Engineering Applications of Analog Computers
Y3.
ANL-6319 Rev
AN L -6 319 ReviEPORis
Do
JAN 14 1963
argonne Bational Xaboratorg
ENGINEERING APPLICATIONS
OF ANALOG COMPUTERS
by
Lawrence T. Bryant, Marion J. Janicke,
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Louis C. Just and Alan
L.
Winiecki
LEGAL
This
work.
person
A.
NOTICE
report was prepared as an account of Government sponsored
Neither the United States, nor the Commission, nor any
acting on behalf of the Commission:
expressed or implied,
any warranty or representation,
or
respect to the accuracy, completeness,
usefulness
of the information contained in this report, or that the use
of any information, apparatus, method, or process disclosed
in this report may not infringe privately owned rights; or
Makes
with
B.
with
respect
to
the
use
any
damages
of any information,
or process disclosed in this report.
method,
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liabilities
Assumes
resulting
from the use
of, or for
apparatus,
As used in the above, "person acting on behalf of the Commission"
includes any employee or contractor of the Commission, or employee
of such contractor, to the extent that such employee or contractor
of the Commission, or employee of such contractor prepares, dis
seminates,
or provides access to, any information pursuant to his
or contract with the Commission, or his employment with
employment
such
contractor.
Price
$1.25
.
Available from the Office of Technical
of Commerce, Washington 25, D. C.
Department
Services,
ANL-6319 Rev.
Mathematics and Computers
(TID-4500, 18th Ed.)
AEC Research and
Development Report
ARGONNE NATIONAL LABORATORY
9700 South Cass Avenue
Argonne,
Illinois
ENGINEERING APPLICATIONS OF
ANALOG COMPUTERS
by
Lawrence T. Bryant,* Marion J. Janicke,**
Louis C. Just,* and Alan L. Winiecki*
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*Applied Mathematics Division
**Reactor Engineering Division
Revised October 1962
(Originally published February 1961)
Operated by The University of Chicago
under
Contract W- 3 1 - 109-eng- 38
with the
U. S.
Atomic Energy Commission
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TABLE OF CONTENTS
Page
INTRODUCTION
I
DECELERATION OF A REACTOR CONTROL ROD
II
PRESSURE
VARIATIONS THROUGH A PACKED BED
13
III
REACTOR KINETICS OVER MANY DECADES WITH THERMAL
FEEDBACK (SIMULATION OF A TREAT TRANSIENT)
18
IV
6
A VIBRATING SYSTEM WITH TWO DEGREES OF
FREEDOM
27
V
TEMPERATURE
DISTRIBUTION IN A RADIATING FIN
32
VI
TEMPERATURE DISTRIBUTION IN AN INFINITE SLAB
CONSIDERING VARIABLE THERMAL PROPERTIES
37
IODINE-XENON BUILDUP IN A REACTOR
49
VII
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5
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LIST OF FIGURES
Title
No.
1.
Circuit Diagram for Solution
Buffer Motion
of Elias'
Page
Equation of
8
2.
Velocity Versus Distance for Various Viscosities
11
3.
Velocity Versus Distance for Various Initial Velocities
11
4.
Distance
5.
Velocity Versus Time for Various Viscosities
12
6.
Circuit Diagram for the Solution of MacFarlane's Equation ...
15
7.
Pressure Versus Distance
17
8.
Circuit Diagram for Duplication of TREAT Transient
23
9-
Kex and
10.
Versus Time for Various Initial Velocities
r\
11
- In nj Versus Time
26
Illustration of a Vibrating System with Two Degrees
of
Freedom
11.
27
Circuit Diagram for the Solution of the Equations Describing
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a Vibrating System with Two Degrees
12.
Distance
of Freedom
29
Versus Time for a System with Two Degrees of
Freedom
31
13.
Geometry of Radiator Fin and Coolant Tubes
32
14.
Circuit Diagram for Solution of Second-order, Fourth-degree
15.
Differential Equation
34
Temperature Versus Length and dT/dx Versus Length for a
0.25-ft Fin (K = 25.0)
36
16.
Model of the Infinite Slab Showing Regions Used for Analysis
17.
Circuit Diagram for an Infinite Slab with Thermal
Conductivity
18.
K
=
Constant
Relay Circuit for the Heat Pulse Used in Experiment VI
.
.
37
40
40
Title
No.
19-
20.
Circuit Diagram for an Infinite Slab with Thermal
Conductivity K = F(Taverage)
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41
Circuit Diagram for an Infinite Slab with Thermal
Conductivity K. = F(Temperature of Region Described
by the Heat Balance)
21.
Page
42
Circuit Diagram for an Infinite Slab with Thermal
Conductivity K - F(Average Temperature Across an
Interface)
42
22.
Temperature Distribution for an Infinite Slab - Case I
47
23.
Temperature Distribution for an Infinite Slab - Case II
47
24.
Temperature Distribution for an Infinite Slab - Case III
48
25.
Temperature Distribution for an Infinite Slab - Case IV
48
ENGINEERING APPLICATIONS OF
ANALOG COMPUTERS
by
Lawrence T. Bryant, Marion J. Janicke,
Louis C. Just, and Alan L. Winiecki
INTRODUCTION
This publication is an extension of Bryant, L. T., Just, L. C., and
Pawlicki, G. S., Introduction to Electronic Analog Computing, ANL-6187
(July I960).
engineering,
Six experiments are presented from the fields of reactor
heat transfer, and dynamics.
