Ornl tm 2571

Contract No. W-7405-eng- 26

Reactor Division

J

ORNL- 'I'M- 2571

THEORETICAL DYNAMIC ANAtYSIS OF THE MSRE WITH 233U FUEL

R. C. StefYy, Jr. P. J. Wood L E G A L N O T I C E

This report m a prepared M m account of oOvenU&nt mponaored work. Nelther the Unltsd , states. nor tbe Commission. nor any pereon mtlw c4 behalf of tbe Commleslon:

A. Makes my ~ M W or represantaUon. expredsed or ImpUsd. with respect to the accuracy. completeness, or ussfulnsss of the LnlormaUon contained in thls report. or that the me of my ~o-uon, apparatus. metbod, or procees ~ f s e l o s d IO thla report nuy not lniringe I

i

privately owned right.; or B. h m m e s my liabilltles rtth respect to the ' ye of. or for d u r w e s resultlllg from the

UM of any information. apparaw. method, or proceds d~10sd in thls report. A. wed in the lhove, "pr- acting on beha4 of the Commission" ~ l u d e s .ng am-

ploy= or contractor of tbe Comrolssloa. or employ of mch contractor. to the m t that much employee or contractor ef the Commission, oItrtployea of much contractor prepares. dsllerntnntes. or DTOVI~~O 8csess to. m y MormaUm pvnunt to hls emplopant or oontuct

OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee

operated by UNION CARBIDE CORPORATION

for the U. S. ATOMIC ENERGY COIvMISSION

iii

CONTENTS

Page

Abstract ........................................................ 1 1. Introduction ................................................ 1 2. Model Description and Verification .......................... 4 3. Transient-Response Analysis ................................. 6 4. Frequency-Response Analysis ................................. 10

4.1 Calculated Frequency Response of Power $0 Reactivity ... 11 4.1.1 For Noncirculating Fuel ......................... 11 4.1.2 For Circulating Fuel ............................ 11

4.3 Sensitivity of Gn/no*6k to Parameter Changes ........... 16 4.2 Effect of Mixing in the Fuel Loop ...................... 14 4.4 Frequency Response of Outlet Temperature to Power ...... 18

5 . Stability Analysis .......................................... 19 5-1 Pole Configuration - Eigenvalues ....................... 19

5, l . l Theoretical Discussions ......................... 19 5.1.2 Eigenvalue Calculation Results .................. 24

5.2 Modified Mikhailov Method .............................. 26 5.2.1 Theoretical Discussion .......................... 26 5.2.2 Results of Applying he Mikhailov Stability

Criterion ....................................... 28 -

6 . Concluding Discussion ....................................... 34 Acknowled@;ments .................................................. 34 References ...................................................... 35 Appendix. Pad6 Approximations ................................... 37

f

II

c

1

k, THEORETICAL DYNAMIC ANALYSIS OF THE MSRE WITH 233U FUEL R. C. Steffy, Jr. and P. J. Wood*

Abstract

A study undertaken to characterize the dynamics of the 233U-fueled MSRE prior to operation revealed that the system is inherently asymptotically stable at all power levels above zero. The motivation for these studies was the expected difference between the MSRE dynamic response with 233U fuel and with 23 5U fuel because of the smaller delayed-neutron fraction of 233U. An existing system model, previously verified for 235U fuel, was modified for use in this work. The reactor system response to-reactivity perturbations is rapid and nonoscillatory at high power, and it becomes sluggish and oscillatory at lower powers. These characteristics were determined by three methods: (1) transient-response analyses, including a check of the validity of the linear model, (2) a frequency-response and sensitivity study, (3 ) stability analyses, both by inspection of the system eigenvalues and by application of the recently developed, modified Mikhailov criterion.

Keywords: MSRE, 233U fuel, stability, eigenvalues, modified Mikhailov criteria, frequency response, sensitivity, time response, Pad6 approximations.

1. Introduction

As a preliminary step in the development of a molten-salt-fueled breeder reactor, the Molten Salt Reactor Experiment (MSRE) was fueled with 233U to perform the necessq physics and chemistry experiments on this first 233U-fueled nuclear power reactor. Before nuclear operation, it was important to anticipate the dynamic behavior and the inherent stability of the system in order to insure safe, orderly operation and to plan safe, definitive experiments.

of the 233U-fueled MSRE after more than 70,000 Mwhr of power operation with 235U fuel is that 233U has a much smaller delayed-neutron fraction

The principal motivation for evaluating the operating characteristics

*Currently Assistant Director of MIT School of Chemical Engineering Practice at Oak Ridge National Laboratory.

2

W (f3) than that of 235U. acter is t ics somewhat different from those of 235U and, i n particular, a

The 233U fue l was expected t o have operating char-

.I fas te r response t o reactivity perturbations. Table 1 l is ts the predicted basic nuclear-kinetics properties of the MSREwith 233U fue l and compares them with those of the reactor with 235U fuel, *

Several different techniques were used i n analyzing the dynamics of the 233U-fueled MSRE, primarily because each technique either gave information unavailable from the others or used different approximate t reat- ments t o describe the system. developed by B a l l and Kerlin was modified s l ight ly and used t o describe the MRE with 233U fuel. reactor control system.) reactivity perturbation at several power levels was calculated first. computer code able nonlinear system with pure time delays, was used i n this study. the system frequency response was determined by using the computer code SFR-3 (Ref. 3). gain) sensi t ivi ty t o changes i n the system variables (Le., the r a t io of the change i n amplitude r a t io t o the corresponding change i n a system variable was determined as a function of frequency). s t ab i l i t y of the system was investigated by two techniques. eigenvalues were calculated at several powers t o determine &ether oscil- lat ions induced i n the system would increase or decrease i n amplitude, and the Mikhailov s t ab i l i t y criterion, as modified by Wight and K e r l i q 4

In performing the dynamics analyses a model

(This model d i d not include the effect of the The time response of the reactor system t o a

The which calculates the time response of a multivari-

Next

This code was also used t o determine amplitude r a t io (or

Finally, the absolute The system

was used t o obtain this same information with fewer approximations i n the mathematical model.

