” .

’ . . a , ., . I

O A K RIDGE NATIONAL LAB0RATOl);rC d c L T , ,5C++ operated by 4 : -I ’-- - --..n-

UNION CARBIDE CORPORATION NUCLEAR DIVISION

for the U.S. ATOMIC ENERGY COMMISSION

ORNL- TM- 1647 f. ’

COPY NO. - f l ”

EXPFXIMENTAL DYNAMIC ANALYSIS OF THE MOLTEN-SAlZ

ASSTRACT

Dynamics t e s t s were performed on the Molten-S It Reactor Experim nt (MSRE) f o r t h e f u l l range of operating power leve ls t o determine t h e power-to-reactivity frequency response. were used: the pseudo-random binary r eac t iv i ty input, t he pulse r eac t iv i ty input, and the s tep r eac t iv i ty input.

Three types of input disturbances

--.

The frequency response of the uncontrolled reactor system displayed resonant behavior i n which t h e frequency of o sc i l l a t ion and t h e damping increased with increasing power leve l . o sc i l l a t ion ranged from t h i r t y minutes at 75 KW t o two minutes a t 7.5 MW. These osc i l l a t ions were l i g h t l y damped at low power, but strongly damped a t higher power.

Measured periods of na tura l

The measured r e su l t s generally were i n good agreement w i t h predictions. The observed na tura l periods of o sc i l l a t ion and the shapes of t he measured frequency response agreed very we l l with predictions. of t h e frequency response d i f fe red from predictions by a fac tor t h a t was approximately constant i n any tes t (though d i f fe ren t tes ts at the same power l e v e l did not have t h e same b ia s ) . due t o equipment l imi ta t ions (standard MSRE control rods were used) and pa r t ly due t o uncertaint ies i n the parameters i n the theo re t i ca l model.

The absolute amplitude f

T h i s b ias d i f f i c u l t y i s apparently pa r t ly

The main conclusion i s that the system has no operational s t a b i l i t y problems and tha t t h e dynamic charac te r i s t ics are e s sen t i a l ly as predicted.

~~ ~

* Research sponsored by the U. S. Atomic Energy Commission under contract w i t h t h e Union Carbide Corporation. For presentation a t the Winter Meeting of the 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 at the Oak Ridge National Laboratory. It is rubiect to revision or correction and therefore does not represent a final report.

LEGAL NOTICE

This 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, wi th respect to 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 th is report may not infringe

privately owned rights; or

B. Assumes any l iabi l i t ies wi th respect to the use of, or for domoges resulting from the use of

any informotion. apparatus, method, or process disclosed in this report.

As used in thm above, “person acting on behalf of the Commission” includes any employee or

contractor of the Commission, or employee of such controctor, to the extent that such employee

or contractor of the Commission, or employee of such contractor prepares, disseminates, or

provides access to, any informotion purrwant to his employment or contract wi th the Commission,

or h is employment wi th such contractor.

iii

V

I

I1

I11

I V

V

V I

V I 1

V I 1 1

I X

CONTENTS

Introduction ............................... Description of t he MSRE .................... Theoretical Predictions .................... B . Results ................................ A . Description of Mathematical Model ......

Select ion of Experimental Methods .......... A . Character is t ics of t h e MSRE

B . Regulating Rod .......................

Pseudo-random Binary Test .......... 2 . Pulse Tests ........................

Test Signals Used i n the Experiments ... 1 .

3 . Step Tests ......................... Experimental Procedures .................... A . Implementation of Pseudo-random

B . Implementation of Pulse and

Analysis Procedures ........................ A .

Binary Tests ......................... Step Tests ...........................

Direct Anzlysis of PRBS Tests .......... B . Indi rec t Analysis of PRBS Tests ........

Step Response Test Analysis ............ Results .................................... A . Transient Responses .................... B . Correlation IiLulctions .................. C . Frequency Responses .................... References .................................

C . D . Pulse Response Test Analysis ...........

In te rpre ta t ion of the Results ..............

Appendix A . Potent ia l Sources of Experimental

Appendix B . The Direct Method f o r Cross-Power Error Due t o Equipment Limitations .....

Page 1 1

4 4 4 6

6 9 9 10

12

13

13

13 13 13 17 2 1

2 1

2 1

22

22

32

32

45

47

-1-

Y

INTRODUCTION

A se r i e s of experiments was performed on t h e Molten Sa l t Reactor

Experiment (MSRE) t o determine t h e frequency response of t he uncontrolled

reactor system.

ranging from zero t o m l l power. Three d i f fe ren t types of input

disturbances were used t o obtain t h e nuclear power t o r eac t iv i ty

frequency response: t h e pseudo-random binary r eac t iv i ty input, t h e

pulse r eac t iv i ty input, and t h e s tep r eac t iv i ty input. Subsequent

sections of t h i s report w i l l give a descr ipt ion of t h e system, a

review of previously published theo re t i ca l predictions, a descr ipt ion

of t h e t e s t i n g procedures, and t h e experimental r e s u l t s .

Tests were performed at eight d i f fe ren t power leve ls

11. DESCRIFTION OF THE MSRF:

The MSRE i s a graphite-moderated, c i rculat ing-fuel reac tor . The

f u e l i s a mixture of t h e molten f luoride salts of uranium, lithium, I beryllium, and zirconium.

1.

calculated f o r operation a t 10 MW, but heat t r ans fe r l imi ta t ions a t

t h e rad ia tor current ly r e s t r i c t maximum power operation t o about

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 ransfer red from the graphite r a i s e the f u e l temperature about

50°F.

the 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 the primary heat exchanger,

where it t r ans fe r s heat t o a non-fueled secondary salt before reenter ing

t h e core. The heated secondary salt t r ave l s t o an air-cooled rad ia tor

before returning t o the primary heat exchanger. The design parameters

of major importance from t h e standpoint of dynamics a re shown i n Table

1. A de ta i led descr ipt ion of t he MSRE appears i n Ref. 1.

The basic flow diagram i s shown i n Fig.

The flows and temperatures shown are nominal values which were

The molten fuel-bearing salt enters t he core matrix a t t h e

When t h e system operates at reduced power, t he flow r a t e i s

-2 -

ORNL- LR- DWG 56070

SPARE FILL AND FLUSH FILL AND DRAIN TANK TANK

DRAIN TANK 1 6 8 c u f t ) ( 6 8 c u f t ) (68 cu f t )

Fig. 1 MSRE Flow Diagram

COOLANT DRAIN TANK

(40 cu ft I

-3 -

Table 1, MSRE Design Data

Nuclear

Temperature coeff ic ient of r eac t iv i ty of t h e fuel , "F-1

Flow -

Temperature coeff ic ient of r eac t iv i ty of t h e graphite, OF-1

Neutron l i fe t ime, see.

