ANALYSIS OF SEVERE REACTIVITY EXCURSIONS
IN FAST REACTORS
by
r
ZACK T. PATE
B.Sc., U.S. Naval Academy
(1958)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF
PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
March, 1970
Signature of Author
Depafm-ent of Nuclear Engineering,
March 12,
1970
Certified by
Thesis Supervisor
Accepted by
Chairman, Departmental Committee on
Graduate Students
ArcTiiVei
AMpSS.
INST TCH
APR A RV7O
ClRp
1 ES
2
ANALYSIS OF SEVERE REACTIVITY EXCURSIONS IN FAST REACTORS
by
ZACK T. PATE
Submitted to the Department of Nuclear Engineering on
March 13, 1970, in partial fulfillment of the requirements
for the degree of Doctor of Philosophy.
ABSTRACT
An investigation of the events which can occur during
and as a result of a severe reactivity accident in a Liquid
Metal Cooled Fast Breeder Reactor is undertaken. Nine
phenomena which can affect reactivity are considered.
Primary emphasis is placed on the sodium voiding and fuel
motion effects which can occur after clad failure in some
region of the core.
An analysis of the energy exchange behavior of fuel
and sodium in the region of clad failure is presented.
Hydrodynamics calculations are employed to estimate the rate
of sodium voiding which takes place subsequent to clad failure. It is shown that the rate of reactivity addition which
results can be as much as an order of magnitude greater than
the maximum rates expected in an intact core.
Fuel motion effects are analyzed employing the ANISN
multigroup code. In addition, a model which allows simple
hand calculated estimates of reactivity perturbations in
fast reactors is developed and employed in assessing fuel
motion effects. It is concluded that fuel motion during the
course of an excursion is of less importance than the sodium
voiding effect unless (and until) pressures sufficient for
wide scale core disassembly are generated.
Thesis Supervisor:
Title:
Michael J. Driscoll
Professor of Nuclear Engineering
Thesis Supervisor:
Title:
Theos J. Thompson
Professor of Nuclear Engineering
(on leave of absence)
3
ACKNOWLEDGEMENTS
The author is indebted to Professor M.J. Driscoll for
his guidance throughout this work.
Rarely did a discussion
with Professor Driscoll end without my having gained new
insight into succeeding steps of the thesis.
Special thanks is extended to Dr. Theos J. Thompson
for his assistance in giving initial direction to this work.
His insight led the author into an interesting and fruitful
area of research.
Appreciation is extended to Professor D.D. Lanning for
his valuable assistance and to Miss Rita Falco for her
patience and competence in typing the final manuscript.
All computer calculations were done at the MIT
Computation Center, the cooperation of the staff is appreciated.
4
TABLE OF CONTENTS
Page
Abstract
2
Acknowledgements
3
Table of Contents
4
List of Figures
6
List of Tables
7
Chapter I: Introduction
1.1 Reactor Type Considered
1.2 Earlier Work
1.3 Objective of the Present Work
9
9
Chapter II: Background
2.1 Categories of Reactivity-Induced Accidents
in LMFBR's
2.2 Description of Typical CATEGORY II Excursions
2.3 Reactivity Effects Considered
Chapter III:
3.1
3.2
3.3
3.4
3.5
4.1
4.2
4.3
Chapter V:
5.1
5.2
5.3
5.4
5.5
13
20
28
Methods of Analysis
Description of Computer Analysis
Spatial and Spectral Effects
Kinetics Model Used
Effect of Severe Transients on the Neutron
Energy Spectrum
Summary
Chapter IV:
11
34
37
0
47
49
Transient Heat Transfer
Considerations of Phenomena in the Region of
Clad Failure, " R. "
Energy Exchange in the Region
Summary
50
55
69
Reactivity Additions from Sodium Voiding
Hydrodynamics of Sodium Voiding
Estimates of Energy Addition Rates to Sodium
Reactivity Addition Rates from Sodium Voiding
Accidents Initiated by Sodium Voiding
Summary
70
79
81
86
89
5
Page
Chapter VI:
6.1
6.2
Effects Leading to Fuel Motion
Reactivity Model for Fast Reactor Core
Perturbations
Applications of the Reactivity Model to
Accident Analysis
Core Compression and Expansion Effects on
Reactivity
Observations from Calculations and ANISN
Results
Reactivity Addition Rates from Fuel Motion
Summary
6.3
6.4
6.5
6.6
6.7
Chapter VII:
7.4
7.5
7.6
7.7
Chapter VII:
8.3
8.4
92
95
105
119
129
137
152
Doppler Effects and Miscellaneous
Reactivity Considerations
Delays in Doppler Feedback
Sources of Delays in Doppler Feedback
Effect of Spectral Shifts (and the parameter
"n")
Doppler Dead Band Due to Heat of Fusion
Positive Doppler Feedback from Fuel Cooling
Homogeneity Effects
Summary
7.1
7.2
7.3
8.1
8.2
Reactivity Effects Resulting from Core
Rearrangements
156
166
170
171
174
175
176
Summary and Conclusions
General
Analytical Models Developed
Previous and Future Work in Accident Analysis
Comparison with Thermal (Water Cooled)
Reactors
References
178
182
184
186
192
Appendix A
1.
Power History Calculations
198
Appendix B
1.
2.
Heat Transfer Correlations Employed
Supplemental Sodium Voiding Analysis
207
208
Appendix C
1.
Spectral Effect Calculations
212
6
LIST OF FIGURES
Figure
Page
2-1
Power Level vs. Time
14
2-2
Reactivity vs. Time
15
2-3
Sketch of Core Following a CATEGORY II Excursion
24
3-1
Neutron Flux vs.
38
3-2
Expanded Plot of Figure 301
39
3-3
Total Group Flux vs. Energy
41
5-1
Schematic of Core Model for Sodium Voiding
Calculations
71
5-2
Schematic of Temperature Behavior for a Fuel
Fragment in Sodium
82
5-3
Pressure vs. Temperature for UO 2
88
6-1
Group III and Group VIII Axial Flux Profile
with Central Sodium Voiding
102
6-2
Reactivity vs. Axial Sodium Voiding (20% Sodium
Removal)
111
6-3
Reactivity vs. Axial Sodium Voiding (Total
Sodium Removal)
112
6-4
Reactivity vs. Radial Sodium Voiding (20% Sodium
Removal)
115
6-5
Differential Void Reactivity vs. Axial Position
of Void
118
6-6
Reactivity vs. Unperturbed Flux (Cylindrical
Geometry)
125
6-7
Reactivity vs. Unperturbed Flux (Spherical
Geometry)
126
6-8
Schematic of Fuel Injection Model
139
7-1
Available Mechanical Work as a Function of
Doppler Feedback
155
Axial Position
7
LIST OF TABLES
Page
Table
1-1
Reactor Model
10
3-1
Multigroup Constants for the Hansen Roach Cross
Section Set
42
4-1
Vapor Pressures of UO 2
52
4-2
Fuel, Sodium, and Clad High Temperature
Properties
60
5-1
Reactivity Addition Rates from Sodium Voiding
84
6-1
Comparison of ANISN and 'PS'
6-2
Reactivity Effects from Core Expansion or
Contraction (5k)
124
6-3
Reactivity Effect of Core Rearrangements
130
6-4
Reactivity Addition Rates Resulting from Fuel
Motion
149
7-1
Maximum Acceptable Delays in Doppler Feedback
165
Model Results
117
(-Cmax)
7-2
Variation of the Magnitude of Doppler Feedback
with the Parameter
172
"n"
8-1
Summary of Reactivity Effects Considered
190
A-1
Approximate Results for Various Reactivity
Insertion Rates
205
C-1
Cross-Sections and Spectral Parameters for
'PS' Model
216
8
Chapter I
INTRODUCTION
The advantages of a practical and safe breeder reactor
have long been recognized.
observed:
As early as 1945, Enrico Fermi
"The country which first develops a breeder reac-
tor will have a great competitive advantage in atomic energy"
(1).
Although a great deal of thought and effort have been
directed toward the breeder concept in recent years, the fact
that much remains to be done is also evident.
An exhaustive
compilation of the tasks requiring completion is given in the
United States Atomic Energy Commission's recently published
Liquid Metal Fast Breeder Reactor (LMFBR) Program Plan
(WASH-ll10) (1).
This ten volume set of documents sets
forth a plan of action for producing "a viable industrial
capability which will provide LMFBR plants on a self-sustaining competitive basis, at minimum cost, and in a timely
manner."
The program plan establishes development of the
LMFBR, with sodium coolant, as the AEC's priority effort
toward large scale acceptance of the breeder.
The present work focuses on the safety aspect of such
reactors.
Specifically, accidents are considered which are
initiated by excessive reactivity insertions, or reactivity
additions at an excessive rate, into a critical reactor.
the course of the study, consideration was given to those
problem areas outlined in Volume 10, SAFETY, of the AEC
Program Plan (1).
In
9 1
M1
1--_
9
1.1
Reactor Type Considered
A series of conceptual design and follow-on studies
have been completed by four U.S. nuclear contractors (2-5).
These studies, under sponsorship of the USAEC, considered
various compositions and configurations for a 1000 megawatt
electric (Mwe) fast breeder reactor.
The present work has
drawn on these studies in arriving at a "typical" LMFBR for
use as a basis for analysis.
1.2
Earlier Work
Numerous earlier studies in the literature have investi-
gated severe reactivity accidents in sodium cooled fast
reactors.
Such studies are frequently referred to as
"Meltdown Analyses" when carried to the point of core meltdown and disassembly.
An important work in 1956 by H.A.
Bethe and J.H. Tait (6),
work.
forms the basis of much of the later
This well known analysis uses perturbation theory to
calculate shutdown reactivity resulting from density changes
as a reactor is disassembled by the high internal pressures
generated in a severe power excursion.
Later investigations
have modified the Bethe-Tait approach and have incorporated
other considerations into the analysis, such as inclusion of
doppler feedback and treatment of zoned reactors (7-10).
In
addition, computer programs such as AX-1 (11) have been
developed to perform coupled neutronic and hydrodynamic calculations.
A common feature of these investigations is their
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Table 1-1
REACTOR MODEL
1.
Fuel Material
PuO 2 /UO
2.
Fertile/Fissile Ratio
6.5/1
3.
Geometry
Cylindrical
4.
Diameter of Core
274 cm
5.
Height of Core
100 cm
6.
Blanket Dimensions
20 cm in thickness
on all sides
7.
Average Specific Power
150 kw/kg fuel
8.
Composition (Vol.
(a)
(b)
2
%)
Core
Fuel
Coolant (sodium)
Structure (stainless steel)
Blanket
U02
Coolant
Structure
9.
35
50
15
35
50
15
Clad/Thickness
316 SS/15 mils
10.
Fuel Pin OD
.25 in
11.
Pitch/Diameter (Triangular Arrays)
1.33
12.
Mean Centerline Na Temperature at
Full Power
1000 0 F (800 0 K)
13.
Maximum Fuel Centerline Temperature
at Full Power
4700 0 F (
2900 0 K)
14.
Mean Fuel Temperature at Full Power
and Location of Peak Flux
2700 0 F (
1700 0 K)
15.
Delayed Neutron Fraction: p
.0033
16.
Full Power
2500 MWt (1000 MWe)
17.
Neutron Mean Generation Time: .A.
3.3 x 10~
sec
11
treatment of core fuel, clad, and structural materials as
homogeneous throughout the excursion.
The principal reac-
tivity effects, and hence the severity of the accident,
have often been analyzed without considering the detailed
progression of events taking place during the course of the
transient.
The reactivity effects usually included are:
an accident "ramp" reactivity insertion from an unspecified
source (or from an unspecified process of sodium boiling);
Doppler feedback*; and disassembly feedback.
1.3
Objective of the Present Work
The objective of the present work is to consider the
progression and interrelation of events taking place during
severe power excursions.
A particular effort is made to
include every phenomena that might affect reactivity and
thus, ultimately, the seriousness of the accident.
A description of the accident and the various reactivity
effects investigated is the subject of Chapter II.
The
detailed study which follows attempts to identify by analytical arguments the more important parameters governing each
reactivity effect and to show clearly the overall importance
of the effects considered.
Chapters III through VII are de-
voted to this effort.
*In reactors of the type under investigation, Doppler
broadening contributes most of the reactor's negative
temperature coefficient and, as such, is the primary inherent mechanism which can provide the negative feedback
necessary to prevent core disassembly in the event of a
severe reactivity excursion.
12
The area of reactivity changes associated with fuel and
other core material movements is the subject of considerable
study in Chapter VI.
The usual form of one group perturba-
tion theory was found entirely inadequate for this purpose.
Thus, the development of a simple analyticalreactivity
model capable of handling local and global perturbations in
an LMFBR, and therefore useful in analyzing accident conditions, is undertaken.
The ANISN/DTR II multigroup code is
employed in analyzing core rearrangements and for determining the accuracy of the simple reactivity model developed.
As a final assessment of the significance of each
phenomenon considered, a qualitative comparison between the
comparable effects in thermal reactors (specifically H20
moderated PWR's and BWR's) and LMFBR's is given in the
concluding chapter of the thesis.
13
Chapter II
BACKGROUND
In this chapter three categories of reactivity excursions in LMFBR's are defined; generally according to the
severity of the accident.
A typical excursion in the category
of principal interest in the present work is described and
reactivity effects which can arise during such an excursion
are discussed.
2.1
Categories of Reactivity-Induced Accidents in LMFBR's
The oscillatory behavior of prompt supercritical excur-
sions in LMFBR's with negative Doppler feedback has been
amply demonstrated. (7)(13)(15)(16)(17)(19)
Figures 2-1
and 2-2 show the time history of power and excess reactivity
for two "cycles" of a typical transient of this type.
If no
corrective action is taken, clearly any excursion induced by
a continuous "ramp" of reactivity will result in clad failure
and eventually in core destruction or disassembly.
For pur-
poses of discussion, it is somewhat arbitrarily assumed that
external corrective action (such as scram or cutback) can
"turn down" or terminate an excursion after about 200 milliseconds (55).
For excursions considered in the present work,
the effects cf phenomena of interest will be seen to take
place well within the first 200 msec after initiation of the
accident; thus the exact value of the scram delay time is
not important.
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In Chapter VII, a ramp rate of 10 $/sec is found to
result in four cycles or power pulses in a 200 msec interval.
Note that 100 msec are required to achieve prompt criticality;
the four super-prompt critical power bursts then occur in the
remaining 100 msec.
If corrective action is taken after the
fourth power oscillation, clad rupture is not expected to
occur for such an excursion in the LMFBR's considered here
(see Appendix A).
For higher reactivity insertion rates,
clad failure can occur within the first 200 msec and may
occur after the first, second, or third power pulse.
For
purposes of discussion, three categories of such excursions
are defined as follows:
CATEGORY 1:
An excursion which is not severe enough
to cause clad rupture (or melt-through) in any
part of the core within 200 msec after initiation
of the accident reactivity ramp.
Present models of transient heat transfer
from intact fuel rods and resulting sodium void
effects, Doppler effects, and the associated
reactor kinetics models appear sufficiently
accurate to predict the threshold of clad failure for such excursions with reasonable precision (8)(12)(13).
An approximate analytical
solution defining this threshold is given in
Appendix A and discussed in Chapters VI and VII.
17
CATEGORY II:
An excursion which leads to clad rupture
in some region of the core but in which the power
pulse causing clad rupture is "turned down" by
Doppler feedback before pressures sufficient for
core disassembly are generated.
Clad failure may
occur at any time within the first 200 msec: after
one or more super-prompt critical power bursts.
To simplify terminology, the power oscillation(s)
occurring prior to clad failure will be referred
to as the initial excursion.
Transients or power
pulses occurring after clad failure will be referred to as secondary excursions.
Core disassembly may or may not occur subsequent to the initial excursion.
For reactors of
the type under consideration, Doppler induced
reactivity changes are thought to be sufficient to
turn down the first power transient before core
disassembly, even for very severe excursions resulting from assumed reactivity insertion rates
in excess of 100 $/sec (13)(15)(16)(17).
(Note
that the time to reach prompt criticality at a
ramp rate of 100 $/sec is 10 msec.
Again from
Chapter VII, the time between successive power
peaks is about 8 milliseconds.)
Thus, CATEGORY
2 covers quite a wide range of reactivity accidents.
18
Clad rupture in this category will be
followed by fuel fragmentation and/or dispersal
into the adjacent coolant channels; a phenomena
which is discussed in some detail in Chapters IV
through VI.
CATEGORY III:
For extremely severe excursions result-
ing from accident ramp rates in excess of approximately 200 $/sec, the first power pulse is not
"turned down" by Doppler prior to core disassembly.
In such a case the entire excursion, from the time
of prompt criticality to shutdown by disassembly,
is over in about five milliseconds or less (7)(18).
The elapsed time from clad rupture until pressures
sufficient to start core disassembly are generated
is of the order of one millisecond (7).
Calcula-
tions in Chapters VI and VII indicate there is insufficient time for the phenomena which takes
place between clad failure and core disassembly
(other than disassembly itself) to alter the
behavior of such a rapid excursion.
Since the
progression of events following clad failure
19
(and prior to core disassembly*)
is of particular
interest in the present work, CATEGORY III accidents will not be treated.
This does not, of
course, limit the severity of accidents studied
but simply rules out of consideration an initial
excursion of the severity here classified as
CATEGORY III.
CATEGORY II accidents, then, are the subject of the
present study.
It must be pointed out that this category
covers the more credible severe excursions, and perhaps the
complete range, since it can be realistically argued that
the rates of reactivity insertion required to override
Doppler and lead to core disassembly during the first power
pulse of an excursion (CATEGORY III) are not credible in
large oxide cores of the type under consideration (see Section 2.2 of the present chapter).
In the important range of
accidents here defined as CATEGORY II, however, the events
following clad rupture may combine to produce a more severe
secondary excursion, or to prevent complete Doppler
*n
, a distinct possibility of core
compaction into a second critical configuration exists for
certain geometries of large fast reactors. Such a compaction
could be accomplished at a very rapid rate, thereby possibly
increasing the severity of the initial excursion or causing a
second excursion just as disassembly is terminating the first.
Indeed, the rates of negative reactivity insertion during disassembly are estimated to be well in excess of 1000 $/sec (7).
If the pressures which cause disassembly at such rates can
conceivably cause re-assembly into a second critical configuration, the serious implications are evident.
Reassemblies of divided sections of a core under the acceleration of gravity have been discussed frequently in the
literature (2)(4)(7)(15). While such an accident appears no
more credible than reassembly by differential internal pressures, the latter could pose a much greater hazard. Neither
type of accident is the subject of the present investigation,
however.
20
"turn down" of an initial
2.2
excursion.
Description of Typical CATEGORY II Excursions
In recent literature on LMFBR accidents, reactivity in-
sertion rates in excess of about 66 $/sec are not given much
credence in that this is the maximum value estimated for
sodium voiding from intact cores (12)(13)(23)(24)(25); for
core collapse under gravity (24); and is the maximum considered in investigating a number of initiating accidents such
as control rod ejection, loss of flow, etc. (25).
In refer-
ence (23), 66 $/sec is used as the reactivity insertion rate
in the Design Basis Accident (DBA) proposed by Atomics
International (Aronstein) and in reference (24), 65 $/sec is
the maximum rate estimated for "hypothetical accidents" considered by Combustion Engineering (Noyes).
Furthermore, in
reference (25), 60 $/sec is established as the "basis of the
design" of the Belgian-Dutch-German Consortum LMFBR; the 300
Mwe SNR reactor.
Thus, in the present work, 66 $/sec will
be used as a basis of comparison for various reactivity
effects considered.
For the LMFBR considered here, a reac-
tivity insertion rate of 66 $/sec will result in a CATEGORY
II accident, as will be seen in succeeding sections of the
present chapter.
In the presence of Doppler feedback of the magnitude
expected in large oxide fueled LMFBR's, as typified by the
reactor of Table 1-1, initial excursions in the category of
21
interest are expected to be "turned down".
That is, power
will be reduced to its level at prompt criticality and
reactivity reduced to less than (1+P) (18)(19).
In fact,
in the limiting case of zero generation time, Hafele has
shown that reactivity is reduced below (l+P) after an initial
transient by an amount equal to the maximum rise above (l+s)
during the transient (19).
As noted, if the initiating reac-
tivity ramp continues, successive power peaks will occur
until the accident is terminated by core disassembly.
References (13),
(15),
(16) and (17) (representing studies
in France, Japan, The United Kingdom, and the United States
respectively) show detailed histories of excursions in which
two or three power peaks occur before disassembly takes
place.
Various "representative" Doppler coefficients are
used in these four studies with reactivity ramp rates of from
40 $/sec to 100 $/sec.
The duration of the excursions, from
prompt criticality until shutdown by disassembly, ranges from
25 to 70 milliseconds; with time intervals between power
peaks of 10 to 20 milliseconds.
Thus, these CATEGORY II ex-
cursions are over in well under 200 msec and insufficient
time is available for external corrective action by presently
known methods.
None of the four studies includes the fuel
fragmentation/dispersal or fuel motion effects considered in
the present work.
During such accidents, clad rupture could
occur after the first or any subsequent power peak.
For
22
purposes of discussion, clad rupture will be assumed to
occur during or immediately following the second power peak
of an excursion typified by the time histories of power and
reactivity shown in Figures 2-1 and 2-2.
The region of
clad failure following such a transient depends primarily
on the accident reactivity ramp rate, the strength of the
Doppler feedback, and the neutron flux shape.
If the
effects following clad rupture are not pre-emptive, and if
the accident reactivity ramp continues, a third power pulse
will occur as discussed above and as shown by the dashed
curves of Figures 2-1 and 2-2.
The excursion represented by Figures 2-1 and 2-2 assumes
an accident reactivity ramp rate of 66 $/sec.
Doppler feed-
back reactivity behavior is assumed to be of the form:
dK
-ADO(
)
,
(2-1)
T
where
ADOP = .003 to .008,
n
=
0.8 to 1.2.
The parameter "n" is determined by the fissile/fertile
ratio, the fuel temperature, and the neutron energy spectrum
of the LMFBR in question.
It is fairly close to unity in
LMFBR of the type considered here.
For the present estimate
n was assumed equal to 1.0 and ADOP was taken as .004; hence:
23
5K
where
DO?
T=
-
oo4n(
,
O)l(-
(2-2)
average fuel temperature at steady state
full power.
The flux shapes for the core considered were calculated by
using the ANISN multigroup
Figure 2-3.
computer code and are shown in
For the ramp rate, Doppler feedback, and flux
shapes described, calculations described in Appendix A indicate that clad rupture and fuel fragmentation/dispersal will
occur over a substantial region of the core ( " 30 volume
percent).
This result is depicted in Figure 2-3.
The growth
of the fuel rupture region, hereafter designated -19,,
tinues until time t2 of Figure 2-1.
con-
The trend in large
LMFBR's is toward designs which produce a relatively flat
radial flux profile with steeper axial flux gradients (through
the use of fuel zoning and pancake geometry) as indicated in
Figure 2-3.
Thus'ucan extend over a large radial area of
the core while its axial propagation remains relatively
small.
The vapor pressure of UO 2 is a strong function of
temperature at pressures near the threshold of clad failure,
hence a transient which produces pressures just sufficient
for clad rupture out to
i z I = 30 cm in the axial direction
could produce pressures several thousand psi higher near the
core center (see Chapter IV for supporting calculations and
further discussion).
Thus, the fuel nearer the center is
F i GuRV
SkE--iL
24
Z- 3
o-E COFCORE FOLLOWiJ& A CA-Te&oaRY rL
-
Exrenoa4O -
4~
zOm.
z
I
I
AX IAL''-RE irLrr-c MQ'
SM i Kts ~ 2o cm.
I
to
/
I
I
137om .
10
CoAE
is VEp c-TED A-r TIME Ag
1:u LvA SHAge
' ?A 10R
-0T-o n%K
or lrl&-2-L.
*2 -
Wi-ir
25
likely to be more finely fragmented or dispersed (20)(32).
Presumably the clad will remain intact outside la; at least
immediately following time t ,
2
The fuel vapor pressure profile within the clad (prior
to clad rupture)
is not expected to be completely symmetric
about the axial centerline of the core due to the fact that
clad temperatures are higher in the upper half of an operating core (21).
The deviation from symmetry and correspond-
ing axial dislocation of regionR is expected to be small
for severe excursions and of little significance in the arguments presented here.
Thus, in Figure 2-3 and for purposes
of discussion, the size and axial position of region IL have
been determined from the axial flux profile and hence for an
axially symmetric pressure distribution.
For an excursion of the type described here and depicted in Figures 2-1 through 2-3, the following characteristics
apply and will be of some importance in subsequent reasoning
and developments:
a)
The time scale of events is of the order shown in
Figures 2-1 and 2-2.
In the absence of pre-
emptive effects following clad rupture, as considered herein, or some external shutdown or cutback action, and if the accident reactivity ramp
continues the time interval between successive
power peaks will be about 2 tp or approximately
26
8 msec for the particular transient illustrated.
Furthermore, the third power peak will occur
about 20 msec after the time of prompt criticality.
b)
For this time scale of events, the heat conduction
through the intact clad is a very small fraction
of that generated in the fuel.
It is, in fact,
generally less than the energy deposited in the
sodium by prompt neutron and gamma heating.
Calculations verifying this assumption are given
in Chapter IV.
c)
No appreciable melting of clad occurs in the regions
where the clad remains intact for 20 milliseconds
or so following the first transient.
in
&R is
Clad rupture
the result of excessive fuel vapor pres-
sures; not clad melting.
(The higher average
temperature of the clad does, however, result in
a significant reduction in stress required for
rupture and this is taken into consideration.)
d)
The fertile constituent of the fuel (UO2 ) contributes
the negative component of the Doppler feedback.
Theory and experiment indicate that the reactivity
contribution from the fissile isotope (PuO ) is
2
quite small by comparison (22).
The Pu0 2 contri-
bution is taken as zero in this work and the small
27
percentage of fissions occurring in the fertile
component is ignored in discussions of Doppler
feedback.
Direct neutron and gamma heating of
the fertile species is also ignored initially;
this is discussed further in Chapter VII.
e)
The power pulses in such transients are of relatively short duration.
In Figure 2-1, where clad
failure is assumed to occur during the second power
pulse, the pulse width is 2 msec.
This is relative-
ly short compared to the time between successive
power pulses (8 msec in the present example).
Thus the growth of region'd& is expected to be
rapid with a distinct termination at time t ,
2
f)
The reactivity above prompt critical is unlikely to
exceed 40,
even for extreme initiating accidents,
before the excursion is turned down by Doppler
feedback or terminated by core disassembly.
(This
observation is discussed further in Chapter III).
The total reactivity available from most of the
phenomena considered in the present work exceeds
40i, however.
Thus, the rate of reactivity addi-
tion rather than the total reactivity available
is found to be the more important consideration.
g)
The transients considered herein are assumed to
start from one of two conditions of reactor
28
operation;
full power or 10 percent power.
In
both cases, the average sodium temperature in
the core is taken to be 800 0 K.
For power levels
below 10%, presumably plant operating procedures
will require that plant temperatures will be
lowered; resulting in a substantial increase in
the energy required to produce sodium voiding and
in a substantial improvement in the magnitude of
Doppler feedback (see Eq.
2.3
(2-1)).
Reactivity Effects Considered
An examination of the possible progression of events in
the excursion describedabove suggests the following reactivity effects should be considered.
2.3.1
Increased Rate of Sodium Voiding Following Clad
Rupture
The fuel fragmentation or dispersal following clad rupture in region-R exposes higher temperature surfaces -
and
potentially a greater surface area - to sodium in region
1.
Analyses which have not considered fuel failure have
been used to calculate maximum reactivity addition rates of
15 to 65 $/sec as a result of sodium voiding (12)(23)(24)(25).
If the heat transfer rate from hot fuel to sodium is appreciably increased in
'6:L
, a substantial increase in the rate
of reactivity addition from this source is possible.
29
Furthermore, high fuel vapor pressures near the core
center, in a region from which sodium has been voided, can
further accelerate the voiding process.
The strong depen-
dence of fuel vapor pressure on temperature in the range of
interest will be seen to make this a plausible effect.
2.3.2
Fuel Injection from Intact Clad
Immediately following time t2 of Figure 2-1, the fuel
outside "
is constrained by its cladding except, for an
"open end" or "split" somewhere in
V.,
(see Figure 2-3).
