PROMPT 1BUTROR LIFETIME
:FROM
TIME DEPENDENT THERMAL DIFFUSION EQUATION
Shwe Hlaing
A Thesis Suhmitted to the Faculty of the
DEPARTMENT OF NUCLEAR ENGINEERING
In Partial Fulfillment of the Requirements
For the Degree of
MASTER OF SCIENCE
In the Graduate College
THE UNIVERSITY OF ARIZONA
1962
STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment
of requirements for an advanced degree at The University of
Arizona and is deposited in the University Library to be
made available to borrowers under rules of the Library.
Brief quotations from this thesis are allowable without
special permission, provided that accurate acknowledgment
of source is made. Requests for permission for extended
quotation from or reproduction of this manuscript in whole
or in part may be granted by the head of the major depart­
ment or the Dean of the Graduate College when in their
judgment the proposed use of the material is in the interests
of scholarship. In all other instances, however, permission
must be obtained from the author.
SIGNED;
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
MONTE V. DAVIS
Professor of Nuclear Engineering
Date
ACKNOWLEDGMENT
The author wishes to express his deepest gratitude
to Professor Monte V. Davis for his help and wise counsel
on many points, and for making this thesis possible0
To the Government of the Union of Burma, the author
wishes to express his gratitude for the financial support
in his studies.
Thanks are further given to Mrs, Rhonda D, Goranson
for her help in the preparation of the manuscript, and to
many friends for their assistance.
ABSTRACT
PROMPT NEUTRON LIFETIME
FROM
TIME DEPENDENT THERMAL DIFFUSION EQUATION
by
Shwe Hlaing
Useful formulas for calculating the prompt neutron
lifetime for reflected reactors are derived from equations
that are applicable to the reactor systems where the diffu­
sion coefficients in the core and the reflector are approxi­
mately equalo In addition an equation is developed that
applies to a reactor system with equal diffusion coeffi­
cients and equal thermal neutron diffusion lengths,Finally,
an analysis is given that states the relation between the
albedo of the reflector and the contribution of the reflec­
tor to the neutron lifetime.
The effect of the neutron reflector on the reactor
at steady state is associated with the effective neutron
lifetime. The discussions are given from the point of view
of steady state reactor kinetics and the augmented neutron
lifetime.
Calculated values and the experimental measurements
of the prompt neutron lifetime in TRIGA, TREAT, and ZEEP
reactors are given for comparison.
An experiment was performed using 9 foil activation
technique that gives the contribution of the reflected
neutrons to the total prompt neutron lifetime. The experi­
mentally determined contribution to the prompt neutron life­
time was 4 9 o7 micro-seconds that is in good agreement with
the calculated results.
CONTENTS
Chapter
I
II
III
Page
INTRODUCTION
FORMULATION OF THE EFFECT OF THE
REFLECTOR ON LIFETIME OF NEUTRONS
A®'The Rumsey Method
A.I. Cylindrical reactor with
infinite side reflector
A.2e Spherical reactor with
finite side reflector
B» Reflector Effects on Kinetics
with One-gronp Delayed Neutronss
The Galanin Method
B.lo Kinetics with one=-group
delayed neutrons
B.2o The effect of the reflector
FOIL ACTIVATION ANALYSIS OF NEUTRON
CURRENT AT THE CORE AND REFLECTOR
BOUNDARY
A® Experimental Method
A.I. Calibration
A.2. Activation
A.3. Calculation of albedo
v
1
4
4
11
15
15
15
17
20
20
20
20
24
Chapter
If
T
APPEHDIX Ac
Page
Bo Formulation of the Effect of the
Reflector on leutron Lifetime
from Albedo
Bolo Analysis
Bo2o Calculation of
COMPARISON OF,EXPERIMENTAL RESULTS
WITH THE CALCULATED VALUES
CONCLUSIONS
Analytical Formulation of Prompt
Neutron Lifetime for Cylindrical
reactor with infinite Side Reflector
25
25
29
30
31
33
APPENDIX Be
APPENDIX Co
Tables
I
II
III
Analytical Formulation of Prompt
Neutron Lifetime for Spherical
reactor with finite Reflector
Formulation of % kc>C^L
for the
Galanines method
Activation experiment data
Values of Fm
n
Neutron Lifetime for TRIGA,
TREATsand ZEEP
38
40
23
26
31
Figures
'1
Neutron cycle in critical thermal
reactor-» with the simplified
reflector effect»
Foil activation experiment set-up
REFERENCES
22
42
vi
CHAPTER I
INTRODUCTION
The major purpose of this paper is the study of the
effect of the neutron reflector on the prompt neutron
lifetime in a reactor. This is essentially the kinetics
effect of the reflected neutrons under steady state
conditions.
