An Analog Computer Model of a Multiple-Region Reactor
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ANL-6482
ANL-6482
argonne Bational Caboratorg
AN ANALOG COMPUTER
MODEL OF A MULTIPLE
REGION REACTOR
by
L. C.
Just, C. N. Kelber,
and
N. F. Morehouse,
Jr.
LEGAL
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of any information, apparatus, method, or process disclosed
in this report may not infringe privately owned rights; or
Makes
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B.
any
damages
resulting
method,
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the
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of any information,
process
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disclosed in this report.
from the use
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apparatus,
As used in the above, "person acting on behalf of the Commission"
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Available from the Office of Technical
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$1.00
Department of Commerce,
Washington 25, D. C.
Services,
ANL-6482
Mathematics
and Computers
(TID-4500, 17th Ed.)
AEC Research and
Development Report
ARGONNE NATIONAL LABORATORY
9700 South Cass Avenue
Argonne,
Illinois
AN ANALOG COMPUTER MODEL OF A
MULTIPLE-REGION REACTOR
by
L. C. Just,* C. N. Kelber,**
and N. F. Morehouse, Jr.*
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*Applied Mathematics Division
**Reactor Engineering Division
February 1962
Operated by The University of Chicago
under
Contract W-31 - 109_eng-38
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TABLE OF CONTENTS
Page
ABSTRACT
I.
INTRODUCTION
II.
THE TIME-DEPENDENT
III.
FLUX EQUATIONS
14
IV.
ANALOG COMPUTER CONSIDERATIONS
17
Flux Equations
Fuel and Poison Equations
Absorption Equation.
Power Equation
Control Poison Simulator
Definition of Machine Variables
Values for Scale Factors
17
A.
B
C.
D.
E
F.
G.
H.
I.
J.
K
V.
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5
VI
7
DIFFUSION EQUATION
7
19
20
20
20
20
20
Scaled Equations
Machine Equations
21
22
Circuit Diagrams
Potentiometer Settings
24
27
ANALOG RESULTS
30
COMPARISON OF ANALOG RESULTS WITH DIGITAL
RESULTS
ACKNOWLEDGMENT
BIBLIOGRAPHY
35
.
36
36
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LIST OF FIGURES
Title
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No.
Page
1.
Three -Region Reactor
2.
Regional
3.
Flux Variation at the Physical Boundary of the Reactor
11
4.
Reactor Cross Section Showing Spatial Division Used
12
5.
Flow Chart
13
6.
Simulator for Fast Flux
24
7.
Simulator for Slow Flux
24
8.
Fuel Burnup and Fission Product Buildup Simulator
25
9.
Iodine -Xenon Simulator
25
10.
Absorption Calculator
25
11.
Control Poison Calculator and Multipliers
26
12.
Fast Flux vs. t
30
13.
Slow Flux vs.
30
14.
Zac vs. t
31
15.
Reactivity vs. t
31
16.
Uranium Concentration vs. t
31
17.
Fission Product Concentration vs. t
32
18.
Xenon Concentration vs. t
32
19-
Xenon -Zac Transient Initiated by A Set-back to lOMw
32
20.
Xenon- Zac Transient Initiated by A Set-back to 20Mw
33
21 .
Xenon- Sac Transient Initiated by A Set-back to 30Mw
33
22.
Xenon-Zac Transient Initiated by A Set-back to 40Mw
33
23.
Xenon- Zac Transient Initiated by A Set-back to 50Mw
34
24.
Xenon -Zac Transient Initiated by 100% Set-back
34
25.
Flux vs. Distance from Center of Reactor at t = 0 and
t
=
Flux Variation
67.5 Days
t
9
9
35
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LIST OF TABLES
No.
Title
1 .
SYMBOLS
5-6
2.
DIMENSIONS
16
3.
REACTOR CONSTANTS
16
4.
COEFFICIENTS FOR FLUX EQUATIONS
18
5.
POTENTIOMETER
SETTINGS
...........
Page
.
.
.
.
.
.
19
AN ANALOG COMPUTER MODEL OF A
MULTIPLE-REGION REACTOR
by
L. C. Just, C. N. Kelber,
and N. F. Morehouse, Jr.
ABSTRACT
This paper presents a technique for solving the time space-dependent diffusion equation for a multiple-region re
actor on an analog computer.
The technique was applied to
a reactor problem involving fuel burnup and poison buildup,
while constant power was maintained by the introduction of
control poison.
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Diffusion across boundaries is computed by putting
the interregion neutron diffusion into the form of a surface
integral. This method allows a good flux plot to be made by
means of only six spatial points.
A sample problem was worked by this technique and
compared to a digital solution.
All computations, circuits,
and operating information necessary to duplicate the exper
iment are given. Also, results are included and compared
with a digital computation.
