....
...t.'.,
ANL-7307
Reactor Technology
(TID-4500)
AEC Research and
Development
Report
/\.:
'
ARGONNE
NATIONAL
LABORATORY
9700 South
Cass
Avenue
Argonne,
Illinois
60439
MONTE
CARLO
ANALYSIS OF
REACTIVITY
COEFFICIENTS IN
FAST
REACTORS;
GENERAL
THEORY
AND
APPLICATIONS
by
Laurence
B.
Mille r
Reactor
Physics
Division
6.
' Submitted
in
partial
fulfillment of
the requirements for
the
degree of
Doctor
of
Philosophy
in
Nuclear
Engineering
in
the
Graduate
College of the
University
of
Illinois, 1967
March 1967
E-
LEGAL
NOTICE
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sponsored
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United
States, nor the Commission, nor
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the
Commission:
t
.A. Makes
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13,7
1'\
3;37RlaUTION OF MIS DOCUMENT
tS
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3
TABLE
OF
CONTENTS
3„r
Page
* CHAPTER
I.'
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 7
A. The.Problem
in
General . . . . . . . . . . . . . . . . 7
B. Limitations
of
Curient
Theory
. . . . . .
...... 8
C. Scope
of
Research . . . . . . . . . . . . . . . . . . . . 12
CHAPTER
II.
MATHEMATICAL
THEORY
FOR
SOLVING THE
SPACE-
AND
ENERGY-DEPENDENT
BOLTZ-
MANN EQUATION
BY
INDIRECT
ANALOGY . . . . . ·1 4
A.
General
Background . . . . . . . . . . . . . . . . . . . 14
1: Theory
of
Resonance
Reactions. . . . . . . . . . 14
2. Theery
of
Stochastic
Analysis. . . . . . . . . . . 17
3. Techniques
for
Space
Transport,
Collision
Mechanics,
and
Estimation
of
Probabilities.
.
23
B.
Perturbation
Methods . . . . . . . . . . . . . . . . . . 26
1.
Source
Perturbation . . . . . . . . . . . . . . . . . 26
2. Computation
of
Temperature
Derivatives
. . .
27
3. Variation
in
Fuel-element
Diameter
. . . . . . .
29
C.
Special
Techniques . . . . . . . . . . . . . . . . . . . . 32
2 1.
An
Absolutely
Random
Number
Generator... 32
2. The
Fractional
Interaction
Model
. . . . . . . .
33
3. The
Source
Distribution.
. . . . . . . . . . . . . .
45
4. Evaluation
of
Cross
Sections
. . . . . . . . . . .
47
CHAPTER
III.
IMPLEMENTATION OF
THE
THEORY . . . . . . . . . 50
A.
Logical
Sequence
and
Description of the
50
Calculation . . . . . . . . · . . . . . . . . . . . . . . . . . .
1.
Logical
Sequence . . . . . . . . . . . . . . . . . . . 50
2. Determination of
the
Absorption
Probability
and
Statistical
E r r o r. . . . . . . . . . . . . . . . . 50
3.
Geometric
Techniques for
the
Full-core
Calculations . . . . . . . . . . . . . . . . . . . . . . 51
4. Tabulation
of
Results . . . . . . . . . . . . . . . , 52
B. Verification of
the
Theory
and
Techniques . : .. 52
1.
Homogeneous
Resonance-integral Test..... 53
2.
Resonance
Integral Test
for
Multiregion
Problems
. . . . . . . . . . . . . . . .
55
Ce)
3.
Test
of
Full-core
Calculation.
. . . . . . . . . .
55
1,7
4
TABLE
OF
CONTENTS
Page
4. Test with
Two
Resonance
Absorbers with
Two Resonances
Each
55
5. Test
of
Perturbation
Technique for Tem-
perature
Derivative
Calculations.
. . . . . . . .
55
6. Test of
the
Variational
Method
for
Calcula-
tion
of
Rod-size
E f f e c t s. . . . . . . . . . . . . . . 56
7. Comparison
with
Other
Numerical and
Monte
Carlo
Methods
. . . . . . . . . . . . . . . .
57
CHAPTER
IV.
VERIFICATION OF
THE
THEORY
BY
COMPARI-
SON
WITH
EXPERIMENTS . . . . . . . . . . . . . . . . . 59
A.
Rod-size
Tests in
a
Critical
Facility........ 59
1. Description of
the
Experiment . . . . . . . . . . .5 9
2.
Mathematical
Model . . . . . . . . . . . . . . . . . 60
3. Results
and
Conclusions . . . . . . . . . . . . . . 61
-WI
B. Tests
with
Various
Materials
Surrounding the
Doppler
Sample . . . . . . . . . . . . . . . . . . . . . . 62
CHAPTER
V.
THE EFFECT
OF
HETEROGENEITY ON THE
DOPPLER
COEFFICIENT OF
A
LARGE FAST
REACTOR ...... . . ............ .. ... , . . 64 .,·. 1
A. Description
of
Reactor . . . . . . . . . . . . . . . . . 64 2
B.
Mathematical
Model . . . . . . . . . . . . . . . . . . . 64
C. R e sults. . . . . . . . . . . . . . . . . . .
65
CHAPTER
VL
SUMMARY ............................ 67
A. Summary of
the
Present W o r k. . . . . . . . . . . . 67
B.
Possible
Future
Applications . . . . . . . . . . . . . 68
APPENDIXES
A. Calculation of
the
Escape
Probability for a
Finite
Cylinder . . . . . . . . . . . . . . . . . . . . . . 70
B. Calculation
of
Multiregion,
Hot-Cold
Inter-
ference
Effects
. . . . . . . . . . . . . . . . . . . . . .
73
A.
Introduction
......................
73
B. AMC Calculations . . . . . . . . . . . . . . . . .
.
74
C. C o n c l u s i o n s. . . . . . . . . . . · . . . . . . . . . . . 76 C
V
ACKNOWLEDGMENTS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
REFERENCES....................................
78