The effect of a reflector on the geometric buckling
of a reactor, and its effect on increasing the average
power output of a reactor is directly associated with the
neutron savings. In the effect of a reflector on reactor
kinetics, we are concerned not only with the neutron savings,
but also with the time the neutrons spend in the reflector
before returning to the core.
In a thermal reactor, a reflector is generally chosen
to help moderate the neutrons. Consequently, the neutrons
that leak out of the core as thermal, epithermal, and fast
neutrons are reflected back into the core as thermalized
neutrons.
In view of the time delay in the process of diffusing
in and out of the reflector, the kinetics effect of a neutron
reflector can be thought of as an effective increase in
the
neutron lifetime in the core under steady state conditions,
or as an additional delayed neutron precursor under transient
operations.
1.
2
The scope of this investigation will be limited to
the studies concerning the effective increase in lifetime
of the neutrons. In order to formulate the kinetics effect
of a neutron reflector in an analytical formg it is essential
to assume a simplified model of the neutron behavior in the
system. As a result of the approximations the answers are
also approximate. More precise results can be obtained only
from the solution of the transport equation or from multi­
group equations.
The life cycle of the neutrons and the effect of
the neutron reflector is qualitatively discussed in Fig.(1).
The neutrons diffusing through the reflector and returning
to the core are designated by R in the diagram.
LlB z
k e eX
'-f
ssonproc/uc Live,
capfurt o-f
fto.r7ria./ nQicfroni
Concjt /<"o >«
l + L2Bz
T h ir m a f n eatran
le a k a .fi
i+ L2-^2
in p o h o n
/
k P ? r kG
^aatroHi
Ttxlr m a . 1 .
Atu.fv'on Q&SoYty-
fcarYna)!^
ed in fu-Q.L.
ro.eLctor
k
fStutron /talcLjrt
t>a/oucerl %trmat
i(i-a)
A/oryoroc/tx c //v£
C a . p i u rQ. in
and m sonance
fa aI.
17
Th(y"mcxl ntsufro^
CapftKTrdd <*1
/V c u l r o n s t o k ic l\
as capo, ra sorianCQ. Ccy>/«ye
f a i l b e .< x d t* \.j i>
fi's s t o n
V
7 f ( ( ' - r ) P
f
4---
A / e t c / v o zi ^
Cci. pta r a c /
m
resonance,
ref ion
/
/
T h Q rm al
f i s s i o n
1
a>>f s y /
a /guL/ro y?^
F ols T yt « u Iron $
produ-Cdd 1*7
€rt/e y I*ny
r?son«^ce reft on
/s/iu tro n
b ifo m
p/*. _
Ihtyynal
(jenketyQ
resonan
cisston
ce raf'on.
T o ta l
fa s t
neatrOTiS frorn
fisslon
Figure (1)
Neutron Cycle in Critical Thermal Reactor
(With the simplified reflector effects)
CHAPTER II
FORMULATION OF THE EFFECT OF THE REFLECTOR
ON
LIFETIME OF NEUTRONS
A. The Rumsey Method [ l]
The two energy group approximation to describe
the behavior
neutrons
of neutrons is developed, assuming that the
are either fast or thermal. With,
p(?,r)dr = the probability that a fast neutron which
*
is born at r will die at r in dr.
t(r,r)
= the average time between
*
birth at
and
death at r- for fast neutrons.
p(r,r)dr = the probability that a thermal neutron which
^
is born at rf will die at r in dr.
t(r,r)
- the average time between
death at
birth at
and
for thermal neutrons.
Note: The fast neutrons die by being thermalized
or
by being captured (resonance capture). Thermal
neutrons die by being captured.
F(r,t)drdt = the total number of fast neutrons produced
in dr at r , during the time dt at t.
T(r,t)drdt= the total number of thermal neutrons produced
in dr at
p(r)
r ,during the time dt
= the resonance
escape probability
at t.
at r.
Note: p(r) = p for points r in the core; p ( r ) = l for
points ? in the reflector,
k
=
multiplication factor for fission reaction;
k = 0 in the reflector.
