figure shows the relative uncertainty of 4 pins in a high (with freshfuel) respective a low (with older fuel) absolute power region plotted with burnup.The high power region corresponds to one assembly in the second column and thecentral assembly (of the subpart) is used as the low power region. The variation withburnup is very little, but there is a distinct difference between the two regions, i.e theinvestigation indicates that the absolute power impacts the relative uncertainty, butother effects are also possible.



1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

0 rel p

ower

den

sity

[%

]

#10-3

High power regionLower power regionMean 0rel

Low power regionHigh power region

-1

Pow

er c

hang

e[%

]

Burnup

FIGURE 2.8: The relative uncertainty in power change (from 2.3) withburnup for 4 pins in a region of high/low absolute power.

Figure 2.4 shows that the relative difference of U235 mass density is most severeat the border of the perturbed assemblies. In figure 2.9 is the relative difference in themass density of 2 pins at each distance from the border plotted for U235, U238, Pu239,Cm244 and, Nd148, by starting from the border of the central assembly and movetowards the center. One realizes that the bowing effects of the isotopic compositionbecome less with the distance from the assembly edge, which are seen for all isotopesplotted. Hence, the bowing affects the border regions of the FAs mostly.


1 2 3 4 5 6 7Pin position from assembly edge

-15

-10

-5

0

5

10

"i m

ass

dens

ity [%

]

U235

U238

Pu239

Cm244

Nd148

FIGURE 2.9: The relative difference in mass density as a function ofthe pin position. Pin 1 is at the assembly edge and pin 7 is at the

center.

The global distribution of the relative mass density difference is also plotted forU238, Pu239, Cm244, and Nd148 at 15MWd/kgU burnup, the results are shown in fig-ure 2.10. Note that the colorbar scales in these plots are not the same since the iso-topic composition does not change equally.

The difference in (A) for U238 is about 100 times less than for U235 in figure 2.4, butthe bowing impacts are still seen. Some random noise is present in the center of theassemblies which may be related to the low rate of neutron-induced fission for U238.Pu239 in (B) shows good statistics and low noise, hence all color transitions betweenadjacent pins are smooth. The impacts at the assembly centres is very little for Pu239

and almost everything occurs at the borders. Nd148 in (C) and Cm244 in (D) exhibitsimilar, but different patterns from what are seen before and the relative differencein mass density is not directly deduced from the bowing maps 2.2. Some randomnoise is also present, especially for (D), which indicates less statistics.


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(C)

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244

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5

10

15

20

(D)

FIGURE 2.10: The relative difference in mass density for U238, Pu239,Nd148, and Cm244 for burnup step 10 at 15 MWd/kgU.

2.2.2 Subpart 2

The corresponding bowing map of subpart 2 is to the right in figure 2.2. The bowingcharacteristics of subpart 2 are different from what are seen for subpart 1. All FAsin subpart 2 perform a collective displacement towards the right with similar butdifferent amplitudes, which is different from subpart 1, where all central assemblieswere displaced away from each other.

The relative power difference between the nominal and the perturbed FAs of sub-part 2 is calculated according to equation 2.4 and the results are plotted in figure2.11. As previously, the yellow arrows show the 2D direction and amplitude of theperturbations from the corresponding bowing map, with the scale noted in the lowerleft corner. The same blue-shift of the central assemblies in subpart 1 is also notedhere, with a relative power decrease in the middle of the assemblies for burnup at15 MWd/kgU. The relative effect of the assembly-border regions become less severeas the fuel is depleted (comparing burnup at 0.1 and 15 MWd/kgU), which is alsoseen for subpart 1.


Rel

ativ

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ange

in p

ower

[%]

-10

-8

-6

-4

-2

0

2

4

6

8

10

10 mm

Rel

ativ

e ch

ange

in p

ower

[%]

-10

-8

-6

-4

-2

0

2

4

6

8

10

10 mm

FIGURE 2.11: The yellow arrows show the direction and amplitude ofthe bowing, the scale is noted in the lower left corner. Left: The rela-tive power difference between the nominal and the bowed assembliesfor burnup at 0.1 MWd/kgU of subpart 2. Right: The relative powerdifference between the nominal and the bowed assemblies for burnup

at 15 MWd/kgU of subpart 2.

Figure 2.12 shows the relative mass density difference of U235 between the nom-inal and the perturbed FAs for the first and last burnup step. The power maps in2.11 are consistent with the mass density difference, except for one FA in the firstburnup step. The middle assembly at the left border shows random (noisy) changesbetween adjacent assemblies with higher amplitudes than seen for the rest. This isnot realized for the higher burnup case and since the effect is quite low for the firstburnup step no major importance is made (thinking of the scale difference betweenthe two figures). As for 2.4 there is a 100 times difference between the first and lastburnup step.

The statistics are less in the simulation of subpart 2 than for subpart 1, which isbelieved to be one reason why more noise is present in figure 2.12, than seen before.Although, Serpent does not provide any uncertainty for the isotopic composition,random changes between adjacent pins, without uniformity and smooth transitions,especially in the central assemblies and less affected regions, may indicate less statis-tics. Thus, implying higher uncertainty in the composition data.


Rel

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assd

ensi

ty fo

r U23

5 [%

]

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

10 mm

Rel

ativ

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ange

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assd

ensi

ty fo

r U23

5 [%

]

-10

-8

-6

-4

-2

0

2

4

6

8

10

10 mm

FIGURE 2.12: The yellow arrows show the direction and amplitudeof the bowing, the scale is noted in the lower left corner. Left: The rel-ative isotopic mass density difference between the nominal and thebowed assemblies for burnup at 0.1 MWd/kgU of subpart 2. Right:The relative isotopic mass density difference between the nominaland the bowed assemblies for burnup at 15 MWd/kgU of subpart

2.

In figure 2.13 is the relative mass density difference with burnup plotted for U235,U238, Pu239, Cm244 and, Nd148. The same behaviour as noted in figure 2.5 is also con-firmed for subpart 2. The mass density of U235 and Pu239 decreases with depletionwhereas the mass density of Cm244 and Nd148 increases, and the mass density of U238

stays roughly the same.


-10

-5

0

5

10

15

"i m

ass

dens

ity [%

]

U235

U238

Pu239

Cm244

Nd148

Burnup

FIGURE 2.13: The relative difference in isotopic mass density for thenominal and the bowed assemblies with burnup for 4 pins in the cen-

tral subpart and the most affected region.

The global relative uncertainty in power change of each pin, corresponding to therelative power change of figure 2.11 (left), is seen in figure 2.14. The noise increase


in the simulation of subpart 2 compared with subpart 1, discussed for figure 2.12,is confirmed in the uncertainty map. Generally, the global uncertainty is increased,compared to figure 2.6. The reason for this is most likely related the less neutronsand number of cycles used in the simulation of subpart 2.

"0 re

l

[%

]

#10 -3

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3-1

FIGURE 2.14: The relative uncertainty in power change for subpart 2in figure 2.11 at 15 MWd/kgU burnup.

