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Reactor Division J

THEORETICAL DYNAMIC ANAtYSIS OF THE MSRE WITH 233U FUEL

R. C. StefYy, Jr.

P. J. Wood

LEGAL NOTICE

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 orW represantaUon. expredsed or ImpUsd. with respect to the accuracy. completeness, or ussfulnsss of the LnlormaUoncontained in thls report. or that the me of m y ~o-uon, apparatus. metbod, or procees ~ f s e l o s d IO thla report nuy not lniringe privately owned right.; or B. h m m e s m y liabilltles rtth respect to the ' y e of. or for d u r w e s resultlllg from the ofany information. apparaw. method, or proceds d ~ 1 0 sin d thls report. A. wed in the lhove, "practing on beha4 of the Commission" ~ l u d e s.ng amploy= 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

........................................................ 1. Introduction ................................................ 2. Model Description and Verification .......................... 3. Transient-Response Analysis ................................. 4. Frequency-Response Analysis ................................. 4.1 Calculated Frequency Response of Power Reactivity ... 4.1.1 For Noncirculating Fuel ......................... 4.1.2 For Circulating Fuel ............................ 4.2 Effect of Mixing in the Fuel Loop ...................... 4.3 Sensitivity of Gn/no*6k to Parameter Changes ........... 4.4 Frequency Response of Outlet Temperature to Power ...... 5 . Stability Analysis .......................................... 5-1 Pole Configuration - Eigenvalues ....................... 5 , l . l Theoretical Discussions ......................... 5.1.2 Eigenvalue Calculation Results .................. 5.2 Modified Mikhailov Method .............................. 5.2.1 Theoretical Discussion .......................... 5.2.2 Results of Applying â&#x201A;Źhe Mikhailov Stability Criterion ....................................... 6 . Concluding Discussion ....................................... Acknowled@;ments.................................................. References ...................................................... Appendix. Pad6 Approximations ................................... Abstract

1 1

4 6 10 11 11 11

14 16

18 19 19 19

24 26 26

28 34

34 35 37

f I I

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-reactivityperturbations 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. The principal motivation for evaluating the operating characteristics 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 *Currently Assistant Director of MIT School of Chemical Engineering Practice at Oak Ridge National Laboratory.

2 than that of 235U. The 233U f u e l was expected t o have operating chara c t e r i s t i c s somewhat different from those of 235U and, i n particular, a f a s t e r response t o reactivity perturbations. Table 1 l i s t s the predicted (f3)

W . I

basic nuclear-kinetics properties of the MSRE’with 233U f u e 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 r e a t ments t o describe the system. In performing the dynamics analyses a model developed by B a l l and Kerlin’ was modified s l i g h t l y and used t o describe the MRE with 233U f’uel. (This model d i d not include the effect of the reactor control system.) The time response of the reactor system t o a reactivity perturbation at several power levels was calculated first. The computer code which calculates the time response of a multivariable nonlinear system with pure time delays, was used i n t h i s study. Next the system frequency response was determined by using the computer code SFR-3 (Ref. 3). This code was also used t o determine amplitude r a t i o (or gain) s e n s i t i v i t y t o changes i n the system variables (Le., the r a t i o of the change i n amplitude r a t i o t o the corresponding change i n a system variable was determined as a f’unction of frequency). Finally, the absolute s t a b i l i t y of the system was investigated by t w o techniques.

The system eigenvalues were calculated at several powers t o determine &ether oscillations induced i n the system would increase or decrease i n amplitude, and the Mikhailov s t a b i l i t y criterion, as modified by Wight and K e r l i q 4 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 s e n s i t i v i t y calculations and in both types of s t a b i l i t y analysis. This was necessary because general calculational methods do not presently exist

, t h a t would permit these types of analysis with a s e t of nonlinear system equations. The time-response calculations u t i l i z e d the nonlinear equations. This was possible because they involved an i t e r a t i v e proce$*e

S If

that provided for.updating t h e nonlinear %erns a f t e r each iteration,

Table 1. Comparison of Mrclear Parameters Used i n Dynamics Analyses of USRE with 233U Fuel and with 235U Fuel 2 3 3 Fuel ~

Group

Ai, Decay Constant (sec-l)

Fraction

x 10-4 3

4 5 6

0.0126 0.0337 0.139 0.325 1.13 2.50 Total f3

Pi, Delayed-Neutron

Pi, Delayed-Neutron .