4
5
pF(rft)
= k tines the rate of dying of thermal neutrons
at r in the core.
= kJ dr T^r, t-t_(r,r)] p_(r,r)
pF(r,t)
(1)
= c o % plus reflector volume.
T(r,t)
= p /dr pfr, t-t(f,f)l p(f,r)
x
/
'A
I2
f
^ = core volume.
(2)
iJ / 2 i
pF(f,t) - k /df p /dr pfr, t-t(f,r)-t(r,r) 1 ^
(3)
From the nature of the neutron fission process, we
seek a solution of the form:
F( r,t) =■ y
F(r) exp.(v.t)
i
t-
for all positive
i .
(4 )
Except during transients , the most positive w,pre­
dominates; denoting this value of w^ by w, we have
F(r,t) =
F(r) exp(wt) .
(5)
Substituting this in the integral equation (3) gives
pF(r) »
k /dr p /dr F(r^) exp f-wt(r,r)-wt^(r,r) J *
{
J
VT
P(^,r) p(r,r)
/ x '
. (6)
-t 1
Suppose that we put F(r) fast neutrons into an otherwise
empty system. Then we can develop the concept of effective
multiplication factor, k^ , as follows:
pk
e
s
k times the fraction of the total number of
fast neutrons put into the system which are
captured as thermal neutrons in the core.
6
If T(rf) is the distribution of thermal neutrons
arising from the fast neutron source distribution F(r),
then*
p fdr F(r) P(r,r)
T(ij') ”
•
vfe
The number of thermal neutrons captured in the
core is then*
d? J i t ' T(r) p(r,r) » j dr /dr p /dr F(r) p(r,r) p(r,r)
z
f.* ‘
t '
vc
vr
v/
Therefore,
_
k / d f / d r p ( jf.P(f) p(c.r) Ptlf.r)
€
_
(7)
pfdr F(r)
r
■
Integrating (6) over the core volume gives:
Comparing (7) and (8 ) gives:
expf-wt(r,r)-vt(r,r) 1 •
g
L
f * ’
r ‘
-1
Since ^ is of the order of unity, for our case of
intrest* we can approximate
exp I -wt-wt|=l - w ( t 4" t ) .
L
f tj
f t
(9)
7
Substituting (9) in (8) gives:
1 = k - Tv
(10)
where.
far fdr p f d r F(r) p(r,r) p(r,r) t(r,r) t(r,r)
= ;y. 4^ ' Vy, *•
*- b 1 ' U ____t JLJ *_!--- $ r j t i r F(r)
v<
(11)
Equation (10) is similar to the relation obtained
for v by one-group theory, where T is the neutron lifetime.
From (11), ve can calculate the lifetime of neutrons.
8
Evaluation of prompt neutron lifetime from equation (11).
Let,
p(r,r) =
the rate of dying of neutrons at f due to a
unit source at
y(r,r)
=
P(r,r)f(r)
s-
the density at r due to a unit source at r.
t(r) s mean lifetime in the medium at r.
e
1
”
t(r,r)
I *
a
= the average time between birth at r and
death at f .
p(r,r) t(r,r) = N
= the total number of neutrons present
in the system at any instant, which are
destined to die at r.
N = / y(r,r) p(r,r) dr
y
« a.
z
%
vr
From the above relations we have,
= P(r,r) t(r,r)
p(r,r)
^dryir.r) y(r,?)/^(?;
Also from the reciprocity theorem:
/.(*> P(r,r) . y(r,r)= y(r,"f) = -f(r) p(r,r)
For a source distribution S(?), we have
/ dr S(r) p(r,r) t(if,?)=/ d f / d f S(r) y(r,r) y(?,r)/f(r)
y
’
*
,x
Vr
J
J
1
=' V df ^ 3 ^
where,
P ^
and
1
•i
yCr.'rJ/^r)
j * » %
(12)
'r
=
8(r) y(r,r)
vy
/.D 7 p - p - ( 8 •
According to equation (11),
T cr
/ d r / d # S(r) p(#,?) t(r,r)
Lik^J--- !---- L_&---- !— 2J i t f i t S(r) p(r,r)
*
vr
/ dr f i f p ( i ) y ( t , r ) / { ( r )
/dr /dr 3 (r) y(f,r)//(r)
4 "4
From which,
T
h dr q
(13)
where.
p = resonance escape probability
steady state thermal neutron density
qr
i t D’,/v ^ +
it(>
p s steady state fast neutron density .
