Numerical Modeling of Neutron Flux

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NUMERICAL MODELLING OFNEUTRON FLUX IN NUCLEAR

REACTORS

Bachelors Thesis

submitted by

Milan Hanus

completed at

Department of MathematicsFaculty of Applied Sciences

University of West Bohemia

under supervision of

Ing. Marek Brandner, Ph.D.

Pilsen, June 2007


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Declaration

I hereby declare that this Bachelors Thesis is the result of my own work and that allexternal sources of information have been duly acknowledged.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Milan Hanus


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Abstract in English

The aim of this thesis is to develop an efficient method that describes neutron distribu-tion within a reactor. The method is based on neutron transport equation which can bederived from physical foundations presented at the beginning of the thesis. By using a

general balance principle, an accurate albeit overly complex neutron transport equation isobtained from the derivation. Simplications are discussed next and lead to a two-groupsystem of steady state diffusion equations. An eigenvalue problem is then formulated forthe established boundary value problem whose dominant eigenpair is the critical numberof the reactor and a corresponding neutron ux distribution. Application of reactor criti-cality calculations in the area of fuel reloading optimization is explained. The eigenvalueproblem is solved by the power method in which the dominant eigenvector is determinedby solving a set of algebraic equations arising from nite-volume discretization scheme. Toobtain a sufficiently accurate solution on a coarse hexagonal assembly mesh, the standardnite-volume method is rened by a nodal method based on a special transverse integrationprocedure. For solving the one-dimensional diffusion problems arising from the procedure,semi-analytic method is employed. The resulting numerical scheme is then benchmarkedon a model VVER-1000 type reactor conguration. Improvement directions are discussedat the end.

Keywords:nuclear reactor, reactor physics, neutron transport equation, neutron balance, neutron dif-fusion equation, two-group approximation, reactor criticality, eigenvalue problem, powermethod, nite volume method, CMFD system, neutron ux, neutron current, nodal method,transverse integration, semi-analytic method, hexagonal assembly, fuel reloading optimiza-

tion.


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Abstrakt v ce stin e

V praci je navrzena efektivn metoda pro popis rozlozen neutronu v jadernem reaktoru,vych azejc z transportn rovnice neutronu. Teoretick y prehled reaktorove fyziky v druhekapitole slouz k pops an velicin potrebn ych pro odvozen teto rovnice. Pro prakticke

vypocty je vsak obecn y transportn model prlis slozit y, a tak pr ace pokracuje jeho difuzndvougrupovou aproximac ve stacion arnm rezimu. Pro vzniklou okrajovou ulohu difuzeneutronu je d ale formulov ana uloha na vlastn csla, jejz dominantn resen popisuje tzv.kriticke cslo reaktoru a prslusne rozlozen neutronov ych toku. V pr aci je blze rozebr anaaplikace tohoto v ysledku na problem optimalizace palivov ych vs azek. Dominantn vlastncslo je hled ano iterativn mocninnou metodou, pricemz prslusn y vlastn vektor je v kazdemkroku zsk an resenm soustavy rovnic sestavene na z aklade diskretizace dane okrajoveulohy. Tu je mozne provest napr. metodou konecn ych objemu, avsak pro potreby conejvernejsho popisu neutronov ych toku v reaktoru s sesti uhelnkov ymi kazetami je nutnoklasicky konecne-objemov y prstup zpresnit. K tomu je v pr aci pouzita tzv. nod aln metodasest ava jc ze speci aln prcne integrace difuznch rovnic a n asledneho semi-analytickehoresen vznikl ych jednorozmern ych problemu. V ysledkem je efektivn a presn a numerick ametoda, jez je v z averu pr ace otestov ana na modelove palivove konguraci reaktoru typuVVER-1000. Pr ace je zakoncena prehledem moznost jejho vylepsen.

Klcova slova: jaderny reaktor, reaktorov a fyzika, transportn rovnice neutronu, neutronov a bilance, di-fuzn rovnice neutronu, dvougrupov a aproximace, kriticke cslo reaktoru, uloha na vlastncsla, mocninn a metoda, metoda konecn ych objemu, CMFD soustava, neutronove toky,neutronove proudy, nod aln metoda, prcn a integrace, semi-analytick a metoda, sesti-

uhelnkov a palivova kazeta, optimalizace palivove vsazky.


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Acknowledgements

I would like to thank my supervisor Ing. Marek Brandner, Ph.D., for his constant sup-port and guidance throughout the course of thesis preparation. Without his insight, myunderstanding of the subject would never reach present levels. I would also like to thank

Ing. Roman Kuzel, Ph.D., for his invaluable advises on implementation of the methods inMATLAB. Finally, I am gratefull to all who had patience with me in times when progressbecame slow and time-consuming.


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i

Contents

List of Symbols iii

List of Figures xviii

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Overview of solution methods . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Physical background 52.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Neutron-nuclei interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Neutron properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Quantifying interactions . . . . . . . . . . . . . . . . . . . . . . . . 82.3.3 Types of reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.4 Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.5 Radiative capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Chain Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Mathematical model 173.1 Neutron transport theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Neutron transport equation . . . . . . . . . . . . . . . . . . . . . . 173.1.2 Steady state formulation . . . . . . . . . . . . . . . . . . . . . . . . 253.1.3 Conditions on angular ux . . . . . . . . . . . . . . . . . . . . . . . 263.1.4 Criticality calculations . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Neutron diffusion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.2 Neutron diffusion equation . . . . . . . . . . . . . . . . . . . . . . . 293.2.3 Diffusion conditions on ux and current . . . . . . . . . . . . . . . 31

3.3 Multigroup approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.1 Two group model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36


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ii CONTENTS

4 Numerical solution 394.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.1 Spatial domain discretization . . . . . . . . . . . . . . . . . . . . . 424.2.2 Discretization of the nodal balance relation . . . . . . . . . . . . . . 46

4.3 Finite volume scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.2 Approximation of integral averages of neutron currents . . . . . . . 504.3.3 Numerical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3.4 Solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4 Nodal methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.4.1 Modication of the FV scheme . . . . . . . . . . . . . . . . . . . . 594.4.2 Two-node subdomain problems . . . . . . . . . . . . . . . . . . . . 614.4.3 Numerical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4.4 Solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Numerical results 795.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2 Solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3.1 = 0.5, at leakage approximation . . . . . . . . . . . . . . . . . . 835.3.2 = 0.5, quadratic leakage approximation . . . . . . . . . . . . . . . 855.3.3 = 0.125, at leakage approximation . . . . . . . . . . . . . . . . . 87

5.3.4 = 0.125, quadratic leakage approximation . . . . . . . . . . . . . 896 Conclusion 91

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.2 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


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iii

List of Symbols

Usage guideline

The list of letters used throughout the text is divided into two categories: those writtenin Roman script and those in Greek script. The entries are sorted alphabetically in eachcategory, and the page where they rst appear in the text is displayed after each item. If some letter is used with different meanings in different chapters, a remark is made just afterits statement. Letters not used beyond rst few lines after their denition are normally notlisted. If the letter denotes some quantity for which there is a numbered denition relationin the text, the equation number is referenced after the description of the quantity.

Dependency of functions is written in terms of variables used in the place of their rstappearance. Transport theory functions are in the text rst presented in a general time-dependent form and this dependence is later removed by the assumption of steady state(section 3.1.2). Basic meaning of both forms of the function is preserved, however, sothe list includes only the general time-dependent functions for the sake of brevity. If forany other reason two different symbols share the same meaning (e.g. due to omission of some indices that became redundant under additional assumptions), they are separated bysemicolon but make one entry in the list.

The table of general non-letter symbols follows. For greater clarity, auxiliary letters areused together with symbols of universal meaning (e.g. of an operation applicable to anyarbitrary function). The letters are:

a , b any arbitrary vectors

q an arbitrary function

W an arbitrary domainIn addition to explanations of indexed letters in the above mentioned lists, commonly

used indices are also enumerated separately in the next two tables (for subscripts, resp.superscripts). Listing of acronyms concludes the summary.