Total delayed neutron f rac t ion

Reactivity loss due t o f u e l c i rculat ion, $ E

K

Flow r a t e i n t h e primary loop, gpm

Flow rate i n t h e secondary loop, gpm

Fuel t r a n s i t t i m e i n t h e core, sec.

Fuel t r a n s i t time i n external primary loop, see.

T o t a l secondary loop t r a n s i t t i m e , sec.

Heat Transfer

Fuel salt heat capacity, MM sec/"F

Graphite heat capacity, MX sec/"F

Heat exchanger heat capacity, MW sec/"F

Bulk graphite - f i e 1 salt heat t r ans fe r coeff ic ient , MW/'F

Fuel salt -heat exchanger metal heat t r ans fe r coeff ic ient , W/"F

Heat exchanger m e t a l , secondary salt-heat t r ans fe r coeff ic ient , MW/"F

Fraction of power generated i n t h e f u e l

-4.7 10-5

-2.6 10-5

.0002 4

.00666

-0.212

12 00

830

8.5

16.7

24.2

4.2

3 -6

.4

0.02

0.36

0 2 7

0 093

-4-

111. THEORETICAL PREDICTIONS

A . Description of Mathematical Model

Throughout the MSm design e f fo r t , a wide var ie ty 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 de ta i led model reported,

referred 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 ibu t ion and t h e nuclear

importances f o r each node were derived from a 2-group neutron d i f fus ion

calculat ion. The flow r a t e s and heat t r ans fe r coef f ic ien ts f o r each

node were determined from calculat ions based on fu l l - sca l e hydraulic

core mockup t e s t s .

ve r i f i ed by t rans ien t t e s t s on the mockup.

The assumed flow mixing cha rac t e r i s t i c s were

The neutron k ine t i c behavior was described by t h e usual space- independent equations w i t h s i x delayed-neutron groups, but w i t h modi-

f i ca t ions t o include the dynamic eyfects of t he circul&lon 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.

iodine production and decay i n t o xenon, xenon decay and burnup, and

xenon absorption i n t o the graphite.

The t ransport of molten salt i n the primary and secondary loop

The xenon poisoning r e a c t i v i t y feedback included

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 the s a l t - t o - a i r

heat exchanger were each represented by a 50-node model.

B. Results of Theoret ical Analysis

Several d i f fe ren t methods of solut ion 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 the experimental r e su l t s , since the l a t t e r a r e readi ly cas t i n

t h i s form.

Fig. 2 shows t h e theo re t i ca l E R E inherent frequency response

L

W

W .

I

I

-5- ORNL-DWG 6579816

5

2

10,000

5

2

2

1 0 0

5

2

10 0.001 2 5 001 2 5 0.1 2 5 1 2 5 10 2 5 loo

FREOUENCV Irodions/rrc)

90

80

70

60

50

40

30

20

- F 'O

a Z 0 -10

-20

-30

-40

-50

- 60 -70

-80

-90 00001 2 5 0001 2 5 001 2 5 01 2 5 1 2 5 1 0

FREQUENCY (radions/sec)

Fig. 2 MSRF: Theoretical Frequency Response

-6-

charac te r i s t ics f o r normalized neutron l eve l response t o r eac t iv i ty

perturbations a t several power l eve l s . It can be seen t h a t t h e system

becomes more osc i l la tory a t progressively lower frequencies as t h e

nominal power l e v e l decreases, though it i s s tab le f o r a l l power leve ls

of i n t e r e s t . An explanation of the inherent s t a b i l i t y charac te r i s t ics

i s given i n R e f . 2 .

I V . SELECTION O F EXPERIMENTAL METHODS

The se lec t ion of t h e experimental methods fo r t he E R E dynamics

tes ts w a s based on t h e information required and on t h e capab i l i t i e s

of t h e avai lable equipment. It may be seen from Fig. 2 t h a t t h e most

s ignif icant par 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 fo r t h e operating power leve ls of i n t e r e s t . This frequency range

corresponds t o long periods of natural osc i l l a t ion (10 min. t o 1 min.).

This emphasis on l o w frequency r e su l t s for tunately made it possible

t o obtain t h e important par t of t he system frequency response using

t h e standard MSRE control rods t o introduce t h e input r e a c t i v i t y

perturbations. In t h i s section, we w i l l examine t h e charac te r i s t ics

of t he MSRE regulating rod and the properties of t he t e s t s ignals used.

A. Character is t ics of t h e MSRE Regulating Rod

The MSRE has three control rods, each with an ac t ive length of

One rod i s normally designated as t h e regulat ing rod and 59.4 inches.

i s used f o r f i n e control. The other two rods a r e shim rods used fo r

coarse adjustments. The rods a re ac tua l ly f lex ib le , s t a in l e s s s t e e l

hoses on which a re strung gadolinium oxide poison cylinders. The rods

a r e mounted i n thimbles which have two 30" offse t t ing bends so t h a t

t he rods can be cent ra l ly located even though the re w a s no room fo r

t h e control rod drive assemblies above t h e cent ra l ax is of t h e core.

The maximum rod speed i s 4 . 5 inches/second.

The three control rods a re ident ica l . Figures 3 and 4 show t h e

control rod and drive assembly.

i s obtained from two synchros geared t o t h e rod dr ive mechanism.

Synchro number 1 i s used fo r coarse posi t ion indicat ion and has a

The posi t ion indicat ion for each rod

V

.

-7-

ORNL-LR-DWG 673II

7 REVERSIBLE DRIVE MOTOR

mING AS INLET

SDLENOIO ACTUATED RELEASE

MOIANW CUlTcH GEbRAmARM

FIXED ORlM SUPPORT AN0 3in. CONTAINMENT TWE

in. 0. a-304 s s- FLEXIU HOSE CABLE

SPRlNo LOADED ANTIBACKL HEAD AND lOLER GEAR

3hx2h.ECCENTRIC4 \\\a\ \ REDUCER 1\11 7 1 6 In. RADIUS x 30. BEND

COOLANT EXHAUST-

GUIDE BARS, 4AT 90.

BEADED POISON ELEMENTS

2 h CONTAINMENT THIMBLE

Fig. 3 Control Rod Drive Assembly

-8-

ORNL-LR-OW0 78806

W

GEAR REDUCER No. i

SYNCHRO NO. 2 600 PER INCH OF ROD MOTION

POTENTIOMETER

SYNCHRO NO.(

OF ROO MOTION GEAR REDUCER N0.2

INPUT SPROCKET

I SPROCKET CHAIN

OVERRUNNING CLUTC

FLEXIBLE TUBULAR ROD SUPPORT

1 REACTd)ERL?ESSEL

CORE

IHl TEMPERATURE,* 44OOOF

POSITION INDICATOR AIR FLOW RES TR ICTOR AIR DISCHARGE - RADIAL PORTS

Fig. 4 D i a g r a m of Control Rod Drive

-9-

Y 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 pos i t ion indicat ion and has a s e n s i t i v i t y of 60" per inch.

