\&3kfLy
HOME
MASTER
-t
w 2
:
HELP
O A K RIDGE N A T I O N A L LABORATORY operated by
UNION CARBIDE CORPORATION NUCLEAR DIVISION
al!iD
for the
c
U.S. ATOMIC ENERGY COMMISSION
ORNL- TM- 3007
A N EXTENDED HYDRAULIC MODEL OF THE MSRE CIRCULATING FUEL SYSTEM (Thesis)
W. C. Ulrich
Submitted to the Graduate Council of the University of Tennessee i n partial fulfillment for the degree of Master of Science. DfSTRlBUTfON OF THIS DOCUmNT IS UN-
4
.
- -~
LEGAL NOTICE
-
--
T h i s 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.
Mokes any warranty or representation, expressed completeness, or ony information,
or implied, w i t h respect t o the occurocy,
usefulness of the infarmotion contained i n t h i s report, or that the use of apparatus,
method,
or process disclosed i n t h i s report may not infringe
privately owned rights; or
8 . Assumes any l i a b i l i t i e s w i t h respect t o the use of, or for damages resulting from the use of any information, opporotus, method, or process disclosed i n t h i s report.
As used i n the above, "person acting on behalf of the Commission"
includes any employee or
contractor of the Commission, or employee of such contractor, t o the extent that such employee or contractor
of the Commission,
or employee of
such controctor prepares,
disseminates,
or
provides access to, any information pursuant t o h i s employment or contract w i t h the Commission, or h i s employment w i t h such contractor.
ORNL-TM- 3007
Contract No. W-7405-eng-26
REACTOR DIVISION
AN EXTENDED HYDRAULIC MODEL OF THE MSRE CIRCULATING FUEL SYSTEM
W. C . U l r i c h
Submitted t o t h e Graduate Council of t h e U n i v e r s i t y o f Tennessee i n p a r t i a l f u l f i l l m e n t for t h e degree of Master o f Science.
JUNE 1970
OAK R I D G E NATIONAL LABORATORY O a k Ridge, Tennessee
o p e r a t e d by UNION CARBIDE CORPORATION for the U. S. ATOMIC ENERGY COMMISSION
ACKNOWLEXEEMENTS The a u t h o r i s i n d e b t e d t o Dr. J. C . Robinson of t h e U n i v e r s i t y of Tennessee f o r t h e o v e r a l l guidance he provided on t h i s work.
The t e a c h i n g
s k i l l , p a t i e n c e , and h e l p he o f f e r e d a t c r i t i c a l p e r i o d s i s g r e a t l y apprec iated. Thanks a r e due a l s o t o Dr. J. E. Mott of t h e U n i v e r s i t y of Tennessee
for s e v e r a l v a l u a b l e s u g g e s t i o n s which he made d u r i n g t h e development of t h e h y d r a u l i c model. The a s s i s t a n c e of t h e management of Oak Ridge N a t i o n a l Laboratory
*
i n g r a n t i n g t h e a u t h o r t h e n e c e s s a r y l e a v e of absence t o pursue h i s s t u d i e s , and i n f u r n i s h i n g t h e u s e of computer f a c i l i t i e s , g r a p h i c
a r t s , and r e p r o -
d u c t i o n s e r v i c e s i n connection w i t h t h e p u b l i c a t i o n of t h i s t h e s i s i s g r a t e f u l l y acknowledged. The a u t h o r a l s o wishes t o t h a n k t h e United S t a t e s Atomic Energy Commission f o r t h e T r a i n e e s h i p he w a s awarded t h a t enabled him t o undert a k e t h e academic work of which t h i s t h e s i s i s a p a r t .
It i s a p l e a s u r e t o t h a n k Mrs. Annabel Legg f o r t h e e f f i c i e n t , c a r e f u l way i n which she typed t h e manuscript. L a s t l y , t h e a u t h o r wishes t o t h a n k h i s w i f e for her s u p p o r t , unders t a n d i n g , and encouragement d u r i n g w h a t must have been a d i f f i c u l t t i m e f o r her.
Without it, none o f t h i s would have been p o s s i b l e .
