ORNL-TM-3007.txt

 

i |
~ OAK RIDGE NATIONAL LABORATORY
operated by
: UNION CARBIDE CORPORATION m
! NUCLEAR DIVISION
. for the

U.S. ATOMIC ENERGY COMMISSION
ORNL- TM~- 3007

 

-5 -

AN EXTENDED HYDRAULIC MODEL OF THE MSRE CIRCULATING FUEL SYSTEM
(Thesis)

W. C. Ulrich

“wr Submitted to the Graduate Council of the University of Tennessee in
partial fulfillment for the degree of Master of Science.

DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED
 

e e e e LEGAL NOTICE - e m e s e e

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, Moakes any warranty or representation, expressed or implied, with respect to the accuracy,
completenass, or usefulness 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, apporotus, method, or process disclosed in this report. :

As used in the obove, '“person acting on behalf of the Commission® includes any employee or

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

or controctor of the Commission, or employee of such contractor prepares, disseminates, or
provides access to, any information pursuant to his employment or contract with the Commission,

or his employment with such contracter,

 
LEGAL NOTICE———

This report was prepared as an account of Government sponsored work. Nelther the United
States, nor the Commission, nor any person acting on behalf of the Comtmisaion:

A. Makes any warranty or representation, expressed or implied, with respect to the accu-
racy, completeness, or usefulness 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 llabilities with respect to the use of, or for damages resulting from the
use of any information, apparatus, method, or process disclosed in this report,

Asp used in the above, '‘person acting on behall of the Commisaion® Includes any em-
ployee or contractor of the Commission, or employee of such contractor, to the extent that
such employee or contractor of the ( or employee of such tractor prepares, ORNL_TM_ 3007
disseminates, or provides acceas to, any information pursnant to his employment or coantract
with the Commigesion, or his employment with such contractor.

 

 

 

 

Contract No. W-TLOS-eng-26

REACTOR DIVISION

AN EXTENDED HYDRAULIC MODEL OF THE MSRE CIRCULATING FUEL SYSTEM

W. C. Ulrich

Submitted to the Graduate Council of the University of Tennessee in
partial fulfillment for the degree of Master of Science.

JUNE 1970

OAK RIDGE NATIONAL LABCRATORY
Oak Ridge, Tennessee
operated by
UNION CARBIDE CORPORATION
for the
U. 5. ATOMIC ENERGY COMMISSION

ver e TS DOCUMENT 15 UNLLIMITES
BISTRIBUTIOH OF TI5 DOCUMENT 15 URLLIALED
(3§
5
ACKNOWLEDGEMENTS

The author is indebted to Dr. J. C. Robinson of the University of
Tennessee for the overall guidance he provided on this work. The teaching
skill, patience, and help he offered at critical periods is greatly
appreciated,

Thanks are due also to Dr, J, E, Mott of the University of Tennessee
for several valuable suggestions which he made during the development of
the hydraulic model,

The assistance of the management of Oak Ridge National Laboratory*
in granting the author the necessary leave of absence to pursue his studies,
and in furnishing the use of computer facilities, graphic arts, and repro-
duction services in connection with the publication of this thesis is
gratefully acknowledged.

The author also wishes to thank the United States Atomic Energy
Commission for the Traineeship he was awarded that enabled him to under-
take the academic work of which this thesis is a part.

It is a pleasure to thank Mrs. Annabel Legg for the efficient,
careful way in which she typed the manuscript.

Lastly, the author wishes to thank his wife for her support, under-
standing, and encouragement during what must have been a difficult time

for her, Without it, none of this would have been possible.

 

*
Operated for the United States Atomic Energy Commission by Union
Carbide Corporation.

iii
ABSTRACT

The hydraulic portion of a combined hydraulic-neutronic mathe-
matical model for determining the effects of helium gas entrained in the
circulating fuel salt of the Molten Salt Reactor Experiment on the neutron
flux-to-pressure frequency response was extended to include effects due
to the fuel pump and helium cover-gas system.

The extended hydraulic model was combined with the original neu-
tronic model and programmed for computations to be made by a high-speed
digital computer,

By comparing the computed results with experimental data, it was
concluded that pressure perturbations introduced by the fuel pump were
the main source of the naturally occurring neutron flux fluctuations in
the frequency range of one to a few cycles per second.

It was also noted that the amplitude of the neutron flux-to-
pressure frequency-response function was directly propcrtional to the
pressure in the fuel-pump bowl; however, further work will be required
before completely satisfactory results are obtained from the extended
model, Recommendstions are proposed which should prove useful in future

modeling of similar hydraulic systems.

iv
TABLE OF CONTENTS

CHAPTER PAGE
INTRODUCTION & v 4 & & o o o o o o o o o s o s o o s o« o o o o o« o 1
1., DESCRIPTION OF THE OPEN-IOOP MODEL . & 4 4 o o o o o s o o« L
2. DESCRIPTION OF THE EXTENDED MODEL & v & & « & o o o o o o o o 9
Physical Characteristics of the Fuel Pump and
Fuel=Pump BOWL . . . v &« v ¢« ¢ o o o o o o o o o o s « o 9
Assumptions Made in the Development of the Model . . . . . 11
Procedures Used in Deriving the Equations for the Model . . 14
3. RESULTS OF CALCULATIONS MADE WITH THE EXTENDED MODEL
COMPARED TO EXPERIMENTAL DATA . . & ¢ ¢ 4« o « o o & o o + & 30
L., CONCIUSIONS AND RECOMMENDATIONS o & v & ¢ « « o « o o o o o« « 37T
LIST OF REFERENCES . & v 4 & 4 ¢ 4 « o o o o o o o « o o« s o o o o H1

APPENDIX - . . -* * * * * 2 * * - . . - . . * . . . . - - - - . . . uB
LIST OF FIGURES

FIGURE

l.

Model Used to Approximate the Molten Salt Reactor
Experiment Fuel Salt TloOP « v ¢ o + o + & o o &+ o o o o o
Extended Hydraulic Model of the Molten Salt Reactor
Experiment Fuel Sall LoOP ¢ v ¢ ¢ o o o o o o o o o o
Molten Salt Reactor Experiment Fuel Pump and
Fuel-Pump BOoWl., . . o & ¢ ¢ o o o o ¢ o o o o o o s a o o
Hydraulic Performance of the Molten Salt Reactor
Experiment Fuel PumpP . ¢ o ¢ ¢ ¢ o« o &+ o o s o o o s+ o
Square of the Modulus of the Pressure in the Molten Salt
Reactor Experiment Fuel-Pump Bowl; Case 1: F = 1.0,
AP14 = 0.0, AP1s = 0.0; Case 2: APja = 1.0, AP35 = 0.0,
= 0.0 o o o o o o o s o o o ¢ o o a o o o o o s s o o
Modulus of the Neutron Flux-to-Pressure Irequency-Response