Table
1
SYMBOLS
B£
axial buckling
D
diffusion coefficient
Df
diffusion coefficient for fast group
Ds
diffusion coefficient for slow group
Dfi
diffusion coefficient for fast group in the itn region
Dsi
diffusion coefficient for slow group in the i"l region
Dfi
Dfi/ri + i - ri
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Table
(Cont'd.)
1
Dsi
Dsi/(ri + i - ri)
*>fi
2
Dfi Dfi+i/(Dfi
Dsi
2
D^i
f
subscript for fast group
FP
fission products
i
subscript for i^h region
J
leakage coefficient (Eq.
n
neutron density
P
power at time
P0
desired power
r
radial distance
s
subscript for slow group
S
neutron sources
Si
surface area between regions i-1 and i
v
neutron velocity
Vi
volume of ith region
-y
fission yield
A.
decay constant
Xt
transport mean free path
11
fission neutrons per fission (average)
+
D^/tDsi
Dii+1)
+
Dsi+i)
17)
t
macroscopic absorption cross section for cladding and moderator
macroscopic absorption cross section for control poison
total macroscopic absorption cross section for the core
aa
microscopic absorption cross section
microscopic fission cross section
T
Fermi age,
0
neutron flux, n/cm2-sec
V
gradient
y2
Laplacian operator
cm2
these are further subscripted in the text to
relate them to a par
ticular element
I. INTRODUCTION
The neutron flux in a reactor may frequently be represented by the
time -dependent solution of the diffusion equation. When it is possible to
characterize the neutron reaction rates in the reactor by their values at
only a few spatial points, an analog computer of reasonable size can be used
to give a detailed history of the neutron flux and reaction rates.
In reactors composed of many regions of differing composition, a
common practice is to compute the flux at many spatial points and for com
paratively few temporal points. Such programs of computation employ a
large digital computer (e.g., the IBM-704) and would require an extremely
large analog machine for their execution.
In the program presented here, spatial resolution is given up in
favor of a detailed flux history. The extreme example of such computations
is the one -point technique. In the one -point technique the spatial dependence
of the flux is completely suppressed, and the average flux and compositions
are used instead.
One -point solutions abound in the literature, but when the reactor has
regions of sharply varying composition and fluxes the one -point solution is
clearly inapplicable. The program presented here applies the one -point ap
proximation in each region; appropriate boundary conditions are used to
estimate neutron diffusion effects, and neutrons are conserved. Thus the
extremely simple one -point method is extended and becomes applicable to
wider range of problems. This extension is brought about with a
relatively small increase in the amount of analog computer equipment used.
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a much
The equations solved are those for fuel depletion, saturating fission
product buildup, poison burnup, xenon and iodine buildup and reactor control
poison. The method here described has been applied to the study of a three region reactor. Six space points were used. Figure 5 is a flow chart for
this problem.
At each space point two fluxes are computed - the fast (0f) and the
slow flux (0s)- The method of calculating the diffusion between regions is
described in Section II. In Section III, the method of computing reaction
rates, conserving neutrons and calculating burnup is given.
Section IV is concerned with details peculiar to the analog computer.
Section V is a presentation of the results of a specific problem, while in
Section VI these results are compared with those of a comparable digital
program.
II. THE TIME -DEPENDENT DIFFUSION EQUATION
Any studies of the time -dependent diffusion equation on an analog
computer of reasonable size will have the following points in common:
8
(l ) The reactor will be divided into a number of regions with pro
vision for a solution in each region;
(2)
the energy
The energy -dependence will be taken into account by dividing
spectrum into n ranges and solving an equation for each energy
group in each region;
(3)
only.
VDV0 must be approximated
as a function
of one space
variable
For a unit volume in a homogeneous region
--
=
VDV0 -Z__a
for one energy group.
0 +S
Since
0-Z0+S
(1)
,
0 = nv and D
is constant in each region,
(2)
.
For an arbitrary volume V,
v
l?dV=
or
fff DV20dV
JJJ
+
(S
-Za0)dV
.
(3)
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But
|//V20dV=//DVn0dS
,
(4}
where Vn 0 is the component of V 0 directed toward the outward normal to
the
surface S.
Since the volume integral of the divergence of the neutron current
can be replaced by the surface integral of the current (i.e. Vn 0), difficulties
inherent in approximating second derivatives are avoided. This manipulation
will also reduce the amount of analog equipment needed.
The solution of Eq. (3) involves an assumption as to the spatial vari
ation of the flux within each region. The assumption used here is that there
is a linear variation in the flux between the mid-point of a region and the
boundary of that region.
Equation (3) is still a three-dimensional problem, but it may be re
duced to one of one dimension. The reactor under study is cylindrical -with
three annular regions (Fig. 1 ); this means that the flux distribution will be
a function of the radius and the height.
If a cosine variation is assumed in
the axial direction, a correction for axial leakage can be added to the absorp
tion loss in each region. Therefore, the calculation of DVn0 can be done with
respect to one space variable.