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The mathematical representation for most of these experiments is
in the form of nonlinear differential equations. In usual practice sim
plifying assumptions are introduced to linearize the equations. This
linearization may alter the mathematical model sufficiently to cast doubt
upon its applicability. If an analog computer is available, the nonlinear
equation may be solved directly.
The presentation of these experiments has been designed to pro
vide insight into physical phenomena and their mathematical representa
tion. The steps required for producing the analog solution will be shown,
as well as complete information for duplicating the solution.
Graphical
results are provided.
The format of each experiment
1.
2.
will be:
Description of the problem
Mathematical statement of the problem including:
a.
b.
Constants
Initial Conditions
3.
Preparation of machine equations
Machine Variables
a.
Scale Factors
b.
4.
Analog
a.
b.
c.
circuit diagram
Flow Sheet
Potentiometer setting sheet
Static Check sheet
5.
Graphical representation of the solution.
6.
Bibliography
I.
1 .
DECELERATION OF A REACTOR CONTROL ROD
Problem Description
When a control rod is suddenly inserted or rejected from the core
of a reactor, the rapid motion is quickly dampened by a dashpot or buffer
mechanism, usually consisting of a hydraulic system which prevents sudden
shock of the control drive mechanisms.
Constant deceleration-type dashpots give the most favorable charac
Essentially, a piston moves
through oil, and the oil is squeezed into small clearances; this process in
turn develops large amounts of frictional resistance. This friction, which
is proportional to the speed of the moving piston, instigates the retarding
force which slowly stops the motion of the control drive.
teristics for protection against shock loads.
This hydraulic drag and the ensuing kinetic energy dissipation are
frequently described by differential equations. Elias1 equation of buffer motionU"!)* is given by
2 (J.TT DpLd
dX
X
W(LdC - CX)2
Various plots of the buffer characteristics
are shown on Figs.
2, 3, 4, and 5.
Many parameters may be investigated before the design conditions
for a particular problem are satisfied. (1-2,1-3)
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2.
Mathematical Statement of the Problem
a.
Equations:
X
(1)
W(LdC - CX)2
b.
Constants and Variables
Description
Symbol
VP
DP
C
velocity into the dashpot
diameter
of the dashpot
dashpot clearance
Value
Units
Variable
f t/ s e c
2
inches
0.03
inch
6
inches
Variable
lb/(ft)(sec)
Ld
dashpot length
M
viscosity of the dashpot fluid
w
weight of the control rod
290
Ib
X
distance into the dashpot
6
inches
*References in each section are given at the end of each section.
Initial Conditions
c.
At t
X
=
0:
=
0
=
70 ft/sec
'
dt
Analysis of Equations
d.
Since
dVp
dX
the
dVp/dt
dX/dt
_"
_~
d2x/dt2
'
dX/dt
(
'
original equation may be restated as
dx
dt
3.
Preparation of Machine Equations
Machine Variables and Scale Factors
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a.
X'
=
bX
;
a
=
1 03
t'
=
at
;
b
=
10
b.
(4)
Scaled Equations
d2X'
b
/2/"TDpLdx' WdX'
dt'2
a2
I
WC2
A b/Vdt'
a
'
i
y
~^&—)r d^ALdb-x')
c.
•
(5)
Machine Equation
When the values of constants and scale factors are introduced
into Eq. (5), the machine equation results:
=
- (5.07
M)
X'
[^-lU-iy.)
.
(6)
The initial conditions (in terms of voltages) are:
X'
dX'
dt'
=
bX(0)
=
0
b dX
a dt
=
7.0 volts
d2X'
dt 12
4.
Analog
a.
dt2
a2
Circuit Diagram
Flow Sheet
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-100 V
Fig.
1.
Circuit Diagram for Solution of Elias'
Equation of Buffer Motion
argonne national laboratorg
APPLIED MATHEMATICS DIVISION
ANALOG
COMPUTER
b. POTENTIOMETER
SETTINGS
REACTOR CONTROL ROD
DECELERATION
POTENTIOMETER
NO.
MATHEMATICAL
VALUE
DRAWING MACHINE
VALUE
CORREC
TION
SETTING
PROBLEM
DRAWING
NO.
NO. .
DATE
PARAMETERS
SET
1
V'p (volts)
v
'"l
7.00
0700
a = 103
2
X' (volts)
-50.00
5000
b = 102
3
0.2
0.2000
2000
4
I^(5-07M)
For
Figs. 2 & 5
Vp(0)
= 70
ft/sec
Mi
=
0.0494
0.0125
0125
Dp = 2 in.
M?
=
0.0795
0.0202
0202
C
M3
=
0.102
0.0258
0258
= 0.03
in.
L , = 6 in.
W = 290 Ib
X
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1
2
Vp(0) (volts)
Yj
= 70
ft/sec
7.00
0700
V2
= 50
ft/sec
5.00
0500
V3
= 20
ft/sec
2.00
0200
X* (volts)
-50.00
5000
0.2
0.2000
2000
I £. C\~? ti \
{-K<)7 ^2)
0.0202
0202
3
4
AMI-2C 18-57)
5
100
For
Fiaq
3 & 4
= 6
in.
