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HELP O A K RIDGE N A T I O N A L LAB0RATOl);rC operated by
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UNION CARBIDE CORPORATION NUCLEAR DIVISION for the U.S. ATOMIC ENERGY COMMISSION
ORNL- TM- 1647 COPY NO.
-
f.
’ f l ”
EXPFXIMENTAL DYNAMIC ANALYSIS OF THE MOLTEN-SAlZ
ASSTRACT Dynamics t e s t s were performed on t h e Molten-S It Reactor Experim nt (MSRE) f o r t h e f u l l range of operating power l e v e l s t o determine t h e power-to-reactivity frequency response. Three types of input disturbances were used: t h e pseudo-random binary r e a c t i v i t y input, t h e pulse r e a c t i v i t y input, and t h e s t e p r e a c t i v i t y input. --.
The frequency response of t h e uncontrolled r e a c t o r system displayed resonant behavior i n which t h e frequency of o s c i l l a t i o n and t h e damping increased with increasing power l e v e l . Measured periods of n a t u r a l o s c i l l a t i o n ranged from t h i r t y minutes at 75 KW t o two minutes a t 7.5 MW. These o s c i l l a t i o n s were l i g h t l y damped at low power, but strongly damped a t higher power. The measured r e s u l t s generally were i n good agreement w i t h predictions. The observed n a t u r a l periods of o s c i l l a t i o n and t h e shapes of t h e measured frequency response agreed very w e l l with predictions. The absolute amplitude of t h e frequency response d i f f e r e d from predictions by a f a c t o r t h a t was approximately constant i n any t e s t (though d i f f e r e n t t e s t s a t t h e same power l e v e l d i d not have t h e same b i a s ) . T h i s b i a s d i f f i c u l t y i s apparently p a r t l y due t o equipment l i m i t a t i o n s (standard MSRE c o n t r o l rods were used) and p a r t l y due t o u n c e r t a i n t i e s i n t h e parameters i n t h e t h e o r e t i c a l model. The main conclusion i s that t h e system has no operational s t a b i l i t y problems and t h a t t h e dynamic c h a r a c t e r i s t i c s a r e e s s e n t i a l l y as predicted. ~~
~
* Research
sponsored by t h e U. S. Atomic Energy Commission under contract w i t h t h e Union Carbide Corporation. For presentation a t t h e Winter Meeting of t h e American Nuclear Society t o be held October 30-November 3 , 1966 i n Pittsburgh, Pa. NOTICE This document contains information of a preliminary nature and was prepared primarily for internal use a t the Oak Ridge National Laboratory. It is rubiect to revision or correction and therefore does not represent a final report.
f
LEGAL NOTICE T h i s report was prepored as on account of Government sponsored work.
Neither the United Stotes,
nor the Commission, nor any person octing on behalf of the Commission:
A. Maker ony worranty or representation. expressed or implied, w i t h respect t o the occurocy, completeness, or usefulness of the informotion contained i n this report, or that the use of any
information,
apparatus,
method, or process disclosed in t h i s report may not infringe
privately owned rights; or
B.
Assumes
any l i a b i l i t i e s w i t h respect t o the use of, or for domoges resulting from the use of
any informotion. apparatus, method, or process disclosed i n this report.
As
used i n
contractor
thm
above, “person
acting on behalf of the Commission�
includes any employee or
of the Commission, or employee of such controctor, t o the extent that such employee
or contractor of the Commission, or employee of such contractor prepares, disseminates, or provides access to, any informotion purrwant t o h i s employment or h i s employment w i t h such contractor.
or
contract w i t h the Commission,
iii
CONTENTS
I I1 I11
IV
............................... Description of t h e MSRE .................... Theoretical Predictions .................... A . Description of Mathematical Model ...... B . Results ................................ S e l e c t i o n of Experimental Methods .......... A . C h a r a c t e r i s t i c s of t h e MSRE Regulating Rod ....................... Introduction
B
.
Test Signals Used i n t h e Experiments
.......... 2 . Pulse T e s t s ........................ 3 . Step Tests ......................... Experimental Procedures .................... A . Implementation of Pseudo-random Binary Tests ......................... B . Implementation of Pulse and .
1
V
...
Pseudo-random Binary Test
........................... Analysis Procedures ........................ A . Direct Anzlysis of PRBS Tests .......... B . I n d i r e c t Analysis of PRBS Tests ........ C . Step Response Test Analysis ............ D . Pulse Response Test Analysis ........... Results .................................... A . Transient Responses .................... B . Correlation IiLulctions .................. C . Frequency Responses .................... Step Tests
VI
VI1
.............. IX References ................................. Appendix A . P o t e n t i a l Sources of Experimental Error Due t o Equipment Limitations ..... Appendix B . The Direct Method f o r Cross-Power VI11
V
I n t e r p r e t a t i o n of t h e Results
Page
1 1
4 4 4 6 6 9 9 10 12
13 13 13 13 13 17 21 21
21 22
22 32 32
45
47
-1Y
INTRODUCTION
A s e r i e s of experiments was performed on t h e Molten S a l t Reactor Experiment (MSRE) t o determine t h e frequency response of t h e uncontrolled r e a c t o r system.
Tests were performed a t eight d i f f e r e n t power l e v e l s
ranging from zero t o
mll
power.
Three d i f f e r e n t types of input
disturbances were used t o obtain t h e nuclear power t o r e a c t i v i t y frequency response:
t h e pseudo-random binary r e a c t i v i t y input, t h e
pulse r e a c t i v i t y input, and t h e s t e p r e a c t i v i t y input.
Subsequent
sections of t h i s report w i l l give a d e s c r i p t i o n of t h e system, a review of previously published t h e o r e t i c a l predictions, a d e s c r i p t i o n of t h e t e s t i n g procedures, and t h e experimental r e s u l t s . 11. DESCRIFTION OF THE MSRF:
The MSRE i s a graphite-moderated, c i r c u l a t i n g - f u e l r e a c t o r .
The
f u e l i s a mixture of t h e molten f l u o r i d e s a l t s of uranium, lithium, beryllium, and zirconium.
I
The basic flow diagram i s shown i n Fig.
1. The flows and temperatures shown a r e nominal values which were
calculated f o r operation a t 10 MW, but heat t r a n s f e r l i m i t a t i o n s a t t h e r a d i a t o r c u r r e n t l y r e s t r i c t maximum power operation t o about The molten fuel-bearing salt e n t e r s t h e core matrix a t t h e
7.5 MW.
bottom and passes up through t h e core i n channels machined out of unclad, 2-inch graphite blocks.
The heat generated i n t h e f u e l and
t h a t t r a n s f e r r e d from t h e graphite r a i s e t h e f u e l temperature about 50째F.
When t h e system operates a t reduced power, t h e flow r a t e i s
t h e same as a t f u l l power and t h e temperature r i s e through t h e core i s smaller.
The heated f u e l salt t r a v e l s t o t h e primary heat exchanger,
where it t r a n s f e r s heat t o a non-fueled secondary salt before r e e n t e r i n g t h e core.
The heated secondary salt t r a v e l s t o an air-cooled r a d i a t o r
before returning t o t h e primary heat exchanger.
The design parameters
of major importance from t h e standpoint of dynamics a r e shown i n Table 1. A d e t a i l e d d e s c r i p t i o n of t h e MSRE appears i n Ref. 1.
-2
-
ORNL- LR- DWG 56070
SPARE FILL AND FILL AND DRAIN TANK DRAIN TANK 1 6 8 c u f t ) (68 cu f t )
FLUSH TANK
(68cuft)
Fig. 1 MSRE Flow D i a g r a m
COOLANT DRAIN TANK ( 4 0 cu ft I
-3Table 1, MSRE Design Data Nuclear Temperature c o e f f i c i e n t of r e a c t i v i t y of t h e f u e l , "F-1
-4.7
10-5
Temperature c o e f f i c i e n t of r e a c t i v i t y of t h e graphite, OF-1
-2.6
10-5
Neutron l i f e t i m e , see.
.0002
T o t a l delayed neutron f r a c t i o n
.00666
Reactivity l o s s due t o f u e l circulation, $ E
-0.212
K Flow Flow r a t e i n t h e primary loop, gpm
1200
Flow rate i n t h e secondary loop, gpm
830
Fuel t r a n s i t t i m e i n t h e core, sec.
8.5
Fuel t r a n s i t time i n e x t e r n a l primary loop, see.
16.7
T o t a l secondary loop t r a n s i t t i m e , sec.