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iv List of Symbols

Roman letters

A atomic mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

A 22 matrix given by R P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71a 0, a1, a2 21 vectors of the rst, second, resp. third expansion coefficients (foreach group) of the particular solution of transverse integrated ODE 69a 3, a4 21 vectors of the integration constants (for each group) in the homo-geneous solution of the transverse integrated ODE . . . . . . . . . . . . . . . . . 69B k 22 matrices used in the semi-analytic method (eq. 4.94) . . . . . . . . 71C ; C i 2 2 auxiliary matrix for the relationship between ux and current atthe same interface ( (4.101a) or (4.101b)) . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2D(r , E ) diffusion coefficient (eq. 3.34) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

CD gi, coupling correction factors for current approximations . . . . . . . . . . . . . 59

C D 2N 2N matrix of nodal coupling correction factors. . . . . . . . . . . . . . . 61D g(r ); Dg group-discretized diffusion coefficient (eq. 3.45) . . . . . . . . . . . . . . . . . . . . 35

D gi discrete value of diffusion coefficient homogenized over node V i . . . . . 51D gi, approximations of diffusion coefficients at interfaces between two nodes

in the directions (eq. (4.394.41)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53D 22 matrix of group diffusion coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 69E neutron energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

E k kinetic energy of a neutron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

E energy of neutrons not belonging to the balance set . . . . . . . . . . . . . . . . 19

e x , eu , ev; e unit vectors in local hexagonal directions . . . . . . . . . . . . . . . . . . . . . . . . . . 45

e 21 vector with even constants for the denition (4.97) of x . . . 72e J 21 vector with even constants for the denition (4.98) of J x . . . . 72f (x) transverse leakage shape function (eq. 4.86) . . . . . . . . . . . . . . . . . . . . . . . 6 7

G 22 diagonal matrix of boundary condition coefficients g . . . . . . . . 74


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List of Symbols v

G number of energy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

h width of a node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

J gi, numerical approximation of positively oriented face-averaged currentacross any arbitrary face of node V i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

J i, nite volume approximation of neutron current averaged over face i, (eq. 4.27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

CJ gi, corrected nite volume approximation of neutron current averaged overface i, (eq. 4.52) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

J gi,x approximation of transversly averaged currents at i,x . . . . . . . . . . . . 66

J x ; J i,x 2 2 auxiliary matrix for the relationship between ux and current atthe same interface ( (4.101a) or (4.101b)) . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2J x ; J i,x 2 1 vectors of transverse averaged currents (for each group) at sidesi,x (eq. 4.98) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 j gi,x , j

gi,u , j

gi,v ; j

gi, positively oriented face-averaged currents across the respective faces of node V i (eq. 4.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

j gi,

positively oriented face-averaged current across any arbitrary face of node V i (eq. 4.9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

j(r , E, , t ) transport theory expression for for neutron current ( angular neutron current ) (eq. 3.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

j(r , E ) diffusion theory expression for neutron current (angularly independentnet neutron current ) (eq. 3.24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

jg(r ) group-discretized neutron current (eq. 4.1) . . . . . . . . . . . . . . . . . . . . . . . . . 4 2

K eff numerically computed value of keff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

keff effective multiplication constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

lgi, face-averaged net leakage across the two edges perp. to direction . 48

lgi total area-averaged neutron leakage across the transverse boundaries of node V i (eq. 4.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

lgt,i (x) transverse leakage term (repr. neutron leakage across transverse bound-aries at point x, integrated over the transverse cross section at thatpoint) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63


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vi List of Symbols

lgi (x) transverse leakage term of the 1D diffusion equation for transverse av-eraged ux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Lgi nite volume approximation of total average leakage out of node V i (anapproximation of lgi ) (eq. 4.42) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

CLgi, corrected nite volume approximation of the face-averaged net neutroncurrent (average leakage) across faces perpendicular to the -direction(eq. (4.564.58)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

CLgi,y t approximate leakage through transverse boundaries yt,i (x), averagedover the area of the node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Lgi (x) approximation of l

gi (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Lgi approximate leakage through transverse boundaries, integrally averagedover all transverse cross sections of the node along the x-axis (Lgi (x)averaged over the width of the node) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Lgi, nite volume approximation of the face-averaged neutron leakage in the -direction (eq. (4.394.41)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

L g N N matrix of average neutron leakages . . . . . . . . . . . . . . . . . . . . . . . . . 57C L g N

N matrix of average neutron leakages, approximated using the

corrected (nodal) scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

L (x); L i(x) 2 1 vector of average transverse leakages in respective groups . . . . 69M 2N 2N neutron migration matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57C M 2N 2N neutron migration matrix with leakage components approxi-mated by corrected (nodal) scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61M number of rows in the nodal mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

M number of core rows in direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

me edge length (eq. 4.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

mN node area (eq. 4.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

N (in Chap. 2) number of nuclei in a unit volume of a sample . . . . . . . . . . . . . . . . . . . . . . 8

N (in Chap. 4) total number of nodes (assemblies) in the reactor . . . . . . . . . . . . . . . . . . 44

N (r , E, , t ) transport theory expression for the elementary number of neutrons ( an-gular neutron density ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18


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viii List of Symbols

sgt,i (x) transverse integrated neutron sources density in group g . . . . . . . . . . . 63

S

gi approximate group sources density, integrally averaged over all trans-verse cross sections of the node along the x-axis (S gi (x) averaged over

the wid th o f the node) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

S f 2N 2N ssion sources matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57T a mesh of hexagonal nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42t time variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

u local spatial variable (direction) associated to a node

V neutron balance domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20v local spatial variable (direction) associated to a node

v speed of a neutron (a scalar) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

v (E ) velocity of a neutron (a vector) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

V i an arbitrary reference node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44x local spatial variable (direction) associated to a node (identied with

the Cartesian x-direction)

x ij point with Cartesian coordinates [ xi , y j ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3

yt,ij (x); yt,i (x); yt (x)a function representing the top boundary of node V i (centered at pointx ij ) (eq. 4.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Z atomic number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5


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List of Symbols ix

Greek letters

albedo constant (a ratio of the number of neutrons escaping the coreand the number entering it through the same point) (eq. 3.20) . . . . . 27

p(E ) energy spectrum of prompt ssion neutrons . . . . . . . . . . . . . . . . . . . . . . . . 12

p(r , E E, ); (r , E E, )transport theory expression for the energetic spectrum of prompt ssionn e u t r o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1

1

4 (E

E ) diffusion theory expression for the energy spectrum of ssion neutrons

(normalized isotropic ssion spectrum ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8

g group-discretized ssion spectrum (eq. 3.43) . . . . . . . . . . . . . . . . . . . . . . . 34

d(E ) energy spectrum of delayed ssion neutrons . . . . . . . . . . . . . . . . . . . . . . . . 12

convergence criterion for the eigenvalue updating method . . . . . . . . . . 58

neutron ux (eq. 2.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

(r , E, , t ) transport theory expression for neutron ux ( angular neutron ux )

(eq. 3.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

g(r ); g group discretized neutron ux (eq. 3.44) . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

(r , E ) diffusion theory expression for neutron ux (angularly independent total ux density ) (eq. 3.23) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

[1(r ), 2(r )]i i-th eigenvector of the two-group criticality problem . . . . . . . . . . . . . . . 37

gi numerical approximation of the integral average of ux over node V i(eq. (4.23)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50gi integral average of ux over node V i (eq. 4.6) . . . . . . . . . . . . . . . . . . . . . . 46gi,x + numerical approximation of the integral average of ux over face i,x +

52

gt,i (x) transverse integrated neutron ux (eq. 4.66) . . . . . . . . . . . . . . . . . . . . . . 63

gi (x) inexact version of gi (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

gi (x) transverse averaged neutron ux (eq. 4.69) . . . . . . . . . . . . . . . . . . . . . . . . 6 4