The relative uncertainty in power change with burnup for a region of higher re-spective lower absolute power distribution is plotted in figure 2.15, for consistencywith the analysis done for subpart 1. Again, the absolute power distribution for onesimulation run is shown in the Serpent power output 2.16 (right) for 15 MWd/kgU.The power differences among the assemblies for subpart 2 are not as distinct as be-fore, only two assemblies (the brighter ones in the right corner) originate from freshfuel at the beginning of the simulation. The high power region is from the top bor-der of the middle assembly and the second column, the low power region is fromthe right border of the fourth assembly and the second last column. From figure 2.16one sees the absolute power differences between the two region, but what is notedfor this subpart is that the two regions also show a distinct relative power difference(seen in figure 2.11). That was not the case for subpart 1, where the relative changewas milder, which could impact the results. However, it is believed that the absolutepower differences has the biggest contribution, or as discussed for subpart 1, the ini-tial fuel composition could also have an impact.

Simulated data with the absolute power distribution for the combined results existswhich can be used as a complement to 2.16 (since only results from one simulationis shown), further increasing the statistics, but it is omitted in this thesis.



2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

0 rel p

ower

den

sity

[%

]

#10-3

High power regionLower power regionMean 0rel

Burnup

-1

Pow

er c

hang

e

FIGURE 2.15: The relative uncertainty of power change for 4 pins ina lower and a higher power region.

Figure 2.16 shows the Serpent generated absolute power distribution for the per-turbed assemblies of one simulation run with 0.1 and 15 MWd/kgU burnup. Nocombining of results are possible for this figure, since the output is generated fromSerpent directly. The power is normalized, darker regions means less fission andthus less power. The different spacings between adjacent assemblies correspond tothe perturbations of the bowing maps in 2.2.

FIGURE 2.16: The Serpent output for the normalized absolute powerdistribution within subpart 2. Darker regions corresponds to lesspower density and less fission, the opposite applies for brighter re-

gions. Left: 0.1 MWd/kgU burnup. Right: 15 MWd/kgU burnup.

In figure 2.17 is the relative difference of the mass density with pin position, start-ing from the edge of one FA, plotted for 2 pins and the isotopes U235, U238, Pu239,


Cm244 and, Nd148. As previously noted for subpart 1, the bowing effects to the iso-topic composition become less in the central part of the assembly, which applies forall isotopes plotted. Meaning that, the bowing impacts on the isotopic distributioncan be thought of as a "border effect", mostly affecting the border regions. This be-haviour is also realized for most assemblies in the figures describing the isotopicmass density change. However, changes are still visible for the central parts of theFAs.

1 2 3 4 5 6 7Pin position from assembly edge

-15

-10

-5

0

5

10

"i m

ass

dens

ity [%

]

U235

U238

Pu239

Cm244

Nd148

FIGURE 2.17: The relative difference of mass density as a function ofthe pin position. Pin 1 at the assembly edge and pin 7 in the center.

The relative difference in mass density are plotted for U238, Pu239, Cm244, andNd148 in figure 2.18 for the last burnup step at 15MWd/kgU. The colorbar scales in(A) - (D) are different from each other but are equivalent to the settings for figure2.10. Pu239 in (B) exhibit less noise than U238 in (A), Nd148 in (C) and Cm244 in (D),which is the same behaviour as noted for subpart 1. The mass density patterns aredirectly related to the assembly displacements in figure 2.2 and the power distribu-tion change in 2.11, beside some statistical noise. No inexplicable pattern is realized,which was the case for (C) and (D) in 2.10.


Rel

ativ

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r U23

8 [%

]

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

(A)

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39 [%

]

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2

4

6

8

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(B)

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48 [%

]

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(C)

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244

[%]

-20

-15

-10

-5

0

5

10

15

20

(D)

FIGURE 2.18: The relative difference of mass density for U238, Pu239,Nd148, and Cm244 at 15 MWd/kgU burnup of subpart 2.

Serpent does not calculate any uncertainty for the isotopic composition data dur-ing the simulation. The statistical uncertainty in mass density is estimated by thestandard deviation from multiple simulations of the same geometry but using dif-ferent seed. Changing the seed is done to ensure that the random numbers used forthe neutron histories in the Monte Carlo simulation come from a different set. Thesame model as previously, with the same number of neutrons and cycles, is used forthe uncertainty estimation. The number of burnup steps are reduced to 3 with thehighest depletion at 1MWd/kgU, to reduce the computational time.

Histograms for the mass density of U235 and Pu239 in two different pins, one inthe lower and the other in the top part of the central assembly, are plotted in 2.19for 50 independent calculations together with a normal fit of the data. The meanvalue, µMdens , and the standard deviation, σ, for the mass density are calculated foreach case. With the standard deviation one gets an estimate of the uncertainty in thedepleted fuel. The standard deviation for the 4 cases in (A) to (D) are similar andequal for Pu239 in (C) and (D). A standard deviation of ∼ 8−6 is considered as low,hence the statistical accuracy for the isotopic data in the simulations are consideredgood enough for the purpose of this thesis. The histograms and the data fitting aredone in Matlab using the histfit function and the mean and the standard deviationare calculated with corresponding functions in Matlab. The statistical differencesare expected to follow a Gaussian (or normal) distribution since random variablesare used in the Monte Carlo calculation. The tendency of a Gaussian distribution isseen in 2.19 but the 50 calculations are too less to prove a pure Gaussian distribution.


0.2239 0.22395 0.224 0.22405 0.2241 0.22415 0.2242 0.22425 0.2243 0.22435 0.2244Mdens U235 [g/cm3]

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Num

ber o

f bin

s

7Mdens

-- 2.24e-01

< -- 7.81e-05

(A)

0.2236 0.2237 0.2238 0.2239 0.224 0.2241 0.2242Mdens U235 [g/cm3]

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Num

ber o

f bin

s

7Mdens

-- 2.24e-01

< -- 8.53e-05

(B)

0.05245 0.052455 0.05246 0.052465 0.05247 0.052475 0.05248 0.052485 0.05249 0.052495 0.0525Mdens Pu239 [g/cm3]

0

0.5

1

1.5

2

2.5

3

3.5

4

Num

ber o

f bin

s

7Mdens

-- 5.25e-02

< -- 7.93e-06

(C)

0.05242 0.052425 0.05243 0.052435 0.05244 0.052445 0.05245 0.052455 0.05246 0.052465 0.05247Mdens Pu239 [g/cm3]

0

0.5

1

1.5

2

2.5

3

3.5

4

Num

ber o

f bin

s

7Mdens

-- 5.24e-02

< -- 7.93e-06

(D)

FIGURE 2.19: Four histograms of 50 independent simulations for themass densities of U235 and Pu239 in two separate pins. (A) for U235 inthe lower, (B) for U235 in the top, (C) for Pu239 in the lower and (D)for Pu239 in the top part of the central assembly. The mean value and

the standard deviation are calculated for each case.

33

Chapter 3

Discussion and conclusions

3.1 Discussion

A full core study the bowing effects would most accurately describe the real be-haviour and the impacts of assembly perturbations. Unfortunately, the bowing dataused in this thesis did not cover the full core and a full core Monte Carlo study isalso very computer memory demanding. To cover the full core, better computationalpower is necessary, such as international computational clusters. Additionally, Ser-pent is not well scaled for parallel calculations on a larger number of computationalnodes. Which is why the study was better suited for subparts of the reactor core.If bowing data exists for the full core together with essential computer capacity, acontinuation of this thesis could be to study the full core and possibly be a thesis ofits own.