24.2
Heat Transfer Fuel s a l t heat capacity, MM sec/"F
4.2
Graphite heat capacity, MX sec/"F
3 -6
Heat exchanger heat capacity, MW sec/"F
.4
-
Bulk graphite f i e 1 salt heat t r a n s f e r c o e f f i c i e n t , MW/'F
0.02
Fuel s a l t -heat exchanger metal heat t r a n s f e r c o e f f i c i e n t , W/"F
0.36
Heat exchanger m e t a l , secondary salt-heat t r a n s f e r c o e f f i c i e n t , MW/"F
02
Fraction of power generated i n t h e f u e l
0 093
7
4
-4L
111. THEORETICAL PREDICTIONS
A.
W
Description of Mathematical Model Throughout t h e
MSm
design e f f o r t , a wide v a r i e t y of mathematical
models w a s used t o predict t h e dynamic behavior.
We w i l l l i m i t our
discussion here t o t h e most up-to-date and d e t a i l e d model reported, r e f e r r e d t o i n Ref. 2 as t h e "complete" model.
The core f l u i d flow
and heat t r a n s f e r equations were represented by 18 f u e l nodes and
9 graphite nodes.
The nuclear power d i s t r i b u t i o n and t h e nuclear
importances f o r each node were derived from a 2-group neutron d i f f u s i o n calculation.
The flow r a t e s and heat t r a n s f e r c o e f f i c i e n t s f o r each
node were determined from c a l c u l a t i o n s based on f u l l - s c a l e hydraulic core mockup t e s t s .
The assumed flow mixing c h a r a c t e r i s t i c s were
v e r i f i e d by t r a n s i e n t t e s t s on t h e mockup. The neutron k i n e t i c behavior was described by t h e usual spaceindependent equations w i t h s i x delayed-neutron groups, but w i t h modif i c a t i o n s t o include t h e dynamic e y f e c t s of t h e c i r c u l & l o n of precursors around t h e p r i m r y loop.
The thermal r e a c t i v i t y feedback w a s computed
by using a weighted nuclear importance for each of t h e 27 f u e l and graphite nodes.
The xenon poisoning r e a c t i v i t y feedback included
iodine production and decay i n t o xenon, xenon decay and burnup, and xenon absorption i n t o t h e graphite. The t r a n s p o r t of molten s a l t i n t h e primary and secondary loop piping w a s described by a plug flow m o d e l , where heat t r a n s f e r t o t h e pipes w a s included.
The primary heat exchanger and t h e s a l t - t o - a i r
heat exchanger were each represented by a 50-node model. B.
Results of T h e o r e t i c a l Analysis
Several d i f f e r e n t methods of s o l u t i o n were used on t h e various MSRE dynamics models, including analog and d i g i t a l computer simulation (time response), frequency response analysis, and pole configuration analysis.
The frequency response analyses can be d i r e c t l y compared
t o t h e experimental r e s u l t s , since t h e l a t t e r a r e r e a d i l y c a s t i n t h i s form. Fig. 2 shows t h e t h e o r e t i c a l E R E inherent frequency response W
.
-5-
I
ORNL-DWG 6579816
5
I
2 10,000
5 2
2
100
5
2 10 0.001
2
001
5
2
5
0.1
2
5
2
1
5
10
5
2
loo
FREOUENCV Irodions/rrc)
90 80
70 60 50 40
30 20
-F Z 0 'O
a -10 -20
-30 -40 -50
- 60 -70 -80
-90 00001
2
5
0001
2
5
001
2
5
01
2
5
1
FREQUENCY (radions/sec)
Fig. 2
MSRF: Theoretical Frequency Response
2
5 1 0
-6c h a r a c t e r i s t i c s f o r normalized neutron l e v e l response t o r e a c t i v i t y perturbations a t s e v e r a l power l e v e l s .
V
It can be seen t h a t t h e system
becomes more o s c i l l a t o r y a t progressively lower frequencies as t h e nominal power l e v e l decreases, though it i s s t a b l e f o r a l l power l e v e l s of i n t e r e s t .
An explanation of t h e inherent s t a b i l i t y c h a r a c t e r i s t i c s
i s given i n R e f . 2 .
IV.
SELECTION O F EXPERIMENTAL METHODS
The s e l e c t i o n of t h e experimental methods f o r t h e E R E dynamics t e s t s w a s based on t h e information required and on t h e c a p a b i l i t i e s of t h e a v a i l a b l e equipment.
It may be seen from Fig. 2 t h a t t h e most
s i g n i f i c a n t p a r t of t h e frequency response i s i n t h e range 0.01 t o 0 . 1 radians per second, since t h e amplitude peaks a r e i n t h i s frequency
range f o r t h e operating power l e v e l s of i n t e r e s t .
This frequency range
corresponds t o long periods of natural o s c i l l a t i o n (10 min. t o 1 min.). This emphasis on l o w frequency r e s u l t s f o r t u n a t e l y made it possible t o obtain t h e important p a r t o f t h e system frequency response using t h e standard MSRE c o n t r o l rods t o introduce t h e input r e a c t i v i t y perturbations.
I n t h i s section, we w i l l examine t h e c h a r a c t e r i s t i c s
of t h e MSRE regulating rod and t h e properties of t h e t e s t s i g n a l s used. A.
C h a r a c t e r i s t i c s of t h e MSRE Regulating Rod The MSRE has t h r e e c o n t r o l rods, each with an a c t i v e length of
59.4 inches.
One rod i s normally designated as t h e r e g u l a t i n g rod and
i s used f o r f i n e control. coarse adjustments.
The other two rods a r e shim rods used f o r
The rods a r e a c t u a l l y f l e x i b l e , s t a i n l e s s s t e e l
hoses on which a r e strung gadolinium oxide poison cylinders.
The rods
a r e mounted i n thimbles which have two 30" o f f s e t t i n g bends s o t h a t t h e rods can be c e n t r a l l y located even though t h e r e w a s no room f o r t h e c o n t r o l rod d r i v e assemblies above t h e c e n t r a l a x i s of t h e core. The maximum rod speed i s
4 . 5 inches/second.
The t h r e e c o n t r o l rods a r e i d e n t i c a l . c o n t r o l rod and d r i v e assembly.
Figures 3 and
4 show
the
The p o s i t i o n i n d i c a t i o n for each rod
i s obtained from two synchros geared t o t h e rod d r i v e mechanism.
Synchro number 1 i s used f o r coarse p o s i t i o n i n d i c a t i o n and has a
.
-7ORNL-LR-DWG 673II
REVERSIBLE DRIVE MOTOR
7 mING AS INLET
SDLENOIO ACTUATED RELEASE
MOIANW CUlTcH GEbRAmARM FIXED OR M l SUPPORT AN0 3in. CONTAINMENT TWE
in.0.a-304 s s- F L E X I U HOSE CABLE
SPRlNo LOADED ANTIBACKL HEAD AND lOLER GEAR
3hx2h.ECCENTRIC4 REDUCER
\\\a\\
1\11
7 1 6 In.RADIUS x 30. BEND
COOLANTEXHAUST-
GUIDE BARS,
4AT 90.
BEADED POISON ELEMENTS
2 h CONTAINMENT THIMBLE
Fig. 3
Control Rod Drive Assembly
-8W
78806
ORNL-LR-OW0
SYNCHRO NO. 2 600 PER INCH OF ROD MOTION
POTENTIOMETER SYNCHRO NO.(
GEAR REDUCER N0.2
OF ROO MOTION
INPUT SPROCKET
GEAR REDUCER No. iI
SPROCKET CHAIN
OVERRUNNING CLUTC
FLEXIBLE TUBULAR ROD SUPPORT
1
REACTd)ERL?ESSEL
CORE
IHl TEMPERATURE,* 44OOOF POSITION INDICATOR AIR FLOW RESTR ICTOR AIR DISCHARGE RADIAL PORTS
-
Fig.
4
D i a g r a m of Control Rod Drive
-9Y
s e n s i t i v i t y of 5" per inch of rod motion.
Synchro number 2 i s used
f o r f i n e p o s i t i o n i n d i c a t i o n and has a s e n s i t i v i t y of 60" per inch. The s i g n a l f r o m t h e coarse p o s i t i o n synchro i s transmitted t o a torque a m p l i f i e r which d r i v e s a single-turn potentiometer feeding a d-c s i g n a l t o t h e MSRE on-line computer.
After t h e system had operated f o r some time, it became d i f f i c u l t t o o b t a i n reproducible r e g u l a t i n g rod p o s i t i o n changes f o r a given time of i n s e r t or withdraw.
This was due t o t h e wearing of t h i s
rod and d r i v e assembly caused by frequent use.
For t h e dynamics
t e s t s , one of t h e rods normally used as a shim rod was used as t h e r e g u l a t i n g rod.