Since the fuel vapor pressure in region
6p.
was sufficient to
rupture the clad, the vapor pressure within the clad can be
considerable some several inches outside 6R (or in a nonvented design, the fission gas pressure could also be considerable).
Since the fuel even farther away from 'R is
cooler and is eventually "tamped" by the axial blanket material in the continuous rod designs usually proposed, the potential exists for fuel movement or "injection" into the region
V
and hence into a higher worth region of the core.
2.3.3
Fuel Movement Under a General Inward Pressure
Gradient
If fuel dispersal and subsequent substantial fuel cooling takes place in region
L,
as discussed in 2.3.1 above,
the potential exists for an inward pressure gradient; independent of restraint by the clad.
During the time interval
between an initial and secondary power excursion, fuel in
30
central regions ofltcould conceivably be subcooled by
several hundred degrees relative to that outside or near the
edge of
"aL.
If this occurs, fuel vapor pressures in'R will
tend to "lag" the vapor pressures generated in adjacent
regions, thus tending to accelerate fuel into higher worth
regions of the core during some portion of a secondary
transient.
It is shown in Chapter IV that the fuel vapor
pressure generated during such a secondary transient can
rapidly exceed the sodium vapor pressure generated between
the initial and secondary excursions.
2.3.4
Positive Doppler Effect
The high temperatures in the fertile material, which
produced the negative Doppler feedback required for turn
down of the initial transient, can be reduced at a rapid rate
in region - as the fertile material comes into more intimate
contact with sodium coolant.
Thus the cooling of fuel in
region -R is a source of positive reactivity insertion and
could significantly change the rate of reactivity insertion
at the time of prompt criticality during the start of a
secondary excursion.
This positive Doppler feedback will
continue until a secondary transient progresses to a point
where the rate of heat addition from fission is sufficient
to overcome the cooling effect described.
2.3.5
Doppler "Dead Band" Due to Heat of Fusion and
Vaporization
During the time UO2 is being melted, little or no Doppler
31
broadening is thought to occur in a macroscopic sense (26).
The heat of fusion of UO 2 is 278 joules/gm0 K; or roughly
equivalent to a temperature "dead band" of 700 0 K.
During a
secondary transient, which starts from a higher temperature
datum, much more fuel could be affected by this "dead band"
than is the case for an initial transient.
If a significant
portion of the fuel is in this dead band during particular
portions of a secondary excursion, the effect could be substantial.
The heat of vaporization can play a similar role but
behavior of the system with regard to this phenomenon is not
clear.
This is discussed further in Chapter VII.
2.3.6
Doppler Reduction Due to Spectral and Temperature
Effects
As temperature increases, a reduction in the magnitude
of Doppler feedback is predicted by Eq. (2-1) for all values
of "n" of current interest.
In addition, effects which
harden the energy spectrum (such as sodium voiding) cause a
reduction in the strength of Doppler feedback.
The reduction
has been represented by a decrease in ADOP (27) or by an
increase in "n" of Eq. (2-1).
Furthermore, an increase in
"n" has been suggested as temperatures increase (28)(29).
All of these effects tend to reduce Doppler effectiveness
during a secondary transient.
2.3.7
Delay in Doppler Feedback
If the fissile and fertile fuel materials are mixed as
32
powders with particle sizes on the order of the mean range
of fission products in fuel ( ^j 10 microns) or larger, a
finite period of time is required for transfer of the
heat
of fission from the fissile isotope to the fertile isotope.
This results in a "delay" or "time lag" for Doppler feedback
(9)(30).
This delay will affect the initial transient, but
could have a stronger effect on a secondary excursion since
the fuel material will be mixed with sodium and structural
materials.
This dispersal or mixing reduces the direct fis-
sion product heating of the fertile species.
Additionally,
if dispersion is fine enough, the introduction of sodium
vapor between fissile and fertile particles and the resulting
film temperature drop could increase the time required for
conductive heat transfer.
Both of these latter effects pro-
mote a greater time delay between the fission event and
Doppler broadening of UO .
2
2.3.8
Transient Induced Spectral Shift
If the neutron distribution as a function of energy is
altered as a result of a strong transient, a reactivity
effect and an influence on the behavior of other phenomena,
such as Doppler feedback and sodium voiding, can be induced.
Specifically, the inclusion of "W/v" terms in the usual multigroup formulation is analogous to the insertion of a "l/v"
absorber.
This in turn can be expected to produce a spectral
perturbation.
I........
..
33
2.3.9
Homogeneity Effects
Although fast reactors are generally considered homogeneous in neutronic spatial calculations, a recent study (31)
indicates the small degree of fuel "lumping" employed in
current LMFBR designs is beneficial (i.e. results in increased
reactivity).
The gain in reactivity arises primarily from
the increased "first flight" neutron flux within the fuel
rods (31).
Thus the tendency towards homogenization which
results from severe accident conditions can induce a negative
reactivity effect.
Effects 2.3.1 through 2.3.9, then, could combine to substantially alter the behavior of a secondary excursion.
The
controlling parameters influencing these effects are discussed in succeeding chapters; with quantitative estimates of
the maximum total reactivity change and reactivity insertion
rates for some of the individual effects.
The nine effects
above are listed in Table 8-1 together with information
developed in succeeding chapters.
34
Chapter III
METHODS OF ANALYSIS
The present chapter gives a description of the computer
analyses employed in this research.
The effect of severe
perturbations or core rearrangements (simulating accident
conditions) on the space and energy distribution of the neutron flux is analyzed, using multigroup computer calculations.
The neutron kinetics equations employed in succeeding
chapters are presented and the question, posed in Chapter II,
as to whether a severe transient produces a significant spectral disturbance, in and of itself, is answered.
Description of Computer Analysis
3.1
The ANISN/YDTF II Multigroup Code (56) was employed for
various reactivity calculations and for investigation of spatial and spectral effects.
The S-8 transport theory approxi-
mation was used for the final values of runs; the diffusion
theory approximation was employed in setting up the various
problems of interest to save running time.
Cross section
sets employed included the Hansen Roach 16 group set (57)
and the Russian ABBN 26 group set (58).
Differences in reac-
tivity effects predicted by the two sets were found to be
quite small; comparative data is given in Chapter VI.
The ANISN code is one dimensional, but buckling values
(B
)
can be specified in one or two additional dimensions to
35
simulate cylindrical or parallelpiped configurations.
The
code then calculates leakage in these additional directions
by including an artificial absorption term, DB , in the
2x
multigroup formulation for each Bx specified. In the present
work, the following steps were employed to obtain a base case
representation of the reactor described in Table 1-1.
1)
Calculations were made in cylindrical geometry with
a first estimate of axial buckling, B , to account
for axial leakage.
The core radius was adjusted
to achieve a critical system.
2)
The code was then modified to calculate the flux
dependence in the axial direction using a value of
B
to characterize radial leakage.
B
chosen was that corresponding to the fundamen-
The value of
tal mode eigenvalue which gave the best fit to the
radial flux shape calculated in step (1).
This
proved to be a very accurate method of estimating
radial leakage because of the predominance of the
fundamental mode shape over most of the core, as
will be seen in the next section of the present
chapter.
3)
The fissile/fertile ratio was then modified
slightly to obtain a critical system for a core
height of exactly 100 cm.
36
4)
Iterations of steps (1) through (3) were carried
out until the changes in radial buckling and the
fissile/fertile ratio were negligible.
The re-
sulting values are those given in Table 1-1.
All subsequent calculations for the reactor of Table 1-1 were
made in this pseudo-cylindrical geometry with the axial
direction as the dimension of primary interest.
In addition, a number of computer runs were made in
spherical geometry for comparison purposes, using a reactor
with the same composition as that of Table 1-1.
The core was divided into 40 intervals of 2.5 cm each in
the axial direction for cylindrical geometry and 20 radial
intervals of equal volume in spherical geometry.
The concen-
tration or density of each core constituent could be varied
in any interval(s) to represent various accident conditions.
For example, sodium voiding in the central 10 cm of the cylindrical core could be represented by specifying zero sodium
density in the four central 2.5 cm intervals.
For convenience, a concentration factor, C, is defined
as the ratio of the perturbed density of a given material to
its density in the unperturbed critical core.
C
px perturbed
px unperturbed ,
Thus,
(3-1)
where the subscript x denotes the core constituent; Na, Fe,
or fuel.
37
3.2
Spatial and Spectral Effects
Calculations with the ANISN code, made by using both
16 and 26 energy groups, indicate that the neutron flux shape
over the central 80% of the core volume corresponds to the
fundamental mode shape.
Furthermore, the results obtained
show that even severe local perturbations do not significantly alter the flux profile.
Figure 3-1 shows a plot of
neutron flux versus axial position in the core for four cases
of interest.
Figure 3-2 is an expanded plot of the same data
showing percent deviation from the unperturbed flux profile.
Curve 1 (in both figures) is the unperturbed profile.
Curve 2 is a cosine curve, representing the axial fundamental
mode,
with an effective core height of 137 cm.
In Figure 3-1,
curves 1 and 2 coincide except for the region within 7 or 8
centimeters of the blanket.
Curve 3 shows the results for
complete sodium voiding (CNA=0) in the central 25 centimeters
axially, and in the entire radial direction.
The reactivity
calculated for this degree of sodium voiding was + $3.70.
Curve 4 shows the results for sodium voiding as above, but
with all fuel (UO2 and PuO 2 ) from the 12.5 to 15 cm region
(Cfuel=0 in interval 15~
) moved inward and smeared uniformly
over the central 25 cm of the core (Cfuel=1.2 in intervals
Ll-4 - ).
served.
Notice that the total mass of fuel material is conIn Figure 3-2, observe that the maximum deviation
from the unperturbed flux profile or from the fundamental
38
Lse..
~
cr
-J
C U ZV
F-
C'
~E'4CEP-
7'
U42t
2r
X
p
I
opl~4-r~-s scjLA- P/-or
Oulr6lbr. z=:40c*, 4
PkU
V2S%-.
f-l
CUAVC
NAzVIDE
u R4 i L 4 0te
31
~)43-2 Ok2 bATA 2Poi.sr)
Tr L,
2to
Not)Iz
IZ
C.vi.
S
e
A~s>
2.
Scz(o
)
Ar16ru~e-)?ADFD
?J-oTr
salb
rZ
oa
I
E(7
F 9 3- 1
514DWiN&
DE,11wrloo 4
0 V: C J Me -5 1 1-3.P MAD
FRO M - HE
NNbAMC-NIML
IF&oAOrm___
,
3-2~~
MODE 5HIAPr=
-Irr
I Yr
<p
REctoN~ OF'
C-o~ uM
tA
FIL A
LoJ
N 0RCuAVt'
R~
CU
o-IA'N
VE
/
/
/ 01
7'
2
0
g
a:
.7
(~J
A
0
A
I
2
U
(~.
/
/
cp
1k
~~-/;~
Cf,
2o
-
N~S
Cu~vcE 3
NA1j,
'4JoIDE
jriif, )Z}lL )2.5c-.
-
UNFOPb1L
Sri~cZ~
10
IC,
20
30
~
~z~j2.c~,&E~1oN
Sb
j
4o
mode shape is about 1%.
Also note that even in the 12.5 to
15 cm region where all fuel is removed, the flux dip is quite
small.
The reactivity calculated for this rearrangement was
+ $3.30; quite a substantial perturbation.
This tendency of
the flux shape to remain very close to the fundamental was
observed to hold for all core perturbations or rearrangements considered.
The flux shown in Figures 3-1 and 3-2 is
the total flux (sum of the flux in all energy groups) at a
given spatial position.
The spatial shape of the flux in
each individual energy group was observed to behave very
much like the total flux, but there are, of course, local
changes in the energy spectrum as will be discussed shortly.
Reactivity perturbations in the hypothetical cases investigated ranged as high as + $6 (see Table 6-3
VI).
of Chapter
Recall from Chapter II that the net positive excess
reactivity cannot realistically exceed about + $1.40.
Thus,
for perturbations of interest, the assumption of a fundamental mode flux shape in the central regions of the core
should be quite valid.
Figure 3-3 is a plot of the total flux in each energy
group for the first nine groups of the Hansen Roach crosssection set (groups 10-16 are in the epithermal and thermal
range and have very little influence on the behavior of the
LMFBR under consideration; Table 3-1 shows the energy and
lethargy range of each group).
Curve 1 shows the unperturbed
3-3
Fua-ForAL
d/o .
3
'I.
CvE13:
Iuoj
30-
I
6&(Zou?
FLJ
WAOJ-
ON
SK c-rqw
C- u Rv s-:
, 44 -
C,,=-I 4), Lv -S E c
Lj = -
0 000 stC
2
N
4
to.
E
0
40
4
0
Af
4
1~0
2.
7
0.0011j77
W. %-.,I tj
I
.
a
W. .4 t
0.
5-
A V C kA(s E t R 0 u -,-' S S C- R Z- Y, Z:..* M C V. --f-
J. I S'
z
2.2.0
I
42
Table 3-1
MULTIGROUP CONSTANTS FOR THE HANSEN ROACH CROSS SECTION SET
Group
Neutron Energy
Range
Avg.
Neutron Velocity,
Au
Fission
Spectrum
108 cm/sec
28.5
0.204
0.762
19.9
0.344
0.9 - 1.4 Mev
o.442
14.7
0.168
4
0.4 - 0.9 Mev
o.811
11.0
0.180
5
0.1 - o.4 Mev
1.386
6.7
0.090
6
17 - 100 key
1.772
2.70
0.014
7
3 - 17 key
1.735
1.14
0
1.696
o.48o
0
1
3 - oMev
2
1.4 - 3 Mev
3
8
0.55 - 3 key
9
100 - 550 ev
1.705
O.206
0
10
30 - 100 ev
1.204
0.101
0
11
10 - 30 ev
1.099
0.0566
0
12
3 - 10 ev
1.204
0.0319
0
13
1 - 3 ev
1.099
0.0179
0
14
o.4 - 1 ev
o.916
0.0109
0
15
0.1 - 0.4 ev
1.386
0.00606
0
0.00218
0
16
Thermal (0.025 ev)
43
energy spectrum calculated by ANISN.
Curve 3 shows the
hardened spectrum which results from voiding the entire core
of sodium.
Notice that the reduction in neutron flux in
group 8 is roughly 50% and that the increase in group 3 is
roughly 20%.
Thus a comparison of Figures 3-1 and 3-3 shows
a much more substantial qualitative change in the spectral
shape than in the spatial shape for analogous perturbations.
These results suggest that reactivity effects can be accurately
calculated by assuming that the spatialflux profile corresponds
to the fundamental mode, provided the spectral shift effect
is properly accounted for.
In particular, perturbation theory
should be useful in dealing with the effects in question.
This conclusion is given further support in Chapter VI and a
simple method for accounting for spectral effects, within the
framework of perturbation theory, is presented.
In addition, since power density and energy density are
directly proportional to neutron flux, the fundamental shape
functions should be useful in analyzing effects associated
with energy density; such as local temperatures and pressures.
Use is made of this observation in Chapters IV and V.
3.3
Kinetics Model Used
The simple spatial flux dependence shown in the previous
section indicates that the point kinetics equations can be
usefully employed in investigating time dependent effects.
Only system behavior in the prompt critical range is considered
since CATEGORY 2 excursions are expected to lead rapidly to
prompt criticality.
For such analyses,
the simple point
kinetics equations given below have been found to give accurate results provided proper initial conditions, calculated
with consideration of delayed neutrons, are specified (7)(8).
With the usual notations:
5
k
dn
(3-2)
or, since in the reactor of interest, power level is everywhere proportional to neutron flux:
dq__
k(t))
-
dt
- qo
A(3-3)
where
q(t) = power density at time t
bk(t) = reactivity above prompt critical
A
= neutron generation time.
Time zero is taken as the time of prompt criticality and the
powerlevel at this time is given by: (7)(8)
0o =ss
where
2a
q ss = delayed critical steady state power density
P
= delayed neutron fraction
a
= reactivity ramp rate in 5K/sec.
45
Because of the short neutron generation times, ramp
reactivity additions are generally considered more realistic
than step additions in fast reactors.
The solution to Eq.
(3-3) for a step insertion is given here, however, to illustrate a point:
q(t) = qoexp(
j) .
(3-5)
Typically, lifetimes in a thermal reactor are about a factor
of 100 longer than in LMFBR's, whereas the delayed neutron
fraction is roughly a factor of 2 larger.
With these compara-
tive values, if a step reactivity input of 40o
above prompt
critical is assumed for both systems, Eq. (3-5) gives:
q(t)
qoe 4 000t
for the LMFBR
and
= qoe 8 0t
q
for the THERMAL system.
In an LMFBR, as will be shown, time intervals of interest
during an excursion are typically of the order of 10 milliseconds.
If the systems described above are not altered for
10 msec following the step insertion, the power level will,
in theory, reach
q
= qoe
10
3q
for the LMFBR
and
q~)= qoe*
8
2.2q 0
for the THERMAL system.
46
Thus, the hypothetical power level achieved in the LMFBR is
more than 1012 times greater than the level reached in the
THERMAL system if an excess reactivity of ~%$1.40 is maintained
in both systems for 10 msec.
This indicates in a very quali-
tative way that, while excess reactivities of several dollars
may be of interest in THERMAL systems, such is not the case
for LMFBR's.
This conclusion is reached in a more rigorous
manner in references (7) and (8) and substantiated by comments
in (22).
For the more realistic ramp input of reactivity as the
accident initiating condition, Eq. (3-3) gives:
E~t)
oexp
2 -
t d
p
+ A15K
36
feedback
The feedback reactivity is frequently written:
SK~t
+ 5K
=5K
Doppler
feedback
Na
voiding
+5SK+..
disassembly
(3-7)
Such an expression may be misleading, however, as some of the
effects may not be linearly additive.
In Chapter VI, in
fact, it is shown that the effect of a given amount of fuel
motion or rearrangement depends strongly on the degree of
sodium voiding present.
The nonlinearity is shown to arise
primarily from spectral effects.
In the present analysis,
feedbacks from the various effects listed in Table 8-1 are
treated as part of the integral term of Eq. (3-6) whenever
possible; they are not assumed to be linearly additive.
3.4
Effect of Severe Transients on the Neutron Energy Spectrum
The question of whether a significant shift in the neutron
energy spectrum is induced by a strong transient, independent
of the effect producing the transient, is now examined.
One
straightforward method of doing this is to include an appropriate "temporal absorption" cross section, W /Vi, in each
energy group of the usual multigroup formulation; where W is
the inverse reactor period and V. is the appropriate velocity
The "/v"
for each group (see Table 3-1).
terms can be intro-
duced into an otherwise critical system which, for a positive
(a , causes the code to calculate an "effective k" of less
than 1.0.
Alternately, the "M4/v" terms can be introduced to
compensate for some other perturbation, such as sodium voiding.
For a given degree of sodium voiding, if the proper W is chosen,
clearly the code will calculate an effective k of 1.0.
approaches were used in the present analysis.
Both
Values of
J
of from 1000 sec1 to 40,000 sec~
were employed and the cal-
culated energy spectrum examined.
The energy spectra genera-
ted were compared with those calculated for an unperturbed
critical system and with the spectrum which results from voiding the entire core of sodium.
The spectral shift associated
with sodium voiding produces relatively well known reactivity
effects and therefore should form a good basis for comparison.
48
The spectral shifts induced by severe transients were found
to be quite small for all values of W considered.
3-3 shows the results for 4a = 40,000 sec~
.
Figure
Curves 1 and 2
show the energy spectrum for an unperturbed system (k = 1.000)
and one with
Wa
=
40,000 sec 1
(k = 0.998) respectively.
Curve 3 shows the spectrum with sodium voided (k = 1.012)
and curve 4 shows the results for sodium voiding with the
simultaneous inclusion of an W' value of 40,000 sec~ ; which
reduced the net k value to 1.000.
Examination of Figure 3-3
shows that the shift introduced by an
3 of 40,000 sec~ 1 is
negligible compared to that induced by sodium voiding.
As
should be expected, values of W of less than 40,000 sec1
produced proportionately smaller spectral shifts.
It should be noted that an
) of
40,000
sec-
corres-
ponds to an asymptotic period of 25 microseconds and, for the
values of neutron generation time and delayed fraction given
in Table 1-1, this corresponds to a (point kinetics model)
total excess reactivity of .013 or about + $3.90. (in
fairly good agreement with that calculated by ANISN; i.e.:
(K-l) = .012 or about + $3.65)
Recall from Chapter II that
the maximum realistic excess reactivity is about $1.40.
Therefore an ta of 40,000 sec
1
corresponds to an appreciably
shorter asymptotic period than can be realistically achieved
in an LMFBR.
Thus, clearly, the transient induced spectral
shift can be considered negligible for purposes of accident
analysis in the present work.
-7
49
3.5
Summary
The application of the multigroup code ANISN to the
present work has been described.
The code was employed in
the present chapter to show the effects of core perturbations (which are exemplary of accident conditions) on the
neutron space and energy distributions.
A very minor influ-
ence on the spatial distribution was observed whereas the
spectral disturbance was found to be quite pronounced.
The minor influence of severe core perturbations on the
spatial flux shape was used to justify employing the point
kinetics equations for analyzing time dependent behavior.
An appropriate form of these equations for use in succeeding
chapters was presented.
Finally, it was shown that the spectral distortion or
shift produced by a severe transient is minor; and negligible
when compared to that produced by other effects of interest
in the present analysis.
50
Chapter IV
TRANSIENT HEAT TRANSFER
The present chapter considers the extent of fuel fragmentation subsequent to clad rupture or failure.
The mechanisms
and rate of energy exchange between the hot fuel fragments
and the relatively colder sodium are examined in detail.
4.1
Considerations of Phenomena in the Region of Clad
Failure "-R"
For rapid (CATEGORY I]
excursions, no clad melting is
expected to occur prior to the time of clad failure.
During
the time interval of interest for such an excursion, more
heat is added to the clad and coolant by fast neutron and
gamma energy deposition than by conduction.
Thus, the time
of clad failure is determined primarily by the rate of pressure buildup within the clad.
Furthermore, since very
little heat conduction takes place within the fuel during
the power transient, the fuel can be considered to be heated
adiabatically for purposes of analysis.
These simplifications
have been employed in earlier works (21) (43); their validity
is also substantiated by results obtained in the present
chapter.
Clad failure during a CATEGORYII excursion may occur by
longitudinal splitting or by general fragmentation of the
clad.
The degree of fuel and clad fragmentation and the
method of clad failure are important in the present work.
-I
51
The threshold of clad failure in a rapid transient corresponds
to the integrated energy addition at which the hottest fuel in
a given core location reaches a temperature high enough to
produce vapor pressures sufficient to overstress the clad.
At
the core hot spot this requires an adiabatic energy addition
of about 1080 joules/gram above the energy density of the fuel
at steady state full power.
As seen from Table 1-1, the maxi-
mum hot spot steady state temperature is taken to be 2900 0K at
the core center.
Equation of state relationships for oxide fuel materials
have been given by Meyer and Wolfe (33), Braess (9), and
others (44), for various ranges of temperature and pressure.
In the range of interest here, the pressure of the vapor in
equilibrium with liquid UO 2 can be described by the following
exponential equation; obtained by curve fitting the data given
by Braess et. al. from work done at Karlsruhe.
p = 8 x 107 exp
L
This gives:
6.7 x 10-
with
p in atmospheres
T in degrees Kelvin.
Valid from the melting point of U0 2 to 5400'K;
roughly accurate to about 6200 0KK.
(4-1)
52
Pressures predicted by this equation over the temperature
range of interest are tabulated below for convenient reference.
Table 4-1
VAPOR PRESSURES OF UO
2
T (K)
p (psia)
3070 (melting point)
0.45
4000
63
4500
485
4800
1130
5000
1760
5500
5850
6000
16,200
The internal pressure required for clad rupture is estimated to be 1200 psia, (21) (62) corresponding to a peak fuel
temperature of 485 0 0K.
This rupture pressure is based on the
clad thickness given in Table 1-1 and the mean clad temperature expected at time t2 of Figure 2-1.
As can be seen from
Table 4-1, however, the energy required to produce clad
failure is relatively insensitive to the estimated internal
pressure required for rupture.
For example, a 47% increase
in internal vapor pressure (to 1760 psia) requires only a 3%
increase in temperature (to 5000 0 K) and therefore only about
a 3% increase in energy density.
53
The energy density required to produce clad failure is
calculated in Appendix A, using 4850 0 K as estimated above
for the peak fuel temperature at the threshold of clad failure.
The value obtained is 1855 joules/grams, in excellent
agreement with an experimentally observed value of 1900
joules/gram (59).
Both of these results take 273 0K as the
"zero energy" reference temperature.
The total energy
density calculated corresponds to an energy addition of 1080
joules/gram to the fuel at the core center, as noted earlier.
As shown in Chapter III, the axial fuel temperature
distribution prior to clad rupture can be accurately described
by a cosine function of the form:
T
= T cos
.
(4-2)
In the absence of axial fuel motion, the fuel vapor pressure
within the clad can be obtained by employing Eqs. (4-1) and
(4-2).
As noted in Chapter II, axial fuel motion is expected
to be of little
influence prior to clad failure for a transient
of the severity considered here.
Thus, inserting Eq. (4-2) in
(4-1) gives the following approximate expression for the axial
fuel vapor pressure distribution prior to clad failure:
p
=
8
x lOexp
P(Z)
107
6.7
cis
x
1j.
VTZ
(4-3)
-
--
ft"A"W"A - -0 idm""W"
54
A key observation in the present work follows from this relationship.
If an excursion occurs which is severe enough to
produce clad rupture within the region
I z I ! 30 cm; a pres-
sure of 1200 psia and a centerline fuel temperature of 4850 0 K
at Iz ;M 30 cm is implied.
Equation (4-2) then predicts a
peak fuel temperature of 6250 0 K at the axial core center and
Eq. (4-3) predicts a potential peak pressure of approximately
26,000 psi.
Of course, clad failure will occur at a much
lower value and a pressure of 26,000 psia cannot actually be
attained.
The intent here, however, is to show the disruptive
potential for fuel near the center of the core when clad failure occurs over a substantial portion of the reactor.
Further
appreciation of this effect can be gained by examination of
Figure 5-3; showing a plot of UO2 vapor pressure versus temperature.
Note the steepness of the curve at temperatures above
50000K.
Early investigations suggest (20), and recent ex-
perimental results indicate more conclusively (32), that the
degree of fuel fragmentation or dispersal depends rather
strongly on the excess energy above that required for clad
failure.
These experimental results and the above calculation
indicate that a high degree of fuel dispersal is likely in the
central regions of an LMFBR core when a severe excursion produces clad failure over a fairly large region.
heretofore referred to as 6l,
This region,
may contain subregions varying
from those with split clad and lightly fragmented fuel to
55
those where fuel, clad, and sodium are intimately mixed in
a fine dispersion.
In the following section behavior of
the system subsequent to clad rupture will be seen to depend
very strongly on this degree of dispersion.
4.2
Energy Exchange in the Region -&
Analysis of the radial temperature profile in a fuel rod
during a severe transient indicates that the percentage of
fuel which is in the molten state at the threshold of clad
failure varies from about 50% for a transient which starts
with the core at full power to about 80% for one which starts
from low ( ~ 10%) power levels (36) (59).
Thus, the fuel
which disperses into adjacent sodium coolant will have a mean
temperature in excess of 3070 0 K at the instant of clad failure
for all excursions considered.
The temperature of the adja-
cent sodium will be about 800 0 K at this time.
It is shown in
Chapter V that the maximum energy addition to the sodium in
region'R during the time interval between an initial and
secondary
excursion is about 3600 joules per gram of sodium.
For this upper limit in energy addition an examination of the
properties of sodium in Appendix B of reference (35) shows
that the heat transfer problem in region It following fuel
fragmentation involves primarily transfer from solid fuel
fragments to saturated liquid or two-phase sodium.
A rough
sketch of the phase diagram for sodium (using the data of
reference (35)) is given below to clearly illustrate this
important point.
56
PHASE DIAGRAM FOR SODIUM
SKETCH
V S.
'PAT
N
3R
cpaT= 365ot.
-PArg 2
.Atoo
(K)
(pP
S
kI94
--
The energy required to heat the sodium to the saturated vapor
(or critical) state along any of the three paths shown exceeds
the 3600 j/gm expected to be available.
found to hold for all possible paths.
This condition was
Thus, sodium in region
-R will remain in the saturated liquid or two-phase state
until a secondary excursion adds additional energy to the
system.