1 2

2 3 5 Fuela ~

Circulating, Effective

.2,28 7.88 6.64 7.36 1.36 0.88

x 10-4 1.091 3.848 4.036 5.962 1.330 0.876

26.40

17.14

Prompt neutron generation time, sec Temperature coefficients of reactivity, O F Fuel salt Graphite &Data from Ref. 1.

4 x 10-4

-6.13 x -3.23 X

lom5

hi, Decay Constant (sec-l )

Fraction Static

O.Ol24 0.0305 0.lll 0.301 1.14 3.01 Total B

Circulating, Effective

x 10-4

.2.23 14.57 13.07 26.28 7.66 2.80

0.52 3.73 4.99 16.98 7.18 2.77

66.61

36.17

2.4 x 10-4

4 . 8 4 x 10-~ -3.70 x

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. The model chosen f o r t h i s study was essentially t h a t called the "complete model" i n Ref. 1, which was developed t o analyze

the dynamics of the 235U-fueled ERE. The j u s t i f i c a t i o n f o r using t h i s model was i t s 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 t h e 233U-fueled system.

The experimental results of the previous t e s t i n g 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

f u e l circulation time of approximately 25 sec. This was attributed t o insufficient mixing of f u e l salt i n the external loop of the theoretical model. To provide t h e model with more m i x i n g , an additional 2-sec firstorder-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 reactorvessel plenums.

The chief features of the 44th-order model, shown i n Fig. 1, are the following: 1. The reactor core was divided into nine regions, each of which was s p l i t i n t o t w o f u e l lumps and one graphite lump.

Consideration was given t o the nuclear importance of thermal disturbances i n each of the

lumps.

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

ment of a physical system that i s considered t o have constant properties throughout and which' interacts with i t s surroundings through only those properties. ) 2. 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 temperat u r e of the f u e l leaving the f i r s t of t w o f u e l lumps and heat being exchanged from the metal lump t o t h e 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 i r s t of two coolant lumps.

ORNL-DWG 68-{0067R

3.77-sec DELAY

I 2-sec

MIXING POT

R :AM L

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

4. A l i n e a r 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 t h a t accounted f o r the dynamic effect of circulating precursors (except for the eigenvalue calculation, which required the use of effect i v e 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 d i f f e r e n t i a l equations (with pure time delays) of the form

ax at = 6 + All(;;) where

H+ f(t)

-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, [thus aA(3;) includes all. nonlinear effects and time delay terms] , f ( t ) = forcing function. The predicted response of the reactor power t o a step r e a c t i v i t y increase of 0.02% 6k/k i s shown i n Fig. 2 for various i n i t i a l powers.

These

response curves point out several important characteristics of the MSRE. f

A t 8 Mw the maximum power l e v e l i s reached during the f i r s t second a f t e r the step reactivity input.

The rapid increase i n reactor power i s accom-

panied by a rapid increase i n f u e l temperature i n the core, which, coupled with the negative temperature coefficient of reactivity, more than

7 ORNL-DWG 68-tO077R

4.2 t.0

0.8

0.6 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 l e v e l begins t o decrease. The temperature of the salt entering the core i s constant during this interval, and when the power has decreased enough f o r the react i v i t y associated with the increased nuclear average temperature t o j u s t cancel the step reactivity input, the power levels f o r a brief time (from -6 t o -17 sec a f t e r the reactivity input). About 17 sec a f t e r the reac-

id I

t i v i t y increase, the hot f l u i d generated i n the i n i t i a l power increase has completed i t s c i r c u i t of t h e loop external t o the core, and the negat i v e temperature coefficient of the salt again reduces the reactivity so that the power l e v e l 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 t h e 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 r s t oscillat i o n before the f i e 1 has completed one c i r c u i t of the external f u e l loop. The plateau therefore does not appear i n the lower power cases. An important characteristic of the MSRF: dynamic response i s that as the power i s decreased the reactor becomes both sluggish (slower responding) and oscillatory; that is, at low powers the time required f o r oscillations t o d i e out i s much larger than at higher powers, and the fractional amplitude of the oscillations (A power/power) i s larger. As part of the time-response analysis, the v a l i d i t y of the linear approximation f o r reactivity perturbations roughly equivalent t o l / 2 in. of control rod movement (0.04s 6k/k) was checked. The r e s u l t s of t h i s analysis are shown i n Figs. 3 and 4 for 8- and 0.1-Mw operation. A t 8 bh the l i n e a r approximation i s f a i r l y good, but a t 0.1 Mw it i s poor. This re-

sult can be understood by considering the general form of the neutronkinetics equations :

9 ORNL-DWG

2.5

68-1006JR

2.c

--LINEAR

0.5

0 0

TIME (sac 1

Fig. 3. Power Response of the 233U-F’ueled M3RE I n i t i a l l y Operating at 8 Mw t o a 0.048 Step Reactivity Insertion as Calculated with Nonlinear and Linearized Kinetics Equations.