The s igna l f r o m t h e coarse posi t ion synchro i s transmitted t o a

torque amplif ier which dr ives a single-turn potentiometer feeding

a d-c s igna 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 obtain reproducible regulat ing rod pos i t ion changes f o r a given

time of i n se r t or withdraw. This was due t o the wearing of t h i s

rod and dr ive 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 the

regulat ing rod. Since it i s moved much less frequently than t h e

normal regulat ing rod, it had experienced l e s s wear and had much

t i g h t e r response charac te r i s t ics .

There a r e a number of f ac to r s which could adversely a f f ec t both

the accurate posit ioning of t he rods and t h e indicat ions of e f fec t ive

rod pos i t ion given by the instruments. Some of t h e po ten 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 indicate t h a t t h e MSRE control rods a r e hardly

idea l ly su i ted f o r dynamic t e s t i n g . However, since no provision w a s

made f o r spec ia l control rods and since t h e main fea tures of i n t e re s t

occur at low frequency, t h e t e s t i n g program w a s car r ied out using

t h e standard MSRE control rods. Fortunately, t h e rods performed

far b e t t e r than 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 ren t types of t e s t s ignals which were used t o obtain

t h e frequency response of t h e system are described i n t h i s sect ion. ( 4) 1) Pseudo-random Binary Test

I n t h i s t e s t , spec ia l ly selected periodic s e r i e s of pos i t ive

and negative r e a c t i v i t y pulses ca l led 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 consis ts of a number of harmonics of approximately

equal s i z e .

at a large number of frequencies i n a s ingle t e s t . The spectrum of

This means t h a t t h e frequency response may be evaluated

-10-

t h e pseudo-random s igna l from one of t he MSRE t e s t s i s shown i n

Fig. 5 . We note t h a t t he s igna l s t rength i s concentrated i n

d i sc re t e harmonic frequencies ra ther than d is t r ibu ted over a

continuous spectrum as i n t h e case of non-periodic (e.g. pulse and

s tep) s igna ls .

to-noise r a t i o a t these frequencies.

T h i s i s he lpfu l since it improves the e f fec t ive s ignal-

A PRES may be generated on-line a t t he t e s t or may be pre-recorded

i n some fashion and played back as a control s igna l . On-line

generation of t he s igna l w a s used i n these t e s t s (see Section V ) . A PRBS i s characterized by the number of b i t s i n the sequence and

t h e b i t duration. A b i t i s defined as the minimum possible pulse

durat ion i n the sequence. All pulses i n a PRES a r e minimum width or i n t eg ra l multiples thereof . Numerous sequences may be generated, but

they a r e r e s t r i c t e d t o ce r t a in spec i f ic numbers of b i t s . I n the MSRE,

PRBS tes ts were run w i t h 19, 63, 127 and 511 b i t s . If t h e number

of b i t s i s Z and the b i t duration i s At , then the PRES has a period

Z A t . The lowest harmonic radian frequency, cu and the spacing of

t h e harmonics, DLD, i s given by LU = DLD = K. The PREE t e s t s which

were run and analyzed a r e shown i n Table 2 . I n each t e s t , t h e rod

motion w a s se lected t o give a r eac t iv i ty 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 per turbat ion was determined on

the basis of keeping t h e resu l t ing power l e v e l perturbations i n t h e

l i nea r range, i . e . 6N/No maximum w a s kept below 0.1.

0’

mt 0

(4) 2 ) Pulse Tests

I n theory, it i s possible t o exc i te a system with a s ingle

pulse-type disturbance and obtain the frequency resonse by numerically

determining t h e r a t i o of t he Fourier transform of the output t o t h e

Fourier transform of the input. The frequency response can theo re t i ca l ly

be evaluated at a l l frequencies s ince t h e input has a continuous

frequency spectrum. I n pract ice , the pulse t e s t i s of ten unsat isfactory

because t h e avai lable s igna l s t rength i s d is t r ibu ted over a l l

frequencies, resu l t ing i n a small amount of s igna l around the analysis

frequency and thus poor e f fec t ive s igna l t o noise r a t i o .

‘( ’

- I I I II I I III

10L2

Frequency (radiana/eec)

ORNL DWG. 66-l 1076

I 1 1 1

I I I I

Fig. 5 Power Spectrum of the Input PFE!S at 0.465 MW

Several t e s t s using approximately square r eac t iv i ty pulses of

between 0.01 and 0.02% were employed fo r t h e MSRE at zero power.

Tests were carr ied out both with c i rcu la t ing f u e l and with s ta t ionary

fue l .

Table 2 . Pseudo-Random Binary Sequence Tests

Power B i t s i n B i t Per iodici ty of Minimum Levels (MM) Sequence Duration (sec) t h e PRBS (sec) Frequency

(rad/sec)

0

0

0075 .465 1 .o 1 .o 2 05 2 05 5 -0 5 00 6 97 7.5 7 95

511 511 =7

6.58 3 035 3 035 3 935 3 935 5.02

3 *32 4 097 3 *33 4 *97 3 -32 3 *33 4 997

2 11

1711 1 7 2 1711 638 1699 631 1701 63 1 1698 1701 63 1

3) Step Tests ( 5 )

If t h e output eventually leve ls off t o some constant value

a f t e r a s tep disturbance, t h e frequency response may be obtained by

a modified Fourier analysis of t he output and t h e input. A s with t h e

pulse t e s t s , t h e s tep input has a continuous spectrum, but i s hampered

by a low ef fec t ive signal-to-noise r a t i o a t any analysis frequency.

Step tes ts were used at power leve ls 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 ig ina l power

after t h e r eac t iv i ty 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

W

-13 - V , EXPERIMENTAL PROCEDURES

U

A. Implementation of Pseudo-random Binary Tests

The pseudo-random binary r eac t iv i ty sequence required a very

precisely controlled ser ies of regulating rod inser t ions and with-

drawals.

t o 1.0 radian/sec., t he rod jogger w a s designed so t h a t sequence with

a b i t time of 3 t o 5 seconds could be run for as long a s 1 hour. The

rod jogger system, which required no special-purpose hardware,

consisted of a hybrid computer cont ro l le r shown schematically i n Fig.

6. The portable EAI-TR-10 analog computer w a s used t o control t he

b i t t i m e and t h e rod dr ive motor “on” times f o r the inser t and withdraw

commands.

sequencing of t h e pulse t r a i n by means of a sh i f t - r eg i s t e r algorithm.

The number of b i t s i n the sequence could be varied over a range between

3 and 33,554,431 b i t s .

