ORNL-TM-2318.txt

RECEIVED BY DTIE APR 1 § 1969

 

OAK RIDGE NATIONAL LABORATORY
operated by

UNION CARBIDE CORPORATION
NUCLEAR DIVISION

for the
U.S. ATOMIC ENERGY COMMISSION

ORNL- TM-2318
copy no. - 1035

DATE - February 6, 1969

  

Instrumentation and Controls Division
DETERMINATION OF THE VOID FRACTION IN THE MSRE
USING SMALL INDUCED PRESSURE PERTURBATIONS

J. C. Robinson® D. N. Fry
ABSTRACT

With the Molten~Salt Reactor Experiment (MSRE) operating at 5 Mw, sawtooth
pressure perfurbations were iniroduced into the fuel-pump bowl to determine the
amount of helium gas enirained in the circulating fuel. The pressure and neutron flux
signals were simultaneously amplified and recorded on magnetic tape. Then the
signals were analyzed using auto-power spectral density, cross-power spectral
density, cross-correlation, and direct Fourier transform techniques to obtain the
neuiron-flux—to—pressure frequency-response function.

An analytical model, developed previously to aid in the interpretation of the
fluctuations of the neutron flux in an unperturbed system, was used to infer from the
experimental data the amount of helium void (interpreted as a void fraction) entrained
in the fuel salt. A description of the analytical model and its experimental verifi-
cation are included in this report.

The void fraction was determined to be between 0.023 and 0.045%. The

uncertainty of this inference is attributed to assumptions made in the model.

 

*Consultant, University of Tennessee, Nuclear Engineering Department.
NOTICE This document contains information of a preliminary nature
and was prepared primarily for internal use at the Oak Ridge National
Laboratory. It is subject to revision or correction and therefore does

not represent a final report.

OIAIRIBULION OF [HiS QCCUMEND & UNUMIED
 

 

 

- —_— LEGAL NOTICE - ———
! This report was prepared as an account of Government sponsored work. Neither the United States,
| nor the Commission, nor any person acting on behalf of the Commission:

A. Maokes any warranty or representation, expressed or implied, with respect to the accuracy,
:‘ completeness, or usefulness of the information contained in this report, or that the use of
! any information, apparotus, method, or process disclosed in this report may not infringe
! privately owned rights; or
{ B. Assumes any liabilities with respect to the use of, or for dumages resulting from the use of

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

! As used in the above, ‘‘person acting on behalf of the Commission'’ includes any employee or
! contractor of the Commission, or employee of such contracter, to the extent that such employee
or contractor of the Commission, or employee of such contractor prepares, disseminutes, or
provides access to, any information pursuant to his employment or contract with the Commission,

or his employment with such contractor,

 
CONTENTS

I. Introduction. . . . . . . .+ . « « . .
2. Theoretical Considerations. . . . . . .

2.1 Introduction . . . . . . . . . .

2.2 Development of the Analytical Model

2.3 Verification of the Analytical Model

3. Experimental Method . . . . . ., . . .

4. Data Acquisition and Reduction. . . . .

4.1 Data Acquisition and its Relationship to the

Significant Quantities . . . .

4.2 Data Reduction. . . . . . . ..

5, Results . .. . . o . . o o .00
5.1 Introduction . . . . . . . . ..
5.2 Results from Analog Analysis . . .

5.3 Results from FFT Analysis. . . . .

*

Physically

5.4 Results from Digital Cross-Correlation Analysis

5.5 The Void Fraction from Experimental and Analytical Results

6. Conclusions . . . . . . . ... ...
/. Recommendations for Future Investigations

8. Appendix . . . . . . o o000

8.1 Interpretation of the APSD of Deterministic Signals in the

Presence of Noise . . . . . . .

This report was prepared as an account of Government sponsored work. Neither the United

States, nor the Commission, nor any person acting on behalf of the Commission:

A. Makes any warranty or representation, expressed or implied, with respect to the accu-
racy, completeness, or usefuiness of the information contained in this report, or that the use
of any information, apparatus, method, or process disclosed in this report may not infringe
privately owned rights; or

B. Assumes any liabilities with respect to the use of, or for damages resulting from the
use of any information, apparatus, method, or process disclosed in this report.

Ag used in the above, “‘person acting on behalf of the Commission® includes any em-
pleyee or contractor of the Commission, or empioyee of such contractor, te the extent that
such employee or contractor of the Commission, or employee of such contractor prepares,
disseminates, or provides access to, any information pursuant io his employment or contract
with the Commission, or his employment with such contracter.

-

-

Page

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43
INTRODUCTION

A technique for determining the amount of helium gas entrained in a nuclear
reactor, circulating fuel-salt system was developed and applied to the Molten-Salt
Reactor Experiment (MSRE) to determine the amount of entrained gas in the system
at any time. This information is useful for calculation of the overall reactivity
balance, since the amount of gas varies and consequently has a varying effect on
the overall reactivity balance. The technique is accomplished by introducing
small pressure perturbations in the fuel-pump bow! and then cross correlating the
resulting neutron~flux fluctuations with the pressure perturbations. Since the
perturbations are small, the technique can be carried out at full reactor power. The
technique is very sensitive; even with a poor signal-to-noise ratio, meaningful data
can still be extracted.

It has been demonstrated' that the amount of helium gas entrained in the MSRE
fuel salt is a function of the system pressure, temperature, and fuel-pump bowl
level. The amount of gas in the system has been estimated from experimental data
from slow power transient 'resfs,2 pressure release ’resfs,3 mass invenfory and zero-
power reactivity balance calculations, * and analysis of the fluctuations (noise

analysis) in neutron density levels.

The noise analysis technique is nonperturbing;
however, it allows determination only of the relative changes in the void fraction.
The other techniques are performed at special reactor conditions; hence, they are

not directly applicable at arbitrary reactor conditions.

 

'D. N. Fry, R, C. Kryter, and J. C. Robinson, Measurement of Helium Void
Fraction in the MSRE Fuel Salt Using Neutron Noise Analysis, ORNL-TM~2315
(Aug. 1968).

J. R. Engel and B. E. Prince, The Reactivity Balance in the MSRE,
ORNL-TM-1796 (March 1967).

SMSRP Semiann. Progr. Rept. Aug. 31, 1966, ORNL-4037, pp. 29-35.

*MSRP Semiann. Progr. Rept. Feb. 29, 1968, ORNL-4254, pp. 3-7.

 

 

 

 

 
The manner in which the amount of void is determined from the technique
developed here is

1.  The neutron-flux—to—pressure frequency-response is obtained from the

analysis of the varying neutron flux and pressure signals.

2, The neutron-flux—to—pressure frequency-response is obtained from an

analytical model, with the amount of void in the system as an unknown.

3.  The moduli of the experimentally and analytically determined frequency

response are forced to be the same through the specification of the amount
of void in the system.

The model is described, and, due to the complexity of the model, experimental
results are presented and compared with predictions from the model for the purpose
of developing confidence in the model.

As stated previously, there is a small amount of void in the system, and we
propose a small pressure perturbation on the system. One would immediately question
the feasibility of the technique. Hence, data were collected and analyzed for the
conditions with and without pressure perturbations. From these tests, we believe
that it is established that the method is feasible. Then the experimental data were
analyzed by several different techniques and compared with predictions from the
model, from which the amount of void was determined.

The authors are grateful to C. B. Stokes for his assistance in performing the
measurements and to J. R. Engel and especially to R. C. Steffy for their assistance

in designing and implementing the experiment.

2. THEORETICAL CONSIDERATIONS

2.1 Introduction

In this section the theoretical model will be described briefly with emphasis
on the basic assumptions. This description will be followed by a comparison of
mode! predictions with experimental data for the purpose of developing confidence

in the model.
2.2 Development of the Analytical Model

Since the technique described herein for determining the helium void consists
of analyzing those fluctuations in the neutron flux signal that are caused by induced
pressure fluctuations in the fuel pumpbowl, a model was required to relate the
neutron flux to pressure. In the development of this model, the compressibility of
the entrained helium gas was postulated as the mechanism having the greatest effect
on that reactivity induced by pressure perturbations. The primary governing equations
are, therefore, the equations of state, conservation of the mass of the gas, of mass
of the fuel salt, of momentum, of energy, of neutrons, and of delayed neutron
precursors. In particular, with the assumption of one~dimensional flow, the governing
S

equations are:

Equation of state for gas,

o = P/RT. (1)
g

Conservation of mass for the gas,

 

a3 T ’ a3 r
0B +____ V :0. 2
2t pga/_] oz i_pg ga_J 2

Conservation of mass for the fuel salt,

-

d

(1=a) +=—

3 r
51 %f

= 0. (3)

-

::_, prf(] - )

Conservation of momentum for the gas-salt mixture,

o _ T8 T o2y 2.,
5T Lpfvf(l a) +nggaJ + 53 Lpfvf (1 - ) +ngg o
3P Wr - W
T9.a WA Ll tegaj

F

 

L. G. Neal and S. M. Zivi, Hydrodynamic Stability of Natural Circulation
Boiling Systems, Vol. 1, STL 372-14 (June 1965).