The increased water gap at the subpart border (see figure 2.1), which allows for theassembly displacements in the simulations model, impacts the power distributioninside the reactor. The increased border of each subpart is extended throughout thesimulation model since a periodic boundary condition is used at the border of thesubparts, thus impacts the reactor globally. However, the relative effects in terms ofpower and isotopic composition are still clarified, where distinct effects are seen andthe increased border is believed to have a minor impact for this purpose. Although,a more close investigation of the pin composition may not be valid at the border ofthe subparts. Thinking of the neutron mean free path, the central part should notbe affected as much by this "border effect". Radial and azimuthal effects in separatepins can thus be studied in detail, if one considers the central assembly of each sub-part. The idea was to cover that in this thesis, but due to time limitation, it is left forlater work.

In both subpart 1(figure 2.3) and subpart 2 (figure 2.11) one sees a red-shift (powerincrease) of the relative power change distribution for the central parts of the as-semblies in the top and bottom row at 15 MWd/kgU. The opposite is seen for themiddle rows of the subpart, where all central parts of the assemblies are blue-shifted(power decrease). This means that the relative power increases of the central partsfor the perturbed assemblies at the top and bottom rows whereas the relative powerdecreases for the middle rows. The phenomenon is clearly seen for subpart 1 but isalso noted for subpart 2. The reason for this might be related to the increased borderand the border effect redistributing the thermal neutron distribution, as previouslydiscussed. However, subpart 1 and 2 have a different bowing characteristics, wherethe top border is decreased for subpart 1 and increased for subpart 2, implying thatthe moderating region at the border is resized differently. The differences shouldprevent that the same shift is noted for the two subparts, especially for the top row,

34 Chapter 3. Discussion and conclusions

if it is only dependent on the increased border. The center-shift of the central as-semblies is believed to be directly related to the bowing. The clearest differencesin power distribution (and isotopic composition) are naturally seen at the borderof each assembly but the red- or blue-shift confirms that the central parts are alsoaffected. The power change affects the isotopic composition but the correspondingred- or blue-shift of the assembly center are not as clearly seen for the isotopic com-position of U235 in figure 2.4 and 2.12 at 15 MWd/kgU, where a red-shift in powershould correspond to a blue-shift (isotopic decrease) of U235 content, because of ura-nium consumption. However, a tendency is noted, especially for 2.12. To conclude,the largest effect of the bowing is seen at the assembly border but it is confirmed thatthe central part of the assemblies are affected in terms of both isotopic compositionand power. Random variations (noise) between adjacent assemblies are also seen,especially for the isotopic distribution. The blue-shift in power for the central partsof the middle assemblies indicate that fewer fission events are present in the centerof these perturbed assemblies. The opposite applies to the red-shifted regions witha relative increase of events. Thus, the perturbations affect the neutron distributionand fission events globally inside each fuel assembly and not only the regions be-tween the assemblies, directly affected by the displacements.

The assembly perturbations redistribute the power dissipation inside the reactor insuch a way that the increased water gaps between adjacent assemblies increases theprobability of fission. More neutrons are moderated into the thermal energy regionbecause of the water excess, which increases the probability of fission and the powerdissipation in that region. This is easily realised in figure 2.3 and figure 2.11. By com-paring with the corresponding displacements, one realizes that an increased distancegives a corresponding power increase and the opposite for decreased distances. Asdiscussed in the result section 2.2, a region of increased power means that more U235

are consumed, which is seen in figure 2.4 and 2.12. The same applies to U238 andPu239 in (A) and (B) of figure 2.10 and 2.18, respectively. The results for Pu239 indi-cates a low statistical uncertainty of the calculations due to the low noise (of randomchanges in relative isotopic concentration). More noise is seen for U238 and the rel-ative change of U238 content is also very low (∼ 0.1%) when comparing with U235

and Pu239, which may be one reason why more noise is realized. The low relativechange is most likely related to the low fission cross section for thermal neutrons andU238. Naturally, Nd148 and Cm244 in (C) and (D) have an opposite relative increasefor the regions of increased power since more isotopes are produced with burnup ofthe nuclear fuel. The relative change pattern has the same analogy with the bowingmaps as previously discussed, but one notes that the central parts of some assem-blies for Cm244 in subpart 1 are more affected, whereas others are very little affected.The central parts in subpart 2 are more affected in terms of Nd148 and Cm244 contentthan subpart 1, but it also contains more noise. The reason why subpart 1 shows abig difference between the assemblies might be related to the initial isotopic compo-sition of the fuel, since the assemblies with more affected central parts contain freshfuel initially. The two assemblies in the lower right corner of subpart 2 also containfresh fuel and exhibit the same behaviour, especially seen in (D). Where one assem-bly shows a distinct relative increase and the other a distinct relative decrease of theisotopic content. The increased noise for subpart 2 is believed to be relative to theless statistics used.

Generally, there is a good consistency between the relative change in power dis-tribution, the bowing maps and the resulting changes in isotopic composition of the

3.2. Conclusion 35

spent fuel introduced by the bowing. The bottom row of pins in the middle assem-bly of the second last row of assemblies in figure 2.4 shows a relative large decreasein U235 content, however, this is not realized from the corresponding power plotwhere a small relative difference is seen. The same is noted for figure 2.12 wherethe first assembly of the middle row and the first burnup step show a big differencefrom what is seen in the corresponding power map. The reasons are not obvious,it could either be due to plotting issues or because of errors in the input files/ data.For subpart 1, it might be because a few unique definitions of fuel pins are missingin the input files, defining the fuel content of each pin, which results in anomalies inthe matrix components of the isotopic composition. The power and isotopic data arenot obtained in the same way. The power distribution of the assemblies is obtainedby the Serpent power detectors and the isotopic data by the evaluation of the Ser-pent output matrices containing composition data with burnup for each pin with itsunique name. Thus, the different approach to collect the results is believed to be thereason why only the anomaly is seen for the isotopic composition. The same is alsoseen for other isotopes in figure 2.18. Since issues are only found for a few places itis not believed to be a bug in the simulation code.

Serpent is still under development and there are still issues found in the code. Oneneeds to be observant with such and investigate unexpected happenings. One ex-ample is that the matrices produced by the power detectors were rotated 90 degrees.Fortunately, this was realized because of an unexpected pattern while plotting thecolormaps of the relative power change. The issue was solved using a Matlab func-tion.

Serpent does not produce any uncertainty for results regarding isotopic data, as forthe power distribution. An estimate of the uncertainty in mass density change isobtained by calculating the standard deviation of the mass density results for 50 in-dependent simulations. The estimate is done for a few isotopes and pins in regionsof specific interest. By plotting the isotopic mass density of each simulation in a his-togram (figure 2.19) one can estimate if the distribution are Gaussian (as expected)or not. In this case more simulations are necessary to prove a Gaussian distribution,however a tendency is seen. The standard deviation calculated is low which indicatethat the statistical accuracy of mass density results in this thesis is sufficient, consid-ering the number of neutron and cycles used in the simulations. Estimations like thisare very time consuming. For the 50 separate simulation runs used, was an equalamount of 50 days needed just in computational time, even though the number ofburnup steps were reduced.

3.2 Conclusion

The purpose of this thesis is to study impacts introduced by bowing effects withburnup in a nuclear reactor and the possibility to simulate this phenomenon usinga Monte Carlo code together with realistic data in terms of nuclear fuel compositionand observed bowing, both from the same nuclear reactor. The bowing, i.e the per-turbations, of the fuel assemblies show significant impacts (up to 10− 20%) on thereactor operation in terms of power and isotopic composition while comparing thenominal (non-perturbed) and displaced (perturbed) cases.