Since it i s moved much less frequently t h a n t h e
normal r e g u l a t i n g rod, it had experienced l e s s wear and had much t i g h t e r response c h a r a c t e r i s t i c s . There a r e a number of f a c t o r s which could adversely a f f e c t both t h e accurate positioning of t h e rods and t h e i n d i c a t i o n s of e f f e c t i v e rod p o s i t i o n given by t h e instruments.
Some of t h e p o t e n t i a l sources
of d i f f i c u l t y a r e l i s t e d i n Appendix A. These observations i n d i c a t e t h a t t h e MSRE c o n t r o l rods a r e hardly i d e a l l y s u i t e d f o r dynamic t e s t i n g .
However, since no provision w a s
made f o r s p e c i a l c o n t r o l rods and since t h e main f e a t u r e s of i n t e r e s t occur at low frequency, t h e t e s t i n g program w a s c a r r i e d out using t h e standard MSRE c o n t r o l rods.
Fortunately, t h e rods performed
far b e t t e r t h a n expected by t h e i r designers, and t h e f i n a l r e s u l t s were only s l i g h t l y degraded by equipment problems. B.
Test Signals Used i n t h e Experiments Three d i f f e r e n t types of t e s t s i g n a l s which were used t o obtain
t h e frequency response of t h e system a r e described i n t h i s s e c t i o n . 1) Pseudo-random Binary Test ( 4)
I n t h i s t e s t , s p e c i a l l y s e l e c t e d periodic s e r i e s of p o s i t i v e and negative r e a c t i v i t y pulses c a l l e d t h e pseudo-random binary sequence (PRBS) were introduced.
The PRBS has t h e advantage t h a t i t s
frequency spectrum c o n s i s t s of a number of harmonics of approximately equal s i z e .
This means t h a t t h e frequency response may be evaluated
a t a l a r g e number of frequencies i n a s i n g l e t e s t .
The spectrum of
-10t h e pseudo-random s i g n a l from one of t h e MSRE t e s t s i s shown i n Fig. 5.
We note t h a t t h e s i g n a l s t r e n g t h i s concentrated i n
d i s c r e t e harmonic frequencies r a t h e r t h a n d i s t r i b u t e d over a continuous spectrum as i n t h e case of non-periodic (e.g. pulse and step) signals.
T h i s i s h e l p f u l since it improves t h e e f f e c t i v e s i g n a l -
to-noise r a t i o a t t h e s e frequencies. A PRES may be generated on-line a t t h e t e s t or may be pre-recorded
i n some fashion and played back as a c o n t r o l s i g n a l .
On-line
generation of t h e s i g n a l w a s used i n t h e s e t e s t s ( s e e Section V )
.
A PRBS i s characterized by t h e number of b i t s i n t h e sequence and
t h e b i t duration.
A b i t i s defined as t h e minimum possible pulse
All pulses i n a PRES a r e minimum width or
d u r a t i o n i n t h e sequence. i n t e g r a l multiples t h e r e o f .
Numerous sequences may be generated, but
they a r e r e s t r i c t e d t o c e r t a i n s p e c i f i c numbers of b i t s .
PRBS t e s t s were run
with
19, 63, 127
and
511
bits.
I n t h e MSRE,
I f t h e number
of b i t s i s Z and t h e b i t duration i s A t , t h e n t h e PRES has a period Z At.
The lowest harmonic r a d i a n frequency, cu
0’
t h e harmonics, DLD, i s given by LU
0
= DLD =
and t h e spacing of
K.The mt
were run and analyzed a r e shown i n Table 2 .
PREE t e s t s which
I n each t e s t , t h e rod
motion w a s s e l e c t e d t o give a r e a c t i v i t y change of 0.02$ t o 0.0346, peak t o peak.
The m a x i m u m r e a c t i v i t y p e r t u r b a t i o n was determined on
t h e basis of keeping t h e r e s u l t i n g power l e v e l perturbations i n t h e l i n e a r range, i . e . 6N/No maximum w a s kept below 0.1. 2)
Pulse Tests (4) I n theory, it i s p o s s i b l e t o e x c i t e a system with a s i n g l e
pulse-type disturbance and obtain t h e frequency resonse by numerically determining t h e r a t i o of t h e Fourier transform of t h e output t o t h e Fourier transform of t h e input.
The frequency response can t h e o r e t i c a l l y
be evaluated at a l l frequencies s i n c e t h e input has a continuous
frequency spectrum.
I n p r a c t i c e , t h e pulse t e s t i s o f t e n u n s a t i s f a c t o r y
because t h e a v a i l a b l e s i g n a l s t r e n g t h i s d i s t r i b u t e d over a l l frequencies, r e s u l t i n g i n a small amount of s i g n a l around t h e a n a l y s i s frequency and t h u s poor e f f e c t i v e s i g n a l t o noise r a t i o .
‘(
ORNL
-
I
I
I II
I
I
III
1
1
1
I
I
I
I
10L2 Frequency
Fig.
DWG.
I
(radiana/eec)
5 Power Spectrum of the Input PFE!S at 0.465 MW
66-l
1076
’
Several t e s t s using approximately square r e a c t i v i t y pulses of between 0.01 and 0.02% were employed f o r t h e MSRE at zero power. Tests were c a r r i e d out both with c i r c u l a t i n g f u e l and with s t a t i o n a r y fuel. Table 2 . Power Levels (MM)
Pseudo-Random Binary Sequence Tests
Bits i n Sequence
Bit Duration ( s e c )
P e r i o d i c i t y of t h e PRBS ( s e c )
0
6.58
0
2 11
1.o
3 035 3 035 3 935 3 935
1.o
5.02
2 05
3 *32 4 097 3 *33 4 *97 3 -32 3 *33 4 997
638 1699 631 1701
0075 .465
2 05
5 -0 5 00 6 97 7.5 7 95
511 511 =7
Minimum Frequency (rad/sec)
1711 172 1711
631 1698 1701 631
3) Step Tests ( 5 ) If t h e output eventually l e v e l s off t o some constant value
a f t e r a s t e p disturbance, t h e frequency response may be obtained by a modified Fourier a n a l y s i s of t h e output and t h e input.
A s with t h e
pulse t e s t s , t h e s t e p input has a continuous spectrum, but i s hampered by a low e f f e c t i v e signal-to-noise r a t i o a t any a n a l y s i s frequency. Step t e s t s were used at power l e v e l s where t h e temperature feedback w a s adequate t o cause t h e power t o l e v e l off near t h e o r i g i n a l power a f t e r t h e r e a c t i v i t y change
of 0.01 t o 0.02%.
These t e s t s used a r e a c t i v i t y perturbation
Step t e s t s were done a t 2.5,
5.0, 6.7, and 7.5 MW.
v
-13V,
W
A.
EXPERIMENTAL PROCEDURES
Implementation of Pseudo-random Binary Tests The pseudo-random binary r e a c t i v i t y sequence required a very
p r e c i s e l y controlled s e r i e s of regulating rod i n s e r t i o n s and withdrawals.
Since t h e frequency range of i n t e r e s t w a s from about 0.002
t o 1.0 radian/sec.,
t h e rod jogger w a s designed s o t h a t sequence with
a b i t time of 3 t o 5 seconds could be run f o r as long a s 1 hour.
The
rod jogger system, which required no special-purpose hardware, consisted of a hybrid computer c o n t r o l l e r shown schematically i n Fig.
6. The portable EAI-TR-10 analog computer w a s used t o c o n t r o l t h e b i t t i m e and t h e rod d r i v e motor “on” times f o r t h e i n s e r t and withdraw commands.
The MSRE d i g i t a l computer was programmed t o c o n t r o l t h e
sequencing of t h e pulse t r a i n by means of a s h i f t - r e g i s t e r algorithm. ( 3 ) The number of b i t s i n t h e sequence could be varied over a range between
3 and 33,554,431 b i t s . The rod jogger system performed extremely well, as it was a b l e t o p o s i t i o n t h e rod with an indicated positioning accuracy of about
+0.01 i n .
(corresponding t o +O.OOO5$ 6k/k) out of 1/2 i n . peak-to-peak
rod t r a v e l f o r over a 500 insert-withdraw operations.
A t y p i c a l PRBS
rod p o s i t i o n s i g n a l i s shown i n Fig. 7 and a t y p i c a l record of neutron flux changes during a PRES t e s t i s shown i n Fig. 8. The analog computer w a s a l s o used t o amplify and f i l t e r t h e rodp o s i t i o n and power-level s i g n a l s p r i o r t o d i g i t i z i n g .
Throughout t h e
dynamic t e s t s t h e MSRE computer was used i n t h e fast scan mode t o d i g i t i z e and s t o r e t h e d a t a on magnetic tape. sampled a t t h e r a t e of
B.
4 per
Each v a r i a b l e w a s
second.