10
argonne National Caboratorg
APPLIED
MATHEMATICS
DIVISION
ANALOG
COMPUTER
c. STATIC
CHECK
PROBLEM
DRAWING
REACTOR CONTROL ROD
DECELERATION
UNIT NUMBER
UNIT
OUTPUT
(VOLTS)
DRAWING MACHINE
1
POT
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AMP
MULT
AM).7ft (8-57)
REMARKS
M2
=
0.0795
7.00
1
+
2
-50.00
5
-10.00
3
- 1.40
4
+
0.02
1
-
7.00
2
+ 10.00
3
+40.00
4
+16.00
5
- 8.75
6
- 0.02
1
-16.00
2
+ 8.75
3
+
0.88
FOR STATIC
CHECK
INTE
GRATOR
NO.
NO..
DATE
INITIAL
CONDI
TION
SET
PARAMETERS
11
Graphical Results
5.
70
o
z
o
60
w
50
»
40
iUJ
iij
".
>-\-
o
°
UJ
>
V\
/«!
30
\'
Fig. 2
w
=.0494 POUND FORCE PER FOOT SECOND
II
"
"
"
»
A =.0795
Velocity Versus Distance
for Various Viscosities
/".= .102
20
10
01
0.2
03
0.4
05
DISTANCE, FEET
Fig.
3
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Velocity Versus Distance for
Various Initial Velocities.
Viscosity is 0.0795 Pound
Force per Foot-Second
02
03
DISTANCE, FEET
Fig.
z
4
Distance Versus Time for
02
Various Initial Velocities
3
4
5
6
TIME, SECONDS X 10-'
12
70
60
Fig.
30
5
Velocity Versus Time for
Various Viscosities
>i, =0494 POUND FORCE PER FOOT SECOND
yU, ■.0795
><, =.102
20
10
2
6.
3
7
Bibliography
1-1.
Freund, G. A. et al., Design Summary Report on the Transient
Reactor Test Facility (TREAT), ANL-6034 (June I960).
1-2.
Koch, Li. J. et al., Hazard Summary Report Experimental Breeder
Reactor II (EBR^II), ANL.-5719 (May 1957) p. 30.
1-3.
Bishop, A. A., and Berringer, R. T., Hydraulic Shock Absorbers for
Reactor Control Rods, YAEC-111 (March 1959).
the Yankee
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4
5
6
TIME, SECONDS X 10-'
13
II. PRESSURE VARIATIONS THROUGH A PACKED BED
1 .
Problem Description
The future applications of nuclear power sources will depend on
whether reactor technology can match the demand for higher power densities
and higher operating temperatures. A popular concept for advanced appli
cation is the packed bed reactor. I""* / Ul-^j Equations of fluid flow and
heat transfer for this concept are dependent upon the particular packed-bed
system, particle shape, and the fluid for which they are developed.
The solutions to problems for this type of reactor design are usually
'
obtained through use of empirical corrections.^
An equation derived by
MacFarlane^"') from the basic Bernoulli equation illustrates a fundamen
tal method for calculating the performance of packed bed arrangements.
This relationship
dP
(K + Hx)P
E + Dx - P2
dx
expresses in differential form the variation of pressure and distance of a
packed bed one square foot in cross section. MacFarlane also indicates
four other general methods used for calculating laminar fluid flow in packed
beds and describes their derivation.
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2.
Mathematical Statement of the Problem
a.
Equations
dP
and Constants
(K + Hx)P
E + Dx - P2
dx
QP0G
--
D
=
G
=
H
=
E
=
G2P0/gcP0
K
=
D + G
-
,
k
1'73x
=
. lb2
3. 439x10* -r^r
ft5
f°po
-
2gcDpPo
-
lbz
=
=
3.44 x
108
10
'
14
=
Volumetric heat generation rate
=
5
G
=
Mass flow rate of helium coolant
=
0.378 lb/(sec)(ft2)
gc
=
Gravitational constant
=
32.17
p
=
Initial density
=
0.083 lb/ft3
T0
=
Initial temperature
=
200°F
P0
=
Initial pressure
=
10
C
=
Specific heat at constant pressure
=
1.25
BTU/(lb)(°F)
D
=
Diameter of the particle
=
200
microns
f
=
friction factor, a + jS (x/L,)
=
285
+230 (x/L)
b.
Initial Conditions
ft/secz
atmospheres
2.12 x 104 lb/ft2
P0
=
x0
= 0
L
=0.2 ft.
Preparation of the Machine Equations
Machine Variables and Scale Factors
a.
x
=
t
t1
=
at
"
P'= bP
P'
=
t'(final)
=
bP0
X(final)
t(final)
dt1 = adt
a
= 102
dP1 = bdP
b
=
=
102(0.2)
10-3
(2)
=21. 2 volts,
(10-3)(2.12 x 104)
at(final)
=
=
20.0
volts
The Scaled Equation
equations
dt1
=
_ [K
f
dP1
_!_
Substituting
obtained:
a
E
+
b.
I
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Mw/ft3
(2)
into equation (l), the scaled equation is
(H/a) t'] P'
(Dt'/a) - (P'2/b2)
+
3.