Since t h e frequency range of i n t e re s t w a s from about 0.002

The MSRE d i g i t a l computer was programmed t o control t h e ( 3 )

The rod jogger system performed extremely well, as it was able t o

posi t ion t h e rod with an indicated posit ioning 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 ave l f o r over a 500 insert-withdraw operations.

rod posi t ion s igna l i s shown i n Fig. 7 and a typ ica l record of neutron

flux changes during a PRES t e s t i s shown i n Fig. 8.

A t y p i c a l PRBS

The analog computer w a s a l so used t o amplify and f i l t e r t h e rod-

posi t ion and power-level s ignals pr ior t o d ig i t iz ing .

dynamic t e s t s t he MSRE computer was used i n t h e fast scan mode t o d i g i t i z e and s to re t h e da ta on magnetic tape.

sampled at t h e r a t e of 4 per second.

B.

Throughout t h e

Each var iable w a s

Implementation of Pulse and Step Tests

The same se t up as described i n ( A ) was used for t h e pulse and

s tep t e s t s with t h e PRBS generator omitted.

V I . ANALYSIS PROCEDURES

A. Direct Analysis of PRBS Tests

The “d i rec t” method of analysis 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 spec t ra l

densi ty functions. ( 6 ) A block diagram of t he analyzer i s shown i n

-14-

I

I I I

1 I I

OANL-DWC 66-4763

I BINARY SEQUENCE ’ PSEUDO-RANDOM

BIT-TIME GENERATOR

I

c INSERT ! MOTOR - 1 1 COMPUTATION WITHDRAW I “ON” MOTOR

PORTABLE ANALOG MSRE DIGITAL COMPUTER COMPUTER

1 I I I I

--------- 1 r B R - 3 4 0 r--------

EAI-TR-40

I I

I I I

I I nn’SITION I 1 ‘ I

I 1

1 , AMPLIFYING - 1 ; - I

AND I MAGNETIC TAPE I I & DIGITIZER AND

A I I t I

FILTERING RECORDER ’ I 6 POWER-LEVEL I

REACTOR

Fig. 6 Block D i a g r a m of MSRE R o d Jogging and Data Logging System

l

A

-4-P

-16-

ORNL-DWO 66- 4764 42

38

P 9

0 34 Bp

0'

8 30 a

c

L

U w A u 3

26

22

I I I I I I

-

0 4.47 8.34 12.51 17.09 21.26 25.43 29.60 TIME ( 30-min FULL SCALE

Fig. 8 Measured Flux Signal During a 511 B i t PRBS T e s t

-.17 - W Fig. 9.

a quadratic l ag and a t r ans fe r function

The narrow band-pass f i l t e r H(s) has t h e charac te r i s t ics of

S 2 H ( s ) =

w2 + 2i wo s + s 0

where coo i s the center frequency of t h e f i l t e r . Typically

t h e damping fac tor 5 i s set equal t o 0.05, giving a f i l t e r frequency

response a5 shown i n Fig. 10.

described by Chang") and Van Deusen(8); t h e main difference i s t h a t

it includes ex t ra cross-multiplication terms resu l t ing i n higher

accuracy.

Appendix B. B.

T h i s analyzer i s s i m i l a r t o that

A mathematical descr ipt ion of the method i s given i n

Indi rec t Analysis of PRBS Tests

The ind i rec t analysis began with a calculat ion of t h e

autocorrelat ion function of t he 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 = t h e l a g t i m e

T* = the in tegra t ion t i m e (a multiple of t he s igna l per iodici ty) r ( t ) = t h e input s igna l

O ( t ) = t h e output s igna l

t = t i m e

-18-

QUAD PMR

R

Fig. 9 Block Diagram of Cross-Parer Spectral Density Analyzer

-19-

10

5

2

5

ORNL-DWG 66-10635

2

0.01 0.01 2 5 0.1 2 5 10 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 2 5 10 2 5 100

W/WO

Fig. 10 Frequency Response of Filter Used in CFSD Analyzer

-20-

Subsequent Fourier analysis of C

power spectrum and t h e cross power .spectrum:

( T ) and C 11 12 ( 7 ) gave t h e input

1 Q, ( ) = J Cl1(7) COS %T dz

0 11 %

T 1 T Q, ( 12% ) = r 1 J C,(z) cos %T dz - j r J C=(z) s i n %z d7

0 0

where

' w z Q, ( ) = input power spectrum at frequency 11 %

wz = cross power spectrum at frequency,

= angular frequency of t he kth harmonic (% = -)

%(%) 2kfi T %

T = per iodic i ty of t he t e s t s igna l

Note t h a t t h e input power spectrum i s a real quantity s ince C

always an even function of 7 .

( 7 ) i s 11

The frequency response, G( ju$), i s given by

2, JTC,(z) cos %z dz T o Re[G(j%)l = 1 T - J c&) cos w.;" az

T o

Jo C,(z) s i n %z dz I m [ G ( j(Lx) 1 = - T . . T 1 - - Jo Cl1(7) cos u y dz T

The magnitude r a t io , MR, and t h e phase, 0, are given by:

~m [G(J%) 3 e(%) = tan-1 {Re [G( j%) ]}

A Fortran computer code f o r the Im-7090 or IBM-360 ca l led CABS ( 9 )

w a s prepared t o carry out these computations.

W

-21-

C. Step Response Test Analysis

The s tep response t e s t s were analyzed using a d i g i t a l computer code (5,101 which implements Samulon's method.

D. Pulse Response Test Analysis

Although pulse response t e s t s were attempted at power leve ls

of - 0, 75 KW, 465 KW, and 1 MW, t he only successful runs were the

ones made at zero power. This w a s because the low frequency random

f luctuat ions i n heat load at low (but non-zero) powers d ra s t i ca l ly

reduced t h e signal-to-noise r a t i o . A t these powers, the rad ia tor i s

cooled primarily by na tura l convection and radiat ion, and i s consequently

sens i t ive t o atmospheric disturbances. A t higher powers, where most

of t he cooling was due t o forced convection, these f luctuat ions were

not apparent.

The zero power t e s t s required extremely accurate core temperature

control and rod posit ioning i n order t o avoid d r i f t of the f lux leve l .

Since a zero power reactor i s an integrat ing system, a pulse r eac t iv i ty

input r e s u l t s i n a change i n steady-state f lux output, and Fourier

transforms are va l id only i f the response function eventually re turns

t o i t s i n i t i a l value. We got around t h i s problem by f irst numerically

f i l t e r i n g t h e output response data through a high-pass network HP( s )

with a t r ans fe r function

then performing t h e numerical Fourier transform, and f i n a l l y compensating

t h e resu l t ing frequency response f o r t he high-pass f i l t e r charac te r i s t ics .