Implementation of Pulse and Step Tests The same s e t up as described i n ( A ) was used for t h e pulse and
s t e p t e s t s with t h e PRBS generator omitted. VI. A.
ANALYSIS PROCEDURES
Direct Analysis of PRBS Tests The “ d i r e c t ” method of a n a l y s i s uses a d i g i t a l computer simulation
of an analog f i l t e r i n g technique f o r obtaining cross-power s p e c t r a l U
d e n s i t y functions. ( 6 ) A block diagram of t h e analyzer i s shown i n
-14-
OANL-DWC 66-4763
PORTABLE ANALOG COMPUTER
r-------EAI-TR-40
1
MSRE DIGITAL COMPUTER
---------
rBR-340
I
I
I
I
I
I c
INSERT WITHDRAW
MOTOR
I 1
!
I
1
I I 1 1
-
MOTOR
PSEUDO-RANDOM BINARY SEQUENCE COMPUTATION
I1 I
AMPLIFYING - 1 AND FILTERING POWER-LEVEL
I
’
;
A’
‘ 1
I
I
I , DIGITIZER AND MAGNETIC TAPE RECORDER
I &
I
I I t
6
REACTOR
Fig. 6
I I
I
“ON”
I nn’SITION
I
I
BIT-TIME GENERATOR
I
1
Block D i a g r a m of MSRE R o d Jogging and Data Logging System
I I I
I I
-4-P
A
l
-16-
ORNL-DWO 66- 4764
42
I
I
4.47
8.34
I
I
I
I
-
38
P
9 c L 0 34
Bp
0'
8
30
a U w A
u
3
26
22 0
12.51 17.09 21.26 TIME ( 30-min FULL SCALE
25.43
Fig. 8 Measured Flux Signal During a 511 B i t PRBS T e s t
29.60
-.17 W
Fig. 9 .
The narrow band-pass f i l t e r H(s) has t h e c h a r a c t e r i s t i c s of
a quadratic l a g and a t r a n s f e r function S
H(s) = w2
0
+
2
2i wo s + s
where coo i s t h e center frequency of t h e f i l t e r .
Typically
t h e damping f a c t o r 5 i s set equal t o 0.05, giving a f i l t e r frequency T h i s analyzer i s s i m i l a r t o t h a t
response a5 shown i n Fig. 10. described by Chang")
and Van Deusen(8); t h e main d i f f e r e n c e i s t h a t
it includes e x t r a cross-multiplication terms r e s u l t i n g i n higher accuracy. A mathematical d e s c r i p t i o n of t h e method i s given i n Appendix B.
B.
I n d i r e c t Analysis of PRBS Tests The i n d i r e c t a n a l y s i s began with a c a l c u l a t i o n of t h e
a u t o c o r r e l a t i o n function of t h e input and t h e cross-correlation function
of t h e input and t h e output
c,(d
-
= ;1 JT:(t) O ( t +
T*
0
where
Cll(s) = t h e autocorrelation function CX(
T)
= t h e cross-correlation function
T
= the lag t i m e
T* = t h e i n t e g r a t i o n t i m e ( a multiple of t h e s i g n a l
r(t) = the
input s i g n a l
O ( t ) = t h e output s i g n a l
t
= time
periodicity)
-18-
QUAD PMR R
Fig. 9
Block Diagram of Cross-Parer Spectral Density Analyzer
-19-
ORNL-DWG 66-10635
10 5
2
5
2
0.01
0.01
2
5
0.1
2
5
10
2
5
10
2
5
100
2
5
10
2
5
100
W?%
t 120
t 90 t 60 t30 0
-30 -60
-90 -120 0.01
2
5
0.1
2
5
1.0 W/WO
Fig. 10 Frequency Response of Filter Used in CFSD Analyzer
-20-
Subsequent Fourier a n a l y s i s of C
(T)
11
and C
(7)
12
gave t h e input
W
power spectrum and t h e cross power .spectrum:
( ) 11 %
Q,
Q,12% ( )
1
=
J
Cl1(7)
COS %T
dz
cos %T
dz
0
rJ
T
= 1
Cz (,)
- j r1 J
0
T C=(z)
s i n %z
d7
0
where
'wz frequency, wz
( ) = input power spectrum at frequency 11 %
Q,
%(%)
% T
= c r o s s power spectrum at
= angular frequency of t h e k t h harmonic =
(% = 2kfi -) T
p e r i o d i c i t y of t h e t e s t s i g n a l
Note t h a t t h e input power spectrum i s a r e a l q u a n t i t y s i n c e C always a n even f u n c t i o n of
11( 7 ) i s
7.
The frequency response, G( ju$), i s given by
2, JTC,(z) T o Re[G(j%)l =
1
-J T
I m [ G ( j(Lx) 1 =
-
T
o
c&)
cos
%z dz
cos
w.;"
az
s i n %z dz T Jo Cz(,) . . -T 1 T Jo Cl1(7) cos u y dz
The magnitude r a t i o , MR, and t h e phase, 0, are given by:
e(%)
= tan-1
~m [G(J%) 3 {Re [G( j%) ]}
A F o r t r a n computer code f o r t h e
Im-7090 o r IBM-360c a l l e d CABS( 9 )
w a s prepared t o c a r r y out t h e s e computations.
-21-
Step Response Test Analysis
C.
The s t e p response t e s t s were analyzed using a d i g i t a l computer code which implements Samulon's method. (5,101 Pulse Response Test Analysis
D.
Although pulse response t e s t s were attempted a t power l e v e l s
-
of 0, 75 KW, 465 KW, and 1 MW, t h e only successful runs were t h e ones made at zero power. T h i s w a s because t h e low frequency random f l u c t u a t i o n s i n heat load at low (but non-zero) powers d r a s t i c a l l y A t t h e s e powers, t h e r a d i a t o r i s
reduced t h e signal-to-noise r a t i o .
cooled primarily by n a t u r a l convection and r a d i a t i o n , and i s consequently s e n s i t i v e t o atmospheric disturbances.
A t higher powers, where most
of t h e cooling was due t o forced convection, t h e s e f l u c t u a t i o n s were not apparent. The zero power t e s t s required extremely accurate core temperature c o n t r o l and rod positioning i n order t o avoid d r i f t of t h e f l u x l e v e l . Since a zero power r e a c t o r i s an i n t e g r a t i n g system, a pulse r e a c t i v i t y input r e s u l t s i n a change i n steady-state f l u x output, and Fourier transforms are v a l i d only i f t h e response function eventually r e t u r n s t o i t s i n i t i a l value.
We got around t h i s problem by f i r s t numerically
f i l t e r i n g t h e output response d a t a through a high-pass network HP( s ) with a t r a n s f e r function
t h e n performing t h e numerical Fourier transform, and f i n a l l y compensating t h e r e s u l t i n g frequency response f o r t h e high-pass f i l t e r c h a r a c t e r i s t i c s . The numerical Fourier transform calculations were done by a d i g i t a l computer code using a novel method
which i s mathematically equivalent
t o t h e standard technique of summing t h e products of f ( t ) f(t)
cos ust and
s i n ut, but which reduces t h e computing time by a f a c t o r of 3. VII.
RESULTS
The experimental data were analyzed t o give c o r r e l a t i o n functions, input power s p e c t r a and cross power spectra, and frequency responses. These r e s u l t s a r e given i n t h i s s e c t i o n along with t h e d i r e c t l y observable
-
-22
t r a n s i e n t response t o system disturbances. A.
U
Transient Responses A t each power l e v e l , a t r a n s i e n t w a s induced by i n s e r t i n g a
r e a c t i v i t y pulse or step, or by simply allowing t h e system t o seek These t r a n s i e n t
equilibrium after t h e completion of some other t e s t .
responses a r e informakive i n themselves since they demonstrate t h e damping and t h e n a t u r a l frequency of o s c i l l a t i o n of t h e system. Figure 11 shows t h e observed t r a n s i e n t response.
At
0.075 Mw,
it took over two hours f o r t h e flux t o r e t u r n t o equilibrium. B.
Correlation Functions The pseudo-random binary t e s t s were analyzed by t h e d i r e c t method
and t h e i n d i r e c t method.
Autocorrelation functions of t h e input and
cross c o r r e l a t i o n functions of t h e input and output were obtained as intermediate r e s u l t s by t h e i n d i r e c t method.
The c o r r e l a t i o n functions
had t h e expected appearance i n a l l t e s t s u n t i l t h e 2-1/2 MW t e s t s . I n t h a t t e s t , spikes appeared i n t h e c o r r e l a t i o n functions which have been present i n a l l t e s t s since then. a t points
432
The spikes always appeared
sec. from each end of a period i n t h e 5ll b i t t e s t s
(- 1700 sec. period), and 34 sec. from each end i n t h e 127 b i t t e s t s (- 630 sec. p e r i o d ) .