*Operated
f o r t h e United S t a t e s Atomic Energy Commission by Union Carbide C o r p o r a t i o n .
iii
ABSTRACT The h y d r a u l i c p o r t i o n of a combined h y d r a u l i c - n e u t r o n i c mathem a t i c a l model f o r determining t h e e f f e c t s of helium gas e n t r a i n e d i n t h e c i r c u l a t i n g f u e l s a l t of t h e Molten S a l t R e a c t o r Experiment on t h e neutron f l u x - t o - p r e s s u r e frequency response w a s extended t o i n c l u d e e f f e c t s due t o t h e f u e l pump and helium cover-gas system. The extended h y d r a u l i c model w a s combined w i t h t h e o r i g i n a l neut r o n i c model and programmed f o r computations t o b e made by a high-speed d i g i t a l computer.
By comparing t h e computed r e s u l t s w i t h e x p e r i m e n t a l data, it was concluded t h a t p r e s s u r e p e r t u r b a t i o n s introduced b y t h e f u e l pump were t h e main s o u r c e of t h e n a t u r a l l y o c c u r r i n g n e u t r o n f l u x f l u c t u a t i o n s i n t h e frequency range of one t o a few c y c l e s per second.
It w a s a l s o noted t h a t t h e amplitude o f t h e n e u t r o n f l u x - t o p r e s s u r e frequency-response f u n c t i o n was d i r e c t l y p r o p o r t i o n a l t o t h e p r e s s u r e i n t h e fuel-pump bowl; however, f u r t h e r work w i l l be r e q u i r e d b e f o r e completely s a t i s f a c t o r y r e s u l t s a r e o b t a i n e d from t h e extended model.
Recommendations a r e proposed which should prove u s e f u l i n f u t u r e
modeling of similar h y d r a u l i c systems.
iv
TABU OF CONTENTS PAGE
CHAPTER
........................... 1. DESCRIFTION OF THE OPEN-LOOP MODEL . . . . . . . . . . . . .
II!I"RODUCTION
2
.
DESCRIFTION OF THE EXTENDED MODEL
..............
1
4 9
P h y s i c a l C h a r a c t e r i s t i c s of t h e F u e l Pwnp and Fuel-Pump Bowl
3.
.....................
..... i n Deriving t h e Equations f o r t h e Model . .
9
Assumptions Made i n t h e Development of t h e Model
11
Procedures Used
14
RESULTS OF CALCULATIONS MADE WITH THE EXTENDED MODEL
............... 4. CONCWSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . . . LIST OF REFEFd3NCES . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . COMPARED TO EXPERIMENTAL DATA
V
30
37
41 43
LIST OF FIGURES PAGE
FIGURE 1.
Model Used t o Approximate t h e Molten S a l t Reactor Experiment F u e l S a l t Loop
2.
..................... ..................
10
12
Square of t h e Modulus of t h e P r e s s u r e i n t h e Molten S a l t Reactor Experiment Fuel-Pump Bowl; Case 1: F AJ?14 =
0.0, a P I 5 = 0.0; Case 2 :
AP14
=
=
1.0,
1.0, aP15 = 0.0,
33
F = O . O . . . . . . . . . . . . . . . . . . . . . . . . .
6.
3
Hydraulic Performance o f t h e Molten S a l t Reactor Experiment F u e l Pump
5.
................
Molten S a l t Reactor Experiment F u e l Pump and Fuel-Pump Bowl.
4.
2
Extended Hydraulic Model of t h e Molten S a l t Reactor Experiment Fuel S a l t Loop
3.
................
Modulus of t h e Neutron Flux-to-Pressure Frequency-Response Function for t h e Molten S a l t Reactor Experiment
vi
.....