36 Chapter 3. Discussion and conclusions

A relative power difference is seen depending on the changes in moderation regionswhich are directly related to the pattern of the perturbations in the correspondingbowing map. The impacts are mostly seen at the border of adjacent assemblies witha ∼ 10% relative power difference, but the central parts are also affected. A slightshift of either decreased or increased relative power is seen for the central parts of theassemblies. This shift may also change sign from an increased power at low burnupto a power decrease for higher depletion, which is seen for the central assemblies ofsubpart 1. The same phenomenon is also seen for the relative change of U235 con-tent. The power and U235 content are to some extent translated into each other (withopposite correlations) due to the power density and U235 burnup relation.

The power distribution is directly associated with the consumption of U235 and theborder regions between adjacent assemblies of increased relative power show a cor-respondingly relative decrease in U235 concentration, the same holds for the concen-tration of Pu239. As confirmed for the power, the global distribution of the relativechange in U235 and Pu239 content are directly related to what is expected from thecorresponding bowing maps. The other isotopes investigated contain more statisti-cal noise and some irregularities in the uniformity of the changes between adjacentassemblies. Some assemblies show a big relative change (up to 10− 20%) whereasothers show very little change and the patterns seen are not directly understood bythe corresponding bowing maps. The reason is not yet fully understood. It might berelated to the fuel composition at the beginning of the simulation, but since relativeeffects are studied, such effects should be prevented.

The relative statistical uncertainties in power change distribution are generally low.Small differences are seen globally of each subpart. The differences might have arelation to the absolute power. The uncertainty in mass density is estimated by thestandard deviation of 50 independent calculations plotted in a histogram. The ten-dency is that the mass density follow a Gaussian distribution and the statistical ac-curacy is considered good enough due to a low standard deviation of the results.The Gaussian (or normal) distribution hypothesis cannot be tested due to lack ofstatistics.

Serpent require substantial amount of memory for separate depletions zones in eachpin and already a 5x5 subpart is computational demanding for the simulation serversused. A full core Monte Carlo study with separate zones for each pin will requirelarger computational clusters.

One purpose of this thesis was a detailed study of a few pins in the central assem-bly of each subpart, where radial and azimuthal differences were covered but thatis omitted due to the computational time limits and the work load. It is left for laterwork.

The biggest accomplishment by this thesis is the proven methodology that shows,based on the results and the findings, how bowing effects and its impacts can bestudied with the Monte Carlo method using realistic data from an operating reactor.

37

Appendix A

The CRAM method

An explanation of a method developed for Serpent to solve the depletion equation.

A.1 CRAM method

The Bateman depletion equation 1.6, discussed in the nuclear reactor physics sec-tion 1.3.3, can be solved either with the CRAM method (Pusa, 2011) or a linear chainmethod.

The CRAM method is applied to a system of first order linear differential equa-tions describing the isotopic evolution by Pusa (Pusa, 2011), which highlight essen-tial parts of the method.

The system of burnup equations are written,

dndt

= An, n(0) = n0, (A.1)

n(t) ∈ R contain nuclide concentrations. A ∈ R is the burnup matrix with decayand transmutation coefficients.

The matrix equation A.1 is formally solved with the matrix exponential method,

n(t) = eAtn0 (A.2)

where eAt is a power series expansion,

eAt =∞

∑k=0

1k!

Atk, (A.3)

and A0 = I.

It is showed that the rational approximation rk,k of order k for equation A.1 can bewritten,

n = α0n0 + 2Re

(k/2

∑j=1

αj(At− θj I)−1n0

), (A.4)

where α0 is the limit at the infinity and αj is the residues of the poles θj.

The CRAM method uses the rational function r(z) as the best approximation for

38 Appendix A. The CRAM method

the exponential function on the negative real axis. If πk,k is a set of rational functionsrk,k(x) and,

δk = supx∈R−| rk,k(x)− ex |, (A.5)

then it is shown that,

limk→∞ δ1/kk = H, (A.6)

where H is the Halphen constant in closed form by elliptic integrals (Goncharand Rakhmanov, 1989). The asymptotic convergence on the negative real axis is fastand the convergence is also shown to cover subsets of the complex plane togetherwith Hankel contours in C \R− (Stahl and Schmelzer, 2009),

limk→∞

(supz∈K | rk,k(z)− ez |

)1/k

= limk→∞

(supz∈Γ | rk,k(z)− ez |

)1/k

= H,

(A.7)

with any compact K ⊂ C and any Hankel contour Γ ⊂ C \R−.

39

Appendix B

Serpent input files

This Appendix shows examples of the inputfiles with simulation settings and con-struction of the reactor geometry.

B.1 Pin set up

The definition for one 15x15-pin fuel assembly is shown in figure B.1 containing fuelpins and control rods.

FIGURE B.1: The assembly-core definition for one assembly.

Figure B.2 shows the pin geometry-definition with the corresponding materialsfor a few pins (A) and an example of how geometry-divisions can be set (B). As seenin (B) each pin can be specified with its own sub-division setting, which correspondsto one depletion zone, or more.

% ---- Pin definitions for the 18-05 assembly pin 18-05.01.01 18-05.01.01 0.4565 void 0.4650 Zircaloy4 0.5375 water pin 18-05.01.02 18-05.01.02 0.4565 void 0.4650 Zircaloy4 0.5375 water pin 18-05.01.03 18-05.01.03 0.4565 void 0.4650 Zircaloy4 0.5375 water pin 18-05.01.04

(A)

% --- Material division for burnup calculation% Treat different pins 1-204 of div 1-204 as separate depletion zones % (sep 1) additionally divide each of those fuel pellets into 10 equal % volume rings between radial coordinates of 0.0 and 0.4565 and 4% sectors of 45.0 degree div 18-05.01.01 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.02 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.03 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.04 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.05 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.06 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.07 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.08 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.09 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.10 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.11 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.12 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.13 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.14 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.01.15 sep 1 subr 10 0.0 0.4565 subs 4 45.0div 18-05.02.01 sep 1 subr 10 0.0 0.4565 subs 4 45.0

(B)

FIGURE B.2: The pin geometrics and the material definitions (A) andthe division setting for a few pins (B).