20 30 TIME t r e c )

Fig. 4. Power Response of‘ the 233U-Fueled hSRF, I n i t i a l l y Operating at 0.1 Mw t o a Step Reactivity Insertion of O.U$ as Calculated with Nonlinear and Linearized Kinetics Equations.

where 6n = deviation of power from i t s i n i t i a l value (no), po = reactivity necessaryto overcome effect of f u e l circulation, f3,

= t o t a l delayed-neutron fraction,

E = neutron generation time i n system, 6p =

deviation of r e a c t i v i t y from i t s i n i t i a l value,

hi = decay constant f o r iE delayed-neutron precursor, 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 i s 56% of t h e initial power and i s s t i l l increasing af%er 50 sec. When 'the 6n term i s this large with respect t o the no term, the nonlinear terms in the kinetics equation play a much larger r o l e than t h e

linear terms.

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

cant error i f the power deviates from i t s i n i t i a l l e v e l by more than a For the time-response analysis, it was necessaryto include the nonlinear terms t o obtain r e a l i s t i c results; however, use of the l i n earized equations i n the frequency-dmain analysis (Section 4 of t h i s re-

few percent.

port) i s acceptable because the analysis i s based on small reactivity and

power perturbations that o s c i l l a t e around t h e i r i n i t i a l values.

Frequency-Response Analy sis

Because a closed-loop frequency-response analysis provides informat i o n about r e l a t i v e system behavior, the linear MSRE model was studied at different power levels from t h i s point of view. The l i n e a r system equations were f i r s t Laplace transformed and then solved f o r the r a t i o 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

U #.

where G(S) = transfer f’unction, O ( S ) = the output variable,

I(S) = input variable, S = Laplace transform variable.

v-1

and w i s the For a stable syst’em, S may be replaced by j w , where j = frequency of an input sine wave. With t h i s substitution, G(jw) i s a com-

. t

plex 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 i o of the amplitude of an input sinusoid t o that of an output sinusoid. The phase of G ( j w ) i s the phase difference between the input and the output sinusoids. A plot of G ( j w ) and the phase of G(jw) versus w i s referred t o as a Bode diagram o r a frequency-response plot. A Bode plot provides qualitative s t a b i l i t y information i n the peaks of the magnitude r a t i o curves. High, narrow peaks indicate lower s t a b i l i t y than f l a t t e r , broader peaks. There are two basic reasons f o r 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 f o r checking t h e validity of a model. When the experimentally determined frequency response of the system i s i n agreement with t h a t of the theoretical model, confidence i s gained i n the conclusions drawn from t h e s t a b i l i t y analysis applied t o t h e model.

4.1 Calculated Frequency Response of Power t o Reactivity

4.1.1 For Noncirculating Fuel. “he calculated frequency response of the MSRE f o r the noncirculating, zero-power, 233U-fueled condition i s shown i n Fig. 5. These curves a r e very similar-bo those of the cJ6LssIc zero-power reactor, and reference curves may be found i n most textbooks on reactor dynamics. 6~ A t zero-power, gemperatme feedback effects are not important, so the calculated response i s that of the neutron-kinetics equations. 4.1.2 For Circulating Fuel. A s e t of frequency-response curves f o r circulating 233U f u e l i s shown i n Fig. 6 . These curves show the effect

ORNL-OWG

68-

40079Rl

40.000

-+ s

4000

400

I I IIIII

I I I I t 1 1

I IIIII

I 1 l 1 1 1 1 1

I I t 1 1

0.004

0.04

0.4

FREOUENCY (radions /set 1

-40

- 20 -30 -40

-50 -60

-70 -80 -90 0.004

0.04

0.4

, 4 FREQUENCY ( r o d i a n d s 8 d

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

1 ORNL-DWG 68-40084R2

FREQUENCY(radiam/sec)

n 10-3

io-*.

io-'

2 5 400 FREQUENCY (radians/sec)

40'

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

io*

. .