Q
(3)
15
c.
The Machine Equation
When
equation
(4)
dP'
dt'
4.
(3.44 + 0.522 t')P'
0.0002 t1 - P'2
(4)
Analog Circuit Diagram
a.
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numerical values are placed into equation (3), the machine
results:
Flow Sheet
ORIGINALPOSITIONOF RELAY
TO HOLDRELAY
+60V
Fig.
6.
Circuit Diagram for the Solution
of MacFarlane's
Equation
16
Slrgonnt Bational laboratorg
APPLIED MATHEMATICS DIVISION
ANALOG
b.
COMPUTER
SETTINGS
POTENTIOMETER
PROBLEM
DRAWING
PRESSURE
NO.
POTENTIOMETER
DATE
REDUCTION THROUGH A PACKED BED
MATHEMATICAL
VALUE
DRAWING MACHINE
CORREC
TION
VALUE
1
O.0la volt
-1.00
0100
2
P0b volt
+21.2
2120
3
b2D/a
0.0002
0002
4
b2H/a2
0.5220
5220
5
b2K/a
3.44
0344
6
VTo
3.162
3162
7
aL
-20.00
2000
PARAMETERS
SET
SETTING
NO.
NO. .
( 1C )
2lrgonne national laboratory
APPLIED
MATHEMATICS
DIVISION
ANALOG
COMPUTER
c. STATIC
CHECK
PROBLEM
DRAWING
NO.
NO..
DATE
PRESSURE
REDUCTION
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UNIT NUMBER
UNIT
DRAWING MACHINE
AMP
POT
MULT
OUTPUT
(VOLTS)
1
+ 1.00
2
0.0
3
-3.96
4
-21.2
5
+66.9
6
44.7
7
-0.188
8
0.0
3
0.0
4
0.52
6
-6.69
1
-0.84
2
-44.7
3
+1.88
THROUGH A PACKED BED
REMARKS
INTE
GRATOR
INITIAL
CONDITION
SET
PARAMETERS
17
5.
Graphical Results
25
O
20
"-
15
IE
m
L±J
S
0.1
DISTANCE, FEET
0.2
Fig. 7. Pressure Versus Distance
6.
Bibliography
II- 1. Leroy, M. M., and Newgard, J. J., Pebble Bed Nuclear Reactor for
Space
Vehicle Propulsion, Aero/Space Engineering, 19.(4), 54-58
(April I960).
T., and Bevenati, R. F., A High Temperature Gas Cycle
Pebble Bed for Central Station Use, TID-7564 (1958).
II-2. Robinson,
S.
R. B., Leyberg, E. A., and Morris, J. F., Heat-transfer
Coefficients for a Full-scale Pebble-Bed Heater, Ind. Eng. Chem.
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II- 3 . Lancashire,
52, 433
(I960).
II-4. Rodin, M. B., Study of the Packed-Bed Fuel Element Concept,
ANL-6193
(August I960).
18
III. REACTOR KINETICS OVER MANY DECADES WITH THERMAL
FEEDBACK (SIMULATION OF A TREAT TRANSIENT)
1.
Problem Description
The TREAT reactor was designed to generate a very large,
transient, thermal flux field of short duration. (HI-1) The maximum in
tegrated flux is greater than 1015 neutrons/cm2.
The core is a dispersion of highly enriched uranium (as the oxide
or carbide) in a graphite matrix. The graphite serves as a moderator, a
heat sink, and a generator of a sizeable negative temperature coefficient.
The latter effect is due to the fact that the energy of the thermal neutrons
increases with graphite temperature thus causing an increase in the leak
age probability.
The purpose of this experiment is to simulate a TREAT transient
initiated by control rod withdrawal and terminated by the negative tempera
ture coefficient. Since a large excursion is expected, the reactor kinetics
equations will be transformed by a substitution. (III-2)
ri
=£n
n(t)/n(0)
The equations describing the neutron kinetics (with
will be solved on the analog computer.
delayed groups)
They will be forced by changes in
Kex-
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2.
Mathematical Statement of the Problem
Equations:
a.
=1
where
T\
= .gn
n(t)/n(0)
6
19
Constants
b.
p
=
0.00755
3,
=
8.6 x 10-4
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M
Pi
1
0.01246
0.00025
2
0.0315
0.00166
3
0.1535
0.00213
4
0.456
0.00241
5
1.612
0.00085
6
14.3
0.00025
c.
Initial Conditions
7](0)
= 0
^i
= 0
Kiex=
3.
i
0
Preparation of Machine Equations
Machine Variables
a.
t1
=
at
T)'
=
bT)
Kiex - c Kiex
Scale Factors
b.
a
=
b
= 2
c
=
25
=
d2
10
.
=
d
= 2
20
c.
Scaled Equations
3
ex
dt'
dt1
d.
The Generation of K
The expression for Kex is made up of two parts: the contribution
of the control rod and the contribution of the negative temperature coeffi
cient, that is,
Kex = K(t) - K(n,t)
where
K(t)
_
fo.04t for 0<t£0.5 sec
for t>0.5 sec
I 0.02
and
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K(n.t)
=
ICT10/ n dt
Since
_c_
IB
then
=
J
K(t) -
j K(n,t)
or, more simply,
=
Kj(t) - Ki(n,t)
0.04-|
-
for 0<t'<0.5a sec
0.02 —for t'>0.5a sec
')
can be generated by means of an integrator and a relay.