Heat transfer calculations in the remainder of the
present chapter are predicated on this observation.
From consideration 2.2.b of Chapter II, the sodium adjacent to intact fuel pins in regions outside R
is not expected
57
to reach the boiling point within the several milliseconds
under consideration, thus liquid sodium will exist in all
regions of the core outside the vapor bubble generated by
rapid heat transfer in 16L .
shows that
.
Furthermore, work in Chapter V
will not expand by more than about 10 cm in
the axial direction before a secondary excursion occurs.
To obtain an estimate of the heat transfer in region
-
under these conditions, it is assumed that the fuel frag-
ments can be represented by spherical particles of some mean
diameter.
Additionally these particles are assumed to be at
a uniform temperature at the instant of dispersion, at which
time they are immersed in the surrounding sodium.
The heat
conduction equation for the fuel fragments with constant fuel
material conductivity is
qs(r,t) = pCp $
-
KV9
(4-4)
where
9 = T(0,t) T(r,t)
and
qs(r,t) = heat source within the fuel fragment.
For the time interval of interest, namely between an initial
and secondary excursion, the heat source (fission) is found
to be negligible compared to the heat conducted to the
surrounding sodium.
For example, the power level at prompt
58
criticality in Figure 2-1 is found to be about 1500 joules/
gram sec of fuel; using Eq. (3-4).
During the time interval
between transients, the power level is approximately at this
level, again as seen in Figure 2-1.
By contrast, in Chapter
V, it is shown that typically about 500 joules/gm fuel of
energy will be transferred from the fuel to the sodium in a
period of 5 milliseconds
or less.
The mean heat transfer
rate during this interval, then, exceeds 100,000 joules/gram
sec of fuel.
Thus, qs of Eq. (4-4) can be set equal to zero
with a negligible loss of accuracy in the present analysis.
With this approximation the solution to Eq. (4-4), obtained
by separation of variables, is:
o
9(r,t) =
sin7nr
Am
r
exp
K
2
pC
) n=1
_nt
,
(4-5)
p
where
nwF
0
R
= mean fuel particle
radius,
and
K, p, and Cp are average properties of the fuel.
For uniform fuel particle temperature as an initial condition, all harmonics are required for the complete solution.
If it is assumed for the moment that the boundary layer or
film heat transfer resistance at the oxide-sodium interface
is negligible, the higher harmonics are seen to die rapidly;
lw
59
giving rapid transfer of that quantity of energy associated
with the higher harmonics.
For aspherical fuel particle
initially at uniform temperature, this is about 69% of the
energy "available" to the surrounding sodium bath.
energy available if
theresidual
words,
In other
the temperature dis-
tribution assumes the fundamental mode profile with a central
temperature of T
of that available from the
is only 3/v
particle at uniform temperature T0 ; as can be verified from
Eq. (4-5).
The time behavior of the fundamental is given by
= G0 e
9
(4-6)
,
where
9(t) =T (r=Ot)
-
T(r=R)
'(4-7)
and
S
1
=
pC
K
R
P(-)
7r
2
2
(4-8)
*
For decay of the higher harmonics:
.
= -y-
; n = 1, 2, 3,
...
.
(4-9)
For the oxide fuel, the following average properties
from Table 4-2 are used to approximate behavior in the 2000K
to 3000 0K temperature range:
p = 10 gm/cm 3
K = 0.02 j/gm K sec
Cp =
0.4Zj,/gm K
60
Table 4-2
FUEL, SODIUM, AND CLAD HIGH TEMPERATURE PROPERTIES
A.
FUEL: (33)(34)
1) Melting point (Tmelt) at atm pressure
3070 0 K
2) Heat of fusion (Ahfusion)
278 j/gm
3) Heat of vaporization (Ahvap)
1850 j/gm
4) Vaporization temperature (Tvap) at atm
6200 0 K
pressure
5)
6)
"Mean" specific heat of solid
(cPs)
"Mean" specific heat of liquid (cP)
7) "Mean" specific heat of vapor,
by 13/2
(cv )
B.
given
0K
0.33 j/gm
0K
0.42 j/gm
0.20 j/gmOK
8) Density of solid (ps)
0. 10 j/cm3
9) Conductivity (k) (at 3000 0 K)
0.02 j/cm 0 K sec
SODIUM: (35)
(properties at 1300 0 K)
1) Heat of vaporization (h ap)
1600 BTU/LB
2) Density of liquid (PL)
43 lb/ft
3
( 0.7 gm/cm )
3) Density of vapor (pV)
0.05 lb/ft 3
4) Viscosity of liquid (1L)
0.34 lbm/fthr
5)
(4070 j/gm)*
)
Specific Heat of liquid (c
0.32 BTU/LBMOF
(1.47 j/gmOK)
6) Conductivity of liquid (KL)
26 BTU/hrftOF
7) Viscosity of vapor ( V)
0.04 BTU/hrft
8) Specific heat of vapor (c
9) Conductivity of vapor (KV)
)
0.30 BTU/LBMOF
(1.4 j/gm0 K)
0.04 BTU/hrft0 F
*Values are given in Metric and British units when both are required for a calculation given in the text.
61
For a mean dispersed fuel particle size of 500 microns
(about one-tenth the intact fuel pellet diameter), which represents fairly fine fuel dispersal,
and which may be a real-
istic particle size in the central regions of
:
t'i = 15 milliseconds
' 2 = 3.75 msec
r3 = 1.67 msec
...
etc.
For comparison, recall that the time interval between
power peaks is on the order of 10 msec (Section 2.3 of Chapter II).
For a mean particle size of 2000 microns (0.2 cm or
about one-third of the intact fuel rod diameter) representing
only minor fragmentation of the fuel pellets;
1
200 msec
Z2 2
50 msec
V3 2
22 msec
... etc.
Clearly, if the assumption of low film or boundary
layer resistance is reasonable, heat transfer of a large
fraction of the energy available to the surrounding sodium
will be extremely rapid; even for the case of relatively
minor fragmentation.
It is interesting to note that if R0
is chosen equal to the intact fuel rod radius, Eq. (4-8)
gives:
62
1 = 1800 milliseconds,
which is very nearly equal to the e-folding time for transient
heat transfer from the intact cylindrical fuel rods; as should
be expected.
This substantiates statement 2.2.b of Chapter
II (negligible heat transfer from intact fuel during the time
of interest) but of more importance shows the effect of higher
harmonics in producing an enormous increase in the initial
rate of heat transfer as a result of only minor fuel fragmentation.
The physical explanation of this "mathematical"
phenomena lies simply in the fact that any degree of fuel
fragmentation exposes extremely high temperature internal
surfaces to cold sodium.
The limitations imposed by boundary layer resistance
and film boiling are now considered.
Prior to Departure from
Nucleate Boiling (DNB), the customary Nusselt number (Nu
hD
-- ) is used to represent the boundary layer heat transfer
coefficient, whereas after DNB a film boiling heat transfer
coefficient (hfb) is employed.
It is not a foregone conclu-
sion that DNB and film boiling will occur around all fuel
fragments in the usual manner for a number of reasons:
(1)
Heat transfer in "R is an exceptionally rapid transient
process; the usual rules for estimating the start of film
boiling may not apply.
(2)
If fuel dispersal is fine,
there may not be "room" for film growth around each fuel
fragment to the film thicknesses usually associated with film
63
boiling until substantial voiding takes place.
It is shown
below, in fact, that about 10% of the sodium in region
1,
must be removed or "voided" before film boiling can predominate due to this film thickness limitation.
of highest heat transfer in 36
present in other regions of
(3)
The region
will determine the pressure
3-.
Thus,
fine dispersal and
high heat transfer rates in one relatively small region of
may lead to sodium vapor pressures sufficient to significantly delay DNB in other regions of 4V.
If q0
represents the energy loss per unit volume for
a given fuel particle; the initial
rate of heat transfer
associated with the fundamental temperature mode for a pure
conduction process is:
fI
where 9
= 3Kf
1K~) Go
(4-10)
,
is defined in Eq. (4-7).
The rate of energy trans-
fer through a boundary layer is given by:
q/A
With q'''
q'' = hBL
=
=
3 q''
R-
(4-11)
for a sphere and using the Nusselt number
0
for heat transfer prior to DNB, Eq. (4-11) can be written
q11' = 3/2KNANu()BL
.
(4-12)
0
For a spherical fuel particle immersed in a pool of
sodium, after all harmonics except the fundamental have
64
decayed, the heat transfer process is limited primarily by
the conductivity of the oxide; as is the case for intact
geometry in an LMFBR.
In this case Q
of Eq. (4-10)
approximates the total temperature drop from the fuel particle center to the surrounding sodium bath.
On the other
hand, at the instant of fuel dispersion, before the harmonics
decay, the entire temperature drop occurs across the boundary
layer, or:
9
=
0
(t
(t=0)
oo)
Thus, as a measure of the initial rate of heat transfer with
respect to the rate associated with the fundamental temperature distribution (or "final" rate for an infinite pool of
sodium) a ratio "m"
Eqs.
(4-10)
M =
may be defined as follows, making use of
and (4-12):
q0BL
Nu KNA
,,C
= - KF.E
q 0COND
(4-13)
FUEL
Note that the ratio m is independent of fuel particle size
when written in terms of Nu.
When m > >1 the process is not
boundary layer limited and the assumption of rapid transfer
of the energy associated with higher harmonics is correct.
Typically, for solid oxide fuel and liquid sodium at the
appropriate temperatures:
KFUEL
0.02 g/cm0 K sec
KNA
0 .50 j/cmoK sec.
65
From reference (41) p. 202, a lower limit on Nu for heat
transfer from solid spheres to a stagnant liquid is 2;
independent of the size of the spheres.
For the process under
discussion, one expects Nu to be greater than 2 until film
boiling starts.
Thus, at the instant of fuel dispersal, Eq.
(4-13) gives:
m
> 25.
Since m >>1, the values of thermal relaxation time, t
,
calculated above appear to provide a realistic picture of the
energy exchange rates in regions of R where DNB has not
occurred.
The start of film boiling requires that a vapor blanket
of sodium be formed around the fuel fragments and may, therefore, require that considerable sodium voiding take place
before film boiling is an important part of the heat transfer
process at the fuel-sodium interface for the majority of
region R
.
It appears reasonable to assume, however, that
the lowest rate of energy exchange between fuel and sodium
in _& , for a given fuel to sodium temperature difference,
will correspond to the occurrence of complete film boiling in
the usual sense.
The degree of voiding required to permit
vapor film growth to a "limiting" thickness, 5, is considered
first; then the limitations imposed on energy exchange rates
after complete film boiling occurs is investigated.
66
Rewriting Eq.
(4-13)
in a form more appropriate for
film boiling;
2KhfbD
FUEL
(4-14)
where,
hfb = film boiling heat transfer coefficient
D
= mean fuel particle diameter.
For a vapor film around each particle of thickness 5,
if 5 << D, the following relationship applies:
hfb
KNA VAPOR .
(4-15)
This simply expresses the film heat transfer coefficient in
terms of the conductivity of the sodium vapor in the film
and comes from the usual relations:
q/A =
K dT = K AT = hAT
Using Eq. (4-15) in (4-14), the relation
M = KNA VAPOR( )
F
KFUEL
is obtained.
(4-16)
Vapor conductivities for sodium are tabulated
extensively in reference (35).
For KNA VAPOR
=
1.0 x 10-
j/gm 0 K cm at a sodium temperature of 15000K (35), Eq. (4-16)
R
gives m = 0.05 -F .
Treating m > 1 as an indication of the
point at which the film temperature drop predominates and at
- -
I ! iiii ii!:
"' -
,
- I ..-
,--
W
- -
67
which the process is therefore limited by film boiling:
5
R
0.05
is required.
(4-17)
For the spherical fuel particles assumed, this
result requires that about 7% of the region 'R consist of
sodium vapor - or that 7% of the region be "voided" of sodium before the average film thickness around all fuel particles is
such that the film boiling process is clearly limiting.
Note
that this vapor fraction is independent of the assumed size of
the spherical fuel fragments.
For very small fuel particles,
the film thickness, however, becomes microscopic.
For 200
micron fuel particles, for example, the film thickness is only
5 microns.
It may be erroneous to equate conductivity of a
sodium vapor film of this thickness with that of a macroscopic
film.
One might expect, however, that molecular exchange
across a microscopic film would increase the energy exchange
rate as compared with a thick film.
If this is the case,
Eq. (4-17) predicts a conservative result for an accident
which results in fine fuel dispersion in the sense that a
larger degree of sodium voiding will take place before film
boiling can predominate.
In any case, it i's difficult to
conceive of a means by which a microscopic vapor film can
result in a lower heat transfer rate than would otherwise be
expected.
In order to examine the limitations imposed on heat
transfer rates by complete film boiling, two cases are considered.
68
First, natural convection of sodium around fixed fuel spheres,
as represented by Eq. (5.64) of reference (42), is treated.
Second, a reasonable sodium velocity across an equivalent
cylinder of fuel material, as described by Eq. (9.30) of
reference (41), is assumed.
The relative velocity between
the sodium and fuel particles is arbitrarily taken to be the
same as that corresponding to normal core sodium flow (lOn/sec).
This should give a reasonable estimate of hfb and thus of the
heat transfer rate for a sodium/fuel mixture undergoing film
boiling.
In both cases, the properties of sodium are taken
from reference (35).
For both correlations, it is found that
m, as defined by Eq. (4-14), is roughly proportional to particle size and that:
m>l
for
D > 250 microns.
The detailed calculations are given in Appendix B.
These
two latter approaches give a crude approximation for hfb at
best and represent only an attempt to obtain a reasonable
estimate of the heat transfer coefficient for a process for
which a suitable correlation is not available (60).
The
results indicate that the energy exchange rate in 1
remains
much higher than the rate outside Rk
even after each fuel
fragment is completely blanketed by sodium vapor.
69
4.3
Summary
The strong dependence of fuel vapor pressure on position
within the core during a CATEGORY 2 excursion has been shown.
The resulting possibility of fine fuel fragmentation in central regions of the core, supported by some experimental evidence, has been indicated.
Extremely rapid heat transfer from fragmented or dispersed fuel to the surrounding sodium has been demonstrated
for regions where DNB has not occurred.
This energy transfer
rate has been shown to depend strongly on fuel fragment size,
12
namely a (R-) dependence.
Furthermore, it was shown that a significant degree of
sodium voiding must take place in the region of clad failure
before a vapor film of the usual thickness around the fuel
fragments can be produced.
Even so, after complete film
boiling in the usual sense occurs, the heat transfer rate
between fuel fragments and sodium was seen to remain high in
region VC as compared to the case for intact LMFBR geometry.
The energy transfer characteristics presented in this
chapter are employed in Chapters V and VI to predict the
effect of subsequent events; particularly for estimating rates
of sodium voiding induced by rapid additions of energy to the
sodium in region
-
.
__ I _WAWAM__
=_
--
I ,-
70
Chapter V
REACTIVITY ADDITIONS FROM SODIUM VOIDING
It
is the purpose of this chapter to consider the magni-
tude of possible sodium void reactivity insertion rates during
a specific accident sequence.
An excursion leading to clad
rupture over a significant portion of the core is assumed to
have occurred and to have been turned down by Doppler feedback
(CATEGORY II excursion) as shown in Figures 2-1 and 2-3.
If
the excursion is initiated by some effect other than sodium
voiding, the fuel in the region of failure mixes with liquid
sodium and the heat transfer behavior considered in the previous chapter applies.
If the accident is initiated by
sodium voiding, fuel dispersal into a previously voided
region may occur.
The former condition is of primary inter-
est in the present chapter; the latter is treated briefly,
however.
The description of energy exchange between hot fuel and
relatively colder sodium developed in Chapter IV is used in
conjunction with the hydrodynamic equations to calculate the
rate of sodium vapor generation (or voiding) and, ultimately,
the reactivity addition rates which result.
5.1
Hydrodynamics of Sodium Voiding
The core described in Table 1-1 is assumed to have a
large volume of sodium at the core exit, with a cover gas at
a pressure of about 2 atmospheres, as depitced in Figure 5-1.
This arrangement is typical of currently proposed LMFBR
71
FICuRE. 5'-l
Sc HE\ATiC OF CooE
r/t oti>Nr
Coyca
&AS
-09 Sobirt
d
lf;ktAr cA~ iin
~~'
F2La SoAFACL
-4o.
OUT Lcf
NA
NA
caitis
/7
L~t~P~ESS"'t4
UPW A&AFLoW
p
. .
CORE
_
L
T
DOWNWA RD
plow
~RG.P ZE
SEWetATr\V9
C ANNEL FOk
F.
SIN&,LF,
$obiUt FI.oW,
LIQUID
TH- ICiANNEL! is THe pLow
AREA ASSOCIA-rED Wi-R
CotrE
So~,,ur4
V=FEL 'Roo.
EXPNOIPtMG
AND vAPoa
L
.
5orN\
0U.b
I
-F
72
designs (2A)(4A).
The core model employed in considering the
hydrodynamic effects associated with sodium voiding is shown
schematically in Figure 5-1.
in Table 1-1.
Appropriate dimensions are given
In the present analysis, the following assump-
tions or simplifications are made.
1)
Only upward flow of sodium is considered.
Very
little radial motion of sodium is expected compared to axial motion because of the close packed
nature of LMFBR fuel rods and the employment of
fuel rod wire wrapping and subassembly boxes in
current designs.
As shown in Figure 5-1, however,
downward flow paths clearly exist.
Further
comment on this aspect is made in the next paragraph.
2)
A constant pressure is assumed to exist at the
core exit.
This pressure consists of the cover
gas pressure and the static liquid head above
the core.
The justification for this assumption
rests on the following observations.
a)
The volume of the cover gas region is much
greater than the size of the vapor bubbles
considered; thus only slight compression of
the cover gas is required to accommodate a
large sodium void.
b)
The velocity of sound in liquid sodium is
high, typically about 7000 ft/sec
(35),
so
73
that pressure waves generated at the core
exit can affect (compress) the cover gas in
time intervals on the order of 1 msec.
c)
The volume of sodium above the core is typically
large compared to the size of the vapor bubble generated and the compressibility of liquid sodium at core exit temperatures is high.
Thus, from the velocity of sound cited in (b)
above,
liquid sodium compression tends to
relieve the pressure at the core exit in time
intervals less than that required for pressure
pulses to reach the cover gas region.
Calculations in Appendix B,
through (c),
employing observations
(a)
further indicate that this assumption
is reasonable.
Downward flow is not included in the hydrodynamic analysis primarily because of the complexities
duced.
intro-
The smaller volume of sodium at the core
entrance and the more restricted access to the cover
gas region can lead to high compressive pressures
in the core inlet region (see Figure 5-1).
The in-
clusion of check valves at the core inlet or main
coolant pump outlets in various designs further
complicate the matter.
For example, the question
-~now
74
of the position of the valves at the start of the
accident as well as their activation time arises.
As seen in Figure 5-1, however, sodium forced from
the bottom of the core by an expanding void can
pass upward through the outer regions of the core
and blanket.
Thus, while the neglect of downward
flow, in conjunction with the present assumption
of a constant pressure at the top of the core,
greatly simplifies the analysis; an underestimate
in the severity of the sodium voiding effect calculated is expected.
The principal conclusion of the
present analysis will not be affected, however.
3)
"Slug" or column flow in subcooled regions outside
the vapor bubble generated by rapid heat transfer
in "R is assumed.
This implies that subcooled liq-
uid sodium moves axially as a uniform slug of liqquid without entrained vapor bubbles.
This assump-
tion is in agreement with that of other authors
(9)(12)(39) and some experimental work (40).
4)
Choked or sonic flow in the core is not considered.
The maximum flow velocity attained in the present
analysis is 103 m/sec.
Since sonic velocity in
liquid sodium is about 2100 m/sec at the temperatures of interest, this assumption is clearly
valid in the liquid regions.
In the two phase
- 09 000111=1 - I t
75
region the matter is not so clear, however, as
insufficient information is available to predict
the velocity at which choking will occur for the
high pressures and low qualities of interest
here (61).
A development of the hydrodynamic equations employed is
given in Appendix B.
An outline of the development is given
here for continuity and to show salient features of the
analysis.
The core friction drop is included by employing the usual
relationships
4A
APfriction
~
e
p
V2
c
(5-1)
and
f = .o46(p-L)
PDe
V
,
(5-2)
where
I = core half height
De = equivalent diameter for the flow area associated
with one fuel rod.
A representative core pressure drop at normal full flow is
taken to be
80
psi.
This value was determined,as explained
in Appendix B, by averaging the pressure drops given by
General Electric and Atomics International for proposed 1000
MWe LMFBR's; given in their respective follow-on-studies,
76
references (2A) and (5A).
Then, since the properties of
sodium in the subcooled region outside the vapor bubble (see
Figure 5-1) change only slightly, the friction pressure drop
can be written, employing Eqs. (5-1) and (5-2), as:
1.8
Apf = 'V'( - Z)(-)
,
(5-3)
where Ap0 = reference half-core pressure drop at full
flow; namely 40 psi
VO = reference velocity for full sodium flow, taken
as 10 m/sec (2A)(5A).
In the present analysis, for the range of z(t) (see Figure
5-1) considered, the fraction ( 'Q-Iz(t))
varies only from
0.57 to .71, and an average value of 0.64 was therefore
employed; that is:
(-
Z) = 0.64 in Eq. (5-3).
The influ-
ence of this approximation is seen to be negligible when the
V 1.8
factor (V-)
is observed to range from 1.0 to 51.5 (see
Table 5-1).
This approach properly includes end or exit effects
since Ap 0 is the total (half) core pressure drop and since
the functional dependence of the exit pressure drop is approximately the same as employed in Eq.
V 1.).
Thus,
(5-3)
(41)
(Apexit~
V2 vice
frictional effects are incorporated in the analy-
sis by a method which is simple but which is in keeping with
the accuracy of the theory on which the parent relations,
Eqs.
(5-1) and (5-2),
are based.
77
Inertial forces are included through the usual conservation of momentum relationship:
2
Apinertial =
p(X-
Combining Eqs. (5-3)
z(t))
dt
and (5-4)
(5-4)
4
and considering the pressures
depicted in Figure 5-1,
2
1.8
p(t) = p(j - z(t)d)z + Apo(.64)(l- Xdz 1
(5-5)
+ p
-
dt
If Q(t) is the energy added to the sodium in
at any
time t,
Q(t)
=
c (Tf
-To) + (h (t)-hf) ,
(5-6)
where
T
= saturation temperature at time t,
h(t)
= enthalpy of two phase mixture at time t
Te
= initial
h
= enthalpy of the saturated liquid at Tf(t)
cP
p
sodium temperature
= mean specific heat of sodium
It was shown in Chapter IV that sodium is expected to remain
in the two phase state, thus the applicability of Eq. (5-6)
is assured.
The Clapeyron equation relating changes in
enthalpy and specific volume for a reversible process in a
two phase mixture is (38)
78
Ah = T
AV
(5-7)
p
dT
The saturation pressure as a function of temperature for
sodium is given by (35)(37)
p = Be-A/T
(5-8)
,
where
B = .332 nts/m
A = 11,950
T in 'K.
A key to the present analysis is the employment of this
empirical relation, Eq. (5-8), in the Clapeyron equation to
eliminate the enthalpy term of Eq. (5-6).
The appropriate
manipulations give the following expression for
Q(t)
=
(T
-TO)
+ ET A
A
)
Q(t):
vo(
(5-9)
Note that
z
is a measure of the extent of the axial void.
zo
Now, using Eq. (5-8),
Bexp(TA )
(t)
Eq. (5-5)
2
--p( t -z )d2
dt
can be written:
1.8
+ Ap (.
))
1.8
+ p(
.
0
(5-10)
These latter two equations give Q(t) and f(z,T) and z = f(T)
respectively.
Thus, it is seen that if Q(t) is specified,
the rate of void growth as a function of time can be determined.
79
The method employed to obtain a solution is described in
Appendix B.
The specification of Q(t) is the subject of the
next section of the present chapter.
Although the hydrodynamic analysis developed in this section is by no means exact, the inaccuracies generated are
small compared to the uncertainties in Q(t) discussed in
Chapter IV.
The importance of Q(t) in the analysis is further
indicated by the results of the next section.
5.2
Estimates of Energy Addition Rates to Sodium
The following average properties of fuel material and
sodium are used in the present analysis (35).
0
1.4 joules/gram K
c
Na
Liquid
=
c
Fuel
= .4 j/gm K
= .8 gm/cm3
PNa
Liquid
PFuel
3
= 10 gm/cm
The heat of fusion for the fuel is assigned an equivalent
temperature change given by:
AT
eq ==
fusion
-.
cp
-
278
4j/no gK=
7000K.
(5-11)
Using this equivalent temperature and the initial fuel temperature distribution for full power operation (see Table 1-1),
61
the
80
mean equivalent fuel temperature at IzI
20 cm is 44000 K
(Tactual = 3700 0 K) when the peak temperature is sufficient to
produce clad failure within the region Jzj 4 20 cm.
temperature in the central regions of R
,
The mean
then, will be
somewhat higher and an underestimate in the energy added to
the sodium results if the mean equivalent temperature throughout the regionR is taken to be 44000K at the instant of clad
rupture.
Furthermore, if the initial excursion starts from a
power level below full power and clad rupture over a region
1R similar in size occurs, the mean temperature of the fuel
in 'A,will be much higher.
This results from the lower
initial fuel rod peak to mean (radial) temperature ratio at
lower power levels.
With a mean equivalent fuel temperature of 4400 0 K and
an initial sodium temperature of 800 0 K, as given in Table 1-1,
the energy transferred to sodium for three assumed cases,
a, b and c, is calculated in Appendix B.
For case (a) it is
assumed that all energy associated with the higher harmonics
of the fuel temperature distribution is transferred in a few
milliseconds.
As shown in Chapter IV, fuel fragmentation in-
to particles of a mean diameter of 500 microns (about onetenth of the intact fuel pellet diameter) gave an e-folding
time for the first higher harmonic of 3.75 milliseconds and
of 1.67 msec or less for the next and higher harmonics.
These times are relatively short compared to the time required
to produce significant sodium voiding.
The e-folding time of
81
the fundamental mode, on the other hand, was 15 msec; a
period somewhat longer than the time required for significant
sodium voiding under certain circumstances; as will be seen.
Thus for mean particle sizes of 500 microns or less,
it
is
reasonable to assume transfer of the energy associated with
the higher harmonics in a few milliseconds.
The temperature
distribution effects associated with case (a) are depicted
in Figure 5-1.
The energy transferred by collapse of the higher harmonics is used as the value of Q(t) in case (a).
As stated
in Chapter IV, this is 69.6% of the total energy available from
the fuel if the mixture is allowed to come to equilibrium at
constant volume.
For cases (b) and (c),
respectively, values
of Q(t) of 50% and 30% of the total energy available are
employed to account for inaccuracies in the analysis and to
show the trend in the behavior of the system for rapid transfer of smaller fractions of the available energy.
5.3
Reactivity Addition Rates from Sodium Voiding
Using the values of Q(t) cited above, Eqs. (5-9) and
(5-10) have been solved (see Appendix B) to obtain the rate
of axial void growth as a function of time.
given in Table 5-1.
The results are
The initial axial extent of region 1. was
assumed to be the region
i
4 20 cm but the resulting void
growth rates were found to be relatively independent of the
axial extent of &Rwithinthe Izi
= 10 cm to jz[= 30 cm range.
82
f% C uR~
SC146-MA-ric
ov:
TECvPAavOzE
A VuEi. PPA&r4r
FO
IN sooluwi
FOCL
PRA&FAENr
103 CA I C Ro t4lb
I
I
oAk~4(kk#L UtWOPNM
FXAGMNT
ryUeAN
Orem?
£N-k&Xm A ssaUATED
rrSii4 Akt4~c HAP-Rmics
INFOEL 5:AAf1EM1r
AT FULEL
MET4 IUCL-rerrit. ApiTi IA-Y
I
T
itMCum
AFGR~ Dr'cA'
OF HPAQMfONtet
NtECAN
I
I
I
I
9
09iGI4NAL
soblurf\ TEMVP.
83
The reactivity addition rates for the given rates of axial
void growth were calculated for regions 'I which extend over
25% (note (2) of Table 5-1) and 50% (note (3) of Table 5-1)
of the radial area of the core.
The reactivity addition rate
and net reactivity calculations were based on the results
shown in Figure 6-2; the curve showing sodium voiding reactivity calculated by ANISN was employed.