Linearized system equations were used i n the frequency-response and sensi t ivi ty calculations and in both types of s t ab i l i t y analysis. was necessary because general calculational methods do not presently exist

This

, that would permit these types of analysis with a se t of nonlinear system equations. tions. that provided for.updating the nonlinear %erns a f te r each i teration,

The time-response calculations ut i l ized the nonlinear equa- This was possible because they involved an i te ra t ive proce$*e

S

If

W

ch #I c

Table 1. Comparison of Mrclear Parameters Used i n Dynamics Analyses of USRE with 233U Fuel and with 235U Fuel

2 3 3 ~ Fuel 2 3 5 ~ Fuela

Pi, Delayed-Neutron . P i , Delayed-Neutron Fraction hi, Decay Fraction Group Ai , Decay

Constant Constant (sec-l) Circulating, (sec-l ) Circulating,

ic Effective Static Effective

x 10-4 1 0.0126 .2,28 2 0.0337 7.88 3 0.139 6.64 4 0.325 7.36 5 1.13 1.36 6 2.50 0.88

x 10-4 1.091 O.Ol24 3.848 0.0305 4.036 0.lll 5.962 0.301 1.330 1.14 0.876 3.01

Total f3 26.40 17.14 Total B

x 10-4 .2.23 14.57 13.07 26.28 7.66 2.80

66.61 -

x 10-4 0.52 3.73 4.99

16.98 7.18 2.77

36.17

Prompt neutron generation 4 x 10-4 time, sec

of reactivity, O F Temperature coefficients

Fuel salt -6.13 x Graphite -3.23 X lom5

2.4 x 10-4

4 . 8 4 x 10-~ -3.70 x

&Data from Ref. 1.

w 4

4

2. b d e l Description and Verification

A mathematical model was required t o describe the dynamic behavior of the 233U-fueled MSRE. t ha t called the "complete model" i n Ref. 1, which was developed t o analyze the dynamics of the 235U-fueled ERE.

The model chosen for this study was essentially

The just i f icat ion for using this model was i ts good agreement with experimentctl.results when applied t o the 235U-fueled system. 5

Some changes were made i n the model of Ref. 1, however, before it was applied t o the 233U-fueled system. previous tes t ing program did not verif'y the dip in the calculated frequency response curve a t approximately 0.25 r a d i d s e c , which. corresponds t o a fuel circulation time of approximately 25 sec. sufficient mixing of fue l salt i n the external loop of the theoretical model. To provide the model with more m i x i n g , an additional 2-sec first- order-time lag (mixing pot) was incorporated at the core outlet. This i s a reasonable approximation f o r the mixing i n the upper and lower reactor- vessel plenums.

The chief features of the 44th-order model, shown i n Fig. 1, are the following:

1.

The experimental results of the

This was attributed t o in-

The reactor core was divided into nine regions, each of which was s p l i t in to t w o fue l lumps and one graphite lump. given t o the nuclear importance of thermal disturbances i n each of the lumps. ment of a physical system that is considered t o have constant properties throughout and which' interacts with i ts surroundings through only those properties. )

2.

Consideration was

(The term "lump" as used i n mathematical modeling refers t o a seg-

A five-lump representation of the fuel-to-coolant heat exchanger was used, with heat being exchanged t o a single metal lump a t the temperature of the fue l leaving the first of two fuel lumps and heat being exchanged from the metal lump t o the coolant at the outlet temperature of the first of two coolant lumps.

3. Athree-lump coolant-to-air radiator was used i n which the coolant transferred heat t o a single metal lump at the temperature of the coolant leaving the f irst of two coolant lumps.

U

i

s

s

d

U'

3.77-sec DELAY

I 2-sec MIXING POT f

t

4

3

- 2

5

ORNL-DWG 68-{0067R

Fig. 1. Schematic Drawing of MSRE Showing System Divisions Used Mathematical Analysis.

R :AM L

in

6

4. A l inear model of the reactor kinetics equations was used i n a l l studies, except the time-response calculations, i n which nonlinear kinetics effects were taken into account..

5. The neutron kinetics equations were represented by a mathematical expression tha t accounted fo r the dynamic effect of circulating precursors (except for the eigenvalue calculation, which required the use of effec- t i ve delayed-neutron fractions).

6. Xenon poisoning was assumed t o be at steady s t a t e and not influ- enced by s m a l l perturbations.

3. Transient-Response Analysis

Time responses were obtained at several power levels t o provide a .physical picture of the reactor response t o reactivity perturbations such as control rod motions. The computer code MA=* applied i n this analysis makes use of the matrix exponentiation technique of solving a system of nonlinear ordinary different ia l equations (with pure time delays) of the form

W

ax = 6 + All(;;) H + f ( t ) , at where

- x = the solution vector (system s t a t e variables), A = system matrix (constant square matrix with r e a l coefficients),

All(;) = a matrix whose elements are deviations from the values i n A, includes all. nonlinear effects and time delay [thus aA(3;)

terms] , f ( t ) = forcing function.

The predicted response of the reactor power t o a step react ivi ty increase of 0.02% 6k/k i s shown i n Fig. 2 for various i n i t i a l powers. response curves point out several important characteristics of the MSRE. A t 8 Mw the m a x i m u m power level i s reached during the f i r s t second a f t e r

These

f

the step reactivity input. panied by a rapid increase i n fue l temperature i n the core, which, coupled

The rapid increase i n reactor power i s accom- - W with the negative temperature coefficient of reactivity, more than W

c

c

7

ORNL-DWG 68-tO077R 4.2

t.0

0.8 I - 0.6 E 0.4

0.2

0

-0.2

-

.

' Fig. 2. Calculated Power Response of the 233U-Fueled mRF: t o a 0.02% &/k Step Reactivity Insertion at Various Power Levels.

8

counterbalances the step reactivity input, so the power level begins t o id decrease. ing this interval, and when the power has decreased enough for the reac- t i v i t y associated with the increased nuclear average temperature t o just cancel the step reactivity input, the power levels f o r a brief time (from -6 t o -17 sec a f te r the reactivity input).

