Table Of ContentA Quasi-Static Polynomial Nodal Method for Nuclear
Reactor Analysis
by
Jess C. Gehin
S.M., Nuclear Engineering, Massachusetts Institute of Technology _ _
B.S., Nuclear Engineering, Kansas State University _ _ ._ _._
(1988) _ _l_ ca 1:_
o-to ._
Submitted to the Department of Nuclear Engineering ",_._,:.A_.,?-__.__'q_._t
in partial fulfilhnent of the requirements for the degree of _ _ _ _ _ '_
at the
DOCTOR OF PHILOSOPHY i_.!!_ !
MASSACHUSETTS INSTITUTE OF TECHNOLOGY __ _ ._ _ d _
(_ Jess C. Gehin, SMepCtMemXbCerII1.992All Rights Reserved. t___¢"a_I__2_ "_ _ 2_
The author hereby grants to MIT permission to reproduce and
to distribute copies of this thesis document in whole or in part.
Au, hor ___'.Z_.... (!' _./_, _4_"
/ Department of Nuclear Engineering
August 18, 1992
Certified by -- _(,_.1..,..
...... f Allan F. Henry
Professor, Department of Nuclear Engineering
Thesis Supervisor
Accep,ed by _? 'T :_/f_*1,,4._ Alia.,, F. Henry
Chairman, Department Committee on Graduate Students
I_STRIBUTtON OFTHIS DOCt,IMEN'r !£ UNLIMITED
A Quasi-Static Polynomial Nodal Method for Nuclear
Reactor Analysis
by ,. EIV 'D
,Jess C. Gehin OCTO7'
08TI
Submitted to the Department of Nuclear Engineering
on August 18, 1992, in partial fulfilhnent of the
requirements for the degree of
DOCTOR OF PHILOSOPHY
Abstract
Modern nodal methods are currently available which can accurately and eificiently
solve the static and transient neutron diffusion equations. Most of the methods,
however, are limited to two energy groups for practical appfication. The objective
of this research is the development of a static and transient, multidimensional nodal
method which allows more than two energy groups and uses a non-linear iterative
method for efficient solution of the nodal equations.
For both the static and transient methods, finite-difference equations which are
corrected by the use of discontinuity factors are derived. The discontinuity factors are
computed from a polynomial nodal method using a non-linear iteration technique.
The polynomial nodal method is based upon a quartic approximation and utilizes
a quadratic transverse-leakage approximation. The solution of the time-dependent
equations is performed by the use of a quasi-static method in which the node-averaged
fluxes are factored into shape and amplitude functions. Since the shape function
generally changes more slowly than the amplitude function it can be computed less
frequently, providing a substantial computational savings. The amplitude function is
obtained by solving point kinetics equations for which the parameters are determined
by precise mathemati,'_d expressions based on the nodal model.
The application of the quasi-static polynomial method to several benchmark prob-
lems demonstrates that the accuracy is consistent with that of other nodal methods.
The use of the quasi-static method is shown to substantially reduce the computation
time over the traditional fully-implicit time-integration method. Problems involv-
ing thermal-hydraulic feedback are accurately, and efficiently, solved by performing
several reactivity/thermal-hydraulic updates per shape calculation.
Thesis Supervisor:: Allan F. Henry
Title: Professor, Department of Nuclear Engineering
ACKNOWLEDGMENTS
I would like to extend my sincere gratitude and appreciation to Professor Alia,,
F. Henry for his unmeasurable guidance and support, throughout this project, and my
education at M.I.T.
Further, I would like to thank my thesis reader, Professor David D. Lanning, for
his comments and suggestions.
During my stay at M.I.T. I have made many friends who have made my gradu-
ate education more enjoyable. These people include Mark Byers, Jonathan Witter,
Santiago Parra, and Chris Owens.
Finally, I would like to thank my wife Ann for the love she has provided and the
sacrifices that she has made in order for me to complete my degree. I am looking
forward to our future together.
DISCLAIMER
This reportwas preparedasanaccount of worksponsoredbyanagencyof theUnited States
Government. Neither the United States Governmentnoranyagencyth,:reof,noranyof their
employees, makesanywarranty,expressor implied, orassumesany legalliability or responsi-
bility fortheaccuracy, completeness,orusefulnes:_el any information,apparatus,product,or
processdisclosed,or representsthat its usewould notinfrhjge privatelyowned rights.Refer-
ence hereintoanyspecific commercialproduct,process,orservicebytrade name,trademark,
manufacturer,orotherwised_s notnecessarilyconstituteor implyitsendorsementr,ecom-
mendation,or favoringby the United States Government orany agency thereof. The views
and opinions of author_expressed herein do not necessarilystate or reflect those of the
UnitedStatesGovernmentoranyagencythereof.