The reactivity
addition rates obtained were found to be relatively independent of the axial extent of 6R in the Izi
cm range.
Beyond values of \ zi
= 10 to
izI
= 20
= 20 cm for the initial size
of '11,, the rates shown in Table 5-1 drop off sharply (see
Figures 6-2 or 6-5).
Observe that the reactivity addition rates estimated
range from 62 $/see to 642 $/sec.
For the case of transient
heat transfer from intact fuel geometry, maximum sodium voiding reactivity rates of from 20 $/sec to 65 $/sec have been
predicted (12)(23)(24)(25).
In view of the heat transfer
rates estimated and the comparisons with the case of intact
geometry given in Chapter IV, the rates shown in Table 5-1 do
not seem too surprising.
Note further that for the larger t 3 (time to achieve
void growth rate tabulated) for each case in Table 5-1, the
friction pressure drop has become a substantial fraction of
the peak pressure generated in -R .
Since the friction
pressure drop is proportional to V1.8 and the reactivity
Table 5-1
REACTIVITY ADDITION RATES FROM SODIUM VOIDING
- Case -
Peak Pressure
C
b
1520ps
a
4850 Ps i
360
s
1 msec
2 msec
2 msec
3 msec
4 msec
7 msec
dz
a at t3
51.5 m/sec
103 m/sec
33 m/sec
49 m/sec
15.2 m/sec
26.5 m/sec
Az at t3
2.5 cm
10 cm
3.3 cm
7.5 cm
3.05 cm
9.3
380 psi
8oo psi
170 psi
355 psi
43 psi
115 psi
219 $/see
418 $/sec
136 $/see
197 $/see 62 $/sec
108 $/sec
328 $/sec
642 $/sec
212 $/sec
309 %/sec 96 $/sec
168 $/sec
+ 110'
+ 430'
+ 140'
+ 170'
+ 68U
t3 (1)
Apfriction at t3
dk (2)
dk (3)
dt
(2)
5k t
(3)
5 knet
+ 131'
I+
210'
cm
+ 4o
+ 649(
NOTES:
1) Time at which void growth rate and reactivity rate shown are reached.
Figure 2-1.
2)
6k
3) R
extends over 25% of radial core area
extends over 50% of radial core area
See also
85
addition rate is roughly proportional to V; the friction
effect quickly becomes important for time intervals larger
than those tabulated.
Furthermore,
as the bubble grows
beyond the 7.5 cm to 10 cm Az value corresponding to the
larger time intervals shown, differential reactivity of sodium
voiding rapidly diminishes (see Figure 6-5).
Thus, the larger
reactivity rates shown for each case are approximately the
maximum values expected within the limits imposed by the
assumptions made.
An examination of the influence of the
more important assumptions follows.
For all cases, the size of -A,grows axially from an
initial dimension of 40 cm to between 42.5 and 50 cm; as seen
in Table 5-1.
Thus, sodium vapor fills between about 6% and
20% of the region -R over the time span considered.
From the
calculations of Chapter IV, then, the lower values of the
reactivity addition rates estimated can be attained before
the film thicknesses associated with complete film boiling
(and therefore slower energy exchange) can be present.
Perhaps of more impact is the observation that for pressures
in region R
below the critical pressure of sodium (
5980
psi), the time required to accelerate sodium outward sufficiently to permit a vapor fraction of about 10% (by volume) is
of the same order as the time required for transfer of a
large fraction of the energy available if the mean dispersed
fuel particle size is 500 microns or less.
Thus, the sodium
is confined by inertial and frictional effects long enough
for extensive energy exchange between fuel and sodium.
86
As stated, only upward axial bubble growth is allowed in
the calculations.
As suggested in Figure 5-1, downward
motion might be significant.
low pressure (-
(If the assumption of a constant
0) at the bottom of the core is employed in
the present analysis; the rates of void growth and reactivity
addition shown in Table 5-1 almost double in some cases an
increase by over 150% in all cases.)
In view of the small
extent of axial bubble growth considered in Table 5-1 (Azmax
=
10 cm), calculations in Appendix B indicate the assumption of
a constant low pressure at the top of the core is realistic.
Finally, for primary excursions which start from a low
power level and result in clad failure over a region of the
size considered here, the mean fuel temperature in the region
of clad failure will be appreciably higher, as will be seen in
the next section.
This again will result in higher sodium
voiding rates than estimated for the present case with full
power as the power level before the initial excursion.
Thus, the net effect of these latter three assumptions
is likely to result in an underestimate of the sodium voiding
reactivity addition rates which can follow CATEGORY II excursions.
5.4
Accidents Initiated by Sodium Voiding
If the initial excursion is initiated by sodium voiding,
it is possible that the region of clad rupture will be largely
enveloped by sodium vapor.
For the particular LMFBR of
87
interest, the specific volume available to the fuel in a
region from which sodium is removed is 65.5 cm 3/gm mole (see
Table 1-1).
Vr
This corresponds to a reduced volume of
=
c
655 = 0.73,
(5-12)
where
V
=
critical volume for UO 2 = 90 cm 3/gm mole (44).
Menzies tabulates the vapor pressure of UO 2 as a function of
temperature for reduced volumes between 0.4 and 1.0 (44).
Applicable data from this work is plotted in Figure 5-3, along
with Eq. (4-1); the latter representing fuel vapor pressure in
the intact clad (Vr :=
.33).
The data given by Menzies for Vr
in the range 0.7 to 1.0 essentially coincide for UO 2 temperatures below 7000 0K.
high power level (
For initial excursions starting from a
--
100%), as assumed in the preceeding sec-
tions of the present chapter, the mean fuel temperature in
region R
(Izt
1E 20 cm) is estimated to be 4000 0 K.
As seen
from Figure 5-3, this results in a fuel vapor pressure of
about 100 psi; a value which is insufficient to substantially
affect the outcome of the excursion.
If, however, the tran-
sient starts with the plant operating at a low power level
(-~l0%), the mean fuel temperature in the region R will be
about 5200 0 K.
The difference, of course, results from the
lower peak to average radial temperatures in a fuel rod at
low power.
For this case, examination of Figure 5-3 shows
88
Fr&oke
S'-3
FOtR UOL
A PPAD. C i-AD
FAtLU(F. TE-MP.
V APOp gR65 !SJR-E
F (zo\ DATip S
PLOT OF
-Q(i4- 1)
ctiove. 3
cu IM e z
112 'PAGE1SSUE P~ioR
TrO e-Ab FAftuRE
ANr
SO
AFr'ER
CLIb
(EALLLI(i
-
IV CoRE REMn96$
10,000
FILLEO
LiauiL
wt-TH
SobiUM)
I
I
/
/
5.000
/;0
MEN
'y uo.'' TO
. zd
' E.sV ACoMac
FOR
1. o
O- 'T Y- '. Y<oETPLO-OR, -T~<?Doao
1000
5
5'If0i VI
1- (*K) -+
1.0
89
that the fuel pressure will be about 1800 psi after clad
failure in region -
.
As seen from Table 5-1, this peak
pressure is comparable to that in case (b) for energy
exchange with sodium.
According to data given by Menzies
(44), only a slight pressure drop will occur as the fuel
expands to fill a volume corresponding to Vr = 1.0; or expansion by a factor of about 1.4.
Thus for excursions which
start from relatively low power levels, the problem examined
in the present chapter is not eliminated even if sodium
voiding is the cause of the initial accident.
Furthermore,
for excursions which start from low power levels or which
result in initial clad rupture over a region larger than
Iz
' 20 cm, the fuel vapor pressure itself can play an impor-
tant role in initiating or increasing the rate of sodium
voiding whether sodium is present in the region of clad failure or not.
Note, however, that fuel vapor pressures were
not considered in arriving at the reactivity addition rates
cited in Table 5-1.
5.5
Summary
To preface the following discussion it is reiterated
that the rate of reactivity addition at approximately the
time of prompt criticality is a major, if not the primary
factor, in determining the consequences of a severe excursion;
as discussed in Chapter II.
Recall also that a reactivity
90
insertion rate of 66 $/sec is taken as a basis of comparison
in the present analysis.
An accident which results in clad rupture over a substantial portion of the core, namely a CATEGORY II excursion,
is considered credible since such an accident can result from
initial reactivity rates below the 66 $/sec basis; as shown
in Chapter II.
Given that such an accident can occur, the
assumptions leading to the reactivity addition rates shown
in Table 5-1 are realistic with one important qualification:
the rate of heat transfer to sodium, which depends on the
degree of fuel dispersal, is an area of great uncertainty.
Insufficient information is available at present to properly
estimate this effect.
Fuel vapor pressure effects must be taken into account
for a complete analysis of CATEGORY II excursions; particularly if the excursion is initiated by sodium voiding itself.
In the present chapter the reactivity effects of fuel motion
were not considered.
As will be seen in Chapter VI, however,
sodium voiding is the predominant effect until fuel motion
becomes quite substantial.
The small amount of outward motion
in a close packed LMFBR lattice may be of little influence
until pressures sufficient for destruction of the lattice
(outside -R ) are generated and, in fact, a mechanism which
produces inward fuel motion (into 6R. ),
discussed in Chapter
VI, may override any outward fuel motion occurring early in
a CATEGORY II excursion.
L
The principal conclusion which must be drawn in the
present chapter, then, is the following:
If a large portion
of the fuel in region _R fragments to particle sizes of mean
diameter equal to one-tenth the intact pellet diameter or
less, the reactivity addition rates resulting from the subsequent sodium voiding can exceed the 66 $/sec basis by as
much as an order of magnitude.
Some additional appreciation for the estimates obtained
in this chapter can be gained by noting that the energy
transferred to the sodium in cases a, b, and c of Table 5-1
is sufficient to produce sodium superheats of 1700 0 K, 1150 0K,
and 610 0 K respectively.
These values are based on a sodium
boiling point temperature of 1000 0 K, typical of normal
operating plant pressures.
92
Chapter VI
REACTIVITY EFFECTS RESULTING FROM CORE REARRANGEMENTS
The reactivity effects resulting from core rearrangements
which can result from a severe accident are explored in the
present chapter.
Reactivity insertion rates which can arise
during such rearrangements are estimated.
A model is developed which successfully predicts the
reactivity effects introduced by sodium voiding and fuel
motion.
The basis of the model is perturbation theory.
In
Chapter II it was shown that the spatial neutron flux shape
remains very close to the unperturbed shape even for quite
large perturbations.
Thus, it is assumed that a function
describing the unperturbed flux shape can be used to account
for spatial effects.
Use is made of the spectral characteri-
zation work done by Shaeffer and Driscoll (45) to include the
all important energy or spectral effects for a fast reactor.
It is shown that removal of a quantity of core fuel
material from any given location produces a stronger negative
reactivity than the positive reactivity which would result
from inserting the same quantity of material at the given
position.
The influence of this effect in ameliorating the
hazard of possible fuel rearrangements during an accident is
demonstrated.
6.1
Effects Leading to Fuel Motion
The rupture of cladding over some region of the core
has been discussed in some detail in Chapters II and IV.
93
Immediately following time "t2 " of Figure 2-1, the fuel in
regions outside $,
is constrained by intact clad except for
the "open end" inside -R
2-3.
,
as shown schematically in Figure
Movement in the axial direction away from blis preven-
ted by colder intact fuel and blanket material.
As noted in
Chapter IV, 50 to 80% of the fuel at the edge of region 'Q
is in the molten state (36).
Since the fuel vapor pressure
was sufficient to rupture the clad in 1R1,
the vapor pressure
within the clad will be considerable some several inches outside P% (in a non-vented design, the fission gas pressure
could also be considerable).
Thus, the potential exists for
axial fuel movement or "injection" into bl, from regions adjacent to
R .
That such axial motion is plausible is indica-
ted by recent experimental evidence (36)(46)
and by calcula-
tions in Section 6 of the present chapter.
A second method by which fuel material can be moved inward or into potentially higher worth regions of the core is
now outlined.
In Chapter II it was shown that appreciable
cooling of the fuel material in region IR could take place
during the several milliseconds following an initial transient.
Thus the mean fuel temperature in -R could be reduced
below that in adjacent regions by several hundred degrees.
Furthermore, fuel fragmentation reduces the ratio of peak
fuel temperature to mean fuel temperature at any given location.
From the developments in Chapters IV and V, it is
-
m
94
clear that the peak temperature in regions adjacent to
could be over 500 K higher than the peak temperature in
by the time a secondary prompt-critical excursion is initiated.
Examination of Figure 5-3 shows that as temperatures
in regions adjacent toA reach about 6500 0 K during a secondary transient, 500 0 K of subcooling in R
leads to a pressure
differential in excess of 5000 psi tending to induce inward
fuel motion.
Only one of the two phenomena described above is likely
to have a significant effect on the overall excursion.
If
substantial cooling of the fuel in-R occurs, high sodium
vapor pressures are generated in 'R
,
preventing the fuel
injection mechanism induced by clad constraint.
This is
shown conclusively in Section 6.6 of this chapter.
If this
fuel cooling does not occur; then the conditions for an inward pressure gradient during a secondary excursion are not
established and the second mechanism of general inward fuel
motion cannot be appreciable.
The second fuel motion phenomenon can take place subsequent to significant sodium voiding, however.
As seen in
Chapter V, the sodium vapor pressure is not expected to reach
the critical pressure (5980 psi) during the time interval of
interest whereas, from Figure 5-3, the fuel vapor pressure
generated during a secondary excursion can quickly exceed
the sodium critical point pressure (and therefore the maximum
95
pressure expected in region _R during the early phase of a
secondary excursion).
Thus, the initiation of a secondary
excursion by sodium voiding can, in affect, lead to further
reactivity additions by this fuel movement mechanism.
Although these two phenomena leading to fuel movement
into higher worth regions of the core may be of short duration and may result in fuel movement of only a few centimeters, the important question is whether such rearrangements
can add on the order of 50
in reactivity at a rapid rate.
The remainder of the chapter is devoted to this question.
6.2
Reactivity Model for Fast Reactor Core Perturbations
Figures 3-1 and 3-2 show that the spatial shape of the
neutron flux remains very close to the fundamental mode
shape over most of the core, even for quite large perturbations.
Conversely, Figure 3-3 shows that the shape of the
energy spectrum shifts appreciably for comparable perturbations.
While the former result suggests that spatial effects
can be accounted for by the usual one-group perturbation
theory formulism, the latter result indicates that the one
or two group approach will be entirely inadequate for assessing such reactivity effects in fast reactors.
Subsequent
developments in the present chapter show that this indeed is
the case.
In recent work at MIT (45) directed toward the development of simple LMFBR core calculational methods, Shaeffer
96
and Driscoll have developed an accurate spectral characterization technique.
The spectral characterization parameters
employed in this technique have been found particularly useful in calculating the influence of spectral shifts on reactivity.
A brief description of this work is given to set
the stage for the present analysis.
In reference (45) a one group method for calculation of neutron balances in the core of fast breeder
reactors is developed and evaluated.
The key feature
of the method is the definition of two spectrum characterization parameters,
S =
(6-1)
f +
tr
and
R
1S
1tr
r
(6-2)
where
7r
=
removal cross section (45).
The former index enables correlation of all required
microscopic cross sections except those for threshold
fission in the form
0i
k 0k
riSk(63
97
where
i = element in question
k = type of cross section
(r- = reference cross section for element i and
cross section type k
=
correlation parameters for element i and
cross section type k.
For the threshold fission elements, the fission cross
section is correlated in the form:
I
(6-4)
R f
A rapidly converging iterative procedure is presented
(45) through which S and R can be determined for any
practical core composition.
Microscopic cross section data has been correlated
employing Eqs. (6-3) and (6-4) for some 43 materials as
of this writing; using the 26 group Russian ABBN multigroup set (58).
(The same set employed with the ANISN
multigroup computer code in much of the present work.)
The one group model developed has been tested for some
45 different fast reactor compositions by comparing
the results of one group calculations to 26-group
fundamental mode calculations.
The results have been
found to agree with an average error of 1.77% in
98
material buckling, 0.218% in enrichment, + 0.588% in
infinite multiplication factor, + 0.69% in reactivity,
+ 2.19% in core conversion ratio, and + 2.17% in the
ratio of fertile to fissile fission.
The present reactivity model is based on the selfadjoint perturbation theory formulation.
Changes in macro-
scopic cross sections are defined as follows:
5,= gAN+
7AS
(6-5)
,
where
N = number density
S = spectral characterization parameter from reference
(45).
The first term of Eq. (6-5) is simply the change in cross
section resulting from a material concentration change, as
usually employed in perturbation theory.
The second term
purports to account for the change in cross sections arising
from a shift in the energy spectrum.
From reference (45),
all cross sections except the threshold fission cross sections
can be written directly in terms of "S" as given by Eq. (6-3).
Thus, the "spectral shift" term of Eq. (6-5) can be evaluated.
For non-threshold cross sections, one obtains:
5Zspectral
-
I
=
3
0
Sg)A
= gZ( -).
(6-6)
99
For the threshold cross sections, the following relation is
used:
=
( R )AS.
(6-7)
spectral
Employing Eqs.
(6-2)
and (6-4) in this relationship and
writing the result for U238, the only threshold fission element of interest here,
gives:
28
6(7,
+
28 V728
Cgf vf)gr
)spectral
+
I-
A
-
(6-8)
where
1 1
=
g
gtr =z
+ 2 2
+gZtrtr
tr tr
(6-9)
tr
The change in diffusion length arising from a spectral shift
is evaluated in a similar manner:
D =
1
1
1
3,tr
S
3(Z
(6-10)
2
S
+ z
tr
+
...
)
tr
Thus :
5Dspectral =
DAS
=
-
3D2(
+ g2
+ ...
)A
or
SD~
spectral
=
-
t D( TD(~)
where gtr is given by Eqs. (6-9).
(6-11)
-U
100
If it is assumed for the moment that the spectral shift
referred to occurs in the perturbed region, that is; in the
region where material concentrations are altered, and does
not occur elsewhere, the desired result follows immediately
from perturbation theory:
+ AJ [g
+ A
dV
+
kAgtr
-
Vzf
gtrD] - (
gi
A
2dV
(6-12)
?)T2 d
where:
A =
F
vz f
dV
core
F
=
fraction of fissions occurring in the core
5 k= reactivity change calculated from the usual onegroup perturbation theory method.
The two terms of Eq. (6-5) are separated in Eq. (6-12)
simply for convenience.
A major significance of the result
achieved thus far is that the quantities in brackets in Eq.
(6-12) need be evaluated only once for a given core.
Only
the quantity "AS" (and the usual 5#) must be determined for
each perturbation of interest.
Thus the quantities in brackets
are evaluated and Eq. (6-12) becomes:
101
Sk =A
a <P 2 + b( V~)2]
dV
+ 5k,
where V1 is the region in which material concentrations are
altered.
The method of evaluating "S"
is given in reference
(45) and "AS" is determined simply by calculating S' for the
perturbed region (AS = St-S).
An example of the calculation-
al procedure, including a simplified method of determining
S' for most cases of interest, is given in Appendix C.
The energy spectrum, of course, does not abruptly change
to the perturbed spectrum in V1 .
Figure 6-1 in conjunction
with Figure 3-3 shows the behavior of the spectral shift as
a function of core position as calculated by ANISN.
Figure
3-3 shows the energy spectrum for the unperturbed core and
for the core with sodium removed.
Figure 6-1 shows the
shift in Group III and Group VIII fluxes as a function of
axial distance from the core centerline for sodium removal
from the central 30 centimeters of the core.
Notice that
the flux in Groups III and VIII is approximately that of a
totally voided core in the central 20 cm, then changes
gradually to the flux characteristic of the unperturbed core
outside approximately Iz J = 30 cm.
Similar behavior was
found for other energy groups and other perturbations.
Thus
the accuracy expected from Eq. (6-12) for localized perturbations will depend on the extent to which the influence of the
C-
Flr~
L
,5c>_r
Soolum
(
T
1- - U rN ? f I Z r RCrQ & D A
W LT-ACI
Ait- MA av'r F
~x A
L
V 4RGLA-n
F -I -V IS
f-o~
N4 O R MC A L Z k o)
~
(;.Rovp
I
it.
ALL N%.A IN
.
3 iJ
IA/
///,
/f/7~~
-
,-
,,
,ALL
NA ip,4FLuy(
-
-
RE-LA-VW E-
'I
Di
FLV)( FO&Z W46.E
OCc,9E~ VOIDIK&
''I
~LL NAO~fr
WA Ovr
A"
I
30
7-
( FJXAL
-O-nON~Vv4
)
FLu~-.J'
FLUX ,T
0
103
shaded areas in Figure 6-1 tend to cancel.
That is, if the
overestimate of the spectral shift in the perturbed region
(area (1) of Figure 6-1) compensates, or nearly so, for the
unaccounted for shift outside the perturbed region (area (3)
of Figure 6-1), the accuracy of Eq. (6-12) in predicting
reactivity changes arising from a spectral shift will be
limited only by the accuracy of the spectral parameters "S"
and "R" in correlating cross sections.
"nneffect",
Even without this
Eq. (6-11) should give good results
for perturbations involving large or "global" regions of the
core.
It is worth noting here that sodium voiding produces
the strongest spectral shift of any of the plausible core
rearrangements or perturbations examined, as will become
clear from subsequent calculations in the present chapter.
The cancellation effect just described is found to be quite
beneficial for all perturbations considered and Eq. (6-12)
has been found to give good results for localized perturbations.
Furthermore, the "spectral" contribution to the
total reactivity change predicted by Eq. (6-12) has been
found to be substantial in most cases of interest.
For the
important case of sodium voiding, the spectral contribution
predominates over much of the core.
Before citing results predicted by Eq. (6-12), it is
interesting to examine the components of the equation.
For
simplification in calculations employing Eq. (6-12) the core
F
104
of interest is assumed to be composed of Pu239
U238,
02,
Fe
ANISN runs comparing calculations using these con-
and Na.
stituents with runs employing the actual core composition
(316 stainless steel instead of Fe and including the small
fractions of U235 and Pu240 expected to be present) show that
this assumption introduces negligible errors for purposes of
the present analysis.
For the core described in Table 1-1, Eq. (6-12) then
gives:
5(vZ f)
6k
5D
a
-4
Core
Con
stituent
+ 8.7xl0~ ]c2
+ 1-.1161( v)LdV
+ 7.4xl0 4 J 92
+ E-.018
0.0
+ 0.25x10 4
+ [+.0101( V<P)2j
+ A
0.0
±
+ A
0.0
+ 0.59xl0
=
A
I110.05x1
-8.4x1o~4
+ Af
+A
+ 6
j
.k
V2
0.26xl0-4B 2 +
J2
+
(
)Z
U28
V
Pu
dV
02
-. 60] ( vp) 2JdV
Na
-. l06( vc)
2:dV
Fe
(6-13)
Note that essentially all of the net contribution from 5(v7f)
and 5(Za), and therefore the F 2 weighting, comes from U2.
For the particular case of total sodium voiding Eq. (6-12)
becomes:
105
i=
A T(4.67 x 1o~4) ? 2 - 0.077 ($)1
5
2d
spect 10
(6-14)
+ AfC(. 24 x 10~ )-4 2
- 0. 48(
2~ dV
usutatlo
Inspection of this equation shows the dominance of the strong
spectral effect in the central regions of the core where the
flux gradient is small.
Equation (6-14) in conjunction with
(6-13) shows the important role of the fertile isotope in
producing strong positive sodium void coefficients in LMFBR's.
It should also be noted that the decrease in absorption by
U 238 produces essentially as strong an influence on void
reactivity as the increase in fission which results from the
spectral hardening accompanying sodium voiding.
The contribu-
tion of each core constituent to the sodium void coefficient
can be readily analyzed with the present theory.
It may thus
be possible to employ the theory to help minimize the positive
sodium void reactivity within a given set of design criteria.
The reactivity model derived in this section will be
referred to hereafter as the " PS Model " for brevity.
6.3
Applications of the Reactivity Model to Accident Analysis
The PS model developed above and expressed in component
form in Eq. (6-13) is written
here in its condensed form for
the core under investigation:
[(18.85 x 10~)9 2
k =A
+
I
.31(
o)p )1
V
(6-15)
1o6
Note that this result is applicable for all perturbations
and in any geometry.
The simplicity of the final result for
a given core composition is one of the nicer features of the
model.
Equation (6-15)
is now solved for a number of plausi-
ble core rearrangements during an accident condition and the
results compared with ANISN runs.
While primary emphasis is
on the pseudocylindrical geometry described in Chapter II, the
equation is also solved and compared with ANISN in spherical
geometry.
The following cases of core rearrangement are considered:
Condition or Rearrangement
Case
I
(percentages shown are relative to the unperturbed core)
20% Sodium Voiding:
Sodium density is reduced by 20%
in the region of interest.
II
Total Sodium Voiding:
Sodium density is reduced to zero
in the region of interest.
III
20% Fuel Addition:
Fuel (U28 , Pu49, 02) density is in-
creased by 20% in the region of interest.
IV
20% Fuel Removal:
Fuel density is reduced by 20% in
the region of interest.
V
Total Na Voiding and 20% Fuel Addition:
Sodium density
is reduced to zero and fuel density simultaneously
increased by 20% in the region of interest.
107
VI
Total Na Voiding and 20% Fuel Removal.
VII
20% Na Voiding and 20% Fuel Addition.
VIII
20% Na Voiding and 20% Fuel Removal.
Equation (6-15) for the eight cases of interest becomes:
20% Na Voiding ( AS = .0484):
Case I:
5k
= A
(.91 x 10- 4 ) y 2 - .015(
spectral
v<r )27 dV
effect
(6-16)
+ A
Case II:
p
L(. 0 5 x 104 )
2 -
AS
(-s
Y
Total Na Voiding
(4.67 x 104) <T 2
k =
.080(
usual
perturbation
vqp7)]dV
.248):
spectral
.077( vp )23dV
effect
(6-17)
+ A
usual
perturbation
r(.24 x 10~4) ( 2 - .48( Vc?)2jdV
As seen from these results and as noted earlier, the
spectral effect strongly dominates in the central regions of
the core for the case of sodium voiding.
Case III:
6
= A
= .055):
20% Fuel Addition($
16
j(l.04 x 10~)
+ AI,(3.00 x 10
2
)
2
.06(V
]dV
<P ) dV
-150( .10(
7rpJdV
-
spectral
effect
{
(6-18)
usual
perturbation
108
20% Fuel Removal (A
Case IV:
=
A
=
- .073):
\-<p)22dV
(- 1.37 x 10~4) cp 2 + .023(
spectral
(6-19)
+ A
3.00 x 10~ ) Q9
-
.190(
V')jdV
erturbation
Two important points are clear from Eqs. (6-18) and (6-19).
First, the spectral shift plays an important role in the reacSecond, the negative
tivity effects produced by fuel motion.
reactivity induced by fuel removal is stronger than the positive effect induced by the same amount of fuel addition.
In
this case the primary reason for the difference is the stronger
spectral shift induced by fuel removal (=AS
-
.073) as com-
= + .055).
pared to that induced by fuel addition (A
This
important phenomenon is discussed further later in the present
chapter.
Case V:
Total Na Voiding with 20% Fuel Addition( .A= .265):
S
AJ
c)
2 - .082(
(4.99 x 1 04
2
dV
spectral
effect
(6-20)
+ A
Case VI:
4
L(3.24 x 10
)
2 - .240(
\7 )2JdV
Total Na Voiding with 20% Fuel Removal (
Sk = A
(4.13 x 10
4
) <
2 - .68(
.'-\
perturbation
= .219):
spectral
) 2 dV
'eJLeffect
(6-21)
+ AJL(- 2.76 x 10~
) cp
2 - .830(
v'e)JdV
usual
perturbation
I-
-9 -
ION-- -1
1--
109
Note for case VI that the overall reactivity effect near
the core center is positive, that is; sodium removal has a
stronger positive influence via the induced spectral shift
than the concurrent negative effect of reducing fuel density
by 20%.
This is indicative of the strong influence sodium
voiding has in such a reactor; a behavior which becomes more
apparent as the present development continues.
Additionally,
for case VI, note the strong negative coefficient of the
leakage (( 'q')2) term as compared with cases II and V.
Section 6.4 of the present chapter considers the significance
of this behavior in detail.
Case VII:
20% Na Voiding with 20% Fuel Addition (
gv)2)dV
7
5= A kc [(1.98 x 10-4) q 2 - .0326(
.