Figures I 2 through
19 show t y p i c a l a u t o c o r r e l a t i o n
functions and cross-correlation functions for t e s t s before t h e 2-1/2 MW t e s t and a f t e r t h e 2-1/2 MW t e s t . The reason f o r t h e appearance of t h e spikes i s not yet known. only s i g n i f i c a n t difference noted i n t h e input s i g n a l a t
0.465 MW
The and
a t 2-1/2 MV was a change i n e f f e c t i v e pulse duration f o r p o s i t i v e and
negative pulses.
At
0.465 MW,
t h e duration of a pulse above t h e m i d -
point w a s t h e same as t h e duration of a pulse below t h e midpoint.
At
2-1/2 MW, t h e duration of pulses below t h e midpoint w a s longer than t h e duration of pulses above t h e midpoint.
This w a s apparently due
t o changes i n t h e coasting c h a r a c t e r i s t i c s of t h e rod.
Calculations
were performed on pseudo-random binary sequences i n which t h e pulse duration w a s changed f o r p o s i t i v e and negative pulses t o determine whether t h i s would cause spikes i n t h e autocorrelation function. These calculations showed spikes f o r some sequence lengths but not f o r others.
I n p a r t i c u l a r , spikes were not found i n t h e s e c a l c u l a t i o n s
for a 5 l l b i t sequence.
The c o n f l i c t i n g i n d i c a t i o n s furnished by
-23-
ORNL-DWG 66-10284
40 20
0
-20
20
0 -20
0
2
4
6
8
10
12
14
18 2 0 TIME (min)
16
22
24 26 2 8
Fie. 11 MSRE Power Level Transients
30 32 34 36
GRNL
DWG.
M-l
. . .
.
I
. e
w
0
ho0
800
Comlation
Fig.
I2
Input
I200
Tim
bee)
Autocorrelation Function PRBS Test at 0.465 MW
for a 511 Bit
lb00
lm8
‘(
OWL
I200
400
col-lFlation
Fig.
The
(sac)
13 Cross-Correlation Function for a 511 Bit PRBS Test at 0.465 MW
DWG.
l600
66-11079
’
0
160
320
Correlation
Fig.
. (
4eo
Time
(set)
14 Input Autocorrelation Function 127 Bit PRBS Test at 1.0 MW
for a
640
‘(
ORNL
DWG.
66-l
1081
640
Correlation
Fig.
Time
(em)
15 Cross-Correlation Function for a I27 Bit PIES Test at 1.0 MW
.
ORNL DWG. 66-11082
0
Corralation % (sac)
Fig. 16 Input Autocorrelation Function f o r a 511 B i t PRBS T e s t a t 2.5 MW
ORNL DWG. 66-11083
h
a
Y
r)
B P E . Y 4
s
Y
C 4 .
3 0
.4 Y
1
I
4k V
800 correlation Tiw (eec)
Fig.
17 Cross-Correlation Function for a 5 1 1 Bit PRBS Test at 2.5 MW
ORNL
DWG.
66-11084
* 1 . .
I
I
*
‘
I 0
I60
oorralatioa
Fig.
18 Input
I
I 480
320
The
6bo
(ccc)
Autocorrelation Function for a I.27 Bit F!ElBSTest at 2.5 MIW
,(
ORNL DWG. 66-1 1085
0
160
Fig. 19 Cross-Correlation Function for a 127 Bit PRBS T e s t at 2.5 MW
610
t h e s e c a l c u l a t i o n s have not yet been resolved.
However, w e should
note t h a t t h i s unexpected f e a t u r e of t h e c o r r e l a t i o n functions does
It simply
not i n v a l i d a t e t h e f i n a l r e s u l t , t h e frequency response.
means t h a t t h e s p e c t r a l p r o p e r t i e s of t h e input and output a r e s l i g h t l y d i f f e r e n t than anticipated, but t h a t t h e r a t i o of cross power spectrum t o input power spectrum s t i l l gives a v a l i d frequency response. C
.
Frequency Responses The r e s u l t s of t h e frequency response analyses are shown i n Figures
20 through 28.
The legend i n each f i g u r e i n d i c a t e s t h e type of t e s t
and t h e a n a l y s i s procedure.
The t h e o r e t i c a l curves were taken from
previously published r e s u l t s ( * ) when t h e experimental power l e v e l s corresponded t o a v a i l a b l e calculations.
The curves f o r t h e other cases
were obtained using t h e same procedures as were used t o obtain t h e published r e s u l t s .
Results obtained by Roux and Fry (
by analyzing
t h e random noise i n t h e neutron f l u x s i g n a l a r e a l s o shown with t h e zero-power r e s u l t s .
These r e s u l t s were normalized t o t h e t h e o r e t i c a l
r e s u l t s a t 9 rad/sec.
These points were obtained by t a k i n g t h e square
root of t h e measured power s p e c t r a l density ( E D ) of t h e neutron f l u x s i g n a l a f t e r subtracting t h e background PSD. This gives a r e s u l t which i s proportional t o t h e amplitude of t h e f l u x - t o - r e a c t i v i t y frequency response, assuming t h a t t h e observed noise i s t h e r e s u l t of a white noise r e a c t i v i t y i n p u t . The n a t u r a l period of o s c i l l a t i o n of t h e system w a s determined e i t h e r by d i r e c t observation of' a t r a n s i e n t or by l o c a t i o n of t h e peak o f t h e frequency response.
These r e s u l t s , along with t h e o r e t i c a l
predictions, a r e shown i n Fig. 29. VIII.
IETERPRFTATION OF THE RESULTS
The objectives of t h e dynamic t e s t i n g program a r e twofold:
1)
provide information on t h e s t a b i l i t y and o p e r a b i l i t y of t h e system; 2)
provide inf'ormation f o r checking procedures f o r making
t h e o r e t i c a l predictions s o t h a t f u t u r e c a l c u l a t i o n on t h i s and other similar systems w i l l be improved.
terms of t h e s e two objectives.
The r e s u l t s a r e i n t e r p r e t e d i n
W
-33v
to4 5
2 103 5
2
102
5
2
.
0.001
0.005 0.01 0.02
0.05 W,
01 0.2 0.5 !I) 2.0 FREQUENCY (rodionskac)
5.0
100 200
ORNL-DWG 65-8907A
20 10
0
-10 -20 0 0)
'D
501) 100.0
-30
Y
w -40 v)
a r -50
a
-60
-70
-80 -90 -100
0.001
0.01
Fig. 20
0.4 w, FREQUENCY (radians/sec)
1
~9
~N/N Frequency Response of 6K KO at Zero Power;
Fuel C i r c u l a t i n g
ORllLDYIi 66-10708
7-Scc Pulse, Teat l o . 1
'/-See Pulse, Test No. 2 3-1!2-Sec
Pulse, Test Ih. 3 4
R i l s e , Test l o . h 0 Nolee Analysis j-1/2-8cc
0.001
0.01
I. 0
0.1
w , FREQUENCY ( radions/sec 1
WNo
Fig. 21 Frequency Response of 6K
at Zero Power; 7 Fuel S t a t i c
lo
-35W
0.001
0.002
0.005
0.01 0.02 0.05 O.! FREOUENCY (rodionr/sec)
0.2
0.5
4.0
70
60
50 40
30 20 c
Y
W
tn
9
Q
0
-io
-20 -30 -4 0
-50 -60 -70 .-
0.OOi
0.002
Fig. 22
0.005
0.01 0.02 a05 0.i FREQUENCY (rad ions /sec 1
Frequency Response of
0.2
0.5
Power = 0.075 MM
i.0
-36W
5
IKI "/ifo
2
to3 5
2
0.001
0.005
0.002
0.01 0.02 0.05 0.1 FREOUENCY (radians /set)
0.5
0.2
1.0
ORNL-OWG 66-4770
60
-_
I
I
I I 1 I l l
I
I
I
I
I
POWER LEVEL 0.465 MW 511 B1T PRES DIRECT ANALYSIS 0 - 511 BIT PRES INDIRECT ANALYSIS 0
50 n
40
I
30
-
I
_-
-__
20 10
b
o " 0 w -10
v)
a * -20 a
-30 -40
- 50 - 60
-70 - 80
0.001
0.002
0.005
0.01
0.02
0.05
0.1
0.2
0.5
FREQUENCY ( radians/sec
~N/NFig. 23
Frequency Response of
Power = 0.465 MW
1.0
-37ORNL- DWG 66-4771
7 B I T PRBS-INDIRECT ANALYSIS 0 I BIT PRBS-DIRECT ANALYSIS V I BIT PRBS-INDIRECT ANALYSIS 0
SAMPLING INTERVALS : INDIRECT ANALYSIS - 4 sec
0.004
0.002
0.005 0.01
0.05
0.02
0.1
0.5
0.2
1.0
FREQUENCY (rodions/sec)
90
.
