ORNL-TM-2571.txt

 

 

 

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ORNL-TM-2571

Contract No. W-7405-eng-26

Reactor Division

THEORETICAT DYNAMIC ANALYSIS OF THE MSRE WITH 233U FUEL

Ro' c. Steffy, JI‘. .Pc Jo WOOd
- LEGAL IQC)TIC:E |

~ This report was prepared 28 an account of Government lponsored work, Nelther the Unltod
States, nor the Commission, nor any person acting ok behalf of the Commission:

A. Makes any warranty or representation, expre!aed or lmplied, with respect to the accu- |
i racy, completeness, or usefulness of the fnformation contained in this report, or that the use ;
! of any Information, apparatus, method, or process diaclosed in this report may not lntrlnga ;
privately owned rights; or !
t . ... B, Assumes any liabilities vlth respect to the n’e of, or for duna.gea resuitlng from the |
L. nse of any information, apparatus, method, or proceds disclosed in this report.
{ As used in the above, ‘‘person acting on beha of the Commission® fncludes any em-
! . ployee or contractor of the Commission, or employ of such contractor, to the extent that
.~ such employee or contractor of the Commission, o employee of such contractor prepares,
§ " diaseminates, or provides access to, any information pursuant to his employment or wntmct
i -with the Commuslon. or hl- employment with such contractor. . :

JULY 1969

 

OAK RIDGE NATIONAL LABORATORY
- Oak Ridge, Tennessee
operated by |
UNION CARBIDE CORPORATION
. for the .
U.S. ATOMIC ENERGY COMMISSION

GRIRIBUTION O gis COCUMENZ & UNTIMTED

 

 

 
 

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CONTENTS

“Abstract.........l......’C.’VO‘.Ol..'...""I.l.l...l.......‘......
-J-l IntrOduCtion QO..'.OO.-.l.O...'..l............f..O..'.O..O.I.'
2. Model Description and Verification ...cc.ecevcecccccceccnnesn

3, Transient-Response'Anglysis .......,.......;...,.............

4. Frequency-Response Analysis_.....;...........,.,............;
4.1 Calculated Frequehcy Response of Power bosReactivify .o
:4.1.1 For ancirculafiinglFuel..........................

4.1.2 For Circulating Fuel «.cecasenccrercccecacencnons

4.2 Effect bf'Mixing*in~the FUEl LOOD seesvocesasescsscnnsns
4.3  Sensitivity of 8n/ng-8k to:Parameter Changes ....eceee..
b4 Frequency'Responsé:of‘Outlet:Temperature-to=Powerr......
5.  Stability Analysis .eececssscsccncscccnccnccrsresoccccconanes
5.1 Pole Configuration — Eigenvalues eeeeeececseccsansanncss
5.1.1 Theoretical DiscusSSiOnS ssccecscsccncenccnnccncne

5.1.2 Eigenvalue Calculation ReSUIES eeeececvececncesos

5.2 Modified MIKNAilov MethOR «eeeeseessenecsescassocnnsoens
5.2.1 Theoretical DiSCUSSION .eceeeevsscsccassacscosnns

5.2.2 -Results-of:Appiying"the Mikhailov Stability
,Criterion:o-_'-.-,....s-.-..-.r-.-..-.....-..........

6. CQHCludingDiscuSSion 8 8 8 8 O 0 0 8 5 T O 0 PSSO ET O PO NEESEPeSHSESEs s

‘ACkflOWledgnentS;...g.ob;..-..a.-.......-..-Q.......'...........-...

References ..............‘lOOQIVO;OO.UOIOCOIO.....-.'...C.Q..'.'......

Appendix. Pa,dé -Approximations.-.....;.C.‘...0...'.'........0'....

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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
233y.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 dif-
ference between the MSRE dynamic response with 232U fuel and
‘with 23 U fuel because of the smaller delayed-neutron frac-
tion of 233y, An existing system model, previously verified
for 2357 fuel, was modified for use in this work. The reac-
tor system response_to reactivity perturbations is rapid and
nonoscillatory at high power, and it becomes sluggish and
oseillatory at lower powers. These characteristics were de-
termined 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 analy-
ses, both by inspection of the system eigenvalues and by ap-
plication of the recently developed modified Mikhailov cri-
terion. .

Keywords: MSRE, 233U fuel, stablllty, elgenvalues,
modified Mikhailov crlterla, frequency response, sens1t1v1ty,
time response, Padé 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 necessary physlcs and chemistry experlments on thls

:flrst 233U-fueled nuclear power reactor. Before nuclear operatlon, it was

1mportant to antlclpate the dynamlc behavior and the. 1nherent stability of

the system in order to insure safe, orderly operatlon and to plan safe,

,deflnltlve experlments.,

The principal motivation. for evaluatlng the operatlng characterlstlcs

of the 233U fueled MSRE after more than 70 000 Mwhr of power operatlon

wuth 235y fuel is that. 233y has & much smaller delayed-neutron fraction

 

*Currently- As31stant Dlrector of MIT School of Chemlcal Englneerlng
Practice at Oak Ridge National Laboratory.

 
 

2

~ (B) than that of 23°u. fhe.?33U'fue1 was expected to have operating char-
acteristics somewhet different from those of 233y and, in particular, a
faster response to feadtivity perturbations{ Table 1 lists the predicted
basic nuclear-kinetics properties of the MSRE with 233y fuel and compares
‘them with those of the reactor with %2°U fuel.

Several different technigues were used in analyzing the dynamiés of
the 23%U-fueled MSRE, primarily because .each technique either gave infor-
‘mation unavailable from the others or used different _approx:i.mate :fireat- _
ments to describe the system. In performing'the-dynamics analyses a model
developed by Ball and Kerlin' was modified slightly and used to -describe
the MSRE with ?32U fuel. (This model did not include the effect of ‘the
réaétor control system.) The tifie'response of the reactorfsystem to a
reactivity perturbatlon at several power levels was calculated first. The
‘computer code.NmEEXP which calculates the time response of a multivari-
able nonlinear system with pure time delays, was used in this study. Next
~the syStem frequency response was determined by using the computér code
'SFR-3 (Ref. 3). This code was also used to .determine amplitude ratio (or
‘gain) sensitivity'to‘changes in the system variables (i.e., the ratio.of

‘the change in amplitude ratio to the correspbnding change-inra'éystem_varif

-able was determined as a function of‘fréquency). Finally, the absolute
stability of the system was investigated by two techniques. The_system
eigenvalues were calculated at several powers to determine whether oscil-
latiqns.in&uced in the system would increase or decrease in amplitude,
and the Mikhailov stability criterion, as modified by Wright ‘and Kerlin,*
wag used to obtain this same information with fewer approxnmatlons in the
mathematlcal model.

Linearized system equations were- used in the frequency-response and
sensitivity calculations and in both types of stability analysis. This
was-ngcessary'because-general calculational methods do not presently exist
.that would permit ‘these types of analysis'with 8 set of nonlinear system
‘equations. The;time-response calculations utilized the nonlinear equa~
tions. This was possible because they involved an iterative procegdure
that provided for -updating the nonlinear terms after each itefation.