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x List of Symbols

gi approximate ux, integrally averaged over all transverse cross sectionsof the node along the x-axis (gi (x) averaged over the width of the node)

66 i 21 column vector of ux averages for node V i and both groups . . 70 g N 1 column vector of ux averages for all nodes and groups . . . . . 57 x ; i,x 2 2 auxiliary matrix for the relationship between ux and current atthe same interface ( (4.101a) or (4.101b)) . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 (x); i(x) 21 vector of transverse averaged uxes in respective groups (the com-plete solutions to the transverse integrated ODE) . . . . . . . . . . . . . . . . . . 69 h (x) 21 vector of the homogeneous solutions (for each group) of the trans-verse integrated ODE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 p(x) 21 vector of the particular solutions (for each group) of the transverseintegrated ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 x ; i,x 21 vectors of transverse averaged uxes (for each group) at sides i,x (eq. 4.97) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 boundary condition coefficient (eq. 3.40) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3

i,x , i,u , i,v ; i, faces of node V i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 criticality parameter (eigenvalue of the associated eigenvalue problem

(4.1)) (eq. 3.21) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

i i-th eigenvalue of the two-group criticality problem . . . . . . . . . . . . . . . . 37

1 smallest eigenvalue of the two-group criticality problem ( principal eigen-value ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

average number of neutrons released by ssion. . . . . . . . . . . . . . . . . . . . . 12

p (resp. d) average number of prompt (resp. delayed) neutrons released by ssion12

p(r , E ); (r , E ) transport theory expression for prompt ssion neutrons yield . . . . . . 20

the whole core domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

unit direction vector of neutron movement . . . . . . . . . . . . . . . . . . . . . . . . . 17

direction vector of neutrons not belonging to the balance set . . . . . . . 19


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List of Symbols xi

macroscopic cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

(in Chap. 4) any arbitrary face of a given node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

(in Chap. 2) microscopic cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

macroscopic cross section for an arbitrary reaction . . . . . . . . . . . . . . . . . . 9

(r , E , t ) transport theory expression for a general macroscopic cross section . 19

g N N matrix of cross sections substituted by . . . . . . . . . . . . . . . . . . . 5 7a (resp. a ) microscopic (resp. macroscopic) cross section for neutron absorption 10

a (r , E , t ) transport theory expression for the macroscopic absorption cross section23

ga (r ); ga group-discretized absorption cross section (eq. 3.45) . . . . . . . . . . . . . . . 35

f (resp. f ) microscopic (resp. macroscopic) cross section for ssion . . . . . . . . . . . . . 8

f (r , E , t ) transport theory expression for the ssion cross section . . . . . . . . . . . . 21

gi,f discrete value of the ssion cross section for node V i (eq. 4.14) . . . . . 49 gf (r );

gf group-discretized ssion cross section (eq. 3.45) . . . . . . . . . . . . . . . . . . . . 3 5

gr (r ); gr total neutron removal cross section (eq. 3.47) . . . . . . . . . . . . . . . . . . . . . . 36

gi,r discrete value of the neutron removal cross section for node V i (eq. 4.12)49s (resp. s ) microscopic (resp. macroscopic) cross section for scattering . . . . . . . . . 9

s (r , E E, , t )transport theory expression for the macroscopic scattering cross section(differential scattering cross section ) (eq. 3.1) . . . . . . . . . . . . . . . . . . . . . . 1 9

s (r ,E , t ) transport theory expression for the macroscopic scattering cross sectionwithout the explicit mentioning of energy and direction change . . . . 23

12i,s discrete value of the neutron in-scatter cross section for node V i (eq. 4.13)49 g gs (r ); g gs cross section for neutron scattering from group g into group g (group-

discretized in-scatter cross section) (eq. 3.45) . . . . . . . . . . . . . . . . . . . . . . 35

gs (r ); gs cross section for neutron scattering from group g into any energy group(group-discretized out-scatter cross section) (eq. 3.45) . . . . . . . . . . . . . 35


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xii List of Symbols

14 s (r , E E ) diffusion theory expression for the differential scattering cross section(normalized isotropic ssion spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 a symbol representing either of the node-wise spatial variables x, u, v

44


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List of Symbols xiii

Symbols

a b dot product of vectors a , b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 operator nabla : formally = ( x , y , z ) in Chap. 3, = ( x , y ) inChap. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

0 N N zero matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 a separator between the integration symbols and the d E d symbolspecifying the balance neutrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20dE elementary (differential) energy range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

dE d differential subset of neutron phase space describing neutrons involvedin the balance relation ( balance neutrons ) . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0

d elementary (differential) solid angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

dr elementary (differential) spatial volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

d integration symbol for line integrals over a face . . . . . . . . . . . . . . . . . . . . 45

dS differential surface element (a scalar value) . . . . . . . . . . . . . . . . . . . . . . . . 19dS oriented differential surface element (a vector) . . . . . . . . . . . . . . . . . . . . . 19

dt elementary (differential) time interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

q k(x), q l(x) inner product def. by eq. (4.90) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

U-235 isotope

AZX nuclide

W boundary of some domain W Q numerical approximation of quantity q

q (r , . . .) W , q (W )limit of function q as r approaches W from inside, resp. outside of W )

q integral average of quantity q over a node

q integral average of quantity q over all transverse cross sections of a nodealong the x-axis


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xiv List of Symbols

q (x) integral average of quantity q over a transverse cross section of a nodeat point x

q integral average of quantity q over a faceqt time partial derivative of function q

qx ,

qy ,

qz spatial partial derivatives of q


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List of Symbols xv

Subscripts

placeholder for an arbitrary reaction type (ssion, absorption, scatter-ing)

a absorption

x, u, v; indices of nodal faces, perpendicular to the x, u and v (coll. ) directions(respectively), with positive (plus sign), resp. negative (minus sign)orientation with respect to that direction

f ssion

h homogeneous solution of ODE

i, k linear index of a node

i 1/ 2, i + 1 / 2 index of point lying on the left, resp. right face of node V ii + 1 index of node adjacent to V i in a positive x-directioni x, i u, i v; i indices of neighbouring nodes of V i j row index of nodal mesh

n index representing relation to normal vector n

p (in Chap. 4) particular solution of ODE

p (in Chap. 2, 3) related to prompt ssion neutrons

s scattering

t for denoting transverse integrated quantities, also used in the symbolfor transverse prole function

x, u, v denote quantities oriented in x, u, resp. v directions

a placeholder for either of the indices x, u, v

denotes the matrix innity norm


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xvi List of Symbols

Superscripts

excited state of a nuclide

C prescript to symbolize quantities approximated by the nodal-correctedscheme

g index of actual energy group

1 related to fast energy group

2 related to thermal energy group

g index of some group different from the actual group g (also summationindex over energy groups)

( ) iteration index


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List of Symbols xvii

Acronyms

ANC-HM A dvanced N odal C ode for H exagonal Geometry using ConformalM apping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

ANM A nalytic N odal M e t h o d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1

CMFD C oarse M esh F inite D ifference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

FDM F inite D ifference M ethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

FV F inite V olume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

GET G eneralized E quivalence T heory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

NEM N odal E xpansion M ethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

ODE ordinary differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

PDE P artial D ifferential E quation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

PWR P ressurized Light- W ater-Moderated and Cooled R eacto r. . . . . . . . . . . 1

VVER V oda-V odyanoi E nergetichesky R eaktor (Russian term for the pressur-

ized water reactor) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2


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xviii List of Symbols


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xix

List of Figures

2.1 Proton and neutron potentials . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Binding energy per nucleon . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Fission spectrum of prompt neutrons for U-235 . . . . . . . . . . . . . . . 122.4 Energetic dependence of ssion cross section . . . . . . . . . . . . . . . . . 14