4Oj
= .105):
fspectral
effect
(6-22)
+ A
[(3.05 x 10-4) q 2
.o8o(wy)2]dV
-
usual
(perturbation
Case VIII:
A
k=
k
20% Na Voiding with 20% Fuel Removal (
(
-
.374 x 10-)
2 + .oo6( p
L
2dV
J
= - .0198:
spectral
effect
--
(6-23)
+ A
(
-
2.95 x 10~)7 2
-
.270( 9q)ldV
usual
perturbation
L
Comparisons between the reactivity predicted by the ANISN
computer code and the reactivity model developed above are
now made.
For the pseudocylindrical geometry described in
110
Chapter II, the axial flux shape is taken to be
cO
q=0cos
7z
As shown by Figures 3-1 and 3-2 this gives an exceptionally
good fit to the axial flux distribution predicted by ANISN
over the core region of interest.
cal geometry a flux shape of
'f=
For comparison in spheri< 0 El
-
(1)2
is assumed.
This was found to give a reasonably good fit to the ANISN
data for runs in spherical geometry.
The reactivity predicted by Eq. (6-16) (case I:
20%
Sodium Voiding) is compared with that predicted by ANISN in
Figure 6-?.
The void extends over the entire radial area of
the core and is assumed to expand symmetrically about the
axial core centerline.
The 26 group Russian (ABBN) cross sec-
tion set in the S-8 transport theory approximation was employed for these ANISN runs.
case is seen to be quite good.
Success of the model in this
In Figure 6-3 a similar
comparison is made for total sodium voiding (case II). In
AS
this case the spectral shift is severe (.1
= .248). Since
S
derivation of the spectral reactivity model employed partial
derivatives with respect to S, the results are not expected
to be accurate for large values of
.
An improvement in
the results for each case investigated was achieved when an
average value is used for the quantity
S.
For example,
for the case of total sodium voiding, the following values
were calculated:
FIGuI~e
GZ
-ZE:A C F IV
-f
DIPP
00
*009
1004DT
.003
0
:Z0
7- CS%7-r= of: 6oWUM '40\6 . 2W-% Sot>%UtA 007;
3.
4'0
112
vs.
Cr~p
N
=.0 ?4'E&IGO
01F
.03O1
-000-Ai
A--
l's't%~L '4IT'A
MVOjL
'010?Sl
(AIrr$ 5
)4AN6E4 JRoMC.e-H
\1okt>
-rzowrt'4 xs
<pmrae
I
X
(5)ZF-
OV:
:~
~SOt )OrA
-30
\JOb; Al-
:Sc2-CV,)AA
9EMfOVO;
W%
~r
113
S = .3530 for the critical unperturbed reactor
St = .4531 for total sodium voiding
= S
+
= .403
AS
.1001
.
AS
.1001
.248
The dashed curve in Figure 6-3 results from using this latter
Points are also shown in Figure 6-3
quantity in Eq. (6-15).
for calculation based on the unperturbed S values.
The im-
provement in accuracy in this case is typical of that for
The use of
other cases calculated.
-$
for large spectral
S
found to give more consistent
perturbations was, in fact,
results than the employment of
AS
- or
AS
.
If the change in
cross sections is evaluated by means of a Taylor expansion
instead of differentials, the use of AS- can be more rigorously
justified.
For example, by taking differentials, it was found
that:
5E = gZ
(6-6)
AS
Employing the Taylor expansion in evaluating 67 gives:
5z= ZO(st)g - z'(S)g
=Zo
(S+AS)g -
S
,
114
where
2
+...
lAS + g(g-2)S-
(S+AS)6 = Sg + gS
By taking the first three terms of the expansion
g~[S r
is obtained.
(g-1) g(AS
+S
2
]I
Notice that the first term of this latter
equation gives the result obtained in Eq. (6-6) by taking
derivatives; as expected.
By including one additional term,
better accuracy for larger values of i
is expected.
As
shown in Appendix C, the "average" value of g for all constituents used in the current LMFBR is considerably less
than unity.
Then if (-
1/2) is taken as the coefficient of
AS 2
(T)
in the above expression for
5z = gz 0SD
AS
=
1 AS 2
-
8
U.; the result becomes:
S-.
+ l/2ASSJ
= az SC(
and the advantage of employing As for large perturbations
in the present analysis is seen not to be entirely fortuitous.
- -S = 0.0484, and for
For the case of 20% sodium voiding, where
other cases where A
is small, the averaging technique is
clearly unnecessary.
Figure 6-4 shows the results for global sodium voiding
in spherical geometry.
The deviation of the ANISN and
115
IE
(2.o~SoD~uH\
RernoviE0
,009
'003,
000
2.0
((ZADV
L
30
t
'SlZrz. OF 401b- 20,2,0 SDt %tjrA
70
~-i1i~1~
116
calculated curves beyond the 50 cm radial position is due in
part to the fact that the parabolic flux shape assumed did
not give a perfect fit to the unperturbed ANISN flux data.
In particular, the fit was not as good as that achieved by
using the simple cosine shape for the axial direction in
cylindrical geometry.
Since the curves in Figure 6-4 are
integral curves and the error in flux shape is weighted by
the core volume; the error generated in spherical geometry
might be expected to be appreciable in the outer regions of
the core.
Table 6-1 shows the reactivity predicted by ANISN and
the spectral reactivity model for several cases of core
rearrangement.
Success of the model in handling the cases
tabulated was good except for RUN 407.
In this particular
run the calculated result is the algebraic difference in the
large positive reactivity calculated for 120% fuel density in
the 0-15 cm interval
=
+ .0278) and the large negative
reactivity calculated for 80% fuel density in the 15-30 cm
interval (5k = -.0240).
RUN 407 was included to show this
characteristic, but expected, weakness of the model.
Each of the comparisons given thus far has involved
perturbations which are uniform over fairly large regions of
the core.
As cited earlier,
the model is not expected to be
highly accurate for localized perturbations.
in handling such perturbations
Its usefulness
is considered in Figure 6-5.
Table 6-1
COMPARISON OF ANISN AND 'PSt MODEL RESULTS
Condition or Rearrangement
For Run
Run
401
CNa
= 0 in
402
CNa
= 0 in
403
CNa (=
405
407
0 in
interval [0-15 cm]
int.
{O-30
cm]
interval 10-15 cm] (2)
Cfuel = 1.2 in
int.
[0-15 cml
Cfuel = 0.8 in
int.
[15-30 cm]
CNa
= 0 in
interval [0-30 cml
Cfuel = 1.2 in
int.
[0-15 cm]
Cfuel = 0.8 in
int.
[15-30 cm]
Cfuel = 1.2 in interval [0-15 cm]
Cfuel = 0.8 in
4o8
.
CNa
= 0 in
int.
ANISN Results
with
with
16 groups 26 groups
I
I
T PS'
Model
Results
.0156
.01520
.0172
.0245
.0239
.0275
.0241
.0236
.0220
.0286
.0282
.0292
.oo625
.0038(3)
[15-30 cmj
interval [0-15 cm
Cfuel = 1.2 in
int.
Cfuel = 0.8 in
int. [7.5-15 cm]
+.01515
[0-7.5 cm3
density after perturbation
unperturbed density
2) Intervals are symmetric about axial centerline. Thus [0-15 cm3 represents
a void in the central 30 cm of the core.
3) The calculated result here is the difference in two large numbers. See text
for additional discussion.
NOTE: 1) C
F-i
118
OF'
Wk-t~4 -=2-S CwA~. Pia
or
. 003
INTER~EST-~
I'ObO-L
rA-rA
'002,
ENISe
Co(I.
CCuvR-
co5
K.
loot
0
-. 001
410S
0
A)(AL
?oP0Sj-r0N
04C c6-eA-IO'
c-
~
.~
mm
119
Local axial sodium voids of 2.5 centimeters in width were
employed in ANISN and in Eq. (6-17).
Figure 6-5 shows that
the success of the model in predicting the magnitude and
shape of the resulting reactivity changes is relatively good.
Observe that this particular series of calculations involves
the limits of the model in two respects; the value of As is
large (0.248) and the size of the region perturbed is quite
small (2.5 cm).
As can be seen from the
values listed
above for cases I-VIII, sodium voiding produces about as
strong a spectral shift as any of the rearrangements considered.
As a result of the reasonable success of the PS model
in handling spectral shifts of such magnitude; its perhaps
fortuitous accuracy for localized perturbations; and, more
importantly, the excellent results achieved for less severe
global perturbations, it has been found highly useful in
analyzing reactivity effects associated with accident conditions.
6.4
Core Compression and Expansion Effects on Reactivity
The ultimate shutdown mechanism for an extremely severe
accident is the negative reactivity inserted as a result of
core expansion or disassembly.
In analyzing such effects it
has frequently been assumed that the core remains homogeneous
(in the sense that the volume fraction of each constituent
remains constant in all locations) throughout the disassembly
process. (7)(8)(9)(13)(18)(47)
In such cases reactivity
mu
120
effects can be realistically expressed as a function of the
homogenized core density for a fast reactor.
This is the
method used in the Bethe-Tait approach employed in references
(7), (8),
(9),
(13), (18), and (47) and in numerous other
places in the literature.
In the present section, a more
general relationship than that given by Bethe-Tait is derived
for predicting reactivity changes as a function of core density changes.
The present result shows the aforementioned
behavior of density reductions in producing a stronger negative reactivity insertion than the positive reactivity added
by a comparable density increase.
A comparison of the
present result and the Bethe-Tait method is given in Appendix
C and it is shown that the Bethe-Tait model cannot predict
the anomaly just cited.
A key to the success of the present
analysis is the observation that the spectral parameter "S"
defined by Eq. (6-1) is independent of homogenized core
density.
Thus a core compression or expansion does not cause
a spectral shift and the second term of Eq. (6-5) is zero.
This implies that the usual one group self-adjoint perturbation theory result should be adequate for analyzing compression and expansion effects.
A negligible spectral effect
has previously been assumed but not demonstrated in various
applications of the Bethe-Tait method.
-U
121
The following relationships are employed in the present
application of perturbation theory:
C= Pf
final or perturbed density
initia1 or unperturbed density
P
(3-1)
(C-1)Za
(6-24)
5z = (C-1)Zf
(6-25)
6,a
D=
(6-26)
-C-)D
With these definitions, the usual perturbation theory result
can be written as follows:
5k = A
(C
(
~)cp2 + D(
7qp)2dV.
(6-27)
In order to see the effect of compressions and expansions
more clearly, it was found convenient to assume a cosine
flux shape.
Note that this does not restrict the applicabil-
ity of the analysis to a particular geometry; the result will
still be valid to the extent that a cosine function adequately
describes the actual shape.
cp = qpocosBx
This simplification gives:
for a slab (or the axial direction in a cylinder),
1
S=
B =
YQcos Br
He
or
e
for a sphere,
.
122
Thus:
(
\7q)2
(
Vq)2 = 1B
= B(P2_
2)
(6-28)
for a slab
and
20
2)
for a sphere
(6-29)
.
Using these results and the one-group relationship:
( v-,f -
in Eq. (6-25)
6=
A
2a) = DB 2
(6-30)
,
we obtain:
Cl
1
+ (
)2(C-1)
dV
(6-31)
in slab geometry and
A
+
1= 1
()
(C
-
)jdV
(6-32)
in spherical geometry, where
A
FDB2
=
l
(6-33)
2f
dV
and
F = fraction of fissions occurring in the core
~ .94 for the case of Table 1; as calculated by ANISN.
Equations (6-31) and (6-32) show clearly the effect of
density changes on reactivity.
If density is reduced to 80%
of the original value in some region of the core, the quantity
=
-
On the other hand, if density is somehow
increased to 120% of its original value,
+ 1=
In
123
addition to the results shown by Eqs. (6-31) and (6-32),
relationships have been worked out for spherical geometry
using the more correct parabolic flux shape, q = <o 1l (r)
and for the pseudocylindrical geometry employed in
ANISN calculations as discussed in Chapter II.
The resulting
equations are:
Sk = AoF4)
2
) + C(
(l-
(6-34)
)dV
in spherical geometry and
Ao
C= 1 + (
)(C
in pseudocylindrical geometry.
- .246)1 dV
(6-35)
This latter expression con-
siders only axial compressions and expansions for the cylindrical core under consideration. Table 6-2 shows the results
predicted by Eqs. (6-32) through (6-35) for localized density
increases or decreases at various positions in the core.
Note that in each case except at the core center in spherical
geometry, removal of a given quantity of material produces a
stronger negative effect than the positive effect produced by
addition of an identical amount of material.
Figures 6-6 and
6-7 compare these results with those predicted by ANISN.
Examination of these figures shows that the ANISN results
substantiate the conclusion drawn from Table 6-2.
From
Figure 6-6, if 20% of homogenized core material is removed
from a small volume element at position (a), it must be
moved inward closer to the core center than position (b)
Table 6-2
REACTIVITY EFFECTS FROM CORE EXPANSION OR CONTRACTION 5
( k)
Axial Expansion
or Contraction
in "Cylindrical"
Spherical Geometry
2
(q
)2
990
Cosine Flux
C=l.2
Parabolic Flux
C=l.2
C=.8
C=.8
1
+ .20A
0
-
.20A
.7
+ .151AO
-
.156A 0
.5
+ .120A
0
-
.130A 0
±
.25
+ .o83A
-
.100A 0
+ .o8l2Ao
1.0
0
+ .20A
0
.121A
0
-
.20A0
-
.131A
Geometry
C=.8
C=l.2
0
- .097A0
+ .33Ao
-
+ .25A
0
- .325A
0
+ .208A
0
- .288A
0
.40AO
ro
125
' I&L
rm.
G- (
uwraarupo3hO FLoyx
( SQuA r.Eb)
FOe
L.OC.ALIZ E D(2,5-e^. INTERVAL)
OF
?RTAGTO4
)-Jt1orsoNizEaD
~12
~.004~
CC (--
TAA
--
.002
((b)
.DOO
001/4
,&
(2.
l'4'
.0031W"
N....or
-.
0He
o
v.....
.....
SA TCLA.A T~ .
126
f f &0RE
r;, -
REiACT-w iTrY
uNprVERU rGE) FLux
FO(P LoLCALtr.6E
OF
)4oMO6ENIZEt
t~ascr
-C..=
rUFSP7'ods
Cotr
1,
orn O
.004j
A- 00Z
A
-
\
N
ATE
C ALC~ L.ATEb
DATA
,001
0
1,0
v.'g
0.,
&.- .,q 03
P,.2
0.1
-oo\
-,002
---
00
-.10o5
-.
R-
I
(
a
m
127
before a positive reactivity effect is produced.
If it is
inserted between (a) and (b), a negative reactivity effect
results from moving core material inward.
It is, of course, unrealistic to speak of adding 20% of
homogenized core material at a given location.
The tendency
shown in the present analysis is exhibited, however, in physically realizable rearrangements.
Consider, for example,
cases II, V and VI above (Eqs. (6-17), (6-20) and (6-21)).
For case II, total sodium voiding, the coefficients of the
leakage term is - .557.
For case V, total sodium voiding
with 20% fuel addition, this term is - .322, while for case
VI, total sodium voiding with 20% fuel removal it is - .898.
Thus if sodium is voided from a region of the core and then
20% of the fuel in part of this region is injected into
another region (removal of sodium now makes room for the
fuel), the difference in leakage coefficients clearly shows
the density effect cited.
Namely for fuel addition we ob-
tain a change in the coefficient from - .577 to - .322 or
+ .235 whereas for fuel removal we obtain a change from
- .557 to -.898 or - .341.
This is identical to the behavior
for the homogenized compression and expansion effects shown
by Eqs. (6-31) through (6-35).
AP
Notice that the coefficient C-1 can be written as
C
Pfinal
The behavior of this quantity in ameliorating the effect of
density shifts on reactivity is analogous to the behavior of
128
the quantity
S
on reactivity.
in ameliorating the effect of spectral shifts
These two apparently independent phenomena
play a highly significant role in reducing the reactivity
additions from all core rearrangements investigated.
An example of the combined influence of these parameters
can be deduced from data in Table 6-1.
Consider the ANISN
results with the 26 group cross-section set (column 2).
In
RUN 407 the reactivity introduced by fuel movement alone is
seen to be + .00625.
In RUN 402 the reactivity introduced
by sodium voiding alone is seen to be + .0239.
In RUN 405,
the same extent ofsodium voiding and fuel movement taken
simultaneously gives a reactivity change of + .0282.
Thus,
if the fuel movement described in RUN 407 occurs after sodium
voiding of the region of interest, the reactivity added by
fuel motion is:
.0282 -
.0239 = + .0043
and not the + .00625 calculated in RUN 407.
The same con-
clusion is readily reached by comparing the hand calculated
results for the same runs in Table 6-1.
A more impressive example can be shown by considering
RUNS 401 and 408 of Table 6-1.
In RUN 401 the reactivity
introduced by voiding sodium in the central 30 cm of the
core is seen to be + .01520.
In RUN 408, fuel is moved in-
ward while the same degree of sodium voiding is present.
-
U-
129
The result is a slight reduction in reactivity to + .01515.
Inward fuel motion results in a slight shutdown effect.
Further examples showing the combined influence of the
density ( AP) and spectral (-S ) effects are given in succeeding sections of the present chapter.
6.5
Observations from Calculations and ANISN Results
The reactivity changes resulting from a number of core
perturbations or rearrangements are shown in Table 6-3.
The
data presented was obtained from ANISN computer program runs.
As noted earlier, the runs listed in Table 6-2 were calculated with the PS Model as well.
The following observations from the data of Table 6-3
are of interest:
The compression of core material in one
(1)
region and expansion in another shown by RUNS 18, 20,
22, 24 and 25 results in an overall negative reactivity effect.
served.
Note that in each case material is con-
Note also that RUNS 24 and 25 are the reverse
of RUNS 18 and 20 respectively.
As would be expected,
when the general material movement is away from the
core center, the shutdown effect is stronger.
(2)
In the runs cited above, such uncompensated
compression of the core is, in general, physically unrealistic.
In RUNS 212, 213 and 214, however, room is
available for fuel inward motion as a result of sodium
130
Table 6-3
REACTIVITY EFFECT OF CORE REARRANGEMENTS
Condition or Rearrangement
for Run
Run
18
20
22
24
25
C = 1.5 in interval [0-17.5 cm]
C = 0.5 in
int. 17.5-35 cm]
C = 1.5 in
int. [7.5-10 cml
C = 0.5 in
int.
C = 1.1 in
int.
C = 0.5 in
int.f12.5-15 cm]
C = 0.5 in
int.
C = 1.5 in
int.117.5-35 cm1
C = 0.5 in
int. [7.5-10 cmJ
[10-12.5
Net
Reactivity
Effect
- .00199
- .00350
cmj
- .00240
O-12.5 cm]
10-17.5 cm] (run 18
reversed)
(run 20
- .03080
- .0044
reversed)
C = 1.5 in
212
= 0
11O-12.5 cm]
in
int.
Cfuel =2.0 in
int.
[20-22.5
in
int.
L22.5-25 cm]
in
int.
E10-12.5 cm]
Cfuel = 2.0 in
int.
f10-12.5 cm]
in
int.
CNa
Cfuel =0
213
int.
CNa
= 0
Cfuel = 0
20-22.5 cm]
- .ooo8
cmj
[12.5-15 cm]
- .0012
-1:
131
Table 6-3
(Continued)
Run
214
CNa
= 0
in
C fuel = 2.0 in
218
int.
[35-37.5 cmI
int.
[35-37.5 cm]
in
int.
in
int.
C fuel = o.8 in
int.
C fuel = 1.2 in
int.
C fuel = 0.8 in
int.
C fuel = 0
217
Net
Reactivity
Effect
Condition or Rearrangement
For Run
C fuel
=
1.2
- .ooo6
p37.5-40 cm)
[20-22.5 cm]i
- .0017
[22.5-25 cm]
E10-12.5 cmI
- .0020
J12.5-15 cm]
219
CNa
= 0
in
int.
[0-12.5 cm]
+ .0131
223
CNa
= 0
in
int.
[0-12.5 cm]
±
C fuel = 1.2 in
int.
T0-12.5 cm]
224
225
C fuel
= 0
in
int.
[12.5-15 cmJ
C Na
= 0
in
int.
'0-12.5
cm]
C fuel
=
1.5 in
int.
{0-12.5
cmj
C fuel = 0.5 in
int.
[12.5-25 cmj
= 0
int.
(0-12.5
C fe1 = 1.5 in
int.
EO-12.5 cml
C fuel = .5
in
int.
112.5-15 cmJ
C fuel = 0
in
int.
[15-20 cmj
CNa
in
cmI
.00170
+ .0271
+ .01863
132
Table 6-3
(Continued)
Condition or Rearrangement
Run
226
227
228
For Run
in
int.
in
int.
[25-27.5 cm]
in
C fuel =1.1
int.
0-45 cm]
C fuel =-0.1 in
int.
£45-50 cm]
1.1
Cfuel
fuel
Cfuel =0
CNa
= 0
in
int.
CNa
= 2.0
in
int.
Cfuel+ss = 2 .0
in
int.
Cfuel+ss = 0
in
int.
[ 0-25 cmj
[0-12.5 cml
Net
Reactivity
Effect
+ .00256
+ .02007
+ .o165
112.5-25 cm]
[ 0-12.5 cm]
[12.5-25 cm]
401
CNa
=0
in
int.
[0-15 cm~
+ .0152
402
C
= 0
in
int.
[0-30 cm]
+ .0239
403
CNa
0
in
int.
EO-15 cmJ
+ .0236
C fue 1 = 1.2 in
int.
[0 -15
C fuel = 0.8 in
int.
[15-30 cml
= 0
in
int.
0-30 cm]
in
int.
0-15 cm]
Cfuel = 0.8 in
int.
15-30 cmJ
405
CNa
Cfuel = 1.2
cmI
± .0282
133
Table 6-3
(Continued)
Condition or Rearrangement
for Run
Run
407
408
int.
[0-15 cm}
Cfuel = 0.8 in
int.
115-30 cm}
in
int.
1.2 in
int.
Cfuel = 0.8 in
int.
[7.5-15 cm]
Cfuel
=
1.2 in
CNa
=0
Cfuel
=
[0-15
cm]i
Net
Reactivity
Effect
+ .00625
+ .01515
0-7.5 cm]
409
CNa
= 0.8 in
int.
£ 0-15 cm}
+ .00346
412
CNa
= 0.8 in
int.
(0-30 cm]
+ .00562
414
CNa
= 0.8 in
int.
0-15 cm]
Cfuel
= 1.2 in
int.
[0-15 cm]
Cfuel
= 0.8 in
int.
15-30 cm]
CNa
= 0.8 in
int.
[ 0-30 cm]
Cfuel
=
1.2 in
int.
L0-15 cm]
Cfuel = 0.8 in
int.
[15-30 cm]
= 0.8 in
int.
[O-15 cm~]
= 1.2 in
int.
E0-15 cmI
Cfuel = o.4 in
int.
[15-20 cmj
415
420
CNa
+
.oo6o
+ .0083
+ .00565
134
Table 6-3
(Continued)
Condition or Rearrangement
for Run
Run
421
422
0.5 in
int.
£0-15 cmj
C fuel = 1.5 in
int.
E5.0-15 cmj
C fuel = 0.0 in
int.
115-20 cm]
= 0.8 in
int.
[0-25 cm]
C fuel = 1.2 in
int.
[17.5-25 cmj
o.4 in
int.
CNa
CNa
=
C fuel =
Net
Reactivity
Effect
+ .00744
+ .00576
[25-27.5 cml
NOTES:
1) C = density after perturbation
unperturbed density
2) C with no subscript implies overall core density (homogenized).
3) In all cases all material except sodium is conserved. That
is, only sodium is allowed to cross the core boundaries.
4) All runs in this table are in pseudocylindrical geometry.
All rearrangements are symmetric with respect to the core
axial centerline.
5)
The Hansen Roach 16 group cross section set was used in all
In this series the Russian 26
runs except the 400 series.
group set was used. The S-8 transport theory approximation
was used in all runs.
-
U-
135
voiding.
The same tendency in producing an overall
shutdown effect is seen in these runs.
(3)
For realistic rearrangements affecting
larger regions of the core, consider 401 and 408;
runs which were compared earlier in Section 6.4 of
the present chapter.
As noted, the overall effect of
inward fuel motion in RUN 408 is a slight reduction in
the reactivity present from sodium voiding alone in
RUN 401.
RUNS 219 and 223 show a second example of
this behavior.
In RUN 223, overall reactivity is re-
duced substantially when inward fuel motion follows
sodium voiding.
(4)
Comparison of RUNS 224 and 225 show the
strong effect of a local fuel "void".
The same degree
of sodium voiding and inward fuel injection is present
in both cases.
In RUN 225, however, the fuel for
inward motion is obtained by completely removing fuel
in the 15-20 cm region.
This results in a substan-
tial reduction in the reactivity inserted as compared
with RUN 224.
(5)
The reactivity induced by fuel motion cannot
be calculated separately and added to that produced
by sodium voiding.
invalid.
In other words,
superposition is
This is a general observation from the pres-
ent study and can be seen by comparing RUNS 402, 405,
136
and 407.
The sum of the reactivities from RUNS 402
and 407, with sodium voiding and fuel motion taken
separately, is + .03015, not the result obtained in
RUN 403 for simultaneous voiding and fuel motion,
+ .02820.
The non-linearity of these effects is
further evidenced by the spectral shifts predicted
in cases I-VIII of Section 6.4 above. For example,
20% sodium voiding produces a AS value of + .0484; 20%
sodium voiding with simultaneous 20% fuel removal
gives a value of
g- of - .0198; whereas 20% fuel re-
moval alone produces a value of - .073.
Then the
sum obtained by adding the separate spectral shifts
is + .o484 - .0730
=
-
.0246; not the more correct
value of - .0198.
(6)
In general, when 20% or less of the fuel
in a given region is moved to some other core location, the reactivity effect appears to be small in
comparison to the effect of sodium removal (voiding)
from a region of comparable size.
Even when a large
fraction of the fuel in a localized region is shifted
inward through an appreciable distance, the reactivity effect is of the same order as that expected
from extensive sodium voiding.
RUNS 212-214, 225 and
228 are indicative of these observations.
137
(7)
Comparison of RUNS 226 and 227 is particu-
larly interesting.
The rearrangement considered for
these runs is intended to simulate a collapse of
fuel within the clad to 100% theoretical density;
approximately 110% of the normal density.
In RUN 226
the central half of the core is collapsed toward the
center.
In RUN 227 the entire core is collapsed
toward the center.
Note that the reactivity added
in RUN 227 is about eight times as great as that added
in RUN 226, although the extent of fuel movement is
only about twice as great.
RUN 226, in essence, is a
rearrangement of the central region of the core and
is somewhat similar to the rearrangements of interest
in the present work.
RUN 227, on the other hand, is
more representative of a core compaction into a
second critical configuration in the sense that
material on the edge of the core is involved in the
compaction.
This result is somewhat incidental to
the present study and its implications were not pursued in detail.
It does suggest, however, that a
small amount of compartmentalization within an LMFBR
fuel element might substantially limit reactivity
insertions due to fuel motion.
6.6
Reactivity Addition Rates from Fuel Motion
In Section 6.1 of the present chapter mechanisms which
can lead to fuel motion or rearrangement were discussed.
In
138
Sections 6.2 through 6.4 methods of predicting overall reactivity changes resulting from various core rearrangements
were investigated.
In Section 6.5 a tabulation (Table 6-3)
of core rearrangements and the resulting reactivity changes
is presented and discussed.
In the present section a method
of estimating reactivity addition rates which can result as
fuel motion (leading to the tabulated rearrangements) takes
place is developed.
The fuel motion mechanism treated is
axial fuel injection from intact clad as discussed in Section 6.1.
The following sequence of events is postulated.
Refer to Figure 6-8:
(a)
A CATEGORY II excursion occurs producing
clad failure as shown in Figures 2-3 and 6-8.
The
region 6R- extends to position z1 of Figure 6-8.
(b)
Immediately after clad failure (time t=O in
the present analysis) the pressure in region R. drops
to a low value, say
-
20 psia.
This value is typical
of the pressure at the core center in normal operation.
Such a pressure can be expected to be reached quickly
as the molten fuel in the center of the fuel rods in
region-R is expanded and cooled.
A relatively small
degree of cooling is required to sharply decrease the
pressure in region -R as can be seen from Figure 5-3.