40 Appendix B. Serpent input files

Figure B.3 shows the extracted and normalized isotopic densities from the de-terministic simulated fuel data in one pin (A) and an example how that is importedand collected for the corresponding pin in one assembly (B). In this way the materialdefinition for each pin is separated. (A) is created with Bash scripting C.1.

mat 18-05.07.15 -10.3070 burn 1 92238.09c 9.60978E-0192235.09c 2.32331E-0294239.09c 5.24654E-0392236.09c 3.74222E-0394240.09c 1.25611E-0355137.09c 8.18283E-0494241.09c 7.00008E-0443099.09c 5.57413E-0440093.09c 5.13902E-0438090.09c 4.17432E-0458144.09c 4.07886E-0492234.09c 3.36639E-0460148.09c 2.47103E-0493237.09c 2.29417E-0455135.09c 2.20775E-0437087.09c 2.01581E-0461147.09c 1.93643E-0494242.09c 1.12045E-0444106.09c 9.86703E-0540095.09c 9.08523E-0546107.09c 9.05508E-0553129.09c 8.33357E-0539091.09c 6.23435E-0541095.09c 5.94494E-0558141.09c 5.24252E-0555134.09c 5.05662E-05

(A)

/***********Data lib***********/ set acelib ”./Libraries/endfb7/sss_endfb7u.xsdata" /************************* Material definitions *************************/ % --- Fuel content imported from fuel data% --- Temperature is set to 600 K include "./pindata/assembly3/cycle18.asm18-05/unzipped/file.18-05.1.1.new"include "./pindata/assembly3/cycle18.asm18-05/unzipped/file.18-05.1.2.new"include "./pindata/assembly3/cycle18.asm18-05/unzipped/file.18-05.1.3.new"include "./pindata/assembly3/cycle18.asm18-05/unzipped/file.18-05.1.4.new"include "./pindata/assembly3/cycle18.asm18-05/unzipped/file.18-05.1.5.new"include "./pindata/assembly3/cycle18.asm18-05/unzipped/file.18-05.1.6.new"include "./pindata/assembly3/cycle18.asm18-05/unzipped/file.18-05.1.7.new"include "./pindata/assembly3/cycle18.asm18-05/unzipped/file.18-05.1.8.new"include "./pindata/assembly3/cycle18.asm18-05/unzipped/file.18-05.1.9.new"include "./pindata/assembly3/cycle18.asm18-05/unzipped/file.18-05.1.10.new"include "./pindata/assembly3/cycle18.asm18-05/unzipped/file.18-05.1.11.new"include "./pindata/assembly3/cycle18.asm18-05/unzipped/file.18-05.1.12.new"

(B)

FIGURE B.3: (A) the fuel material card definition with normalizedisotopic densities for one pin and (B) the material definitions for a

few pins.

Listing B.1 is an outline of the main Serpent input file, where all secondary filesand all setting are collected. Note that the script is stripped and does not covereverything, but all the main features are covered.

LISTING B.1: Outline of main Serpent input1 %%% Main input file for 2D 5x5 assembly with burn up %%%%%

3 /************************ Material definitions*

5 ************************/

7 % --- Including material definitioninclude "asm16 -31nom/asm16 -31mat"

9 include "asm18 -14nom/asm18 -14mat"include "asm16 -32nom/asm16 -32mat"

11 include "asm16 -14nom/asm16 -14mat"include "asm18 -12nom/asm18 -12mat"

13% --- Material definitions for water and cladding and thermal scattering library

15% --- Cladding material Zircaloy -4

17 % [Composition from PNNL -15870 , Rev. 1]mat Zircaloy4 -6.56000E+00 tmp 610

19 8016.03c -1.19276E-0324050.03c -4.16117E-05

21 24052.03c -8.34483E-0424053.03c -9.64457E-05

23 24054.03c -2.44600E-0526054.03c -1.12572E-04

25 26056.03c -1.83252E-0326057.03c -4.30778E-05

27 26058.03c -5.83334E-0640090.03c -4.97862E-01

29 40091.03c -1.09780E-01

B.1. Pin set up 41

40092.03c -1.69646E-0131 40094.03c -1.75665E-01

40096.03c -2.89038E-0233 50112.03c -1.27604E-04

50114.03c -8.83732E-0535 50115.03c -4.59255E-05

50116.03c -1.98105E-0337 50117.03c -1.05543E-03

50118.03c -3.35688E-0339 50119.03c -1.20069E-03

50120.03c -4.59220E-0341 50122.03c -6.63497E-04

50124.03c -8.43355E-0443

% --- Coolant is water with 650 ppm soluble boric acid added45 % The temperature of water is 600 K

% Density is calculated based on a pressure of 15.5 MPa47 mat water -0.70602 moder lwtr 1001 rgb 200 200 252

O -16.03c 3.330861e-0149 H-1.03c 6.663259e-01

B -10.03c 7.186970e-0551 B -11.03c 2.892846e-04

53 % --- Define thermal scattering library associated with hydrogen in light watertherm lwtr lwe7 .12t

55/************************

57 * Geometry definitions *************************/

59% --- Including pindefinitions

61 include "asm16 -31nom/pin16 -31"include "asm18 -14nom/pin18 -14"

63 include "asm16 -32nom/pin16 -32"include "asm16 -14nom/pin16 -14"

65 include "asm18 -12nom/pin18 -12"

67 % --- Adding pin definition for control rodpin wwwwwwwwwww

69 water 0.6200Zircaloy4 0.6900

71 water

73 % --- Empty position in the middle of the corepin xxxxxxxxxxx

75 water

77 % --- Including assembly cores with coordinates for 5x5 matrixinclude "asm16 -31nom/asm16 -31 core"

79 include "asm18 -14nom/asm18 -14 core"include "asm16 -32nom/asm16 -32 core"

81 include "asm16 -14nom/asm16 -14 core"include "asm18 -12nom/asm18 -12 core"

83% --- Secondary lattice definition where pin assemblies are defined within a 5x5 lattice

85 lat 5x5 1 0.0 0.0 5 5 21.4516-31 18-14 16-32 16-14 18-12

87 18-20 14-15 18-16 18-13 16-0116-30 18-18 16-29 16-19 18-05

89 16-13 18-15 16-16 15-34 19-0118-07 16-02 18-06 19-05 15-49

91% --- Surface definition for each assembly at its corresponding position

93 surf s1 sqc -43.12 43.12 10.725surf s2 sqc -21.56 43.12 10.725

95 surf s3 sqc 0.000 43.12 10.725surf s4 sqc 21.56 43.12 10.725

97 surf s5 sqc 43.12 43.12 10.725

99 % --- Surface definition to containing all assembliessurf Score sqc 0.0 0.0 55.245

101

42 Appendix B. Serpent input files

% --- Cell definition , each assembly is filled in corresponding surf definition103 cell c1 0 fill 16-31 -s1

cell c2 0 fill 18-14 -s2105 cell c3 0 fill 16-32 -s3

cell c4 0 fill 16-14 -s4107 cell c5 0 fill 18-12 -s5

109 % --- Cell definition water inside Score and outside boundary condition "outside"cell Ccore 0 water s1 s2 s3 s4 s5 -Score

111 cell out 0 outside Score

113 /******************* Run parameters *

115 ******************/

117 % --- Assembly linear powerset powdens 38.6E-3

119% --- Boundary condition

121 set bc 3

123 % --- Neutron population and seedset pop 500000 2000 400

125 set seed 6942179190

127 % --- XY-plot (3)plot 3 1200 1200

129% --- XY-meshplot (3),

131 mesh 3 1200 1200

133 /**************************************** Settings for the burnup calculation *

135 ***************************************/

137 % --- Burnup points for the burnup calculation (in MWd/kgU)dep butot 0.1 0.5 1 3 5 7 9 11 13 15

139% --- Material divisions for burn up calculation

141 include "asm16 -31nom/div16 -31"include "asm18 -14nom/div18 -14"

143 include "asm16 -32nom/div16 -32"include "asm16 -14nom/div16 -14"

145 include "asm18 -12nom/div18 -12"

147 % --- Calculate material volumes before simulation by% sampling 10 million random points in the geometry