A= INTERNAL
REFLECTOR
(H20)
8= CORE
C = EXTERNAL
REFLECTOR (Be)
Fig.
Three-region Reactor
1.
It is now possible to determine the function that will describe the
diffusion between two regions. Two assumptions are made:
(l )
Within
a
region,
a linear
variation in flux exists from the center
to the boundary;
(2)
At an interface,
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(a)
(b)
lim
— o
0(r + c ) = lim 0(r - e )
e — o
lim
DI V0(r + £) = lim D2V0(r-e)
£ — 0
£
£
— 0
Fig.
2.
Regional
Flux Variation
10
In the right semi -region of I,
-
0V,
and
01
(5a)
in the left semi -region of II,
V0(r)
=
-
(5b)
.
- r2
r3
-
According to assumption
(0,
-0,)
2D,— b r2
2(b) and Eqs . (5a) and (5b),
(0, _ 0b)
il=2D2iJr3 - r25i
rl
(6)
,
is the diffusion coefficient in the i ^n region.
-where Di
If (6) is solved for the flux at the boundary,
D
r3
_t
- r2 02
,
+
A
r2
(7)
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'b
r2
- r1
r3
- r2
To simplify these relationships, let
Di
D,'
1
(8)
r i + i - ri
so that
D. + D.
It is now possible to express Dl V0 as
of Eqs. (5a) and (9).
D
0
+
D
0 .
\
a
function of
01
and 02 by use
2 D
(10)
11
Equation (10) is valid in any interior region for half of the region.
The total diffusion L, across the interregional boundaries (for an interior
region) is
L
'2D'
=
|
|
DV n
v
ill
Dn-i
2D' D'
D'
+
(0n- 0n.1)Sn+
DA/
II/
" *
"
)
Dn
(0n- 0n+1)Sn+1
(ID
where Sn is the total surface bounding the left side of region n and
the total surface bounding the right side of region n.
is
Sn+!
At an outer boundary (see Fig. 3), the inward neutron current is
<2>,
X
4
6
b
, .
t d0
(12)
dr
The following substitutions
Dn
=
(1)
3
(2)
Multiply Xt by 1 .0656 (correction from transport theory);
(3)
dr
Xt
;
,
n
^
- r
n'
If n = 6, these assumptions
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can be made in Eq. (12):
and Eq. (12)
yield
4.2624D^
1
+
(13)
4.2624
o
J»
Region n-1
Region n
n
(r)
/
0
^^^.^ t
Q
J_ = 0
rn+i
Fig. 3. Flux Variation at the Physical Boundary of the Reactor
12
The neutron current out of the reactor is
J+ =
"
4
6
(14)
"dr
From previous results, if n = 6,
2.1312D6
J+ =
(15)
The total radial leakage out of the reactor is
L = J+ S7 = J 06 S7
(16)
,
where
J
=
2.1312 D6
+
(17)
4.2624D6
The left boundary of region 1 (see Fig. 4) is a special case. In the
left semiregion a zero flux gradient is assumed, so that diffusion must be
only considered across 82 (see Eq. (19) and (23).
r,=
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H20
C
1
2
Be
Be
5
6
OJRE
34
*
4>
5
C)
l1
I
7. 6
i
14.86 1977 2 3.68
r (cm) —*-
38.68
5 5.6
«
Si=
Fig. 4.
Reactor Cross Section Showing
Spatial Division Used
13
Now that it is possible to calculate DV0 in any region for the geonv
etry in question, equations can be written that will yield 0 as a function of
time. Two energy groups are used so that a subscript s represents slow
and a subscript f represents fast. The division between the two energy
groups is at 0.6Z5 ev.
The reactor shown in Fig. 1 was subdivided into regions as shown
Two flux equations were solved in each of the regions. In each
core region (2, 3, 4), equations were solved for power, fuel depletion, fis
sion products concentration, iodine concentration, xenon concentration,
total absorption cross section and control poison cross section.
in Fig. 4.
The analog computer block diagram is shown in Fig.
5.
BOUNDARY CONDITIONS
POWER
0
^
fn
:
* 4
1
1
I
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3
;
*3
F3
1
CI+Mod— t.
'
M
U
L
I
9
.
ROD
POISON
m
r—*
U3 ^S3
_
!
°c+mS3 £-»
i*
kr .'\ .
Fig. 5.
BOUNDARY CONDITIONS
•
r
U
M
1
-P(C
la I
E
R
Flow Chart
p
14
For this analysis, the core was divided into three shells of equal
volume and the reflector into two shells of equal thickness.
III.
FLUX EQUATIONS
Define
2D' D'
D.1
i
08)
+ D1
i+i
The equations that describe the fast flux are
lSz
(0fi" 0fz) "
/D
+
DfiBz)
vi*fi
;
09)
- 0fl) -jBf2S3 (0f2 - 0f3)
V20f2 +v af U235 V2 0s2
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(The differential equations for 0f3 and 0f4 are
c^f4S5 (0f5 - 0f4)
Df5B|
6
(0f6
V5
- 0f5)
similar to Eq.