21
The generation of Ki(n,t) is more complicated: a function
generator is needed. If the machine variables and scale factors are sub
stituted into
K(n,t)
the
=
/ ndt
10"10
result is
Kj(n,t)
~
10"10
/ndt'
.
But
n = ni n(0)
;
therefore
/ Bldt
Kf(n,t) = filler10
pa
The analog computer
variable)
and since e^n ni
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Kf(n,t) = £^i£l 10-io
=
.
will supply
7)
=
in n^ (due to a change in
n^
J er?dt,
After the terms are collected,
Ki(n,t)
=
c
/ eadt'
where
a = t) +£n n(0) - Zn 1010 - £n ap
For the values given the constants,
a= i) - 15.836.
and
for n(0) - 102,
Then 25 ea will be generated with a diode-function generator
(DFG).'111"3) In order to decrease the slope of the function, the DFG will
be driven by 107] - 100.
22
In the actual experiment, the input to integrator
means
of
a relay) until 25 ea = 0.01 volt.
(by
DFG DATA
e.
dr
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dt'
7]
ion -
0.0
a
25 ea
-100
-15.836
-
8.012
- 19.88
-
7.821
00.01
9.938
-
0.62
- 6.438
00.04
10.42
+
4.2
- 5.416
00.11
12.0
+
20.0
-
3.836
00.54
13.0
+
30.0
- 2.836
1.47
14.0
+
40.0
-
1.836
3.96
15.0
+
50.0
- 0.836
10.59
16.0
+
60.0
+
0.164
29.45
17.0
+
70
+
1.164
80.75
100
Machine Equations
=
0.0697 K^ex +0.3512 (0.0331^/{
+
0.2821^
+
0.3192^4
+
0.1126^^
dt'
+
0.2199^1
+
0.0331^)
3
is removed
23
=
0.0001
- 1.43^
=
-0.5
KJ(t) - Kj(n,t)
fl3.25t' volts for 0<t'<5 sec
,
Ki(n,t) = 25 exp
4.
Analog
- 15.836)
Circuit Diagram
Flow Sheet
a.
-100
3-K'iex
EXPERIMENT
Fig.
8.
-
?
EXPERIMENT
3
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(7]
" ' ' 1 66.23 volts for t'>5 sec
= iU -n,
Circuit Diagram for Duplication of TREAT Transient
24
argonnt national taooidtorji
APPLIED MATHEMATICS DIVISION
ANALOG
COMPUTER
b. POTENTIOMETER
SETTINGS
PROBLEM
DRAWING
DATE
THE TREAT REACTOR
POTENTIOMETER
NO.
MATHEMATICAL
VALUE
DRAWING MACHINE
0697
S, = 8.6x10-
2
0/a2
0.8779
8779
/3 = 0.00755
3
b/VdiJ3
0.0331
0331
4
b/Vd^
0.2199
2199
X, = 0.0315
5
b/Vdj/3
0.2821
2821
X3
= 0.1535
6
bA,/d4/3
0.3192
3192
X4
= 0.456
b/Bj/dsjS
0.1126
1126
X5
b/36/d6,3
0.0331
0331
\
= 1.612
V'a
0.0012
0012
0j = 0.00025
Va
Va
Va
Va
Va
0.0032
0032
/3Z = 0.00166
0.0154
0154
|33
= 0.00213
0.0456
0456
/34
= 0.00241
0.1612
1612
/35
= 0.00085
1.43
1430
/36
PARAMETERS
7
SET
8
SETTING
0.0697
= 0.00025
15
d5X5|3/ac
0.0001
0001
a = 10
16
d6X6(3/ac
0.0009
0009
17
FOR
TO
STATIC
24
CHECK
25
- 5.0 volts
0500
26
1.0 volts
0100
27
28
AtO-2C (8-571
+
-0.04c//3a
volt
-19.88 volts
= 25
=
n(0)
1325
1988
2
b
=
c
1(
-- 14.3
dt
14
(
13
)
12
0
11
0.01246
0
10
\-
0
1
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CORREC
TION
9
6b(l - 0)/a^c
VALUE
NO.
NO. .
..=d6=2
= 102
25
Slrgonnc .Rational laboratory
APPLIED MATHEMATICS
ANALOG
c STATIC
DIVISION
COMPUTER
CHECK
NO.
PROBLEM
DRAWING NO..
DATE
THE TREAT REACTOR
UNIT NUMBER
DRAWING MACHINE
POT
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AMP
OUTPUT
(VOLTS)
REMARKS
INTE
GRATOR
CONDI
TION
17
-10
1
+10
18
+ 10
2
-10
19
+ 10
3
-10
20
+ 10
4
-10
21
+ 10
5
-10
22
+ 10
6
-10
23
+ 10
7
-10
24
-50
23
+50
9
-7.0
10
-10
11
-10
12
+50
13
+50
14
+ 50
15
+50
16
+50
17
+ 50
18
+5.0
20
Negative
25
Positive
ALL M ULTIPl ,IER CI JANNELS = +5 volts
AMD-?A(8-57)
INITIAL
SET
PARAMETERS
26
5.