36
LIST OF SYMBOLS
Description
Symbol A
area
cf
C
g
‘i
orifice coefficient for fuel salt o r i f i c e c o e f f i c i e n t f o r gas c o n c e n t r a t i o n of i t h ’ group of delayed neutron p r e c u r s o r s
‘k
conversion c o n s t a n t from f i s s i o n r a t e t o t h e d e s i r e d u n i t s f o r power d e n s i t y
C
h e a t c a p a c i t y of t h e moderator a t c o n s t a n t p r e s s u r e
PM
D
neutron d i f f u s i o n c o e f f i c i e n t
F
f u e l pump f o r c i n g f u n c t i o n (head d e l i v e r e d )
g
a c c e l e r a t i o n due t o g r a v i t y
gC
gravitational constant heat transfer coefficient
h
hf
e n t h a l p y of t h e f u e l s a l t
h
e n t h a l p y of t h e gas
H
f u e l pump head a t normal o p e r a t i n g c o n d i t i o n s
54
thermal c o n d u c t i v i t y of t h e moderator
K
d i s t r i b u t i o n , o r flow, parameter f o r two phase f l o w
m
mass f l o w rate
m
f u e l s a l t mass flow rate
g
f
m
Q
M Mf
gas mass flow r a t e mass mass of f u e l s a l t
v ii
viii
Symbol
M
g
0
PI1
PW
P
Description mass of gas
o r i g i n a l or s t e a d y - s t a t e c o n d i t i o n ( s u b s c r i p t ) heated p e r i m e t e r we t t e d p e r i m e t e r pressure heat flux
Q
f u e l pump v o l u m e t r i c flow r a t e
Q
power d e n s i t y
R
u n i v e r s a l gas c o n s t a n t
S
Laplace-transformed time v a r i a b l e
S
s l i p v e l o c i t y r a t i o ( r a t i o of gas v e l o c i t y t o f u e l s a l t velocity)
t
tine variable
T
a b s o l u t e temperature
Tf
f u e l s a l t temperature
T
gas temperature
...
63
TM Tw Uf U
g
v
f
v
g
vf
v n'
g
moderator temperature temperature of wetted p e r i m e t e r ( w a l l t e m p e r a t u r e ) i n t e r n a l energy f o r t h e f u e l s a l t i n t e r n a l energy f o r t h e gas volume of f u e l salt volume of gas v e l o c i t y of t h e f c e l s a l t v e l o c i t y of t h e gas v e l o c i t y of t h e n e u t r o n s (one group)
ix Symbol
Description space v a r i a b l e space v a r i a b l e gas void f r a c t i o n delayed neutron f r a c t i o n delayed neutron f r a c t i o n f o r t h e i t h group f r a c t i o n of t h e " u n i t c e l l " power d e n s i t y g e n e r a t e d i n the f u e l s a l t d e v i a t i o n about t h e mean, i n c r e m e n t a l q u a n t i t y , o r p e r turbed quantity decay c o n s t a n t f o r t h e i t h group of delayed n e u t r o n precursors number of neutrons produced p e r f i s s i o n d e n s i t y of t h e f u e l s a l t d e n s i t y of t h e gas d e n s i t y of t h e moderator a b s o l u t e v a l u e of t h e s l o p e of t h e f u e l pump head c a p a c i t y curve a t t h e normal v o l u m e t r i c f l o w r a t e of t h e pump macroscopic neutron a b s o r p t i o n c r o s s s e c t i o n macroscopic n e u t r o n f i s s i o n c r o s s s e c t i o n t o t a l w a l l shear s t r e s s due t o f r i c t i o n neutron f l u x f u e l pump bowl volume t h e Laplacian, o r d i f f e r e n t i a l ,
Superscript
-
-
average
operator
IIYIROIIJCTION
An a n a l y t i c a l model f o r determining t h e v o i d f r a c t i o n of helium c i r c u l a t i n g i n t h e Molten S a l t R e a c t o r Experiment (MSRE) f u e l s a l t l o o p by r e l a t i n g n e u t r o n f l u x t o p r e s s u r e u s i n g frequency response t e c h n i q u e s w a s f o r m u l a t e d by Robinson and Fry.'
The h y d r a u l i c p o r t i o n of t h i s com-
b i n e d n e u t r o n i c - h y d r a u l i c a n a l y t i c a l model w a s n o t complete, however, i n t h a t it d i d n o t c o n t a i n a s p e c i f i c r e p r e s e n t a t i o n o f t h e f u e l pump and
fuel-pump bowl.
(See F i g . 1.) Omission of t h e s e two items r e s u l t e d i n
a n open-loop h y d r a u l i c system which was c l o s e d by a p p l y i n g boundary cond i t i o n s * which approximated t h e i r e f f e c t s on t h e system. The scope or purpose of t h e work d e s c r i b e d below i n c l u d e s (1) ext e n d i n g t h e h y d r a u l i c p o r t i o n of t h e o r i g i n a l model by e x p l i c i t l y i n c l u d i n g t h e f u e l pump and fuel-pump bowl t o c l o s e t h e loop (see F i g . 2 ) , ( 2 ) combining t h e extended h y d r a u l i c model w i t h Robinson and F r y ' s o r i g i n a l
n e u t r o n i c model t o c a l c u l a t e t h e frequency response of n e u t r o n f l u x t o p r e s s u r e , and (3) a t t e m p t i n g t o v a l i d a t e t h e mathematical models by comp a r i n g r e s u l t s o b t a i n e d from experimental measurements w i t h p r e d i c t i o n s made by t h e two models.