14 of power over the range from zero t o 8 Mw.

The primary reason f o r the

change i n curve shape as the power l e v e l i s changed i s the change i n magnitude of the temperature-feedback effect. A t higher power levels a smaller percentage change in power l e v e l causes a larger absolute temperat u r e change which, i n turn, affects reactivity through the negative temperature coefficient. 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 t e r a reactivity perturbation, the r e l a t i v e power re.

sponse Sn/no w i l l be of smaller magnitude and w i l l tend t o return t o zero f a s t e r a t high power levels than it w i l l at lower power levels. (Tkis observation was also made i n the the-response calculations of the preceding section. ) The dip i n the higher power curves at approximately 0.25 radian/sec (corresponding t o the f u e l loop transit time of -25 sec) r e s u l t s from temperature feedback from the external. loop. During a periodic reactivity perturbation a t a frequency of about 0.25 radian/sec, the f u e l i n the core during one cycle wuuld return t o the core one period l a t e r and produce a reactivity feedback effect that would partially cancel t h e extern a l perturbation. With the 235U fuel the amplitude r a t i o of the frequencyresponse curves was relatively low at t h i s frequency, and the dip was not as pronounced.

However, the amplitude r a t i o of the frequency-response

curves f o r the 233U f u e l loading me relatively high out t o greater than 1radian/sec, so the loop feedback i s emphasized. The more negative temperature coefficient of reactivity f o r the 233U f u e l also tends t o emphas i z e the effect of loop feedback. 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. The choice of a 2-sec mixing pot at the core outlet was based on the estimated mixing t h a t occurs i n the reactor vess e l plenums but was somewhat arbitrary, and it was therefore desirable t o explore the effect of different mixing approximations on the frequencyresponse curves. Figure 7 shows the effect on the 8-Mw frequency-response

c, ORNL-DWG 68-fO080R

2000

I 00 40-2

to-’

2 5 IO0 FREQUENCY (radians/sec)

IO!

I 0-2

10-’

2 5 ‘FREQUENCY (radians/sec)

I 00

IO’

Fig. 7. Theoretical Frequency-Response Plot of Gn/no-Gk for the 233U-Fueled MSRE Operating at 8 Mw for Vmious Amounts of Mixing in the Circulating Loop.

curve of varying the lag time of the mixing pot from 0 t o 5 sec. (Note 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.) "he use of pure time delays (i.e., no mixing) t o describe flow i n l i n e s external t o the core gives a plot with sharp curves. As the time i n the mixing pot increases, the curve becomes smoother. A t low frequencies ( 4 . 0 6 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. A t low frequencies the f u e l temperature

bi L

changes slowly enough for the effects of additional mixing i n the l i n e s t o become insignificant, and at high frequencies,the neutron-kinetics eff e c t s dlrminate and temperature feedback from the external loop i s unimportant. The amount of mixing i n a circulating fuel, such as i n the

IGm,

not easily determined analytically. The quantity of interest i s not r e a l l y 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.

Physical mixing of t h e salt, conductive heat

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 r e s u l t i n Itmixing" the salt temperature. The experimental r e s u l t s of the 233U t e s t i n g program may permit a determination of the size of the mixing pot t h a t best

describes the actual physical situation i n the ISRE. 4.3

Sensitivity of Gn/no*6k t o Parameter Changes

The equations used t o model the ISRF,contain many independent parameters. Most-of these are well known, but conditions may be postulated under which t h e i r values m i g h t change. The effects on the magnitude r a t i o of changes in each individual parameter were determined by performing a sensitivity "he r e s u l t s of this analysis f o r selected m i ables are shown i n Fig. 8. The frequency range in which each variable i s most important i s apparent from the curves.

In principle the theoretical frequency response

could be compared with the response determined experimentally and the parameter t h a t m i g h t be changing could be selected, or it could be determined

ORNL

4 .O

0.42 0.8

0.08 0.6

0.04 0.4

0 0.2

-0.W 0

-0.08 -n

1 ;

- DWG €8- 4 0082

t h a t the value of a certain parameter was estimated incorrectly i n the theoretical model. I n practice, it i s more l i k e l y that an actual change i n system performance would affect more than one parameter. For example,

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. 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 r a 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 r a t e that m i g h t have an effect on only one or two parameters. For these changes, it m i g h t be possible t o determine the exact cause of the anomalous behavior. Under any conditions, the sensit i v i t y analysis gives valuable information as t o t h e frequency range i n which each parameter affects the system response and the r e l a t i v e 5mportance of each parameter.