Graphical Results
Fig.
9
=
Kex and T) J>n n,
Versus Time
345
TIME,Seconds
6.
Bibliography
III-l.
Okrent, D., The Reactor Kinetics of the Transient Reactor Test
Facility (TREAT), ANL-6174 (Sept. I960)
III-2. Bryant, L. T., Just, L. C., and Pawlicki,
G. S., Introduction To
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Electronic Analog Computing, ANL-6187 (July I960).
III-3. Scott, W. E., Fundamental Components of~ the "PACE" Analog
Computer,
ANL-6075 (Nov. 1959).
27
A VIBRATING SYSTEM WITH TWO DEGREES OF FREEDOM
IV.
1.
Problem Description
Problems of vibration must be considered in the design of power
plants using fissionable fuel. Fuel elements, control rods, and structural
supporting members are capable of vibrating; their characteristics must
be analyzed, for vibration problems prove to be of importance to eliminate
concern for the safe operation of the power plant. Good representations
of the true situation usually involve systems with several degrees of
freedom.
A typical vibration problem which will serve as an introduction to
multi-degree-of-freedom systems is shown in Fig. 10. The two masses
mj and m2 are suspended vertically by springs kj and k2. The masses are
constrained such that they only move vertically. The displacements xj and
x2, taken positive for a downward motion, are measured using static equi
librium as reference. The elongation of the upper spring is Xj and the
elongation of the lower spring is (x2 - xj). The restoring force acting on
mi is [-kjXj + k2(xz - xj], and on m2 the restoring force is -k2(x2 - xj),
where ki and k2 are the spring constants of the respective springs.
Effects due to energy dissipation in the elastic spring, wind friction,
and springs that have appreciable mass have been neglected in the equations
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of motion given below.
Fig.
10
Illustration of a Vibrating System
with Two Degrees of Freedom
m-,
x axis
2.
Mathematical Statement of the Problem.
a.
Equations
+
k2(x2
-
(1)
28
m2
d2x2
—
—
dt
b.
-kz(X2 - XiJ
=
(2)
Equation Constants
mj: mass (ib) (i = 1,
2)
A:
spring constant (ib force/ft) (i = 1, 2)
Initial displacement of the springs (feet)
c.
Initial Conditions
kj:
It is obvious that with the springs displaced a certain distance,
A, the initial conditions will have the following values
Xl(0)
=
=
dt
dX2(0)
dt
= 0
d2Xl(0)
d2X2(0)
dt2
dt2
dXl(0)
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3.
A
x2(0)
(3)
= 0
Preparation of Machine Equations:
In transforming to the machine equations, the folio-wing relationships
are made. Let
jq - bXi
(i
=
1, 2)
and
(4)
t1 =
Substitution
at
of equations (4) into equations (1) and (2) yields
2
ix
dt
a iri
~k?
dt'2
a2mz
a
'fc-xj)
(x2-x{)
(5)
(6)
29
where
xi(0)
=
bXl(0)
=
bA
x£(0)
=
bx2(0)
=
bA
and
The solution to the equations will vary with m1( m2, kj, kz, and A.
solution given here, we consider the following physical constants:
ki
=
k2
=
0.2 lb force/ft
mj - m2 = 1 lb mass
A = 1 ft
4.
Analog
a.
Circuit Diagram
Flow Sheet
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-IOOV
Fig. 11. Circuit Diagram for the Solution of the Equations
Describing a Vibrating System with Two Degrees
of Freedom
In the
30
argonne national Xaboratorg
APPLIED MATHEMATICS DIVISION
ANALOG
COMPUTER
b. POTENTIOMETER
SETTINGS
A VIBRATION SYSTEM WITH
TWO DEGREES OF FREEDOM
POTENTIOMETER
NO.
MATHEMATICAL
VALUE
DRAWING MACHINE
VALUE
CORREC
TION
PROBLEM
NO.
DRAWING
NO. .
DATE
PARAMETERS
SET
SETTING
1
bA
-50.00
-5000
a = 1
2
bA
-5000
-5000
b = 50
3
k^/a2^!j
0.2
2000
4
k2/a2mj
0.2
2000
5
k2/azm2
0.2
2000
argonne Rational laboratory
APPLIED
DIVISION
MATHEMATICS
ANALOG COMPUTER
c. STATIC
CHECK
PROBLEM
DRAWING
NO.
NO..
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DATE
UNIT NUMBER
UNIT
DRAWING MACHINE
POT
AMP
OUTPUT
(VOLTS)
3
10.00
4
0.0
5
0.0
1
0.0
2
+50.0
3
-50.0
4
0.0
5
-10.0
6
0.0
7
50.0
REMARKS
INTE
GRATOR
INITIAL
CONDI
TION
SET
PARAMETERS
31
Graphical Results
5.
10
-1.0
Fig.
12
Distance Versus Time for a System
with Two Degrees of Freedom
10
-i.o
0
20
40
60
80
TIME, SECONDS
6.
Bibliography
IV- 1. Timoshenko, S., Vibration Problems in Engineering,
D. Van Nostrand
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Company, Inc., Princeton, New Jersey, (1955) 3rd. Ed.