149 set mcvol 100000000

151 % --- Nuclide inventory , nuclides in depletion outputset inventory

153 922340922350

155 922360922370

157 922380922390

159 932360932370

161 932380932390

163 942360942380

165 942390942400

167 942410942420

169 942430952410

171 952420952430

173 952440

B.1. Pin set up 43

952421175 962420

962430177 962440

962450179 962460

962470181 962480

962490183 972490

972500185 982490

982500187 982510

982520189 360830

451030191 451050

471090193 531350

541310195 541350

551330197 551340

551350199 551370

561400201 571400

601430203 601450

601480205 611470

611480207 611490

611481209 621470

621490211 621500

621510213 621520

631530215 631540

631550217 631560

641520219 641540

641550221 641560

641570223 641600

225 % --- Predictor corrector method for the depletion solutionset pcc leli 10 10

227% --- Decay data library

229 set declib "/Libraries/endfb7/sss_endfb7.dec"

231 % --- Neutron induced fission yield libraryset nfylib "/Libraries/endfb7/sss_endfb7.nfy"

233% --- Detectors that calculates pin power distribution in each assembly lattice

235 det pinpowers1 dr -8 void dl 16-31det pinpowers2 dr -8 void dl 18-14

237 det pinpowers3 dr -8 void dl 16-32det pinpowers4 dr -8 void dl 16-14

239 det pinpowers5 dr -8 void dl 18-12

45

Appendix C

Scripting and written codes

Some of the written codes and the scripts used in this thesis are presented here. Gen-erally, the bash scripts are used for pre- and post-processing of data using mathemat-ical calculations. Matlab is used to plot the results. No Matlab code is presented.

C.1 Preprocessing of data

Listings C.1 is the general script for pre-processing, formatting and generating fileswith fuel content using the deterministic simulated data. One file is generated foreach pin and an example is seen in (A) of B.3.

LISTING C.1: A Bash script for pre-processing the isotopic fuel con-tent for the Serpent input files.

#!/bin/sh2

# --- Script for extracting isotropic consentration in each pincell4 # and prepare it for serpent input

# Data from deterministic code simulating real power plant and specific cycles6

for fileDATA in ‘ls | grep cycle1 ‘8 do

cd $fileDATA10 mkdir -p unzipped

12 i=1for fileA in ‘ls pincell.out*.gz ‘

14 docd unzipped

16 echo "$i $fileA"cp ../ $fileA .

18 gunzip -f $fileAfile=‘echo $fileA | sed ’s/.gz//g’‘

20startPattern="ISOTOPE CPHIST CINIT"

22start=‘grep -n "$startPattern" $file | sed ’s/:/ /’ | awk ’print $1’ | tail -1‘

24 total=‘wc $file | awk ’print $1’‘

26 keep=‘echo $total $start | awk ’print $1 -$2’‘tail -$keep $file > file.end

28end=‘grep -n ’CF252 ’ file.end | sed ’s/:/ /’ | awk ’print $1’ | tail -1‘

30

32 pin=‘echo $file | sed ’s/pin/ /g’ | sed ’s/.heigh/ /g’ | awk ’print $2’‘asm=‘echo $file | sed ’s/assembly/ /g’ | sed ’s/.time/ /g’ | awk ’print $2’‘

34 head -$end file.end | awk ’if ($4 >50) print $1,$4’> file.$asm.$pinrm file.end

36for element in CM SN EU KR RU TC RB SR ZR PD CS CE PM SM ND NP PU AM NB I U Y

38 doif [ $element == RU ];then Z=44

46 Appendix C. Scripting and written codes

40 elif [ $element == TC ];then Z=43elif [ $element == RB ];then Z=37

42 elif [ $element == SR ];then Z=38elif [ $element == ZR ];then Z=40

44 elif [ $element == PD ];then Z=46elif [ $element == CS ];then Z=55

46 elif [ $element == CE ];then Z=58elif [ $element == PM ];then Z=61

48 elif [ $element == SM ];then Z=62elif [ $element == ND ];then Z=60

50 elif [ $element == NP ];then Z=93elif [ $element == PU ];then Z=94

52 elif [ $element == AM ];then Z=95elif [ $element == SN ];then Z=50

54 elif [ $element == EU ];then Z=64elif [ $element == KR ];then Z=36

56 elif [ $element == CM ];then Z=96elif [ $element == NB ];then Z=41

58 elif [ $element == I ] ;then Z=53elif [ $element == U ] ;then Z=92

60 elif [ $element == Y ] ;then Z=39else

62 echo "Z not defined for $element"exit

64 fised "s/$element/$element /g" file.$asm.$pin | awk ’if (NF==2) print $0;

66 else if ($2 <100) print $1"0"$2".09c",$3;else print $1$2".09c",$3’ > file.$asm.$pin.new

68 sed "s/$element/$Z/g" file.$asm.$pin.new > file.$asm.$pindone

70sum=‘awk ’sum+=$2 END print sum’ file.$asm.$pin ‘

72 awk -v summ=$sum ’printf "%s %4.5E\n", $1 ,$2/summ’ file.$asm.$pin |sort -k2 -rg > file.$asm.$pin.new

74pinNR=‘echo $asm.$pin.new | sed ’s/.new/ /g’|sed ’s/\./ /g’ | awk ’

76 if (($2 <10 && $3 >=10)) print "0"$2"."$3; else if (($2 <10) && ($3 <10)) print "0"$2".0"$3;else if (($2 >=10)&& ($3 <10)) print $2".0"$3; else print $2"."$3’‘

78 sed -i "1i mat $asm.$pinNR -10.3070 burn 1" file.$asm.$pin.new

80 rm file.$asm.$pinrm $file

82i=‘expr $i + 1‘

84 cd ..done

86 cd ..done

C.2 Post-processing of results

Listing C.2 is one example of how two separate calculations are weighted and com-bined in terms of power distribution and power uncertainty. The script calculates theabsolute uncertainties from the relative uncertainties, which are given by Serpent. Itis done for both the bowed and the nominal calculations and files are created for eachburnup step. The absolute uncertainty is used to calculate the relative uncertainty ofthe relative change in power between the bowed and the nominal simulation. Boththe propagation of the relative uncertainty and the calculation of the relative powerdifference are done with C.3. One file is generated for each burnup step.

LISTING C.2: A Bash script for weighting and combining the powerdistribution and the power uncertainty results from separate simula-tions (here 2). The absolute power uncertainty is calculated from the

relative, for later use. One file is created for each burnup step.

C.2. Post-processing of results 47

1 #!/bin/sh

3 # Script for combining several results (here 2) from serpent with different# seeds and number of cycles and making a wieghted avarage regardning power

5 # distribution. Uncertainty in power distribution is propagated.# Serpent output give relative uncertainty , here it will changed to absolute

7 # to use it again and calculate the relative uncertainty when making the relative# difference between bowed and nominal assemblies.