This low pressure will then persist until sufficient
energy is added to the sodium in region IR to produce
139
MAooc-
fr:-E6LDr
FOE
i
-
01P INCR06AEv,
IVE.N~ir 4
<D
0
I 0,:
T-
Moe
(1')CPN T-EmP.
r
00'*V,
E~
(:t
~ol
e
14r
l- 'J C.
-T
-P-PICAL
YPILALLS
A'TOLTrzN
1:0 EL
&WND'1ua
P~tSSI~E-~I
f~l EtAN L.
00
I F ~
z
RZO F .F
oP1 e-.A
P(14
Goo
-
ASsurme
PtzL-ssuc
spj
I
mMFTELyr
2007-
0 -fl
DI1MGN6IO")$ [L'&Pr.Nt ON"
Rom~
CONS%_tAjqb
(AC-VAL
14o
high sodium vapor pressures; as discussed in Chapter
V.
Quantitative comparisons of the fuel motion process
and the sodium voiding process are given later in the
present section.
(c)
The fuel vapor pressure in region (B) of
Figure 6-8 remains high initially.
In the case shown
the pressure at z2 in the center of region (B) is 600
psia at t=0.
(d)
As a result of (b) and (c) above, a pres-
sure gradient is established between points in region
(B) and iR ; resulting in fuel motion into-R .
Fuel
motion away fromR
3is prevented by the cooler solid
fuel and eventually by the much cooler blanket material
as shown in Figure 6-8.
(e)
These events produce a decrease in the fuel
density in region (B) and an increase in (A); resulting in a reactivity change.
The hydrodynamics and kinetics analysis of the events
described in (a) through (d) above are clearly complex.
Some
considerable insight into the significance of the fuel injection mechanism can be gained, however, by invoking a few
reasonable assumptions about the time dependent behavior of
the process and by making use of the fact that the overall
reactivity change for a given static rearrangement can be
accurately determined.
141
The hydrodynamic analysis employed is somewhat similar
to that developed in Chapter V for investigation of the
sodium voiding process.
The necessary equations are listed
here and discussed momentarily.
The relation:
2
7
(6-36)
d(z,t)
,7p~.,t)== pPdt
7
expresses conservation of momentum for the fuel material in
region (a) and (b) of Figure 6-8 in the absence of frictional effects.
From Chapter V:
Q(t) = ~c'P(T(t)-TO) + (h(t)-hf)
,
(5-9)
which is the energy conservation equation for the two-phase
fuel in region (B).
Equation (4-1), repeated here for con-
venience, gives the vapor pressure of fuel material as a
function of temperature:
7 exp
p = 8 x 10
6.7 x 10 4
T( K)
(4-1)
Figure 5-3 is a plot of this
was emEquation (4-2), T(Z) = Tocos w2 ,17
where p is in atmospheres.
relationship.
ployed in Eq. (4-1) to obtain the axial pressure profile in
the central core regions.
In Figure 6-8, the dashed curve
in the "p vs. z" plot shows this dependence schematically.
The Clapeyron relation
Ah
Tdp
TV= T
(57)
U U
~-
142
is used to eliminate the enthalpy term in Eq.
(5-9)
as in
chapter V and as discussed below.
The following steps, including pertinent assumptions,
apply the above equations to the problem described and depicted in Figure 6-8.
(1)
In Eq. (5-9), Q(t) is the fission heat source
and is found to be negligible in the present application.
In Chapter VII it is shown that the average power
level between an initial and secondary excursion is of
For c = .42 j/gm 0 K for
p
the fuel (Table 4-2); the rate of temperature rise of
the order of 1500 j/gm sec.
the fuel is seen to be about 3.60 K/msec.
In the pres-
ent analysis the maximum time interval considered is 8
msec; corresponding to the time between power peaks in
the typical excursion of Figures 2-1 and 2-2.
the approximation Q(t) -
Thus,
0 is quite reasonable in that
the maximum temperature error which can result is about
290 K.
From Eq. (5-9) this approximation gives:
po(T~OT(t) = Ah
(2)
.
(6-37)
The expansion process in region (B) of Fig-
ure 6-8 is assumed to be reversible; generally a reasonable approximation in evaluating the expansion work
done by a two phase substance.
With this assumption
Eq. (4-1) can be used in (5-7) to evaluate the enthalpy
143
The result obtained is:
term in Eq. (6-37).
S(T-T
p
)
(tW
=
B()
rJo
l
-
1)
(6-38)
,
where
B
= 8 x lo
A
= 6.7 x 104oK
atm=
8 x 10
dynes/cm2
3
= volume of region of interest (cM)
(3)
Two cases for the condition of fuel in region
(B) are employed.
(B) is taken as
In the first case 10% of the fuel in
V-0 of Eq. (6-38) and the mean tempera-
ture of the region is estimated to be 4600 0 K.
From Eq.
(4-1) this gives an initial "mean" pressure of 600
psia.
In the second case 20% of the fuel in region (B)
is taken as
Tr 0.
The mean temperature of the region is
then found to be 4400 0 K; leading to a mean initial
pressure of 290 psia.
If smaller values of
V-0 are
assumed (and therefore, very slightly higher mean
temperatures), it is found that the pressure in region
(B) drops rapidly as expansion takes place; resulting
in a lower effective pressure for producing fuel motion
over the time interval of interest.
If volumes of
larger than 20% are assumed; the initial mean temperature is lower and the result again is a lower "time
average" driving force for fuel motion.
Thus, the two
- U -
144
cases employed result in "worst case" behavior for a
number of combinations considered.
Recall also that
the clad rupture pressure is about 1200 psia so that,
in view of Figure 6-8,
a pressure of 600 psia some
several centimeters outside
6-R
is about as high an
initial pressure as can be reasonably expected.
(4)
No frictional effects are included.
In five
of the seven cases of rearrangement considered only
motion of molten fuel in region (B) is assumed to
occur.
Of these five, fractions of from 20% to 60%
of the fuel in region (B) is assumed to be injected
inward.
In the two remaining cases essentially all
of the fuel in region (B) is assumed to be injected.
(For purposes of calculation a minimum of 10% of the
fuel in (B), that is;
9
,
is retained in the region.
Thus the wide range of cases considered allows, in a
sense,
for a wide range of influence of "friction" in
determining how much fuel moves inward.
method of including this effect,
An analytical
for which no experi-
mental guidance is available, could not be realistically
envisioned.
The short distance travelled by the fuel
material inside the clad rods (see Figure 6-8) and the
very low velocities attained by the fuel while within
the clad (a maximum of about 10 m/sec) indicate, however, that the frictional effect in the usual sense
145
(see Eq. (5-1) et. seq. in Chapter V) is a minor consideration compared to the question of how much fuel
in region (B) actually moves inward as the pressure
in (B) is relieved.
(5)
In Eq. (6-36) the pressure gradient is
assumed to be linear as shown by the solid curve in
Figure 6-8.
The gradient is established by assuming
a pressure of zero at some location in (A)
and a pres-
sure in (B) determined by inserting the appropriate
(time dependent) temperature in Eq. (4-1).
The dis-
tance over which the pressure is dissipated, Az, corresponds to the "width" of region (A).
For example, in
RUN 420 of Table 6-3; Az = 15 cm and the pressure
gradient is written as:
Vp
Az0(6-39)
The following events occur in regions (A) and (B):
The vapor pressure in (B) forces molten fuel in that
region to expand toward the open end of the clad with
the result that fuel from (B) is accelerated by the
pressure gradient of Eq. (6-39) until the extent of
axial motion specified for the RUN of interest is
accomplished.
Employing Eq. (6-39) in Eq. (6-36)
gives:
2
dz
~()=p(Az) p.,(6-40)
i46
and with Eq. (4-1) one obtains:
B 1 exp(-
A1
-T)
= p(Az)
d z
t
dt
( -1
(6-41)
Equations (6-38) and (6-41) thus
give T(t)
f(
and T(t) = f(z(t)) respectively. The relation between
\
and z(t) is linear and is determined by conserva-
tion of material for each particular RUN considered.
of the fuel in (B) : when
corresponding to 10%
\T
In Run 420, for example, with
= 6,
region (B) remains; as specified.
15 cm is required.
40% of the fuel in
At this time, z(t)
In other words, the fuel which is
forced from (B) initially must be accelerated 15 cm
into (A).
The solution to Eqs. (6-38) and (6-40) is obtained
in a manner similar to that described in Appendix B.2
The hand calculational procedure is quite simple in
the present case, however, since the pressure in
region (B) does not change by more than about 10% in
the time interval of interest.
Thus, from Eq. (6-41)
one obtains:
2
constant =C
dt
dz
=tC1
and
t2
z(t)
=Cl
2
(6-42)
147
as a first approximation.
Relatively minor corrections
are required as the fuel motion progresses.
(6)
The reactivity change for a given RUN is ob-
tained from Table 6-3 and Figures 6-2 and 6-3.
Again
using RUN 420 as an example; the total reactivity addition for the rearrangement, consisting of sodium voiding plus inward fuel motion, is
5k
= + .00565.
In the
present analysis fuel motion is assumed to occur after
the specified degree of sodium voiding has taken place.
Thus to obtain the reactivity addition from the fuel
motion specified, the reactivity change wrought by
sodium voiding must be subtracted.
In this case, from
Figure 6-2, the reactivity addition from 20% sodium
voiding in the region i z
\
4 15 cm is
5 k=
+ .0034.
The net change is therefore:
knet
ktotal
rearrangement
=
0.00565 - 0.00340
Na
voiding
=
0.00125.*
Since the reactivity change given by this method is
for motion over the entire radial area of the core;
statistical weighting was invoked to obtain the reactivity effect for fuel motion over more reasonable
*Note that this method does not violate the caution in
Section 7.4 against indiscriminately adding and subtracting reactivity effects. Here the motion of fuel
is superimposed on an existing degree of sodium voiding and, therefore, so is the resulting reactivity
effect.
U.
148
radial areas, namely 25% and 50% as employed in Chapter V.
Selected rearrangements representative of the fuel
injection mechanism, were chosen from Table 6-3 and steps
(1) through (6) above were applied to calculate the data
presented in Table 6-4.
In this table, the appropriate RUN 3
from Table 6-3 is given in column 1; the radial core area
affected is listed in column 2; and the total reactivity change
which results when the rearrangement is complete is given in
column 3.
In column 4 the time required to complete the
specified rearrangement is given.
Note that in many cases
this exceeds the 8 msec assumed to be available.
Columns 5-7
give the reactivity insertion rates at time intervals of 2,
5 and 8 msec after clad failure in region 7R..
given are rounded off to the nearest $/sec.
The values
Pertinent obser-
vations from the data in Table 6-4 include:
(a)
The RUNS with
the fuel in (B) at T
%0 corresponding to 10% of
= 4600 0 K produce the highest
reactivity addition rates in all cases.
(b)
RUNS 224, 225, 403 and 420 all involve inward
fuel motion through a distance between 10 and 15 cm
(see Table 6-3).
RUNS 4o8, 421 and 422 involve inward
motion between 7 and 10 cm.
The latter RUNS produce
substantially smaller reactivity insertion rates.
RUN
421, in fact, results in a negative reactivity insertion.
Table 6-4
REACTIVITY ADDITION RATES RESULTING FROM FUEL MOTION
Fuel Density in (A) Increased by 20%
Mean Fuel Temp. in Central 10% of (B) = 4 6000K
Run
Core
Radial
Area
Affected
403
25%
50%
408
420
422
403
408
420
422
Max.
Avail.
6k
$1.04
$1.65
Time
tl;
Rearrangement is
Complete (msec)
10.4 msec
10.4
Reactivity Insertion Rate
After t = ($/sec)
2 msec
38 $/sec
61
5 msec
93 $/sec
149
8 msec
146 $/sec
232
-(1)
5.2
4
10
50%
30'
4.80'
5.2
7
17
25%
27s'
10.5
10
50%
430'
10.5
15
24
28
25%
5.3
50%
9.60'
15.5'
14
22
25%
Fuel Density in (A) Increased by 2o%
Mean Fuel Temp. in Central 20% of (B) = 44 000K
41 $/sec
18 $/sec
$1.04
15.2 msec
50%
$1.65
15.2
28
66
25%
30'
2
50%
4.80'
7.6
7.6
3
5
7
25%
27'
15.1
5
11
50%
430'
15.1
7
18
17
28
25%
9.60'
7.8
7.8
7
10
17
23
-(1)
-(1)
25%
50%
15.5'
5.3
34
54
36
55
-(l)
68 $/sec
104
Table 6-4
(Continued)
Fuel Density in (A) Increased by 50%'
Mean Fuel Temp. in Central 10% of (B) = 4 6000K
Run
Core
Radial
Area
A ffected
224
25%
50%
225 (2)
25%
50%
421 (2)
25%
50%
224
25%
50%
225 (2)
25%
50%
421 (2
25%
tl;
Max.
Avail.
5k
Time
Rearrange-
ment is
Complete (msec)
$1.77
$2.80
9.2 msec
741'
$1.17
8.8
8.8
(-)7.2i
-11.51
9.2
7.0
7.0
Reactivity Insertion Rate
After t = ($/sec)
2 msec
8 msec
5 msec
81 $/s ec
130
39
61
(-)6
(-)9
180 $/sec
290
89
147
(-)14
280 $/sec
420 $/sec
142
223
-(1)
- (1)
(-)23
Fuel Density in (A) Increased by 50%'
Mean Fuel Temp. in Central 20% of (B) = 4 4000K
42 $/sec[
13.6 msec
$1.77
102 $/sec 154 $/see
$2.80
66
13.6
232
157
74'
$1.17
12.7
18
42
12.7
29
71
(-)7.21
10.1
-3
68
102
-7
-9
10.1
-4
50%
-10
-11.-5,1
-16
NOTES: 1) The new configuration is reached in less than 8 msec for these cases
2) In these two RUNS the complete rearrangement require zero fuel in part
of region (B). In the calculation the fuel expansion was not allowed
to proceed beyond
10% fuel in (B). No rates are cited for the complete
rearrangement when t = t;
hence the rates cited are valid.
H4J
-
U
151
This observation is in agreement with the behavior discussed in Section 6.4 of the present chapter; namely,
that slight inward motion toward presumably higher
worth regions of the core can induce a shutdown effect
and that for rearrangements where a positive reactivity
is induced the effect is strongly ameliorated.
(c)
Only RUNS 224, 225 and 403 result in reactiv-
ity addition rates greater than the 66 $/sec basis discussed in Chapter II.
As noted and as seen from Table
6-3, the rearrangements postulated for these RUNS involve
large axial regions of the core.
In particular, the
axial region affected,considering that identical events
are assumed to occur on each side of the core axial
centerline, is 50 cm, 40 cm and 60 cm for RUNS 224, 225
and 403 respectively.
The axial motion postulated is
assumed to occur over the entire radial region of
interest (25% or 50%) simultaneously.
For the perhaps more realistic axial fuel motions
assumed in the remaining RUNS, the maximum reactivity
addition rate is seen to be about 55 $/sec occurring
after 8 msec in RUN 420 or 54 $/sec occurring after 5
msec in RUN 422.
If the postulated region of fuel
motion is further limited to 25% of the radial area of
the core, these maximums are seen to be 36 $/sec and
152
34 $/sec respectively.
(d)
The time intervals required to produce high
rates of reactivity additions are on the same order as
those required to produce high rates of sodium voiding,
as determined in Chapter V (see Table 5-1).
Thus, the
events following clad rupture cannot produce both the
fuel injection mechanism considered here and the sodium
voiding process studied in Chapter V.
Specifically, if
energy exchange between the fuel and sodium in region
is sufficiently rapid to produce high sodium vapor
pressures and rapid sodium voiding; the present process
cannot occur.
Note that the sodium vapor pressures in
Table 5-1 are generally higher than the 600 psia maximum
fuel vapor pressure employed in the present analysis.
Conversely, if the fuel/sodium energy exchange rate in
e is a relatively slow process, the present mechanism
can be more significant than sodium voiding.
6.7
Summary
Mechanisms which can lead to fuel motion into higher
worth regions of the core have been discussed and the net
reactivity change induced by a number of rearrangements which
could result from such motion has been determined.
The 'PS' Model for predicting the reactivity change
arising from such core rearrangements or perturbations was
developed and compared with ANISN multigroup computer calculations.
The model employs one group perturbation theory in
-
U
~.-
153
conjunction with recent fast reactor spectral characterization work (45).
It was shown to be quite accurate for
small perturbations involving global regions of the core and
reasonably accurate for large localized perturbations as
severe as total sodium voiding from a region with simultaneous alterations in local fuel densities by as much as 20%.
The model should prove useful in sodium voiding analyses.
Use of the model is quite simple once the parameters in brackets in Eq. (6-12) are evaluated for a particular core.
The importance of the parameters (
) and (AS),
the so-
called density and spectral effects, in ameliorating the reactivity affect of various internal core rearrangements has
been amply demonstrated.
Finally, the reactivity addition rates which can take
place as fuel motion occurs have been roughly estimated for
the fuel-injection process described in Section 6.6.
In
general, the rates obtained are less than those estimated in
Chapter V from sodium voiding which occur after a CATEGORY
II excursion.
If the analysis in Section 6.6 is limited to
the more plausible cases or RUNS considered, the fuel injection process results in much lower reactivity addition rates
than those estimated for the sodium voiding process.
154
Chapter VII
DOPPLER EFFECTS AND MISCELLANEOUS REACTIVITY CONSIDERATIONS
As discussed in Chapter II, Doppler broadening is the
source of the major inherent negative feedback mechanism in
As such, a great deal of attention has been given
an LMFBR.
to this important phenomenon.
(13),
(15),
(18),
(16),
(19),
References (7), (9),
(30),
(10),
(47), (48) and (49)
report the results of a number of the more recent investigations in this area.
Equation (2-1) from Chapter II, which describes the
temperature dependence of the Doppler reactivity insertion,
is repeated here for convenience:
dk
ADOP(
=
)
(2-1)
T
(7-1)
As stated earlier, a value of "n" near unity is generally
employed in LMFBR analyses.
Using this form of the equation,
the importance of a Doppler coefficient, "ADOP", of about
.003 or greater has been well established. (9)(13)(15)(22)
(47)
Doppler feedback of this magnitude or stronger has
been shown to have a pronounced effect in reducing the
energy released as a result of a given accident.
Figure
7-1, taken from reference (22) shows typical examples of
this behavior.
The calculations on which Figure 7-1 is
based assume that Doppler feedback is instantaneous, namely
'55
AVAA~LF*QLrE ((I)cARWCAL
F%)NC1\Or4
or-
i00fPLER.
RV~tKSG:(RnoN.
AS
\.oAV,
rEt~ac-w
A
IFQO^ NC)RtnL
~,p~gfriN4-coN~r~r~s.(-o1 ~22,)
to
40 bse
qj
73
3e
0
0O00z?
156
that Doppler broadening of the absorption resonances of
the appropriate isotopes, and the resulting increase in
neutron capture, occurs just as rapidly as the integrated
neutron flux level (or integrated power level) increases.
Sections 7.1 and 7.2 of the present chapter deal with the
validity of this assumption.
Additional reactivity effects are treated, primarily
qualitatively, in Sections 7.3 through 7.6.
respectively:
The influence of the parameter
These include,
"n" of Eq.
(7-1); the significance of the heat of fusion of UO 2 in
producing a Doppler "dead band".; the possibility of positive Doppler feedback during certain portions of an excurfinally, the reactivity effect induced by core
sion; and,
homogenization which can occur during a severe accident.
7.1
Delays in Doppler Feedback
In Chapter III, Eq. (3-6) was derived from the point
kinetics model to show the time history of reactor power
during a ramp reactivity input with appropriate feedbacks.
When only Doppler feedback is considered, this equation
becomes:
q
=
oexp [yt2 +
6kDOPdt'
L_1
,
(7-2)
- 1
2U
I - . -
--
--
-
-
-
--
-
157
where
7=
a=
reactivity ramp rate,
A-= neutron generation time.
Inserting Eq. (2-1) into Eq. (7-1) with n=l, one obtains:
q(t) =oe
[
APf
A
2
ln(T
))dt'
(7-3)
An equation for q(t) as a function of time only can be
readily obtained by assuming a constant specific heat over
the temperature range of interest, so that:
T(t) _
) + T
,
(7-4)
and by noting that:
=
Q(t)
q(t')dt'.
(7-5)
The direct substitution of Eqs.
(7-4) and (7-5) into (7-3)
gives a double integral expression which can be written in
differential form as:
d
dt
2Q.(t)
+ A
ln(_
+ 1) .
(7-6)
cPT0
This equation cannot be solved analytically in closed form.
-
Uw
158
Even the assumption of values of "n" near but not equal to
unity, thus eliminating the logarithmic term from Eqs. (7-3)
and (7-6), will not simplify the problem.
tial Doppler feedback, $,
If the differen-
is written as a constant average
value, however, Eq. (7-6) becomes much more manageable.
This approximation takes the following form for the temperature range of interest:
T
dk
(m)
Ak DOP
DOP
A DOpln(7
=-
T
- T0
f
where
Tf = final or maximum temperature considered
T
= initial
temperature.
Employing a final temperature of 39000K and an initial
temperature of 17000K in Eq. (7-7)
gives an average Doppler
feedback of:
(d) =-
D
= -
1.5 x 10-6 5k/ 0K.
(7-8)
The final temperature employed is the mean fuel temperature
at a location (normally near the core center) where the peak
fuel temperature is sufficient to reach the threshold of
clad failure.
The initial temperature employed is the mean
fuel temperature at the core center during steady state full
power operation.
With ()
=
-
, Eq. (7-3) can be written:
159
ft=
qoexp[7t2
-
b
(7-9)
Q(t')dt'
where
b=
D
max
(q(t))
power
peak
valid
until
solution,
an
approximate
and
is reached, can be obtained.
The result, derived in Appen-
dix A, is:
1
q(t)
1
2
=q exp -7yt
bq
-7-
22
2
1 2
+ (7xt2 )+-.-T-
2
12
( 2t
3
+
+0
(7-10)
(valid only from time zero to the time of peak
power, tp.)
The reader
should be advised that while the series in
brackets can be shown to converge (by the ratio test, for
example), it converges very slowly.
For ramp rates in the
10 $/sec to 200 $/sec range, a minimum of seven terms is
needed for reasonable accuracy (i.e. + 10%).
The result
expressed by Eq. (7-10) can be employed in a straightforward
way in determining the effect of a delay in Doppler feedback,
however, and an extremely useful result obtained.
The time of peak power is determined by requiring
=
0 and solving for tp.
as shown in Appendix A:
From Eq. (7-10) this gives,
160
1Yt 2
(2?tr
ln(
)
)
) + ln(t
1
bq0
(7_11)
,
7-1
2
where
tp = time of peak power.
For all cases of interest, it is found that
>>
(2)
bq0
(1y
2)
P
and therefore an excellent approximation for Eq. (7-11) is:
)
( 7t
=
ln(
) + 1nn( 2)
(7-12)
.
Suppose, now, that the time interval between the production of energy in the reactor and the introduction of reactivity from Doppler feedback can be represented by an "average" or "characteristic" time delay, C'
Then for Eq. (7-9)
.
one can write:
~
t
q t)
=
7t2 - b
The requirement that
(-)
7y(t-'C)2 = ln( 2
2 p
Q(t--'
-oexp
2
bqo
)dt
(7-13)
.
0
-0
) + ln
in this relation leads to:
7
2
L
(t --C')2(l
P
_
J
(7-14)
where tp is the time of peak power when a delay is present.
161
) is ignored for the moment in this
,
t -t
equation, the result is identical to that of Eq. (7-11)
If the quantity (
but with:
tp
=
(t -V
)
(7-15)
is found to be on the order of 10-2 for
Typically ( -5)
t I-'c
delay times Bf interest, and it will be shown that dropping
this term from Eq.(7-14) has a negligible influence on the
result obtained.
Equation (7-15) can thus be used in Eq.
(7-9) to determine the ratio of peak power with a delay to
that without a delay.
qpeak
e
with
delay
This substitution gives:
(7-16)
.
qpeak
without
delay
Presumably, a characteristic feedback delay which allows a
significantly higher peak power is unacceptable; thus from
Eq. (7-16) 7t-C << 1 is required.
This condition can be
rewritten, using Eq. (7-12), as:
1
52
FSqo
<(7-17)
+ lnln(
)y
0
on the quantity
This relationship, dependence of
2(
) is seen to be extremely weak. Recall from
(bq
-
Mu-
162
Chapter III that the power level at prompt criticality, q,
is related to the steady state power level, qss, by:
q
=
(7-18)
2&c a.
qss
(3-4)
Substitution of this expression into (7-17) gives:
1
.707 L2
lc
3
1
3 1
-22
1
2 2
+ lnln(l.77qss p
ln( .77qs_
[ ~
l
s
PD
ss
_&
D
(7-19)
Equation (7-19) then shows the extremely weak influence of
initial power level and Doppler coefficient in determining
whether a given delay is acceptable.
(Doppler, of course,
plays a major role in determining the peak power itself.)
The reactor lifetime and reactivity insertion rate are seen
to be the dominant parameters with respect to the affect of
a given delay time.
For the reactor of Table 1-1 with reactivity ramp
rates of from 20 $/sec to 200 $/sec, the quantity in brackets in Eq. (7-19) was found to vary from 2.06 to 2.68; for
an 'average" value of 2.37.
(7-19) becomes:
With this average value, Eq.
U U w-
163
1
0. 3 (-)
(7-20)
In Eq. (7-16), if the more specific requirement is imposed
that power overshoot with a delay not exceed 120% of the
value without a delay, then 'ytV-'
0.182 is required.
-
p
Equations (7-19) and (7-20) then become, respectively:
.129(L)
3
ln(.77qss P _
L
and
s
1
3
1
) + lnln(l.77qss P -2
2
(
+75
s c~(7-21)
1
'' < .0545
(-)
(7-22)
.
This latter result is almost identical to that given in
recently published work by Kohler (49); in which the limiting relation
cited,
'Z:4 .05(A) 1/2,
was based on the require-
ment that energy release with a delay not exceed 110% of the
energy release without a delay.
Examination of Eqs.
(4) and
(5) of reference (49), however, show that a delay which leads
to a 20% power overshoot (as employed here) is exactly
equivalent to the 10% energy increase employed by Kohler.
The parametric relationship, ()1/2,
given in reference
(49) was obtained by using approximate equivalency relationships for ramp and step induced excursions;
and the coeffici-
ent (.05) was determined by "comparing numerical solutions
164
for step and ramp induced super-prompt-critical excursion
In the present analysis, Eq. (7-21)
in fast reactors."
was obtained by an analytical technique, and thus shows the
influence of all parameters involved.
For reactivity ramp rates in the range of interest,
limiting values of the allowable delay time, C , from Eq.
(7-22) (for a 20% power overshoot or, roughly, for a 10%
increase in energy release) are given in Table 7-1.
The
time of peak power, tp, and the maximum reactivity inserted
before Doppler "turn down" are also tabulated.
Recall from
Chapter II that the time of peak power as employed here is
the time interval between prompt criticality and turn down
of the transient ( - time tp of Figure 2-1).
Note that
't
varies from 13 to 120 microseconds and is less than 2%
of t
in all cases.
I= 11
-
Then the validity of the inequality
is confirmed and Eq. (7-14) can be written
t -c
tp
iR the form:
1Y ( t
- ' ')2
2?bq
= ln ( 2 Y ) + l n
0
1 7( t -
')2
_
Z'
t
p
(7-23)
Comparison of this result with Eq. (7-11) shows that the
real test of the accuracy of the above analysis is the
validityof the inequality:
165
Table 7-1
MAXIMUM ACCEPTABLE DELAYS IN DOPPLER FEEDBACK ('c'mx
max)
(2)
(1)
edmax
cmax(msec)
5kmax
t
t
p(msec)
~7t
(
a $/sec
20
4.40
6.55
.120
.0183
66
16s
4.81
4.oo
.070
.0175
100
23
5.76
3.40
.054
.0159
200
42d
7.00
2.04
.037
.0179
$1.10
9.90
1.40
.013
.0093
l000(3)
NOTES:
1) The time of peak power for an excursion with the maximum
acceptable delay is simply t' = t + 'm
mx
p
p
2) The maximum acceptable delay is defined as that which
allows a maximum increase of 10% in the energy release
calculated with a zero delay (or, equivalently, a 20%
increase in peak power, see text.)