>
;
(20)
(20):
-^f5S6 (0f5 -
0f5^
;
(21)
- Jf6S7 0f6
The equations for the slow flux are
1
d0sl
Vl
_77~d7~
f"
=
V^siS2 (0si
-
0s2)
-
Dfl
Vl(t>ai+—
(2ai+DslB|)
(23)
15
dt
vs
V2
+—
T, V2 0f2
0s2
(The differential equations for 0s3 and
vs
S45
4S5 (0 s5
dt
0s4
s4 )
(24)
are
(0 s5 -
-JB
D fs
V,
dt
VA =.<-
similar to Eq. (24):
T,
s6
V5 0f5
s6
Df6
—
V6 0f6
In the equation for
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2 ai,
is introduced.
ai - a au Ui
The quantities
time and flux.
+
aaf
Ui,
0s2,
0s3 and
+
(25)
a ax
0s4,
a.
time -varying absorption term,
i + Dsi B| + 2' ac
FPi, X^ and Z ac are calculated as functions
i Vi 0si) -
(26)
dt
(27)
of
(28)
1= 2
2ac(t) represents the amount of control poison necessary to hold the
reactor critical at a given power level P0. That is, the power level,
Zy^f
U^
v^
0si, is compared continuously
with the reference power, P0, and
integral of the resulting error signal is removed from the initial value
of 2ac, thus maintaining the power at the reference level. This is done via
the conventional feedback loop. The presence of this loop is indicated by
the occurrence of the term 2 ac on the right hand side of Eq. (27).
the
16
Table
2
DIMENSIONS
Sub
script
ri.
Si,
Vi,
(cm)
(cm2)
(cm3)
8,053
26,633
1
0
0
2
7.16
14.86
2249
4668
19-77
23.68
38.68
53.68
6211
3
4
5
6
7
26,708
26,686
146,932
7439
12152
16864
Table
217,618
3
REACTOR CONSTANTS
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Property
Beryllium
Core
H2O
1.1
0.7
39-06
72.0
Df
(cm)
T
(cm2)
Ds
(cm)
0.1377
Ea
(cm"1)
0.01591
Zaclad+mod
(cm-1)
0.0
0.08426
0.0
(cm"2)
0.00226
0.00226
0.00226
/Including^
B|
Reflector)
\Savings /
v
=
P0
=
1.19
30.0
0.1684
0.3456
changes with time
0.00987
2.47
108 watts
Element
Concentration
at t = 0 (a/cm3)
af(cm2)
Uranium
6.407 x 1020
Iodine
Xenon
Fis sion
Products
X
Oa(cm2)
7
3.98 x 10"22
4.72 x 10_22
0.0
0.0
0.0
0.0
1 .2
0.064
0.1033
3600
0.0
0.0
3.0 x 10"18
0.0
0.0
c
? .3
£
x ID"24
X Jlf)-23
U
0.003
cm2
fission
0.0
0.0752
3600
0.0
17
IV. ANALOG COMPUTER CONSIDERATIONS
A.
Flux Equations
The time constants in the flux equations are much smaller than the
time constants in the equations for control, burnup or poison buildup.
Therefore, the solution of the flux equations represents a steady-state
solution to the slower equations. In practice, however, the flux equations
are almost steady state but are modified by some slowly varying coeffi
cients contributed by the other equations.
Consider Eq. (19),
vf
dt
V,
= <
|
\
/Dfl D,
-&f,S, (0,^j - •>fzy
r a
Vv yj0,r >. = Y
B*
0..1\ - i ~
c
-TP- i+ LJ fi
** * \
\ T,
z/ i^fi
J-(T
Then
d0fl
VfY
dt
V,
(29)
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Since vf is a large number, the actual value of vf cannot be used.
Instead a value G is used, where G is as large as possible. This can only
affect the solution in the case of a very rapid transient. For the equations
that follow, coefficients will be found in Table 4 together with the auxiliary
quantities in Table 5.