9file1 =../../ longerCalculation1/Bowing/asm5x5burn_det10.m

11 file2 =../../ longerCalculation2/Bowing/asm5x5burn_det10.m

13 wc=2000w1 =500000

15 w2 =500000

17 paste $file1 $file2 | awk -v wc=$wc -v w1=$w1 -v w2=$w2 ’if (NF == 2) print $1;else if (NF == 6) print $1,$2,$3;

19 else if ($11 >0) printf "%5s%5s%5s%5s%5s%5s%5s%5s%5s%5s %6.5E %6.5f\n",$1 ,$2,$3,$4 ,$5 ,$6,$7,$8 ,$9,$10 ,(1/(wc*w1+wc*w2))*(wc*w1*$11+w2*wc*$23),

21 (1/(wc*w1+wc*w2))*( sqrt((wc*w1*$12*$11 )**2+( wc*w2*$24*$23 )**2))else printf "%5s%5s%5s%5s%5s%5s%5s%5s%5s%5s %6.5E %6.5f\n",

23 $1 ,$2,$3,$4 ,$5 ,$6,$7,$8 ,$9,$10 ,0, sqrt($12 **2+ $24 **2)’ >powercombineBstep10.msed "s/ 0.00000E+00 0.00000//g"

25 powercombineBstep10.m > tmv t powercombineBstep10.m

LISTING C.3: A Bash script for calculating the relative change inpower of the combined results between the bowed and the nominal

case and to calculate the relative uncertainty of this change.#!/bin/sh

2# Script for calculating the relative power distribution change in procent

4 # between bowed and nominal assemblies. The relative uncertainty in procent# is calculated. Last "else printf ..." prints zeros in the 2 last collumns

6 # if previous statement dont hold.

8 file1 =../ powercombineBowing/absoluteerror/powercombineBstep8.mfile2 =../ powercombineAsm5x5nom2/absoluteerror/powercombineBstep8.m

10paste $file1 $file2 | awk ’if (NF == 2) print $1;

12 else if (NF ==6) print $1,$2 ,$3;else if ($11 >0) printf "%5s%5s%5s%5s%5s%5s%5s%5s%5s%5s %6.5E %6.5f\n",

14 $1 ,$2,$3,$4 ,$5 ,$6,$7,$8 ,$9,$10 ,100*($11 -$23)/$11 ,sqrt ((($23*$12 )**2+( $11*$24 )**2)/( $11 **4));

16 else printf "%5s%5s%5s%5s%5s%5s%5s%5s%5s%5s %6.5E %6.5f\n",$1 ,$2,$3,$4 ,$5 ,$6,$7,$8 ,$9,$10 ,0,0’ >powerdiffBstep8.m

18sed "s/ 0.00000E+00 0.00000//g"

20 powerdiffBstep8.m > tmv t powerdiffBstep8.m

As for the power results, results with isotopic composition data from differentsimulations are weighted and combined, for the bowed and the nominal case, re-spectively. C.4 shows the script for the bowed case. New files with the combinedresults are generated for the two cases. With C.5 is the relative change between thebowed and the nominal case calculated, together with some formatting for Matlab-readability.


LISTING C.4: A Bash script for weighting and combining isotopiccomposition results from separate simulations.

1 #!/bin/sh

3 # Script for combining results from serpent with different seeds and number of cycles into one ,# making a wieghted avarage regardning isotopic composition.

5for fileA in ‘ls ../ longerCalculation2/asm*Bowingcompositionnew.m‘

7 doasm=‘echo $fileA | sed ’s/asm/ /g’ | sed ’s/Bowing */ /g’ | awk ’print $2’‘

9 echo $asm

11 file1 =../ longerCalculation1/asm$asmBowingcompositionnew.mfile2 =../ longerCalculation2/asm$asmBowingcompositionnew.m

13wc=2000

15 w1 =500000w2 =500000

17 zeros = ’0.000000000 ’

19 paste $file1 $file2 | awk -v wc=$wc -v w1=$w1 -v w2=$w2 -v zeros=$zeros ’if (NF == 6) print $1,$2 ,$3 ;

21 else if (NF == 2) print $1;else if ($1 >0 || $2 >0 || $3 >0 || $4 >0 || $5 >0 || $6 >0 || $7 >0 || $8 >0 || $9 >0 || $10 >0 || $11 >0)

23 printf "%6.5E %6.5E %6.5E %6.5E %6.5E %6.5E %6.5E %6.5E %6.5E %6.5E %6.5E\n",(1/(wc*w1+wc*w2))*(wc*w1*$1+w2*wc*$12), (1/(wc*w1+wc*w2))*(wc*w1*$2+w2*wc*$13),

25 (1/(wc*w1+wc*w2))*(wc*w1*$3+w2*wc*$14), (1/(wc*w1+wc*w2))*(wc*w1*$4+w2*wc*$15),(1/(wc*w1+wc*w2))*(wc*w1*$5+w2*wc*$16), (1/(wc*w1+wc*w2))*(wc*w1*$6+w2*wc*$17),

27 (1/(wc*w1+wc*w2))*(wc*w1*$7+w2*wc*$18), (1/(wc*w1+wc*w2))*(wc*w1*$8+w2*wc*$19),(1/(wc*w1+wc*w2))*(wc*w1*$9+w2*wc*$20), (1/(wc*w1+wc*w2))*(wc*w1*$10+w2*wc*$21),

29 (1/(wc*w1+wc*w2))*(wc*w1*$11+w2*wc*$22);else printf "%6.5E %6.5E %6.5E %6.5E %6.5E %6.5E %6.5E %6.5E %6.5E %6.5E %6.5E\n" ,

31 $zeros ,$zeros ,$zeros ,$zeros ,$zeros ,$zeros ,$zeros ,$zeros ,$zeros ,$zeros ,$zeros’> composition$asmBowingcombine.m

33done

LISTING C.5: A bash script for post-possessing files from C.4 andmake them Matlab-readable. The relative difference of the bowed and

the nominal case is calculated.#!/bin/sh

2# Script for preprocessing files with isotopic mass density

4 # for each assembly and make them MATLAB readable.# The script also accunts for the control rods and insert

6 # zero matrices at their place - for plotting reasons.# When the files are generated the relative differance between

8 # nominal and bowed case is calculated and stored for each assembly

10 inputfile=asm5x5burn_deptest.m

12 n=1for n in 1..25

14 dofor count2 in 1..24

16 docd ../ Calculations/longerCalculation8/asm5x5nom2

18 count=‘echo $count2 | awk ’print 205*$1’‘assembly=‘ grep -m $count "_VOLUME =" $inputfile | tail -1 | sed ’s/MAT_/ /g’ |

20 sed ’s/\./ /g’ | awk ’print $1’‘#assembly =19 -63

22 assemblySort=‘echo $assembly | sed ’s/-//g’‘if [ ! -f /comp*analysis/longer*n8/asm$assemblySortAsm5x5nom2composition.m ];

24 thenecho $count $assembly $assemblySort $PWD

26 i=1#if [ -f ../ asm$assemblySortAsm5x5nom2composition.m ] ;


28 #then rm ../ asm$assemblySortAsm5x5nom2composition.m;fifor i in ‘seq 1 205‘

30 dogrep -A 73 -m $i "MAT_$assembly .* _MDENS" $inputfile | tail -74 |

32 awk ’print $1,$2,$3 ,$4,$5,$6,$7 ,$8,$9,$10 ,$11’ | sed ’s/--/ /g’ |sed "s/MAT_$assembly .*/ asm$assemblySortMDENS$i = \

34 [/" >> ../../../ comp*analysis/longer*n8/asm$assemblySortAsm5x5nom2composition.mdone

36 fi

38 cd ../ Bowingif [ ! -f /comp*analysis/longer*n8/asm$assemblySortBowingcomposition.m ];