Function for the Molten Salt Reactor Experiment e s o o o

vi

PAGE

PO

10

12

33

36
Symbol

LIST OF SYMBOLS

Description
area
orifice coefficient for fuel salt
orifice coefficient for gas
concentration of ith'group of delayed neutron precursors

conversion constant from fission rate to the desired
units for power density

heat capacity of the moderator at constant pressure
neutron diffusion coefficient

fuel pump forcing function (head delivered)
acceleration due to gravity

gravitational constant

heat transfer coefficient

enthalpy of the fuel salt

enthalpy of the gas

fuel pump head at normal operating conditions
thermal conductivity of the moderator

distribution, or flow, parameter for two phase flow
mass flow rate

fuel salt mass flow rate

gas mass flow rate

mass

mass of fuel salt

vii
Symbol

viii

Description
mass of gas
original or steady-state condition (subscript)
heated perimeter
wetted perimeter
pressure
heat flux
fuel pump veoclumetric flow rate
power density
universal gas constant
Iaplace-transformed time variable

slip velocity ratio (ratio of gas velocity to fuel salt
velocity)

time variable

absclute temperature

fuel salt temperature

gas temperature

moderator temperature

temperature of wetted perimeter (wall temperature)
internal energy for the fuel salt
internal energy for the gas
volume of fuel salt

volume of gas

velocity of the fuel salt
veloclity of the gas

velocity of the neutrons (one group)
Symbol

Superscript

ix

Description
space variable
space variable
gas void fraction
delayed neutron fraction
delayed neutron fraction for the ith group

fraction of the "unit cell" power density generated in
the fuel salt

deviation about the mean, incremental quantity, or per-
turbed guantity

decay constant for the ith group of delayed neutron
precursors

number of neutrons produced per fission
density of the fuel salt

density of the gas

density of the moderator

absolute value of the slope of the fuel pump head —
capacity curve at the normal volumetric flow rate of the pump

macroscoplic neutron absorption cross section
macroscopic neutron fission cross section
total wall shear stress due to friction
neutron flux

fuel pump bowl volume

the Laplaclan, or differential, operator

average
INTROTDUCTION

An analytical model for determining the void fraction of helium
circulating in the Molten Salt Reactor Experiment (MSRE) fuel salt loop
by relating neutron flux to pressure using frequency response techniques
was formulated by Robinson and Fry.! The hydraulic portion of this com-
bined neutronic-hydraulic analytical model was not complete, however, in
that it did not contain a specific representation of the fuel pump and
fuel-pump bowl. (See Fig, 1.) Omission of these two items resulted in
an open=-loop hydraulic system which was closed by applying boundary con-
ditions® which approximated their effects on the system,

The scope or purpose of the work described below includes (l) ex-
tending the hydraulic portion of the original model by explicitly in-
cluding the fuel pump and fuel-pump bowl to close the loop (see Fig. 2),
(2) combining the extended hydraulic model with Robinson and Fry's original
neutronic model to calculate the frequency response of neutron flux to
pressure, and (3) attempting to validate the mathematical models by com-
paring results obtained from experimental measurements with predictions

made by the two models,
ORNL—DWG 68 -8417

OQUTLET

 

INLET T

 

A !

L, UPPER PLENUM Lg

 

 

 

 

el

 

. — . — e —————— e dma m——— e

 

 

CHANNELED
REGION

 

 

 

 

 

 

 

 

T
#Ls L/ LOWER PLENUM \J

FIGURE 1. Model Used to Approximate the Molten Salt Reactor
Experiment Fuel Salt Loop

 
ORNL-DWG 70-1376

He (9) 0 e

 
     

FUEL-PUMP
BOWL

    

13

    

FUEL PUMP

G

     

@

  
   

  

J\

UPPER PLENUM

   

 

 
 

CHANNELED
REGION

   

LOWER PLENUM

  

FIGURE 2. Extended Hydraulic Model of the Molten Salt Reactor
Experiment Fuel Salt Loop
CHAPTER 1

DESCRIPTION OF THE OPEN-LOOP MODEL™

The technique for determining the wvoid fraction in the MSRE
circulating fuel salt consists of analyzing fluctuations in the neutron
flux signal caused by induced pressure fluctuations in the fuel-pump bowl,
In the development of the analytical model, the compressibility of the
entrained 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-dimensicnal flow, the governing egquations are:?
Equation of state for the gas,
o = P/RT, (1)
g
Conservation of mass for the gas,
o 2
ST (00 + 35 (o 70) = o (2)
Conservation of mass for the fuel salt,
5 Toe(l - Q)] + 5 [opVp(i - @] = O (3)
ot LPr 9z Pf'f '
Conservation of momentum for the gas-salt mixture,

3
3t legVe(l ~a) + oV 0] + 7?; [ogVE(1 = @) + p Vi) =

@

oP _ W
"8 % T Wwd [l - @)+ pale.

The assumed relationship between Vf and Vg is

 

Conservation of energy in the salt-gas mixture,

<
St [quf (l - a) + Dguga] + Dz f f £ (l @) + ngghgoc]

ap
h on
A TR

Conservation of energy in the graphite moderator,

55IM

o Oy <2 = (- 9) Wiy M

 

()

(5)

(6)

(7)

(8)
Coulomb's law of ccoling:

Q2
{1

h(TW ~ Tf). (9)

Power density,
= C I ¢ . (10)

Conservation of neutrons (one-group diffusion model),

&

V-l'@g = V-DVo + [v(l - B)Z -z J¢ +y %, C (]_1)
n ot f a Lo MYy
1=1
Precursor balance equations,

ic—i-— Zcb-?xC“"‘é'“(VC) (12)

t = Piver i¥1 T Jz i)

fori=1,2, . . . . 6.

Since the interest is in small deviations about steady state, it
was assumed that linearized representation of the governing equations (1)
through (12) would adequately describe the system. It was further assumed
that velocity fluctuations would not significantly affect the precursor
balance, and that the fluctuations in the density of the gas are propor-
tional to fluctuations in the pressure., This latter assumption is based
on the linearized version of Eq. (1}, i.e.

bo, = o, (BT (13)

The last term of Egq. (13) was dropped because of the larger energy input
necessary to change the temperature compared to that required to change

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

The dependent variables in Egs. (1) through (6) are Vs Vg, Py
P, and S. 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 these studies were ANf, N, and AP.

Therefore, the equations defining the hydraulic model were transformed to

the frequency domain and written as

dX{z,s
A(z,s) ——éz*"l - B(z,s) X(z,s8) = 0, (1k)
where X(z,s) is the column matrix
AVf(Z) S)

X(z,s) = |2a(z,s) (15)

AP (z,8)

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

The solution to Eq. (1k) is

X( z + Az,s) = exp[Q(s) Amflix(z,s) , ' (16)
where
Z2+A\7
) = 5 et @, (27)
and Q(z’,8) = A"1(z,5) B(z,8) .

The matrix expfi@(s) Az can be evaluated using matrix exponential
techniques similar to those described in Reference 6. Before the solution
can be completed, the boundary conditions appropriate to the system must

be specified,
To assign boundary conditions, a physical description (model) of
the actual system must be considered. The model chosen to represent the
more complex actual system is presented in Fig. 1, page 2. In particular,
six regions were chosen:

1. the region from the primary pump to the inlet of the down-

comer, Lj:

2., the downcomer, Lg;

3. the lower plenum, Lz;

L., a large number of identical parallel fuel channels*, Lg:

5. the upper plenum, Lg: and

6. the region from the reactor vessel to the primary pump, Le.

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 1t is believed that this would not affect the
hydraulic model, The effects of the pump bowl on the system were approxi-
mated by the boundary conditions between regions 1 and 6.

The matrix represented by the exponential term of Eq. (16) was gene-
rated for each region., Then, continuity equations were applied between
each region, along with the pressure fluctuations inserted at the pump bowl,
to permit the solution of 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 egquations describing the neutronic model.

 

*
The reactor actually consists of hydraulically different parallel
channels, but to date no attempt has been made to model them,
CHAPTER 2
DEVELOPMENT OF THE EXTENDED MODEL

Physical Characteristics of the Fuel Pump and Fuel-Pump Bowl?”

In discussing the extended hydraulic model of the MSRE circulating
fuel system, it may be helpful to begin with a brief description of the
physical characteristics of the fuel pump and fuel-pump bowl which are
shown in Fig. 3.