4.4

Frequency Response of Outlet Temperature t o Power

Analyses of experimental data taken at the end of operation with 235U f u e l showed that temperature perturbations were being introduced into the core as a r e s u l t of heat transfer fluctuations at the radiator. These were significant enough in some cases t o appreciably a f f e c t t h e experimental frequency-response determination.

This, together with the mixing 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. From a theoretical standpoint, many temperature-to-power frequency responses may be calculated. However, only those that may be experiment a l l y v e r i f i e d are of.use i n comparing a theoretical model with a physic a 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. This position was

chosen because it corresponds t o the location of a particular thermocouple

The response of a thermocouple located on t h e

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. A schematic diagram of the model t h a t incorporates these changes i s shown i n Fig. 9. Figure 10 gives the frequency response of the salt temperature t o power changes and the thermocouple response t o power changes. A t low frequencies these are the same, but a t high frequencies there i s an increasing difference between the frequency response of salt temperature t o power changes and t h a t 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 . The effect of f u e l mixing on the 6T/6n frequency response may also

be seen i n Fig. 10. In addition t o the 2-sec mixing pot at the reactor outlet, a 5-sec mixing pot was included between the heat exchanger and the core. This caused t h e frequency-response plot t o be smoothed between 0.1 and 0.3 radianslsec but had no noticeable effect a t other frequencies. !The differences i n magnitude of the 6T/6n curves a t low frequencies f o r different power levels i s caused by heat transfer changes a t the radiator. The tube-to-air heat transfer coefficient was assumed t o vary with t h e a i r flow r a t e t o the 0.6 power.

The a i r flow rate was assumed

t o vary linearly with the power level.

5. 5.1 Pole Configuration

5.1.1

Stability Analysis

- Eigenvalues

Theoretical Discussions.

I n 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 1+G"

I n t h i s application, G is t h e transfer function of the reactor kinetics and H is the transfer function f o 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

ORNL-WG 68-40083R

*THERMOCOUPLE RESPONSE

k e c PURE DELAY

RELATIVE THERMOCOUPLE

PosinoN ON OUNT PIPE

2 5 - m PURE DELAY +(/

t I

REACTOR CORE I

t ~~

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

ss C

tK W

cz3 0

40-5

40-4

(0-2 FREQUENCY (rodions/sec)

to-'

Fig. 10. Frequency-Response Plots of Salt Temperature (in the Outlet 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.

ORNL-DWG 68-1f684

REACTOR KINETICS TRANSFER FUNCTION

POWER

8k FEEDBACK1

REACTIVITY TO NUCLEAR POWER TRANSFER FUNCTION

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

. U

23 transfer function, 1 + GH, i s a polynomial (called the characteristic polynomial) i n the Laplace transform variable S. The roots of t h i s polynomial are the poles of the system transfer function.

These roots are equal t o the eigenvalues of the system coefficient matrix, A. A necessary and sufficient condition f o r linear asymptotic s t a b i l i t y i s that the sys-

tem poles (and therefore the eigenvalues of A ) have negative real parts. This c r i t e r i o n gives a definite answer t o the question of linear s t a b i l i t y . If t h e physical system has an N

N coefficient matrix (N linear equations

i n N system s t a t e variables), it w i l l have N eigenvalues. 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. F i r s 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 i o i s determined by the angle B a vector from the origin of the complex plane t o t h e dominant eigenvalue makes with the imaginary axis:

As t a n 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 characteristic of a. zero-power reactor i s a pole of the system transfer function a t the intemection of the r e a l and imaginary axes. Since the system eigenvalues correspond t o t h e system poles, t h i s also implies an eigenvalue at the origin.

The output (for a reactor, the pwer) of a system of t h i s type w i l l increase as long as there i s a positive input (reactivity) but w i l l l e v e l out when there i s no input.