10
We can also express the neutron lifetime In terms
of the
one-group neutron density as follows.
We shall assume that the fast neutrons slow down
to thermal, with an average slowing down time of j?, giving
rise to a thermal neutron source distribution T(f),
Equation (11) can be written in terms of the thermal
neutron source term, as follows.
since /«t(r,r)
'f i r f t *
; using (12) and assuming
^ d r q(r)
/d?p(r)//(r)
f & r p (?)
\
K W - -
4
\
e - l T(rJ
= i, ~ K ?
T = T + f f including slowing down time
|
$
is constant,
?(f) y(r,r)/^(f)
Jdr q(r)/^(r)
where \
I
(14)
A.I.Cylindrical reactor with infinite side reflector.
A mono—energetic equation is adopted with a minor
correction for the reflector flux. The one-group model
underestimates the reflector flux, but can be corrected
by multiplying the reflector flux by a factor Q. If we
assume the equal diffusion coefficients for the core and
the reflector [ 2 ],
QS
B Lc
where,
k s infinite multiplication factor = 1.2
B = buckling
1^= thermal diffusion length in the core.
The steady state neutron densities for the core
and the reflector for this case are given by:
n
-
pc =
A Cos -mi
I (\r)
C Cos w 2 K, ^ A Kr)
■m = JL
h
=
for the core
for the reflector
the height of the cylinder
Hz
a.
^ = B _ ry, ^
.1
Then,
h*
f * p CV ^
^
Pc
p*
— ^
*
/7A
< D 7 ^ » < ? - - c P
for the core
for the reflector.
12
Using the variation of parameters method, and
assuming the same z-dependence of the neutron densities,
we have £ for derivations see Appendix a J
T, = t
KR.<?
i
I - (“'A,)
(15)
1!
c ez
T = T-h I
where,
mean neutron lifetime in the core
O
Lc =
6=
C
771 =
thermal diffusion length in the core
buckling
e*- rtl*
Vh
&o = radius of the core
%
.
2_
X =
-f- rn
K
IT
1^* thermal diffusion length in the reflector
p
K*
■
rn
Ka ( \ V
the Bessel functions
K, = K (
= I
The first term in (15) represents the neutron
lifetime for bare finite reactor; the second term represents
the contribution from the reflector. Rumsey^z] found, in
analysis of ZEEP experiments, that this contribution
amounts to an increase in the prompt neutron lifetime
of approximately
20#.
13
A.2.Spherical reactor with finite reflector.
As In the previous case, we shall assume the onegroup theory with a corrected neutron density in the
reflector. The diffusion coefficients in the core and the
reflector are assumed to be the same.
The neutron densities for this case are given by,
£
=
p
- Q
-Syq.B.r.
a
for the core
Si-nh K ( R , ~ Y )
for the reflector
6 = buckling
t t 2z _L_
R (= extrapolated outside boundry
Then,
c
c
2
for the core
for the reflector
Using the variation of parameters method, we
have the following formula for the lifetime T:
T
- I
\
| -h L g 2
C
dp 6
H
*
~ KG
G ‘ H<j '-
h '<3
(16)
14
where,
(c= mean neutron lifetime In the core
L -thermal diffusion length in the core
B = buckling
Q =#
^ ‘B ZL
« =T
‘
< ’T t
L = thermal diffusion length in the reflector
i? d sinh r
dH
H' =
dr
r
IF r=R
r=R
e
d sinh (R - r)
dG
G 1 r.
IF
Tr
r
r=Re
r=R»
K 1- d cosh
dr
(R - r)
dK
r=R
a?
r=rR
R c core radius •
o
The first term in (16) represents the finite bare
core neutron lifetime; the second term represents the
contribution of the reflector on the neutron lifetime in
the core.|For derivation see Appendix
B.J
15
Be
Reflector Effects on Kinetics with One-group
Delayed Neutrons: The Galanin Method. £3]
B.l. Kinetics with one-group delayed neutrons.
Let the number of delayed neutrons with period
equal to — be such that the fraction of them in the
A
multiplication factor is / u , that is,
k - ko (1 -y.)
k^ = multiplication factor without delayed neutrons.
The steady state one-group equation can be
written as,
p (r) =
C
0
'C
Z
2
C
M *L
migration area
f = Fermi age
I^= thermal diffusion length in the core.