The temperature of the salt entering the core i s constant dur- I

*

About 17 sec a f t e r the reac- t i v i t y increase, the hot f luid generated i n the i n i t i a l power increase has completed i ts circui t of the loop external t o the core, and the nega- t i ve temperature coefficient of the salt again reduces the reactivity so that the power level starts down again. A t large times the reactor power has returned t o i t s i n i t i a l level, and the step reactivity input has been counterbalanced by an increase i n the nuclear average temperature i n the core. The short plateau observed i n the time-response curve at 8 Mw was also noted i n the 5-Mw case. A t lower powers, however, the slower system response preverrts the reactor from reaching the peak of i t s f i rs t oscilla- t ion before the f i e 1 has completed one circui t of the external fue l loop. The plateau therefore does not appear i n the lower power cases.

the power is decreased the reactor becomes both sluggish (slower respond- ing) and oscillatory; that is, at low powers the time required fo r oscil- lat ions t o die out i s much larger than at higher powers, and the fractional amplitude of the oscillations (A power/power) i s larger.

proximation for reactivity perturbations roughly equivalent t o l / 2 in. of control rod movement (0.04s 6k/k) was checked. sis are shown i n Figs. 3 and 4 for 8- and 0.1-Mw operation. l inear approximation i s f a i r ly good, but at 0.1 Mw it is poor. sult can be understood by considering the general form of the neutron-

- An important characterist ic of the MSRF: dynamic response is that as b

As part of the time-response analysis, the val idi ty of the l inear ap-

The resul ts of th i s analy- A t 8 bh the

This re-

kinetics equations :

9

ORNL-DWG 68-1006JR 2.5

2.c

0.5

0

--LINEAR

0 40 20 30 40 50 TIME (sac 1

Fig. 3. Power Response of the 233U-Fueled M3RE In i t i a l ly Operating at 8 Mw t o a 0.048 Step Reactivity Insertion as Calculated with Nonlinear and Linearized Kinetics Equations.

0 40 20 30 40 50 TIME trec)

Fig. at 0.1 Mw Nonlinear

4. t o a Step Reactivity Insertion of O.U$ as Calculated with and Linearized Kinetics Equations.

Power Response of the 233U-Fueled hSRF, I n i t i a l ly Operating

10

where

6n = deviation of power from i ts i n i t i a l value (no), po = reactivity necessaryto overcome effect of fue l circulation, f3, = t o t a l delayed-neutron fraction,

6p = deviation of react ivi ty from i t s i n i t i a l value, hi = decay constant for iE delayed-neutron precursor,

E = neutron generation time i n system,

6Ci = concentration of i t h - delayed-neutron precursor. The l a s t term i n Eq. (2), 6p6n/Z, i s the nonlinear term. In the

8-MW case the maxirmun deviation of the power f r o m the i n i t i a l power i s only about 3O$ of the i n i t i a l power, whereas i n the 0.1-Mw case the m a x i - mum deviation is 56% of the initial power and i s s t i l l increasing af%er 50 sec. nonlinear terms in the kinetics equation play a much larger role than the l inear terms. cant error i f the power deviates from i t s i n i t i a l level by more than a few percent. For the time-response analysis, it was necessaryto include the nonlinear terms t o obtain r ea l i s t i c results; however, use of the l in- earized equations i n the frequency-dmain analysis (Section 4 of th i s report) i s acceptable because the analysis i s based on small reactivity and power perturbations that osci l la te around the i r i n i t i a l values.

When 'the 6n term is this large with respect t o the no term, the

Thus, neglectingthe nonlinear terms m a y lead t o signifi-

4 . Frequency-Response Analy sis

Because a closed-loop frequency-response analysis provides informa- t ion about re la t ive system behavior, the l inear MSRE model was studied at different power levels from th i s point of view. The l inear system equa- t ions were f irst Laplace transformed and then solved for the r a t io of an output variable (such as power or temperature) t o an input variable (such as reactivity). his ratio, called a transfer function, i s

11

U

#.

c

. t

* t

f

(d

where

G(S) = transfer function, O ( S ) = the output variable, I(S) = input variable,

S = Laplace transform variable.

For a stable system, S may be replaced by j w , where j = v-1 and w i s the frequency of an input sine wave. With th i s substitution, G(jw) i s a complex number; the magnitude of G(jw) i s called the gain or the magnitude r a t i o and i s the r a t io of the amplitude of an input sinusoid t o that of an output sinusoid. The phase of G(jw) i s the phase difference between the input and the output sinusoids. G(jw) versus w i s referred t o as a Bode diagram or a frequency-response plot. of the magnitude r a t io curves. than f l a t t e r , broader peaks.

A plot of G(jw) and the phase of

A Bode plot provides qualitative s t ab i l i t y information i n the peaks High, narrow peaks indicate lower s t ab i l i t y

There are two basic reasons for calculating the frequency response of a system. First , the frequency-response curves are good indicators of system performance, and second, the frequency response of a system may be experimentally determined. The l a t t e r consideration i s important because it provides a means for checking the validity of a model. perimentally determined frequency response of the system i s i n agreement with tha t of the theoretical model, confidence i s gained i n the conclusions drawn from the stabil i ty analysis applied t o the model.

When the ex-

4.1 Calculated Frequency Response of Power t o Reactivity

4.1.1 For Noncirculating Fuel. he calculated frequency response of the MSRE for the noncirculating, zero-power, 233U-fueled condition i s shown i n Fig. 5. These curves are very similar-bo those of the cJ6LssIc zero-power reactor, and reference curves may be found i n most textbooks on reactor dynamics. 6~ not important, so the calculated response is that of the neutron-kinetics equations.

circulating 233U fue l i s shown i n Fig. 6 .

A t zero-power, gemperatme feedback effects are

4.1.2 For Circulating Fuel. A se t of frequency-response curves f o r These curves show the effect

i

12

ORNL-OWG 68- 40079Rl 40.000

- s + 4 0 0 0 -

400

1 I I I I I I I I I I I I I I t 1 1 I I I I I I I I I I I 1 l 1 1 1 1 1 I I I I t 1 1

40 0.004

0

-40

- 20 -30

-40

-50

-60

- 70 -80

-90 0.004

0.04 0.4 4 40 FREOUENCY (radions /set 1

0.04 0.4 , 4 FREQUENCY ( rod iands8d

40

Fig. 5 . Theoretical Frequency-Response Plots of Gri/ntyGk for the 233U-Fueled MSRF: at Zero-Power and with No Fuel Circulation.