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Table 1. Comparison of Nuclear Parameters Used in Dynamics Analyses
of MSRE with 273U Fuel and with 235U Fuel

 

233y Fuel

23 5U Fuela

 

‘Bi,=Delayed~Néutron.

Group Ay Deéay | Fraction

 

Aj, Decay

By, D61ayed—NEutron

 

 

 

 

 

 

Fraction
C?nsta?§ . — C?nstagg —
sec” i Circulating, (sec” s Circulating,
| - Steble  Chprective | Static  rrective
- | X 1074 x 1074 . X 10™% X 10™4
1 . 0.0126 2,28 11,091 0.0124  -2.23 - 0.52
2 - 0.0337 = 7.88 3.848 0.0305 - 14.57 3,73
3 0.139 . 6,64 4.036 0.111 - 13.07 4,99
4 0.325 - 7.36 5,962 0. 301 26. 28 16.98
5 1.13 : 1.36 1.330 1.14 7.66 7.18
6 2.50 0.88 0.876 3.01 2.80 2.77
Total B  26.40 17.14 Total B  66.61 36.17
Prompt neutron generation 4 X 10~% 2.4 x 1074
time, sec : ,
Temperature coefficients
of reactivity, °F | -
Fuel salt - —6,13 X 10°° —4.84 X 1075
Graphite | -3.23 x 1077 -3,70 x 10>

 

&Dats from Ref. 1.

”

 

 
 

 

 

4
2. Model Description and Verification

A mgthematical model was reguired-to:describe the dynamic behavior
of the ???U-fueled MSRE. The model chosen for ‘this study was essentially
that called the "complete model" in Ref. 1, which was developed to analyze
the dynamics of the 235y. fueled MBRE. The'justification for using this
model was its good agreement with experlmental results when applled to
the 23°y-fueled system.’ -

Some changes were made in the model of Ref. 1, howéver, before it
was applied to the *32U-fueled system. The experimental results of the
previous testing program did not verify the dip in the calculated frequency
response cfirve at approximately 0.25 radian/sec, .which. corresponds to a
fuel circulation time of approximately 25 sec. This was attributed to .in-
sufficient'mixing of fuel salt in the .external loop of the .theoretical
model. To provide the model with more mixing, an additiopai 2-sec first-
order time lag (mixing pot) was incorporated at the core outlet. This is
a reasonsble approximation for the mixing in the upper and lower reactor-
vessel plenums. - ;

The chief features of the 44th-order model shown in Fig. 1, are the
fbllow1ng.

B 1. The reactor core was divided into nine regions, each of which
was split into two fuel lumps and one graphite lump. Consideration was
given to the nuclear importance of thermal disturbances in each of the
lumps. (The term "lump" as used in mathematical modeling refers to a seg-
ment'of a physical system that is considered.to have constant\prdperties
throughout and which interacts with its surroundings through only those
properties. ) : |

‘2. A five-lump representatipn of the fuel-to-coolant heat exchanger
was used, with heat being exchanged to a single metal lump at the tempera-
ture of the fuel leaving the first of two fuel lumps and heat being ex-
-changed from the metal lump to the coolant at:fhe‘outlet;temperature of
“the flrst of two coolant lumps.

3, A three-lump coolant-to-air radiator. was used in which the coolant
-transferred heat to a single metal lump at the temperature of the coolant
leaving the first of two  coolant lumps.

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ORNL-DWG 68—10067R
3.77-sec DELAY
1
1 4.74sec DELAY
l
2-sec MIXING POT | '
| : ] AIR
‘ 4 7 9 - ' - , STREAM
FUEL COOLANT COOLANT 3
LUMP LUMP LUMP
3 6 §
MSRE GCORE
(9 REGIONS) ,
8 - --| METAL }--+ -~ = [METAL} --1
FUEL GOOLANT COOLANT
. LUMP LUMP LUMP
i } ] ] ‘ - i
FUEL-SALT HEAT 1 rebiaton
|
| | EXCHANGER 8.24-sec DELAY
8.67-sec DELLAY
Fig. 1. Schemeatic Drawing of MSRE Showing System Divisions Used in
Mathematical Analysis. = .
%

 

 

 
 

4. A linear model of the reactor kinetics equations was used in all
studies, except the time-response calculations,‘in.which.nonlinear'kinetics
effects were taken into account. . |

5. The neutron kinetics equatiohs were represented by a mathematical
~expression that accounted for the dynamic effect of circulating precursors
(except for the eigenvalue calculatlon, which required the use of effec-
tive delayed-neutron fractlons)
6. Xenon poisoning was assumed to be at steady state and not influ-

‘enced by small perturbations.

3. Transient-Respbnse Analysis

Time responses were obtained at several power levels to provide a -
-physical picture of the reactor response to réactivity perturbations such
as control rod motions. The computer code MATEXP? applied in this analy-
sis makes use of the matrix exponentiation technique of solving a.syStem'
of nonlinear ordinary differential equations (with pure time delays) of
the form | |

= Ax + MA(x) x + £(t) , (1)

?f-|§‘|

where

x = the solution vector (system state variables), =
A = system matrix (constant square metrix with real coefficients),

AM(X) = a matrix whose elements are deviations from the values in A,
[thus AA(%) x includes ‘a1l nonlinear effects and time delay
terms],

f(t) = foreing function.

The predicted response of the reactor power to a step reactivity in-

erease of 0.02% 8k/k is shown in Fig., 2 for various initial powers. These

response curves point out several important characteristics of the MSRE.
At 8 Mw the maximum power level is reached during the first second after
-tfie step reactivity input. The rapid increase in reactor power is accom-
-panied by a rapid increase in fuel température in the coré, which, coupled
with the negative temperature coefficient of reactivity, more than

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& POWER (Mw)

A POWER (Mw)

13 ORNL-DWG 68-{0077R

  

1.0
o8
0.6

04

A PONER {(Mw)

02

-0.2
08

0.6
0.4
0.2

 

~0.2 |

A POWER (Mw)

A POWER (Mw)

 

~04
-0.20

0.16
042
0.08
0.04

-0.04

 

-0.08 -
O 100 200 300 400 500 £00 TOO 800 ! 800 - 1000
B ' “TIME (sec) ' '

' Fig. 2. Calculated Power Response of the 233U-Fueled MSRE to a
- 0.02% Bk/k Step Reactivity Insertion at Various Power Levels.

 
 

 

8

counterbalances the step reactivity input, so the power lével_begins'to . kfi’
decrease. The temperature of the salt entering the core is constant -dur- |
‘ing this interval, and when the power has decreased enough for the reac- *
-‘tivity associated with the increased nuclear average temperature to just
cancel the step reactivity input, the power levels for a brief time (from
~6 to ~17 sec after the reactivity input). About 17 sec after the reac-
tivity increase, the hot fluid generated in the ihitial'power\increase
has completed its circuit of the loop external to the core, and the nega-
tive temperature coefficient of the salt again reduces the reactivity so
that the power level starts down again. At large times the reactor power
‘has returned to its initial level, and the step reactivity input has been
-:counterbalanced»by an increase in the nuclear average temperature in the
core. The short plateau observed in the time-response .curve at.8 My was
also noted in the 5-Mw case. - At lower powers, however, the élower system
response-prevents the reactor from reaching the peak of its first oscilla-
tion before the fuel has completed one circuit of the external fuel loop.
The plateau therefore does not appear in the lower pdwer cases.