=
G(-A,«f, + A20f2)
=
G(A30fl - A40f2 + A50f3
A6U20s2)
(31)
=
G(A70f2 - A80f3 + A90f4 + A10U30s3)
(32)
=
G (An0f3 - A120f4 + A130f5
(33)
=
G (A150f4
=
G(+ A180f5 - A190f6)
dt
dt
-
A160f5
(30)
+
+
A170f6)
+
A14U40s4)
(34)
(35)
18
~^L=
G(- B!0s1 + B20s2 + B30f1)
(36)
-***"=
G(B40s1 - B50s2 + B60s3 + B70f2 - Xa20s2)
(37)
dt
-***- = G(B80s2 - B90s3 + B100s4 + Bn0f3 -2a30s3)
dt
d0s4
dt
d0s5
dt
d0s6
_
r
}(B120s3 - B130s4 + B140s5
+
B150f4
{
5(B160s4
- B170s5
+
Bi80s6
+
B190f5)
^(B200s5
- BZ10s6
+
B220f6)
(38)
(39)
-2a40s4)
(40)
(41)
Table4
COEFFICIENTS
FORFLUXEQUATIONS
v-^
BU-T
$5
^
+
S
^"^
S
B18
Ds5Bz
S5
V5
.,,-|
**-%
S
2
S7
.»'^
6
,
V2
*s5^tZa5
S6
'
B10
"
6
66
f6
z
f6
6
+— +OB2
-JD—
+J—
V,
f5V,
T,
'
V *UV^
• J2-
S3
B6
s2
=
*3
V3 s4
Vvofu
A
19
V^
2
-.-^
DfJ
V-4
S4
^"V1
—^
°a"^
s
"
V^V/^V/T/Vz
B?
2
Df5
S5
B5- ^siv;-
S4
\^
S3
S2
V ^SIV^
^4^
B15
f^"
B3
B14
'
D
4
'
A15
"
VCTfu
£)
f3
V
=
'
V^-^
Vk\
s
s
fl
V
*R^
Sj
A7
A6
A14-vafu
'
44
£t^
A5
'
3A
A13
v^*t^-vJ
9A
Generated on 2015-10-13 02:14 GMT / http://hdl.handle.net/2027/mdp.39015078510792
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
= «S —
'
S5 +
B16
S4 ,#
A = JB —
fl V
2
2
V
B12^S3^
B2
-
S2
"'I
+
D
* D B2
S6
/"
S5
8 $2
19
Table5
CONSTANTSNEEDEDFOR TABLE4
Ds4
2
53
°;4
D;3
54
2
DS5 D'
+
D'
Ds4+Ds6
D'
Df6
DS6
D'
s6
+ 4.2624
so
v
V3
'
-?°aV
v
V4
V2
Js6
1
2.1312
D(6
1
+
-'V
a^^fu
D;5
D;4DM
D;5 Di5
D;6Dia
f6
c
C3
c
C4
4.2624
"
D's3+E)u
a2a3ufu
D'
Dsl+Ds2
2
D'
" 20a4
°lu
a2a3
2
2
D|3
+ °b
2
D|4
Di4
+
f5
°S4
JB
51
2
D;2 DI2
Db DJ3
'8f2"Df2
Dsi
2
Dh
+ Dil
•as2-D^DS;3
2
+ D}2
,
9
n
2.1312
D;2
°«
,„
°S5
Df5
,,
°S6
.,
VVr6
B.
r4-r3
,;
°.4
Vr4
j
D
DB
Ds3
M
,
R
„,
Ds2-r3-r2
Vr3
2
D;5
+ Df5
f4
n,
,
°s2
°<2
v
-.3
Dsl"Vri
j
"S5
B
.,
For "Slow" Equations
For "Fast" Equations
C2
,
Dn
DflVri
For "Slow" Equations
Dsl
For "Fast" Equations
Fuel and Poison Equations (i = 2,3,4)
U0 s
=
0
Ii(0)
=
(43)
(44)
0
'si
Xi(0)
=
0
Generated on 2015-10-13 02:14 GMT / http://hdl.handle.net/2027/mdp.39015078510792
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
(42)
(45)
20
Absorption Equation (i = 2,3,4)
=
2ai
°axXei
+
°aFP FPi + 2ac +2a(c+m)
+
DsiBl
(46)
Power Equation
=
P0
108
megawatts
(47)
/PO)
+
I
^fuViUi^8i
=
2
^
\ (-
J\
i-2
dt
1
"fuViUi*.i
+
Z'
=
2ac
IPO)
Control Poison Simulator
+
E.
+
I
D.
aau Ui
/(-
C.
(48)
serves as an initial
Z2
t'
=
a1t
U1
=
a2U
FP1
=
a2FP
0'
=
a3
0
P1
=
a4
P
I1
=
a5I
Xe
=
a5Xe
10"17
=
10"15
a5
3
=
a4
1
10"14
hours
U
=
.6407 x 1021a/cm3
0
=
a3
=
=
1016
P0
=
3.1 x
I
10"19
=
1017
a/cm3
Xe
=
1016
a/cm3
:
=
machine second
:
a2
10,800
:
=
:
a,
:
Values for Scale Factors
:
G.
is unity.
Definition of Machine Variables
1
Generated on 2015-10-13 02:14 GMT / http://hdl.handle.net/2027/mdp.39015078510792
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
F.
Z2
t
0
where Zl is approximately 100 and
after
source and Z2 =
>e.
100
1018
»
64.07
volts
volts
fissions/sec
100
•• 10
volts
volts
* 31 volts
21
H.
Scaled Equations
=
(i
2,3,4)
1
I
, 1
' J- /.
d0f
-> _ /".
/
| A /f\ ,.