The fuel-salt circulation pump is a centrifugal sump-type pump with
an overhung impeller. It is driven by an electric motor at ~ 1160 rpm and
has & capacity of about 1200 gpm when operating at a head of 48.5 ft., The
36-in.-diameter pump bowl, or tank, in which the pump volute and impeller
are located, serves as the surge volume and expansion tank in the primary
circulation system.

The pump bowl, or tank, is about 36 in. in diameter and 17 in. high
at the centerline. The normal fuel salt level in the bowl is about 11 in.
from the bottom, measured at the centerline,

A bypass flow of about 60 gpm is taken from the pump-bowl discharge
nozzle into a ring of 2-in.-diameter pipe encircling the vapor space inside
the pump bowl, This distributor has regularly spaced holes pointing down-
ward toward the center of the pump bowl, The bypass flow is sprayed from
these holes into the bowl to promote the release of fission product gases
to the vapor space.

The bypass flow circulates downward through the pump bowl and re-

enters the impeller. The spray contains a considerable volume of cover
ORNL-DWG 69- 1017241

SHAFT
PURGE

OFFGAS
LINE

 

 

 

 

 

 

 

 

 

 

 

 

 

BUBBLER
' / = | SPRAY RING
1OOE I!J / LI
F —_Z
E -'L_'—'~;~- -
: _
7 DISCHARGE
i MPELLER
VOLUTE o {}_ - peL
T PUMP BOWL
SUCTION

FIGURE 3. Molten Salt Reactor Experiment Fuel Pump and Fuel-Pump Bowl

0T
11

gas in the liquid, and the tendency for this entrainment to enter the pump
is controlled by a baffle on the volute. Tests indicated that the liquid
returning to the impeller will contain 1 to 2 volume percent of gas.

A purge flow of about 4.2 std liters/min of helium is maintained
through the pump bowl to act as a carrier for removing fission-product

gases and to control the pressure in the system.

Assumptions Made in the Development of the Model

To extend the hydraulic model to include the fuel pump and fuel-
pump bowl, it was necessary to make some assumptions concerning the various
regions involved as shown in Fig., 2, page 3.

Including the fuel pump and fuel-pump bowl in the model actually
introduced four components into the system: (1) the fuel pump itself,
(2) the bypass flow connections, (3) the fuel-pump bowl, and (4) the helium
supply and removal system, The assumptions made for each of these com-

ponents are listed below.

1. Fuel Pump

Assume that the relative concentration of gas to fuel salt does not
change as the fluid passes through the fuel pump, This assumption was made
because the residence time of material in the pump is very short, i.e.,
there is practically no holdup of circulating fluid in the pump.

Further assume that the 1150-rpm fuel pump head-capacity curve

(Fig. 4) can be represented by an equation such as

é_g - H- oq. (18)
H ,TOTAL HEAD (ft)

60

50

40

30

20

10

ORNL-DOWG 64-3998 R

 

 

 

 

IMPELLER DIAM: 11 Y5 in. / CONSTANT
MOLTEN-SALT TEMPERATURE: 1200°F ~¢ RESISTANCE )
iy, ey /
e

y

\q\/
> \HSOrpm

 

I’

- /
~Y 7
~ ’,

7

/41030

 

 

 

 

 

 

[~
L / ==
9= —— 1L S L
/ / L7 ~dro0 T
'h-fh .
_ ™~ =~ 600 s
=~ 4\./ ™~
PUMP HYDRAULIC / A

BALANCE LINE -+

 

 

 

 

 

 

 

200

FIGURE L.

400 600 800 1000 1200 1400

Q , FLOW RATE (gpm)

Hydraulic Performance of the Molten Salt Reactor Experiment

Fuel Pump

1600

ol
13

2. Bypass Flow Connections

For the 60-gpm bypass flow that is diverted from the pump discharge
and sprayed into the fuel-pump bowl, assume that this connection (region 11)
can be considered as an orifice and the flow rale represented by an equation

of the form®

Myy; = A13C13./20(Ps - P13) . (19)

At steady-state conditions (no change of level in the fuel-pump
bowl) the amount of fluid returning to the main loop flow across the baffle
on the volute at the pump suction is equal to the amount of bypass fluid
leaving the main loop at the pump discharge through region 11, Therefore,
assume that the bypass flow connection identified as region 12 can also be
considered as an orifice and the flow rate represented by an equation of

the form

My = AjLis,./20(P13 - Pg) . (20)

It is further assumed that Egs. (19} and (20) can each be written
as two separate equations: one for the fuel salt mass flow rate, and one
for the gas mass flow rate. The orifice coefficients will be different

depending on which material is being considered. Thus we have

 

 

 

Bryy T Allcfll\/gpfll(P9 - Pis) (21)
m = AllC

g1l gllx/épgli(Pe - P13) , (22)
Men = Alchlz\/épfle(Pl3 - Pg) (23)
1k

and

 

me = M, Vo, (Piz- o) - (L)

3. Fuel-Pump Bowl

The fuel-pump bowl (region 13) is a two-component (gas and fluid)
region, and it is assumed that each region is a separate, well-mixed
volume because the residence time of helium in the gas space is on the

order of five minutes, and the fluid is agitated by the spray.

4. Helium System

The helium inlet and outlet pressures are either known or assumed
to be known, and the assumption is made that the flow of helium into and
out of the fuel-pump bowl gas space can be represented by orifice-type

equations such as

 

 

mg14 = Al4cg14\/20g14 (P14 - Pl3) (25)
for the inlet (region 1k),
and

mgl5 = A15Cg15 \/20%15 (Plj - P15) - (26)

for the outlet (region 15),

Procedures Used in Deriving the Eguations for the Model

Using the assumptions discussed above, and writing mass balance
equations for the gas and fuel salt across the various regions, a set of
13 simultaneous equations was cobtained.

1. Fuel salt mass balance for the fuel-pump bowl:

fas _
at = Tri1 0 Meap 0 (27)

———"
15

or expressing the mass flow rates in terms of fuel salt density, void

fraction, fuel salt velocity and flow areas, this becomes

delB

i = [ep(1 - o)V, Al - [op(1 - )V, Ay (28)

2. Gas mass balance for the fuel-pump bowl:

aMm
—E1l3 _ - -
dt - mg11 mg12 * mg14 mng ? (29)
or
s
= oV _ A - oV _ A + V A - V A . 0
dt (pg g )11 (pg g )12 (pg g )14 (pg g )15 (3 )
3. Pressure-head relationship for the fuel pump:
From Equation (7)
Po - Pg
_— = H - OQ . (31)
P
But
Tp
Q = _—..:a » (32)
o
SO
or

[ep(1 - @) + p alg + [pa(1 - @) + p ] |
Po - By - H—Z R B (P A)a, (3)

 
16

4, TFuel salt mass balance for the fuel pump:

me, = Mg (35)
or
[pp(1 - @) VAL, = [ep(l - @) VAT . (36)
Assuming that Prg Pro and since A, = Ag, this reduces to
(L-a) V], = [(1-0) Ve, (37)

5. Gas mass balance for the fuel pump:

Mo = Tgg s (38)
or
(o @V, 8, = (og @V A)g (39)
which reduces *to
= . 40
(pg @ V. )g (og @ V.)o (40)

6. Fuel salt mass balance at the junction of regions 1, 9, and 11:

Meg = Tp, T Meyy (41)
or
[op(1 = Q) VLAY, = [po(l - ) VLAY + [pp(1 - @) VAT, . (42)
Assuming that Opy Pro = Ppiq 2 this becomes
17

[(1-0a) VoA, = [(1- o) Vel + [(1 - o) VAT, (43)