-An important characteristic of the MSRE dynamic response is that as

-the power is decreased the reactor becomes both sluggish .(slower respond-

)

~ ing) and oscillatory; that is, at low powers the time required for oscil-
lations to die out is much larger than at higher powers, and the fractional
amplitude of the oscillations (A‘power/power) is larger.

As part of the time-response analysis, the validity of the linear ap-
proximation for reactivity perturbations roughly equivalent:to 1/2.in. of
‘control rod movement (0.04% 8k/k) was checked. The results of this analy-
sis are shown in Figs. 3 and 4 for 8- and 0.1-Mw operation. At 8 Mw the
linear approximation is fairly good, but at 0.1 Mw it is;pocr. This re-
sult can be understood by considering the general form of the neutron-

kinetics equations:

 

ddn  pg ~ Br ng 5p & Spdn
— e &n + -+ A0, + =, = ’

dt l z i=1 I

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u B ORNL —DWG 68—10065R
' 2.5

* | ' — 2.0\

'\ ' ——NONLINEAR
.| —=——LINEAR

 

 

]

 

|

 

A POWER (Mw)
»
-
(7

 

 

 

 

 

 

 

 

1.0 _
\
N -
0.5 \\
. BE \'
0
0 BRI [+ ] 20 30 40 50
TIME(sec)

Fig. 3. Power Response of the 233U-Fueled MSRE Ini’cially Opereting
~at 8 Mw to a 0.04% Step Reactivity Insertion as Calculated with anlinear
and Iinearized Kinetics Equations.

n

~ ORNL—DWG 68—{00€6R

]

 

 

0.5 : NONLINEAR /
| | =——LinEAR /
0.4 ' /

/

0.2 _ ' |

 

 

 

 

 

 

A POWER {(Mw)
o
w
\\

 

 

 

 

 

 

 

 

 

1=
|
0 —L |
__ 0 10 20 30 40 50
'_' ' TIME (sec) '
% | SR Fig. 4. Power Response of the 233y-Fueled MSRE Initially Operating

at 0.1 Mw to a Step Reactivity Insertion of 0.04% as Calculated m.th
g Nonlinear and Linearized Kinetics Equations.

 
 

10

where

on = deviation of power from its initial value (ng),

po = reactivity necessary to overcome effect of'fuel circulation,
fiT = total delayed-neutron fraction,

! = neutron generation time in system,

Op = deviation of reactivity from its initial value,

A\; = decay constant for ith delayed-neutron precursor,
Sci = concentration of ith delayed-neutron precursor.

The last term in Eq. (2), 5p8n/I, is the nonlinear term. In the
8-Mw case the maximum deviation of the power from the initial power is
only about 30% of the initial power, whereas in the 0.1-Mw case the maxi-
mum deviation is 560% of the initial power and is still increasing after
50 sec. When the dn term is this iarge-with respect to:the'no'term, the
nonlinear terms in the kinetics equation play a much larger fole.than-the
linear terms. Thus, neglecting the nonlinear terms may lead to signifi-
cant ‘error if the power -deviates from its initial level by more than a
few percent. For the time-response analysis, it was neceséary'to include
the nonlinear terms to obtain realistic resfilts; however, use of the lin-
earized equations in the frequency-domain analysis (Section 4 of this re-
port) is acceptable because the analysis is based on small reactivity and

power perturbations that oscillate around their initial values.

4. Frequency-Response Analysis

Because ‘a closed-loop frequency-response analysis provides informa-
tion about relative system behavior, the linear MSRE model was studied at
different power levels from this point of view. The linear system equa-
tions were first Laplace transformed and then solved for the ratio of an
output variable (such as power or temperature) to an input varisble (such

as reactivity). This ratio, called a transfer function, is

 

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where
G(S) = transfer function,
0(S) = the output variable,
I(S) = input variable,
S = Laplace.transformeariable.

For & stable system, S may be replaced by jw, where J = V-1 and w is the

ifrequency of an input sine wave.  With this substitution, G(jw) is a com-
plex number; the magnitude of G(jw) is called the gain or the magnitude

ratio and is the ratio of ‘the amplitude of an input sinusoid to that of

an output sinusoid. The phase of G{jw) is the phase difference between

the input and the output sinusoids. A plot of G(jw) and the phase of

G(Jw) versus .w is referred to as a Bode diagram or a frequency-response

plot. A Bode plot provides qualitative stability information in the peaks -

of the magnitude ratio curves. - High, narrow peaks indicate lower stability.

“than flatter, broader peaks.

There are two basic reasons for calculating the frequency response

of a system. First, the frequency-response curves are good 1ndlcators-of

system performance, and seeohd,:the frequency response of a system may be

~experimentally determined. The latter consideration is important because

it provides a means for checking the validity of a model. - When the ex-

- perimentally determined frequency response of the system is in agreement

with that of the theoretical model, confidence is gained in the conclusions

‘drawn from the stability analysis applied to the model.

4.1 Calculated Frequency'ReepenSe of Power to Reactivity

4;1.1 .For aneirculatifinguel.' The calculated'frequeney‘response

-of the MSRE for‘the‘noncirculating,~zero-power,.233Uqueled condition is

, shown in- F1g.-5. These . curves are very 31milar t0 those of the c15881c

zero-power reactor, and reference curves may be found in most textbooks

" on reactor dynamics.®57 AtfzerOTpower, temperature feedback effects are
not ‘important, so.the calculated response is that of ‘the neutron-kinetics

‘equations.

4.1.2 For Circulating'Fuel; A.setiof'frequency—response'curvesvfor

 

“circulating 223U fuel is shown in Fig. 6. These curves show the effect

 
 

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ORNL—DWG 68 — {00T9R1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o
©
5 .
e 1000
S
o
5
g
w
Q
S
B
z
g 00
=
10
0.00¢ 0.0 ) 4 ; 10 100
FREQUENCY (radians /sec)
0
={0
-20
o 1 T \\
g =30 ] h,
= n/ \\
W -a0 '
o \
: \
< \
w ""’50
” N i
g N
T / / \\
-70 » \
-w w
// ]
-390
0.004 0.04 04, { 10

Fig. 5. Theoretical Frequency-Response Plots of Bn'/ ng* 8k for the

FREQUENCY { radians / sec)

233y.Fueled MSRE at Zero-Power and with No Fuel Circula.tiqn.

100
 

 

13

ORNL-DWG €8-1Q08'R2
10,000 o S

- 5000

.