3) The quantity in brackets in Eq. (7-21) is recalculated
for a = 1000 $/sec and found to be 3.06. The 2.37 average value was employed in the 20 $/sec to 200 $/sec range.
166
1
~2~
2
2 4
.
(7-24)
p
From Table 7-1 the left hand side of this equation is seen
to vary from 4.2 x 10~
$/sec.
at 20 $/sec to 9.8 x 10~4 at 1000
Clearly Eq. (7-24) is satisfied in the range of
interest.
7.2
Sources of Delays in Doppler Feedback
In Chapter II it was noted that while the fissile iso-
tope is the primary source of fission energy in an LMFBR,
it contributes a relatively small portion of the Doppler
feedback.
To obtain feedback of the magnitude expressed by
Eq. (7-1) fission energy must be transferred to the fertile
isotope.
Additionally, neutrons from a given generation
must slow to the energy range associated with the resonance
region in order for increased resonance absorption to occur.
Thus, mechanisms can be envisioned which can contribute to
a delay in Doppler feedback.
Those considered in the pres-
ent study include:
(1)
The delay associated with the slowing down
time of fission products.
Energy for Doppler broaden-
ing is not available until fission products of a given
generation slow sufficiently to give up most of their
kinetic energy.
167
(2)
The delay associated with the time required
for fission neutrons to slow from their velocity at
birth to velocities at which capture by Doppler broadened resonances can occur.
(3)
The delay associated with heat transfer from
the fissile to the fertile isotope.
If the fuel con-
stituents are mixed in powder form and the individual
powder particles are larger in diameter than the mean
range of fission products in the fuel ( ~
10 microns),
heat must be transferred to the fertile species primarily by conduction.
Concern for the first two mechanisms can be dispelled
quickly, with the aid of Table 7-1.
The range of fission
products in oxide fuel materials is about 10 microns (9).
From Figure 6.1 of reference (50) and for the 10 micron
fission product range, it is evident that fission products
lose essentially all kinetic energy to the surrounding fuel
material in 10-12 seconds or less.
Clearly, from the
values of 't cited in Table 7-1, this process has no influence in delaying Doppler feedback.
The time required for an average fission neutron to
slow down to the average velocity associated with the U-238
Doppler resonance region has been calculated to be about
2 x 10-6 seconds (49).
While this is substantially longer
168
than the neutron generation time of 3.3 x 10~
seconds, it
is again much shorter than the values of t
in Table 7-1,
even for ramp rates as high as 1000 $/sec.
This this
second mechanism can be ignored in considerations of Doppler
feedback.
With respect to the third mechanism, the thermal relaxation time for an average size particle of the fertile fuel
material can be considered as a measure of the Doppler delay
associated with conductive heat transfer.
This assumes
that the mean size of the fuel particles is substantially
larger than the range of fission products in fuel and, as
mentioned earlier, it ignores prompt neutron and gamma heating and the small fraction of fissions occurring in the
fertile species.
With these assumptions invoked, the
characteristic heat transfer time needed has already been
calculated.
In Chapter IV (Eq. (4-8)) it was found that the
relaxation time for a spherical U02 particle with no heat
sources is given by:
R
2
'= kL(~-)
(7-25)
(4-8)
where R
is the mean particle radius.
For fuel particles
in the size range of interest, this relationship gives the
values of T" listed below:
169
Particle Diameter
(Microns)
'C
(Milliseconds)
10 4t
.005 msec
20
.020
30
.045
4o
.080
50
.125
100
.500
14o
1.00
Comparison of these results with the values of
' max
given in Table 7-1 indicates that fuel powder particle sizes
must be less than about 40 microns in diameter if the maximum credible reactivity insertion rate is taken to be 66
$/sec.
If rates up to 1000 $/sec are credible, then the
fuel particle sizes must be on the order of the range of
fission products in fuel.
These latter results corroborate
a recent study of the heat transfer delay mechanism at
General Electric, Sunnyvale, (48) in which machine calculations were employed to follow the time history of transients
with and without feedback delays.
In the General Electric
study, the fissile/fertile fraction employed was 0.43 (vice
the 0.15 used in the present work) and direct fission
product heating of the UO 2 was assumed to be 14.5% of the
total fission energy generated.
Both differences tend to
170
reduce the effect of a given delay.
Figure 1 of the
General Electric study indicates that particle sizes of
about 50 microns or less are acceptable for ramp rates up
to 100 $/sec.
In addition, the thermal relaxation times calculated
from Eq. (7-25) and listed above corroborate time constants
given in work by Braess et. al. (9).
For example, reference
(9) gives time constants of 0.13 msec and 0.53 msec for 60
and 110 micron particles respectively.
These values are
seen to compare quite favorably with those listed above.
To illustrate the importance of these considerations,
suppose a system is constructed with a mean fuel particle
size of 140 microns (-t'
= 1 msec and an excursion is initi-
ated by an accident ramp rate of 100 $/sec.)
For the LMFBR
described in Table 1-1, Eq. (7-16) gives a peak power 35
times greater than the peak power with no delay in feedback.
7.3
Effect of Spectral Shifts (and the Parameter "n") on
Doppler Feedback
A great deal of attention in the literature has been
given to the calculation of the Doppler coefficient, ADOP'
of Eq. (7-1).
Much less attention seems to have been focused
on accurately determining "n" of Eq. (7-1), particularly as
regards the behavior of n at high temperatures.
Spectral hardening is known to result in an effective
increase in n and thus in weaker Doppler feedback (18)(51).
171
This effect has been estimated to result in a change in n
of from 0.8 to 1.05 for the spectral shift accompanying
total sodium voiding (51).
Some accident analyses have
accounted for the change in Doppler feedback with sodium
voiding by a change in ADOP or a change in To (13).
For
severe excursions, however, a change in n is of overriding
importance.
Table 7-2 shows the total Doppler feedback
available for excursions resulting in maximum temperature
of 6000 0 K and l0,000 0 K, with values of n between 0.8 and
1.2.
The tabulated values of feedback reactivity were cal-
culated using Eq.
(7-1) with ADOP = -
.004 and T 0 = 3000 0 K.
Note that for a maximum temperature of 6000 0 K the total
feedback available varies from - 14.5d for n = 1.2 to - $4.25
for n = 0.8.
An increase in ADOP of 100% is negligible by
comparison.
For example, for the case of n = 1.2, increas-
ing ADOP by a factor of two changes the total reactivity
available to - 29' vice - 14.5'
Thus the arbitrary extrapola-
tion of Doppler behavior determined at low temperatures into
the range of temperatures of interest in severe accident
analyses must be given more attention.
7.4
Doppler Dead Band Due to Heat of Fusion
The heat of fusion of UO 2 was given in Chapter IV as
278 j/gm and the melting temperature as 3070 0 K.
From the
time the fuel at a given location reaches 3070 0 K until melting at that location is complete, no Doppler broadening is
expected to occur (26).
As shown in Appendix A, this dead
172
Table 7-2
VARIATION OF THE MAGNITUDE OF DOPPLER FEEDBACK
WITH THE PARAMETER
"n"
T
max
5k
n
P
dT
T0
Tmax
60000 K
.8
A
- .014
- $4.25)
- 851)
1.0
6000 0 K
-
1.2
6ooo0 K
- .00048
-
14.5 )
-
.026
-
$7.90)
- .0048
-
$1.45)
- .00082
-
25e)
.0028
00K
.8
10,0000 K
1.0
10,0000 K
1.2
A DOP =
-.oo4
T
=0
.0033
T0
=
3000 0K
band is reached at the core hot spot after an energy addition of about 56 j/gm to the fuel.
Equation (4-4) of
reference (8) can be used to calculate the time required to
add the 278 j/gm necessary to heat the fuel at that location
through the Doppler dead band.
For an accident ramp rate of
100 $/sec, a time increment of 0.65 milliseconds is obtained.
173
(This is the time interval between the addition of 56 j/gm
and 278 + 56 = 334 j/gm.)
If this time increment is
thought of as a delay in Doppler feedback for fuel at the
location in question, comparison with the results of Table
7-1 shows 0.65 msec to be an unacceptable delay.
Fortunately,
not all the fuel material in the core is in this dead band
at the same time.
Additionally, the time required to add
278 j/gm depends strongly on the initial temperature of the
fuel in question for a given reactivity insertion rate.
For example, fuel material which is at 2600 0 K (vice the
2900 K employed above for fuel at the hot spot) is heated
through the dead band in about .34 msec for the 100 $/sec
reactivity insertion rate employed in the preceeding calculation.
(While the reason for this behavior is not imme-
diately apparent from a physical standpoint, it is readily
apparent from Eq. (4-4) of reference (8)).
The 278 j/gm "dead band" is roughly equivalent to a
temperature rise of 700 0 K.
Because of the relatively flat
flux profile in a large LMFBR, a significant fraction of
the fuel can be in this dead band during somephase of a
severe accident.
Additionally, as seen in Chapter IV, fuel
dispersal leads to more uniform "whole core" fuel temperatures.
Thus, if an initial excursion produces fuel disper-
sal over a large region of the core, a substantially higher
fraction of the fuel material could be in the dead band
during a secondary excursion.
174
Finally, excursions which start from low power levels,
and consequently more uniform radial temperatures in the
fuel rods, will suffer more from the dead band affect.
A
combination of the latter two effects could conceivably
place the bulk of the fuel material in a large LMFBR in the
dead band at the start of a secondary excursion.
7.5
Positive Doppler Feedback from Fuel Cooling
Recall from Chapters II and IV that the fuel temperature
in regions outside lR remains essentially constant during
the interval between an initial and a secondary transient.
The fuel in 'R, however, will be cooling rapidly, as seen
in Chapter IV.
This cooling introduces a positive reactivity
due to the Doppler effect which may have an important bearing
on the rate of reactivity addition at the start of a secondary
transient and thus on the cumulative severity of the overall
accident.
In Chapter II it was seen that fuel dispersal into
mean fragment sizes of 500 microns
(
--
1/10 intact fuel
pellet diameter) could result in a 700 0K reduction in the
mean fuel temperature in region 'R in about 3.75 msec (a
decrease from a mean temperature of 4400 0 K to about 3700 0 K).
If the importance weighted effect of region -9 is 0.5
( 69
extant over about 30% of the core), Eq. (7-1) shows
that this cooling effect will result in reactivity additions
of + ll
for n = 1.0 and + 7611 for n = .8.
If these
175
reactivity changes are assumed to be introduced linearly
over the 3.75 msec cooling period, the resulting reactivity
addition rates are 33 $/sec and 202 $/sec respectively.
These rates, of course, are substantially increased for
finer fuel dispersion and strongly decreased for a smaller
degree of fuel fragmentation.
7.6
Homogeneity Effects
Recent calculations at MIT (31) indicate that a gain in
reactivity of about 70$' results from the core arrangement
employed in the typical LMFBR considered here, as compared
with a completely homogeneous system.
The gain arises pri-
marily from the increased "first flight" neutron flux within
the fuel rods, although other effects contribute (31).
Thus,
some reactivity effect arises as a result of the homogenization which takes place during fuel dispersal.
If the region
1R. extends over 30% of the core, a maximum negative reactivity insertion of about 351 could accompany complete mixing with sodium in region V.
-
-'
176
7.7
Summary
A limiting expression for the effect of delays in
Doppler feedback has been derived analytically and found to
be in good agreement with a recently published result
based on a more empirical analysis (49).
The expression
obtained shows the relationship of all parameters involved
in determining an acceptable delay time.
Of the possible delay mechanisms considered, only conductive heat transfer times have been found significant.
The effect of this delay mechanism can be eliminated or
minimized by chemical co-precipitation of the fuel materials
or by mechanical processing which insures that fuel particle
sizes are sufficiently small (see Table 7-1 and page
and that the powders are uniformly mixed (52).
Survey calculations have been applied to three additional effects, primarily in an effort to show whether such
effects should be included in analyses of severe reactivity
excursions.
The need for more accurate knowledge of the magnitude
of Doppler feedback, particularly at high temperatures
(possibly by more accurate specification of "n" of Eq. (7-11))
is indicated by the results in Table 7-2.
The influence of the Doppler dead band and of positive
Doppler feedback has been indicated.
These effects should
be included in accident analyses involving a secondary
transient in LMFBR's.
177
Homogenization effects in LMFBR's which can produce
reactivity effects opposite to those encountered in thermal
reactors, should be considered in core meltdown studies.
The influence of a number of the effects considered in
the present chapter depend strongly on the degree of fuel
dispersal or fragmentation during the accident.
The impor-
tance of accurate, and presently unavailable, knowledge in
this area is again indicated.
178
Chapter VIII
SUMMARY AND CONCLUSIONS
8.1
General
The progression of events during a severe excursion in
an LMFBR has been considered in some detail.
The importance
of distinguishing between initial and "secondary" excursions
has been established.
(A "secondary" excursion was defined
in Chapter II as one which occurs after, and possibly as a
result of, substantial cladding failure in the core.)
Nine
phenomena affecting reactivity, primarily during secondary
excursions, have been identified as pertinent and investigated.
A summary of these investigations showing the influ-
ence of the phenomena considered is given in Table 8-1.
The following observations from the results shown in
Table 8-1 are noteworthy.
The letters below correspond to
those in columns (a)-(f) of the table:
(a)
Most of the effects considered can become
significant in a time interval of less than 10 milliseconds.
As seen in Chapter II and Figure 2-1, the
time interval between an initial and secondary transient, in the absence of pre-emptive reactivity effects,
is typically about 10 milliseconds.
Thus, sufficient
time is available for the reactivity effects considered to exert their influence.
179
(b)
Of the effects which can produce rapid reac-
tivity insertions, all but one (item 2) depend on the
degree of fuel fragmentation or dispersal in the region
of clad failure.
Item 2 (fuel injection from intact
clad) depends, as noted, on the type of cladding failure and on frictional effects within intact clad.
Both of the latter topics are areas in which present
knowledge is inadequate.
(c)
The maximum total reactivity change avail-
able from effects 1 through 4 is sufficient to produce
a severe secondary accident if the total available
change is added rapidly enough.
As previously indica-
ted, in the absence of external action an LMFBR remains
near the prompt critical reactivity level after an
initial prompt critical transient.
With the system in
this condition, if more than about 40og
in reactivity
is available, the rate of addition rather than the
total change available is of greater importance.
(d)
At least three and possibly as many as
five of the effects considered can produce reactivity
insertion rates greater than the 66 $/sec "basis"
discussed in Chapter II.
(e)
Of the six effects estimated to have a
significant influence on a severe accident, all
except the Doppler dead band effect (item 5) are of
importance primarily during a secondary transient.
180
(f)
Two effects (items 7 and 8) are concluded
to be insignificant.
Numerous calculations in the
thesis, in fact, hinge on the conclusion that the
energy spectrum is independent of the severity of the
transient.
More specifically, phenomena such as
sodium voiding and fuel motion were found to have a
much stronger influence on the energy spectrum than
the transient itself.
All other Doppler delay mechanisms were found to
be of less importance than the delay associated with
conductive heat transfer; and the latter was found to
be insignificant for properly processed fuel (52).
The importance of quality control in insuring that
this effect is insignificant in mechanically mixed
fuels is clearly indicated in Chapter VII, however.
The need for accurate knowledge of the degree of fuel
fragmentation under severe accident conditions is evident
from Table 8-1 and paragraph (b) above.
This is perhaps
the most compelling conclusion of the present work.
Recall
from Chapter IV that the energy exchange rate between mixed
fuel and sodium was shown to be proportional to the inverse
square of fuel fragment diameter prior to the incidence of
film boiling and roughly proportional to the inverse of
fragment diameter after film boiling is dominant.
For
181
refined analyses, accurate values of the Nusselt Number for
the heat transfer process between fuel and sodium under
accident conditions is needed.
In view of the complex
nature of this heat transfer problem, particularly with
respect to the fact that environmental conditions (including fuel fragment size) can change rapidly during the
course of an accident, it may not be possible to obtain an
acceptable solution analytically.
Experimentation with
actual fuel clusters and sodium coolant in a severe overpower condition may well be mandatory.
The primary argument in favor of the so-called "pancake
geometry" is a reduction in the severity of the sodium voiding problem by promoting increased axial leakage.
This
necessarily increases axial buckling, however, which
increases the severity of the fuel injection effect (item 2).
Thus, at some point, pancaking may actually increase the
susceptibility of a design to severe accident effects.
While experimental work to date indicates that significant
fuel motion within intact clad can take place during power
excursions,
(36)(46) much additional knowledge of the rates
and extent of fuel motion is required before the effect can
be incorporated in accident analyses.
and clad are continuous
So long as fuel rods
(i.e. non-compartmented), as employ-
ed in current designs, and clad rupture pressures are relatively high, this effect must be dealt with, however.
In
-U
182
particular, proposals for high strength clad designs to
allow high fuel burnups should be examined in light of this
phenomena since the reactivity addition rates estimated
for this effect depend almost linearly on the internal clad
failure pressure.
It must be added, however, that the sodium voiding
effect remains as the area of major concern with respect
to LMFBR safety.
The work in Chapters V and VI points
inexorably to this conclusion.
The reactivity addition
rates estimated to result from fuel motion (Table 6-4)
were smaller in general than those estimated to result
from sodium voiding (Table 5-1).
For cases involving more
realistic assumptions with regard to fuel motion, the potential effect is much less severe than is found for sodium
voiding.
spectral
The strong influence of the density ( A)
P
AS
(-s-)
and
behavior in reducing the reactivity effects
of fuel motion within the core is an essential factor in
this observation.
8.2
Analytical Models Developed
Two new analytical models were developed in the course
of the present work which should prove useful in other
applications and in future work in LMFBR accident analysis.
In Chapter VI the
tPS'
model for predicting reactivity
effects in fast reactors is derived.
The model is based on
183
perturbation theory and recent spectral characterization
work (45).
It allows hand calculations of reactivity
effects with a high degree of accuracy for moderate perturbations which extend over substantial portions of the core.
Fairly accurate results are obtained even for strong localized perturbations.
The model is useful in analyzing the various factors
which contribute to a given reactivity change.
For example,
the core constituents or isotopes which provide the greatest
contribution to a spectral shift can be identified.
Furthermore, for a given perturbation, the contribution of
the spectral shift to the overall reactivity charge can be
determined.
The PS model provides a relatively accurate method for
calculating sodium void reactivities.
More noteworthy,
this is the first simple analytical method for calculating
the sodium void effect which is derived directly from basic
principles, and is not merely a curve-fit to multi-group
calculations.
In Chapter VII, an expression was developed for the
maximum Doppler feedback delay acceptable under accident
conditions.
The expression, Eq. (7-19), is a limiting
relation for Doppler delay in terms of well known core
parameters and the accident reactivity ramp rate assumed.
It is not limited to an LMFBR or, for that matter, to fast
184
reactors.
It is, however, limited to situations where Dop-
pler feedback over the temperature range of interest can
be approximated by a constant value for ( 1)
8.3
.
DOP
Previous and Future Work in Accident Analysis
Specific areas for future work have been enumerated in
the preceding two sections.
The present section comments
on past work in fast reactor accident analysis and concludes
with a general recommendation for future work.
Eleven recent studies, which were reviewed extensively
in developing the present work*, have considered severe
transients in LMFBRs.
Numerous earlier studies have consid-
ered severe transients in fast reactors of various types,
some including the LMFBR.
In general the phenomena presen-
ted here and reviewed in Table 8-1 have not been included in
these works (item 7, delay in Doppler feedback, is an exception:
In the present work, this effect is found to be
insignificant.).
A number of the recent works cited have taken "ADOP
of Eq. (7-1) as a variable of primary interest.
Several
have compared the results of using various equations of
state for the fuel material.
The effect of zoning (two or
more regions of different fuel enrichments) has been considered.
Some of the studies have compared the results of
employing different geometries.
*References (7),
(9),
In cylindrical geometry,
(10),(12), (13),
(15-18),
(24) and (47).
185
the effect of various L/D ratios has been investigated.
In addition, the effect of varying enrichments and varying
sodium/fuel volume fractions have been considered.
the exception of the parameter
these considerations
"ADOp",
With
the influence of
is relatively small.
Typically,
for
a given quantity of fuel and a given reactivity ramp rate,
the change in energy release resulting from varying the
other parameters listed above has been less than about 40%.
By contrast, several of the effects considered in the
present work could alter the energy release by a substantial
factor.
A combination of these effects could completely
alter the character of the accident.
Unfortunately, the
difficulty of formulating a precise description of these
effects appears to increase with their possible importance
to the accident sequence.
Thus,
in the past many phenomena
of lesser importance may have been included more because
they are more amenable to analysis than because of any inherent priority
of importance.
In general, no attempt has been made in previous works
to deal with the secondary excursion as defined herein.
Ultimately, the phenomena identified and found significant
in the present work must be considered both singly and in
combination.
A related question which naturally arises with regard
to the effects cited in the present study is whether any
are of comparable significance in thermal reactors.
The
186
next and final section is addressed to this topic.
8.4
Comparison with Thermal (Water Cooled) Reactors
In the course of the present study each phenomena in
Table 8-1 was also considered with respect to its influence in a water cooled (thermal) reactor.
Most of the
phenomena considered are expected to result in an increase
in the severity of an LMFBR accident.
By contrast, all
nine effects are clearly insignificant or actually produce
an opposite (shutdown) effect in water cooled reactors.
In the case of item 1 (increased coolant voiding due
to clad failure) "voiding" due to the formation of steam
or radiolytic gas produces a substantial shutdown effect in
water cooled reactors, primarily due to the corresponding
increase in leakage and increased resonance capture (53).
This difference in thermal and fast systems is being reduced
however,by the use of soluble poisons in PWR's.
The effect of fuel motion in large PWR's and BWR's
(comparable in output to the 1000 MWe LMFBR considered) is
substantially reduced by the much lower fuel "worth" in
such reactors.
The axial buckling is typically so small
that fuel movements of a few centimeters produce a negligible
reactivity change in comparison with such movements in an
LMFBR (item 2, fuel injection from intact clad and item 3,
fuel movement under a general inward pressure gradient).
In reference (53) the rate of energy exchange between
hot fuel and water under accident conditions was found to be
187
inherently limited by the physical properties of water.
Thus, the rate is substantially reduced in comparison with
the present sodium cooled system.
This reduces the effect
of item 4 (positive Doppler feedback due to fuel cooling)
and tends to further reduce the significance of item 3
(fuel movement under a general inward pressure gradient).
When fuel is dispersed or substantially fragmented, a
large increase in the fuel surface to volume ratio takes
place.
In a thermal reactor, this strongly reduces the
self-shielding effect and produces a corresponding increase
in resonance capture.
Thus, this "homogenization" provides
an important shutdown mechanism for thermal reactors (53)
but is a minor effect in an LMFBR (item 9).
In the LWR
reactor, the negative reactivity introduced by this reduction in self-shielding is, in fact, expected to override
the positive Doppler effect which results from fuel cooling
in the region of fuel dispersal (54),
further ameliorating
the effect of item 4 in a thermal reactor.
Item 5 (Doppler Dead Band Effect) is expected to play
a similar role in both reactor types.
Its significance in
a thermal reactor is considerably reduced, however, because
of the less important role of Doppler in such reactors.
As noted earlier, Doppler feedback is the only inherent
negative feedback mechanism of consequence in an LMFBR.
188
The formation of steam or gas voids in a thermal reactor results in spectral hardening, as in the case for sodium
voiding in an LMFBR.
In a thermal system, the spectral shift
is "toward" the resonance region, however, whereas the shift
is "away" from the resonance region in an LMFBR.
Thus,
while sodium voiding results in a decrease in the magnitude
of Doppler feedback in an LMFBR (item 6), the formation of
steam bubbles in a water cooled system actually improves
the Doppler shutdown effectiveness (53) by enhancing absorption in the resonance region.
The problem of delays in Doppler feedback (item 7) is
minimized in thermal reactors by the longer prompt neutron
generation times of such systems. In Eq. (7-26), Vm
max is
1/2
propotionl to
, thus in the typical thermal system,
to a_
proportional
acceptable delays in Doppler feedback are about an order of
magnitude longer.
Additionally, in uranium fueled thermal
reactors any heat transfer delay from the fissile to the
fertile species is eliminated by the atomic scale mixing of
the U2 3 5 and U2 3 8 isotopes.
The effect of item 8 (transient induced spectral shift)
is found to produce slight spectral hardening in both
thermal (54) and fast (Chapter III) reactors.
Although the
overall influence is small in both cases, the effect is the
same as that produced by "voiding" in both systems as discussed above.
The thermal system benefits from the induced
189
spectral shifts, again in contrast to the effect in an LMFBR.
Finally, in view of the results shown in column (c) of
Table 8-1, the fact that the total reactivity change
necessary to produce severe results in a thermal system is
much larger than that required in a fast system, as discussed in Chapter III, must be reiterated.
Rapid addition of
several dollars of reactivity to a thermal system which is
near prompt criticality is required to produce an excursion
of similar severity to that which results from the rapid
addition of about 40
critical level.
to a fast system near the prompt
Recall also from Chapter III that acci-
dents which exceed about 40' in excess reactivity above
prompt critical are extremely severe in an LMFBR.
Note from
column (a) of Table 8-1 that most of the maximum reactivity
estimates given fall between these limits for thermal and
fast systems.
Thus, from this standpoint, the phenomena
considered are of more crucial importancein an LMFBR than
in thermal systems.
Table 8-1
SUMMARY OF REACTIVITY EFFECTS CONSIDERED
Effect
Brief Description
Pertinent
Chapter
a) When
Significant
b) Principal Parameter
on Which Dependent(2)
1
Increased sodium voiding
rate due to clad failure
V
1-10 msec after
initial transient
Degree of fuel fragmentation
2
Fuel injection inZ from
intact clad rods
VI
2-8 msec after
initial transient
Type of clad failure;
frictional effects w/in
clad
3
Fuel movement under general inward pressure
gradient
VI
2-6 msec after
start of secondary transient
Time interval between
initial and secondary
transients
4
Positive Doppler feedback
due to fuel cooling
VII
1-10msec after
initial transient
"n" of Eq. (7-1)
5
Doppler dead band
VII
During initial
Fraction of core fuel
secondary transiin the
700 0 K dead
ent. Likely more
band at a given time
important during
secondary transient
6
Doppler reduction at high
temps. and as result of'
sodium voiding
VII
During secondary
transient
"n" of Eq.
7
Delay in Doppler feedback
VII
any transient
fuel grain or powder
particle size
8
Transient induced spectral
shift
III
insignificant
Homogenization effect
VII
9
(7-1)
0
During secondary
transient
Size of region of
fuel dispersal,
Table 8-1
(Continued)
c) Max. Estimated
6k Change
d) Max. Estimated
dk/dt
e)
Principally
Secondary
Transient Effect
1
1$
>-66 $/sec
Yes
Major
2
50 -2$
66 $/sec
Yes
Possibly substantial
3
500-2$
Yes
Not obtained
Yes
Possibly substantial
Effect
4
501 but strongly
dependent on "n
not ascertained
'-
66 $/sec
5
N.A.
N.A.
6
N.A.
N.A.
7
N.A.
N.A.
8
negligible
N.A.
9
(-) 30'
Unknown but
possibly quite
high
f) Estimated Overall
Significance
Substantial
Likely More
Important During
Secondary Transient
Yes
To Slight Extent
Substantial
Insignificant if fuel
co-precipitated minor
if fuel properly processed mechanically
No
Insignificant
Yes
Possibly substantial
NOTES: 1) N.A. implies Not Applicable
2) Effects (1) through (4) depend strongly on the size of the region of clad
failure., R, as well as the parameters listed.
H
.-
_400NOW4 .
..............
192
REFERENCES
1.
WASH-illO, U.S. Atomic Energy Commission Liquid Metal
Fast Breeder Reactor Program Plan, August 196d.
2.
General Electric Co
Liquid Metal Fast Breeder Reactor
Design Study, GEAP-4 4 18, January 19b4
2A*.
3.
3A*.
4.
General Electric Co., Summary Description of 1000 MWe
LMFBR, G.E., Sunnyvale, November 1968.
Westinghouse Electric Corp., Liquid Metal Fast Breeder
Reactor Design Study, USAEC Report WCAP-3251-1, January
1964.
Westinghouse Electric Corp., Summary Description of
Conceptual Plant Design, November 9, 19b6.
Combustion Engineering, Inc., Liquid Metal Fast Breeder
Reactor Design Study, USAEC Report CEND-200, January 1964.