t
/
80
ORNL-OWG 66
\ \I
1
I
I
I I l l 1
- 4772
I
70
60
50 40 0,
aI
Z W
U I
a
30
20 40
0 -10
-20 -30 -40
-50 -O.CO4
0.002
0.005
0.01 0.02 0.05 0.t FREQUENCY (radions/sec)
0.2
0.5
~N/N Fig. 24
Frequency Response of
%3
; Power = 1.0MW
LO
Y
0001
0002
000s
002
001
02
01
005
05
1.0
FREOUENCY (mdianshs)
ORNL-DWG 66-100
90
$
80 0 127 BIT PRBS - DIRECT ANALYSIS 127 BIT PRBS - INDIRECT ANALYSIS A 511 BIT PRBS DIRECT ANALYSIS A
70 60
-
50
40
-3 3020 Y
10
a E
O -10 -20
-30 -40
-50
-60 -70 0.001
0.002
0.005 0.01
0.02 0.05 0.1 FREQUENCY (rodians/sec)
0.2
0.5
6M/N Fig. 25
Frequency Response of
Power q’ O
*
= 2.5 MN
1.0
-392
127 BIT PRBS -DIRECT ANALYSIS 127 BIT PRES INDIRECT ANALYSIS A PABS DIRECT ANALYSIS A
-
d
2
K? .-
0.-
0.002
0.01
0.05
0.02
0.1
02
1.0
0.3
FREOUENCY (rodianr/sec) ORNL-DWG 66-10035
I
1
I
t
0.001
0.002
0.005
0.01 0.02 0.05 0.1 FREQUENCY (radianslsec)
~N/N Fig. 26 Frequency Response of 6K‘ O
0.2
0.5
= 5 MW 7. Power ’
10
-40ORNLOrCI 66-IOO39
2
io3
2
lo2 a002
0.005
001
0.02
0.05
01
0.5
02
FREQUENCY hdionshec)
~N/N
Fig. 27
Frequency Response of
E q Oq ’*
Power =
6.7
MW
1.o
W
-41ORNL-DWG €640040
2
-
POWER LEVEL 7.5 Mw STEP TEST 127 BIT PRBS -DIRECT ANALYSIS
lo3
0
127 BIT PRBS- INDIRECT ANALYSIS A 511 BIT PRBS DIRECT ANALYSIS A
-
2
I .-d 0.002
0.005
0.01
0.02 FREQUENCY
0.05
0.1
0.5
0.2
10
(radions/sec) ORNL-DWG 66-10036
80 70
60 50
40 30
a
-10
-20 -30
- 40 -50 -60 -70 0.001
0.002
0.005
0.01 0.02 0.05 0.I FREQUENCY (radianshec)
Fig. 28 Frequency Response of
02
Power = 7.5 MW
0.5
I .o
-42-
5
2
2
1
0102
0.05
0.1
0.2
0.5
1 .o
2 .o
FOWER LEVEL (MW)
Fig. 29 MSRE Natural Periods of Oscillation
5.0
10.0
-43 I
The l i n e a r s t a b i l i t y of t h e system i s c e r t a i n l y adequate.
The
frequency response shows a resonance which s h i f t s t o higher frequencies and lower amplitudes as power increases.
This means t h a t t h e t r a n s i e n t
response t o a disturbance a t low power w i l l display a l i g h t l y damped, low frequency r e t u r n t o equilibrium (period g r e a t e r than t e n minutes f o r powers l e s s than 500 KW).
A t higher power t h e system response
i s much more strongly damped and much f a s t e r .
For instance, a
disturbance at 7.5 MM causes a t r a n s i e n t which i s e s s e n t i a l l y completed i n 1-112 minutes.
These observations are i n good agreement with p r i o r
predictions. A d e t a i l e d q u a n t i t a t i v e check of t h e t h e o r e t i c a l predictions by
experimental t e s t s i s much more d i f f i c u l t than a comparison of more general dynamic f e a t u r e s such as s t a b i l i t y , l o c a t i o n of resonance peaks and t h e changes expected i n t h e s e performance measures with power l e v e l .
Early attempts t o f i t parameters i n t h e t h e o r e t i c a l
model t o give agreement i n t h e absolute amplitude of t h e frequency response were abandoned bacause of u n c e r t a i n t i e s i n t h e measured amplitudes caused by equipment l i m i t a t i o n s .
While a l l of t h e t e s t s
at a given power l e v e l give r e s u l t s with t h e same shape, t h e r e i s a
d i f f e r e n c e i n t h e absolute magnitude r a t i o s . this bias effect. abwe
Figure 27 c l e a r l y shows
Furthermore, t h e portion of t h e frequency response
0.3 rad/sec should be t h e same f o r a l l power l e v e l s since
feedback e f f e c t s a r e small i n t h i s frequency range and t h e zero power frequency response should dominate.
The experimental r e s u l t s f o r
various power l e v e l s show t h e same shape i n t h i s frequency region, but d i f f e r e n t absolute amplitudes. problem.
T h i s f u r t h e r i n d i c a t e s a bias
This b i a s problem i s not s u r p r i s i n g i n view of t h e equipment
c h a r a c t e r i s t i c s discussed i n Section I V . I n s p i t e of t h e bias d i f f i c u l t i e s , one feature of t h e t h e o r e t i c a l model i s shown t o be incorrect by experimental r e s u l t s . power ( g r e a t e r than
A t high
5 MM) t h e t h e o r e t i c a l magnitude r a t i o curve has
a d i p at 0.2 rad/sec.
T h i s i s due t o t h e reappearance of a f u e l
salt temperature slug i n t h e core af'ter t r a v e l i n g around t h e primary W
loop.
Since t h i s d i p was not observed experimentally, t h e r e must be
-44more mixing and heat t r a n s f e r i n t h e primary loop t h a n w a s included i n t h e t h e o r e t i c a l model. Because t h e predicted frequency response has t h e c o r r e c t shape and l o c a t i o n on t h e frequency axis, we f e e l t h a t t h e model used f o r t h e E R E dynamic a n a l y s i s i s a good representation of t h e system. The only discrepancy observed i n predicted and observed shapes i s t h e predicted dip a t 0.2 rad/sec,
j u s t discussed.
The apparent
b i a s i n t h e measured amplitude unfortunately p r o h l b i t s d e t a i l e d parameter f i t t i n g .
-45REFERENCES 1. R. C. Robertson, MSRE Design and Operations Report, Part 1: Description of Reactor &sign, USAEC Report om-TM-728,Oak Ridge National Laboratory, January 1965. 2.
S. J. Ball and T. W. Kerlin, Stability Analysis of the Molten-Salt Reactor Experiment, USAM: Report ORNL-TM-lO7O, Oak Ridge National Laboratory, December 1965.
3. T. W. Kerlin, The Pseudo-Random Signal f o r Frequency Response Testing, USAM: Report om-TM-1662,Oak Ridge National Laboratory, September 1966.
4. J.
0. Hougen and P. A. Walsh, Pulse Testing Method, Chem. E!ng. 57( 3 ) : 69-79 (March 1961).
Prog.,
5. H.
A. SamuLon, Spectrum Analysis of Transient Response Curves, Proc. IRE, 39, 173-186 (1951).
6. S. J. Ball, Instrumentation and Controls Div, Annual Progr. Rept. Sept. 1, 1965, USAEC Report ORNL-3875,pp. 126-7,Oak Ridge National Laboratory.
7. S.S.L. Chang, Synthesis of Optimum Control Systems, Chap. 3, McGraw-Hill, New York, 1961.
8. B. D. Van Deusen, Analysis of Vehicle Vibration, ISA Trans. 3, No. 2, 138-L48, 1964. 9. T. W. Kerlin and J. L. Lucius, CABS - A Fortran Computer Program for Calculating Correlation Functions, Power Spectra, and the Frequency Response from Fxperimental Data, USAEC Report ORNL-TM1663, O a k Ridge National Laboratory, September 1966. 10. E. M. Grabbe, S. Ramo, and D. E. Wooldridge (Editors), Handbook of Automation, Computation, and Control, Vol. 1, Chap. 22, J. Wiley and Sons, New York, 1958.
11. S. J. Ball, A Digital Filtering Technique f o r Efficient Fourier Transform Calculations, USAM: ORNL-TM report (in preparation)
.
12. D. P. Roux, and D. N. Fry, ORNL Instrumentation and Controls Division, personal communication.
13.