2000
1000
500
200
100

50

MAGNITUDE RATIO (31 /2 34)

20

 

0
103 2 5 02 2 5 o' 2 5 00 2 5 1o 2 s 10°
FREQUENCY (radians/sec) L

 

%
o
<
£
103 2 5 102 2 5 107! 2 5 100 2 5 0 2 5 102
- - S -FREOUENCY(radions/sec) : ' -
Fig. 6. Theoret1cal Frequency-Response Plots of Bn/noobk for the
233y-Fueled MSRE at Various Power Levels with Fuel Clrcula.tion.
s

 
 

 

14

of power over the range from zero to 8 Mw. The primery reason for the
change in curve shape as the power level is changed is the_change in mag-
nitude of the temperature-feedback effect. At higher power levels a
smaller percentage change in,power71evgl causes & larger absolute tempera-
~ture change which, in tfirn, affects reactivity through the negative tem-
- perature coefficient. | '
| At higher power levels, the maximum peaks of the magnitude-ratio
plots decrease, and these maximums are reached at higher frequencies.
This implies that after a reactivity perturbation, the relative power re-
- sponse Sn/no will be of smaller magnitude and will tend to return to zero
faster at high power levels than it will at lower power levels. (This
observation was also made in the time-response calculationé of the pre-
ceding section.) '

The dip in the higher power curves at approximately 0.25 radian/sec
(corresponding to the fuel loop transit time of ~25 sec) results from tem-
perature feedback from the external loop. During a periodic reactivity
perturbation at a frequency of about 0.25 radian/éeq;'the-fuel in the
core during one cycle would return to the core one period later and pro-
‘duce a reactivity feedback effect that would partially cancel the exter-
nal perturbation. With the 22°U fuel the amplitude ratio of'the_frequenCy-
response curves was relatively low at this frequency, and the,dip‘fias not
as pronounced. However, the amplitude ratio of the frequency-response
curves for the 233U‘fuel loading are relatively high out to greater than
1 radian/sec, so the loop feedback is emphasized. The more negative tem-
perature coefficient of reactivity for the 233y fuel also tends to -empha-
size the effect of loop feedback.

4,2 Effect of Mixing in the Fuel Loop

i-The curves in Fig. 6 were calculated with the model discussed in
Section 2 of this report. The choice of a 2-sec mixing pot at the core
outlet was based on the estimated mixing that occurs in the reactor ves-
sel plenums but was scmewhat arbitrary, and it was therefore desirable to
explore the effect of different mixing approximations on the frequency-

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

 

 

 

 

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2000

5 1000

i

500

200

MAGNITUDE RATIO ( 8

100 L
1072 2

 

15

* ORNL=DWG . 68-10080R

————————— 5-sec MIXING POT
= = e em = 2-g8C MIXING POT
o o e o e PURE DELAY

1o 2 B 10° 2 5 10!
'FREQUENCY (radians/sec)

 

80

 

60 \

 

40

 

20

7
7

 

PHASE ANGLE (deg)

]
_L/ - Ny
N

AN

 

-20

N

 

 

 

 

 

“\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-60
10 2

5 107" 2 . 5 10° 2 5 10"
'FREQUENCY (radians/sec) -

Fig. 7. Theoretit;al ;F-‘IU-eQuencyQRes;ponse Plot of 5n/ ho° Bk for the
233y.Fueled MSRE Operating at 8 Mw for Various Amounts of Mixing in the

- Circulating Loop.

 
 

 

16

curve of varying the lag time of the mixing pot from O to 5 sec. (Note
that mixing was still assumed to occur in each "lump” of the reactor and
heat exchanger and that in_thié—analysis theveffect of additional mixing
was considered.) The use of pure time delays (i.e., no mixing) to describe
flow in lines external to the core gives a plot with sharp curves. As the
time in the mixing pot increases, the curve becomes smoother. At low fre-
quencies (<0.06 radian/sec)'and at high frequencies (>1.0 radian/sec)‘fihe
effect of mixing is negligible. At 10w-fréquencies the fuel temperature
changes slowly enough for the effects of additional mixing in the lines
to become insignificant, and at high.freéuencieSgthe neutron-kinetics ef-,.
fects déminate and temperature feedback from the external loop is unimpor-
tant. | |

The amount of mixing in a circulating fuel, such as in the MSRE, is
not easily determined analytically. The quantity of'interest'is not réally
the amount of salt "mixed" with other salt but, rather, the extent to which
the temperature of a slug of salt is affected by its surroundings during
its journey around the loop. Physical mixing of the salt, conduétive’heét
transfer with adjoining slugs of salt, convective heat transfer to the
pipe wall, and heat transfer in the heat exchanger all result in "mixing"
the salt temperature. The experimental results of the 33U testing pro-
‘gram may permit a determination of the size of the mixing pot that best
describes the actual physical situation in the MSRE. '

4.3 Sensitivity of 5n/ng* 8k to Parameter Changes

, The equations used to model the MSRE contain many independent parame-r
ters. Most of these are well known, but.conditionS‘may’be postulated un-
:der which their values might change. The effects on the magnifiude ratio
of changes in each individual parameter'were determined by performingna
sensitivity analysis.3 The results of this analysis for selected vari-
ables are shown in Fig. 8. | | .

The frequency range in which each variagble is most important is &
parent from-the curves. In principle the theoretical frequency response
could be compared with the response determined experimentally and the pa-
rameter that might be changing could be selected, or it could be determined
 

[ 1]

a

17

 

10

08 ‘/”/ . \J

06 - \

 

oty

 

 

04

 

0.2 | FUEL TEMPERATURE
' COEFFICIENT

 

N gy f

I T

03 S GRAPHITE TEMPERATURE
\ "| | COEFFICIENT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.2]

 

 

 

 

 

=03 —
03 K

 

 

02 \ GRAPHITE HEAT CAPACITY

 

 

 

 

-O.'I - A 1
{0 P \

08 _ S
' / FUEL HEAT,
06 7 ‘ CAPACITY

 

 

 

 

04 LTl

 

 

0.2

 

o
w
.—
1
=
<
@
a
Z
w
&
Z
<
T
o
2
4
=
o
-
Q
<
o
[T
e.
e
g
o
w
o
S
L
=
a
=
<
z -
w
o
g
=
=
z
o
E
o
-
@«
w
w
o
o
g.

-0.2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FREQUENCY (md:onslsec)

T 002

-=0.02

-0.2

, : _ L b 1.0
000¢ OO .  of \ 407 . 100 -

o042
0.08

0.04

~-0.04
-0.08
0.08

0.06

004

-0.04

0.2

0.4

-0

 

 

ORNL — DWG 68— 10082

T
HEAT EXCHANGER
HEAT TRANSFER
COEFFICIENT,

-SALT
HEAT EXCHANGER
HEAT TRANSFER
COEFFICIENT

MIXING
CHARACTERISTIC
TIME

 

 

0.2

—

 

 

 

NEUTRON

 

 

8
=
m
=
=
&
=
—
g_,_.__
m
Pl

 

 

 

 

 

 

 

Bofl

 

LT |

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\

 

 

 

 

 

 

 

 

 

 

 

 

 

0.001

004

 

04 T 100
FREQUENCY {radians/sec)

Fig. 8. Curves Show1ng the Sen81t1v1ty of the Amplltude Ratio to

Change in Various System Parameters.