T. Gas mass balance at the junction of regions 1, 9, and 11:

m = m 4 om (4h)
g9 g1 g11
or
oV A = aV A + oV A . L
(og @V A, = (pg @V A), + (o 0V A), | (45)
8. Fuel salt mass balance at the junction of regions 6, 8, and 12:
Trg = Mg * Tf1z s (46)
or
Top(l = ) VAL, = [pp(1 - @) VAT + [0p(1 - 0) VAT, . (7)
Assuming that Pre = Prg Pryo 2 this becomes
[(1- a) VA, = [(1-q VAT + [(1- a) VAT - (L8)
9. Gas mass balance at the junction of regions 6, 8, and 12:
mga = mge + mg12 » ()4-9)
or
aV A = aV A) + aV A . 0
(pg @ V. A), (pg @V A)g (pg @ VA, , (50)
10. Volume balance for the fuel-pump bowl;
Mf M
— + | B = v + v = const. = V¥, (51)

Pr 1= gl
18

(pp My + 0, Mp) 5 = ¥lop pg), 5

Void fraction relation between regions 1 and 9:

the junction of regions 1 and 9 it is assumed that

Void fraction relation between regions 9 and 1l:

At the Junction of regions 9 and 11 it is assumed that

ng = (111

Void fraction relation between regions 6, 8, and 12:

The void fraction in region 8 may be written as

or
11,
At
120
130
But
and
SO

or

\
2

  

 

 

v
o /88
8 v + v T fm m \ ‘
gs Ts & 1t
Dg Qf
8
m = m + m
g8 g6 g1z
ju!1 = m + m
fa fe fiz "’
m _oom
gl L g6
o = Pga  Pgs
g8 ~ m___ m Moo Moo ’
g1 2 + 26 + + 1

(52)

(53)

(54)

(55)

(56)

(57)

(58)
19

 

(o V,08) . (p V 0h),
0

0
o8
o " T m . G, Tl VAL o0 VAT,
Pga Peg Pfg Pre

(59)

 

The next step was to write these 13 equations in linearized, per-
turbed form, as shown by the following examples.
Referring to Eq, (40), page 16, the equation representing the gas

mass balance for the fuel pump written in perturbed form is

[(pgo +.Apg)(ab + Aa)(vgo + Ayg)]s =
oy * A0 ) (@, + 00)(V_ + AV )],.(60)

Carrying out the indicafed multiplication, the following result
is obtained. (Since both sides are identical except for the regions in-
volved, only the general procedure will be described.) The left-hand

side thus becomes

(v

+ V 0

g0 Pgo a, g0 N+ a Vgo Apg + Q Poo /_\.Vg +

g0

0 OAOlAVg + Q. Apg AVg)8 .

g

Because interest is restricted to deviations about the mean, the first
term, which contains only steady-state quantities, cancels out.
Furthermore, the last two terms which contain the product of two perturbed
quantities, and are assumed small compared to the remaining terms, were

dropped. This gives the linearized form of Eg. (L) as

+ =
(VgO pgo N+ Q!O VgO Apg Odo pgo AVg)B
20

(V.. »
go "go A + Vgo Apg + 00 AVg)g . (61)

The linearized version of Eq. (1), page L, is

ADg = fi?__{%_ ’ (62)
o
and the linearized form of the relationship between Vg and Vf »
Eq. (5), page 5, is
1 - o Vfo(l - K)

AN = T N+ N . (63)
e Koa ot (x-0a)°

Now referring to Fig. 2, page 3, it is noted that at the junction

of regions 1, 9, and 11 the pressures P, = Pg = P13, and at the
junction of regions 6, 8, and 12, the pressures Pg = Pg = Pio.
Linearized, these equalities become AP; = APo = AP;; and APs =
APs = AP,z, respectively.

Substituting Egs. (62) and (63) and the pressure equalities into
Eq. (61) gives the final linearized perturbed form of the gas mass balance

for the fuel pump:

1l -« 1l - ¢«
0 , o (1 - X)
pgo Vfo K-« oo+ % pgo K-« AVf T ao pgovfo _ 5fi1
0 o (k - o)
8
advfo 1 - e e - 1 - a -
RT K- o & = |Pgo 'folK -
0 o 0
8
1 -
0 1L - K
— | AV
* Otopg;o K-« & T Oflopgo fo (K - o )2 £
o o
o Vf 1 -
+ _—— 2 JAV-E T (6)4)

 

RT K-«
0 0

a
21

For another example, consider Eq. (27), the fuel salt mass balance

for the fuel-pump bowl, page 1k, Linearized, this becomes

A - Am. - Am

dat i3 f11 fio (65)

Linearizing the orifice relationships for the fuel salt mass flow
rates for regions 11 and 12 as given by Eqs. (21) and (23), respectively,

page 13, gives
2 w2
11 Cfll e

Amfll = m (AP9 = APl:,)) 2 (66)
foi1

and

A2 2 5
Am .ol (AP15 - APg) . (67)

a2z foro

Substituting Egs. (66) and (67) into Eq. (65) and using the line-
arized pressure equalities mentioned above, the linearized fuel salt mass

balance equation for the fuel-pump bowl becomes

 

2 2 2
AZ p A= C 0
d 11 fll r fiz "f
o, ] - APy + —E—==— AP
fo1i: foiz
.t (A2 CE _+ A®_CZ2_) AP (68)
Mooy, & 1% fi1 12 “fie 13 -

The next step was to Laplace transform the time variable to the
frequency domain. Thus, the linearized, perturbed, Laplace-transformed

version of Eq. (68) is
22

 

2 A2 > A2
A7y Cri1 Pr Al Crio P
Sy, = T AR T e AR
foi1: fole
2 (A2 c2 + A®_CZ ) AP (69)
Mooy 11°f11 12 ‘T2 13 -

As a result of performing the manipulations described in the above
examples on the remaining 11 equations, a set of 13 linearized, perturbed,

ILaplace-transformed equations containing 16 unknowns was obtained,

Method Used to Solve the Bquations Derived for the Model

In order to solve the set of 13 equations containing 16 unknowns
Just derived, three more eqguations were needed, These three additional
equations were obtained from the original hydraulic model.

In the original model, the variables in region 6 are related to those
in region 1 by means of a transfer function. In matrix form, this re-

lationship can be expressed as

[X(z,s)]e = B(s)[X(z,8)]1 - (70)

where X(z,s) is the column matrix

X(z,8) = [ &V, (2,9)]
o (z,8)
NP (z,8) ’ (71)

and B(s) is the 3 x 3 square transfer function matrix which relates the
state vector X(z,s) at the outlet of region & to the state vector X(z,s)

at the inlet to region 1.
23

Expanding the matrix form of Eq. (TO) gave the necessary three
additional equations to complete the set of 16 equations containing 16
unknowns,

The 16 equations constituting the extended hydraulic model are as
follows:

1. Fuel salt mass balance for the fuel-pump bowl:

 

2 2]
13 1
13 P Ofyo 11 mfo
11
2 ’
- (CfA)lz m | & = O (72)
: fo
12
2. Gas mass balance for the fuel-pump bowl:
TO : P Q o P O
s + Rl = (cA)2, Eg_ + (C_A) ag“
g g o0 g "12 20
13 11 12
pgo pgo
+ (c A)Z + (C_A)Z AP =
g ‘14 |m g ‘1simnm
g0 go
14 , 15
T (), - (P) o
_ o _0 0’9 0’13 g0
(CgA)ll R v R{m T ) T APy
g go o’11 go
13 11
T
+ (c_4)2_ Rl Bis— (Bo)y - (g
g "1z \V R{m_T) m APg =
&/ 13 go "o’z go

12
 

 

o Ts Pao (Po) 14 (PO)13
(CgA)14 e m * Rm_ T) AP14
g go g0 0O'14
13 14
T P - (P
P enz f=2| [[=e) - Folis ™ Fols f (73)
g 15 |V m R(m T) 15
g g0 g0 0O'1s
13 15