 
 

—
' The assumed relationship between Vf and Vg is
V =SV, (5)
g f
where S, the slip relationship, is given by’
S= O (6)
K-«
Conservation of energy in salt-gas mixture,
d 1 %Ph |
. + + h { = (7)
St [pfuf(] ) pUdJ Sz [pf ff ) Pv Q’J F YQI
where v is the fraction of the "unit cell" power density generated in the liquid.
Conservation of energy in the graphite moderator,
- aT ¥\
MM T ST Q ko (8)
Coulomb's law of cooling,
q=h(TW-TF) . (?)
Power density,

Q=Ce%0, (10)
where CK is the conversion constant from fission rate to the desired units for power
density.

-
Conservation of neutrons (one-group diffusion model),

6
-1 0% _ . - - 11
Vil =@ Dv¢+[:v(l B) fEe zc}p +Z A
=1
Precursor balance equations,
3 C,
e - .9 ' (12)

fori=1, 2, ... 6.

Since the interest is in small deviations about steady state, it was assumed that
a linearized representation of Eqs. (1) through (12) would adequately describe the
system. Furthermore, it was assumed (a) that the velocity fluctuations would not
significantly affect the precursor balance, and (b) that the fluctuations in the density
of the gas are proportional to fluctuations in the pressure. This latter assumption is

based on the linearized version of Eq. (1), i.e.,

A"gzgo[%g"?%l]' (13)

where °4 is the mean density of gas, and the 4 quantities represent deviations about
the mean. The ratio of the mean temperature T0 to the mean pressure P0 is in the
range of 40; therefore, the last term in Eq. (13) was ignored.

With the assumptions set forth above, the linearized equations generated from
Eqgs. (1) through (6) can be solved independently of those obtained from Egs. (7)
through (12). The former set of equations is referred to as the hydraulic model and

the latter set as the neutronic model.
The dependent variables in Eqs. (1) through (6) are VF’Vg’ oy P and P,
This set can be reduced to a set of three coupled differential equations with three
dependent variables in their linearized version. The dependent variables retained

in this study were AV, , Aa, and AP. Therefore, the equations defining the hydraulic

,F I
model were transformed to the frequency domain and written as

dX(z,s)

Az, s) =

+ B(z,s) X(z,s) =0, (14)

where X(z,s) is the column matrix

aV(z, s)
X(z,s) = | aalz,s) |, (15)
A P,(Z' 5)

and A(z,s) and B(z, s) are 3 x 3 square matrices.

The solution to Eq. (14) is

Z
X(z,5) =exp | [ 2(z',5)dz’ | X(z,,9) , (15)
z, |

where

Qlz,s) =A™ (z,5) B(z,s) , (16)

yd
and the matrix exp[ IQ(Z ! s'dz '} can be evaluated using maitrix exponential
z. '
i
techniques similar to those described in ref. 6. Before the solution can be com-

pleted, the boundary conditions appropriate to the system must be specified.

 

65, J. Ball and R. K. Adams, MATEXP, A General Purpose Digital Computer
Program for Solving Ordinary Differential Equations by the Matrix Exponential
Method, ORNL-TM=-1933 (Aug. 1967).
10

To assign boundary conditions, a physical description (model) of the actual
system must be considered. The mode! chosen to represent the more complex actual
system is presented in Fig. 1. In particular, six regions (identified as Ly through Lg

in Fig. 1) were chosen:

1. the region from the primary pump to the inlet of the downcomer, Ly;

the downcomer, Ly;

the lower plenum, L3;
a large number of identical parallel fuel channels,” Ly;

the upper plenum, Ls;

O\U'I-PLS.JN

the region from the reactor vessel to the primary pump, Lg.
The, perhaps, significant features left out of the physical model are the
heat exchanger and details of the pump bowl. The omission of the heat exchanger
will certainly restrict the lower frequency of applicability of the neutronic model,
but we do not believe that this would affect the hydaulic model. The effects of the
pump bowl on the system were approximated by the boundary conditions between
regions 1 and 6.

The matrix represented by the exponential term of Eq. (15) was generated
for each region. Then, continuity equations were applied between each region,
along with the pressure fluctuations inserted at the pump bowl, o permit the solu-
tion to the closed loop system; i.e., the output of region 6 was the input to region 1.
This permitted the evaluation of the void fraction distribution up through the MSRE
core, which will be required for the solution of the equations describing the neutronic
model.

For the neutronic model, the equations of interest are Eqs. (7) through (12).

The solution to this set of equations could be generated using techniques analogous

 

7The reactor actually consists of hydraulically different parallel channels, but

to date, no attempt has been made to model them.
ORNL—DWG 68-8417

OUTLET

INLET

 

 

 

 

 

 

 

 

[ i e  —— — — — — — — A —  — — — —

CHANNELED 4
REGION

 

 

 

 

/ ?
¢L3 L_/ LOWER PLENUM \J

Fig. 1. Model Used to Approximate the MSRE Fuel Salt Loop.

 

 

 

 
12

to those used in the hydraulic model, but a simpler scheme has been pursured here.
To demonstrate, consider Eq. (12) after it has been linearized and fransformed into
the frequency domain,

Vv _d ACAz,s) + (n +s) AC.(z, s) = B.vZfACD(Z, 5) . . (17)
o dz i i i l

The assumption is made that the flux is separable in space and time, i.e.,
3z, t) = HZ)N() , (18)

and fluctuations occur only in the time-dependent coefficient N(t); therefore,

rg (z,5) is given by

&g (z,s)= H(z) AN(s) . (19)

By use of Eq. (19) and since the precursors leaving the upper plenum return at a
later time (determined by external loop transport time) to the lower plenum, Eq. (17)
was solved for s C. (z,s) as a function of z and aAN(s).

A scheme similar to that used for the precursor equations was applied to the
energy conservation Egs. (7) and (8) to obtain a solution to ATF (z,s) and
AT\, (z,y,s) asa function of axial position z and AN(s).

Now, attention is given to the neutron balance Eq. (11), and a series of
operations is performed: substitute Eq. (18) into Eq. (11); multiply through by
H*(z)N*(t), where H'(2) is a weighting function taken to be the steady-state
adjoint and N (1) is the assumed adjoint time dependence; integrate over the volume;
and require variations of the resultant with N*('r) to be zero (the restricted variational

principle).® This series of operations leads to:

8). C. Robinson, Approximate Solution to the Time Dependent Multigroup

 

Neutron Diffusion Equations Using a Restricted Variational Principle, Ph.D. thesis,
Univ. Tenn. (Dec. 1966).

 
13

<-vH +DvH-H" % H +TH*vsz > N(t)
6
*
) B <HVI H>N()
=]

* — _ * -] dN
+Z < H )\ifDECi( r,t)> =<H V 'H > (20)
i=1

where

6
- _ 5
Fe(l-8)f+) 8.

i=1

(21)

and < > indicates integrals over the reactor volume. The reason for introducing

f will be clarified below.