24 The computer code of h a d and Van Nessâ&#x20AC;?

I;f

was used t o determine the

eigenvalues of the system matrix, but before the eigenvalues of Ishe syst e m could be determined, an approximate treatment of the pure time delays

i n the system model was necessary. An eighth-order Paad approximation (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 a c t that the f u e l was circulating, a s e t of effective delayed-neutron fractions (Beff) was used i n the eigenvalue calculation. 5.1.2 ,

Eigenvalue Calculation Results.

Figure l.2 shows several of

the dominant eigenvalues plotted f o r various power levels.

The r e a l part of each calculated eigenvalue i s negative, with the general trend being

toward smaller absolute values with decreasing power level. The negative r e a l p a r t s insure l i n e a r asymptotic s t a b i l i t y , and the trend toward smaller absolute values with decreasing power l e v e l again p i n t s t o a higher degree of s t a b i l i t y at higher po.wer levels. The eigenvalues that l i e relat i v e l y fax removed from the r e a l axis (those with positive imaginary components between 0.20 and 0.30) are a r e s u l 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 a t the radiator with power level. The physic a l implication of these eigenvalues i s that any temperature disturbance i n the system would tend t o cause s l i g h t temperature oscillations around the system, even i f the nuclear power were zero. The eigenvalues f o r the nuclear kinetics equations were a l l on the r e a l axis f o r the zero-power condition. The calculation f o r zero power resulted i n a dominant eigenvalue with a r e a l pazt equal t o -0.36 A secâ&#x20AC;? and an imaginary part equal t o 0.0. Byproper manipulation of one coefficient i n the system equations i n the f i f t h significant d i g i t (the coefficients are generally not known t o b e t t e r than four places) the value of the r e a l part of the dominant eigen-

value could be changed from a s m a l l negative value t o a s m a l l positive value. While from a mathematical standpoint the positive sign would imply

instability, it must be realized that within the accuracy of t h e calculation, both these values are zero.

A calculated value of identically zero

L1 ORNL-DWG 69-467R

0.3

0.2

- POWER FOR EIGENVAWE -

CALCULATION 8Mw

5Mw

4Mw OdMw ZERO SOLID LlNES INDICATE POWER DEPENDENCE OF A PARTICULAR EIGENVALUE A

0.4

-0.i

-0.2

-0.3 0.44

0.12

0.10

0.08 0.06 0.04 REAL AXIS (seci)

0.02

Fig. 12. Plots of Several of t h e More Dominant Eigenvalues of the 233U-F’ueled MSRE f o r Various Power Levels.

(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 s e t of system equations with an

sec” implies that a perturbed eigenvalue with r e a l part equal t o -1 x system, a f t e 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); simisec” would sponlarly, an eigenvalue with r e a l part equal t o +1 X taneously deviate away from i t s original s t a t e with an 11.5-day time constant.

While the difference i n s t a b i l i t y between these two cases i s ob-

vious, neither would represent a difficult problem t o ’ t h e control system designer, and either case would represent acceptable system behavior3 or, conversely, f o r a particular application, if one of these was unacceptable, both would probably be unacceptable. The eigenvalues t h a t form the curve which goes t o the origin a t zero power (see Fig. 12) were used t o calculate the damping ratio, E. Results of t h i s calculation are l i s t e d below:

0.1 1.0 5.0 8.0

0.16 0.37 0.96 0.99

Since there are eigenvalues on the r e a l a x i s t h a t 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 oscillatory behavior; however, it i s useful as a r e l a t i v e indicator. The damping r a t i o again emphasizes the E R E characteristic of being l e s s oscillatory at high power levels than a t low power. 5.2

Modified Mikhailov Method 5.2.1

Theoretical Discussion.

An alternate technique f o r determin-

ing absolute s t a b i l i t y of a l i n e a r system was recently ‘developed by Wright and Kerlin4 along the l i n e s of the method of Mkhailov.

The chief advan-

tage of t h i s method over the eigenvalue technique i s t h a t no approximations

. W

27 are necessary t o t r e a t pure time delays. The technique of Wright and Kerlin i s based on a l i n e a r constant-parameter system of order N, which can be written as a s e t of coupled, first-order d i f f e r e n t i a l equations with constant coefficients:

where F ( t ) = column vector of system s t a t e variables, A = constant N X N coefficient matrix, f ( t ) = forcing function, E = column vector of coefficients of f ( t ) , R(t,.r) = matrix of time delay terms, each of the form r

isxs ( t - Tis)

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,

r. x . ( t 13 J

Tij)

The transform of one of the

where L(S) i s a matrix composed of the r i j i n t o Eq. ( 5 ) m a manipulation yields

terms.