J. E. Gibson, Nonlinear Automatic Control, Chap. 1, McGraw-Hill, New York, 1963.
-46 14.. G. C. Newton, L. A. Gould, and J. F. Kaiser, Analytical Design of Linear Feedback Controls, pp. 366-381, J. Wiley and Sons, New York,
1957. 15. C. T. Morrow, Averaging Time and Data-Reducing Time for Random Vibration Spectra, J. Acoust. SOC. Am., 30, 456 (1958).
16.
T. J. Karras, Equivalent Noise Bandwidth Analysis from Transfer Functions, NASA-IITN-D2842, November 1967.
17. L,
D. Ehochson, Frequency Response Functions and Coherence Functions f o r Multiple Input Linear Systems, NASA-CR-32, A p r i l l S 4 .
18. H. M. Paynter and J. Suez, Automatic D i g i t a l Setup and Scaling of Analog Computers, Trans. ISA, 3 , 5544, January 1964. lg.
S. J. B a l l and R. K. Adams, MATED, A General Purpose D i g i t a l Computer Program f o r Solving Nonlinear Ordinary D i f f e r e n t i a l Equations by t h e Matrix Exponential Method, USAM: ORNL r e p o r t ( i n p r e p a r a t i o n ) , O a k Ridge National Laboratory.
Y
-47APPENDIX A
Y
P o t e n t i a l Sources of Experimental Error Due t o Equipment Limitations A s discussed i n Section IV.A,
t h e r e were a number of f a c t o r s t h a t
could have had adverse e f f e c t s on the experimental r e s u l t s , and they a r e l i s t e d below: 1) F r i c t i o n i n t h e bends i n the thimbles. Rollers a r e mounted i n t h e bends of the thimble, but t h e r e i s s t i l l considerable f r i c t i o n .
(Tests show t h a t t h e rods f a l l with an acceleration of only about 0.4 g . ) This suggests t h a t p a r t of t h e motion a t t h e rod d r i v e might go i n t o
bowing o f the f l e x i b l e hose r a t h e r than i n t o motion of the bottom of t h e poison section. 2)
Bends i n t h e hose.
It w a s observed t h a t an old MSFE c o n t r o l
rod used f o r out-of-pile t e s t i n g d i d not hang s t r a i g h t when suspended from t h e top.
It had gradual bends t h a t could be worked out by hand,
b u t which were not pulled out by t h e weight of t h e rod ( 6 , t o 8 l b ) .
If
such bends e x i s t i n t h e MSRF: rod used f o r t h e t e s t s , then t h e motion of t h e bottom of the rod w i l l not be the same a s the motion a t t h e top of the rod i f a bend i n the hose i s passing over a r o l l e r i n the bend of t h e thimble.
3 ) Restricted twisting.
The t e s t rod showed a tendency t o t u r n
when i n s e r t e d i n t o a mockup of t h e MSRF: thimble.
This t w i s t i n g i s pre-
vented i n t h e r e a c t o r since the top of t h e rod i s r i g i d l y connected t o t h e chain drive (see. Fig. 3).
If a tendency t o t w i s t i s prevented, t h e
hose w i l l bow and cause a difference i n a x i a l motion between upper and lower sections.
4)
Sprocket chain meshing.
The a c t i o n of t h e d r i v e motor i s t r a n s -
mitted t o the d r i v e chain by a sprocket.
This sprocket has a diameter
of 1.282 i n . and the l e n g t h of the links i n t h e chain i s
1/4 i n .
The
f a c t t h a t t h e f l a t links cannot exactly follow the c i r c u l a r contour of t h e sprocket means t h a t some of t h e sprocket motion i s taken up by a l a t e r a l motion of the chain as well as the desired v e r t i c a l motion.
5 ) Sticking o f poison beads.
The control rod thimble contains
-48vanes f o r centering the control rod. c i r c u l a r spacers located 4 i n . a p a r t .
The vanes a r e held i n place by If the vanes became warped, they
could touch l i n k s i n the poison chain i n c e r t a i n sections without touching l i n k s i n nearby sections.
Since t h e r e i s slack i n t h e threading of the
poison cylinders on t h e c e n t r a l hose, t h i s f r i c t i o n could h o l d up t h e movement of c e r t a i n poison elements.
6 ) Indicated p o i s i t i o n e r r o r s . It was necessary t o use t h e coarse synchro s i g n a l ( 5 deg of t u r n p e r inch of rod motion) f o r logging on t h e MSRE computer and f o r subsequent d a t a a n a l y s i s .
Since t h e 1/2-in. rod
motion used i n the tests corresponds t o only 2.5 deg r o t a t i o n of t h e synchro, sizeable percentage e r r o r could be caused by only a few t e n t h s of a degree of deadband i n the gears leading t o the synchro.
Periodic
c a l i b r a t i o n s of t h e logged rod p o s i t i o n a g a i n s t t h e f i n e synchro, however, indicated t h a t t h e e r r o r i s l e s s than Pj$ f o r a 1/2-in. rod t r a v e l , The maximum deadband i n the logger s i g n a l corresponded t o about k0.008 i n . o f rod motion.
-49APPENDIX B
Y
The Direct Method for Cross-Power Spectrum Analysis Each of the terms used i n t h e cross-power spectrum a n a l y s i s i s computed i n t h e following manner:
W
where HI and H,
are either H(jw) o r
which combination of H1 and H,
0 H( j w ) ; jw
and f($IO) (depending on
a r e used) i s r e l a t e d e i t h e r t o t h e r e a l
(COPOWER) o r t h e imaginary (QUAD POmR) p a r t of t h e cross-power s p e c t r a l
density (CPSD)
410.
Four combinations of t h e b a s i c computation shown
above a r e used i n the CPSD a n a l y s i s a s shown i n Fig.
9.
To convert f i l t e r e d t i m e domain functions t o frequency domain
functions, we make use of Parseval's theorem,13 which i s
-00
'CO
where B ( j w ) = Fourier transform of b ( t ) , and A(-jw) of the Fourier transform of a ( t ) .
= complex conjugate
-50
-
-.
Considering that only a f i n i t e i n t e g r a t i n g t i m e T i s a v a i l a b l e t o us :
Noting that
(3) B(j4 =
H2(jU)
o(j4 ,
(4)
where I(jw) = Fourier transform of input, i ( t ) , O ( j w ) = Fourier transform of output, o ( t ) .
From the d e f i n i t i o n of the complex conjugate, it can be shown t h a t A(-jw)
= H1(-jo) I(-jw)
,
(5)
B(-jw)
= H2(-jw)
0(-jw)
.
(6)
Hence ( 2 ) can be rewritten i n terms o f t h e Fourier transforms of the input and output s i g n a l s
s
0
T l a(t) b(t) dt Z 5
m
H 2 ( j w ) Hl(-jw)
I(-jw)
O(jw) dw
.
-0Q
Since the cross-power s p e c t r a l density i s defined as7
it behooves us t o operate on (7) i n order t o be able t o incorporate
#Io(ju).
Taking the l i m i t of both s i d e s of
T, we g e t
S u b s t i t u t i n g (8) i n t o (9):
( 7 ) as W and dividing by
(7)
-51W
Case 1 For the case where
HI( j w )
=
H2( jw)
=
H( jw)
and defining
T -EwT
Eq.
lim
a(t) b(t) dt
0
,
(10) becomes
Since t h e input and output s i g n a l s a r e f i l t e r e d i d e n t i c a l l y , it should be evident t h a t this operation w i l l y i e l d information only about t h e in-phase relationship, o r t h e r e a l p a r t of $Io( j w ) ,
We can show
t h i s by noting t h a t since
and
the two i n t e g r a l s i n Eqs. (11)and (12) must be equal; thus
and $
(jw) do not change much over t h e 01 But e f f e c t i v e bandwidth of the f i l t e r , then $Io(jw) must equal $oI(jw). If we assume that $Io(jw)
since
-52t h i s means t h a t the imaginary p a r t of '$ jw)
no information about
IO
( j w ) must be zero, o r a t l e a s t
] i s present i n t h e output.
For case 1,
then
If we assume t h a t Re['$ ( j w ) ] does not change much over t h e e f f e c t i v e IO bandwidth of H( j w )
,
For t h e present study, we used a f i l t e r with t h e following t r a n s f e r function: jw
H(jw) = w2
0
+
- w2
j w 2(w0
The f i l t e r "area" term can be evaluated using a t a b l e of integrals14
l J 2s
M IH( j w )
1'
dw =
-m
4b0
(rad/sec)
Thus Re['$IO(jw)]
4(w0
Case 2
For t h i s case
and H2(jw) = H ( j w )
.
Since w
H1(-jw)
=
0
H(-jw)
-jw
,
.
.