The static reactivity is defined as

 

VoSV ’
p_= =, (22)
s v
which is the algebraically largest eigenvalue of the equation’
Dy - y +{1=-p)F =0 . 3
[v-Dv Za] b ( ps)FvZf\bs 0 (23)
We furthermore consider the solution to the equation
- (24)

Sk oK Tk R
[v-Dv-ZGJ 1!;5*(1 ps)(FvZF)ws 0;

this equation is defined to be the adjoint to Eq. (23). Then Eq. (23) is multiplied

through by Qrt and integrated over the volume to obtain

 

’B. E. Prince, Period Measurements on the Molten-Salt Reactor Experiment

During Fuel Circulation: Theory and Experiment, CRNL-TM-1626 (Cct. 1966).
14

 

* 2 )
_ <_v1l's . D\'?ws l's Zqc: s Y Fy Zf¢s>
= ca (25)
<$s f\)zfvs ”
In Eq. (20), by choice
H(r) = t v, (26a)
and H (M) = ¢ (7). (26b)

At this point it is possible to calculate the value of . which is required for a

critical system. This is the procedure that is normally pursued in criticality cal-

culations; therefore, the quantity f was introduced in Eq. (20) so that the formulism

could be reduced easily to conventional static formulation. Accordingly, we
introduce the definition

<-vH -DtH - H*ZGH + H*’f"vsz >

p(t) = (27)

 

H FLoo H>
< .\v»;f A

where the reactivity ¢(t) is, in general, a function of time, since the nuclear
parameters will be changing in time due to feedback, rod motion, etc. It follows
from Eq. (27) that p(o) is the static reactivity if the reactor were "just" critical at

t = 0. We now infroduce the definitions
<-9H" DyH>=<-H DB?H >, (28)

h=<HVIH> /<H T ygH >, (29)

f

and write Eq. (20) as
15

6
PN -y (8.5 /FING
i=1

+§ <<H*)\F . C.(r r)>\/<H*_f_v2H>=A-§-N- . (30)
iDi i p f dt

i=1 -

-

Since the spatial mode H(r) was chosen to be the flux distribution at critical

[eigenfunction of Eq. (23)], N(0) is unity. Then p (0) from Eq. (30) is

6 a.f. <HLELC(F,O0)>
p(O) :z i Di _ i Di i . (31)
=

1 f_ <H*-'F\) ZFH>

 

 

At this point, Eq. (30) is linearized by introduction of

I

N =1+N) , (32a)
o(t) =0(0) + o' (1), (32b)

and
C,(r, 1) =C(r,0 +Cl7,1), (32¢)

where the primed quantities will be assumed to be small deviations about the mean.
Equation (32) is introduced into Eq. (30) and the products of small quantities are

ignored to obtain

6 * -
<H A, 'cDiCi(r’ Q) >

ity =Y N +

o - — % —
. <H fv ZH> o <H fvZIH>
i=1 f i=1 f

6 <H*AiFDiCi'(7, f >

 
16

To reduce Eq. (33) to a more useful form, we define

. { -Da? - E, +Ffy I }
D (rft)E.é . r
L <H*fquH>/<H*H>

 

where the & represents deviations about the mean. Then o’(t) becomes

<H*p (F,i') H>
‘ L
p(f): . !
<H H>

 

and Eq. (33) can be written as (dropping primes and transforming)

 

 

 

% —_
6 <H A f..C(r,0)>
s ANG) +) LD N(s)
1 <H fvEfH>
6 <HWf .C(F s) > <H'p(r,s)
) iDici Y T L
i1 <H*FvZFH > < H* H>

Although oL(?,s) could be evaluated directly from Eq. (34), a somewhat simpler

approach is to expand in a Taylor series, as

- eo, - P S opL -
DL(r’S) = aTF L\Tf-(ris) TX_T—’\-A— ATM(I’,S)"“-—B--&—— AO!(T’S)

 

where AT.(r,s) is the local fluctuation in the temperature of the fuel (]

moderator (j =m), Aa (F’,s) is the local fluctuation in the void fraction

(34)
(35)
H>
. (36)
+ ..., (37)

f) or
of gas,

and the "etc." are assumed to be a deterministic input reactivity that can be

(s)

grouped as P ot
17

Equations(36) and (37) make up the neutronic model. The solution of these
equations and Eq. (15), the hydraulic model, leads to the desired transfer function,

e.g., the neutron-flux—to—pressure transfer function.

2.3 Verification of the Analytical Model

As noted in Sect. 2.2, several assumptions were made in the development of
the model; therefore, we believed that confidence could best be established in the
analytical predictions by direct comparisons with experimental data. In this section
comparisons are presented of analytical results with available experimental data.

The experimental frequency-response function obtained at zero power by
Kerlin and Ball'® is presented in Fig. 2. Since their data extends to about 0.2 cps,
other data obtained by noise analysis by Fry et al. Mis included in Fig. 2a. The
data obtained from noise analysis extends from 0. 14 to 15 cps, but no phase infor-
mation is readily available from the noise data, since an autopower spectral density
(APSD) analysis was formed. Along with the experimental data, the results obtained
from the neutronic model are also presented. We conclude from Fig. 2 that the
analytical model describing the system is satisfactory at zero power. The "hump" in
the calculated frequency response function at about 0.04 cps is attributed to the
assumption of plug flow for the fuel salt around the loop; i.e., there must be mixing
of the delayed precursors, which the model ignores. To check the analytical model
further, the calculated effective delayed neutron fraction Beff’ which is used in
conjunction with the in-hour equation for rod calibration, is compared with the
measured Bofre Experimentally, the decrease in reactivity due to circulating fuel

relative to static fuel was 0.212 # 0.004% sk/k.? An assumed static Beff of 0.00666

 

T, W. Kerlin and S. J. Ball, Experimental Dynamic Analysis of the Molten-
Salt Reactor Experiment, ORNL-TM=-1647 (Oct. 1966).

 

 

"D. N. Fry, et al., "Neutron-Fluctuation Measurements at Oak Ridge
National Laboratory,’ pp. 463-74, in Neutron Noise, Waves, and Pulse Propagation,
Proc. 9th AEC Symp. Ser., Gainesville, Fla., February 1966, CONF-660206
(May 1967).

 

 
18

would lead to a circulating Bt of 0.00454. The circulating B ¢f calculated from
the mode! was 0.00443. Prince’ had calculated a circulating BeFF of 0.00444,

The experimental data'® for the neutron-flux—to-reactivity frequency-response
function for the reactor operating at 5 Mw are presented in Fig. 3 along with the
calculated power—to—reactivity frequency-response function for the same conditions.

In Fig. 3a there is a difference between the calculated and the observed
modulus of the power~to~reactivity frequency-response function below 0.008 cps
because the mode! ignored the heat exchanger; i.e., fluctuations in the fuel-salt
outlet temperature were transported around the loop and back to the inlet of the core
where they affected reactivity directly. As in Fig. 2, the discrepancy in the
0.04 cps region is atiributed to the plug flow model.

The difference between the experimental and calculated phase information in
Fig. 3b at the lower frequencies is atiributed to the heat exchanger assumption. We
do not understand the difference at the higher frequencies.

We conclude from Fig. 3 that the analytical prediction of the power-to-
reactivity frequency-response function is acceptable for frequencies above 0.008 cps
at a power level of 5 Mw.

The calculated modulus of the reactivity-to-pressure frequency-response func-
tion and the available experimental data? are presented in Fig. 4. The calculated
magnitude of the modulus of the frequency-response function is proportional to the
void fraction for each model.

It was stated in Sect. 2.2 that the pump bowl was not explicitly accounted for
in the model, but an attempt was made to account for its effect on the system by the
use of boundary conditions between regions 1 and 6 (see Fig. 1). The difference in
the calculated modulus of the frequency-response function between curves labeled
Model A and Model B in Fig. 4 is attributed to the assumed boundary conditions

between regions 1 and 6. There must be two boundary conditions. The first boundary

 

"ZMSRP Semiann. Progr. Rept. Aug. 31, 1965, ORNL-3872, pp. 22-24.

 
ORNL DWG. 69-3T7h41

10 T 11 1 T TtTt[ T T TT1[ T T TT]

 

— CALCULATED
x EXPERIMENT (REF. 9)

® EXPERIMENT (REF. 10)

    

 

 

 

0
0.001 00l 0l 1.0 100

FREQUENCY (cps)

CRNL DWG. €9-37hZ
0 T T T T T T7T] T T T1
- — CALCULATED —

 

x EXPERIMENTAL

PHASE (deq)

 

 

 

 

FREQUENCY (cps)

Fig. 2. Modulus and Phase of the Neutron-Flux-to-Reactivity Frequency-
Response Function for the MSRE at Zero Power. (a) Modulus and (b) Phase.
20

ORNL DWG. 69-37L3

 

 

 

 

 

 

 

 

 

 

3
ax10 T T T T T T T T
- — CALCULATED B
x EXPERIMENTAL
10> L— .
o x _
o | x ]
x
02 l L 11| | L1 1 L L
0.00I 0.0 ol 0
FREQUENCY (cps)
60 ORNL DWG. 69-3Thk
T T T T T T
B " — CALCULATED —
40— X x EXPERIMENTAL ]
e X B
20 _
3 L , )
2
w O ' _
< B —
I X
a
-20 — B
—40.._.__ .
-60
0.00!1 0.0I ol

FREQUENCY (cps)

Fig. 3. Modulus and Phase of the Neutron-Flux-to-Reactivity Frequency-
Response Function for the MSRE at 5 Mw. (a) Modulus and (b) Phase.
21

condition was that the pressure fluctuations at the exit of region 6 are the same as

the inlet pressure fluctuations to region 1. This condition seems physically reason-
able; therefore, it was used for both Model A and Model B. For the second boundary
condition, we assumed for Model A that the void-fraction fluctuation at the exit of
region 6 was equal to the void-fraction fluctuation at the inlet of region 1. For
Model B, the second boundary condition was that the fluctuation in the total mass

velocity to region 1 was zero.