1 4

t

n io*

13

FREQUENCY (radiam/sec)

ORNL-DWG 68-40084R2

10-3 2 5 io-*. 2 5 io-' 2 5 400 2 5 40' 2 5 FREQUENCY (radians/sec)

Fig. 6. Theoretical Frequency-Response Plots of Sn/no=Gk for the 233U-l?ueled MSRE at Various Power Levels with Fuel Circulation.

. .

14

of power over the range from zero t o 8 Mw. change i n curve shape as the power level i s changed i s the change i n magnitude of the temperature-feedback effect. smaller percentage change in power level causes a larger absolute temperature change which, i n turn, affects reactivity through the negative temperature coefficient.

The primary reason fo r the

A t higher power levels a

A t higher power levels, the maximum peaks of the magnitude-ratio plots decrease, and these maximums are reached a t higher frequencies. This implies *hat a f te r a reactivity perturbation, the relat ive power response Sn/no w i l l be of smaller magnitude and w i l l tend t o return t o zero fas te r at high power levels than it wi l l at lower power levels. observation was also made i n the the-response calculations of the pre- ceding section. )

(corresponding t o the fue l loop transit time of -25 sec) resul ts from temperature feedback from the external. loop. perturbation a t a frequency of about 0.25 radian/sec, the fue l i n the

.

(Tkis

The dip i n the higher power curves at approximately 0.25 radian/sec

During a periodic reactivity

core during one cycle wuuld return t o the core one period l a t e r and pro- duce a reactivity feedback effect that would partially cancel the external perturbation. response curves was relatively low at this frequency, and the dip was not as pronounced. curves for the 233U fue l loading me relatively high out t o greater than 1 radian/sec, so the loop feedback i s emphasized. The more negative temperature coefficient of reactivity for the 233U fue l also tends t o empha- s ize the effect of loop feedback.

With the 235U fuel the amplitude r a t i o of the frequency-

However, the amplitude r a t io of the frequency-response

4.2 Effect of Mixing i n the Fuel Loo2

The curves i n Fig. 6 were calculated with the model discussed i n Section 2 of this report. outlet was based on the estimated mixing that occurs i n the reactor ves- s e l plenums but was somewhat arbitrary, and it was therefore desirable t o explore the effect of different mixing approximations on the frequency- response curves.

The choice of a 2-sec mixing pot at the core

Figure 7 shows the effect on the 8-Mw frequency-response

Y

U

15

c,

! o I

2000

I 00

ORNL-DWG 68-fO080R

40-2 2 5 to- 2 5 IO0 2 FREQUENCY (radians/sec)

I 0-2 2 5 10- 2 5 FREQUENCY (radians/sec)

5 IO!

I 00 2 5 IO

Fig. 7. 233U-Fueled MSRE Operating at 8 Mw for Vmious Amounts of Mixing in the Circulating Loop.

Theoretical Frequency-Response Plot of Gn/no-Gk for the

bi curve of varying the lag time of the mixing pot from 0 t o 5 sec. that mixing was s t i l l assumed t o occur in each "lump" of the reactor and heat exchanger and that i n this analysis the effect of additional mixing was considered.) flow i n l ines external t o the core gives a plot with sharp curves. time i n the mixing pot increases, the curve becomes smoother. quencies (4 .06 r a d i d s e c ) and at high frequencies (>LO r a d i d s e c ) the effect of mixing i s negligible. changes slowly enough for the effects of additional mixing i n the l ines t o become insignificant, and at high frequencies,the neutron-kinetics effec ts dlrminate and temperature feedback from the external loop i s unimpor- tant.

(Note

L

"he use of pure time delays (i.e., no mixing) t o describe As the

A t low fre-

m

A t low frequencies the fue l temperature

The amount of mixing i n a circulating fuel, such as i n the IGm, i s not easily determined analytically. The quantity of interest i s not rea l ly the amount of salt "mixed" with other salt but, rather, the extent t o which the temperature of a slug of salt i s affected by i t s surroundings during i t s journey around the loop. transfer with adjoining slugs of salt, convective heat transfer t o the

pipe wall, and heat transfer i n the heat exchanger a l l resul t i n mixing" the salt temperature. gram may permit a determination of the size of the mixing pot tha t best describes the actual physical si tuation i n the ISRE.

Physical mixing of the salt, conductive heat

It

The experimental resul ts of the 233U t es t ing pro-

4.3 Sensit ivity of Gn/no*6k t o Parameter Changes

The equations used t o model the ISRF, contain many independent parame- Most-of these are well known, but conditions may be postulated un- ters.

der which the i r values m i g h t change. of changes in each individual parameter were determined by performing a sensi t ivi ty "he resul ts of this analysis for selected m i - ables are shown i n Fig. 8.

The effects on the magnitude r a t i o

The frequency range in which each variable i s most important i s ap- In principle the theoretical frequency response parent from the curves.

could be compared with the response determined experimentally and the pa-

f

LJ rameter that m i g h t be changing could be selected, or it could be determined

W

1 ;

' ?

4 .O

0.8

0.6

0.4

0.2

0

-n 7

17

0.42

0.08

0.04

0

-0.W

-0.08

ORNL - DWG 8- 4 0082

18

W that the value of a certain parameter was estimated incorrectly i n the theoretical model. i n system performance would affect more than one parameter. a change i n fuel-salt flow rate would affect the temperature feedback, effective delayed-neutron concentration, prompt temperature effects i n the core, and heat transfer i n the heat exchanger.