*
-53. t
Eq. (10) becomes
Since t h e input s i g n a l ' s f i l t e r has 90 deg more phase l a g than t h e output s i g n a l ' s f i l t e r , we should expect t h a t t h i s operation w i l l y i e l d icformation only about t h e quadrature relationship, o r t h e imaginary p a r t of $ I o ( j w ) .
We can show t h i s a s follows:
I n t h i s case, revising the order of i n t e g r a t i o n of t h e inputs makes a difference, i.e.,
from ( 2 )
Again, since
=
'm-, from (22) and (24) we can conclude
that
410(jw)
-
4oI(jw)
-jw
jw
i f we use t h e same argument as we did f o r case 1. Thus
which i s t r u e only i f the real p a r t of
4 IO
= 0, or a t least i f no
i s present i n t h e output. Hence, we can i n f o r $,,(jw) i n Eq. (22). For case 2, then
information about Re[d,,(jw)] substitute jIm[dIo(jw)]
If we assume that
Im[4 IO( j w ) ]
does not change much over t h e
e f f e c t i v e bandwidth of e i t h e r H ( j w ) or
WO H( j w ) , Jw
then
-54-
-. For t h e p a r t i c u l a r f i l t e r used (Eq.
17):
Thus
Case
3 The case where W
HI( j w ) = H2( j w ) =
0 H( j w ) jw
can be developed s i m i l a r l y t o case 1 as f a r as Eq.
(31)
(16), since
we could
redefine H ( j w ) a s being equal t o t h e expression i n (31). The i n t e g r a l t o be evaluated i s
Fcr t h e f i l t e r of Eq. (lT), t h i s i n t e g r a l i s again equal t o
assuming i n t h i s case, however, t h a t Re[$ ever t h e e f f e c t i v e bandwidth of
wO
- H(jw).
I , so 4b0
( j w ) ] does not change much
IO
jo
Case 4 The case where
can be developed s i m i l a r l y t o case 2.
Equation (10) becomes
-53 -
The expression on the right hand s i d e i s t h e negative of t h a t f o r Eq. ( 2 2 ) , case 2; hence f o r case
4, we
g e t t h e expression corresponding
t o Eq. (3) when ,the f i l t e r of Eq. (17) i s used:
assuming again that Im[$,,(
j w ) ] does not change much i n t h e e f f e c t i v e
bandwidths of e i t h e r H ( j w ) o r
w0 H( j w ) . JW
The power s p e c t r a l d e n s i t y (PSD) of t h e input function i s required
f o r c a l c u l a t i n g the system t r a n s f e r function G ( j u ) .
This i s obtained
by squaring t h e outputs of both t h e in-phase and quadrature f i l t e r s and i n t e g r a t i n g the sum of t h e squares.
I n both cases, f o r t h e f i l t e r of
m* (17): assuming t h a t
4 I1( j w )
does not change much i n t h e e f f e c t i v e pass bands
of H ( j w ) and wo H( j w ) / j w . The system t r a n s f e r function G ( j w ) i s then computed from
(39) where each of t h e t h r e e terms on t h e r i g h t hand s i d e of (39) a r e computed using t h e sum of two estimates. f a c t o r , 45w .) 0
(Note t h a t a l l terms have the same gain
The reason f o r the b e t t e r accuracy of t h i s method as
compared t o using a s i n g l e estimate of each term l i e s i n t h e f a c t t h a t since t h e e f f e c t i v e pass-bands of t h e in-phase and quadrature f i l t e r s
are d i f f e r e n t , t h e r e i s a b i a s i n each of t h e quadrature, o r imaginary term,
estimates.
Since t h e two imaginary term estimates a r e of d i f f e r e n t
sign, t h i s b i a s tends t o be cancelled out. Calculations of t h e percent standard deviations of both input and output (PSD) estimates a r e made using (40) :I5
-56 -
where
u = standard deviation of mean square value, x2 = mean square value,
B T
=
equivalent noise bandwidth, rad/sec,
=
integration time, sec.
The equivalent noise bandwidthi6 for the H ( j w ) filter used in this study is
B
=
nfw
0 ’
rad/sec.
(41)
The coherence function y 2 is also computed:
Id&. I “ The coherence function is useful for estimating expected errors in transfer function calculations when the input and output signals are random.17 For periodic signals, however, such as the PRBS, the expressions for error estimates in the literature have been found to be wildly pessimistic. The calculation of the response of the digital filters is based on Paynterts matrix exponential m e t h ~ d , ~ and ~ ,gives ~ ~ virtually exact time-response solutions very efficiently.
-57ORNL-TM-1647 Internal Distribution
1. R. K. Adams 2.
3. 4. 5. 6. 7. 8. 9-18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
28. 29. 30.
31. 32.
33. 34. 35.
40.
41. 42.
43. 44. 45. 46. 47. 48. 49. 50.
51. 52.
53. 54. 55. 56.
M. G. G. F. F. M. A.
Adamson Affel Alexander Apple Baes Baker Baker S. J. B a l l S. E. Beall E. S. E e t t i s F. F. Blankenship R. Blunberg E. G. Bohlmann C . J. Borkowski J. 3. Bullock G. H. Burger 0. W. Burke S. Cantor W. L. Carter G. I. Cathers E. L. Compere J. A. Conlin W. H. Cook L. T. Corbin G. A. C r i s t y J. L. Crowley F. L. C u l l e r R. A. Dandl H. P. Danforth D. G. Davis S. J. D i t t o R. G. Donnelly F. A. Doss N. E. Dunwoody J. R. Engel E. P. Epler W. K. Ergen D. E. Ferguson A. P. Fraas H. A. Friedman D. N. FV J. H. Frye, Jr. C . H. Gabbard R. B. Gallaher W. R. Grimes A. G. Grindell
9. 37. 38. 39.
G. R. L. R. C. J. M.
57. 58. 59. 60. 61.
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H. Guymon W. Hagen H. Harley S. E a r r i l l N. Haubenreich M. Herbert G. Herndon D. Holt P. X O ~ Z L. Hudson R. Kasten J. K e d l J. Kelly W. Kerlin S. K i r s l i s J. Knowles I. Krakoviak W. Krewson C . Kryter G. Lawson B. Lindauer I. Lundin N . Lyon H . Marable D. Martin E. Mathews G. MacPherson E. MacPherson C . McCurdy F. McDuffie
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E. McNeese J. M i l l e r V . Miskell ( Y - 1 2 ) R. Mixon L. Moore G. O'Brien Patriarca R. Payne M. Perry B. Piper E. Prince L. Redford Richardson C . Robertson C . Roller
R. H. 133 P. 134. H. 135 A. 136. H . 137 B. 138. J. 139. M. 140: R. 141. H. 142. D. P. Roux
132.
-58a
143. 144. 145. 146. 147. 148. 149. 150. 151. 152.
153 154. 155' 156. 157 158. 159 160. 161. 162.
H. C. Savage A. W. Savolainen D. Scott H. E. Seagren J. H. Shaffer M. J. Skinner A. N. Smith P. G. Smith W. F. Spencer I. Spiewak B. Squires R. C. Steffy H. H. Stone R. S. Stone A. Taboada J. R. Tallackson R. E. Thoma G. M. Tolson D. B. Trauger J. R. Trinko
163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173 174. 175 176-177. 178-179. 180-184. 185.
W. C. Ulrich B. H. Webster A. M. Weinberg K. W. West M. E. Whatley 5. C. White G. D. Whitman R. E. Whitt H. D. Wills J. V. Wilson L. V. Wilson M. M. Yarosh MSRP Director's Office, Rm. 219, 9204-1 Central Research Library Y-12 Document Reference Section Laboratory Records Department Laboratory Records Department, LRD-RC
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186-200*
Division of Technical Information Extension (DTIE) 201. Research and Development Division, OR0 202-203. Reactor Division, OR0 204. S. J. Gage, University of Texas, Austin, Texas 205. R. P. Gardner, Research Triangle Institute, Durham, N. C. 206. R. W. Garrison, AEC, Washington, D. C. 207. S. H. Hanauer, University of Tennessee, Knoxville 208. P. F. Pasqua, University of Tennessee, Knoxville 209. J. W. Prados, University of Tennessee, Knoxville 210. J. C . Robinson, University of Tennessee, Knoxville 211. R. F. Saxe, North Carolina Stat,eUniversity, Raleigh, N. C. 212. W. L. Smalley, AEC, OR0 213. S. E. Stephenson, University of Arkansas, Fayetteville, Ark. 214. R. E. Uhrig, University of Florida, Gainesville, Fla. 215. Otis Updike, Universitjr of Virginia, Charlottesville 216. A. A. Wassemn, Westinghouse Astronuclear Laboratory, P. 0. Box 10864, Pittsburgh, Pa. 217. M. J. Whitman, AEC, Washington, D. C.
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