We now apply a perturbation to the reactor, such
that the multiplication factor changes by «£k ( either
positive, or negative). The new multiplication factor is
then,
krk
Cf.t
k= k (
0
'
Now,
dt
16
where the integral term represents the total
number of neutrons produced per unit time at time t as
a result of neutron emitting nuclei before time t.
with,
p(r, t) - ^(r) exp( tw)
t.-'
(r) +
■Z4 #
M
M z
CO
l
x/oj
w
=
Dc
M*
M
f 1* / A
(t) =o
C.d
1 + VoJ
-I
k0 - * ^
l<o/J
M2
From which, we have
W
For & k
For
(17)
------------ I ;■>/ W
> 0
<0
w-w^>0
inverse runaway time,
w« w
inverse period T,
P
transient effect.
< 0
w = w^ < 0
transient effect.
or
17
B.2.
The effect of the reflector.
The neutron density in the reflector can be given
t>y,
vV
H
=
I f
•
•
Let
(>(r,t) = p (?) exp(vt) •
R
P
ro
Then,
O
■fZ
.
(18)
*
This is the same as the steady state distribution,
with.
L = L
K tjl
^L.
*
(19)
R
j L #1 small perturbation in thermal diffusion length..
From (18) and (19) we have,
,<L
Here we assumed that I
I \
T.
2T « 1.
18
From the steady state one-group diffusion equation
7 p (r)
n(^) =
and with perturbed
0
k = k + J k - ^k* =
; Ha= LZ + t:
k
o ;t
»
x
where, i k fociLp and y/ is the delayed neutron contribution;
we have as in the bare case
kj !+>«) - & k + & k ' - 1
ke- 1
- — RT-f
k6>4
>/w
X^/v
w
- IF
9 (20)
The value of &k* can be obtained from the onegroup equation giving the critical size of areactor with
a reflector. For the case of a spherical reactor, assuming
the same diffusion coefficients in the core and the
reflector we find that |see Appendix C for derivationj
Ak'
Tcv
"
r/l ( J T « ti ^ T r
6 R
K
(21)
• r
where,
R = core radius
o
c<» buckling
2
Ty core mean neutron lifetime =
c
c
k
D = D ( 1-H t t ” )
o
ii
0
effective diffusion coefficient.
19
From (20) and (21) we can arrive at the inverse
runaway time, w, formula exactly the same as in (17), but
with an effective thermal neutron lifetime,viz.
T * T
eU
°
i
1 z
^ _L_/?________________
R/I (
+ 1
(22)
Thus the effect of the reflected neutrons on the
kinetics of the reactor amounts to an effective increase
in the lifetime of the thermal neutrons
.
CHAPTER III
FOIL ACTIVATION ANALYSIS OF NEUTRON CURRENT
AT THE CORE AND REFLECTOR BOUNDARY
A. Experimental Method
A.I. Calibration
iq?
s ygAu
foils were placed around the edge of a plastie cup 9 then activated at the top of the TRIG A reactor
core. By rotating the plastic cup the foils were then exposed
to the same integrated neutron flux. The activities of the
foils were then monitored using a scintillation counter.
This gives the foil activities normalized to aftconstant
flux.
A.2. Activation
After several days decay time the same foils were
then activated
at the core-reflector boundary. To account
for the respective fraction of the thermal and epithermal
neutrons standard cadmium filter
The set-up of the ^ A u 3^
techniques were used.
foils and Gd. is given in Fig. (2).
The activities were then counted in a scintillation
counter. With the data from the calibration run, the
normalized activities »?were calculated and expressed in
terms of a standard foil having the highest activity.
Assuming that the neutrons enter the reflector at
epithermal energies and that the fraction which return to
the core are at thermal energy, the fractional neutron
current in and out of the reflector ,> were then calculated.
20
21
The albedo, ^ , was then calculated from these
fractions. The albedo obtained from the data is 0.820 .
The data given in Table I. are
believed to be reliable,
since large Cd. plates and Cd. rings were, used to protect
the gold foils from any neutron streaming or moderation
in the plastic.
The albedo was also calculated from the one-group
equations for the TRIGA reactor, and a value of 0.76
was obtained.
22
Figure (2)
Foil Activation Experiment Set-up
23
Table I
Calibration of An foils: Data
Foil#
x c/m
x/x,
1
2
3
4
5
66?