I n practice, it i s more l ikely that an actual change For example, +

- The frequency response

would be expectedto change i n the region where each of these variables was important. Certainly any appreciable change i n fuel-salt flow ra t e could be detected by an experimental freqliency-response determination, but t o pinpoint the exact cause m i g h t be difficult. There are changes other than fuel-salt flow ra te that m i g h t have an effect on only one or two parameters. exact cause of the anomalous behavior. t i v i t y analysis gives valuable information as t o the frequency range i n which each parameter affects the system response and the relat ive 5mpor- tance of each parameter.

For these changes, it m i g h t be possible t o determine the Under any conditions, the sensi-

.

4.4 Frequency Response of Outlet Temperature t o Power

Analyses of experimental data taken at the end of operation with 235U fue l showed that temperature perturbations were being introduced into the core as a resul t of heat transfer fluctuations at the radiator. These were significant enough in some cases t o appreciably affect the experimental frequency-response determination. proQlem, led t o a calculation o f t h e frequency response of the reactor outlet temperature with respect t o power (ST/Sn) t o provide a means of determining the effect of extraneous temperature disturbances.

This, together with the mixing

From a theoretical standpoint, many temperature-to-power frequency responses may be calculated. However, only those that may be experimen- t a l l y ver i f ied are of.use i n comparing a theoretical model with a physi- ca l system or in determining the effect of outside influences (such as wind velocity affecting the radiator heat transfer) during an actual test. The model discussed previously was modified t o include a term that represented the fuel-salt temperature, as recorded by a thermocouple, 2.5 sec downstream of the outlet of the 2-sec mixing pot. chosen because it corresponds t o the location of a particular thermocouple

This position was

f

. ..

19

gi

.

.

with a convenient output. outside of a fuel-salt pipe t o a change i n fuel-salt temperature was found8 t o be adequately approximated by a 1-sec pure time delay plus a 5-sec first-order lag. changes is shown i n Fig. 9.

The response of a thermocouple located on the

A schematic diagram of the model that incorporates these

Figure 10 gives the frequency response of the salt temperature t o power changes and the thermocouple response t o power changes. quencies these are the same, but at high frequencies there i s an increasing difference between the frequency response of salt temperature t o power changes and that of the indicated temperature t o power changes because of the f i n i t e time required t o transfer heat through the pipe w a l l .

A t low fre-

The effect of fue l mixing on the 6T/6n frequency response may also In addition t o the 2-sec mixing pot at the reactor be seen i n Fig. 10.

outlet, a 5-sec mixing pot was included between the heat exchanger and the core. and 0.3 radianslsec but had no noticeable effect a t other frequencies.

This caused the frequency-response plot t o be smoothed between 0.1

!The differences i n magnitude of the 6T/6n curves at low frequencies for different power levels i s caused by heat transfer changes a t the radiator. with the air flow ra te t o the 0.6 power. t o vary l inearly with the power level.

The tube-to-air heat transfer coefficient was assumed t o vary The air flow rate was assumed

5. Stabil i ty Analysis

5.1 Pole Configuration - Eigenvalues 5.1.1 Theoretical Discussions. In general, i f a reactor system can

be represented by the block diagram shown i n Fig. ll, i t s closed-loop power-to-reactivity transfer function can be written as

6n G 6k 1+G" - =

In t h i s application, G is the transfer function of the reactor kinetics and H is the transfer function fo r the system feedback, which includes the effect of prompt temperature changes i n the core, as w e l l as delayed temperature effects caused by the salt circulation. The denominator of t h i s

20

ORNL-WG 68-40083R *THERMOCOUPLE RESPONSE

RELATIVE THERMOCOUPLE k e c PURE DELAY PosinoN ON OUNT PIPE

2 5 - m PURE DELAY +(/

t I

REACTOR CORE

t

1

I

I I

~~ ~

Fig. 9. Schematic Diagram of hSRE h d e l Including the Model Repre- sentation of a Thermocouple's Response t o a Change i n Sa l t Temperature.

21

5

f

- C s s 0 t K W

3

z 0

I

n c a

40-5 40-4 (0-2 FREQUENCY (rodions/sec)

to-'

Fig. 10. Frequency-Response Plots of Salt Temperature (in the Out- let Pipe) to Power Changes and Thermocouple (on the Outlet Pipe) Response to Power Changes for Various Amounts of Salt Mixing at Power Levels of 1 Mw and 8 Mw.

8k FEEDBACK1

22

ORNL-DWG 68-1f684

REACTOR KINETICS TRANSFER FUNCTION

POWER

I

REACTIVITY TO NUCLEAR POWER

I TRANSFER FUNCTION I

Fig. ll. General Block D i a g r a m of a Nuclear Reactor. P

. U

23

transfer function, 1 + GH, is a polynomial (called the characteristic polynomial) i n the Laplace transform variable S. nomial are the poles of the system transfer function. equal t o the eigenvalues of the system coefficient matrix, A. and sufficient condition fo r l inear asymptotic s t ab i l i t y i s that the system poles (and therefore the eigenvalues of A) have negative real parts. This cr i ter ion gives a definite answer t o the question of l inear s tabi l i ty . If the physical system has an N X N coefficient matrix (N l inear equations i n N system s t a t e variables), it w i l l have N eigenvalues.

The roots of th i s poly- These roots are

A necessary

The dominant eigenvalues of a stable system are the complex eigenvalues nearest t o the imaginary axis. These eigenvalues dominate the time response of a system i f the other eigenvalues are relatively far removed from the imaginary axis. Two things can be learned about the behavior of a system from the location i n the complex plane of the dominant eigenvalues. Firs t , a f t e r a stable system has been perturbed, the amplitude of the oscillations i n a system variable decreases exponentially w i t h a time constant equal t o the inverse of the distance of the dominant eigenvalue from the imaginary axis. Second, the transient overshoot of the system i s determined by the effective damping ratio, E (Ref. 9). This damping r a t io i s determined by the angle B a vector from the origin of the complex plane t o the dominant eigenvalue makes with the imaginary axis:

As tan B increases from O t o 00, f increases from 0 t o 1. The value of f determines how large the overshoot of the system i s during a transient. Small values of f indicate larger overshoot than large values.