 
 

 

18

thatlthe value of a certain parameter was estimated incorrectly in the
theoretical model. In practice, it is more likely that an actual change
in system performance would affect more than one parameter. For example,
& change in fuel-salt flow rate would affect the temperature feedback,
effective delayed-neutron concentration, prompt temperature effects in the
core, and heat transfer in the heat exchanger. The frequency response .
would be expected to change-in-the region where each of these variables
was important. Certainly any appreciable change in fuel-salt flow rate
| could be detected by an:experiméntal.freqfiency-responée determination,
but to pinpoint the exact cause might be difficult. There are changes
other than fuel-salt flow rate that might have an effect on only cne or
two parameters. For these changes, it might be possible to determine the
exact cause of the anomalous behavior. Under any conditibns,.the sensi-
tivity analysis giVes valuable informatiofi as to the frequency range in -
which each parameter affects the system response and the relafive impor-'

tance of each parameter.

4.4 Frequency Response of Outlet Temperature to Power

Analyses of experimental datdvtaken.at'the end of operation with,235U

fuel showed that temperature perturbations were being introduced into the
core as a result of heat transfer fluctuations at the radiator. These
. were significant enough in some cases to appreciably affect the experimen-
 -tal frequency-response determination. This, together with the mixing
problem, led to a calculation of the frequency response of the reactor
outlet temperature with respect to power (8T/®n) to provide a means of
determining the effect of extraneous temperature disturbances.

From a theoretical standpoint, manytempefature-to-power'frequency
responses may be calculated. However, only those that may be experimen- '
tally verified are ofi’use,in,comparihg a theoretical model with a physi-
cal 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 to include a term that rep-
-resented the fuel-salt temperature, as recorded by a thermocouple, 2.5
sec downstream of the outlet of the 2-sec mixing pot. This,positiofi‘was

chosen because it corresponds to the location of a particular thermocouple.
 

 

»

"

L)

19

with a convenient output. The fésponsé of & thermocouple located on the

outside of a fuel-salt pipé to a Change in fuel-salt temperature was found®

to be adeQuately approximated by a l-sec pure time delay plus & 5-sec
first-order lag. A schematic diagram of the model that incorporates these
changes is shown in Fig.. 9. |

‘=Figure'10 gives the frequency reSponse,of'the salt temperature to
power changes and the thermocouple response to power changes. At low fre-

quencies these are the same, but at high frequencies there is an increas-

ing difference between the frequency response of salt temperature to power

changes and that of the indicated temperature to power changes because of
the finite time required to transfer heat through the pipe wall.
The effect of fuel mixing on the 6T/6n frequency response may also

- be seen in Fig., 10. In addition to 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‘the.frequency-response plot to be smoothed between 0.1
and 0.3 radians/sec bufi'had'h9 noticeablé:effect at other frequencies.
Therdifferences in magnitude of theAST/Sn curves at low frequencies
for different power levels is caused by heat transfer changes at the ra-

diator. The:tube—to-airiheaf-transfer ¢oefficient was assumed to vary

with the air flow rate to the 0.6 power; The air flow rate was assumed

~ to vary linearly with the power level.

5. Stability Analysis

5.1 Pole Configuration-é Eigéfivalues

S5.1.1 Theoretical Discussions. In general, if a reactor system can

;bézrepresented;by'the,block~diégram shbwnzin'Fig.vll,.its closedfloop

power—to-reactivity transfer function can be written as

tn_ G | - |
ETT @ - (3)

In~fihis_application,,G»i$ the,transfer function of the reactor kinetics

 an&'H-is_the.transfer-function.for=the system feedback, which includes the -

.effect of promptrtemperature'changes in the core, as well as delayed tem-

perature effects caused'by'the salt circulation. The denominator of this

 
 

 

20

ORNL-DWG 68-10083R
THERMOCOUPLE RESPONSE :

 

RELATIVE THERMOCOUPLE
tsec PURE DELAY POSITION ON OUTLET PIPE

i .

25-sec PURE DELAY

 

 

2-sec

 

 

 

MIXING “SALT FLORN
POT
(4s-ORDER 1 I
LAG)
HEAT COLOO"O?,NT
REACTOR EXCHANGER —_——
CORE

 

 

 

 

 

 

 

 

 

Fig. 9. Schematic Diagram of MSRE Model Including the Model Repre-
sentation of s Thermocouple's Response to a Change in Salt Temperature.
 

L

21

ORNL-DWG 68-10078R
{000

500 —.— SALT TEMPERATURE RESPONSE,2-sec SALT MIXING POT
——— TG RESPONSE, 2-sec SALT MIXING PQT
—=== TG RESPONSE, 2-sec AND 5-sec SALT MIXING POTS

200
100

50

- N
w o Q

MAGNITUDE RATIO (3T/8n)
n

0.5

 

108 074 - 4073 1072 ‘ 10! 10

PHASE (deq)

 

1075 104 _ 1073 - 4072 T 10°
- FREQUENCY(radluns/sec) '

- Fig. 10. Frequency-Response Plots of Sa.lt Tempera‘bure (1n 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.

 
 

 

22

ORNL-DWG 68- 11681

 

 

 

 

 

 

G
REACTOR KINETICS —e=— 3n
TRANSFER FUNCTION . (NUCLEAR)
POWER
H
REACTIVITY TO

 

 

 

NUCLEAR POWER
TRANSFER FUNCTION

 

 

 

Fig. 11. - General Block Diagram of a Nuclear Reactor.
 

 

23

transfer function, 1 + GH,,ié'a polynomial (called the characteristic
polynomial) in:the.Laplace.transform‘variable,s. The roots of this poly-
nomial are the poles of'thé,system-transfér function. These-roots-are
equal to .the eigenvalues of the system coefficient matrix,. A. . A necessary
and sufficient condition for linear asymptotic stablllty is that the sys—
tem poles (and therefore the. elgenvalues of A) have negative real parts.
This criterion gives a definite answer to the question of linear stability.
If the physical system has an N X N coefficient matrix (N'linear equations
in N system state variables), it will have N eigenvalues. The dominant
-eigenvalues of a stable system are the éomplex eigenvalues nearest to the
imaginary axis. -These'eigefiv&lues dominate the time response of\alsystem'
-iflthe'other eigenvalues are relatively'far‘rem0ved fromtthe'imaginary'
axis. Two things can be learned about the behavior of a system from the
location inthe complex'piane of the dominant eigenvalues. First, after

a stable system has been pertufbed,.the amplitude of the oscillstions in
a system variable decreases exponentially with a time constant equal to
the inverse of the distance of the dominant eigenvalue from.the imaginary
axis. Second, the transient overshoot of the system:is determined by the
effective damping ratio, & (Ref..9). This damping ratio is determined by
the.angle B a vector from the origin,of:the complex plane to the dominant

eigenvalue makes with the imaginary axis:

fan B

o
V1 +tan? B

:As tan B increases from:O;torfly.g,increases from O to 1. The.value.of &

- (4)

_determines~how_large'the oversho9t of the system:is during a transient.
Small.values-of-g_indicatellarger'overshoot than large values. .