2 CERL DWG. &9-374%

 

 

 

 

S T TT T 11 T T T T 1T ]
N X X T
Xt
[ x/x/ \x i // ]
/ M x
B . ~ /\\,,/ N N
§ P /E..o—o-”"’“'r'fi'_’—;' ;&W"w\:o—o-—*';:::xf
o o e x a «* X
20— o g2 :
2 8 x ° B -X
= e —
s
o | A~ ]
L -
“a_ A ® CALCULATED MODEL A
X x CALCULATED MODEL B
P D EXPERIMENTAL
X
o Lot vk
0.00! 0.0l 0.l 10 10.0

FREQUENCY (cps)

Fig. 4. Modulus of the Reactivity-to-Pressure Frequency-Response Function

for the MSRE.

The experimental data in Fig. 4 was obtained by suddenly releasing the pres-
sure of helium cover gas in the primary pump from 9 to 5 psi and analyzing the result-
ing control rod motion required for constant power. The amount of void present at
the time of the experiment was estimated to be from 2 to 3% by volume. ? From a
comparison of Model A predictions with the experimental data, we concluded that
there was a 2.5% void fraction, whereas from a similar comparison of Mode!l B
predictions, we concluded 1.6% void fraction.

It appears that the analytical predictions are nominally correct, but we do

not have enough experimental evidence to definitely select either model (from the
22

shape of the predicted frequency-response function it appears that Model A is more
nearly correct). Hence, the analytical results from each model were used in the

reduction of the experimental data.
3. EXPERIMENTAL METHOD

As discussed previously, it had been anticipated that the signal-to-noise ratio
for this test would be poor; therefore, a test signal was desired which had its maxi-
mum power concentrated in a small increment of frequencies (in a narrow band)
about the frequency being analyzed. For this purpose the ideal test signal would
have been a pressure "sine wave." The problem was that we could not, without
excessive difficulty, generate a pressure sine wave because of the limitations of the
system; i.e., the required manipulation of the valves would have been very difficult.

Due to practical considerations, a train of sawtooth pulses, with a period of
40 sec for each pulse, was chosen as the test signal. The scheme employed for the
generation of this signal is explained as follows (see Fig. 5). After valves HV-5228,

HCV-544, and HVC-545 were closed and FCV=-516 was fully opened, the pressure in
the pump bowl increased about 0.3 psi over a period of about 40 sec. At this point,
equalizing valve HCV~544 was opened momentarily to bleed off helium, with a pres-
sure decrease of approximately 0.3 psi. The mean pressure perturbation was held to
approximately zero throughout the duration of the test. The time required to release
the pressure was insignificant relative to the time required for the pressure to rise,

Equalizing valve HCV-544 was controlled by use of the circuit® shown in

Fig. 6, which is a one-shot multivibrator that caused the valve to open when the
pressure exceeded a preset value and controlled the amount of time the valve

remained open. The period of the sawtooth waves was reproducible to within +2%

throughout the test.

 

BThis circuit was suggested by S. J. Ball, ORNL Instrumentation and Controls

Division.
23

ORNL DWG. 69-3Th6E

PRESSURE
SENSOR FCV-516
Hv-5228
He SUPPLY

     
    
 
 
    

 

OFF_GAS

 

FUEL PUMP
BOWL

 

 

HCV-544 HCV-545

 

VENT
HCV-57 HCV-575

 

Fig. 5. The Portion of the MSRE System Used in the Generation of the
Pressure Sawtooth Test Signal.

4. DATA ACQUISITION AND REDUCTION

4,1 Data Acquisition and lts Relationship to the Physically Significant Quantities

The continuous signals obtained from a neutron-sensitive ionization chamber
and a pressure transmitter (located ~15 ft from the pump bowl in a helium-supply
line) were amplified and recorded on magnetic tape (Fig. 7) for a period of ~1 hr.
The schemes used for the reduction of the data will be discussed below, but first
it will be instructive to relate the electrical signals V| and V;, which were recorded
on magnetic tape, to the actual fluctuations in the system flux and pressure.

The instantaneous current I](f) from the neutron-sensitive ionization chamber

can be written as

I](f) = IDC'] + IAC’](f) , (38)
24

where IAC’](t) represents deviation about the mean current, | DC,1° The subaudio
amplifier rejects the mean, or DC, voltage; therefore the output of the amplifier

V] with gain G] is
v](r) = R]G]IAC’](’r) , (39)

where R] is the input resistor. The fluctuating current | (t) from the neutron-

AC,1

sensitive ionization chamber is related to neutron flux fluctuations by'*

Iac1® =Toe aNG/Ng (40)

where §N(t) is the instantaneous deviation of the flux about the mean, and NO is

the mean flux level. Now Eq. (39) can be written as
7 - N
ViR =(RGyTpe ¢ sN(/N, (41)

where all the terms in the bracket can be determined experimentally.
The output of the pressure sensor is a voltage that is proportional to the

pressure, i.e.,

— o _ ]
VP(t) a P(t) C{PO + &5 P(t) 1 (42)
where «is a proportionality constant and §P(t) represents the deviations of the

pressure about the mean pressure PO. As before, the output voltage Vz(f) is

related to the input current Iz(’r) by

V2(’r) = R,G,) Ac'z(t) . (43)

 

4D, P. Roux, D. N, Fry, and J. C. Robinson, Application of Gamma-Ray
Detection for Reactor Diagnosis, ORNL-TM-2144 (March 1968).

 

 
ORKL DWG. HAO-3747

P
---SeLPL___OPEN RELAY WHEN P>P, ot
Pbowl /\/\/ LIMIT

 

 

 

 

 

 

 

 

+ pr—
te =2 JL
NG Pbowh 1
l/
SET POINT C
FOR MAX P g, | YO
~10
17z,

 

 

0 [FON TIME
—O—}k-ams

 

 

COMPARATORf---- ‘T‘

 

 

 

 

ENERGIZE DRIVING
10 RELAY WHICH IN
Ve TURN ENERGIZES
2 RY-1 OR RY-2_TO
OPEN EQUALIZING
VALVE

Fig. 6. Circuit Used for the Generation and Control of the Desired Test
Signal.
26

But
Vo () )
=P (44
ac2 TRTR,
therefore, V2(’r) can be written
R3 <
O L. 4
Vo(t) ‘“\R3+ % G, ) 5P(t) . (45)

Equations (41) and (45) relate the physical quantifies of interest 6N(f)/NO
and 8P(t) to the observed quantities \/] (t) and Vz(’r).

ORNL DWG. 69-3748

 

 

 

 

 

i G
1 4 Vv
[—— 1 = susaupio !
LOW-NOISE
NEUTRON SENSITIVE IONIZATION R [AMPLIFIER

 

CHAMBER INSTALLED IN INSTRUMENT
PENETRATION NO.7 =

 

 

FM TAPE
RECORDER

 

 

 

 

 

 

 

 

 

 

I G
:}P I J\?\, suaw%no va
! LOW-NOISE
PRESSURE TRANSMITTER INSTALLED r. [AMPLIFIER
3

 

IN A HELIUM SUPPLY LINE TO THE
PRIMARY PUMP BOWL.

Fig. 7. Representation of the System Used for the Accumulation and Storage
of the Experimental Data.
27

4.2 Data Reduction

The signals V](’r) and V2(1') were analyzed by use of five different techniques.
We will not describe the details of these various schemes; however, a brief dis~

cussion of each technique follows.

4.2.1 Analog Analysis

Data were recorded on analog tape (Fig. 8) at a tape speed of 3.75 in./sec.
For analysis of the data using the 10-channel analog power spectral density analyzer,”
the tape was played back at a speed of 30.0 in./sec (a tape speedup factor of 8).
This increased speed was necessary so that the fundamental frequency of the pressure
sawtooth wave, 0.025 cps, would appear to be at a frequency of 2.0 cps, which
corresponded to a center frequency of one of the available pretuned filters. The
effective frequency range covered by the 10-channel analyzer was from 0.017 to
6.3 cps with a bandwidth of 0.0125 cps. Only one of the 10 channels was useful
for data reduction, because the center frequency of 9 of the 10 filters did not
correspond with a harmonic of the test signal. (In the Appendix, Sect. 8.1, it is
shown that the center frequency of the filter must closely coincide with a harmonic
of the test signal for a meaningful interpretation of a periodic test signal.) The
primary reason for the use of the analog analyzer was that we wanted to obtain an
absolute value of the pressure power spectral density (PPSD) which could be com~
pared with theoretical predictions as well as with the absolute neutron power spectral

density (NPSD).