One characterist ic of a. zero-power reactor i s a pole of the system transfer function at the intemection of the r ea l and imaginary axes. Since the system eigenvalues correspond t o the system poles, th i s also implies an eigenvalue at the origin. of a system of this type wi l l increase as long as there i s a positive input (reactivity) but wi l l level out when there i s no input.

The output (for a reactor, the pwer)

t

24

The computer code of h a d and Van Ness was used t o determine the eigenvalues of the system matrix, but before the eigenvalues of Ishe sys- t e m could be determined, an approximate treatment of the pure time delays in the system model was necessary. (see Appendix) was used for a l l time delays except the one representing the circulation of the delayed-neutron precursors around the external loop. To account for the f ac t that the fue l was circulating, a se t of effective delayed-neutron fractions (Beff) was used i n the eigenvalue calculation.

An eighth-order Paad approximation

5.1.2 Eigenvalue Calculation Results. Figure l.2 shows several of ,

the dominant eigenvalues plotted for various power levels. of each calculated eigenvalue i s negative, with the general trend being toward smaller absolute values with decreasing power level.

The rea l part

The negative r e a l p a r t s insure l inear asymptotic s tabi l i ty , and the trend toward smaller absolute values with decreasing power level again p i n t s t o a higher degree of s t ab i l i t y at higher po.wer levels. The eigenvalues that l i e rela- t ive ly fax removed from the r ea l axis (those with positive imaginary com- ponents between 0.20 and 0.30) are a resul t of the coupled energy-balance equations and are relatively independent of nucleax power. They do, however, exhibit some power dependence, primarily because of the variation i n heat transfer coefficient at the radiator with power level. ca l implication of these eigenvalues is that any temperature disturbance i n the system would tend t o cause s l ight temperature oscillations around the system, even i f the nuclear power were zero.

*

The physi-

The eigenvalues for the nuclear kinetics equations were a l l on the r e a l axis for the zero-power condition.

The calculation for zero power resulted i n a dominant eigenvalue with a r e a l pazt equal t o -0.36 A 0.0. the f i f t h significant d i g i t (the coefficients are generally not known t o bet ter than four places) the value of the r e a l part of the dominant eigenvalue could be changed from a s m a l l negative value t o a s m a l l positive value. instabil i ty, it must be realized that within the accuracy of the calculation, both these values are zero. A calculated value of identically zero

sec and an imaginary part equal t o Byproper manipulation of one coefficient i n the system equations i n

While from a mathematical standpoint the positive sign would imply

I;f

t

t

7

. hp

L1

f

ti

0.3

0.2

0.4

-0.i

-0.2

25

ORNL-DWG 69-467R

- POWER FOR EIGENVAWE - CALCULATION

8Mw . 5 M w A 4Mw

OdMw * ZERO SOLID LlNES INDICATE POWER DEPENDENCE OF A PARTICULAR EIGENVALUE - -

-0.3 0.44 0.12 0.10 0.08 0.06 0.04 0.02 0

REAL AXIS (seci)

Fig. 12. Plots of Several of the More Dominant Eigenvalues of the 233U-Fueled MSRE for Various Power Levels.

26

u (the expected value a t zero power) would have resulted only if absolute precision had been maintained throughout the calculation.

As a point of general interest, a se t of system equations with an t

eigenvalue with r e a l part equal t o -1 x system, a f te r the input perturbations has vanished, spontaneously ret&ns t o i t s original s t a t e with a time constant of lo6 sec (11.5 days); simi- larly, an eigenvalue with r ea l part equal t o +1 X sec would spontaneously deviate away from i t s original s t a t e with an 11.5-day time constant. While the difference in s t ab i l i t y between these two cases i s ob- vious, neither would represent a difficult problem to the control system designer, and either case would represent acceptable system behavior3 or, conversely, for a particular application, if one of these was unacceptable, both would probably be unacceptable.

power (see Fig. 12) were used t o calculate the damping ratio, E. of t h i s calculation are l i s t e d below:

sec implies that a perturbed -

The eigenvalues tha t form the curve which goes t o the origin a t zero Results

f

I c

0.1 0.16 1.0 0.37 5.0 0.96 8.0 0.99

Since there are eigenvalues on the r ea l a x i s that have r e a l parts of about the same magnitude as those used i n the calculation, the system damping r a t i o cannot alone be used t o determine the system oscil latory behavior; however, it i s useful as a relat ive indicator. emphasizes the E R E characteristic of being less oscil latory at high power levels than a t low power.

The damping r a t i o again

5.2 Modified Mikhailov Method

5.2.1 Theoretical Discussion. An alternate technique fo r determin- F

ing absolute s t ab i l i t y of a l inear system was recently developed by Wright and Kerlin4 along the l ines of the method of Mkhailov. tage of t h i s method over the eigenvalue technique i s that no approximations

The chief advan- . W

27

are necessary t o t r e a t pure time delays. Kerlin i s based on a l inear constant-parameter system of order N, which

The technique of Wright and

can be written as a set of coupled, first-order different ia l equations with constant coefficients:

where

F( t ) = column vector of system s ta te variables,

f ( t ) = forcing function, A = constant N X N coefficient matrix,

E = column vector of coefficients of f ( t ) , R(t,.r) = matrix of time delay terms, each of the form

r x ( t - Tis) . i s s A Laplace transform of Eq. (5) yields

S E ( S ) = A G(S) + R(S,T) + fj. f (S) , where S i s the Laplace transform variable. time-lagged elements,

The transform of one of the

r. x . ( t - Tij) , 13 J i s

where L(S) i s a matrix composed of the rij into Eq. ( 5 ) m a manipulation yields

terms. Substitution

where I i s the identity matrix, and super -1 indicates the inverse matrix.

28

W The absolute s t ab i l i t y of the system i s determined by the location of the roots of t h e equation

det (A - S I + L) = 0 . The values of S for which the above equation i s sat isf ied are the system eigenvalues oi characteristic roots.