32
TEMPERATURE DISTRIBUTION IN A RADIATING FIN
V.
1.
Problem Description
The only economical method for rejecting heat from an outer-space
power plant is by thermal radiation. (V - 1) if the working fluid of the power
plant passes through tubes, the additions of extended surfaces to the tubes
in the form of fins reduces the number of tubes required. This reduction
decreases the probability that a meteor will puncture a vital coolantcarrying passageway. (The puncture of a fin is of lesser concern for the
continued operation of the power plant.) An analysis of the temperature
distribution of these extended surfaces is very important in calculating the
effectiveness (and, indirectly, the safety of the plant) of various fin
geometries.
2.
Mathematical Statement of the Problem
In the development of a differential equation for conduction, (V-2)
dq = d/dx
f2kWYx-|^dx j
(l)
A general heat balance requires this differential equation (l) to be equal
to
dq = 2ae (T4 -
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the heat rejected
Ts) dA
(2)
by radiation.
By assuming that the arc length (ds) on the arbitrary surface is
equivalent to dx on the abscissa and assigning Yx equal to a constant thick
ness for a straight fin geometry, the differential equation for temperature
is
d2T
dx'
ae
Hk
Tl)
(3)
A constant heat source will be assumed at one end of the fin and
dT
dx
0
x=L
at the other end.v""^)
This will correspond to the fin in Fig. 13.
-TEMPERATURE
Fig.
13
Geometry of Radiation Fin
and Coolant Tubes
33
a.
Constants and Variables
T
=
Absolute temperature along the fin
Ts
=
Absolute temperature of the sink
a
=
Stefan-Boltzmann constant
e
=
Emissivity
W
- Width of the fin in the z-direction
L
=
Total length of the fin in the x-direction
q
=
Heat dissipated
H
=
Half -thickness of the fin
k
=
Thermal conductivity
of the fin material
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Typical values are:
Ts
=
0°R
O
=
0.173 x ID-8
c
=
0.9
W
=
1.0 ft
L
=
0.25 ft
H
=
1.250 x 10"3 ft
k
=
25.0
b.
Initial Conditions
T(0)
=
BTU/(hr)(ft2)(°R4)
BTU/(hr)(ft)(°R)
2000°R
The most efficient use of radiator material weight dictates the
arrangement of the finned tubes in a straight bank. The general tempera
ture distribution, of this arrangement, along the fin is given in Fig. 15.
3.
Preparation of Machine Equations
a.
Machine Variables and Scale Factors.
bT
ax
= x1
adx
=
dx'
bdT
=
dT'
a2dx2
=
dx'2
bd2T
=
d2T'
a
=
102
b
= 5
TT
x 10"2
.
.
34
Scaled Equation
b.
Substituting Equations
aeT'4
(Note Ts
a2b\H
dt'
(5) into Equation
=
(3) we get,
0.
Machine Equation
c.
'
0.03986
•=
f
T '4
dt |2
d.
Initial Conditions
T1
=
bT
=
Y volts
dT1
dt1
=
100
volts
so that
dT1
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dt'
4.
Analog
a.
= 0
t=aL
Circuit Diagram
Flow Sheet
— 100v
i )
i
loov
T'
D>
Fig. 14.
Circuit Diagram for Solution of Secondorder, Fourth-degree Differential Equation
35
argonne Bational laboratory
APPLIED MATHEMATICS DIVISION
ANALOG
COMPUTER
b. POTENTIOMETER
SETTINGS
PROBLEM
DRAWING
NO.
NO. .
DATE
POTENTIOMETER
NO.
MATHEMATICAL
VALUE
DRAWING MACHINE
dT1
CORREC
TION
VALUE
1
7—r volts
Ht1
*
2
106ae/a2b3kH
0.03986
* Variabl e to
produce
SETTING
0399
-dT
— = 0 at X :-- L
ax.
argonne Bational Xaboratoru.
APPLIED
MATHEMATICS
DIVISION
ANALOG
COMPUTER
c. STATIC
CHECK
PROBLEM
DRAWING
NO.
NO..
DATE
UNIT NUMBER
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UNIT
DRAWING MACHINE
OUTPUT
(VOLTS)
POT
2
-3.99
MUH
1 AB
-lOOv
AB
-100v
3
+ 3.99
2
AMP
REMARKS
INTE
GRATOR
2
INITIAL
CONDI
TION
+100
PARAMETERS
36
5.
Graphical Results
2000
1500
1000
500
0.10
0.15
FIN LENGTH,FEET
Fig. 15. Temperature Versus Length and dT/dx
Versus Length for a 0.25-ft Fin (K = 25.0)
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dT
/
*dT/dx was plotted to show when boundary condition - —
is satisfied.
6.
= 0
Bibliography
V-l. Corliss, W. R., Nuclear Power in Outer Space, Nucleonics, 18 (8),
59-63(1960).
V-2.
Schneider, P. J., Conduction Heat Transfer, Addison - Wesley Publish
ing Company, Inc., Reading Massachusetts (1955).
V-3.
Lieblein, S., Analysis of Temperature Distribution and Radiant Heat
Transfer Along a Rectangular Fin of Constant Thickness, NASA
Technical Note D-196. National Aeronautics and Space Administra
tion, Washington (November
V-4.
1959).