This research was performed under appointment to the Nuclear Engi-
neering & Health Physics Fellowship Program administered by the Oak
Ridge Institute for Science and Education for the U.S. Department of
Energy.
CONTENTS
Abstract 2
Acknowledgements 3
Table of Contents 4
List of Figures 10
List of Tables 12
Chapter 1 Introduction and Background 13
i t,1 Overview .................................. 13
1.'2 Background ................................ 14
1.3 Research Objectives ............................ 16
1.4 Thesis Organization ............................ 17
Chapter 2 Derivation of the Static Nodal Equations 18
2.1 Introduction ................................ 18
2.2 Notation and the Nodal Balance Equation ............... 18
2.3 Corrected Finite-Difference Coupling Equations ............ '21
'2.3.1 Boundary Conditions ....................... 25
'2.3.2 Evaluation of the Discontinuity Factors ............. 26
2.4 Polynomial Coupling Equations ..................... 27
'2.4.1 The Transverse-lntegration Procedure .............. 27
2.4,2 Tile Polynomial Expansion .................... 29
'2.4.3 The Two-Node Problem ..................... 30
2.4.4 The Weighted Residual Procedure ............... 31
2.,1.5 Expansion Coefficient Solution .................. 35
'2.4.6 Boundary Conditions ....................... 37
2.5 Tlle Non-Linear Iteration Procedure ................... 38
2.6 Summary ................................. 40
Chapter 3 Derivation of the Transient Nodal Equations 42
3.1 Introduction ................................ 42
3,2 Notation .................................. 43
3.3 The Time-Dependent, Corrected Finite-Difference Equations ..... 44
3.4 The Time.Dependent Polynomial Nodal Equations .......... 46
3,5 Time-Integration of the Corrected Finite-Difference Equation ..... 48
3.6 The Quasi-Static Method ......................... 50
3.6,1 The Amplitude Function Equation ............... 51
3.6.2 Shape Function Equation .................... 53
3.6,3 (!hoice of Weight Function .................... 54
3,7 Thermal-ttydraulic and Feedback Models ................ 56
3.7,1 The WIGL Model ......................... 56
3,7,2 The Cabral-IPM Model ..................... 58
3,7.3 (',ross Section Feedback ...................... 58
3.8 Transient Control Mechanisms ...................... 59
3.9 Summary ................................. 60
I Chapter 4 Static and Transient Numerical Solution Methods 62
4,1 Introduction ................................ 62
4.2 Static Solution Methods ......................... 62
4,2.1 Numerical Properties ....................... 63
4,2,2 Discontinuity Factor Iterations .................. 64
4,2,3 Outer Iterations .......................... 65
4.2,4 Inner Iterations .......................... 69
4,.,5 General lterative Strategy .................... 72
4,2,6 Criticality Search Problems ................... 74
4.2.7 Source Problenls ......................... 74
4,2,8 Mathematical Adjoint Problems ................. 75
,1,3 Transient Solution Methods ....................... 76
4,,3.1 Numerical Properties ....................... 76
4,3.2 Iterative Solution of the Transient Equations .......... 77
4.3.3 Frequency Estimation ...................... 78
4,3,4 Solution of the Point Kinetics Equations ............ 79
4.3.5 General Transient Calculational Procedure ........... 81
4.4 Summary ................................. 83
Chapter 5 Application of the Transient Nodal Method 84
5.1 Introduction ................................ 84
5.2 Forward to Transient Problems ..................... 84
5,2.1 Computer Code .......................... 8,5
5.2.2 Transverse-Leakage Approximations ............... 86
,5,2.3 Power Distribution Errors .................... 86
5,2.4 Execution Times ......................... 87
5.3 '['he 2-D TWIGL Seed-Blanket Reactor Problems ........... 88
,5.3.1 The Static Solution ........................ 88
5,3.2 The Step Transient ......................... 92
5.3,3 The Ramp Transient ....................... 94
5.,i The 3-D LMW Operational Transient .................. !)6
5.4.1 '['he 3.D LMW Problem Withottt Feedback ........... 98
5,4.2 The 3-D LMW Problem with Thermal-liydraulic Feedback , . 103
i
5,5 The LRA BWR Transient Pr,Jblems ................... 108
5.5.1 The 2-D LRA Problem ...................... 110
5.5.2 The3.D LRA Problem ....................... 112
5,6 The PWR Operational Transient .................... 117
5,7 The PWR Coolant Inlet.Temperature Transient ............ 125
5.8 Summary ................................. 129
Chapter 0 Summary, Conclusions and Recommendations 130
6.1 Overview of the Investigation ...................... 130
6.2 Conclusions ................................ 131
6,3 Recommendations for Future Research ................. 132
6.3,1 Diagonal Dominance Required by the Iterative Methods , . . 1132
6.3.2 Application to Multi-Group Analyses .............. 1311
6.3.3 Study of the Thermal-Iiydrauljc/Neutron Coupling ...... 134
6.3.4 Study of the Time Dependence of the Discontinuity Factors . 1,34
References 135
Appendix A The Quadratic Transverse Leakage Moments and Coeffl.