Wright and Kerlin defined a quantity M(S) as In the i r development of the modified Mikhailov s t ab i l i t y crierion,

where

D(S) = det (A- S I + L) , P(S) = det (T - S I ) ,

(10)

(W and T i s an N X N diagonal matrix, each element of which i s the negative of the absolute value of the corresponding elements of the A matrix. quantity P(S) can be considered t o be a factor that normalizes D(S) i n both magnitude and phase. The s t ab i l i t y cri terion associated with M(S) is that "the number of zeros of the system determinant det (A - S I + L) i n the right half plane equals the negative of the number of. counterclockwise encirclements of the origin by the vector M(jw) as u goes from 4 0 t o

The

While not explicit ly stated i n this quotation the important parameter i s the - net number of encirclements of the origin, with clockwise encirclements counting negative and counterclockwise encirclements counting positive. t o plot M(jw) for w running from 0 t o SOO, approaches 1.0 as u becomes large, so a f i n i t e number of w values can corn-- pletely describe the mot ion of M( j w ).

Because of symmetry about the r ea l axis it is necessary only It can also be shown that M(jw)

5.2.2 Results of Applying the Mikhailov Stabi l i ty Criterion. By using a c-ter program developed by Wright and Kerlin, plots were generated of M ( j w ) fo r w running from 0 t o $40, these plots for several power levels. els there are no encirclements of the origin as w goes from - t o a.

Figures 13 through 17 show As shown, a t significant power lev-

As

a

ht

29

t.0

0.8

u)

SZ 0.6 a & 2 0.4 a - W

0.2

0

-0.2 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 4.0 t.2

REAL AXIS

ORNL-OWG 68-10069

REAL AXIS

Fig. 13. Modified Mikhailov Plot for MSRE Operating with 233U Fuel at Zero Power. (a) Complete plot. (b) Near origin.

4.2

4.0

0.8

v, 2 Q6 a a

-P

> z g a4

0

-0.2

30

-06 -0.4 *-02 0 0.2 0.4 0 6 Q8 4.0 4.2 REAL AXS

(I 40-=1

8

6

4

u, a

z a 5

X

$ 2

c)

0

-2

-4

( x 4 ~ - 4 ) -2.0 -4.5 -4.0 -0.5 0 0.5 REAL AXIS

?

Fig. 14. Modified Mikhailov-Plot for mRE Operating with 233U Fuel at 1 kw. (a) Complete plot. (b) Near origin.

31

ORNL-OWG 68-10073R

. t

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 REAL AXIS

ORNL-DWG 60-MO12

( x t ~ - 3 )

4

s? 4 x 0 > K a 5 - 4 a E a

-2

( b ) -3 ( x ~ O - ~ ) -6 -5 -4 -3 -2 - 1 0 f

REAL AXIS

Fig. 15. Mdified Mikhailov Plot fo r &ERE Operating with 233U Fuel at 100 kw. (a) Complete plot. (b) Near origin.

32

ORNL- DWG 68-10071

(0)

-0.5 I -0.50 -0.25 0 0.25 0.50 0.75 4.0 4.25

REAL AXIS

?

Fig. 16. Modified Mikhailov Plot fo r %RE Operating with 233U Fuel at 1 I&. (a) Complete plot. (b) Ne= origin.

.

33

f

c

-2 - 4 0 4 2 3 REAL AXIS

Fig. 17. at 8 Mw.

Modified Mikhailov Plot for NRE Operating with 233U Fuel

0

34

i n the eigenvalue calculation, the results of zero-power calculations were highly sensit ive t o s m a l l changes i n coefficients i n the system ma- t r ix . origin when the t o t a l plot from - t o $40 i s considered, the intercept at -6.5 X

gin.

While the zero-power plots actually show an encirclement of the

should, of course, be interpreted as passing through the ori- his i s typical fo r a zero-power reactor.

Application of the modified Mikhailov s t a b i l i t y c r i te r ion t o the se t of system equations that describe the MRE verifies that the WRE i s inherently stable at a l l power levels.

6. Concluding Discussion

The overall resu l t of this study i s a determination that the MRE fueled with 233U i s an inherently stable system at a l l power levels of interest. The response of the uncontrolled system at powers above zero

power, as seen i n the t ransient analysis and corroborated by the eigenvalues, i s rapid and nonoscillatory at high powers and becomes sluggish and osci l la tory at low powers, MSRE be subjected t o a tes t ing program at powers from zero t o f u l l power t o experimentally verify the predicted response. be performed i n such a way that both the 6n/6k and 6T/6n t ransfer functions

It is recommended that the 233U-fueled

These experiments should

can be determined, since both can now be compared with theoret ical predic- t ions.

Acknowledgments

hd

The authors are grateful t o T. W. Kerlin of the University of . Tennessee and S. J. B a l l f o r their assistance and advice. The help of

J. L. Lucius i n assisting with programming d i f f i cu l t i e s i s a l so acknowl- edged.

P

1.

2.

3.

4.

5.

* 6.

7.

8.

9.

10.

11.

35

References

S. J. B a l l and T. W. Kerlin, Stabil i ty Analysis of the Molten-Salt Reactor Experiment, USAEX Report ORNLTM-1070, Oak Ridpe National Laboratory, December 1965.

S. J. B a l l and R. K. Adams, MATEXP, A General Purpose Digital Com- puter Program for Solving Ordinary Differential Equations by the Ma- t r i x Exponential Method, USAEC Report ORNL-TM-1933, Oak Ridge National Laboratory, Aug. 30, 1967.

T. W. Kerlin and J. L. Lucius, The SFR-3 Code - A Fortran.Program for Calculating the Frequency Response of a Multivariable System and Its Sensitivity to. Parameter (?4anges, USAEC Report ORNL-TM-1575, Oak Ridge National Laboratory, June 1966.

W. C. Wright and T. W. Kerlin, An Efficient, Computer-Oriented Method for Stabi l i ty Analysis of Large Multivariable Systems, Report NEUT 2806-3, Nuclear Engineering Department, University of Tennessee, July 1968.

T. W. Kerlin and S. J. Ball, Experimental Dynamic Analysis of the Molten-Salt Reactor Experiment, USAEC Report ORNL-TM-1647, Oak Ridge National Laboratory, October 1966.