1
ORNL- DWG 68 - 8 4 1 7 OUTLET
f
L
UPPER PLENUM
L,
-if
t
Y-
t t CHANNELED REGION
L4
L
t LOWER PLENUM
FIGURE 1. Model Used t o Approximate t h e Molten S a l t Reactor Experinent F u e l S a l t Loop
3 ORNL- DWG 70-1376
He
@
@
He
UPPER PLENUM
-f
t
Y
t t t CHANNELED
REGION
9 FIGURE 2 .
I f f 3 + LOWER PLENUM
Extended Hydraulic Model of t h e Molten S a l t Reactor Experiment Fuel S a l t Loop
CHAPTER 1
DESCRIPTION OF THE OPEN-LOOP MODEL3 The technique f o r determining t h e v o i d f r a c t i o n i n t h e MSRE c i r c u l a t i n g f u e l s a l t c o n s i s t s of a n a l y z i n g f l u c t u a t i o n s i n t h e n e u t r o n f l u x s i g n a l caused by induced p r e s s u r e f l u c t u a t i o n s i n t h e fuel-pump bowl. I n t h e development of t h e a n a l y t i c a l model, t h e c o m p r e s s i b i l i t y of t h e e n t r a i n e d g a s was p o s t u l a t e d as t h e mechanism having t h e g r e a t e s t e f f e c t on t h a t r e a c t i v i t y induced by p r e s s u r e p e r t u r b a t i o n s .
The primary
governing e q u a t i o n s a r e , t h e r e f o r e , t h e e q u a t i o n s of s t a t e , c o n s e r v a t i o n of t h e mass of t h e gas, of mass of t h e f u e l s a l t , of momentum, of energy,
of neutrons, and of delayed neutron p r e c u r s o r s .
In particular, with the
*
assumption of one- dimensional flow, t h e governing e q u a t i o n s are : Equation of s t a t e f o r t h e gas,
%
P/RT.
=
Conservation of mass for t h e g a s ,
Conservation of mass f o r t h e f u e l s a l t ,
4
5 W
Conservation of momentum f o r t h e g a s - s a l t mixture,
aat c P f v f ( l
- a)
+
- gc
Pv g g
aP
-
+
a Ipfvf2(1 pW
T~
A
- bf(l - a)
The assumed r e l a t i o n s h i p between V
where S, t h e s l i p r e l a t i o n s h i p ,
=
SVf
+
p v2aj
g g
pga~ g.
=
(4)
is
and V
Q
f
vQ
a) +
,
(5)
i s given by5
s = 1 - a K-cx
.
Conservation of energy i n t h e s a l t - g a s mixture,
A "h + j i i .
Conservation of energy i n t h e g r a p h i t e moderator,
(7)
6 Coulomb's l a w of c o o l i n g :
q
Power dens i t y
=
h(T
W
-
Tf).
(9)
,
Conservation of n e u t r o n s (one-group d i f f u s i o n model),
= v - D V @ + [v(l
at
- B)cf
-
+),
cal@
c
Xi C i
.
1=1
Precursor balance equations,
f o r i = l , 2,
..
.
.6.
S i n c e t h e i n t e r e s t i s i n small d e v i a t i o n s a b o u t s t e a d y s t a t e , it w a s assumed t h a t l i n e a r i z e d r e p r e s e n t a t i o n of t h e governing e q u a t i o n s (1)
through ( 1 2 ) would a d e q u a t e l y d e s c r i b e t h e system.
It w a s f u r t h e r assumed
t h a t v e l o c i t y f l u c t u a t i o n s would n o t s i g n i f i c a n t l y a f f e c t t h e p r e c u r s o r b a l a n c e , and t h a t t h e f l u c t u a t i o n s i n t h e d e n s i t y of t h e gas a r e proport i o n a l t o fluctuations i n the pressure.