cients 139
3,.1 The Quadratic Transverse.Leakage ,Approximation ........... 140
A,2 LHS-Biased Quadratic Transverse-Leakage Approximation ...... 142
l
A.3 RHS-Biased Quadratic Transverse-Leakage Approximation ...... 144
A.4 The Flat Transverse-Leakage Approximation .............. 146
Appendix B Problem Specifications 147
B.1 Tile TWIGL 2-D Seed-Blattket Reactor Kinetics Problem ....... 148
B.2 The LMW LWR Transient Problem ................... i50
B.3 The LRA BWR Kinetics Benchmark Problem ............. t54
B.4 The PWR Transient Problems ...................... 158
Appendix C Selected Results of Problem Analyses 165
LIST OF FIGURES
2-1 Diagram showing the the surface anti node labeling conventions .... 22
2-2 Diagram showing the orientation of the two-node problem ....... 30
2-3 A flow diagram of the non-linear iteration procedure for the static
problem ................................... 39
3-1 Diagram showing the subdivision of the time steps in the quasi-static
method ................................... 51
4.1 Flow diagram of the quasi-static transient solution procedure ..... 82
5-1 The group 2, x-direction transversely-integrated fluxes (j = 1) for the
TWIGL problem .............................. 90
5-2 The group 2, x-direction, quadratic transversely-integrated currents
(j = 1) for tile TWIGL problem ..................... 90
5-3 The group 2, x-direction, cubic transversely-integrated currents (j = l)
for the TWIGL problem .......................... 91
5-4 The group 2, x-direction, quartic transversely-integrated currents (j =-
I) for the TWIGL problem ........................ 91
5-5 Power density vs, time for the 3-D LMW problem without feedback, . 101
5-6 Reactivity vs, time for the 3-D LMW problem without feedback .... 101
5-7 Power density vs, time for lhe 3-D LMW problem without feedback
demonstrating the cusping correction ................... 104
5.8 Reactivity vs. time for the 3-D LMW problem without feedback demon-
strating the cusping correction ...................... !04
5.9 Total power vs. time for the 3.D LMW problem with feedback ..... 106
5-10 Reactivity vs, time for the 3-D LMW problem with feedback ...... 106
5-1i The 3-D LMW transient with feedback using 5 second shape and
reac: ivity /thernlal-hydraulic steps .................... 107
5-12 Power density vs, time for the 3-D LMW problem with feedbttck clemon-
st,rating the cusping correction ...................... 109
5-I3 Reactivity vs, time for the 3-D LMW problem with feedback, demon.
strating the cusping correction ...................... t09
5-14 Power vs, time for the 2-D LRA transient problem ........... 114
5-15 Fuel temperature vs, time for the 2-D LRA transient problem ..... 114
5-16 Power vs, time for the 3-D LRA transieut problem ........... 119
5-17 Fuel temperature vs, time for the 3.D LRA transient problem ..... 119
5-18 Control rod motions for the PWR operational transient ........ 121
5-19 Power vs. time for the PWR operational transient demonstrtttin_; the
temporal convergence of the solution ................... 1.23
5-20 Reactivity vs. time for the PWR operational transient demonstrating
the temporal convergence of the solution ................. 123
5-21 Power vs. time for the PWR operational transient, large time-step
quasi-static solution ............................ 124
5-22 Reactivity vs, time for the PWR operational transient, large time-step
quasi-static solution ............................ 124
5-23 Power vs. time for the PWR coolant inlet-temperature transient demon-
strating the temporal convergence .................... 127
5-24 Reactivity vs. time for the PWR coolant inlet-temperature transient
demonstrating the temporal convergence ................. 127
5-25 Power vs. time for the PWR coolant inlet-temperature transient, large
time-step quasi.static sohttion ....................... t28
5-26 Reactivity vs, time for the PWR coolant inlet-temperature transient,
large time-step quasi.static solution .................... 128
C-1 3-D LMW problem without feedback, comparison of initial static so.
lutions .................................... 166
C-2 3.D LMW problem with feedback, comparison of initial str_tic solutions, 1.67
C-3 2.D LRA problem, comparison of initial static solutions ........ 168
('-4 2-D LRA transient problem, normalized power distributions r,.ud fuel
temperatures ................................ 169
(!-5 3-D LRA problem, comparison of initial sin.tic solutions ........ 176
C-6 PWR operational transient, comparison of initial static solutions. . . 184
C-7 PWR coolant inlet-temperature transient, comparison of initial static
solutions .................................. 185
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