"
,,,
«1 V-^-S^f 1
at
—
, I
A CDx
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d0£4
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13
4
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T1
1
u-/-')
IT
az
,'
- r" ( A rf r _ A (f)r
ul V-'i-lbV)f4 A16^I5
','
A14
5
(32')
\
(33')
a2
(34')
rf)-f ^
'1 A
A17^I6/
- G, (A180f5 - A190f6)
1
d0si
,-, /
,-,,1
O J ^ - .D(psi T
dt
,|
,
d0g2
l
dt1
1
1^
^
c
TK,,(T)
JL'
,'
70f2
.1
- B50s2 +
60s3
I
|
T (R.{7)
^1 V-0 12^s3
(36')
T .03^1^
j-f-'R
-D1JV^s4 T
j'
(Ti,
T-,
i
y-
.'
\
(37')
a20s2;
'
I
^7
JL'
— 'v
Zj ^.^u),-.
r*i tefn A
IJ14Vs5 "r J-J15V'l4
\
Ja4V-'s4/
(38')
(39')
- B170s5
B180s6
+
G1(B160s4
B190f5)
(40')
/—
,+.
p>
v-Jl\-D20V;sb
- 4- RL1(fir/)
R
-'-'Zl^sfi T "ZZ^lb/
,
f/i
_
1
Vs
j.1
6
1
—— — =
+
1
r.
,,i
,T-IJ'A
.1
£32(/Jg2
1
d0S4
—
._
(35')
,1
40s1
1
Q<^J_
0.-5.
(41')
(49)
rlt!
QL
dFPi .
dt'
a
a
a
2C*3
afu
i
1
Generated on 2015-10-13 02:14 GMT / http://hdl.handle.net/2027/mdp.39015078510792
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
4-
(31')
J.'
!
'
CPr
A
\
(7)
)
U2Vs2/
_
i
)\
,
,
J_i5
,
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a2
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j
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,
P,
j1
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'
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j 1
mr
A
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C
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dt'
(30')
-rt-|St'fl T ^2^.U/
V-TI V
,
i
,
.d.
s1
(42')
(43')
22
dli
XT,
as^fu
_
dt'
dXei _-
.
dt'
..
ai si
Note:
i
-
p; .
•->
/->
BtC
I.
—
i si
a2
2,3,4
=
aj.
""
*1 */
\
\^v
i si
a5
u
Equations (i
Machine
.
Ui*si +-^- Ii - T^X^ -
a,a2a3
""
=
0
O/
^^
^ Xe^
,
.
(45')
.
i si
a2
(4r)
.
7
6
V
"™"^
\^/
Q/
2,3,4)
=
I
n1
=
(30")
G2(- .8527 0^ + .4291 0f2)
Generated on 2015-10-13 02:26 GMT / http://hdl.handle.net/2027/mdp.39015078510792
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
- .742 0f2 + .3058
0f3
+
f- = G2(.30490f2 - 1.19150f3 + .5801 0f4 + 1.966
\
/
=
dt'
s- =
G2
"
G2(. 04053 0f4 - .19220f5
+
+
.03860f6)
1.966
3
I
/
,
.58050i3 - 1.11020f4 + .22320f5
(31")
1.966-^^J
(32")
,\
-^^ }
(33")
(34")
1
G2(. 02606 0f5 - .20330[6)
(35")
=
G2(- -21940s!
(36")
=
G2(. 0172803! - -06410s2
-=
dt
+
.05716032 + .3967010
+
.04681033 + .28160J2 - 102a20s2
(37")
1
G2(. 04668 032 - .1355033
=
+
.08880s4
-1-
23
.28l60f3
- 10Za30s3)
- .1041034+ .0837035
s3
+
(38")
.28160f4 -
=
G2(.01290s5 - -1540036
-0191036
.0972 0f6)
+
G2(,01520s4 - .1408035
+
=
+
f
f-
(39")
-0972 0f5)
(40")
(41")
^-^i-
0.0086
=
, ..2579
(44")
0si
l{
0si)
Ui
\
dt1
'
'
(Ui
^L= 5.502
+.3099li- .2256X^
20
- 6.48
(45")
-^-^-
2
»
-
=
100
(P'(t) - Pi) dt'
(46")
(47")
+
/
20
/
c^si
i
=
Ea
p-
Ci
^f- +0.050
'
-
Z
's1
+.0600
0
0.944
'
=
V
10Zai0'si
Pi
Generated on 2015-10-13 02:26 GMT / http://hdl.handle.net/2027/mdp.39015078510792
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
- .3099
f-5Zi_20
\
dt1
(42")
20
\
dt'
I
(49')
(P'(t) - P0)
(48")
Generated on 2015-10-13 02:26 GMT / http://hdl.handle.net/2027/mdp.39015078510792
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
24
J. Circuit Diagram
Fig.
6.
K/>f6
©
Fig.
7.