3.1 Position and movement direction of neutrons . . . . . . . . . . . . . . . . . 18

4.1 Hexagonal node with its dimensions . . . . . . . . . . . . . . . . . . . . . . 434.2 A core domain composed of 163 nodes . . . . . . . . . . . . . . . . . . . 444.3 Nodal neighbourhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1 Reactor loading map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2 Table of material constants (from ref. [Wag89]) . . . . . . . . . . . . . . . 805.3 Sparsity of power iteration matrix . . . . . . . . . . . . . . . . . . . . . . . 815.4 Legend for power densities maps . . . . . . . . . . . . . . . . . . . . . . . . 825.5 Normalized power densities and relative errors ( = 0.5, at leakage) . . . 835.6 Node-wise power distribution prole ( = 0.5, at leakage) . . . . . . . . . 835.7 Eigenvalue convergence ( = 0.5, at leakage) . . . . . . . . . . . . . . . . 845.8 Convergence of correction factors ( = 0.5, at leakage) . . . . . . . . . . . 845.9 Normalized power densities and relative errors ( = 0.5, quadratic leakage) 855.10 Node-wise power distribution prole ( = 0.5, quadratic leakage) . . . . . . 855.11 Eigenvalue convergence ( = 0.5, quadratic leakage) . . . . . . . . . . . . . 865.12 Convergence of correction factors ( = 0.5, quadratic leakage) . . . . . . . 865.13 Normalized power densities and relative errors ( = 0.125, at leakage) . . 87

5.14 Node-wise power distribution prole ( = 0.125, at leakage) . . . . . . . . 875.15 Eigenvalue convergence ( = 0.125, at leakage) . . . . . . . . . . . . . . . 885.16 Convergence of correction factors ( = 0.125, at leakage) . . . . . . . . . . 885.17 Normalized power densities and relative errors ( = 0.125, quadratic leakage) 895.18 Node-wise power distribution prole ( = 0.125, quadratic leakage) . . . . 895.19 Eigenvalue convergence ( = 0.125, quadratic leakage) . . . . . . . . . . . . 905.20 Convergence of correction factors ( = 0.125, quadratic leakage) . . . . . . 90


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1

Chapter 1

Introduction

1.1 Motivation

The eld of nuclear power plant engineering is a good example of usefulness of mathematicalmodelling. Reactors nd their use not only in power plants but also in naval and spacetransportation, medical industry or materials testing, therefore a thorough understandingof processes occuring in them is of great concern. Mathematical models play an importantrole in effective nuclear reactor design, control of its smooth operation or assessment of various malfunctions and accidents. Specically, this thesis chooses as a model problem theoptimal fuel reloading strategy, which requires many long-term test runs of the reactor for

different fuel loading patterns a task clearly needing computer simulations to accomplish.The most important part of the reactor is its core. It consists of a set of assembliesloaded with fuel, control mechanisms and cooling material. There, during an event called ssion , a great amount of energy is released. To produce more energy, each ssion eventcan under suitable conditions trigger another in a process called chain reaction . Sincethe essential ingredients in ssion chain reaction are the neutrons , the exact knowledge of their distribution and movement throughout the reactor core is sought. The mathematicalmodel that provides this information is based on the neutron transport equation .

1.2 Problem descriptionThe fuel reloading problem will be studied for pressurized water reactors (PWR). Fuelassemblies powering these reactors consist mainly of rods loaded with slightly enricheduranium. During a run of the reactor, ssions continually occur in the assemblies and resultin conversion of uranium into products that cant be ssioned anymore the assembliesare burning up. The process of fuel depletion doesnt occur spatially uniformly within thecore and is directly dependent on neutron distribution in each assembly. After a periodof a so called fuel cycle , the core will contain assemblies with various levels of depletion some will have to be replaced completely by fresh fuel while the others will have to be


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2 1 Introduction

moved to places where they would contribute to power production more effectively 1.This operation changes the overall reactivity of the reactor and adjustment of the distri-

bution of neutron absorbers becomes necessary as well. The new core conguration mustpreserve the best possible performance and meet various operational and safety regulationsfor the course of next fuel cycle. Fuel reloading has thus form of a constrained discreteoptimization problem.

To evaluate reactor effectivity for a particular assembly arrangement, a tunable param-eter is introduced to the transport equation. As we wish to know a long-term behaviour of a reactor congured according to the selected strategy, a steady-state solution is sought. Itturns out that there exists only one value of the parameter for which a physically plausible,non-trivial solution exists. This value species how the quantity varied by the parameterhas to be changed in order to ensure the steady state operation. The associated solutiondescribes the neutron distribution (or ux ) within the steady core and may then be furtherused to calculate power production of each assembly. The uniformity of power distributionis a crucial safety issue since high power levels may result in temperature increases beyondthe cooling ability of the surrounding coolant.

Mathematically, previous paragraph describes an eigenvalue problem for the steady-state neutron transport equation. It must be solved for each loading pattern found bythe optimization method. Since there is a huge number of arrangement possibilities 2,the solving eigenpair must be found rapidly. Development of a method that satises thisrequirement is subject of this thesis.

1.3 Overview of solution methods

The problem as described in previous section features a solution of neutron transportequation. In its full generality, it is a complicated integrodifferential equation which isextremely difficult to solve. Hence for practical computations some simplications must bemade. Often used is the diffusion approximation in which the accurate transport theoryis replaced by approximate diffusion theory. Diffusion formulation consists of a parabolicpartial differential equation describing the neutron balance at each point in the core anda constitutive relation called Ficks law. In general, the energetic dependence of neutrondistribution is discretized into so called groups and a system of multigroup diffusion equa-tions arises as a result. In this work, a two group model is considered and a steady state,time-independent solution is sought, hence two coupled elliptic differential equations willbe solved.

Analytical solution to these equations may be found only in some special cases. For realsituations, numerical methods have to be employed. Apart from statistical Monte-Carloapproach based on simulating collision probabilities for individual neutrons and atoms,

1for instance, fuel cycle for the Temelin power plant spans one year; 1 / 4th of assemblies are replacedduring a refuelling process ([ CEZ])

2e.g. the PWR reactor of type VVER-1000 used in Temelin contains 163 assemblies ([ CEZ])


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1.4 Organization of the thesis 3

deterministic methods are also popular for reactor analyses. Their representatives are e.g.the nite-difference method, the nite-elements method or the nodal method .

Monte-Carlo methods are still too computationally expensive for the given problem.Methods utilizing standard nite differences pose strict conditions on discretization gran-ularity in order to preserve sufficient accuracy. The nal choice of the nodal approach overthat based on nite elements was driven by the ease of implementation. In a nodal method,the multigroup diffusion equations are solved on a coarse grid of nodes (with each nodecorresponding to a single assembly), using the nite-volume method . To solve the equationsarising from such discretization, quantities dened for nodal surfaces must be related tothose dened inside. A set of local two-node coupling problems in one dimension, obtainedby the transverse integration method, is solved for this purpose since it allows a higherorder representation of the surface quantities than the traditional nite-volume technique.To achieve reasonable accuracy and efficiency alike, the non-linear iterative scheme ties thetransverse integration procedure with the whole-core nite-volume method ([SA96]).

1.4 Organization of the thesis

The following chapter serves as an overview of the most important concepts of reactorphysics. Quantities dened in this chapter are then used in chapter 3 to derive the math-ematical model based on transport theory. That chapter further outlines transition tomultigroup diffusion equations. In chapter 4, a nite-volume method is described andprovides a basis for development of a nodal method. Transverse integration procedure is

needed for this second step and is described next, together with an effective method forsolving the associated equations. Implementation of this method in MATLAB is brieydiscussed in chapter 5 and compared with veried test results. The nal chapter sum-marizes the whole work and concludes with possible future improvements and researchdirections.