Substitution

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

28 The absolute s t a b 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 f o r which the above equation i s s a t i s f i e d are the system eigenvalues oi characteristic roots. In t h e i r development of the modified Mikhailov s t a b i l i t y criâ&#x201A;Źerion, Wright and Kerlin defined a quantity M(S) as

where D(S) = det (A- S I P(S) = det (T

+ L)

(10)

- SI) ,

and T i s an N X N diagonal matrix, each element of which i s the negative of t h e absolute value of the corresponding elements of the A matrix. The

quantity P(S) can be considered t o be a factor t h a t normalizes D(S) i n both magnitude and phase. The s t a b i l i t y criterion associated with M(S) i s 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. counterclock-

wise encirclements of the origin by the vector M(jw) as u goes from

4 0

While not explicitly stated i n t h i s quotation the important parame-

ter i s the net number of encirclements of the origin, with clockwise encirclements counting negative and counterclockwise encirclements counting positive. Because of symmetry about the r e a l axis it i s necessary only t o plot M(jw) for w running from 0 t o SOO, It can also be shown t h a t M(jw) 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 ). 5.2.2 Results of Applying the Mikhailov S t a b i l i t y Criterion. By using a c-ter program developed by Wright and Kerlin, plots were generated of M ( j w ) f o r w running from 0 t o $40, Figures 13 through 17 show these plots for several power levels. As shown, a t significant power levels there are no encirclements of the origin as w goes from - t o a . As

t.0

0.8 u)

SZ 0.6

2 - 0.4 W

a 0.2

0 -0.2 -0.6

-0.4

-0.2

0.2

0.4 REAL AXIS

0.6

0.8

ORNL-OWG

4.0

t.2

68-10069

REAL AXIS

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

4.2 4.0

0.8

2 Q6 >

a a

gz a4 -P

-0.2 -06

-0.4 *-02

0.2

0.4

4.0

4.2

REAL A X S

(I40-=1

8 ?

u, X

a $ 2

zc) a

-2

-4 (x4~-4)

-2.0

-4.5

-4.0 -0.5 REAL AXIS

0.5

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

ORNL-OWG 68-10073R

-0.6

-0.4

-0.2

0.2 0.4 REAL AXIS

0.6

0.8

1.0

1.2

ORNL-DWG 60-MO12

(xt~-3) 4

s? 4x0 K >

a 5 -4 a a E

-2

-3

(b)

(x~O-~)

-6

-5

-4 -3 REAL AXIS

-2

-1

Fig. 15. Mdified Mikhailov Plot f o r &ERE Operating with 233UFuel at 100 kw. (a) Complete plot. (b) Near origin.

ORNL- DWG 68-10071

(0)

-0.5 -0.50

-0.25

0.25 0.50 REAL AXIS

0.75

4.0

4.25

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

-2

Fig. 17. at 8 Mw.

-4

0 4 REAL AXIS

Modified Mikhailov Plot f o r NRE Operating with 233U Fuel

i n the eigenvalue calculation, t h e results of zero-power calculations

were highly sensitive t o s m a l l changes i n coefficients i n the system ma-

t r i x . While t h e zero-power plots actually show an encirclement of the origin when the t o t a l p l o t from - t o $40 i s considered, t h e intercept at -6.5

should, of course, be interpreted as passing through the ori-

gin.

â&#x20AC;&#x153;his i s t y p i c a l f o r a zero-power reactor. Application of t h e modified Mikhailov s t a b i l i t y c r i t e r i o n t o t h e s e t of system equations that describe the MRE verifies that t h e WRE i s inherently stable at a l l power levels. 6.

Concluding Discussion

The overall r e s u l t of t h i s study i s a determination that t h e MRE fueled with 233U i s an inherently stable system a t a l l power levels of interest. The response of t h e uncontrolled system a t p o w e r s above zero power, as seen i n the t r a n s i e n t analysis and corroborated by the eigenvalues, i s rapid and nonoscillatory at high powers and becomes sluggish and o s c i l l a t o r y at low powers, It i s recommended that t h e 233U-fueled MSRE be subjected t o a t e s t i n g program a t powers from zero t o f u l l power t o experimentally verify the predicted response. These experiments should be performed i n such a way that both the 6n/6k and 6T/6n t r a n s f e r fâ&#x20AC;&#x2122;unctions can be determined, since both can now be compared with t h e o r e t i c a l predict ions. Acknowledgments The authors are grateful t o T. W. Kerlin of the University of .