Simulator for Fast Flux
-101
Simulator for Slow Flux
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Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
25
-100
53/20
Fig.
Fig. 98.
— -i
(82)
|—
Iodine-Xenon Simulator
@H
Fuel Burnout and
Fission Product
Buildup Simulator
-1oio
Fig.
10.
Absorption
Calculator
Generated on 2015-10-13 02:26 GMT / http://hdl.handle.net/2027/mdp.39015078510792
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
26
+ 100
+ 100
U2<£s2/20
-200 Iac
OUT PUT
LIMITED
NOT TO GO
POSITIVE
ASWITCH
YSWITCH
Fig.
11.
CLOSED AT t<6
OPENED AT t>e
Control Poison Calculator and Multipliers
27
K.
Potentiometer Settings
Potentiometer
Input
Generated on 2015-10-13 02:27 GMT / http://hdl.handle.net/2027/mdp.39015078510792
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
No.
Math. Value
Value
Setting
1
-0fi
10
AI
.8527
8527
2
0fz
10
A2
.4291
4291
3
0fl
10 A3
.1298
1298
4
-0f2
10
A4
.742
7420
5
0f3
10
A5
.3058
3058
6
U20s2/20
200 A6/a2
1.9662
1966
7
0£z
10
A6
.3049
3049
8
-0f3
10 A8
1.1915
1192
9
0f4
10 A9
.5801
5801
10
U30s3/20
200 A10/a2
1.966
1966
11
0f3
10
Au
.5805
5805
1Z
"0f4
10 A12
1.1102
1110
13
0f5
10 A13
.2232
2232
14
U40s4/20
200 A14/a2
1.996
1966
15
0f4
10 A15
.04053
0405
16
-0fs
10 A16
.1922
1922
17
0f6
10 A17
.0386
0386
18
0f5
10 A18
.02606
0261
19
-0f6
10 A19
.2033
2033
20
-0s1
10 B1
.2194
2194
21
0s2
10 B2
.05716
0572
22
0fi
10 B3
.3967
3967
23
0s i
10
.01728
0173
24
-0S2
10 B5
.0641
0128
25
0s3
10 B6
.04681
0468
26
0f2
10 B7
.2816
0563
27
0s2
10 B8
.04668
0467
28
-0s3
10 B9
.1355
0271
B4
G
10
10
10
10
10
5
5
5
28
Potentiometer
Input
Generated on 2015-10-13 02:27 GMT / http://hdl.handle.net/2027/mdp.39015078510792
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
No.
Math. Value
Value
Setting
29
0s4
10 B10
.0888
0888
30
0f3
10
Bn
.2816
0563
31
0s3
10 B12
.0889
0889
32
-0s4
10 B13
.1041
0208
33
0s5
10 B14
.0837
0837
34
0f4
10 B15
.2816
0563
35
0s4
10 B16
.0152
0152
36
-0S5
10 B17
.1408
0282
37
0s 6
10
B18
.0191
0191
38
0f5
10 B19
.0972
0972
39
0f5
10 B20
.0129
0129
40
-0s6
10 B21
.1540
1540
41
0f6
10 B22
.0972
0972
42
+U20s2/20
20 aau/aia3
.0102
0102
43
U30s3/20
.0102
0102
44
U40s4/20
.0102
0102
45
U20s2/20
.0086
0086
46
U30s3/20
.0086
0086
47
U40s4/20
.0086
0086
48
U20s2/20
49
20afu/aia3
G
5
5
5
5
5.502
5502
10
U30s3/20
5.502
5502
10
50
U40s4/20
5.502
5502
10
51
-I2
52
20a57iafu/a1a2a3
Xj/a,
.3099
3099
-Is
.3099
3099
53
-I4
.3099
3099
54
-I2
.3099
3099
55
-I3
.3099
3099
56
-I4
.3099
3099
57
-U20s2/20
.2579
2579
58
-U30s3/20
.2579
2579
2 0 a "Y
CTf
/a a a
29
Potentiometer
Generated on 2015-10-13 02:27 GMT / http://hdl.handle.net/2027/mdp.39015078510792
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
Math. Value
Input
No.
59
-U40s4/20
60
Xe20s2/2
61
Value
Setting
.2579
2579
G
6.48
6480
10
Xe30s3/20
6.48
6480
10
62
Xe40s4/20
6.48
6480
10
63
X2
64
2aax/aia3
Xx/a,
.2256
2256
X3
.2256
2256
65
X4
.2256
2256
66
U20s2/20
.944
1888
5
67
U3 033/20
.944
1888
5
68
U40s4/20
.944
1888
5
69
X20s2/2
.06
0600
70
X30s3/2
.06
0600
71
X40s4/2
.06
0600
72
FP20s2/20
.05
0500
73
FP30s3/20
.05
0500
74
FP40s4/20
.05
0500
75
U20s2/20
C2
2.12
2120
10
76
U30s3/20
C3
2. 126
2126
10
77
U40s4/20
4
2.124
2124
10
78
-100
200 CTauAz
20aax/a5
200aaFP/a2
79
Limit
80
Limit
3100
31.00
~p
9135
dial reading
9500
dial reading
81
-100
U2(0)
64.07
6407
82
-100
uj(o)
64.07
6407
83
-100
U4(0)
64.07
6407
84
0s2
10(Za(c+m) +DS2B|)
85
86
101 to 122
.8652
1730
5
0s3
.8652
1730
5
0s4
.8652
1730
5
used for initial conditions
when
starting from
t
/
0
30
V.