~ One cfiéracteristic Qf_é;zero-pqwer-reactor-is a pole of the system
:transfer‘fuhction:atfthe:interSéétion of:the:realfandimaginary axes.
:Since,the-system eigenvalues.éorre5pond-to-the system poles, this also
‘implies an eigenvalue at fhe-origin. -The output. (for a reactor, the power)
_ of a system of this type will increase as long as there is a p081t1ve in-

put (reactivity) but will level out when there is no input.

.

 
 

24

10 a5 used to determine the

- Thé.camputer code of Imed and Van Ness
eigenvalues of the system matrix; but before.the,eigehvalues ofithe,sys-
tem could be determined, an approximate treatment of the pure time delays
~in .the system model was necessary. - An eighth-order Padé approximation
(see Appendix) was used for all time delays except the one representing
the . c1rcu1at10n of the- delayed-neutron precursors around the external
loop. To account for: the fact that the fuel was c1rculat1ng, a set of
effectlve.delayed-neutrqn_fractlons (Beff)‘was used 1n‘the,e1genvalue
calculation. . - B | |

~ 5.1.2 Eigenvalue Calculation Results. Figure 12 shows several of
the-doménantfeigenvalues plotted,for'various-power'levels. The ‘real part
of each calculated eigenvalue is'negative,,with,the general trgnd being -

 

toward smaller absolute valueS‘with decreasing power level. The negative
real parts insure linear asymptotic stabillty, and the trend toward smaller
‘absolute values with decreasing power level again points to a higher -de-
gree of stability at higher power levels. The eigenvalues that lie rela-
tively far removed from the real axis (those with positive imaginary com-
ponents between 0.20 and 0.30) are a result of the coupled energy-balance
feéuatibns and are relatively independent of‘nuclear power. - They do, how-
ever, exhibit some power dependence, primarily because of the variation
in heat transfer coefficient at the radistor with power level. The phys1-
cal implication of these eigenvalues is that any temperature disturbance
in the system would tend to cause slight-temperature oscillations around
the system, even if the nuclear power were zero. The eigenvalues for the
-nuclear'kinetics‘equationS‘were all on the real axis for the zero-power
condition.

The calculation for zero power resulted in a dominant eigenvalue-fiith
" a real part equal to —0.36 X 1077 sec”! and an-imaginary'part-equal to
0. O:, By proper manipulation of one coefficient in the system equations in
the fifth significant digit (the coefficients are generally not known to
better than four places) the value of the real part of the dominant eigen-
value could be changed from.a-small negative value to & small positive ;
value. . While from a mathematical standpoint .the positive sign would imply
“instebility, it must be realized that within the accuracy of the calcula-
tion, both these values are zero. A calculated value of identically zero
 

25

ORNL-DWG 69 ~{67R

 

 

 

 

 

 

 

 

 

 

= ' ' - 0.3
® Av=¢ ) ; i
N\,
: 1 M
0.2 |— POWER FOR EIGENVALUE
CALCULATION -
e 8 Mw
8 5Mw
A { Mw
¥ O4Mw
¢ ZERO
SOLID LINES INDICATE
POWER DEPENDENCE OF
0.4 }—— A PARTICULAR EIGENVALUE
- _
8 . |
0 \
> Y
< e T oo
). e
% /, .“-_A
P i
- Z o DEPENDENCE
& -0.4
-0.2
. ‘7e ._ .I./‘

 

 

 

 

 

 

 

 

 

 

-0.3 ,
044 042 OO 008 006 004 002 O
: REAL AXIS (sec“)

Fig. 12. Plots of Several of the More Dominant Elgenvalues of the
233g. Fueled MSRE for Various Power Levels. ,

 
 

26

(the expected value at zero power) would have resfilted only if absolute
precision had been maintained throughout the calculation. |

As a point of general interest, a set of system equations with an
eigenvalue with real part equal to =1 X 10-6 sec~?! implies fhat a perturbed
system, after the inpfit perturbations has vanished, spontaneously returns
to its original state with a time constant of 10° sec (11.5 daYs); simi-
larly, an eigenvalue with real part equal to +1 X 10°% sec”?! would spon-
taneously deviate away from its original state with an-11.5-day'time con-
stant. While the difference in stability between these two cases is ob-
vious, neither would represent a difficult problem to;thercontrol system
designer, and either case would represent acceptable system behavier; or,
conversely, for a particular application, if one of these was unacceptable,
both would probably be unacceptable. _ | |

The eigenvalues that form the curve which goes to the ofigin at zero
power (see Fig. 12) were used to calculate the damping ratio, £. Results
of this calculation are listed below: |

Power ¢, Damping

 

(Mw) Ratio
0.1 0.16
5.0 0.96
8.0 0.99

Since there are eigenvalues on the real axis that have real parts of about
the same magnitude as those used in the calculation, the system damping
ratio cannot alone be used to determine the system oscillatory behavior;
however, it is useful as a relative indicator. The damping ratio again
emphasizes the MSRE characteristic of being less oscillatory at high power

‘levels than at low power.

5.2 Modified Mikhailov Method

5.2.1 Theoretical Discussion. An alternate technique for determin-
-ing absolute stability of a linear system was recently developed by Wright
and Kerlin® along the lines of the method of Mikhailov. The chief advan-

‘tage of this method over the eigenvalue technique is that no approximations
 

_27

are necessary to treat pure time delays. The technique of Wright and
Kerlin is based on a linear constant-parameter system of order N, which
can be written as a set of coupled, first-order differential equations
with constant coefficients: |
ax(t) _ ., =, Sy = - |
= = A x(t) +R(t,7) + B £(E), | (5)

where

x(t) = column vector of syétem state variables,

‘A = constent N X N coefflclent matrix,
£(t) = forcing functlon,
B = column vector of coefficients of £(t),
R(t,r) = matrix of time delay terms, each of the form
r x. t - .
JJ( i:i)

A Taplace transform of Eq. (5) yields
s x(S) = A x(8) + R(S,T) + B £(8) , | (6)
where S is the Laplace transform variable. The transform of one of the

time-lagged elements,

rijxj(t —_ 'rid) s
is
'ri;} , (S)e._sm ,
80 'R(s-,-'r') mey be written B |
R(s,7) = ws) ®s), (7)

where L(S) is & matrix composed of the ri4 e~STY terms. Substitution
’into Eq. (5) and manipulation yields

.x(S) |
—_— —[A - ST +1]°' B, - (8)

where I is the identity matrix, énd super —1 indicates the inverse matrix.

 
 

28

The absolute stability of the system is determined by the location of the

roots of ‘the equation
det (A—SI+1L)=0.

The ‘values of S for which the above equation is satisfied are the system
eigenvalues or characteristic roots.