3. Pressure-head relationship for the fuel pump:

a

 

 

 

 

 

 

 

H | o - H i o H B,
1 7R T APy = WL+ g T | 2P *+ 5 (op = pgo) A%
i - -
+ 5 (pf ng)Q + .062)+ (pfA)B AVfB = I ., (T’-l-)
4., TFuel salt mass balance for the fuel pump:
(1 - ao)a Avfs - (Vfo)s ma - (- ao)ea Avfg * (Vfo)9 mg = 0 . (75)
5. Gass mass balance for the fuel pump:
1- oo cxo(l - X) aVe, 1-a
o V + Lo+ —1 AP
- - &
go fo | K o (X - ao)g 8 RTO K a
8 8
1-a 1-a oaO(l-K)
+ ]l p ~ ANV, - oV - + —3 A0
o"go K Odo 8 go fol K ao (X - o) 1
8 o
aovfo 1 - Oio 1l - Odo
TR, K-a) YT | %% k-a | N T 0 (76)
25
6. Fuel salt mass balance at the junction of regions 1, 9, and 11:

oo = M - .0061h N. = 0 (1)

7. Gas mass balance at the junction of regions 1, 9, and 1l:

 

Lo g 1 - K Voo 17 9%
pgo K-« Ayfg * pgovfo Z| Mdo + RT K «
oo (K - o) 0 o
o S
Vv l -« v 1 -«
fo o _ fo 0
k- o .0061L T " o AP,
0 of, 0 of
1 -« 1 - K
- .0061k4 Peo K = o AVery ™ .0061k4 Pao Vfo Z 11
o (K - o)
11 o 11
1l -« 1 -K
- ip = AV - 0 I = 0 (78)
go K o f1 go fo (X ao) 1

1

8. Fuel salt mass balance at the junction of regions 6, 8, and 12:

Mo - N, = 12T N, = O (79)

9. Gas mass balance at the junction of regions 6, 8, and 12:

‘_l
§
Q
<
I
Q

0 1l - K fo 0
0 - AV 0 Do+ ~
go K a fa go fo (K - o )2 8 RTD K a
o
8
Yoo 1%\ | (Yo IT%) g o %) 4 -
RTO K - RTO K - Qb 6 go K - ab fe
lo.

13

ll.

26

 

1l - K L- ao
P N - 127 o — AV
go fo (K - o )2 6 go K a f
s =
1 - K
L27 Jp, Vo, T M. = O
go fo (K - o )2 12

o
12

 

 

Volume balance for the fuel-pump bowl:
10,541 Poo
MM + 0 — + (C A)2 —£0 + (C A)
fi13 fi13 (RTO)13 g /11 mgo g
11
o o 7085
€ o]+ M\ - my
14 g0 & go 0’13
14 15
, P - (P
] e "0 (B, - (B) .
Pfrs Vel g, m (Bm T )
go g0 0711
11
P (C A)2 (PO);B ] (PO)8 - *_pg.g__
m
fiz Vf /a2 (ngOTo)lg go /.,
(c A)g pgo . (Po)14 - (PO)lB
Pris VYo /s | fm (Bm T )
go g0 0714
14
P - (P
Pris Vg 1s] | m (Rm T )
go go 0’18
15
Void fraction relation between regions 1 and 9:

Aag ~ Aal = 0

1z

&

Pgo

l12in

AP

1

go

13

iz

(80)

(81)

(82)
27

12, Void fraction relation between regions 9 and 11:

oy - bo o= 0 (83)

13. Void fraction relation between regions 6, 8, and 12;

 

 

o
(A(1 - Oéo)]lz (Qb)a Tk -oab 1o (1 - Odo)ea Ayfle
odO
+ [A(1 - ) 1. (oxo)8 "l . (L - 050)8 Vo
a (K - 1) 1 - o
* (Avfo)lg (1 - ao)a (X - o )2 T K- a - (Qb)a AalE
O 12
aO(K - 1) 1-a L
+ (AVfO)6 (1 - o), ——-—--—-——(K T " F oo o). - (o)) 700, = O (84)
0O

1L, Fuel salt velocity relationship between regions 1 and 6 (from

original hydraulic model transfer function):
Npg = 0y &Vp) = b - b AP = 0 (85)

15. Void fraction relationship between regions 1 and 6 (from
original hydraulic model transfer function):
Mot = b WV - bezml b AP = 0 (86)

16. Pressure relationship between regions 1 and 6 (from original

hydraulic model transfer function):

AP, - b3lfiyf1‘ b32Aa1 b AP = O (87)
28

The solution of the resulting 16 equations was carried out by
*
first writing them in matrix form as follows:
AY = C (88)

where

=<
i

the column vector consisting of the 16 perturbed variables,

ordered as follows

b
1

[AO!B &0 APl AP /_wfa APlB AMf13 AVfg AVfl Aocl Avfs AO(S

T
A’Vfll Aall Aalz Ayflz I

where T is the transpose;

A = the coefficient matrix consisting of the constants in the
16 equations, and ordered identically to Y,
and
C = the forcing function column vector consisting of the elements

making up the right-hand sides of the 16 equations.

Before the solution of these equations could be formally carried
out however, it was necessary to consider the forcing function vector C
in Eq. (88).

Referring to Egs. (73), (74), and (81), pages 23, 2L, and 26,
respectively, it is noted that the variables involved in vector C are the
perturbations in the helium inlet pressure, AP,4; perturbations in the
helium outlet pressure, AP;.; and the fuel-pump input (head delivered)

fluctuation, F.

 

*
Examples showing how some of the constants in the A and C matrices

were obtained are given in the Appendix, Specific equaticns of the ex-

tended model are discussed, and a table of the constants is presented.
29

From this it is seen that the extended hydraulic model postulates
that the helium system is but one natural source or cause of pressure
changes in the circulating fuel system; fluctuations in input (head de-
1ivered) from the fuel pump also contribute to the naturally occurring
pressure perturbations around the loop.

The solution obtained in this work was arrived at by specifying
these varisbles individually. That is, one variable was given a value and
the other two were made identically equal to zero. In this way the effect
of one variable on pressure perturbations in the circulating fuel system
was separated from the effects of the others.

Furthermore, this method permitted the use of the concept of a
purposely perturbed system. For example, experimental data were obtained
by introducing a train of sawtooth pressure pulses into the fuel-pump
bowl and analyzing the resulting changes in the neutron flux.'® The
capability of introducing a similar kind of pressure perturbation into
the extended hydraullic model and then using the neutronic model to calcu-
late the effects on the neutron flux exists. Thus, a comparison of
theoretical and experimental resulis is possible,

Once the elements of the A and C matrices were determined,
standard computer techniques® for solving complex matrix equations were
applied to Eg. (88) to obtain the desired values of the 16 variables in

vector Y.
CHAPTER 3

RESULTS OF CAICULATIONS MADE WITH THE EXTENDED MODEL

COMPARED TO EXPERIMENTAL DATA

As stated previously in Chapter 2, page 29, the solution to the
matrix equation representing the extended hydraulic model, Eq. (88),
page 28, was accomplished by assigning values to the forcing function
variables one at a time,

Two cases were considered:

Case 1

The fuel pump forcing function variable, F, in Eq. (Th), page 2k,
was set equal to unity; the fuel-pump bowl helium inlet and outlet pres-
sure fluctuations, AP;, and AP,s, respectively, in Eg. (73), page 23,

and Eq. (81), page 26, were set equal to zero, i.e.,

F = 1.0, (89)
flPlg__ = 0.0 3 and 7 (90)
APis = 0.0 (91)

Case 2

The fuel-pump bowl helium inlet pressure fluctuation, AP;4, was
made equal to unity; the fuel-pump Powl helium outlet pressure fluctuation,
AP{s, and the fuel-pump forcing function variable, F, were sel equal to

zero, 1l.e.,

AP14 = 1.0, (92)
APy = 0.0, and (93)
F = 0.0 . (9k)

30
31

The fuel-pump bowl helium ocutlet pressure fluctuation, AP;s, is
zero in both cases because of the assumption that the helium system dis;
charge pressure, P;s5, is constant (atmospheric),

The reason F and APy4 are unity in Case 1 and Case 2, respectively,
is the result of the way in which the frequency response of a system is
computed. The frequency response of a stable system can be directly ex-
pressed by an equation relating the Fourier transform, F, of the system

output to the input:i?