Tennessee and S. J. B a l l f o r t h e i r assistance and advice. The help of J. L. Lucius i n assisting with programming d i f f i c u l t i e s i s a l s o acknowledged. P

35 References 1. S. J. B a l l and T. W. Kerlin, Stability 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 Computer Program f o r Solving Ordinary Differential Equations by the Mat 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 f o r Calculating t h e Frequency Response of a Multivariable System and Its Sensitivity to. Parameter (?4anges, USAEC Report ORNL-TM-1575, Oak Ridge National Laboratory, June 1966.

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.

C. Wright and T. W. Kerlin, An Efficient, Computer-Oriented Method f o r S t a b i l i t y Analysis of Large Multivariable Systems, Report NEUT 2806-3, Nuclear Engineering Department, University of Tennessee, July 1968.

6. M. A. Schultz, Control of Nuclear Reacâ&#x201A;Źors and Power Plants, p. 116, McGraw-Hill, New York, 1961. 7.

G. Robert Keepin, Physics of Nuclear Kinetics, pp. 328-329, AddisonWesley, 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-

trol 10.

, 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.

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

Appendix PAD; APPROXIMATIOWS

In the analysis of physical systems it frequently becomes necessary

Such a treatment, f o r

t o mathematically approximate pure time delays.

instance, would be necessary t o relate the outlet temperature t o the inl e t temperature of a f l u i d flowing i n plug flow through an 4diabatic pipe if t h e a x i a l heat conduction were negligible.

If the i n l e t temperature

i s x ( t ) and t h e 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 i f a system eontainingtime delays i s t o be analyzed simply, a l i n e a r approximation must be substituted f o r e-Ts. One technique f o r approximating

e-TS i s as an i n f i n i t e series:

-TS =

l - T S +T2S2 7 - - T3S3 + 7 - T4S4 2. 3! 4.

...

An approximation t o this expansion i s

e TS

2 S-E--

T = 1-TS

S ,+- 2

T2S’ T3S3 + . . . . +.z-4

Unfortunately t h i s approximation does not converge rapidly f o r large values of TS. A b e t t e r approximate treatment i s the Pad6 approxi~nation:~ -TS e.

n I

- 2 y r s + TES~

Er=l T1.+ 2 r r ~ r S+ T2 ~2 S’

where n

T = Delay .time =

r=l t

k.0

The order of t h i s approximation i s 2n. t h a t t h e criterion f o r selection of shift,

Stubbs and Single’’ and

(4)

have suggested

be such that the actual phase-

38 3

i s as close as possible t o the ideal phase shift f o r 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 t h e Pad6 approxi-

mation i s l e s s than one degree a t (jY = 13.5 radians. A t higher frequencies this error .increases, and at lower frequencies it decreases. The following l i n e a r equations have been developed f o r the eighth-

order Pad6 approximation, where % i s the variable input t o t h e 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:

dY2

- = -10.5g5gxi - - y1 - 19.5858xo 10

dy3

-4S.8441Xi

- - y2

+ 45.844lxo

(6 1

(7)

T c

dY4 10 = -65.8O83Xi - - y3 - 65.8083~0

39 w5

- - y4 + 6 4 . l l 3 5 ~ 0 1 -

-64.1135~.

dY6

- = - 4 3 . 8 1 6 9 ~-~- y 5 - 43.8169~0

w 8

- = -6.3308xi - - y 7 - 6 . 3 3 0 8 ~ 0

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

(9 1

41 ORNL-TM-2571

Internal Distribution 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. P. N. Haubekeich, 33 A. Houtzeel 34. T. L. Hudson 35. P. R. a s t e n ~ 3 6 . R. 3. Ked1 37. T. W. Kerlin 38. H. T. Kerr 39. A. I. Krakoviak

T. S. JCress 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. M. E. whatley 82. J. C. White 83. G. D. Whitman 84-93. P. J. Wood 94. G a l e Young

40.

41.

95-96. 97-98. 99-101. 102.

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External â&#x20AC;&#x2DC;Distribution \

103. 1@-105. 106-120. 121.

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