The numerical Fourier transform calculations were done by a d i g i t a l

computer code using a novel method

t o t h e standard technique of summing the products of f ( t ) cos ust and

f ( t ) s i n ut, but which reduces t h e computing time by a fac tor of 3.

which i s mathematically equivalent

V I I . RESULTS

The experimental data were analyzed t o give cor re la t ion functions,

input power spectra 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 sect ion along with the d i r ec t ly observable

-22 - t rans ien t response t o system disturbances.

A. Transient Responses

A t each power level , a t rans ien t w a s induced by inser t ing a

r eac t iv i ty pulse or step, or by simply allowing t h e system t o seek

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 na tura l frequency of o sc i l l a t ion of t h e system.

Figure 11 shows t h e observed t rans ien t response. A t 0.075 Mw,

it took over two hours fo r t he flux t o re turn t o equilibrium.

B. Correlation Functions

These t rans ien t

The pseudo-random binary t e s t s were analyzed by the d i r ec t method

and t h e indirect method. Autocorrelation functions of t he input and

cross correlat ion functions of the input and output were obtained as

intermediate r e s u l t s by t h e ind i rec t method. The cor re la t ion 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 correlat ion functions which

have been present i n a l l tes ts since then. The spikes always appeared

a t points 432 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. per iod) .

functions and cross-correlation functions for t es 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 .

Figures I 2 through 19 show t y p i c a l autocorrelat ion

The reason f o r the appearance of t h e spikes i s not yet known. The

only s igni f icant difference noted i n the input s igna l a t 0.465 MW and

at 2-1/2 MV was a change i n e f fec t ive pulse duration f o r pos i t ive and

negative pulses. A t 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 the duration of a pulse below the midpoint. A t

2-1/2 MW, t he 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 charac te r i s t ics 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 posi t ive and negative pulses t o determine

whether t h i s would cause spikes i n the autocorrelation function.

These calculations showed spikes fo r some sequence lengths but not

f o r others. I n par t icu lar , spikes were not found i n these calculat ions

for a 5 l l b i t sequence. The confl ic t ing indicat ions furnished by

U

-23-

ORNL-DWG 66-10284 40

20

0

-20

2 0

0

- 2 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

TIME (min)

Fie. 11 MSRE Power Level Transients

0 ho0

GRNL DWG. M-l lm8

. .

. .

. .

. .

I I

. .

e e w w

800 I200 lb00

Comlation Tim bee)

Fig. I2 Input Autocorrelation Function for a 511 Bit PRBS Test at 0.465 MW

‘( ’

OWL DWG. 66-11079

400 I200 l600

col-lFlation The (sac)

Fig. 13 Cross-Correlation Function for a 511 Bit PRBS Test at 0.465 MW

0 160 320 4eo 640

Correlation Time (set)

Fig. 14 Input Autocorrelation Function for a 127 Bit PRBS Test at 1.0 MW

. (

‘( .

ORNL DWG. 66-l 1081

Correlation Time (em)

640

Fig. 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 for a 511 B i t PRBS T e s t a t 2.5 MW

h

a Y r )

B P E Y .4

s Y

C .4

3 0 .4 Y

1 4 k V

800

correlation Tiw (eec)

Fig. 17 Cross-Correlation Function for a 5 1 1 Bit PRBS Test at 2.5 MW

ORNL DWG. 66-11083

I

ORNL DWG. 66-11084

* 1 .

.

I I

* ‘

I I I 0 I60 320 480 6bo

oorralatioa The (ccc)

Fig. 18 Input Autocorrelation Function for a I.27 Bit F!ElBS Test 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

these calculat ions have not yet been resolved. However, w e should

note t h a t t h i s unexpected feature of t h e cor re la t ion functions does

not inval idate t h e f ina l r e su l t , t h e frequency response. It simply

means t h a t t h e spec t ra l propert ies of t he input and output a r e s l i g h t l y

different than anticipated, but t ha 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 va l id 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

The legend i n each f igure indicates t he type of t e s t 20 through 28.

and the analysis procedure.

previously published resu l t s (*) when the experimental power leve ls

corresponded t o avai lable 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 the

published r e su l t s . Results obtained by Roux and Fry ( t h e random noise i n t h e neutron f lux s igna l a r e a l s o shown with t h e

zero-power r e su l t s .

r e s u l t s at 9 rad/sec.

root of t h e measured power spec t ra l density ( E D ) of t h e neutron f lux

s igna l a f t e r subtracting t h e background PSD. i s proportional t o the amplitude of t h e f lux-to-react ivi ty frequency

response, assuming t h a t t h e observed noise i s the r e su l t of a white

noise r e a c t i v i t y input .

The theo re t i ca l curves were taken from

by analyzing

These r e s u l t s were normalized t o t h e theo re t i ca l

These points were obtained by taking t h e square

This gives a r e s u l t which

The na tura l period of o sc i l l a t ion of t h e system w a s determined

e i the r by d i r ec t observation of' a t rans ien t or by locat ion of t h e peak

of t h e frequency response. These r e su l t s , along with theo re t i ca l

predictions, a r e shown i n Fig. 29.

V I I I . IETERPRFTATION OF THE RESULTS

The objectives of t he dynamic t e s t i n g program a r e twofold:

provide information on the s t a b i l i t y and operabi l i ty of t he system;

2 ) t heo re t i ca l predictions so tha t fu ture calculat ion on t h i s and other

similar systems w i l l be improved. The r e s u l t s a r e interpreted i n

terms of these two objectives.

1)

provide inf'ormation f o r checking procedures fo r making

W

-33-

v to4

5

2

103

5

2

102

5

2

... 0.001 0.005 0.01 0.02 0.05 01 0.2 0.5 !I) 2.0 5.0 100 200 501) 100.0

W, FREQUENCY (rodionskac)

20

10

0

- 10 -20

0) -30

w -40 a r -50 a

0

'D Y

v)

-60

-70

-80

-90

-100

ORNL-DWG 65-8907A

0.001 0.01 0.4 w, FREQUENCY (radians/sec)

~ N / N Fig. 20 Frequency Response of ~9 6K KO at

Fuel Circulat ing

1

Zero Power;

ORllLDYIi 66-10708

7-Scc Pulse, Teat l o . 1 '/-See Pulse, Test No. 2

3-1!2-Sec Pulse, Test Ih. 3 4

j-1/2-8cc Rilse , Test lo . h 0 Nolee Analysis

0.001 0.01 0.1 I . 0

w , FREQUENCY ( radions/sec 1

W N o 7 Fig. 21 Frequency Response of 6K at Zero Power;

Fuel S t a t i c

lo

W

-35-

70

60

50

40

30

20 c

Y

0 W tn

Q 9 -io

- 20 - 30 - 40

- 50

-60

0.001 0.002 0.005 0.01 0.02 0.05 O.! 0.2 0.5 4.0 FREOUENCY (rodionr/sec)