F foutput
F{ Toputf = Frequency response. (95)

To obtain the neutron flux-to-pressure frequency response of the
system described by the combined hydraulic-neutronic model, it was neces-
sary to divide the outputs (left-hand sides of the equations) by the in-
puts (forcing function variables) for each case where the variables were
non-zero,

In Case 1, for example, dividing Eq. (74), page 24, by the fuel-
pfimp forcing function variable, F, makes the right-hand side unity. Thus
the element of the forcing function vector C in Eg. (88), page 28, corre-
sponding to F becomes unity, and all other elements of C are zero. 1In
Case 2, the element of vector C corresponding to AP,4 becomes a constant,
and all other elements of vector C are zero.

In this way it was possible to obtain information about the fre-
quency response even though the absolute values of the forcing function
variables were not known. In addition to information about the frequency
response, resulis concerning the modulus of the pressure in the fuel-pump

bowl, AP,=, were also determined for Cases 1 and 2.
32

Even though this caleculational procedure did not produce absolute
values for the frequency response and fuel-pump bowl pressure modulus,
the results can still be compared to the experimental data.

The pertinent experimental data and the bases for comparison with
calculated results are as follows:

1. Pressure Power Spectral Density Datal® for the MSRE

These data represent measurements of the naturally occurring
pressure noise in the fuel-pump bowl during normal reactor operation.

The pressure power spectral density is proportional to the fuel-
pump bowl pressure freguency response function modulus squared if the
driving function is '"white,” e,g., random and Gaussian.

Therefore, by postulating certain white driving functions, it is
possible to see if the resultant calculated fuel-pump bowl pressure power
spectral density curve is similar in shape compared with that which has
been observed.

2. The Modulus of the Neutron Flux-to-Pressure Frequency Response

Function for the MSRE

These data represent measurements taken during the saw-tooth pres-
sure pulse tests mentioned in Chapter 2,

In Fig. 5, the square of the modulus of the pressure in the fuel-
pump bowl calculated for Cases 1 and 2 is compared with the experimental
pressure power spectral density data. The units of the square of the
pressure modulus are indicated as being arbitrary because the results
computed by the extended model were scaled for comparison purposes.

The calculated square of the fuel-pump bowl pressure modulus for

Case 2 (APy, = 1.0, F = 0.0, AP15 = 0.0) over the range of frequencies
10'
@
g

5 5
ey
2
E
8

2

9 100
-
=2
o
g

5
s
@
o
oy
[12]
W
x

o 2
-
=
Q
m

a 1o
s
-
a
4

W 5
w
U
o
Ll
o

< 2
o
o
W)

10-2

5

2

10~3

0
FIGURE 5.

 

33

CORNL-DWG 70-1374

LEGEND

A CASE
5 |- © CASE 2
V EXPERIMENTAL DATA

0.5 10 1.5 20 2.5 3.0 35

FREQUENCY (cps)

Square of the Modulus of the Pressure in the Molten Salt
Reactor Experiment Fuel-Pump Bowl

Case 1: F=1,0, APy4 = 0.0, APy = 0.0
Case 2: APiy4 1.0, APy 0.0, F = 0.0.

I
I
3L

shown in Fig, 5 agrees reasonably well with the general trend of the ex-
perimental data, also given in Fig. 5, for the same range of frequencies.

The computation of the square of the fuel-pump bowl pressure
modulus for Case 1 (F = 1.0, AP14 = 0.0, AP35 = 0.0) gave results
that are also shown in Fig. 5. This curve showed fluctuations mofe simi-
lar to those actually observed, except in the frequency range of about
0.5 to 2,0 ecps. Here it appears that the extended hydraulic model ex-
hibits extreme under-damping or resonance effects.

From the comparison of results shown in Fig. 5, it seems reasonable
to postulate that the fuel-pump bowl pressure modulus is basically related
to the fuel-pump bowl helium inlet pressure perturbations. Superimposed
on this relationship, however, are the random effects of the fuel pump
input fluctuations.

Congideration of the information presented in Fig. 5 thus leads
to the expectation that pressure fluctuations in the circulating fuel
system of the MSRE are also a result of the combination of these two per-
turbations, It appears that the rescnant structure due to the fuel-pump
input perturbation dominates the effect of the fuel-pump bowl helium in-
let pressure perturbation, however,

Other experimental evidence to support this conclusion is available:
It was noted that when the fuel-pump bowl offgas line became plugged and
the fuel-pump bowl pressure remained essentially constant over a long
period of time, the pressure noice in the fuel-pump bowl increased ap-
preciably, but the neutron nolse remained essentially constant.”
The modulus of the neutron flux-to-pressure frequency response

function for Case 2 (APy4 = 1.0, F = 0.0, APy = 0.0) is shown in -
35

Fig. 6 for two different fuel-pump bowl pressures. Also shown is the
modulus computed by the original open-loop mathematical model., The
modulus of the neutron flux-to-pressure frequency reéponse function de-
termined from experimental results'? obtained from a series of saw-tooth
pressure pulse tests is included for comparison.

Units of the modulus of the neutron flux-to-pressure frequency
response function are shown as being arbitrary because the results cal-
culated by the mathematical models were scaled for the purpose of com-
parison,

The open-loop model results more nearly approximate the experi-
mental data, but both mathematical models appear to be attenuated too
rapidly at frequencies above about 0.1 cps. Unfortunately there are no
experimental data below about 0.02 cps or above about 0.5 cps with which
to compare the two models,

The effect of different fuel-pump bowl pressures on the modulus
of the neutron flux4t0~pressure frequency response function calculated
by the extended model is apparent in Fig. 6. It is noted that the higher
fuel-pump bowl pressure gave a greater amplication of the pressure

perturbation.
36

ORNL-DWG 70-1375
107!

w

AN/No
AP

10°2

> 5

€

3

o

2 LEGEND

2 CASE 2, Py3 = 22.7 psia
= 2 CASE 2, Pz = 20.6 psia

ORIGINAL MODEL
EXPERIMENTAL DATA

1073

MODULUS OF THE NEUTRON FLUX-TO-PRESSURE FREQUENCY RESPONSE FUNCTION,

 

1074
0.01 0.02 005 0.1 0.2 05 1.0

FREQUENCY (cps)

FIGURE 6. Modulus of the Neutron Flux-to-Pressure Frequency-Response
Function for the Molten Salt Reactor Experiment
CHAPTER L
CONCIUSTIONS AND RECOMMENDATIONS

The hydraulic portion of the mathematical model originally formu-
lated by Robinson and Fry was extended by explicitly including the fuel
pump and fuel-pump bowl to close the loop.