614
613
623
600
1.000
0.922
0.920
0.935
0.900
Activation of Au foils:
Data
Foil#
y c/m
y/y(
1
2
3
4
5
2864
1474
2665
2312
1270
1.000
0.516
0.933
0.808
0.444
(y/y()/(x/x)
1.000
0.588
1.010
0.864
0.493
normalized ratio of Au foil activities
Foil#
Normalized ratio
1
2
3
4
5
1.000
0.558
1.010
0.864
0.493
i 0.009
± 0.006
r 0.007
t 0.002
±0.005
24
A.3* Calculation of albedo
Assume that neutrons leave the reactor core at epithermal energies, and that the fraction returning to the
core from the reflector are at thermal energy. This assump­
tion has been based upon the preliminary calculations.
Foil locations
Reflector
Core
#5
#1
#2
#3
#4
w
1
(1
1
| Au.
1/
I Cd.
Foil#
Activity
Contributions
1
2
3
4
5
1.000
0.558
1.010
0.864
0.493
a!
0.009
0.006
0.007
0.002
0.005
+
A*
£
a; +
a;
A‘
t
t
4
AL
t
Epicadmium activity from core
Thermal activity from reflector
K *
<
a;
A1
.
#i
. 1.000
4
- #3
= 1.010
t
* #4 = 0.864
> #5 * 0.493
Average A* -t- A1 » 0.955
e
t
Average A
0.525
#2 = 0.558
e =
= 0..43Q
0.525
0.820
25
B e Formulation of the Effect of the Reflector on
Neutron Lifetime from Albedo J4J
B.l. Analysis
Frankel and Nelson [sjgive the following analysis
in their " Albedo of an Isotropic Surface Source" concept:
Assuming that the scattering medium produce only
isotropic and elastic scattering, the fraction of the
incident neutrons returning across the boundary after any
specified number of collisions can be calculated.
Using the probability of finite sets and mathemati­
cal induction, the fraction of neutrons returning after n
collisions is given by:
Fn = _r
-f"
~o~
io~
••• ■ f r r r r y
(23)
or,
F_ =
n
c2n
Cn_______
22n
(n-*-1)
(2n)i
22n (n-t-1) (n!)2
We shall use this formula to formulate the
reflector effect on neutron lifetime as follows:
Since the effect of the reflector is due to the
time delay in the diffusion process in and out of the
reflector, the time spent by the neutrons in the reflector
until return to the core multiplied by a weighting factor
is essentially the contribution of the reflector to the
total prompt neutron lifetime.
26
Table II
Values of F
___________ n
Collisions n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
Z h
0.2500
0.1250
0.0780
0.0546
0.0397
0.0343
0.0280
0.0232
0.0198
0:0171
0.0150
0.0132
0.0118
0.0106
0.0097
0.0088
0.0078
0.0071
0.0066
0.0061
0.0057
0.0053
0.0050
0.0047
0.0044
0.0041
0.0039
0.0037
0.0035
0.0033
0.0031
0.0030
0.0029
0.0028
0.2500
0.3750
0.4530
0.5076
0.5473
0.5816
0.6090
0.6322
0.6520
0.6691
0.6840
0.6972
0.7090
0.7196
0.7293
0.7381
0.7459
0.7530
0.7596
0.7657
0.7714
0.7767
0.7310
0.7857
0.7901
0.7942
0.7981
0.8018
0.8053
0.8086
0.8117
0.8147
0.8176
0.8204
F,
From the foil activation analysis, at the reflector
and core boundary, the albedo is calculated. This can be
used to calculate the number of collisions in the reflector
until return to the core, from equation (23). That is,
5-1
Fn
=
£
(24)
N = the number of collisions ,
The time the neutrons spent in the reflector until
return to the core is given by:
(25)
As(v) sr scattering mean free path at an average
velocity v-
In order to determine
v
and
A s(v), we shall
assume that the neutron flux leaving the core is inversely
proportional to energy. That is,
Since the reflector acts as a moderator, the
neutron spectrum returning to the core will be at thermal
energy
i.e., Maxwellian about 0.025 electron volts.
Thus, it can be concluded that the average neutron
energy in the reflector can be given by:
E =
0.025 + E 7
(27)
From (24), (25), and (27)
can be determined.
The effective contribution of the reflected neutrons to
the total prompt neutron lifetime, T , can be given by;
T = T l (1 - e8^($
where,
e
(1 - e )|3 = fraction of neutrons leaving the
core at epithermal energies, and
return to the core at thermal
energy.
i
ft
B.2. Calculation of T
Assuming that the reflector has 80# graphite,
10# water, and 10# aluminum,
1(e)
= 0.385. 0.8
Z ( H g O ) , 1.4
• 0.1
£(*1) = 0.084 , 0.1
5
=
0.307
=
0.140
_
0.0084
£. 0.4554
S
^ r 2.2
T*
cm.