-70 . - 0.OOi 0.002 0.005 0.01 0.02 a05 0.i 0.2 0.5 i.0

FREQUENCY (rad ions /sec 1

Fig. 22 Frequency Response of Power = 0.075 MM

-36-

5

2 "/ifo IKI

to3

5

2

0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5 1.0 FREOUENCY (radians /set)

60

50

40

30

20

10 b - o " 0

w -10 a * -20

- 30

v)

a

-40

- 50

- 60

- 70

- 80

ORNL-OWG 66-4770 I I I I I 1 I l l I I I I I I -

-_ POWER LEVEL 0.465 MW _- 511 B1T PRES DIRECT ANALYSIS 0

- 511 BIT PRES INDIRECT ANALYSIS 0 -- n

_ _

0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 FREQUENCY ( radians/sec

~ N / N -

W

0.5 1.0

Fig. 23 Frequency Response of Power = 0.465 MW

-37-

ORNL- DWG 66-4771

7 BIT 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 0.02 0.05 0.1 0.2 0.5 1.0 FREQUENCY (rodions/sec)

ORNL-OWG 66 - 4772 90 t

80 - / .

\ 1 I I I I l l 1

\ I I 70

60

50

40

Z 30

20

a 40

0

-10

-20

-30

-40

- 50

0, aI

W

U I

-- O.CO4 0.002 0.005 0.01 0.02 0.05 0.t 0.2 0.5 LO

FREQUENCY (radions/sec)

~ N / N Fig. 24 Frequency Response of 3% ; Power = 1.0 MW

Y

0001

90

80

70

60

5 0

4 0

30 - 3 20

E O

Y

10 a

-10

-20

-30

-40

-50

-60

-70

0002 000s 001 0 02 005 01 02 05 FREOUENCY (mdianshs)

ORNL-DWG 66-100

127 BIT PRBS - DIRECT ANALYSIS 127 BIT PRBS - INDIRECT ANALYSIS 511 BIT PRBS - DIRECT ANALYSIS

0

A A

0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5 1.0 FREQUENCY (rodians/sec)

6M/N O * Power = 2.5 MN q’ Fig. 25 Frequency Response of

1.0

$

-39-

2

127 BIT PRBS -DIRECT ANALYSIS 127 BIT PRES - INDIRECT ANALYSIS A

PABS - DIRECT ANALYSIS A d

2

K? .- 0.002 0.- 0.01

0.001 0.002 0.005

0.02 0.05 0.1 02 0.3 FREOUENCY (rodianr/sec)

ORNL-DWG 66-10035

I I 1

0.2 0.5 10 0.01 0.02 0.05 0.1 FREQUENCY (radianslsec)

~ N / N

1.0

t

Fig. 26 Frequency Response of 6K ‘ O . Power = 5 MW 7’

-40-

ORNLOrCI 66-IOO39 2

io3

2

0.005 001 0.02 0.05 01 02 0.5 1 .o lo2 a002

FREQUENCY hdionshec)

~ N / N Fig. 27 Frequency Response of E q q ’ O * Power = 6.7 MW

W

-41-

2

lo3

2

Id

ORNL-DWG €640040

POWER LEVEL - 7.5 Mw STEP TEST 0

127 BIT PRBS -DIRECT ANALYSIS 127 BIT PRBS- INDIRECT ANALYSIS A 511 BIT PRBS - DIRECT ANALYSIS A

.- 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5 10

FREQUENCY (radions/sec)

80

70

60

50

40

30

-10 a

-20

-30

- 40 -50

-60

-70

ORNL-DWG 66-10036

0.001 0.002 0.005 0.01 0.02 0.05 0. I 02 0.5 I .o FREQUENCY (radianshec)

Fig. 28 Frequency Response of Power = 7.5 MW

-42-

5

2

2

1 0 102 0.05 0.1 0.2 0.5 1 .o 2 .o 5.0 1 0.0

FOWER LEVEL (MW)

Fig. 29 MSRE Natural Periods of Oscillation

-43 - I

W

The l inea r s t a b i l i t y of t he system i s cer ta in ly 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 tha t t h e t rans ien 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 re turn t o equilibrium (period greater than t e n minutes

f o r powers l e s s than 500 KW).

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 rans ien t which i s essent ia l ly completed

i n 1-112 minutes.

predict ions.

A t higher power t h e system response

These observations are i n good agreement with p r io r

A de ta i led quant i ta t ive check of t he theo re t i ca 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 features such as s t a b i l i t y , locat ion of resonance

peaks and t h e changes expected i n these performance measures with

power leve l . Early attempts t o f i t parameters i n t h e theo re t i ca l

model t o give agreement i n t h e absolute amplitude of t h e frequency

response were abandoned bacause of uncertaint ies i n t h e measured

amplitudes caused by equipment l imitat ions. While a l l of t h e tests

at a given power l e v e l give r e s u l t s with the same shape, there i s a

difference i n t h e absolute magnitude r a t io s . Figure 27 c lea r ly shows

t h i s b i a s e f f ec t . Furthermore, t h e portion of t h e frequency response

abwe 0.3 rad/sec should be the same fo r a l l power leve ls since

feedback e f f ec 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 leve ls show t h e same shape i n t h i s frequency region,

but d i f fe ren t absolute amplitudes. Th i s fur ther indicates a bias

problem.

cha rac t e r i s t i c s discussed i n Section I V .

This b i a s problem i s not surpr is ing i n view of t h e equipment

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 theo re t i ca l

model i s shown t o be incorrect by experimental r e s u l t s . A t high

power (grea te r than 5 MM) t h e theo re t i ca l magnitude r a t i o curve has

a d ip at 0.2 rad/sec.

salt temperature slug i n t h e core af'ter t rave l ing around the primary

loop. Since t h i s d ip was not observed experimentally, there must be

Th i s i s due t o t h e reappearance of a f u e l

-44-

more mixing and heat t r ans fe r i n t h e primary loop than w a s included

i n the theo re t i ca l model.

Because t h e predicted frequency response has the correct shape

and locat ion on t h e frequency axis, we f e e l t h a t t he model used f o r

t h e E R E dynamic analysis 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

bias i n t h e measured amplitude unfortunately prohlbi ts de ta i led

parameter f i t t i n g .

-45 -

REFERENCES

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. Prog., 57( 3 ) : 69-79 (March 1961).

5. H. A. SamuLon, Spectrum Analysis of Transient Response Curves, Proc. IRE, 39, 173-186 (1951).

S. J. Ball, Instrumentation and Controls Div, Annual Progr. Rept. Sept. 1, 1965, USAEC Report ORNL-3875, pp. 126-7, Oak Ridge National Laboratory.

6.

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-TM- 1663, 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).

T. J. Karras, Equivalent Noise Bandwidth Analysis from Transfer Functions, NASA-IITN-D2842, November 1967.

16.