4.2.2 BR-340 FFT Analysis

 

A Bunker-Ramo, mode! 340 digital computer at the MSRE and a program
which had been developed previously for NPSD calculations of noise signals obtained
from a neutron-sensitive ionization chamber were used for FFT analysis. The basic

procedures involved in this calculation were to (@) Fourier transform the time signal

 

15C. W. Ricker, S. H. Hanaver, and E. R. Mann, "Measurement of Reactor

 

Fluctuation Spectra and Subcritical Reactivity, " Nucl. Sci. and Engr. 33 (1), 56
(July 1968).
28

SRWL DWG. 69-3749

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

S (1) F, 1, (0
—INARROW-BANDI
FILTER
FM TAPE | ] LT TIME o(t)
RECORDER| _ ULTIPLIER AVERAGE :
Fa
—{NARROW-BAND—
S2(1) FILTER  {l2(1)

 

 

 

Fig. 8. Representation of the System Used for Analog Analysis of the Data.

and (b) construct the NPSD from the transformed signal.'® This is a very fast tech-
nique, made so because the Fourier transform is obtained by an algorithm proposed
by Tukey, 7" which has become identified as the FFT (fast Fourier transform) technique.

Samples taken from the tape-recorded analog signals from the experiment
had to be digitized before digital analysis was possible. The convenient sampling
rate was 60 samples/sec, but, since this was much higher than necessary, the tape
was played back at 60 in./sec which gave an effective sampling rate of 3.75
samples/sec. Then the power spectral densities of the pressure (PPSD) and neutron
signals (NPSD) were obtained for the frequency range of 0.00366 to 0.937 cps with a
bandwidth of 0.00366 cps.

 

'R. C. Kryter, "Application of the Fast Fourier Transform Algorithm to
On-Line Reactor Malfunction Detection, " paper presented at the [EEE 15th Nuclear
Science Symposium, Montreal, Canada (Oct. 1968).

7). W. Coolen and J. W. Tukey, "An Algorithm for the Machine Cal-
culation of Complex Fourier Series, " Math. Comput. 2, 297-301 (April 1965).
29

4.2.3 Digital CPSD Analysis

 

The tape was played back at a speedup factor of 4 and the analog signals
were digitized using an analog-to-digital converter which gave an effective
sampling rate of 5 samples/sec. These digitized data were analyzed using a digital
computer simulation of an analog filtering technique to obtain the cross-power
spectral density (CPSD) function.'® The calculated results of interest were (a) the
ratios of the CPSD of flux to pressure to the PSD of the pressure at various fre-
quencies, and (b) the coherence function. The frequencies selected for analysis

were the harmonics determined by the period of the input pressure wave.

4,2.4 FOURCO Analysis

 

The ratio of the CPSD of the flux to pressure to the PSD of the pressure is,
in theory, the frequency response function of the flux to pressure. The classical
definition of the frequency=-response function is that it is the ratio of the Fourier
transform of the output to the Fourier transform of the input. Therefore, the code

FOURCO, ¥ which calculates the ratio of the Fourier transforms, was used to reduce

the data.

4,2.5 CABS Analysis
Another way to obtain the system frequency response function is to
(a) calculate the cross-correlation function of the output to the input and the auto-

correlation function of the input, and (b) calculate the Fourier transform of the

 

185, J. Ball, Instrumentation and Controls Div. Ann. Progr. Rept. Sept. 1,
1965, CRNL-3875, pp. 126-7.

 

s. J. Ball, A Digital Filtering Technique for Efficient Fourier Transform
Calculations, ORNL-TM-1778 (July 1967).

 
30

auto-correlation function. The ratio of the transformed functions (cross=to auto-

correlated) is the frequency response function. This analysis scheme was carried

out using CABS.??
5. RESULTS

5.1 Introduction

The flux and pressure signals were recorded on magnetic tape simultan-
eously for a period of approximately 1 hr for the conditions of (a) no perturbations
to the system (for noise background calibration purposes) and (b} pressure perturba-
tions infroduced as a train of continuous sawtooth pulses with a 40-sec period and
a magnitude of 0.3 psi for each pulse. The results obtained from these tests are

discussed in the following sections for each analysis scheme.

5.2 Results From Analog Analysis

The auto-power spectral density of the neutron flux [NPSD(f)] is related
to the auto-power spectral density of the pressure [PPSD(f)] by?!

NPSD(f) = | G(f) |2 PPSD(f) , (46)

where | G(f)|? is the square modulus of the frequency-response function of the neutron
flux to the pump bowl pressure. The implicit assumption for the validity of Eq. (46)
is that the NPSD is due to pressure perturbations only, i.e., that the observed
NPSD(f) has been corrected for background noise. Furthermore, the NPSD(f)

 

207, W. Kerlin and J. L. Lucius, CABS-A Fortran Computer Program for

 

Calculating Correlation Functions, Power Spectra, and the Frequency from

Experimental Data, ORNL-TM-1663 (Sept. 1966).

 

 

21J, C. Robinson, Analysis of Neutron Fluctuation Spectra in the Oak
Ridge Research Reactor and the High Flux lsotope Reactor, ORNL-4149 (Oct. 1967).

 

 
31

and PPSD(f) must be absolute quantities (or at least proportional to the absolute

quantities with the same proportionality constant). Then Eq. (46) can be written as
1/
|G(N| = [NPSD(H) / PPSD(R)] . (47)

In Sect. 4.2 we stated that only 1 of the available 10 filters had a center
frequency that corresponded to a harmonic frequency of the input signal and that
particular filter was centered at an effective frequency of 0.025 cps, which is the
fundamental frequency of the 40~sec-period pressure wave. Therefore, we were able
to evaluate | G(f)| from Eq. (47) only af a frequency of 0.025 cps.

From Eq. (102) of Sect. 8 we note that, if the noise is insignificant in
the pressure signal (which was the case for this experiment in the vicinity of
0.025 cps), the observed PPSD(fo), which is O(T, FO) of Eq. (102), should be given
by

_1 2
PPSD (fo) =5 (ao + bo ) . (48)

After expansion of the sawtooth wave in a Fourier series and evaluation of the right
side of Eq. (48), the calculated PPSD(.FO) was 0.0045. The value of PPSD(fO)
obtained from the calibrated analog spectral density analyzer was 0.0050. A 109
deviation in the PPSD is well within experimental uncertainties.

Although the noise was insignificant for the pressure signal, this was not
the case for the neutron signal, as can be seen in Fig. 9 by comparing curve A
(NPSD obtained from the neutron flux signal recorded during the pressure test)
with curve 8 (NPSD obtained from the neutron flux signal recorded immediately
following the pressure test). The ordinate in Fig. 9 is the output of the analyzer,
corrected for system gains and the square of the mean neutron ionization chamber
current and normalized to the filter area AF’ i.e., the ordinate is O(T,fo)/AF of
Eq. (95). Therefore, the desired NPSD(FO) is obfained from
O(T, FO) ; _of(T,f)

Ar Ar -p

A (49)

 

NPSD(f ) = | [
32

-5 ORNL DWG. 09-3750

 

T T T T T T T IO
—~ _ —
. o—WI|TH PRESSURE FLUCTUATION
— / A-—-NO PRESSURE FLUCTUATION —
o
A
6 s |
10 A —
\
- \\ —
_ ) _
B
\
- \\
— —

NPSD/ 1DC2

 

 

 

 

0.0l O 1.0 0.0
FREQUENCY (cps)

Fig. 9. The Neutron-Flux Auto-Spectral Density (NPSD) from the Analog
Analysis.
33

at fo = 0.025 cps, where subscripts A and B refer to the similarly identified curves
in Fig. 9. By use of Eq. (49), the value of NPSD (0.025) is calculated to be
1.652 x 1078, From Eq. (47), the modulus of the frequency-response function of

the fractional change in neutron flux to the change in pressure (units of psi) is

-9 _1/2
1G(0.025) | =F§—_;>3§-;‘—:%_3-J = 0.00128 psi-! .

This value for the modulus is compared in Sect. 5.5 with results obtained from other
techniques. Furthermore, the procedure used to infer the void fraction from |G |

is also presented in Sect. 5.5.