2.2x 34
_
109
micro seconds.
6.82x 10^
,2- [t
)‘ -
T=
30 cm2
°-M 6 9 6
;
(1 - e* () = 0.555
I =(109).(0.555)-(0.82)
T r 10 + 91.5 + 49.7 «
=
151.2
49.7
micro sec.
micro sec.
CHAPTER IV
COMPARISON OF EXPERIMENTAL RESULTS
WITH THE CALCULATED VALUES
Calculated values of neutron lifetime for TRIGA,
TREAT, and ZEEP reactors are given in Table III. The
corresponding reflector contributions and the experimen­
tal values are also given.
The agreement between the calculated values and
experimental values are good.Nevertheless, the calcula­
tions have been based upon the assumed values of various
reactor parameters. Better comparison will be obtained
when more accurate reactor parameters are available.
Both spherical approximation and cylindrical forms
are used for TRIGA giving the same results. The reflector
contributes a lifetime of 59=5
sec. The result from
the foil activation experiment is 49 =7 y« sec.
The calculated result of 141 jul sec. for the prompt
neutron lifetime for TRIGA is in good agreement with the
foil activation experiment's result of 151 =2 yy sec.
50
31
Table III
Reactors
TRIGA
Experimental T
150 yu sec. *
Calculated T
141 yti sec.
Reflector T
28$
Reactors
Calculated T
R
Reactors
Experimental T
Calculated T
Reflector T_
*
(heavy water reflector)
86 a sec. **
/(sec.
20$
ZEEP
Experimental T
Reflector T
(graphite reflector)
(graphite reflector)***
860 y,sec. (ii) 880y* sec. (iii)
870
sec.
TREAT
900y<sec,
Proc0 Intal. Conf. on Peaceful Uses of Atomic Energy
10 , 282 (1958)
** V.H.Rumseys Canadian Journal of Physics. J2 9 443(1954)
*** (i) PICPUAE 10, 471(1958); AK1» 6174
(ii) AIL- 6173p p40.
(iii) AIL- 6173p p41.
CHAPTER ¥
CONCLUSIONS
The agreement between the theoretical results and
the exprimental values of neutron lifetime and the contri­
butions of the reflector has been shown to be good*
In view of the fact that diversity of results
exists even in the experimental determinations, the formulas
given are recommended for useable approximate values,
provided that reliable estimate.or experimental values of
the reactor parameters are used9
The Rumsey formulation is quite powerful and can be
extended to multi-group approach. The problem, however, is
to solve the resulting differential equations.
The solutions given in this paper make use of the
Variation of Parameters method, a useful tool for nonhomogeneous differential equations with variable coeffi­
cients.
The standard foil activation technique has been
shown to be very useful.
It is recommended that the foil activation be
extended to reduce the experimental errors and to examine
any systematic errors in the experimental technique.
32
APPENDICES
APPENDIX A
Analytical Formulation of Prompt Neutron Lifetime
for Cylindrical reactor with Infinite Side Reflector
^6
Steady state neutron densities for the core and
the reflector for this case are given by:
=
A
Cos m
2 J -(xr)
c
(1 )
m
= —
x2
Also
Thus,
H
2
2
34
From (2) and (1),
V / -
Cos
f
C
*
C
/(*r)
c
= A CosTrtz? J'fAr).
I
o
t Z
Then (5) becomes,
d_X
J _
Jrz +
dX
r
^ _L ii
<Jr
0<z =
(4)
z
A Cos
The homogeneous equation is then,
/x
+
dvz
’
(5)
L A.
y
dy
Let
d 2x
dx
and
/
Z
(6 )
z3
=
assuming the same
2
d;?
=
— m
z-dependence.
d / 2*
The solutions of (6) are then,
*
“
O1
^
j
i,
1
=
\
=
1
2
*< + -™
^ £
t
=
±
£
+. vn .
=
Coi-m*
(7)
35
'
From (7), (6), and (4) the nonhomogeneous equation in X is
I ri ? .
X
or
d Y 1'
*
r*
2
dx
.