ANALOG RESULTS
The answers to the test problem discussed in Section IV are given
graphically in this section.
and 13 give the fast and slow flux levels at the points
subscript
1-6. Note that the slow flux at point 1 is shown
indicated
with the fast fluxes in Fig. 12. This comes about because of the much
higher level of flux at this point, which is representative of the internal
thermal column.
Figures
12
by the
4XI015
FAST FLUX
3XI016
2XI015
10
20
Fig.
30
DAYS
12.
40
50
60
Fast Flux vs t
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4XI014
3XI01'
2XI014
IO'4
60
Fig.
by
ac
,
the indicated
are given
13.
Slo\v Flux vs t
control poison,
and
p, the reactivity held down
and 15.
Isotope number densities in the core regions are given for U235,
fission products, and xenon in Figs. 16-18.
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31
CONTROLRODPOISON
INDICATEDLIFETIME 67.5 DAYS
E .050
.029
Fig.
0
5
Fig.
10
15
16.
20
Fig.
14.
25
15.
30
Z
35
vs t
40
45
50
55
Uranium Concentration
60
65
Reactivity vs t
vs t
70
32
60
Fig.
17.
Fission Product Concentration vs t
60
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Fig.
18.
Xenon Concentration vs t
The control poison (2ac) and the xenon number densities in
region 3 following a set back in power after 60 hours of steady operation
are given in Figs. 19-23. The title "Set-Back to 10%" means the reactor
power (PQ) is reduced to 0. 1 P0 (= 10 Mw in the test problem). Figure 24
represents the effect of a complete shutdown (P0 reduced to 0) at 60 hours.
Fig.
19-
Transient Initiated by
Set-Back to 10 Mw
Xenon-2ac
a
33
Fig.
20
Xenon-Zac Tran
sient Initiated by a
Set-Back to 20 Mw
Fig.
21
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Xenon-Zac Tran
sient Initiated by a
Set -Back to 30 Mw
60 61
63
65
Fig.
22
Xenon- Zac Tran
sient Initiated by a
Set-Back to 40 Mw
60 61
63
65
34
.075
o 20
.050
.025
70
HOURS-
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Fig. 23.
80
Xenon-^c Transient Initiated by
a Set-Back to 50 Mw
Fig. 24. Xenon- £ cLC Transient Initiated by
100% Set-Back at 60 hrs. Full
Power Demanded at 70 hrs.
At 70 hours P0 was restored to its initial level. The reactor could not go
critical until approximately 85 hours. The initial use and sharp decline in
2
represents a transient voltage in the problem caused by switching P0
to zero. It is not a real effect.
35
VI.
OF ANALOG AND DIGITAL RESULTS
COMPARISON
The flux levels computed at t = 0 by the analog program are com
pared with those computed by a standard Argonne digital code, RE 122, in
Fig. 25. The flux levels computed after 67.5 days by the analog program
are also shown in Fig. 25.
3.0
1.0
©
DIGITAL ANALOG
0.8
•
A
+
O
4>f;t=o
D
<£s,t=67.5 DAYS
<£s;t=o
06
©
D
0.2
A
D
A A
0.01
Fig. 25.
Flux vs. Distance from Center
of Reactor at t = 0 and
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t
=
67.5 days
The agreement of the flux levels calculated by the two methods is
good, especially in view of the small number of spatial points used in the
analog computer solution and the large flux gradients observed. As would
be expected with so crude a description of the flux shape, the reactivity
calculated by the analog program does not agree with the result of the
digital program.
A detailed examination of the discrepancy reveals that the relatively
crude estimate of the leakage in the analog program causes the reactivity
decrement from leakage to be over-estimated. Since no burnup occurs in
the outer region, the flux gradient here -would be better estimated by the
insertion of another spatial point near the outer boundary.
The test problem is admittedly an extreme case -with large flux
gradients. Problems typical of power reactor designs do not usually ex
hibit such large gradients and the error in the leakage term will then be
much smaller.
36
ACKNOWLEDGMENT
of The
The authors gratefully acknowledge the assistance of G. S. Pawlicki
International Institute of Nuclear Science and Engineering.
BIBLIOGRAPHY
3
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Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
Generated on 2015-10-13 02:28 GMT / http://hdl.handle.net/2027/mdp.39015078510792
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google