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4 1 Introduction


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5

Chapter 2

Physical background

2.1 Notation

I begin the exposition by introducing some nomenclature used throughout this chapter.Atomic nuclei consist of nucleons the protons and the neutrons. Their total numberis called atomic mass and is denoted by A, the number of protons only is called atomic number and is denoted by Z (thus the number of neutrons can be obtained as the differenceA Z ). When I want to write about a particular nucleus with specic number of protonsand neutrons, I will call that nucleus a nuclide . Nuclides will be denoted by symbol AZX (andneutrons by 10n). Nuclides of the same element, differing only in the number of neutrons (i.e.with same Z but different A), are called isotopes . Usually, I will write already mentionednuclides and their isotopes using only their atomic mass and omitting their atomic number,for instance U-235 and U-238, respectively, will refer to 23592U and 23892U, respectively.

2.2 Basic concepts

Nucleus structure. Within a stable nucleus, nucleons are kept together by attractivestrong nuclear forces they exert on each other. They reside in a well of negative potentialenergy, as schematically depicted in gure 2.2. Further away from the center of the nu-cleus, neutrons are not held by any force and thus their potential energy approaches zero

(assuming that they are not moving). Protons receding from the nucleus rst obtain somepositive kinetic energy due to repulsive electrostatic forces which take over the short-rangestrong nuclear force. They repell the protons from the nucleus, but innitely far away fromits center, even these electrostatic forces vanish.

Binding energy of a nucleon to a nucleus is dened as a difference between the energyof the nucleon innitely far away from the nucleus and its energy inside the nucleus. It isreleased when the nucleon joins other nucleons in the event of nucleus creation, i.e. whenit descends into the potential well. Equivalently, the same amount of energy must besupplied in order to free it from the nucleus. Nucleons always ll the lowest unoccupied


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6 2 Physical background

0 r

U

proton

(a) Proton potential

0 r

U

neutron

(b) Neutron potential

Figure 2.1: Nucleons inside a stable nucleus and their potential energy U with increasingdistance r from the centre of the nucleus at r = 0

energy levels the lower they can descend when creating the nucleus, the higher is theirbinding energy.

The average quantity used when describing actual nuclides from energetic point of view isthe binding energy of nucleus per nucleon which is dened as the sum of binding energies of all the nucleons comprising the nuclide divided by their number (that is, by atomic mass).It is the binding energy of an average nucleon to the nucleus.

Energy release from nuclear reactions. As discussed above, in order to gain energywe need to convert nuclides with lower binding energy per nucleon (imagine a shallowerpotential well for nucleons to fall into) to those with higher (deeper potential well). Lookingat gure 2.2, one easily spots that most tightly bound are the nuclides with atomic massaround 60. Both lighter and heavier nuclides have lower binding energies per nucleon.Therefore a nucleus heavier than

A 60 will release energy when it splits apart (in a

reaction called ssion ), while two nuclei lighter than A 60 will release energy when theyfuse together ( fusion ). Actual energy yield from either ssion or fusion corresponds to thedifference of binding energies per nucleon of the original nucleus and the reaction productsmultiplied by their respective atomic masses.

By comparing the steepness of the fusion part of the graph in gure 2.2 with that of thession part, one can deduce that much more energy is released by fusion however, theactual construction of a commercially suitable fusion device is accompanied by lots of tech-nical difficulties and to this day only ssion-employing power generators are economicallyeffective.


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2.3 Neutron-nuclei interactions 7

Figure 2.2: Binding energy per nucleon (adapted from http://en.wikipedia.org/wiki/Image:Binding_energy_curve_-_common_isotopes.svg , courtesy of WikimediaCommons)

2.3 Neutron-nuclei interactions

2.3.1 Neutron properties

Neutron is a subatomic particle which does not carry any electric charge and has a slightlyhigher mass than proton. Therefore when free, neutrons are unstable and decay into

proton, electron and antineutrino with mean lifetime about 15 minutes ([Y+

06]). Thistime is much longer than the mean lifetime of neutrons freed during reactions in a reactorand therefore the neutron decay does not inuence these reactions.

Concerning energy, neutrons may be divided roughly into three classes:

fast, with energies E > 1MeV,

intermediate, with energies in the range 1 keV < E < 1MeV,

slow, with energies under 1 keV.


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Neutrons moving with a slowest possible speed in a given environment are put into aspecial category and are called thermal neutrons . They are in a thermal equilibrium with

surrounding atoms and have energies of the order of 10 1

eV. Energy of the interactingneutron strongly inuences behaviour of the interaction, as described in next section.

2.3.2 Quantifying interactions

Microscopic cross section. Probability of a specic nuclear reaction is commonly quan-tied by cross sections. Microscopic cross section, , is the probability that a neutroncoming perpendicularly to a unit area interacts with a nucleus on that area. The units of cross section are areal, most often of the order of 1 barn : 10 24 cm2. This is the reasonwhy we usually percieve cross section as the area presented to an incoming neutron by

the nucleus, for a particular reaction process. It deserevs mentioning, however, that thereaction probability generally does not have any relation to the geometry of the nuclei.

Macroscopic cross section. If microscopic cross section is imagined as the area for aspecic interaction presented by one nucleus, the macroscopic cross section would be theinteraction area made up by all the nuclei in a unit volume of material. More precisely, it isthe probability that a specic interaction occurs when the neutron travels a unit distancethrough a homogeneous 1 sample of material. Using the microscopic cross section of eachof the nuclei in the sample, it can be expressed as

= N , (2.1)

where N is the number of nuclei per cm3 of the sample (atomic density ). Its units arecm 1.

Properties of cross sections. Besides their dependence on nucleus structure, crosssections also exhibit a very complex dependence on energy of the reacting neutron. Fur-thermore, they are related to a particular reaction. The reaction will be specied in asubscript, e.g. f (resp. f ) denotes the cross section for ssion.

It is also worth noting that the probability of neutron-neutron interactions is negligiblecomparing to probabilities of their interaction with atomic nuclei, since the density of neutrons per unit volume is negligible when compared to the atomic density. This has animportant implication on linearity of the governing equation, as will be shown in chapter 3.

Although the microscopic cross section is the basic material property actually measuredby experiments, only the macroscopic cross sections will be important for constructing theequations. Eq. (2.1) explains how they can be obtained from microscopic cross sectionsfor a given material.

1 if several types of nuclides are present in the sample, the macroscopic cross section would be the sumof N ( i ) ( i ) over all the different nuclides


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Reaction rate. Lets consider a beam of neutrons with density n neutrons per cm 3 allmoving at speed v through a unit volume 2. The number of interactions per unit time they

will have with nuclei in that volume can be expressed asr = nv

and represents the reaction rate . The asterisks in the equation replace any specic kindof reaction. Furthermore, it is expedient for the purposes of derivation of the transportequation in chapter 3 to dene the neutron ux as

= nv (2.2)

and reaction rate as

r = . (2.3)Note that in such a denition, the ux is a scalar quantity, since it is the product of thescalar neutron density and the scalar magnitude of their velocity vectors. The term uxoriginates again from the units (cm 2s 1) which correspond to units of (vector) uxes inmany other disciplines (e.g. heat transfer or electromagnetic theory).

2.3.3 Types of reactions

There are two classes of possible interactions between neutron and nucleus: scattering andabsorption .

Scattering. When the neutron is subject to a scattering collision, the direction of itsmovement is changed and its energy (or, equivalently, speed) lowered. Scattering reactionscome in two fashions elastic and inelastic . The former can be imagined as a classic colli-sion of two perfectly elastic balls3. The energy of the neutron-nucleus system is conserved,all the energy that the neutron loses in the moment of collision is transferred to the targetnucleus. The latter does not conserve the energy of the system some of the transferredenergy escapes in the form of gamma photons. It happens only when the neutron has en-ergy in the MeV range and thus affects only neutrons during the rst few moments of theirlife in the reactor. Therefore mainly elastic scattering is interesting for reactor modellingpurposes. Scattering probability is measured by the scattering cross section, s (resp. s ).