17. L, D. Ehochson, Frequency Response Functions and Coherence Functions fo r Multiple Input Linear Systems, NASA-CR-32, A p r i l l S 4 .

18. H. M. Paynter and J. Suez, Automatic Dig i ta l Setup and Scaling of Analog Computers, Trans. ISA, 3 , 5544, January 1964.

Y

lg. S. J. B a l l and R. K. Adams, MATED, A General Purpose Dig i ta l Computer Program f o r Solving Nonlinear Ordinary Dif fe ren t ia l Equations by the Matrix Exponential Method, USAM: ORNL report ( i n preparation), O a k Ridge National Laboratory.

Y

-47-

APPENDIX A

Poten t ia l Sources of Experimental Error Due t o Equipment Limitations

A s discussed i n Section IV.A, there were a number of fac tors t h a t

could have had adverse e f f ec t s on the experimental resu l t s , and they

a r e l i s t e d below:

1) Frict ion i n the bends i n the thimbles. Rollers a r e mounted i n

the bends of the thimble, but there i s s t i l l considerable f r i c t ion .

(Tests show tha t the rods f a l l with an acceleration of only about 0.4 g . )

This suggests t h a t pa r t of t he motion a t the rod dr ive might go in to

bowing o f the f l ex ib l e hose ra ther than i n t o motion of the bottom of the

poison section.

2) Bends i n the hose. It w a s observed t h a t an old MSFE control

rod used f o r out-of-pile t e s t ing d id not hang s t r a igh t when suspended

from the top. It had gradual bends t h a t could be worked out by hand,

bu t which were not pulled out by the weight of the rod (6 , to 8 l b ) . If

such bends e x i s t i n the MSRF: rod used f o r the t e s t s , then the motion of

the bottom of the rod w i l l not be the same as the motion a t the 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 the

thimble.

3 ) Restricted twisting. The tes t rod showed a tendency t o turn

when inser ted in to a mockup of the MSRF: thimble.

vented i n the reactor since the top of the rod i s r ig id ly connected t o

the chain drive (see. Fig. 3). hose w i l l bow and cause a difference i n a x i a l motion between upper and

lower sections.

This twist ing i s pre-

If a tendency t o t w i s t i s prevented, t he

4) Sprocket chain meshing. The act ion of the dr ive motor i s t rans-

mitted t o the dr ive chain by a sprocket. This sprocket has a diameter

of 1.282 in . and the length of the links i n the chain i s 1/4 i n .

f a c t t h a t the f l a t links cannot exactly follow the c i r cu la r contour of

the sprocket means t h a t some of the sprocket motion i s taken up by a

l a t e r a l motion of the chain as well as the desired ve r t i ca l motion.

The

5 ) Sticking of poison beads. The control rod thimble contains

-48-

vanes f o r centering the control rod.

c i rcu lar spacers located 4 in . apar t .

could touch l i nks i n the poison chain i n cer ta in sections without touching

l i nks i n nearby sections.

poison cylinders on the central hose, t h i s f r i c t i o n could hold up the

movement of cer ta in poison elements.

6 ) Indicated pois i t ion e r rors .

The vanes a re held i n place by

If the vanes became warped, they

Since there i s slack in the threading of the

It was necessary t o use the coarse

synchro s igna l ( 5 deg of turn per inch of rod motion) f o r logging on the

MSRE computer and f o r subsequent data analysis .

motion used i n the tests corresponds t o only 2.5 deg ro ta t ion of the

synchro, sizeable percentage e r ro r could be caused by only a few tenths

of a degree of deadband i n the gears leading t o the synchro. Periodic

ca l ibra t ions of the logged rod posi t ion against the f i n e synchro, how-

ever, indicated t h a t the e r ro r i s l e s s than Pj$ f o r a 1/2-in. rod t r ave l ,

The maximum deadband i n the logger s ignal corresponded t o about k0.008

in . o f rod motion.

Since the 1/2-in. rod

-49-

Y APPENDIX B

The Direct Method for Cross-Power Spectrum Analysis

Each of the terms used i n the cross-power spectrum analysis i s

computed i n the following manner:

W 0 where HI and H, a r e e i the r H ( j w ) o r - H( j w ) ; and f($IO) (depending on

which combination of H1 and H, are used) i s re la ted e i t h e r t o the r e a l

(COPOWER) o r the imaginary (QUAD POmR) p a r t of the cross-power spec t ra l

density (CPSD) 410. Four combinations of the basic computation shown

above are used i n the CPSD analysis a s shown i n Fig. 9.

j w

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) = complex conjugate

of the Fourier transform of a ( t ) .

-50 -

Considering that only a f i n i t e integrat ing t i m e T i s avai lable t o

us :

Noting that

B ( j 4 = H 2 ( j U ) o ( j 4 , where

I(jw) = Fourier transform of input, i ( t) , O ( j w ) = Fourier transform of output, o ( t ) .

From the def in i t ion of the complex conjugate, it can be shown t h a t

(3)

(4)

A(-jw) = H1(-jo) I(-jw) , ( 5 )

B(-jw) = H2(-jw) 0(-jw) . ( 6 )

Hence ( 2 ) can be rewrit ten i n terms of the Fourier transforms of the

input and output s ignals

( 7 ) l m T s a ( t ) b ( t ) d t Z 5 H 2 ( j w ) Hl(-jw) I(-jw) O ( j w ) d w . 0 -0Q

Since the cross-power spectral 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). T, we ge t

Taking the l i m i t of both s ides of (7) as W and dividing by

-.

Subst i tut ing (8) in to (9):

-51-

W

Case 1

For the case where

HI( j w ) = H2( jw) = H( jw)

and defining

T - E w T lim a ( t ) b ( t ) d t ,

0

Eq. (10) becomes

Since the input and output s ignals a re f i l t e r e d ident ica l ly , it

should be evident that this operation w i l l y ie ld information only about

t he in-phase relationship, o r the r e a l p a r t of $Io( j w ) ,

t h i s by noting tha t since

We can show

and

the two in tegra ls i n Eqs. (11) and (12) must be equal; thus

If we assume that $Io ( jw) and $ (jw) do not change much over the 01 effect ive bandwidth of the f i l t e r , then $Io(jw) must equal $oI(jw). since

But

-52-

t h i s means that the imaginary p a r t of '$ no information about

then

( j w ) must be zero, o r a t l e a s t

For case 1, IO

j w ) ] i s present i n the output.

If we assume tha t Re['$

bandwidth of H( j w ) , ( j w ) ] does not change much over the e f fec t ive I O

For the present study, we used a f i l t e r with the following t r ans fe r

function:

j w H ( j w ) =

w2 + j w 2(w0 - w2 0

* .