\ y)
a
(8 )
Y dr
+
±
r
d*
dy
-/S^X =
A ^ (xr) m
Z
a o
c
We shall make use of the variation of parameters
method to solve for the general solution of (8).
The variation of parameters method states that;
if X and X are the solutions of the homogeneous
2
1
equation
k>]
= f(r)
then, the general solution of the differential equation
is given by;
X = X
+ X D +
’ J
m
m
\
w(#J
CL
where a,b, D^, and
W(r) =
X
' dr
are arbitrary constants; and
-
X —
z dr
W(r)=fijpr) K ‘yr)
.
K ^ <0> I’^ yj
(9)
36
R + J Az - Z ( ^ )
a
r
=
f s J W
-
(SJ(Xj)K(pr)
(3V
^
AI(pr)^Ar)
Since X is finite at the origin, the coefficients
of the terms containing K should be zero; then we have,
o
X
= A I (pr) _
3 o
_ A _
3"( x r)
/5% A 2
The same procedure can be applied to find q from
%
(2), remembering that q is zero at infinity; that is, I
e
°
terms should not enter.
with,
=. Y ( r) Z (*)
y
z£(?) =
J
Cos Tr? £
f
X W
=
A ^ K a(Kr)
^
=
-L
L
+
r K'(nr)
rv?2 =
A*
*
Using the boundary conditions of p
and
£)
-
at the core and reflector boundary- and extending these
conditions to q
and
q : viz.
c
X{R) =
A
/(#?)
xYe) =
A
/ Y
r
).
v
c
Note that ^
°
and
K
^
are included, since q/^ is
the density of neutrons that are destined to die in°the
core, and q/i^is for that of reflector.
*
0
Finally, it can be shown that.
38
APPENDIX B
Analytical Formulation of Prompt Neutron Lifetime
for Spherical reactor with Finite Reflector
In this case:
X72<7 -
Q = - AK
S m Bf _ ^ Sfngr
c
_z
-
5L
ft ^
_ _
5i»/i n ( R i ~ y )
c-f0
Lr
r
c
Solutions of homogeneous equation are,
r
T
1
«r
{
' - T
- T
I
1
■>
V
V
'
•
j.
f " =
f ' [ R ‘ ^
/ ' s : - 8 f i -1^ ]
^
_
—
/\
o
Sin li
tr
A
-+■
^
Sin&r
*
39
f ( r) ~
X Y.
J^F +
~
rf
y
z.
-X>
« ( R -^J g
'
_ x rr
+ _£
r
-
p
Sinl* n (%, 0
1
v
^
p C o ^li x
4
(R-r)
Finally, it can be shown that - as in the previous
case:
T
H
=
=
I
0
H
L 14-^6"
Zffarr
6'
i
Si'llk
k 'G
*" K 6
HG'-
h'
g
^
y=^o
G
=
Sir?li it ( R , - y )
K
=
Cosh* ( R - r )
R.
The derivatives are taken at r= R
( core radius)
40
APPENDIX C
Formulation of Ak '*c il*? for Galanin'a Approach
7 2
o(2 _
k -1
k - I
_
L % r
^
2=
(L
k — (k-l)^L
(tv t r
For spherical reactor with infinite reflector with
equal diffusion coefficients, we have
t a n o( £
=
6
_ L =
*
^o<
-
^
L
R
tan <
_ ^ r _
2
( I+ tan ■* I?,) — t"en o< (?„
a<1
( n- °( £?)•*- °<L«
°<
41
A
SL,
_
=
s l
<x
(i +
L
__
4^:
=
k
2o< <So( +
/ L a_________
2_UlL»
L1
L«
^
M
=
+ 2
lALsj
fk-l)
I L I M+
k-l «
-St<
-
I
SLr
2 ° < £ a< =
i
<<
£L-z./Lf>
- ^
i t
1,
‘R
I
iiL
M
-
5k1 =
u
h
/
q:
>
M
-
r c co
f
-i-
0
REFERENCES
Y 0HoRiaiaseyg Journal of Applied PhysicsP
gSt 1395-13999 Nov. (1954)
YoH.Rmasisyp Canadian Journal of Physics 9
2£s 435-449p(1954)
A»D0Galanin? Thermal Reactor Theory9
Pergamon Press,(I960)
MoVoDaviSj Personal communication
SoPrankel and 2.Eelson9 AECD-3947
MeoLachlanj, Bessel Functions for Engineersp
Oxford Univ. Press, 2 M.Ed. (1955)