The extended model was then combined with their original neutronic
model and used to perform calculations yielding results such as the square
of the modulus of the pressure in the fuel-pump bowl, and the modulus of
the neutron-flux-to-pressure frequency response function for the MSRE,

Results calculated by the extended and open-loop models were com-
pared to those obtained from experimental measurements to determine the
validity of the models.,

It was found that the extended model in its present form cannot
calculate the modulus of the pressure in the fuel-pump bowl in an absolute
sense, On the other hand, examination of this parameter as calculated by
the extended model lead to some interesting qualitative conclusions which
are discussed below,

Fluctuations in pressure, and thus in neutron flux, in the circu-
lating fuel system of the MSRE result from perturbations inserted by the
fuel pump and the helium cover-gas system.

The effects of those perturbations introduced by the fuel pump are
dominant, however, and are responsible for the resonant structure seen in

the square of the modulus of the pressure in the fuel-pump bowl shown in

Fig. 5, page 33.

37
38

The modulus of the neutron-flux-to-pressure frequency response
function calculated by the original open-loop model agrees much better
with the experimental results than does that calculated by the extended
model.

The large resonances in the modulus of the neutron flux-to-pressure
frequency response function noted in Fig. 6, page 36, indicate that the
extended model is not sufficiently damped,

It was further found that the extended model also cannot calculate
the frequency response of the MSRE absolutely.

Tt was stated earlier (page 35) that a scale factor was applied
to the modulus of the neutron flux-to-pressure frequency response function
calculated by the extended model in order to compare it to the experi-
mental data in Fig. 6, page 36. The use of this scale factor was necessary
because the calculated modulus was much smaller than the experimentally
determined modulus. This suggests that there is too much attenuation
(pressure drop) in the extended model between the pressure in the fuel-
pump bowl and the core.

Similarly, the sguare of the modulus of the pressure in the fuel-
pump bowl was much smaller than the experimental results shown in Fig, 5,
page 33. This indicates that there was also too much attenuation in the
portion of the loop (fuel pump and fuel-pump bowl) added by the extended
model,

The following recommendations, based on the above conclusions, are
made with the hope that they might prove helpful in improving the extended

model, or in future modeling of similar hydraulic systems.
39

Although the extended model is not able to generate absolute values
for the modulus of the pressure in the fuel-pump bowl and the modulus of
the neutron flux-to-pressure frequency response function, it could be put
to use in the following way. By normalizing the results to give correct
values based on experimental data, the normalized model could then be
applied in performing parameter studies on the MSRE circulating fuel system.

The attenuation in the portion of the loop between the fuel pump
and fuel-pump bowl must be reduced. In this case, it appears that the
choice of orifice-type equations to represent the bypass flow restrictions
at the pump discharge and pump suction was not a sound assumption.
Furthermore, the use of separate orifice-~type equations for the gas and
fuel salt was also unfortunate., The 1 to 2 volume percent of gas in the
liguid observed returning to the impeller during pump tests could be an
important effect.

These observations are pointed up by the good agreement of the re-
sults obtained from the original model with the experimental data, In
place of orifice-type equations to close the loop, the original model
used as boundary conditions (1) pressure fluctuations at the exit of
region 6, Fig., 1, page 2, are the same as inlet pressure fluctuations to
region 1, and (2) the fluctuations in the total mass flow rate to
region 1 was zero.

A better selection forrthe extended model would have been to use
a momentum balance in the fuel-pump bowl combined with orifice-type
equations which accounted for momentum transfer between the gas and fuel
salt in the bypass loop. These equations would replace the separate gas

and fuel salt orifice relationships that were applied.
Lo

Another recommendation that may be beneficial is to remove the
assumption that there is no holdup volume in the fuel pump, i.e,, consider
the pump to represent a well-mixed volume, This would help account for
the fact that mass can be stored in the fuel pump as it is in the fuel-
pump bowl,

Finally, the neutronic portion of the model, although not directly
addressed by this thesis, could be improved by simulating the effects of

the heat exchanger on the circulating fuel system.
LIST OF REFERENCES
lo.

11,

12,

13.
1k,

J. C.

LIST OF REFERENCES

Robinson and D, N. Fry, '"Determination of the Void Fraction in
the MSRE Using Small Induced Pressure Perturbations,' USAEC
Report ORNL-TM-2318, Oak Ridge National Laboratory (February 6,

1969),

Robinson and Fry, ORNL-TM-2318, pp. 18 - 21.

Robinson and Fry, ORNL-TM-2318, pp. 6 - 10,

L. G.

Neal

Neal and S. M, Zivi, "Hydrodynamic Stability of Natural Circu-
lation Boiling Systems," Volume 1, "A Comparative Study of Ana-
lytical Models and Experimental Data," USAEC Research and De-
velopment Report, STL 372-1h4 (1), Thomson Ramo Woolridge, Inc.,
(June 30, 1965).

and zivi, STL 372-14 (1), p. 85.

. Ball and R. K. Adams, "MATEXP, A General Purpose Digital Computer

Program for Solving Ordinary Differential Equations by the Matrix
Exponential Method, " USAEC Report ORNL-TM-1933, Oak Ridge
National Iaboratory, (August 1967).

. Robertson, "MSRE Design and Operations Report, Part I, Description

of Reactor Design," USAEC Report ORNI-TM-728, pp. 133 - 1k2, 325,
(Tanuary 1965).

. Bird, W. E. Stewart, and E, N, Lightfoot, Transport Phenomena,

p. 226, John Wiley and Sons, Inc., New York, 1960.

. Funderlich, Editor, Programmer's Handbook, USAEC Report K-1T729,

"CMATEQ, Program ORD-9067, Gaussian Elimination for Real and
Complex Systems," (February 9, 1968).

Robinson and Fry, ORNL-TM-2318, p. 22.

M. A.

Schultz, Control of Nuclear Reactors and Power Plants, Second
Edition, p. 70, McGraw-Hill Book Company, New York, 1961.

D. N. Fry, personal communication to W, C. Ulrich, November 26, 1969,

D. N. Fry, personal communication to W. €. Ulrich, November 26, 1969.

J. C.

Robinson, personal communication to W. C. Ulrich, November 26,
1969. (To be published in Nuclear Science and Engineering.)

Lo
APPENDIX
APPENDIX

The various constants in the A and C matrices in Eq. (88), page 28,
were generated using information obtained from three sources:

1. observable gquantities such as system temperatures, pressures,
areas, and volumes;

2. physical properties, such as fuel salt density, based on con-
ditions described by the observable guantities; and

3. calculafions of other constants using flow rates, pressures,
velocities, and void fractions obtained from the solution of
the original hydraulic model in conjunction with items 1 and 2.

The following examples show how the information described above

was applied to two specific equations of the extended hydraulic model.

Example 1 — Consider Eq. (72), page 23, the fuel salt mass balance
for the fuel-pump bowl:

a. An observable quantity is the temperature of the fuel salt

in regions 11 and 12, (Tfo)ll and (Tfo)lg’ respectively:
(Tfo)ll = 1670°R , (A1)
(Te) 1o T 1670°R . (A2)

The areas A,; and A,- were obtained from detail pump design
drawings, The area for region 11 was taken as the size of the connection
between the pump discharge line and the 2-in.-dia. pipe spray ring; the
area of the opening at the pump suction was calculated. Specifically,

these are

Ly
L5
Ay; = 0.48 in? | (43)

9.98 in% , (Ak)

i

Ao

As stated on page 9, the volumetric rate of flow through the by-
pass region is 60 gpm. Since the void fraction of gas is less than
1 percent at normal operating conditions, it was assumed that the bypass

flow consisted entirely of fuel salt. Thus

1 min 231 in?