Absorption. In an absorption event, a freely ying neutron hits a nucleus and is cap-tured. Such a neutron is lost for further interactions but (via ssion, a special case of absorption) can liberate new neutrons. When the neutron is absorbed, the nucleus (A,Z) turns into an unstable compound nucleus (A + 1, Z). The compound nucleus obtains

2 if each neutron had different speed (a typical case in real situations), the following equations had tobe integrated over v

3however, the exact course of that reaction is determined by quantum mechanical laws


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the kinetic and the binding energy of the neutron and this additional energy allow nu-cleons to get to higher energy levels the nucleus is said to attain an excited state. In

order to get back to its stable ground state (with nucleons in positions of lowest energy),it has to deexcite . Deexcitation has the form of particle emission where the actual par-ticle emitted is determined by the type of the absorption reaction. The two absorptionreactions that mostly affect neutron distribution in a reactor ssion (cross section f and radiative capture (cross section ) are described in the next two sections in moredetail. However, there are more types of absorption events happening when a neutronstrikes the nucleus (interested reader is referred to publications devoted to nuclear physics,such as [Sta01]). The total probability that a neutron will be captured by any of them isexpressed by the absorption cross section, a (thus a = f + + . . . where the ellipsisrepresents any unmentioned capture events). Macroscopic absorption cross section a isdened accordingly.

2.3.4 Fission

Fission is the reaction that makes power production in nuclear energy stations possible.For ssion to start, the neutron and proton potential barriers must be surpassed in order toseparate the nucleons from each other and let them reorganize into different nuclides withhigher binding energies per nucleon (recall paragraph about nucleus structure on page 5).There are several possibilities how to overcome the barrier, though only two are viable ina nuclear reactor 4

1. quantum tunneling allows penetration of the barrier without much effort, but happensextremely rarely. Its probability becomes measurable only at high atomic masses.Nevertheless, certain of those heavy nuclides like 25298Cf exhibit this so called sponta-neous ssion quite often and are even used as additional neutron sources in nuclearreactors.

2. neutron induced ssion , on the other hand, is common in a nuclear reactor and is theprincipal source of neutrons in the reactor.

Neutron induced ssion. As noted in section 2.3.1, neutron is uncharged which meansthat it is not repelled by protons inside the nucleus. As a consequence it can easily comeclose enough to other nucleons inside the nucleus to be bound by strong nuclear forces.The compound nucleus in an excited state is formed in the process. If there is sufficientexcitation (above a threshold called activation energy of the nucleus ), ssioning may start.

There are nuclides, called ssile , whose activation energy is not much higher than thebinding energy of an additional neutron to that nuclide. The neutron doesnt need to havemuch kinetic energy in order to initiate ssion of these nuclides. They are represented e.g.by 23592U or 23994Pu generally heavy nuclides with odd atomic mass number ([Sta01, p. 6]).

4 the others include e.g. bombarding the nucleus by highly-energetic protons, which requires a particleaccelerator.


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Fission of ssile nuclides is utilized in thermal reactors , so called because ssion-inducingneutrons can be slow, even thermal (section 2.4 explains why having thermal neutrons

initiate ssion is so effective).The other kind of nuclides must also utilize high kinetic energies of impacting neutronsin order to split apart. These are said to be ssionable (e.g. 23892U or 24094Pu, heavy nuclideswith even masses) 5. Fast reactors utilizing fast moving neutrons ( E k 1 MeV) to ssionssionable material will not be considered since the PWR reactors studied in this work fallinto the former category.

Scheme of the reaction. A ssion of 23592U (U-235) has the following general form (theasterisk denotes the excited state of the nuclide):

235

92U + 1

0n

(236

92U)

2 ssion fragments + recoil energy 2 ssion products + 10n + ( + ) + E .

(2.4)

The heavy uranium nucleus splits into two medium sized nuclei called ssion fragments .When split, the nucleons of the fragments are in a range of dominating electrostatic re-pulsion and thus the fragments recoil at very high speed. However, due to their limitedmovement possibility in solid fuel (only a few centimeters), all their kinetic energy (about170 MeV) gets quickly converted to heat. This comprises about 85% of the total energyproduction by ssion, the rest being released during subsequent decays of the fragments( E ).

Fission fragments are highly unstable and begin stabilizing almost instantly (10 12 s)by emission of the so called prompt neutrons and rays of prompt gamma photons. Thisprocess is governed by statistical rules, usually two or three prompt neutrons are freedper ssion. The result of this initial decay is a conversion of ssion fragments into morestable ssion products . Again, the precise pair of ssion products cannot be determined,but usually nuclides with atomic mass around 95, respectively 139, come out of a U-235ssion event (e.g. Ba+Kr or Xe+Sr, see e.g. [Her81, p. 23]).

Delayed neutrons. Although more stable than ssion fragments, the initial ssion prod-ucts are typically still radioactive and undergo a series of electron emissions ( decay)followed by gamma radiation to nally end up as stable isotopes. Some nuclides in this se-quence may also decay by emitting neutrons instead of electrons. Such neutrons are calleddelayed because they may be released as late as 101 s after ssion. Nuclides exposing thisbehaviour are called delayed neutron precursors and are typically divided into six groupsaccording to their half-lives.

Emission of a delayed neutron is a rare event only about 1% of neutrons liberated byssion are delayed and the rest are prompt. However, even such a small fraction increases

5note that from the point of view of additional neutron energy needed to trigger the ssion process,every ssile nuclide is also ssionable


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the average neutron lifetime from about 10 4 s (for only prompt neutrons) to about 10 1 s(taking into account also the delayed neutrons), depending on the half-life of the precursor.

By increasing the neutron lifetime, delayed neutrons make it possible to control the rateof power generation by adjusting the neutron production rate without them, a smallchange in neutron production rate would result in an unmanageably fast change in powerproduction rate.

The effect of delayed neutrons becomes important in reactor dynamics calculations. Asthis thesis deals with a steady state problem, it is not necessary to be concerned withthem in much detail. A more thorough treatment of this topic can be found e.g. in[Sta01, Chap. 5] or [Her81, Chap. 6].

Properties of neutrons released by ssion. I will denote the average number of prompt and delayed neutrons, respectively, by

p and

d, respectively These counts depend

on the energy of the neutron responsible for the reaction and on the nucleus being ssioned.The average number of both prompt and delayed neutrons obtained is then denoted by , = p + d. For U-235 ssioned by thermal neutrons, its value is = 2.5 neutrons.

The energy with which are the neutrons emitted is of the order of 1 MeV for promptneutrons and 0.1 MeV for the delayed ones. The probability of prompt (respectivelydelayed) neutron being released with energy E is measured by the ssion spectrum , p(E )and d,i (E ), respectively, where the index i represents the particular precursor group fromwhich the delayed neutron originated. Empirical behaviours of these functions of energyare known, gure 2.3 shows an example of the spectrum of prompt neutrons freed by U-235ssion event. After setting free, neutrons diffuse through the surrounding environment and

Figure 2.3: Fission spectrum of prompt neutrons for U-235 (from [Sta01, p. 12])

lose their energy by scattering, until they are captured by some nucleus, escape the core


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or even up their energy with the nuclei around.

Fission cross section. The probability that a neutron causes ssion is expressed bythe ssion cross section , f . As pointed out in section 2.3.2, the cross section is a verycomplicated function of neutron energy. For instance, microscopic ssion cross sections of two main isotopes that are contained in natural uranium are plotted in g. 2.4 against theenergy of the neutron inducing that reaction. The ssion neutron is born in the rightmostpart of the graph (MeV energy range) and slows down during its life to the leftmost part.The difference between ssile U-235 and ssionable U-238 can be clearly spotted from thegraphs. Apart from resonances in the intermediate energy range, the probability that U-238 will be ssioned by a neutron with energy below 1 MeV is negligible. On the otherhand, note that the less energetic a neutron is, the greater chance it has to ssion a U-235nucleus.