5.3 Results from FFT Aralysis

We concluded from the analog spectral density analysis (see Fig. 9) that
the neutron flux signal did contain information at a frequency of 0.025 cps which
was related to the pressure driving function. However, we could not determine
if information was present in the neutron flux signals at other harmonics of the
fundamental of the pressure signal. This was because no filters were available with
the proper center frequency. In principle, there is an infinite number of harmonies
present in the input sawtooth, but it is known that the power associated with the nth

harmonic Pn is related to the power associated with the fundamental PO by
Pn 1

PO n? ‘

wheren=1, 2, 3, .... Therefore, we expected that there would be, at most, a
few harmonics from which we could extract useful information. To determine the
number of harmonics that we could analyze with confidence, the NPSD(f) and
PPSD(f) were obtained using the BR-340. From these results (Fig. 10) we concluded

that we could analyze the fundamental and its first three harmonics. No attempt
34

 

ORNL-DWG 6B-12532
s

0 i T T T T T T ]
i i

L
o* i -
»T et
| _lnjL! !—[{/NSPD ~
v ]
IOS ] J‘ a L 1 —

 

 

 

AUTO POWER SPECTRAL DENSITY (arbitrary units)
5 3,
T AT T
‘_L_'L___ -
é _ —
E{—mm'l_'_f__j
\i_
= ’:.::::1{{_4
ii? o
5

T
1

 

 

0

o L L | !

| |
Q 00366 00732 01098 0.464 01830 0.2196 02562
FREQUENCY (cps)

 

Fig. 10. The Neutron-Flux Auto-Spectral Density (NPSD) and the Pressure
Auto-Spectral Density (PPSD) from the BR-340 FFT Analysis.

was made to obtain the modulus | G(f) | of Eq. (47), from Fig. 10, since these results
were not presented in absolute units; i.e., this speciral density analyzer has not

been calibrated.

5.4 Results from Digital Cross-Correlation Analysis

Three different digital techniques were applied to obtain the modulus
of the neutron-flux—to—pressure-frequency response function directly: (1) the
cross-power spectrum, (2) the Fourier transform of the input and output signals,
and (3) the Fourier transform of the cross- and auto-correlation functions. Each
of these techniques will be discussed briefly and the resulis tabulated. In each case,
the voltage signals V](f) and V2('r) (Fig. 7) were related to the fractional change in
neutron flux 5N(’r)/NO and the change in pressure & P(t) as dictated by Eqs. (41)
and (45).
35

5.4.1 CPSD Analysis

With this technique the objective was to calculate the CPSD of the
output to the input signals, the PPSD of the input signal, and the NPSD of the out~
put signal. Then G(f) was obtained from?

_ CPSD(f)
G(f) = FPSD( (51)

and the coherence function v2(f) was obtained from

20 | CPSD()|?
v ) = PSDmPPSDT (52)

The coherence function, which has numerical values between zero and unity, is

25 e., its value

used as a quantitive indication of the signal-to-noise ratio;
approaches unity for a high signal-to-noise ratio.

Since the scheme used for this analysis was a digital simulation of analog
techniques (see Sect, 4.2), we will refer to this scheme as the D-analog CPSD
analysis. The results are tabulated in Table 1.

We had concluded in Sect. 5.3 that we could accept results up through
the third harmonic, but since examination of the coherence in Table 1 indicates

that the fourth harmonic is equally acceptable, we included this harmonic in our

analysis of the void fraction (see Sect. 5.5).

5.4.2 Fourier Transform Analysis

 

With this technique the procedure was to obtain the ratio of the Fourier
transform of the output signal & N(f)/NO to the Fourier transform of the input
6 P(t) signal. Then we stated

 

22), S. Bendat and A. G. Piersol, Measurement and Analysis of Random
Data, Wiley, New York, 1966.

 
36

 

G(f) =‘3{ é'r:'o(f)} /’5 {6P(r)} ' (53)

where the operator indicates the Fourier transform.

Inasmuch as the c;oIde used for the data analysis scheme is called
"FOURCO", we refer to this analysis scheme as the FOURCO analysis. The results
are tabulated in Table 2.

5.4.3 Cross-Correlation Techniques

 

The cross correlation function ¥ 2(fr) between two continuous signals can be
I

defined by

+
- Lim ]J' S (DS, (t+ 1) dt, (54)

where S](f) and Sz(f) represent the continuous functions. If S](f) and Sz(f) are the
same, { is identified as the auto-correlation function. A program (CABS?®) was
available which computed the cross-correlation function, the auto-correlation func-
tion of each signal, and their Fourier transforms. Since the desired information for
this study was the frequency-response function, we were interested in the Fourier
transforms of the cross-correlation and auto-correlation functions, because G(f) is

given by

G =3/ 'lf]’z('r)} /3?{¢]' @}, (55)

where subscript 1 refers to the pressure signal and subscript 2 refers to the neutron

flux signal. We obtained the coherence function from

13{ ‘v],z("')} ‘2
'3-’{\3], : (T)}'}'{ \U2'2(T )}

 

vi(f) = (56)
37

As stated previously, the code used for the data reduction is identified as
CABS; hence, this analysis is identified as the CABS analysis.

From the results of this analysis (Table 3), we note that the coherence begins
to increase for the fourth and fifth harmonic, but this is physically unreal since the
signal-to-noise ratio decreases with increasing harmonic number (see Fig. 10).
Therefore, we accept the results obtained from the fundamental and the first two

harmonics by the CABS analysis.

5.5 The Void Fraction from Experimental and Analytical Results

The modulus of the fractional change in the neutron-flux—to—pressure
frequency-response function as obtained from the experiment was discussed in
Sect. 5.4. The objective of this experiment was to determine the amount of cir-
culating void (the void fraction, VF) in the fuel salt. At present, there are no
experimental data available that relate the modulus of the frequency-response
function | G(f)| to the amount of void present; therefore, we calculated a lG(f)‘
to the amount of void present; therefore, we calculated a IG(F)I using the model
discussed in Sect. 2 (the calculated | G(f)| is @ function of the assumed void
fraction in the model). Analytically, we determine that (a) in the frequency range
of interest there was no frequency dependence of |G(f}| on VF, and (b) the magni-
tude of | G(f)| was directly proportional to VF. Therefore, the actual void fraction

VF was obtained from
act

1500,

VFac’r: —Tg(mc— VFref’ (57)

ale

where the subscripts exp and calc refer to the | G(f) |'s obtained experimentally and
analytically respectively, and VFref is the value of the void fraction used in the

analytical model for generation of | G(f) ‘calc
38

Table 1. Results from D~Analog CPSD Analysis

 

 

 

 

 

 

 

 

Frequency | G(f}] Coherence
(cps)
0.025 0.00121 0.944
0.050 0.00088 0.522
0.075 0.00128 0.484
0.100 0.00085 0.340
0.125 0.00112 0.340
0.150 0.00013 0.002
Table 2. Results from FOURCO Analysis
Frequency |G|
(cps)
0.025 0.00126
0.050 0.00080
0.075 0.00220
0. 100 0.00104
Table 3. Results from CABS Analysis
Frequency |G(f)] Coherence
(cps)
0.025 0.00117 0. 61
0.050 0.00103 0.31
0.075 0.00130 0.2/
0.100 0.00105 0.10
0.125 0.00102 0.12
0.150 0.00099 0.13

 
39

Two different assumptions were used in the calculation of the pressure-to-
reactivity transfer function, referred to as Models A and B in Sect. 2.3. The values
of IG(f) ‘calc from each model are presented in Table 4 for a VFref of 0.064%.

The objective now is to infer the actual void fraction from Eq. (57). This was
done by using the data in Sect. 5.4 and Table 4. The results obtained from each
analysis scheme (Table 5) show that the fundamental is consistent for all data reduc-
tion schemes. Furthermore, the scatter, which increases with increasing frequency,
is attributed to two factors: (1) that the power in each harmonic of the experimental
test signal was proportional to the inverse harmonic number squared [see Eq. (50)7;
and (2) that the total loop time of the fuel salt was about 25 sec, which corresponds
to a frequency of 0.05 cps. The analytical model used to determine | G(f) ‘ccalc of
Eq. (57) was based on one-dimensional flow. This caused some humps in the calcu-
lated frequency-response function at frequencies of 0.05 cps and above (Fig. 4).
Examination of |G(f) ‘exp (Sect. 5.4 and Fig. 11) indicates that G(f) varies smoothly
in this frequency range; hence one is led to suspect that mixing occurs which the

model does not account for.

Based on the results obtained at the fundamental (0.025 cps) a mean void
fraction of 0.045% was obtained for Model A and 0.023% for Model B. It is interest-
ing to note that calculation of a weighted average void fraction of all the data pre-
sented in Table 5 gave the same results. This weighted average was obtained by
assigning o weighting factor, a confidence factor, to each harmonic equal to the
fraction of the total input signal power associated with that harmonic, i.e., an
inverse square of the harmonic number.