M. A. Schultz, Control of Nuclear Reacors and Power Plants, p. 116, McGraw-Hill, New York, 1961.

G. Robert Keepin, Physics of Nuclear Kinetics, pp. 328-329, Addison- Wesley, Reading, Mass., 1965.

S. J. B a l l , Oak Ridge National Laboratory, personal communication t o R. D. Steff'y, Jr., July'24, 1968.

D. R. Coughanowr and L. B. Koppel, Process Systems Analysis and Con- t r o l , pp. 86, 187, McGraw-Hill, New York, 1965. F. P. h a d and J. E. Van Ness, Eigenvalues by the QR Transform, Share-1578, Share Distribution kency, I B M Program Distribution, White Plains, N. Y., October 1963.

G. S. Stubbs and C. H. Single, Transport Delay Simulation Circuits, USAEC Report WAPD-T-38 and Supplement, Westinghouse Atomic ,Power Division, 1954.

t

k.0

37

Appendix

PAD; APPROXIMATIOWS

In the analysis of physical systems it frequently becomes necessary t o mathematically approximate pure time delays. Such a treatment, for instance, would be necessary t o relate the outlet temperature t o the inlet temperature of a f lu id flowing i n plug flow through an 4diabatic pipe if the a x i a l heat conduction were negligible. is x( t ) and the delay time i s T, the outlet temperature will be x( t - T). The Laplace transform of a delayed variable x ( t - T ) i s x(S) e-TS, and if a system eontainingtime delays i s t o be analyzed simply, a l inear approximation must be substituted fo r e-Ts. e i s as an inf in i te series:

t

If the inlet temperature

One technique fo r approximating -TS

. . . T2S2 T3S3 T4S4 2. 3! 4. = l - T S + 7 - - + 7 - - TS e An approximation t o this expansion i s

2 S - - T 2 4 S ,+- T

+ . . . . 1 - T S +.z-- E-- = T2S T3S3 - TS e

Unfortunately th i s approximation does not converge rapidly fo r large values of TS. A bet ter approximate treatment i s the Pad6 approxi~nation:~

-TS n I - 2 y r s + T E S ~ 0 e . E T . r=l 1 + 2r r~ rS + T ~ S 2 2

where

T = Delay .time =

The order of t h i s approximation i s 2n. t h a t t h e cri terion fo r selection of rr shif t ,

n m

t ~ r ~ . r r r=l ( 4 )

Stubbs and Single have suggested

and T~ be such that the actual phase-

3

38

is as close as possible t o the ideal phase shift for a pure time delay,

f#)o = 4177~ . To meet this criterion,. they have suggested (for an eighth-order Pad6 approximation) that the following values be used:

rl == r2 = 0.40 r3 = 1.0 r4 = 0.5

71/72 = 1.68 73 /74 = 1.14

For -this choice of constants the phase error ( 4 - $ 0 ) i n the Pad6 approximation i s less than one degree at (jY = 13.5 radians. cies this error .increases, and at lower frequencies it decreases.

A t higher frequen-

The following l inear equations have been developed for the eighth- order Pad6 approximation, where % is the variable input t o the time delay, xo is the delayed variable output of the time delay, T i s the delay time, and y1 through y8 are Pad4 variables:

e

dY2 10 - = -10.5g5gxi - - y1 - 19.5858xo (6 1 at T

d y 3 10 - 7 -4S.8441Xi - - y2 + 45.844lxo (7) at T dY4 10 - = -65.8O83Xi - - y3 - 65.8083~0 a t T

f

W

c

f

39

w5 10

a t . T - = -64 .1135~. - - y4 + 6 4 . l l 3 5 ~ 0

1 - (9 1

dY6 10 - = - 4 3 . 8 1 6 9 ~ ~ - - y 5 - 43.8169~0 dt T

w 8 10 - = -6.3308xi - - y 7 - 6.3308~0 at T

For a further discussion of this method see Stubbs and Single.

,

t

6

c

1. J. L. Anderson 2. R. K. Adams

-3. C. F. Baes 4. S. J. B a l l 5. H. F. Bauman 6. S. E. Beall 7. E. S. Bettis 8. R. Blumberg 9. E. G. Bohlmarh

10. C. J. Borkovski ll. R. B. Briggs 12. W. R. Cobb 13. E. L. Compere 14. W. B. Cottrell 15. J. L. Crowley 16. F. L. Culler 17. S. J.. Ditto 18. W. P. Eatherly 19. J. R. Engel 20. E. P. Epler 21. D. E. Ferguson 22. L. M. Ferris 23. A. P. Fraas 24. J. K. Franzreb 25. D. N. Fry 26. C. H. Gabbard 27. R. B. Gallaher 28. W. R. Grimes 29. A. G. Grindell 30. R. H. Guymon 31. P. H. Harley 32. 33 . 34. 35. ~36 . 37. 38. 39.

P. N. Haubekeich, A. Houtzeel T. L. Hudson P. R. a s t e n R. 3. Ked1 T. W. Kerlin H. T. Kerr A. I. Krakoviak

I

41

ORNL-TM-2571

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40. T. S. JCress 41. R. C. Xryter 42. M. I. Lundin 43. R. N. won 44. R. E. MacPherson 45. H. McClain 46. H. E. McCoy 47. H. C. McCurdy 48. A. J. Miller 49. R. L. b o r e 50. E. I,. Nicholson 51. L. C. Oakes 52. A. M. Perry 53. H. B. Piper 54. B. E. Prince 55. G. L. Ragan 56. J. L. Redford n. M. Richardson 58. J. C. Robinson

59-60. M. W. Rosenthal 61. A. W. Savolainen 62. Dunlap Scott 63. M. J. Skinner 6s. A. N. Smith 65. I. S p i e d

66-75. R. C. Steff'y '76. D. A. Sundberg 77. J. R. Tallackson 78. R. E. Thoma ,79. D. B. Trauger 80. J. R. Weir 81. 82. 83.

84-93. 94.

95-96. 1 97-98. 99-101.

102.

M. E. whatley J. C. White G. D. Whitman P. J. Wood G a l e Young Central Research Library Y-12 Document Reference Section Laboratory Records Department Laboratory Records (LRD-RC)

42

\

103. 1@-105. 106-120.

121.

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