2.3.5 Radiative capture

The most common type of interaction that can happen when a neutron hits a nucleus of virtually any element is the radiative capture. The excited compound nucleus achievesits ground state by emission of a photon of gamma radiation. This kind of absorption isparasitic since the absorbed neutron can no longer cause any other reaction and no newneutron is freed as a compensation. For some kind of nuclei (called fertile , e.g. U-238), thisform of decay may result in conversion of the nucleus to a ssile one. Reactors utilizingthis process to breed new fuel on the course of their run are called breeder reactors (e.g.[Sta01, chap. 7]).


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-910 -810 -710 -610 -510 -410 -310 -210 -110 010 110Energy (MeV)

10-1

10 0

10 1

10 2

10 3

C r o s s

S e c

t i o n

( b )

(a) Microscopic cross section of U-235 ssion

-810 -710 -610 -510 -410 -310 -210 -110 010Energy (MeV)

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

10 0

10 1

C r o s s

S e c

t i o n

( b )

(b) Microscopic cross section of U-238 ssion

Figure 2.4: Energetic dependence of ssion cross section (from reference [Kor](http://atom.kaeri.re.kr/cgi-bin/endfplot.pl ))


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2.4 Chain Reaction 15

2.4 Chain Reaction

Every ssion reaction consumes one free-moving neutron, but releases a few more in ex-change. Each of these liberated neutrons might induce new ssion reactions if they hit afuel element. However, not all manage to hit that fuel nucleus in a way that makes it split.To sustain ssion in the reactor, at least one of the liberated neutrons must, on average,survive to induce new ssion. This process of sustaining ssion is called chain reaction .

Neutron lifecycle. Chain reaction takes place in the reactor core, sometimes also calledactive zone . The active zone contains fuel assemblies, coolant, moderating and absorbingmaterial and some structural material like cladding and spacers.

The fuel has typically a form of uranium-lled rods plugged in the assemblies. Natural

uranium, however, contains more than 99% of ssionable isotope U-238 and only about0.7% of ssile U-2356. Only neutrons with high energy (on MeV levels) can induce thession of U-238 (see gure 2.4(b)). Such neutrons are present only near the origin of theirliberation since with increasing time they gradually lose their energy due to collisions withother nuclei.

Isotope U-235, on the other hand, may be ssioned by neutrons of any energy but theprobability is much higher for thermal neutrons (see gure 2.4(a)). In thermal reactors,most ssion events are caused by thermal neutrons hitting U-235 nuclides. Since thereis so little of U-235 in natural uranium, the effort is to slow down the neutrons as muchas possible to facilitate the reaction. This is the job for the moderator nuclei. However,as neutron energy sinks to the keV range, the probability of non-productive radiativecapture (see section 2.3.5) on U-238 becomes quite high. Therefore, a good moderatorshould slow down the neutrons as quickly as possible in order to avoid these unwantedcaptures. Naturally, it should also capture the neutrons by itself as little as possible.Efficient moderators having both these properties are e.g. heavy water or graphite; PWRreactors use ordinary water both to cool down the fuel assemblies and to moderate fastssion neutrons.

Freely moving neutron can also be non-productively absorbed by other nuclides insidethe active zone, e.g. by coolant, moderator and structural atoms or even by U-235 (thoughmore often this capture event ends by productive ssion). Except being absorbed, theneutron can also leak entirely out of the core through its boundary 7. This may be remediedby placing a special material around the active zone that reects a portion of the neutronsback (hence the commonly used term reector ).

Effective multiplication constant. The conclusion of the previous paragraph is thatonly a small fraction of neutrons initially created by ssion events in fuel assemblies will

6PWR designs utilize slightly enriched natural uranium with concentration of ssile material raised toabout 3% ([ CEZ], [Her81, p. 210]).

7 this is possible since neutrons do not interact with the electron clouds of atoms comprising the boundarywalls; only the minuscule nuclei stand in the way of the escaping neutron.


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have the opportunity to induce further ssions of U-235. Such neutrons give birth to anew generation of neutrons. If their number is ng 1 and the number of neutrons in this

new generation is denoted by ng then the ratiokeff =

ng 1ng

(2.5)

is called effective multiplication constant . Depending on its value, the reactor can berunning in three different states:

keff < 1, subcritical state. The population of neutrons in the active zone decreases;chain reaction is dying out.

keff = 1, critical state. The population of neutrons in the active zone remains con-stant; chain reaction is self-sustained.

keff > 1, supercritical state. The population of neutrons increases; chain reactiongrows.

The critical state describes the steady operation of the reactor number of neutronswithin the core doesnt change and power production is constant. The aim is to alwaysbe as close to this state as possible. The condition for criticality depends on the sizeand geometry of the reactor (affecting leakage) and on the material composition (affectingnumber of absorptions and ssions). Usually, the geometrical properties are given andcant be changed, so in order to maintain neutron equilibrium, material properties mustbe adjusted.

During reactor operation, changes in multiplication coefficent are handled by manipu-lating absorption devices. In PWR reactors, absorption is controlled by two main mecha-nisms. In the rst, absorbing rods made of boron steel are moved into the fuel assemblies tocontrol the momentary uctuations in neutron population density (usually during startupand shutdown operations). To compensate for the slow changes in long-term chain reac-tion, absorbing boric acid is dissolved in the cooling water and its concentration is variedappropriately.

In the fuel management area, the multiplication coefficient is interesting as a measureof long-term stability of the chain reaction for a proposed fuel pattern. The primary ob-

jective of fuel loading optimization is to nd a conguration that supports a self-sustainedchain reaction of the core and hence the multiplication coefficient serves as an optimalityindicator. As various models are obtained by adjusting the parameter of the governingequations (as mentioned earlier in section 1.2), this parameter is directly related to themultiplication factor. The connection will be revealed in more detail in section 3.1.4.


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17

Chapter 3

Mathematical model

Neutron distribution inside the reactor core is mathematically best described by trans-port theory, originally developed by Boltzmann to explain distribution of particles in gas.However, it has been observed that the complicated equations constituting the theory canbe simplied to well-examined diffusion equations which still retain sufficient amount of accuracy.

3.1 Neutron transport theory

3.1.1 Neutron transport equation

The transport equation governs neutron balance inside the active zone of a reactor. In orderto formulate it in its full generality, one must take into account all the dependencies of quantities mentioned in chapter 2. As an important assumption, this chapter will study thethree dimensional reactor core on a macroscopic scale, i.e. as a continuous environment, andalso all involved time-dependent quantities will be assumed continuous in time. Moreover,all external forces (like gravity) that may act on the particles and nuclei will be neglected.

3.1.1.1 Basic quantities

Description of neutrons state. Lets think of a neutron moving throughout the corein a direction of a unit vector . If v is the speed of its movement, then its velocity vector,v = v (E ), is given by

v = v .Denoting m the mass of the neutron, its energy 1 is

E = 12

mv2.

The state of the neutron at any time t is completely described by its position r (r = ( x,y,z )),energy E , and direction in which it is moving. These are the fundamental quantities

1 in the form of kinetic energy, since no other force affects the free neutral particle


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18 3 Mathematical model

uniquely characterizing each neutron and all others may be expressed in their terms 2. Theranges of all the possible values they can take form the neutron phase space .

Fundamental quantities of reactor modelling. The most important physical quan-tity used to quantify the processes caused by neutrons is the angular neutron density ,denoted 3 by N :

N = N (r , E, , t ).

It is dened in a statistical sense, such that N (r , E, , t ) dr dE d dt expresses the ex-pected number of neutrons located in a time interval [ t, t + d t ] in a volume element draround r , having energy in an interval [ E, E + dE ] and moving within a cone formed bya solid angle d around a unit direction vector ([Wil71, p. 12]). The velocity vectorwithin solid angle d of one of the neutrons from an element dr is depicted in gure 3.1.

x

y

z

v x

v y

v z

r

d r

v

d

Figure 3.1: Position and movement direction of neutrons (adapted from [Wil71])

To describe the reactions occuring within the reactor, their cross sections need to bedetermined. Because of the macroscopic nature of the model, only macroscopic crosssections have to be considered. Writing explicitly their dependence on energy (as mentionedin section 2.3.2) and time (since cross section is a material property and material attributes

2note that this choice of quantities is not unique, e.g. the set r , v and could have been used insteadlto describe the neutrons.