4A*.
Combustion Engineering, Inc., 1000 MWe Liquid Metal Fast
Breeder Reactor Follow-On Study, November 19b6.
5.
Atomics International, Liquid Metal Fast Breeder Reactor
Design Study, January 1964.
5A*.
Atomics International, 1000 MWe LMFBR Follow-On Study,
November 1968.
6.
H.A. Bethe and J.H. Tait, "An Estimate of the Order of
Magnitude of the Explosion When the Core of a Fast
Reactor Collapses", British Report UKAEA-RHM (56)/113
(1956).
*References (2A) through (4A) were presented at the Int.
Conference on Sodium Technology and Large Fast Reactor
Design, Argonne National Laboratory, ANL-7520, November
1968.
193
7.
Nicholson, R.B., Methods for Determining the Energy
Release in Hypothetical Fast Reactor Meltdown Accidents,
NSE 18, 207-219 (1964).
8. McCarthy, W.J. and Okrent, D., The Technology of Nuclear
Reactor Safety, Vol. I, Ch. 10, 1964, Ed: Thompson and
Beckerley.
9.
Braess, D. et. al., Improvements in Second Excursion
Calculations, Proc. of Inter. Conf. on Safety of Fast
Reactors, Aie-en-Province, Article 111-2, September 1967.
10.
Sha, W.T. and Nicholson, R.B., Maximum Accident of Zoned
Fast Reactors, ANL-7410, pp. 28b-2b9, January 19b9.
11.
Okrent, D. et. al., AX-1, A Computing Program for
Coupled Neutronic-Hydronamics Calculations, USAEC Report
ANL-5977,
(1959).
12.
McFarlane, D.R. et. al., Theoretical Studies of the
Response of Fast Reactors During Sodium Boiling Accidents, ANL-7310, pp. 310-317, January 1966.
13.
Renard, A. and Stievernart, M., Evaluation of the Energy
Release in Case of a Severe Accident for a Fast Reactor
with Important Feedbacks, Proc. of Int. Conf. on the
Safety of Fast Reactors, Aix-en-Province, pp. III-1-1 to
11, September 1967.
14.
Carter, J.C., The Phenomenology of Fast Reactor Accidents,
ANL-7410, pp. 290-292, January 19b9.
15.
Oyama, A., et. al., Analysis of Fast Reactor Core Meltdown Accidents, Univ. of Tokyo, Proc. of Int. Conf. on
Safety of Fast Reactors, Aix-en-Province, pp. 111-4-1
to 15, September 1967.
16.
Hicks, E.P. and Menzies, D.C., Theoretical Studies on
the Fast Reactor Maximum Accident, Dounreay Exp. Reactor
Establishment; Thurso, Caithness, Scotland, ANL-7120,
pp. 654-670, October 1965.
17.
Meyer, R.A. et. al., A Parameter Study of Large Fast
Reactor Meltdown Accidents, General Electric Co., ibid,
pp. 671-bb5.
18.
Storrer, F. et. al., Quasi-Static Model for the Analysis
of Reactivity Accidents in Fast Neutron Reactors, Proc.
of Int. Conf. on Safety of Fast Reactors, Aix-en-Province,
pp. 111-5-1 to 10 (1967).
194
19.
Hafele, VonWolf, Prompt Uberkritische Leistungsekwisionen
in Schnellen Reaktores, Nucleonik, 5, Band 5, Heft, pp.
201-20d (1963).
20.
Swift, D.L. and Baker, L., Experimental Studies of the
High Temperature Interaction of Fuel and Cladding
Materials with Liquid Sodium, ANL-7120, p. b39, October
1965.
21.
Sanathanan, C.K. and Carter, J.C., Phenomena Leading to
Fuel Casing Rupture During Transients, ANL-7410, pp.
293-304, January 1969.
22.
Okrent, D., Design and Safety in Large Fast Power Reactors, review article prepared for publication in Atomic
Energy Review.
To be published.
23.
Aronstein, R.E. et. al., Summary Description of Reference
Plant, Proc. of Conference on Sodium Technology and Large
Fast Reactor Design, ANL-7520, November 1968.
24.
Noyes, R.C. et. al., Parametric Studies and Core Performance, ibid.
25.
Harde, R., Design Considerations and Experimental Program for the Common Development of a 300 MWe Sodium
Cooled Fast Breeder Prototype SNR by a Belgium-Dutch,
German Consortum, ibid.
26.
Schenter, G.E. and Gibbs, A.G., Binding Effects on
Reactivity Change for Transitions from Solid to Liquid
Phase in a Fast Reactor System, BNWL-717, April 19b6.
27.
COO-279, An Evaluation of Four Design Studies of a 1000
MWe Ceramic Fueled Fast Breeder Reactor, Chicago Operations Office, USAEC, December 1964.
28.
Hwang, R.N. and Ott, K., Comparison and Analysis of
Theoretical Doppler Coefficient Results for Fast Reactors, ANL-7269 (19b6).
29.
Nicholson, R.B. and Fischer, E.A., The Doppler Effect
in Fast Reactors, Advances in Nuclear Science and
Technology, Academic Press, New York (1968).
30.
Petersen, R.F., Goldsmith, S., Fast Reactor Safety
Considerations Related to Fuel Macro-Structure, GNWLSA-346, October 1965.
195
31.
Westlake, W.J., Heterogeneous Effects in LMFBR Fuel
Elements, M.S. Thesis, MIT, to be published.
32.
Barghusen, J.J. et. al., Behavior of Zircaloy Clad UO0
Fuel During Nuclear Transients in TREAT, TANS 12-2,
November 19b9.
33.
Meyer, R.A. and Wolfe, B., High Temperature Equation
of State of UO2 , TANS, Vol. 7, No. 1, p. 111, June
19b4.
34.
Etherington, H., Ed. Nuclear Engineering Handbook, p.
11-7 (1958).
35.
Golden, G.H. and Tokar, J.V., Thermophysical Properties of Sodium, ANL-7323, August 1967.
36.
Dickerman, C.E. et. al., First TREAT Loop Experiment
on Oxide Fuel Meltdown, TANS, 12-2, November 1969.
37.
Judd, A.M., Sodium Boiling and Fast Reactor Safety
Analysis, UKAEA, AEEW-R561 (1967).
38.
Hatsopoulos and Keenan, Principles of General Thermodynamics, Wiley and Sons, New York (1964).
39.
Descamps, C. et. al., Analysis of Thermal and Hydraulic
Behavior of Sodium During a Prompt Power Insertion in
a Fuel Assembly, Belgonucleaire, Bruxelles, Proc. of
Inter. Conf. on Safety of Fast Reactors, Aix-enProvince, pp. 1-3-11 to 18, September 1967.
40.
LeGonidec, B. et. al., Experimental Studies on Sodium
Boiling, ibid, Article 1-3.
41.
Rohsenow, W.M. and Choi, H., Heat, Mass, and Momentum
Transfer, Prentice Hall Inc. (1961).
42.
Vance, R.W., Ed. Cryogenic Technology, Wiley and Sons,
New York (1963).
43.
Horst, K.M., Fast Oxide Breeder-Stress Considerations
in Fuel Rod Design, GEAP-3347.
44.
Menzies, D.C., The Equation of State of Uranium Dioxide
at High Temperatures and Pressures, TRG Report 1119(D),
February 1966.
U U
196
45.
Sheaffer, M.K. and Driscoll, M.J., A One Group Method
for Fast Reactor Calculations, MITNE-lO and Sheaffer,
M.K., PhD thesis (MIT) to be published.
46.
Hikido, T., Field, J.H., Molten Fuel Movement in Transient Overpower Tests of Irradiated Oxide Fuel, TANS
12-2, November 1969.
47.
Lord, R.M., Effect of Core Configuration on the
Explosive Yields from Large Sodium Cooled Fast Reactors,
UKAEA, Int. Conf. on Safety of Fast Reactors, Aix-enProvince, Art. 111-3, September 1967.
48.
Bailey, H.S. et. al., Effect of Fast Reactor Fuel
Homogeneity on Transient Behavior, TANS 12-2 (G.E. Sunnyvale), November 1969.
49.
Kohler, W.H., The Effect of Short Delay Times in SuperPrompt-Critical Excursions in Fast Reactors, TANS 12-2,
November 1969.
50.
Evans, R.D., The Atomic Nucleus, McGraw-Hill (1955).
51.
Till, C.E. et. al., Doppler Coefficient Temperature
Dependence and the Effect of Sodium Voiding, TANS
10(1), p. 335, June 1967.
52.
HW-81603 Quarterly Report, October-December 1964,
Hanford Atomic Products Operation, Richland, Washington.
53.
Elbaum, G.J., Rapid Excursions in Water Reactors Involving Fuel Element Rupture, PhD Thesis, MIT, June 1967.
54.
Buckman, F., Severe Reactivity Transients in Boiling
Water Reactors, ScD Thesis, MIT, February 1970.
56.
ANISN/DTF II Conversion to the IBM System/360 performed
by Atomics International, AI-66, Memo 171, undated.
57.
ANL-5800, Reactor Physics Constants, July 1963.
58.
et. al. (ABBN) Group Constants for
Bondarenko, I.I.
Nuclear Reactor Calculations, pub. Consultants Bureau,
New York (1964).
59.
Agrawal, A.K., On the Analysis of Fuel Meltdown Studies
with TREAT, TANS 12-2, November 19b9.
197
60.
Comment based on private communication with Professor
W.M. Rohsenow of MIT, co-author of reference (41).
The two heat transfer correlations (from references
(41) and (42)) employed in Section IV.2 were suggested
by Professor Rohsenow.
61.
Fauske, H.K., Two-Phase Compressibility in Liquid
Metal Systems, ANL, Int. Conf. on the Safety of Fast
Reactors, Aix-en-Province, Art. IVa-l, September 1967.
62.
Horst, K.M., Fast Oxide Breeder - Stress Considerations
in Fuel Rod Design, GEAP-3347, March 19b0.
198
APPENDIX A
1.
Power History Calculations
An approximate solution to Eq. (7-9), valid until the
time peak power is reached, is derived.
Equation (7-9),
repeated here for convenience is:
1
q(t)
=oexp
7t
2
- b
(A-1)
Q(tl)dt'J
where
b=
cp
a = accident ramp rate in 5k/sec
= neutron generation time
D= Doppler feedback defined in Eq. (7-8).
As seen in Table 7-1, the inequality yt 2>>
times of interest.
1 holds for all
For this condition, Eq. (4-2) et. seq.
of reference (8) gives the result for power level growth with
a ramp reactivity input and no feedback.
It is found that a
good approximation for power level behavior with feedback up
until the time of peak power can be obtained from Eq. (A-1)
by employing an expression for power level without feedback
in the integral term of Eq. (A-1).
This is analogous to
approximations made in references (7) and (8).
term of Eq.
(A-1)
thus becomes:
The integral
199
t
0Q(t')dt'
b
O q(t'
= b
)dt
t
b
t'=
g qee p(}
'2 )dt.
The first integral of this result has been evaluated in series
form (8).
A good approximation for times when 17t2
obtained by taking the first term of the series.
t
b
valid for
This gives:
t
(t I ) dt ' ~'--bqo
b fQ
1 is
7-t2>>
1.
A2
1p}7 '2)d
O
The desired expression for the condi-
tion of peak power can now be obtained.
By requiring d
= 0
in Eq. (A-1) one obtains the condition:
d
_
2
r
o
A Fp
-7t
0
2)
(t
,
dt'
=
0
0,
or
7 t2
-exp(yt
)
(A-3)
p
Rearranging and employing t
for the time of peak power, as
done in Chapter VII, Eq. (A-3) becomes:
(l7t)
=
+ ln(7t ) ,
ln
Dq0
identical with Eq. (7-11), as desired.
(A-4)
200
Equation (A-2) can be integrated analytically by substituting the appropriate series for the exponential term (i.e.:
el/27t2 = 1 + 1
(/2Yt 2)
2
and integrating term by term.
+ ... ),
dividing by 7t as shown,
This approach, while not ex-
plicit, gives a usable result as follows:
2
2
1
q(t)
= qoexp
t
(
o 1 2
27 274 + 2.2.
3
+
3.31
)
+
(A-5)
valid for (
2
)> *t1 and only until the time of peak power.
If the power burst is assumed to be symmetric about the time
of peak power, as in references (18) and (19), Eq. (A-5) can
be employed to trace the power history and thus determine
the energy release for one "cycle" of an excursion.
Equation
(A-5) has been graphically integrated for several cases of
interest and found to be in good agreement with the QuasiStatic Model presented in reference (19).
Figure 2-1, in
fact, is a plot of Eq. (A-5) for a reactivity insertion rate
of 66 $/sec and using the averaged parameters for fuel properties given in Table A-1.
In the Quasi-Static Model (19), a
method for calculating the time of the start of a power rise
(the quantity At1 of Figures 2-1 and 2-2) is given.
A
method for calculating the power pulse width is not given,
however, and this is cited as a deficiency in the QuasiStatic model.
From Figure 2-1 or 2-2, clearly if the time
of peak power, tP, is known and At1 is known, the pulse width
201
is determined.
A method for calculating t was C
given in
p
Chapter VII of the present work. On rearrangement of Eq. 14
of reference (19) one obtains:
1
At
=
(A-6)
P)
2ln(c
On rearranging Eq. (7-12) of the present work there results:
1
2
tp=S~)
2c
in _ F)
2 ac
nn(A-7)
Dq 0
D-q
Thus the parametric dependence of tp and At
are identical but
tp is larger due to the factor 2 in the "ln" term and the
additional "ln ln" term.
For the example of Figure 2-1, t
is 4 msec and At 1 is 3.05 msec.
As can be seen from the
figure, At 1 is the time at which the power level reaches the
asymptotic power level, qAS, a quantity representing a mean
of the oscillating level, and which is also the value to
which the power level trends as the oscillations are
damped (19).
In each of the several cases investigated,
good correlation between At1 (calculated from the Quasi-Static
vJA3
o6+Aa,vec1.
Model) and tp (calculated from Eq. (A-7),(see, for example,
Figure 2-1).
This provides further substantiation of the
Doppler delay analysis, Eq. (7-20) et. seq.,
of Chapter VII.
The Quasi-Static Model, with calculations of tp by the
methods of the present analysis, was employed to estimate
the behavior of a number of transients of interest for the
LMFBR of Table 1-1 and, consequently, to form the basis for
202
some of the qualitative reasoning, particularly in Chapters
II and III, of the present work.
The results of calculations
employing this method are presented in Table A-1.
The quan-
tities qAS and 5kmax were computed directly from the appropriate equations in reference (19), namely:
ac
qAS=
and
ac
P+
5k =qf-al?2 ln
max
Dq0
The quantity t
21n
was computed from Eq. (A-7) above.
The
total fuel temperature rise associated with one power burst,
AT.., was calculated from Eq. (13) of reference (19) and from
the observation (Figure 2 of reference (19)) that the
WT
given by Eq. (13) is linearly related to the temperature rise
in the interval required for one power cycle; namely 2tP as
shown in Figure 2-1.
Finally, the time to clad failure is
estimated by employing the asymptotic power level to calculate the time required to reach the threshold of clad failure.
As shown in Chapter III, the peak fuel temperature at
clad failure is about 4850 0 K.
For fuel initially at room
temperature (293 0 K, internal energy taken as zero), the
energy addition required to reach the clad failure threshold
can be estimated as follows:
203
AT = .30(3070-293)
(to melting n c p3(00C
point)
AQ(heat of
=
Ah
j/gm
=
831
=
278 j/gm
-
fusion)
AQ(melting point = cPLAT = .42(4850-3070)
=
747 j/gm
=
1855 i/gm
to clad failure)
AQtotal
(A-8)
As noted in Chapter II, this calculated value is in good
agreement with an experimentally observed value of 1900 j/gm.
For 100% reactor power, the initial fuel temperature in the
center of fuel rods near the core center is 29000K (see Table
1-1).
Thus, the energy addition, starting from this power
level, is 1080 j/gm, as is readily verified from Eq. (A-8).
Similarly, from a steady state power level of 10%, the
required addition is estimated to be 1590 j/gm.
These values,
then, were used in conjunction with the tabulated values of
qAS to estimate the time to clad failure, tF, in Table A-i.
The reader must be cautioned that while the results of Table
A-1 are useful for purposes of discussion and in discerning a
number of important behavioral characteristics of the LMFBR
excursions of interest, the uniform cyclic behavior on which
the results are based cannot be expected to persist for more
than a few cycles.
It is pointed out in reference (19) that
204
even in the absence of pre-emptive effects, each successive
power oscillation will be damped.
The present work, of
course, also discusses a number of potential pre-emptive
effects.
f
Table A-i
APPROXIMATE RESULTS FOR VARIOUS REACTIVITY INSERTION RATES
c' ($/sec)
10 $/sec
66 $/sec
20 $/sec
100 $/sec
200 $/sec
1000 $/sec
full
10
100
24
16.2
13.1
AT (OK)
410 0
720'
qAS (j/gm msec)
6.5
6.5
13
tF (msec)
240
166
122
kmax (g)
At1 (msec)
7s'
s s(power)
(2tP)
msec
10
100
10'
10
100
10
10
100
100
10
100
10
8
8.1
6.8
6.2
4.1
3.1
2.8
13500
9500
19000
13400
28oo0
2500'
75000
60000
13
40
4o
66
66
132
132
660
660
83
39.7
27
24
16.3
12
8.2
2.4
1.64
23U'
165
23,1
48g'
421'
$1.35
$1.10
3.05
3.25
2.90
(2t ) = time interval between successive power pulses as shown in Figure 2-1.
AT
=
q AS=
tF
total fuel temperature rise resulting from one power pulse.
"asymptotic" power level.
= time from prompt criticality until clad failure.
-1. This value is the same for each uniform power pulse for a given initial
= k
5k m
p39er and reactivity insertion rate.
max
R)
0
206
Table A-1
(Continued)
Average properties used in calculating the results tabulated
were:
D = 1.5 x 10-6 5k/sec
= 0.3 joules /gmK
C
=
150 j/gm sec of fuel at 100% power
qss = 15 j/gm sec of fuel at 10% power
'=
3.3 x 10~
sec
= .0033
Notes:
(1)
D is based on the average value given by Eq. (7-7)
between the mean fuel temperature at steady state
and that at clad failure.
Tf
=
T
= 12000 K
47000oK
These values are:
~at 10% power
Tf
= 3900 K
f=
T0 = 17000K
at 100% power
Purely by coincidence Eq.
(7-7)
gives the same D
for both calculations.
(2)
The missing data in the table corresponds to conditions
at which the Quasi-Static Model is not applicable.
207
APPENDIX B
1.
Heat Transfer Correlations Employed
The heat transfer correlations employed for the film
boiling conditions described in Chapter IV were taken from
references (41) and (42).
correlations.*
Both equations are experimental
These are:
1
h fb
=
2 .7
V Kf pq (h fg+ o. 4c
0
D 0 ATf
f
f 0
AT)
AT )
where
(B-1)
(Eq. 9.30 of
reference 41)
V0o
t
2 TJgD
,
and
hf =0.14
fb
p(p0-p.)g cPL
2
-Po
L42
f
h
)(1 + 0.5
p AT0
c AT
h
)
fg
K ,
(B-2)
(Eq. 5.64 of
reference 42)
where the subscripts .9 and f refer to the liquid and vapor
states respectively and the sodium properties used in evaluating hfb in each case were taken from reference (35) at a "mean"
temperature of 1760OR:
*The use of these two equations was recommended by Professor
W.M. Rohsenow (MIT).
- im
208
P1 = 43 lb/ft
3
[f
Pf = 0.05 lb/ft 3
= 0.05 lbm/ft-hr
Kf = 0.04 BTU/
D
= 2R0 = fuel particle diameter
cp
= 0.28 BTU/lbm0 F
4
= 0.34 lbm/ft-hr
AT
= 10000 F
cp
= 0.32 BTU/lbm0 F
K
= 26 BTU/hr-ft0 F
g = 32 ft/sec
= 1600 BTU/lb
h
In Eq. (B-1) a value of V
corresponding to the normal flow
of sodium through the core is assumed ( -'10 m/sec).
The
results obtained for hfb were employed in Chapter IV in the
hfbD
.
relationship m = 2K
Equation (B-1) gives m = 2 for
fuel
D = 250 microns and Eq. (B-2) gives m ~ 12 for D = 25 microns.
This substantiates the conclusion in Chapter IV that m >
1
for D > 250 microns.
2.
Supplemental Sodium Voiding Analysis
Figure 5-1 is a sketch of the core and channel model
employed here and in Chapter V for a hydrodynamic analysis
of the sodium voiding process.
From Figure 5-1, P(t) is
taken as the pressure in the vapor bubble as a function of
time and is assumed to be uniform throughout the vapor
bubble.
In the present analysis, the axial extent of the
vapor bubble never exceeds
z
=
30 cm.
As stated, p 0 is
the pressure at the core exit (top) and is assumed to be
constant.
The reasonable nature of this assumption can be
MMWA
209
justified as follows.
Using the General Electric LMFBR design
as an example (2A), the primary system sodium volume is
45,300 ft 3 , about 23,000 ft 3 of which is in the reactor vessel (see Table 1 and Figure 4 of reference 2A).
The bulk of
the reactor vessel volume in typical current designs is the
region containing a "pool" of sodium above the core and the
cover gas at the top of the vessel (see Figure 5-1).
The
volume of this pool above the core is assumed to be 15,000
ft 3 ( -- 420 m 3 ) for purposes of the present analysis.
From
Table 5-1 the maximum size of the sodium void generated in
the present analysis is about 0.3 m.
The velocity of sound in
liquid sodium at the core exit temperature is 2250 m/sec (35).
Thus, if 3 msec (a time interval typical of the values calculated in Table 5-1) is allowed for generation of the 0.3 m3
vapor bubble, a compression wave originating at the core
exit can affect sodium at a radial distance of 6.7 m, or
essentially all sodium in the "pool" above the core.
The
constant temperature compressibility of liquid sodium at the
core exit temperature is
PT
AV
(35):
2.7 x 10-5 -/atm
(B-3)
For the present analysis, we have AV
hence Eq. (B-3) gives AP = 26 atm.
0.3 m3 and V = 420 m3
Notice that this pressure
is relatively low compared to the pressures generated inside
the core vapor bubble.
Thus, even without access to the
210
cover gas region, compression of the large volume of liquid
sodium at the core exit tends to relieve the pressure buildup in this region.
The present analysis is intended only to
show that the assumption of a low constant pressure at the
core exit (top) is not unreasonable.
To compensate for
errors introduced by this assumption, only upward flow of
sodium from the core was allowed.
Then, from Figure 5-1, with p0 as the core exit pressure, the applicable hydrodynamics equation, in its simplest
form, is:
(t)
in vapor
bubble
APfriction + APinertial + po
(B-4)
.
The appropriate relations for Apfriction and Apinertial are
given in Chapter V and lead immediately to the result:
Bexp(T)
(t))
dz 1.8
d2
d22 + Ap (.64)(V-)
+ p0
dt2
.
(B-5)
- 1)
(B-6)
000
The relation:
Q(t)
=
FP(Tt)-TO) + B( A)exp(- T
was obtained in Chapter V.
)'()
It is found that an approximate
solution to these equations can readily be obtained by hand
calculations according to the following steps.
Q(t) employed are given in Chapter V):
(Values of
211
At t=O, z(t) = Zo,
(1)
hence since Q(0 ) is specified
(in Chapter V) T
is determined from Eq.
(B-6).
(2)
This value of T
d z ( with (-)
(3)
(4)
is used in Eq. (B-5) to obtain
=
0 at t=0).
2
d
d2
Using this value of d
dand
z
are deterdtz
T
dt
d2 2 Z(t) aedtr
mined at time t1 by taking d
as a constant
dt
over the next time step.
Steps (l)-(3) are then repeated until the times
shown in Table 5-1 are reached.
The procedure is simplified considerably by the fact
2
that dz)
does not change appreciably from its initial
dt
value for the time scale of events considered in the present
analysis.
of Tt,
(t)
As a check on the method of solution, the values
z, , and
z
-
\~)-'d
dt2
obtained at the times cited in
Table 5-1 were substituted in Eqs. (B-5) and (B-6) and
corrections made to the iterative procedure, if necessary,
to achieve + 10% accuracy in d
dt
212
APPENDIX C
1.
Spectral Effect Calculations
The method for calculating the spectral parameters
"S" and "R" is given in reference (45).
A sample calculation
for the LMFBR of Table 1-1 is given below.
Equations (6-1)
and (6-2) are repeated here for convenience:
1)
tr
S =V
(C-2)
tr
S
R = 1-S7
(6-2)
Step 1:
(45)
Estimate S and calculate R.
S is shown in reference
to be in the 0.3 to 0.5 range for oxide fueled LMFBR's.
An initial estimate of Sl = 0.4 is chosen.
As will be seen,
the initial choice is not too important.
On simple substitution of the values from Table C-1:
ZR =
1
Ni 5R. = 0.03652
1
With S1 = 0.4,
Vtr
=
:
*
tr
i~ tr
is
(C-3)
given by:
by:
N, (o. 4)
= 0.012042
(C-4)
from Eq. (C-2)
Thus,
R
= 0.2126.
1-1-S1 ( 067)=0 )012042
(C-5)
213
Step 2:
With the estimated S1 and R 1 from Eq. (C-5), calcu-
late Yf :
TZ f=
(vN a 0S1 D) for
+ (vN
R
(C-6)
)
for
U28
f = 0.0067115 is obtained.
Substituting from Table (C-1),
Thus, from Eq. (C-1)
S
0+.067115
.
2 = 0.0067115 + 0.0120T2
Step 3:
estimate.
Repeat steps (1)
=
and (2)
This will give S3
=
0.3578 .
(C-7)
using S2 as the initial
0.354 and R3
=
0.1855.
As
noted in reference (45), only one iteration is required for
reasonable initial estimates of S.
At most, two iterations
are necessary.
The final values of S and R thus obtained are then used
to calculate cross sections for the core of interest.
For
the present LMFBR the values obtained are:
Macroscopic Cross Sections
Element
vzf
a
tr
0.005709
0.00241
0.00854
0.000937
0.00261
0.05400
0
3.7 x 10-5
0.04825
Na
2.4 x l0-5
0.04893
Fe
10.8 x 10-5
0.04814
214
For calculations of S', for use in the quantity
in
the PS Model, the following simplified technique can be
used.
For total sodium voiding, for example, the major
parameter involved in the change is
Ztr
.
The re-
Ztr' , can be obtained by subtracting
quired new value,
as given by Eq. (C-4).
z Na from
The value ob-
tained is:
z tr
Ftr ~
N7trNa = 0.008014
Thus:
S=
=
v2;:+
0.453
(Y
and
AS
S
.099 0.283
o7.3547
AS
.099 = 0.248.
If the iterative procedure given by Steps (1) through
(3) above is repeated, the value of S' calculated is less
than 4% larger.
This is readily seen to be small compared
to the uncertainty generated by the difference in the quantities AS and AS
Recall from Chapter VI that total sodium
voiding is an extremely strong spectral perturbation compared
to the other cases of interest.
For the less severe cases,
215
the error introduced in S' by not repeating the iterative
procedure is even smaller, including those cases where fuel
rearrangement is involved.
Thus, once S is calculated for a given core composiAS
AS
tion,
AS and the quantity
-5-
(or --
for estimating the influ-
S
ence of large perturbations) can be quickly obtained for
perturbations of interest.
-4
Table C-1
CROSS SECTIONS AND SPECTRAL PARAMETERS FOR 'PS' MODEL
(taken from reference 45)
Cross Sections and "g" Values
Number
24
Density(xo~
)
(x102 4 )(cm2)
Absorption
Fission
Transport
Go
g
Low
mean
energy
decrement
Removal
cross
section
R
Element
N(-/cm3 )
PU49
0.0010
1.713
-. 1472
1.813
-0.3067
6.661
-0.2725
0.0083
3.040
U28
0.0065
0.1972
0.7911
0.2147
-o.6249
6.331
-0.280
o.oo84
2.236
0
0.0150
0.002468
o.6728
3.307
-0.0269
0.120
0.1859
Na
0.0127
0.00064
-1.073
3.273
-0.1585
0.0845
0.3875
Fe
0.0127
0.00490
-0.5456
2.829
-0.2844
0.0353
0.6584
g
g
Notes
1)
For Pu 9,
v-= 2.953.
For U
,
y = 2.806.
N)