. 3
5 ses X ) x 0.087 1b/ins

(mfo)ll €0 gpm X 6

20.1 1b/sec. (45)

Furthermore, with no change of mass of fuel salt in the pump bowl, the

mass rate of flow through region 12 equals that through region 11, or

(mfo)ll = (me)lg = 20.1 1b/sec. (A6)

b. The temperature in regions 11 and 12 determines the fuel salt

density; thus

Ppyq = 0.087 lb/il’l‘?’ s (AT)
Prin = 0.087 1b/in? . (A8)
c. The orifice coefficients C and C were calculated by
fia fiz

applying some of the above information to Eqs. (21) and (23), page 13,

as follows:
L6

From Eq. (21),

(meo)
f
c _ 0’11 ,

B A Ve, (o), - (BQ),,]

 

(A9)

where (Po)9 and (PO)13 are given in the solution obtained by the original

hydraulic model, namely

(P ) = 7T0.0 psia,
O g
P = 20.6 psia.
( 0)13
Therefore, Eq. (A9) becomes
20.1

 

 

f2a 0.48 2 x 0.087 x 49.k

Cfll

Il

14.6 inT% - gec™?

Similarly, Eq. (23) gives

(mfo)lz

AlE\/EDflE [(PO)13 - (PO)BJ

Cf12 =

 

where (Po) is also obtained from the original hydraulic model
8
(p) = 19.6 psia,
'8

Thus Eq. (AlLk) is now

20.1

 

iz 9.98 /2 x 0.087 x 1.0

2,76 in>% sec”™ ! ,

C
fie

(A10)

(A11)

(A12)

(A13)

(ALY

solution:

(A15)

(A16)

(817)
b7

Now all the numbers required to determine the constants in Eq. (72)
are available; substituting them into the equation and carrying out the

indicated operations will give the desired values,

Example 2 — Now consider Eq. (73), pages 23 and 2k, the gas mass
balance for the fuel-pump bowl,

a, The observable quantities for this example include:

- = - ° Al8

(1), = (2) = (T) _ = 1670°R, (418)
= = ° Al

(), = (T) _ = 530°R, (A19)

(P ) = 14,7 psia , (A20)

0" 1s

Veis = 3456 in? , (A21)

R = Wb635.4 in,/°R, (A22)

All = O.)-I-B in? 3 (A23)

A = 9.98 in? (A2k)

b. There are no physical properties such as the fuel salt density
which depend only on the observable quantities, That is, the gas density
is dependent also on the system pressure which must be obtained from the
original model solution, Therefore it is necessary to use the original
model to calculate the remainder of the constants,

c. The gas densities may be calculated by using Eq. (1), page k.

For example,

) = 2D, - (825)
L8

where (Po) is obtained from the original hydraulic model solution,
11

i.e.,

() . = 70.0 psia .
Thus Egq. (A25) becomes
o 70.0
(ogo),, = T835.% * 2670 -

9.24 x 107° 1b/in?

(p..)

fi

g0 11
Using the wvalues of

(P) = 27.7 psia (specified),

0 14
(Po)l = (Po)g = (Po)ll = T0.0 psia 3
(2 ) = 20.6 psia ,

0713

P = P = P = 1 .6 sia .
(B,), (B (). 9.6 p

from the solution by the original hydraulic model, the remaining gas

density terms can be calculated.

The gas mass flow rate for region 11 was calculated as follows:

The voild fraction in region 9 may be expressed as

pgg Pfo

(A26)

(427)

(428)

(429)

(430)

(A31)

(A32)

(A33)
49

But from Egs. (53) and (54), page 18, @ = « . Therefore,
1 9

solving for mgg, we get

G Mg
m = — —— . (A3L)
g° 1 Qa Prg g9

Now the fuel pump volumetric flow rate is 1200 gpm, all of which passes
through region 9; the bypass flow rate is 60 gpm, or 5 percent of this,
Because the pressures and temperatures of the gas in regions 9 and 11 are
assumed to be equal, the density of the gas in these two regions is also
the same, Therefore, it was assumed that 5 percent of the total gas

mass flow rate also flowed through the bypass, i.e.,

(M), = 0.05 (m) . (435)

Substituting Bq. (A34) into Eg. (A35) gives

oL m
0 fo

m = 0,05 {—/—/—/ —

( 80)11 1-al . \Pr (

A
] pgo)g , (A36)

where (Ob) is obtained from the solution presented by the original hy-
1
draulic model, Numerically, Eq. (A36) is

0.004 X 381L.4
- 0.004 & 0.087

I

(m ) x 9.2k x 107® (A37)

g0" 11

(m_.)

g0 11

0.05 x T

8.12 x 107° 1b/sec. (A38)

Furthermore, when there is no mass accumulation in the fuel-pump

bowl,

(mgo) = (mgo) - (A39)
50

The orifice coefficients for the gas are determined by using Eqgs,

(22) and (2L4), pages 13 and 1k, Thus

g1l

gl1

£11

Similarly,

glz

giz

C
giz

(m )

g0 11

 

 

0.48 /2(p,) _ [(R)) - (B)) ]

13

8.12 x 10~

 

 

0.48 /2 x 9.24 x 107° x ho.lL

5.61 x 10”% in7* sec -7

(m_.)

80712

 

 

9.982(0,) _ [(B) = (B)) ]

8.12 x 10

 

 

9.98 /2 x 2.6 x 107° x 1.0

3.57 x 1072 inT% sec™?

Similar calculations were made to obtain the remaining coef-

(ALO)

(A1)

(Ak2)

(Ak3)

(AkL)

(AL5)

ficients for the 16 eguations (Egs. (72) through (87)) which make up the

extended hydraulic model; the results are presented in Table Al and the

following list.
 

Constants Used to Calculate the Coefficients Used in the A and C Matrices for Eg. (88)

TABLE Al

 

 

Region

1 6 8 9 11 12 13 1k 15
Constant
A, in% -- 78.L 78.4 78.L 0.48 9.98 | -- 0.30k 0.30k4
Cpr inT* sec™? -- -- -- -~ 1.6 2.8 -- -- --
Cg’ in,”% sec™?t -~ -- -~ -- 5.6x107% 3,6x107°| -- | 1.5x10"%} 1.9x10"%
Mo s 1b/sec -- -- - -- 20.1 20.1 -- -- --
m_, 1lb/sec -- ~- -- -- 8.1x107°| 8.1x107° -- | 4.8x107°| L.8x107°
P , psia 70.0 19.6 19.6 70.0 T0.0 19.6 20.6 27.7 4.7
T, °R 1670 1670 1670 1670 1670 1670 1670 530 530
Veor 1n./sec 55.5 55.9 55.9 55.5 181 23.2 -- -- --
a, 4.0x1077| 1.2x10-%| 1.2x107%| 4.0x10">| L,ox10™>| 1.2x107° -- -- --
Ops 1b/in? -- -- 0.087 0.087 0.087 0.08710.087 -- --
o, 1b/in? 9.2x1078| 2.5x107%| 2.5x107%| 9.2x10"%| 9.2x107%| 2.5x107°| -- | 1.1x10-5] 7.2x10°°

 

 

 

 

 

 

 

 

 

 

 

15
52

The remaining constants are:

s = Jjw, where @ is specified;
H = 582 in.,
K = specified value between 0.8 and 1.0,
v = 3456 in?
g13 3% ’
and v = 6184 in? ,

fi3
-

\C o310V W

.

11.
12.
13.
1k,
15.
16.
17.

18.

19.
20.
21.
22,
23.
2k,

. - - » »
- .

-
-

-

»
.

U rrQUuGagaEs Dot N @Tn=S
Eron-aoaydm=a2dYitbwerpdaEAES g

0 a3

53

ORNL-TM-3007

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EXTERNAL DISTRIBUTION

 

S. J. Hanauer, University of Tennessee, Knoxville,
Tennessee 37916

J. E. Mott, University of Tennessee, Knoxville,
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303 Power Building, Chattanooga, Tennessee 3740l
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Division of Technical Information Extension, OR