Carslaw, H. S., and Jaeger, J. C., Conduction of Heat in Solids,
Second Edition, Oxford University Press, Amen House, London
E. C.4, 1959.
37
VI.
1.
TEMPERATURE DISTRIBUTION IN AN INFINITE SLAB
CONSIDERING VARIABLE THERMAL PROPERTIES
Problem Description
When the thermal properties of various materials are studied,
thermal conductivity, specific heat and density are usually considered as
constants; they are, however, dependent upon temperature. (VI-1) In this
experiment, an insulated zirconium slab is studied. Four cases are
considered:
2.
Diffusivity
(2)
«.F(i^Ti)
(3)
K.
(4)
K.
( K.
;
- F (Temperature of the region described by the heat
balance);
- F (Average temperature across an interface).
Mathematical Statement of the
a.
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- k/p c) is constant;
(1)
Equations
Fig.
16.
Model of the Infinite Slab Showing
Regions Used for Analysis
38
/T
'
(
d
dT
dt
-=
Ax" v
4
pcAx
Constants
b.
(1)
Constant case
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k = thermal conductivity
=
11
BTU/(hr)(ft)(°F)
c
=
specific heat
=
0.066
p
=
density
=
0.397 lb/ft3
/c
=
k/cp = diffusivity
=
0.4198
S
=
heat source
=
18.3
€
=
emissivity
a
=
Stephan-Boltzmann
constant
Ax = 1/60 ft
(Z)
As a function of temperature
T, °F
100
ZOO
300
400
500
600
700
800
900
K(T)
0.4198
0.4
0.38
0.364
0.35Z
0.336
0.3ZZ
0.309
0.Z98
*Radiation heat loss will be neglected.
BTU/(lb)(°F)
ft2/hr
BTU/(ft2)(sec)
39
Initial Conditions
c.
Ti
3.
=
T3
=
T4
=
T5
=
100°F
Preparation of Machine Equations
Machine Variables
a.
t1
=
at
T1
=
bT
S"
=
AxSb/k
Scale Factors
b.
a
=
1.0
b
=
0.1
Machine Equations
c.
KS"
dTl
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=
T2
_jc
,
~
z>
Initial Conditions
d.
T' = TZ = ... TS = 10 volts
S"
=
10
volts [S = 18.3 BTU/(ft2)(sec)] for 50 sec
40
4.
Analog
a.
Circuit Diagrams
Flow Sheets
Case I - ( K. = constant)
Fig.
17.
Circuit Diagram for
an Infinite Slab with
Thermal Conductivity
Generated on 2015-10-14 14:14 GMT / http://hdl.handle.net/2027/mdp.39015078510917
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
K.
- Constant
R-RELAY CIRCUIT
-TO INTEGRATOR
Fig.
18.
5000
Relay Circuit for
the Heat Pulse Used
in Experiment VI
Generated on 2015-10-14 14:14 GMT / http://hdl.handle.net/2027/mdp.39015078510917
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
41
Case
Fig.
19.
II - [K = F(Tav)]
Circuit Diagram for an Infinite Slab with Thermal
Conductivity /c = F(Taverage)
42
Case
III - [K - F( Temperature of the Region Described by the
Heat Balance)]
Circuit Diagram for an Infinite Slab with
Thermal Conductivity K = F(Temperature
of Region Described by the Heat Balance)
Generated on 2015-10-14 14:14 GMT / http://hdl.handle.net/2027/mdp.39015078510917
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
Case IV -
Fig. 21.
=
F(Average Temperature Across an Interface)]
Circuit Diagram for an Infinite Slab with
= F(Average
Thermal Conductivity
Temperature Across an Interface)
K.
20.
[/c
Fig.
43
argonne Bational Caboratorg
APPLIED MATHEMATICS DIVISION
ANALOG
b.
COMPUTER
POTENTIOMETER
SETTINGS
TEMPERATURE VARIATION IN A ONE FACE INSULATED SLAB
CONSIDERING VARIABLE THERMAL PROPERTIES
CASE I
POTENTIOMETER
NO
MATHEMATICAL
VALUE
Generated on 2015-10-14 14:14 GMT / http://hdl.handle.net/2027/mdp.39015078510917
Public Domain, Google-digitized / http://www.hathitrust.org/access_use#pd-google
DRAWING MACHINE
CORREC
TION
VALUE
SETTING
SET
PROBLEM
DRAWING
NO..
NO. -
DATE.
PARAMETERS
I
0.01T]
0.1
1000
S = 18.3
2
O.OlTj
0.1
1000
S" = 10
3
0.0ITJ
0.1
1000
a = 1
4
0.01T4
0.1
1000
b = 0.1
5
O.0lTj
0.1
1000
6
(0.01/cS"/Ax2a)
0.042
0420
Ax
7
/c/Ax2a
0.4198
4198
Ti(0) = 100°
8
/c/Ax2a
0.4198
4198
9
/c/Ax2a
0.4198
4198
10
K/Ax2a
0.4198
4198
11
/c /A x2a
0.4198
4198
12
/c/Ax2a
0.4198
4198
13
/c/Ax2a
0.4198
4198
14
/c /A x2a
0.4198
4198
C ASE
0.4198
3600
II
1
0.0lTj
0.1
1000
2
0.01