40 theni=1

42 #if [ -f ../ asm$assemblySortBowingcomposition.m ] ;#then rm ../ asm$assemblySortBowingcomposition.m;fi

44 echo $count $assembly $assemblySort $PWDfor i in ‘seq 1 205‘

46 dogrep -A 73 -m $i "MAT_$assembly .* _MDENS" $inputfile | tail -74 |

48 awk ’print $1,$2,$3 ,$4,$5,$6,$7 ,$8,$9,$10 ,$11’ | sed ’s/--/ /g’ |sed "s/MAT_$assembly .*/ asm$assemblySortMDENS$i = \

50 [/" >> ../../../ comp*analysis/longer*n8/asm$assemblySortBowingcomposition.mdone

52 fidone

54cd ../ compositionanalysis/longerCalculation8

56# Include zeros at the position of the controlrods - for plotting reasons

58 for file in ‘ls asm*Bowingcomposition.m‘do

60 assemblyname=‘echo $file | sed ’s/asm/ /g’ | sed ’s/Bowing/ /g’ | awk ’print $1’‘

echo $assemblyname62

if [ ! -f /comp*analysis/longer*n8/asm$assemblynameBowingcompositionnew.m ];64 then

cp asm$assemblynameBowingcomposition.m tmp66

#Positions > 33 is the position number minus # of zero assemblies incerted before68 for pos in 33 35 38 40 61 63 65 71 80 101 106 127 136 142 144 146 167 169 172 174

do70 asmpos=‘grep -n "MDENS$pos " tmp | sed ’s/:/ /g’ | awk ’print $1 -1’‘

head -$asmpos tmp > tmp$asmpos72 cat tmp$asmpos ../ compositionanalysis/longer*n8/rodzeros > tmp2

mv tmp2 tmp$asmpos74 P=‘wc tmp | awk -v asmpos=$asmpos ’print $1-asmpos’‘

tail -$P tmp >> tmp$asmpos76 mv tmp$asmpos tmp

done78

i=180 while [ $i -le 225 ]

do82 asmpos=‘grep -n -m $i "MDENS" tmp | sed ’s/:/ /g’ | tail -1 |awk ’print $1’‘

sed -i "$asmposs/.* MDENS .*/ asm$assemblynameMDENS$i = [/" tmp84 i=‘expr $i + 1‘

done86 mv tmp asm$assemblynameBowingcompositionnew.m

88 fidone

90# Ending for 25 loop

92 done

94 # Adding zeros for control rodsif [ -f /compositionanalysis/longer*n8/asm1906Bowingcompositionnew.m ];

96 then

98 for file in ‘ls asm*Asm5x5nom2composition.m‘


do100 assemblyname=‘echo $file | sed ’s/asm/ /g’ | sed ’s/Asm5x5nom2/ /’ | awk ’print $1’ ‘

echo $assemblyname102

if [ ! -f /comp*analysis/longer*n8/asm$assemblynameAsm5x5nom2compositionnew.m ];104 then

cp asm$assemblynameAsm5x5nom2composition.m tmp106

#Positions > 33 is the position number minus # of zero assemblies incerted before108 for pos in 33 35 38 40 61 63 65 71 80 101 106 127 136 142 144 146 167 169 172 174

do110 asmpos=‘grep -n "MDENS$pos " tmp | sed ’s/:/ /g’ | awk ’print $1 -1’‘

head -$asmpos tmp > tmp$asmpos112 cat tmp$asmpos ../ compositionanalysis/longer*n8/rodzeros > tmp2

mv tmp2 tmp$asmpos114 P=‘wc tmp | awk -v asmpos=$asmpos ’print $1-asmpos’‘

tail -$P tmp >> tmp$asmpos116 mv tmp$asmpos tmp

done118

i=1120 while [ $i -le 225 ]

do122 asmpos=‘grep -n -m $i "MDENS" tmp | sed ’s/:/ /g’ | tail -1 |awk ’print $1’‘

sed -i "$asmposs/.* MDENS .*/ asm$assemblynameMDENS$i = [/" tmp124 i=‘expr $i + 1‘

done126 mv tmp asm$assemblynameAsm5x5nom2compositionnew.m

fi128

# Taking the relative differance in massdensity of isotopic composition130 # between nomonal and bowed case

file1=asm$assemblynameBowingcompositionnew.m132 file2=asm$assemblynameAsm5x5nom2compositionnew.m

if [ ! -d compositiondiff ];then134 mkdir compositiondiff

fi136 paste $file1 $file2 | awk ’if (NF == 2) print $1; else if (NF ==6) print $1,$2,$3;

else if($1 >0) printf "%6.5E\n" ,100*($1-$12)/$1; else printf "%6.5E\n", $1’ > *diff/t1138 paste $file1 $file2 | awk ’if (NF == 2) print ""; else if (NF == 6) print "";

else if($2 >0) printf "%6.5E\n" ,100*($2-$13)/$2; else printf "%6.5E\n", $2’ > *diff/t2140 paste *diff/t1 *diff/t2 > *diff/t3

mv *diff/t3 *diff/t1142

paste $file1 $file2 | awk ’if (NF == 2) print ""; else if (NF == 6) print "" ;144 else if($3 >0) printf "%6.5E\n" ,100*($3-$14)/$3; else printf "%6.5E\n", $3’ > *diff/t2

paste *diff/t1 *diff/t2 > *diff/t3146 mv *diff/t3 *diff/t1

148 paste $file1 $file2 | awk ’if (NF == 2) print ""; else if (NF == 6) print "" ;else if($4 >0) printf "%6.5E\n" ,100*($4-$15)/$4; else printf "%6.5E\n", $4’ > *diff/t2

150 paste *diff/t1 *diff/t2 > *diff/t3mv *diff/t3 *diff/t1

152paste $file1 $file2 | awk ’if (NF == 2) print ""; else if (NF == 6) print "" ;

154 else if($5 >0) printf "%6.5E\n" ,100*($5-$16)/$5; else printf "%6.5E\n", $5’ > *diff/t2paste *diff/t1 *diff/t2 > *diff/t3

156 mv *diff/t3 *diff/t1




164 else if($7 >0) printf "%6.5E\n" ,100*($7-$18)/$7; else printf "%6.5E\n", $7’ > *diff/t2paste *diff/t1 *diff/t2 > *diff/t3

166 mv *diff/t3 *diff/t1


170 paste *diff/t1 *diff/t2 > *diff/t3


mv *diff/t3 *diff/t1172

paste $file1 $file2 | awk ’if (NF == 2) print ""; else if (NF == 6) print "" ;174 else if($9 >0) printf "%6.5E\n" ,100*($9-$20)/$9; else printf "%6.5E\n", $9’ > *diff/t2

paste *diff/t1 *diff/t2 > *diff/t3176 mv *diff/t3 *diff/t1

178 paste $file1 $file2 | awk ’if (NF == 2) print ""; else if (NF == 6) print "" ;else if($10 >0) printf "%6.5E\n" ,100*($10 -$21)/$10; else printf "%6.5E\n", $10’ > *diff/t2



184 else if($11 >0) printf "%6.5E\n" ,100*($11 -$22)/$11; else printf "%6.5E\n", $11’ > *diff/t2paste *diff/t1 *diff/t2 > *diff/t3

186 mv *diff/t3 *diff/asm$assemblynamecompositiondiffnew.m# cd asm5x5nom2

188done

190fi

192#done

53

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