In their development of the modified Mikhailov stability crit'ericn,
.Wright and Kerlin defined a quantity M(S) as | ‘

D(s) | _
M(S) = —— , (9)
-~ P(8) | | -
where
D(S) = det . (A— SI + L) , (10)
"P(8) = det (T — 8I), (11) .

and T is an N X N disgonal matrix, each element of which is the negative
of the absolute value of the corresponding elements of 'the A matrix. The
quantity P(8) can be considered to be a factor that normalizes D(S) in
both magnitude and phase. The stability criterion associated with M(S)
~is that "the number of zeros of the system determinant det (A — SI + L)

in the right half plane cquals the negative of the mumber of counterclock-
‘wise encirclements of the origin by the vector M(jw) as w goes from —o to
400, "4 yhile hot explicitly stated in this quotationthe important parame-
ter is the net number of encirclements of the origin, with clockwise .en=
circlements counting negative and counterclockwise encirclements counting
positive. Because of symmetry about the real axis it .is necessary only
to plot M(jw) for w running from O to 4. It can also be shown -that M(jw)
approaches 1.0 as.w becomes large, so -é, finite number of w values can com--
pletely describe the motion of M(jw).

5.2.2 Results of Applying the Mikhailov Sta.bility(Criterion. By
using a computer program developed by Wright and Kerlin, plots were gen-
-erated of ‘M(;jw) for w running from O to 4. Figures 13 through 17 show
these plots for several power levels. As shown, at significa.rrb power lev-

els there are no .encirclements of the origin as w goes from —© to « .As
 

 

 

29

ORNL-DWG 68-10076¢

 

[_

|

 

o

INCREASING
FREQUENCY

i |
ZERC POWER

 

o
o

o
o

A

)

™
N\

 

 

LN

(L N/

P )

 

* IMAGINARY AXIS
o
»

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.2 :
: \ \ : 1
0 / »
(a) I
-0.2
-06 =~04 -~0.2 0 0.2 0.4 0686 0.8 1.0 1.2
REAL AXIS ® -
. ORNL—DWG 68 —10069
(x40
o ,
ZERO /
» POWER INTERCEPT
% —0.5 . -6.5344 x 10~
« / w=0
>
[+
<T
z - /
8 4.0
3 /
-1.5 /
(b)- -
~2, 3
(x40°%) -0.8 -06 -0.4 -0.2 0

REAL AXIS -

Fig. 13. Modified Mikhailov Plot for MSRE Opera.tlng with %33U Fuel

at Zero Power. (a) Complete plot.

(o) Near origin.

 
 

30

CRNL-DWG 68-10075

-~
n

POWER =1 kw

o -
2T 88 8 s

IMAGINARY AXIS

R

 

-02

 

-06 ~-04 .-02 0 02 04 06 08 10 1..2
_ REAL AXIS
ORNL—DWG 68—10074
-6
(x40 ) L~
e N

 

 

6
INCREASING
FREQUENCY

 

 

 

 

 

 

 

 

 

 

 

 

 

4
0 /
>
L
> 4
g 2
5 POWER = 1 kw
<1
=
¢ w=0
-2 I
{5
-4
(x107%H -20 -5 -0 -05 0 05
REAL AXIS

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

 

31

ORNL-DWG 68-10073R

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

| T
, : POWER = 100 kw
1.0 =
INCREASING _
FREQUENCY \
. / ~ N
wn
3 0.6 / :‘15'“
: || [ LN
x _
<1
= I : N
T v B
=
0 \ A s _
]
op @ . |
-06 =04 -02 O 02 04 06 08 10 42
REAL AXIS
ORNL-DWG 88-10072
(x10°3)
1
POWER = {00 kw
" . | petmresmm——
% o - B
w=0
x - /
< , A
5 -{ / A
z :
CREASING
z /‘/FfiREQJENCY _
= ]
/,
L@
(xt0°03) -6 -5 -4 -3 -2 =4 0 1

REAL AXiS

| Fig. 15. Modified Mikhailov Plot for MSRE Operating with 2?°U Fuel
et 100 kw. (a) Complete plot. (b) Near origin.

 
 

o

IMAGINARY AXIS
o
v

32

ORNL~-DWG 68~{00714

 

| INCREASING

POWER = {Mw

|

 

 

FREQU%, T —
/ ~ e =N

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

{ )\\
\ W)
0 < / "m
w=0 w=
\__’/
{ag)
-0.5 '
-050 -025 © 025 050 075 1.0 125
REAL AXIS
. ORNL—-DWG 68~ {0070
(x10°2)
1.2 \\‘-.
\\
w
% 08 \\
.« INCREASING
z FREQUENCY |
Z 04 N
&
=1 .
= POWER = | Mw \
0 w=0
(&)
-0.4
-06 -04 =02 0 02 04 06 (x10°)

REAL AXIS

Fig. 16. Modified M:Lkhallov Plot for MSRE Operat:mg with 233U Fuel
at 1 Mw. (a) Complete plot.

(b) Near origin.

Wl

w
 

 

oy

ot

Fig. 17.

33

ORNL-DWG 68-40068R

 

P

 

FREQUENCY

 

1.5 : N
_POWER =8 Mw //NCREASING \

 

 

05

 

 

Y

 

O
o
o

/

 

 

 

 

 

-1.0 | | \\/

 

 

 

-1.5
-2 -{ 0 {
REAL AXIS

Modified Mikhailov Plot for MSRE Operating with *33U Fuel

2

3

 
 

34

in the eigenvalue calculation, the results of zero-power calculations
were highly sensitive to small changes in coefficients in the system ma-
trix. While the zero-power plots actually show an encirclement of the
origin when the total plot from —o to +» is considered, the intercept at
—6.5 x 1078 should, of course, be interpreted as passing through the ori-
gin. This is typical for a zero-power reactor. ' |

Application of the modified Mikhailov stability criterion to the set
of system equations that describe the MSRE verlfles that the MBRE is in-
herently stable at all power levels.

6. Concluding Discussion

The overall result of this study is a determination that the MSRE
fueled with 233U is an inherently stable system at all pcwer levels of
interest. The response of the uncontrolled system at powers. above zero
power, as seen in the transient analysis and corroborated by the eigen-
values, is rapid and nonoscillatory at high powers and becomes sluggish
and oscillatory at low powers. It -is recommended that the 233{3_'—fue1ed
MSRE be subjected to a testing program at powers from zero to full power
to experimentally verify the predictéd response. These experiments should
be performed in such a way that both the 5n/k and 6T/6n transfer functions
can be determined, since both can now be compared with theoretical predic-

tions.

Acknowledgments

The authors are grateful to T. W. Kerlin of the University of
Tennessee and S. J. Ball for their assistance and advice. The help of
J. L. Lucius in assisting with programming difficulties is also acknowl-

edged.
 

 

 

 

wt

e

4.

35
References

S. J. Ball and T. W. Kerlin, Stability Analysis of the Molten-Salt
Reactor Experiment, USAEC Report: ORNL-TM%lO?O Oak Rldge National
Laboratory, December 1965,

S. J. Ball and R. K..Adams, MATEX'P, ‘A General Purpose Digital Com-

‘puter Program for Solving Ordinary Differential Equations by the Ma-
-trix Exponential Method, USAEC Report ORNL-TM-1933, Oak Ridge National

Laboratory,  Aug. 30, 1967.

T. W. Kerlin and J. L. Luclus, The SFR-3 Code — A Fortran- Program for
Calculating the Frequency Response of a Multivariable System and Its

-Sensitivity to Parameter Changes, USAEC Report ORNL-TM;I575 Oak
-Rldge National Laboratory, June 1966.