The f i l t e r "area" term can be evaluated using a t ab le of integrals14

Thus

Case 2

For t h i s case

and

Since

- l M J IH( j w ) 1' dw = - (rad/sec) . 2s 4b0

-m

Re['$IO(jw)] 4(w0

H 2 ( j w ) = H ( j w ) .

w H(-jw) 0

H1(-jw) = - j w ,

.t

-53-

Eq. (10) becomes

Since the input s ignal ' s f i l t e r has 90 deg more phase l a g than the

output s igna 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 ie ld

icformation only about the quadrature relationship, o r the imaginary

p a r t of $ I o ( j w ) .

a difference, i.e., from (2)

We can show t h i s a s follows:

I n t h i s case, revising the order of integrat ion of the inputs makes

Again, since = 'm-, from (22) and (24) we can conclude

that

410(jw) 4 o I ( j w ) - - - jw j w

i f we use t h e same argument as we did fo r case 1. Thus

which i s t rue only i f the real p a r t of 4 information about Re[d,,(jw)] i s present i n the output.

subs t i tu te j I m [ d I o ( j w ) ] i n f o r $,,(jw) i n Eq. (22).

= 0, or a t least i f no IO Hence, we can

For case 2, then

If we assume that Im[4 ( j w ) ] does not change much over the WO

ef fec t ive bandwidth of e i t h e r H ( j w ) or - H( j w ) , then Jw

IO

-54-

For the pa r t i cu la r f i l t e r used (Eq. 17):

Thus

Case 3 The case where

W

(31) 0

HI( j w ) = H2( j w ) = - H( j w ) j w

can be developed s imi la r ly t o case 1 as f a r as Eq. (16), since we could

redefine H ( j w ) a s being equal t o the expression i n (31). The in t eg ra l t o be evaluated i s

I Fcr the f i l t e r of Eq. (lT), t h i s i n t eg ra l i s again equal t o - , so 4b0

assuming i n t h i s case, however, t h a t Re[$

ever the e f fec t ive bandwidth of - H ( j w ) .

( j w ) ] does not change much IO wO

j o

Case 4 The case where

can be developed s imilar ly t o case 2. Equation (10) becomes

-.

-53 -

The expression on the right hand s ide i s the negative of t h a t f o r

Eq. ( 2 2 ) , case 2; hence f o r case 4, we get the 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 the e f fec t ive

bandwidths of e i t h e r H ( j w ) o r - H( j w ) . w0 J W

The power spec t ra l densi ty (PSD) of the input function i s required

f o r calculat ing the system t ransfer function G ( j u ) .

by squaring the outputs of both the in-phase and quadrature f i l t e rs and

in tegra t ing the sum of the squares. In both cases, f o r t he f i l t e r of

This i s obtained

m* (17):

assuming

of H ( j w )

The

t h a t 4 and wo H( j w ) / j w .

system t r ans fe r function G ( j w ) i s then computed from

( j w ) does not change much i n the e f fec t ive pass bands I1

(39)

where each of the three terms on the r igh t hand s ide of (39) a re computed

using the sum of two estimates.

factor , 45w .) compared t o using a s ingle estimate of each term l i e s i n the f a c t t ha t

since the e f fec t ive pass-bands of the in-phase and quadrature f i l t e rs

are d i f fe ren t , there i s a b ias i n each of the quadrature, o r imaginary

term, estimates.

sign, this b i a s tends t o be cancelled out.

(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 th i s method as 0

Since the two imaginary term estimates a re of d i f f e ren t

Calculations of t he percent standard deviations of both input and

output (PSD) estimates a re made using (40) :I5

-56 -

where u = standard deviation of mean square value,

x2 = mean square value, B = equivalent noise bandwidth, rad/sec, T = integration time, sec. The equivalent noise bandwidthi6 for the H ( j w ) filter used in this

study is

B = nfw rad/sec. (41) 0 ’

The coherence function y2 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 pressions for error estimates in the literature have been found to be

wildly pessimistic.

For periodic signals, however, such as the PRBS, the ex-

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.

-57-

ORNL- TM- 16 47

In te rna l Distribution

1. R. K. Adams 2. G. M. Adamson 3. R. G. Affel 4. L. G. Alexander 5. R. F. Apple 6. C . F. Baes 7. J. M. Baker 8. M. A. Baker

19. S. E. Beall 20. E. S. Ee t t i s 21. F. F. Blankenship 22. R. Blunberg 23. E. G. Bohlmann 24. C . J. Borkowski 25. J. 3. Bullock 26. G. H. Burger 27. 0. W. Burke 28. S. Cantor 29. W. L. Carter 30. G. I. Cathers 31. E. L. Compere 32. J. A. Conlin 33. W. H. Cook 34. L. T. Corbin 35. G. A. Cris ty 9. J. L. Crowley 37. F. L. Culler 38. R. A. Dandl 39. H. P. Danforth 40. D. G. Davis 41. S. J. Ditto 42. R. G. Donnelly 43. F. A. Doss 44. N. E. Dunwoody 45. J. R. Engel 46. E. P. Epler 47. W. K. Ergen 48. D. E. Ferguson 49. A. P. Fraas 50. H. A. Friedman

52. J. H. Frye, Jr. 53. C . H. Gabbard 54. R. B. Gallaher 55. W . R. Grimes 56. A. G . Grindell

9-18. S. J. Ball

51. D. N. FV

57. R. H. Guymon 58. E. W. Hagen 59. P. H. Harley 60. C . S. E a r r i l l 61. P. N. Haubenreich 62. G. M. Herbert 63. P. G. Herndon 64. V. D. Holt 65. P. P . X O ~ Z 66. T. L. Hudson 67. P. R. Kasten 68. 69

70 -109. 110. 111. 112.

114. 113.

115. 116. 117. 118. 119. 120. 121. 122.

124. 123.

125. 126. 127. 128. 129. 130. 131. 132. 133 134. 135 136. 137 138. 139. 140 : 141. 142.

R. J. K e d l M. J. Kelly T. W. Kerlin S. S. K i r s l i s D. J. Knowles A. I. Krakoviak J. W. Krewson R. C . Kryter C . G. Lawson R. B. Lindauer M. I. Lundin R. N . Lyon J. H . Marable C . D. Martin C . E. Mathews H. G. MacPherson R. E. MacPherson H. C . McCurdy H. F. McDuffie C . K. McGlothlan L. E. McNeese A. J. Mil ler R. V . Miskell ( Y - 1 2 ) W. R. Mixon R. L. Moore H. G. O'Brien P. Patr iarca H. R. Payne A. M. Perry H . B. Piper B. E. Prince J. L. Redford M. Richardson R. C . Robertson H. C . Roller D. P. Roux

-58-

143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153 154. 155 ' 156. 157 158. 159 160. 161. 162.

186 -200 * 201.

202-203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216.

217.

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.

165. 166. 167. 168. 169. 170. 171. 172.

164.

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

External Di s t ribut ion

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