3 the atomic density for which the same letter was used in chapter 2 will not be needed anymore


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3.1 Neutron transport theory 19

change during the run of a reactor):

= (r ,E , t ),

we get the probability of neutron with energy E interacting with a nucleus at point r .Reactions that may be substituted for the asterisk were discussed in section 2.3.3.

In the case of scattering, the differential scattering cross section is usually introduced([Wil71, p. 13], [Sch97, p. 10]) to emphasize the fact that scattering changes energy anddirection of the inciding neutron:

s (r , E E, , t ) = s (r , E , t )P (r , E E, ), (3.1)where P (r , E E, ) dE d has the meaning of probability that a collision at pointr will change neutrons energy and direction from d E d to dE d .

The angular neutron ux is dened by using equation (2.2):(r , E, , t ) = v(E )N (r , E, , t ). (3.2)

As noted in section 2.3.2, the angular neutron ux is a scalar quantity. Instead of repre-senting a ow of neutrons through some surface, it rather characterizes the rate at whichthe neutrons are passing through that surface regardless of its orientation 4.

Neutron ux is the unknown quantity which is actually solved for in the neutron trans-port equation, since it allows to quantify reaction rates. Using the denition of angularneutron ux, the reaction rate density is given as in (2.3) by the product:

r (r , E, , t )dE d = (r , E , t )(r , E, , t )dE d . (3.3)

It represents the expected rate per unit time about t at which neutrons characterized byenergies and directions within d E d are interacting with nuclei in a unit volume aroundr .

In deriving the transport equation, a vector quantity corresponding to neutron ux isneeded to specify the directional ow of neutrons. Angular neutron current obtained by

j(r , E, , t ) = (r , E, , t ) = v (E )N (r , E, , t ) (3.4)

provides such a tool. Indeed, lets take an arbitrary volume V enclosed by oriented surfaceS . Let dS be the element of that surface located at point r with a unit outward normal n

and let dS

= n

dS . Then5

j(r , E, , t )dS dE d = (r , E, , t )( n )dS dE d (3.5)stands for the expected number of neutrons energetically distributed within d E around E ,owing per unit time through surface element d S in a direction within d around andwith orientation given by n : in the case of n > 0, neutrons leak out of V , in the caseof n < 0, neutrons ow in.

4perhaps ow rate would be a more appropriate name but ux has already become well-established inreactor physics eld

5 the dot () between vector quantities signies a dot product


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3.1.1.2 Neutron balance

Let V

R 3 be an arbitrary balance domain inside the core and [ t1, t2]

[0, +

) an

arbitrary time interval. According to the general balance principle the change in neutronpopulation in V between times t1 and t2 must be equal to the amount of neutrons producedby sources inside the domain less the amount of them lost due to sinks in the domain andtheir net outow (leakage) across the domain boundary, during the time interval [ t1, t2].

In addition to position, energy and movement direction must be determined in order tofully characterize each neutron at any given time. To complete the conservation statement,I will treat the remaining two independent variables locally in the sense that balance involume V will be affected only by neutrons having energy within some differential elementdE and moving in a direction of some vector from a differential solid angle d . Both theseelements may be chosen arbitrarily and the differential area dened by them in the neutron

phase space will be denoted as dE d . I will ocassionally refer to these neutrons as tobalance neutrons , i.e. neutrons with direct inuence on the balance. Integration over theappropriate energy and direction ranges would yield balance for a general set of neutrons.

Changes in neutron population

The expression for the change of the amount of neutrons with energies and directionsfrom balance range dE d , in volume V and between times t1 and t2, has the followingform:

V

[N (r , E, , t2) N (r , E, , t1)] dr dE d (3.6)

(the times sign has been included to separate the d r symbol having the role of anintegration symbol and the d E d symbol originating from the statistical foundations of the variables and specifying the energetic and directional range of balance neutrons).

Sources

The primary sources of new neutrons of given energy and direction distribution are thession reactions and particular forms of decay. The secondary means of increasing thepopulation of such neutrons is scattering of existing neutrons originally having differentenergies and moving in different directions into the balance range.

Fission. In a ssion reaction, an average number of p neutrons is emitted promptlyin the process and d neutrons are, on average, released during the subsequent decay of ssion products (see section 2.3.4).

Prompt ssion neutrons. Let us consider only the prompt neutrons for the moment. p is a function of energy of the ssion-initiating neutron and, to be exact, depends alsoon what nuclide is being ssioned, i.e. on r . Thus p = p(r , E ).

Consistently with the discussion on neutron properties on p. 12, the distribution of promptly released neutrons in energy will be measured by their ssion spectrum p. As


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3.1 Neutron transport theory 21

in the case of differential scattering, a rigorous treatment of ssion requires to considernot only the energy of the outgoing neutron, but also in which direction it is emitted as

well as the energy and incoming direction of the neutron that actually caused the reaction.Fission spectrum is usually used in place of the probability function P that appeared indenition (3.1) of scattering cross section ([Wil71, p. 16]). So p = p(r , E E, )is the probability that a prompt neutron is released with energy E and direction as aresult of ssion caused by a neutron with energy E coming from .

Number of ssions caused by neutrons of energy E is characterized by the ssion cross section , f = f (r , E , t ). Note that differential ssion cross section could be dened as in(3.1), though it is more common to keep the two terms separated.

In thermal reactors, it is reasonable to expect the same ssile material distributedthroughout the core (within fuel bundles). Therefore, the number of neutrons comingfrom a ssion and their energetic and directional distribution will be the same (withinstatistical margins) for all ssions regardless of the point at which they occur. Spatialdependence can thus be omitted from variables characterizing ssion neutrons.

Applying denition (3.3) to ssion and integrating over all initial energies and directions 6,the overall expected ssion rate in a unit volume around point r per unit time at t is givenby

4

0

0 f (r , E , t )(r , E , , t ) dE d .As each ssion triggered by a neutron with energy E (incoming from ) follows by therelease of p(E ) p(E

E,

) prompt neutrons with energy E and moving in the

direction of unit vector ,

s f (r , E, , t ) =4

0

0 p(E E, ) p(E ) f (r , E , t )(r , E , , t ) dE d (3.7)species the total expected yield of prompt ssion neutrons with energy E and direction in a unit volume around r at t.

Delayed ssion neutrons. The yield of delayed neutrons is proportional to theconcentration of their precursors. In a general case, the concentration of the precursornuclei from each of the six precursor groups would at any point in the core change in timeand depend on neutron ux. In order to include the effects of delayed neutrons, the balanceequation must be modied to account for them and must be solved simultaneously withsix additional partial differential equations governing the concentration changes. However,in the case of a two-group model of the reactor and steady state calculations, the termsconcerning delayed neutrons essentially disappear from the balance equation and there willbe no contribution of delayed neutrons to the total ssion neutron production ([Gar05]).Therefore, for the purposes of derivation of the nal set of equations (steady state, two-group, diffusion equations) which will be further discussed in this thesis, taking delayed

6using the fact that solid angle spans from 0 to 4 and letting energy span interval [0 , + )


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neutrons into account is not necessary. Hence I also omit the redundant indexing, namely p and p .

External sources. There may be other mechanisms of neutron production besidesssion (for instance, neutrons are spontaneously released from Cf-252, as described insection 2.3.4), though their yield is usually negligible comparing to ssion. It is sufficientto treat all those additional sources by a general term sex (r , E, , t ).

So the total source of new neutrons is the sum sf + sex and the sources density

[s f (r , E, , t ) + sex (r , E, , t )] dE d (3.8)

describes the amount of new balance neutrons introduced per unit time at t to a unitvolume element around r .

In-scatter. The rate at which collisions between neutrons and nuclei result in changingthe colliding

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Numerical Modeling of Neutron Flux