The problem now is to determine which mode! more nearly represents the
actual physical system. Since we do not have enough experimental evidence to
do this with greater precision, we can only state that the void fraction is between

0.023 and 0.045%.
40

Table 4. Calculated Values of |G(f)| for a VF = 0.0647

F

 

 

 

 

 

 

 

 

Frequency | G(f) |
_ (eps) Model A Model B
0.025 0.00176 0.00343
0.050 0.00163 0.00530
0.075 0.00146 0.00185
0.100 0.000596 0.00406
0.125 0.000496 0.00503
0.150 0.000710 0.00279
Table 5. Calculated Values of the Actual Void Fraction
Vac’r (%) for each Analysis Scheme and Models A and B
Frequency Analog D-Analog CPSD FOURCO CABS
(cps) A B A B A B A B
0.025  0.045 0.024 0.044 0.023 0.046 0.024 0.042 0.022
0.050 - — 0.035 0.011 0.031 0.010 0.040 0.012
0.075 - — 0.056 0.044 0.096 0.076 0.057 0.045
0.100 - - 0.091 0.013 0.111 0.016 ~ -
0.125 ~ - 0.144 0.014 - — — -

 
41

 

|O-2 ORNL DWG. 69-3751
T T T T T T
a ANALOG
- © D-ANALOG CPSD—
0 FOURCO
| v CABS —
»n
= - _
O v\ v.
S5 o} N
&l 103 — &7 v v °__
| 8 © ]
L b

 

 

 

O 0025 005 0075 0.0 0.025
FREQUENCY (cps)

Fig. 11. The Experimentally Determined Modulus of the Neutron-Flux—to—
Pressure Frequency-Response Function from the Various Analysis Schemes.

6. CONCLUSIONS

The primary objective of this experiment was to determine the amount of
helium void entrained in the MSRE fuel salt for the condition of the reactor operating
at power. This was accomplished by forcing the modulus of the power-to-pressure
frequency-response function obtained experimentally and analytically to be the same.
Therefore, considerable study was made to verify the analytical results. We conclude
that the analytical prediction of the power-to-reactivity frequency-response function
was adequate, but the analytical prediction of the reactivity-to-pressure frequency-

response function was only nominally correct.
42

The response of neutron flux to small induced pressure perturbations was
significantly larger than the nominal background response; therefore, a meaningful
frequency-response function of the neutron flux to pressure signals can be experimen-
tally obtained.

At the fundamental frequency of the input pressure wave, the signal-to-noise
ratio of the neutron flux signal was approximately 10; this ratio decreased with
increasing harmonic number. At larger signal-to-noise ratios, the modulus of the
frequency-response function can be obtained by either APSD, CPSD, cross-correla-
tion, or direct Fourier transform techniques. As the signal-to~noise ratio decreases,
the APSD technique becomes unsatisfactory. The direct Fourier transform technique
becomes less desirable than the CPSD or cross-correlation techniques as the signal-
to-noise ratio decreases.

The void fraction, at the time of the experiment, was determined to be
between 0.023 and 0.0457. This large spread is attributed to assumpfions made in

the modeling of the fuel pump bowl.

7. RECOMMENDATIONS FOR FUTURE INVESTIGATIONS

Since the major cause of the uncertainty in the void fraction reported herein
is the model, we recommend that an experiment, analogous to that which we per-
formed, be performed at zero power. The on-line reactivity balance could be used
to determine the void fraction, which in turn would yield a reference point to
permit the selection of either Model A or B, or to indicate that additional work is
required to devise a model.

We further recommend that the experiment described in this report be repeated
for different void conditions during operation of the MSRE fueled with 22U, The
results could be combined with the results obtained from noise analysis.
Finally, we recommend that the CPSD analysis technique be pursued for

extraction of information from neutron fluctuations and background pressure fluctua-

tions that usually occur in an unperturbed reactor.
43
8. APPENDIX

8.1 Interpretation of the APSD of Deterministic Signals

in the Presence of Noise

The direct method,” filtering and time averaging, for APSD analysis will be
considered for the purpose of aiding in the interpretation of noisy periodic signals
with poor signal-to-noise ratios. Consider the output signal from two detectors S;(t)
and Sy(t) which are filtered, multiplied together, and time averaged, and start with
the data analysis scheme shown in Fig. 8. In particular, start at the output Oft),
and move backwards to the inputs Si(t) and S;(t). With the multiplier and time

averager considered, QO(t) can be written as

o) =1L, (58)

where the bar represents a time average. Assume that the fime averaging is carried

out using a unity weighting function and write

¢
Oft) = i'l J [1(y) L,(y) dy . (59)
0

Now the problem is to relate 11(y) and (y) in Eq. (59) to Si{t) and Sy(t). Assume

that the filters are linear and write

Y
Ly = | Fily - x) Si(x) dx (60a)
0
and
Y
I, (y) =j Fo(y = x) Sp(x)dx . (60b)

0
44

Further, assume that Fy and F, are very narrow-band filters so that 1; and I will be
a narrow-band-limited signal; i.e., the Fourier transform of these signals will be
nonzero only for a narrow band of frequencies about the filter center frequency,
even though Sy(t) and S;(t) may have been unlimited in the frequency domain.

Before substituting Eq. (60) into Eq. (59), it will be advantageous to examine

Parseval's theorem in the form%
+oc +x0
" Gi() Gyl df = [ g1 arl-et , (61)

where GE(F) is the Fourier transform of gi(’r). Consider gi(’r) to be nonzero only in the

interval 0 <t < T; then Eq. (49) can be reduced to (see Ref. 23)

T-7 +es
JV (Dot + 7)dt = f’ G1 ()Gy(F) exp [—ZTrFT .J df, (62)
0 -

where the asterisk denotes conjugate complex. For v =0, zero lag time, Eq. (62)

reduces to
T +x
[ alhg dt= [ G () G, of, (63)
O -

where the integrand on the left is analogous to the integrand of Eq. (59).

Let Ii(f) be the Fourier transform of I;(‘r), then

T +eo
O =+ Thiy) Lty dy =+ [ 170 16 of . (64)
0 -co

 

55.0. Rice, "Mathematical Analysis of Random Noise, " pp. 133-294 in
Selected Papers on Noise and Stochastic Processes, ed. by Nelson Wax, Dover

Publications, New York, 1954.

 
45

From Eq. (60),

L{f) = Fi(f) Si(F) , (65a)

and

l2(f) = Fa(f) S(f) ; (65b)

therefore, O(t) can be written as

+

oM == [ F () Falf) S0 5,0) o . (66)

-0

Since Fi(f) and F5(f) are narrow=band filters, S;(f) and S;(f) can be taken outside the
integrand of Eq. (66) if they are smoothly varying functions over the filter widths,
e.g., if $i(f) and Sy(f) are near-white noise signals. It will now be shown that

Si(f) and Sy(f) are not smoothly varying if Si(t) and S;(f) contain a periodic signal.

Let
5.1 =5, B +S P(f) ' (67)
where i = 1 or 2, subscript N refers to a nonperiodic component (assumed to be a

smoothly varying function in the frequency domain), and subscript p refers to a

periodic signal of period TO. Then Si P(’r) can be expanded in a Fourier series as

 

 

 

“oi - 21n 271N

5. (=—— +) la. cos T t+b . sinTrt) (68)
ip 2 ni T ni T
n=1 0 0

where a and bn are the Fourier coefficients, i.e.,

0
S. () cos
P

2mn

To

 

a :“—.i.z— tdt , (6%9a)

ni 0

o b
46

 

and
T
2 r‘o : 21 n
b =— ’ S. (B sin t dt . (69b)
ni T . ip T

Substitute Eq. (55) into Eq. (66) to obtain

+

o(T) = f)F, () I sT®'S, (f)+S] ()SZP(F) (70)

Tl N

+ STP(F) NGE s;‘p(f)szp(f) } df.

An alternative, but useful, form of Eq. (70) is

+ o
(£) S,
_ * 1N
om=[ FrmF,m o —E

L
-

o sSHS B e Sy (S, (f)
- e @ TSZp P Ip TSZN df

JF S]p(g) Szp(g)
T

0

+ fF (FF, () d dg

 

 

0 S]p(g) Szp(g)
J T
-f

0
+JF E, (~f) d dg | , (71)
0

*
]

 

 

where the last two integrals are to be regarded as Stieltjes' integrals.