W. C. Wright and T.. W. Kerlin,. An Efficient, Computer-Oriented Method

for Stability Analysis of Large Multivariable Systems, Report NEUT
2806-3, Nuclear Englneerlng Department, Uhlver31ty of Tennessee, July
1968,

‘T, . W. Kerlin aend S. J." Ball, Experimental Dynamic Ana_lysis of the
‘Molten-Salt Reactor Experiment, USAEC Report ORNL-TM-1647, Oak Ridge

Natlonal Laboratory, October 1966.

M. AL Schultz, Control of Nuclear Reacfors and Power Plants, P. 116
-MCGrawaHlll, New York, 1961. ,

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

S. J. Ball,.Oak Ridge National Laboratory, personal communication to.

R. D..Steffy, Jr., July 24, 1968.

D. R.,Coughanowriande. B. Koppel, Process Systems Analysis and Con-

-trol, pp. 86, 187, McGraw-Hill, New York, 1965.

. F. P. ITmad and J. E. Van :Ness, Eigenvalues by the QR Transform,
. Share-1578, Share Distribution Agency, IBM‘Program Dlstrlbutlon,
-White Plalns, N. Y., October 1963

G. S. Stubbs and C.,H.;Slngle, Transport Delay Simulation Circuits,
USAEC Report WAPD-T—BS and Supplement, Wbstlnghouse Atomlc ‘Power

- Division,.1954.

 
 

 

—_—

 

e

“

 
 

B

ft6~mathematically'approximate=pure.time delays. Such'axtreatment, for

‘ues of TS. A better approximate treatment is the Padé approximation:

37

Appendix

PADE APPROXIMATIONS

'In the analysis of phySical,systemS'it:frequentlyfbecomeS'necessary ’

‘instance, would be necessary to relate the outlet temperature to the in--

let temperature of a fluid flowing in plug flow through an adiabatic pipe

- if ‘the axial heat conductibn*were-negligible. If the inlet temperature
is x(t) and the delay time is ‘T, the outlet temperature will be x(t - T).
- The Laplace transform of a delayed variable x(t-— T) is x(8) e ', and if

a system containing time delays is to be analyzed simply, a linear approxi-

 

mation must be substituted for:e'TS. 'Oné3technique for approximating
'é_Ts.is as an infinite series:
TS T282 73g3 gt |

- An approximation to this expansion is

om
|
Hin

 

 

-8 -1 7457 T3s3 -
e - .2-_1fTs,+—é—— N _(2)
S'F-E -

Unfortunately this approximation does not converge rapidly for large val-

9 .

;TS n 1- 2FrT£S,+-T§82 e |
e. = 1Tfl — ’ . | (3)
r=1 1 + 2r 7S + 128° ' | |
, rr T

 

~ where

T Delay time = Z} AT T o ) (4)
r=1

The order of this approximation is 2n.  Stubbs and Singlel! have suggested

that ‘the criterion for selection of I, and 7, be such that the actual phase -
shift,

 
 

38

2r1m
¢ =2 tan~?t -———————5 .

1 - T W

is as close as possible-to,theAideal phase shift for a pure time delay,

To meet this criterion,.they have suggested (for an- elghth-order Padé
approxlmatlon) that the follow1ng*values be used: '

, .rl = \f57—

T2 = 0.40

T3 = 1.0

'r4 = 0.5
11f12 = 1.68
1314 = 1.14

For ‘this choice of constants the phase error (¢ — do) in the Pad€ approxi-
mation is less than one degree at wl = 13.5 radians. A# higher frEQuen-
cies this error ‘increases, and at lower frequencies it decreases.

The followifig linear equations have been developed for the eighth-
order Padf approximation, where x; is the variable input to the time de-
lay, xo is the delayed variasble output of the time delay, T is the delay
’time,.and'yi'throfigh yg are Padé varisbles: ‘

dnn |
— = —=3,91715x, + 3.91715x%g | (5)
at . |
[ J
— = =19.5858x, — — y1 — 19.5858x¢ : (6)
dt 1
dys iO '
= = 45.8441x, — — y2 + 45.8441x0 (7)
at | T - | | '
ay s 10 ,
— = —65.8083x, — — y3 — 65.8083x0 (8)

dt ' T
 

 

39

dys 10

at ' T
dye ' 10
—_— = =43,8169x, — — y5 = 43.8169x%¢ (10)
dt T
dyy ; 10 - |
— = =20.6831x, — — yg + 20.6831x¢ (11)
dat | T |
dyg 10 A
— = ,_6_3308xi - — Y7 — 6.3308xy (12)
at ST | -
X4 = X, + 5 ys - (13)

For a further discussion of this method see Stubbs and Single. 1

 
 
 

 

 

 

 

 

At
 

1.
2.

4.
5.

8.
‘9.

10.

.u. -

12,
RER

14,

15,

16,
7.
18.

19,

20.

21.
22,

23. .

24,
25,
26.

27, .

28,
29,
30.

31. .
- 32.
33, .

34.

35,

‘.36.

37.

.38.

39. -

L.

’K.

F.
Je

l'F.

El
S.

Anderson

. Adams

Baes
Ball
Bauman
Beall
Bettis

Blumberg

. G.
. do

B

B.
L.
L.

_Jo'

P'

R.
Jd.

W..

T.
I,

R.v
. L.

Bohlmanh -
Borkowski
Briggs
Cobb
Compere
Cottrell
Crowley
Culler
Ditto
Eatherly

- Epler

Ferguson

. Ferris
. Fraas

Franzreb
Fry
Gabbard
Gallaher

.Grimes

Grindell

- Guymon

Harley

‘Haubenreich
- Houtzeel
-L.

Hudson
Kasten
Kedl .
Kerlin
Kerr

Krakoviak

41

ORNL-TM-2571

 

InternalfiDistribution

400

41.

42,

- 43,
b,
45,
46.
47,
48,
49,
.50.
21,
- 52.
53,
54,
- 55.
56.
5.
58.
59-60.
61.
62.
63,
64,
65,
66-75.
76,
7.

78..

579.

-80.

-81.

82,

| 83.
84-93,

-95.96,

k. 97-98.

- 99-101.
102.

-H. C.
A, J.

. A.. M‘

: G. L.
J. L,

’A.'NO
. I. Spiewak

: D‘ - A.

-Je Re..

-G. D.

Kress

R. C. Kryter

M. I. Lundin
R.  N. Lyon

R. E. MacPherson
H. MeClain

H. E. McCoy
MceCurdy
‘Miller
Moore
Nicholson
-Qakes

. Perry
Piper
Prince
Ragan
Redf'ord
M. Richardson

J. C. Robinson

T. S.

R. L.
E. L.
L. Co

H. B‘
Bo ! Eo

‘M. W. Rosenthal
A, W.

. Savolainen
Dunlap Scott

M. J. Skinner
Smith

R. C. Steffy
-Sundberg
Tallackson
Thoma
Trauger
Weir
Whatley
White
.Whitman
Wood

J. R.
R. E.

M. E.
J. C.

P. Jd.

- Gale Young

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