To simplify Eq. (71) further, consider the cross-correlation function, defined
0522'23
+T1/2
60 = Limit —— | g glt+odr, 72
T=e -1/2
47

where the limit is assumed to exist. Then the cross power spectral density Pjp(f) and

the correlation function are related by

oo
Pl'z(f) =j mlz(fr) exp [-ZTrifT] dr, (73a)
and
+ o
047 -:_L P (F) exp [2wif¢] df (73b)
where
. *(f) g, (1)
p (F):LimnL : g]___g_g___ . (74)

Tow T

Returning to Eq. (71) assume that the observation, or averaging, time T is of
sufficient length so that O(T) has reached its limiting value. Then, Eq. (71) can be

written as

+eo
oM = [ F(HF (P (f) df

1N, 2N

+oo + oo
+f FIOE P N 2p(f)df+J F7 () Fy (f) Pro, o of
o f o 0
F[F MR ) [P o (@de it [F(AF(-Nd { [P ) (@dg ) . 75
J Je T,
0 0 s

In general, the noise and periodic components of the signal will be uncorrelated;
therefore, from Eq. (73), the second and third terms in Eq. (75) will be zero (the
first term would be zero in some cases, but it is retained for generality) and Eq. (75)

becomes
48

 

 

 

 

O(T)—j Fr () F, () ]NZN(f)df
® * f 0 . 0
SRNUE: :[P]przp(g)d + [ F(-NE(-d J' p2p @48 - 78
0 0 0
We define
+ e
AGIGIN e
Anan o) =3 ; (77)
[F (f) F, () d

_CXJ

where f_ is the center frequency of the filters Fy and F;, and write

0

+

o(T) = 1N2N(f0)jF (f) F, () df

o f
+JFF (F) £, () d {t‘fp p(g)dgt
0

 

JF (- (- d 3]} 1p.2p (g) dg
0

To reduce Eq. (78) further, the following terms are examined:

f,0

o .

| P]p,2p(g) dg . (78)
0, -f

Equation (72) is used to obtain the correlation function for periodic signals and then

Eq. (73a) is applied for ( P]p,2p(g) dg. Write Eq. (68) explicitly for S]p(f) and
t+ 1), i.e.,

SZp( T), i
49

 

 

 

“1 2rn 2mn
Slp(t)= 5 +ZI (an] cos —3 ’r+bn}sm T f) , (79a)

and

 

 

 

Cl02 § —2 Th 2tn
(t+7)= = [a cos (t+ 1) +b Asin = (’r+"r)]. (79b)

Multiply the series in Eqs. (79a) and (79b) together and integrate over t to obtain

T a..d
] 01702
—T—J S!p(t) Szp(f+«r)-——-z-——~
0

@

1 N | 27N
* Z §<Gnlcn2+ bnlan/cos TO T

s - a b ) sinED 14 80
+z _é_Ccnlan °n2bn1 sin—— T+e, (80)
0

=1
where ¢ is an error term that approaches zeroas T = = . The error term is also
dependent on the ratio of the integration fime T to the period of the periodic signal TO;
e.g. the error term will be a minimum when T/T0 is an integer for finite T. By

taking the limit of Eq. (80) as T - =, the correlation function () is obtained for

periodic signal of peried T_, i.e.,

0
50

 

a..a o
_ 01702 T 27n
%P:ZP(T) 4 +Z] 2 (\Gn]anZ * bnl bn2> cos T0
=

[s o]
21n

1/ N L.
+E 2 (C'n] bnZ - Oann'I/’ S0 T (81)
1 0

where the error term ¢ of Eq. (80) goes to zero in the limit.

Integrate Eq. (73a) over frequencies f from O to f to obtain

f 4+
T 7 sin 21f T
"—.._-. ) i ————
ur P‘p,Zp(f)) df* = 27 e () T dr
0 -
. +o 1 - 2 f ,
-5 [ () —=Edr (82)

Substitute Eq. (81) into Eq. (82) and integrate to obtain®
f

 

a0
( ;01702
JEP] ,2p(mdf =5
s Gn]cn2+bnlbn2 e 21n N\
+ —_— Y f - -
[ 4 n'- T. ./
e 1 0

 

 

® o b - )
+iL T n24 CIann] [UnO' 2

=1

=11, (83)

N
TO

 

24K, S. Miller, Engineering Mathematics, Rinehart and Company, New York,

 

1956.
51

 

 

 

where Un(f - 2.|.Tm) is the unit step function which is zero for f < .Fm and unity
0 0
for f > 2Trrn Also integrate Eq. (73a) over frequencies from 0 to -f, i.e.,
0

sin2m fr dr

T

0 | + o
-
_jf Pip2p 18F = | 1o

4+
i 1 - 2mf
+_21F_L¢12(T>L__;£_lfldq-. (84)

A development analogous to that leading to Eq. (83) yields

0 [ PRNe
;01702
_7[ P p’2p(f')df =

 

 

 

 

 

 

 

+io dn]°n2+bn]bn2 U (f- 21n N\
"4 T/
=1 0
@ a.,b.~a
_ |Z nl ni négn] [UnG _ 2Trm > _ ]J . (85)
n=1 0
We define
co b .b
1| G012 %1%2 P12 . 2mn N
X == — ) ? Uy fmg— (- &
n=1 0
and
b ,-a b
] YPn n2 n2 271N 7
X0 == ) 2 (U= -1, (86b)

0
52

Write Eq. (78) as

+
om =P N2N (fo)-i FY (F) Fp(f) df

+ | FTOF0 d [XG(0 + X ()
0 .

+ j Y (-f)Fy(-fd X - X ()} .
0

The input and output signals to the filters are real; therefore,
* *
Fi (-F(-f) = Fy (f) Fa(f) ,

which is the condition of reality. By use of Eq. (88), Eq. 87 is reduced to

+o
(f) [ FY (A Fo() of

-0

O =R 2N

+2 J R (F) Fylf) dX,(f) .
0 .

The term XR(F) is a step function which is discontinuous at frequencies Fn' where

Fn = 2rrn/T0

(88)

(89)

(90)

forn=0, 1,2, *"* . Assume that F1 () Eo(f) is a continuous function and let the

step changes in XR(f) at f = Fn be denoted by hn (where hn >0), then Eq. (89) can

be written
53

4+
o(T) = P]N (A j Fy () Fp(f)dif
+ 2 ). F; (fn) Fz(fn) hn ; (9])
n=0 -

where the last summation is the value of Stieltjes in’regrcul.24 From Eq. (74a), we

determine that 2hn is given by

 

 

., Q
2h0 = Ol 02‘ , (92a)
and
a .a.*tb b
Zhn _ nl n22 nl n2 ’ (92b)

forn=1, 2, 3,
At this point, we introduce the nomenclature that the area of the filter AF

is defined as

4o

A= j Fy (F) Fp(P)f . (93)

-0

Furthermore, we assume (a) that the signals Sy(t) and Sy(t) are the same and (b) that

if the center frequency of the filter f. is near a periodic signal's harmonic frequency

0

fn’ the filter's response at all other harmonics will be zero. Then the output signal

O(T) for this filter setting of FO [denote the output by O(T,FO)] will be given by

O(Tf) ___2___n_'

(F) A + |F(f) (94)

1N 2N

and, in particular, for filters which have unity gain at their center frequencies,
(fO) A+ —n__n_ (95)

o, ) = 2

PIN,2N

Consider the evaluation of the area of the filter AF as defined by Eq. (93).
This area is usually determined by the analysis of a noiseless sine wave at several
frequencies (constant amplitude) about the filter center frequency, which permits the
evaluation of the integrand of Eq. (93). Then a straightforward integration permits

the evaluation of the integral

JF F1 () F(f)df . (96)
0

Now Eq. (88) permits evaluation of the area, i.e.,

Az =2 j F]* () Fp(f) dFf . (97)
0

Actually, the area A_ is not directly observable since we cannot generate signals

F

with negative frequencies; therefore, another area term AR is introduced which will

be referred to as the physically realizable area. This area will be defined by

A= [ FT (AP af o (98)
0

which is a directly observable quantity. Equation (95) is written as

ofT, f) = 2F F)A +1

T2 2
iN,2N ) Ag g la + b0 (79)

Although Eq. (99) is consistent with the definitions presented in Eqs. (73a)
and (73b), there is an alternative form which is generally used when working with
the auto-power spectral density. Since this alternative form was used by Ricker, '°

we will proceed to develop it here. We define Yy {f) by?
55

Y1(f) = 2P (f) (100)

for 0 < f < = and zero otherwise. From Eq. (73a),

Foo

Yii(f) = 2 { y11(7) exp(- 211 FT)dT , (101a)
and, conversely,
o . N
ynlr) = [ Y1i(f) exp<2fl|f'r/df : (101b)

0

Use of Egs. (101a) and (101b) instead of Eqs. (73a) and (73b) leads to

_ Vo2 o2
AT, FO) = Yn(fo) AR 3 (an + bn) , (102)

which is the desired result for this study.
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