T h i s l a t t e r assumption i s based
on t h e l i n e a r i z e d v e r s i o n of Eq. (l), i . e . ,
The l a s t term of Eq. (13) w a s dropped because of t h e l a r g e r - e n e r g y i n p u t necessary t o change t h e temperature compared t o t h a t r e q u i r e d t o change the pressure.
7 With t h e assumptions s e t f o r t h above, t h e l i n e a r i z e d e q u a t i o n s g e n e r a t e d from Eqs. (1)through (6) can be s o l v e d independently of t h o s e o b t a i n e d from Eqs. (7) through ( 1 2 ) .
The former s e t of e q u a t i o n s i s r e -
f e r r e d t o as t h e h y d r a u l i c model and t h e l a t t e r s e t as t h e n e u t r o n i c model. The dependent v a r i a b l e s i n E q s . (1) through (6) a r e Vf, Vg, P, and S.
a,
pg,
T h i s s e t can be reduced t o a s e t of t h r e e coupled d i f f e r e n t i a l
e q u a t i o n s w i t h t h r e e dependent v a r i a b l e s i n t h e i r l i n e a r i z e d v e r s i o n . The dependent v a r i a b l e s r e t a i n e d i n these s t u d i e s were AV
f’ La, and AP.
Therefore, t h e e q u a t i o n s d e f i n i n g t h e h y d r a u l i c model were transformed t o t h e frequency domain and w r i t t e n as
dxo dz where X ( Z , S )
B(z,s) X(z,s)
=
0
,
i s t h e column m a t r i x
and A(z,s) and B(z,s) a r e 3 x 3 s q u a r e m a t r i c e s . The s o l u t i o n t o Eq.
(14) i s
where
. Z
and Q(z’,s) = A’l(z,s) B(z,s) The m a t r i x exp[q(s) Az] can b e e v a l u a t e d u s i n g m a t r i x e x p o n e n t i a l t e c h n i q u e s similar t o t h o s e d e s c r i b e d i n Reference 6.
Before t h e s o l u t i o n
can b e completed, t h e boundary c o n d i t i o n s a p p r o p r i a t e t o t h e system must b e spec i f i e d .
8 1
To a s s i g n boundary c o n d i t i o n s , a p h y s i c a l d e s c r i p t i o n (model) of t h e a c t u a l system must be c o n s i d e r e d .
The model chosen t o r e p r e s e n t t h e
more complex a c t u a l system i s p r e s e n t e d i n F i g . 1, page 2.
In particular,
s i x r e g i o n s were chosen :
1. t h e r e g i o n from t h e primary pump t o t h e i n l e t of t h e downcomer, L1: 2.
t h e downcomer, L2;
3.
t h e lower plenum, L3;
4. a
l a r g e number of i d e n t i c a l p a r a l l e l f u e l channels
*, L4:
5.
t h e upper plenum, L5: and
6.
t h e r e g i o n from t h e r e a c t o r vessel t o t h e primary pump,
Lg.
The, perhaps, s i g n i f i c a n t f e a t u r e s l e f t o u t of t h e p h y s i c a l model
are t h e h e a t exchanger and d e t a i l s of t h e pump bowl.
The omission of t h e
h e a t exchanger w i l l c e r t a i n l y r e s t r i c t t h e lower frequency of a p p l i c a b i l i t y
of t h e n e u t r o n i c model, b u t it i s b e l i e v e d t h a t t h i s would n o t a f f e c t t h e h y d r a u l i c model.
The e f f e c t s of t h e pump bowl on t h e system were approxi-
mated by t h e boundary c o n d i t i o n s between r e g i o n s 1 and
6.
The m a t r i x r e p r e s e n t e d by t h e e x p o n e n t i a l term of Eq. r a t e d f o r each r e g i o n .
(16) w a s gene-
Then, c o n t i n u i t y e q u a t i o n s were a p p l i e d between
each r e g i o n , a l o n g w i t h t h e p r e s s u r e f l u c t u a t i o n s i n s e r t e d a t t h e pump bowl, t o p e r m i t t h e s o l u t i o n of t h e c l o s e d - l o o p system; i . e . ,
t h e o u t p u t of
r e g i o n 6 w a s t h e i n p u t t o r e g i o n 1. T h i s p e r m i t t e d t h e e v a l u a t i o n of t h e v o i d f r a c t i o n d i s t r i b u t i o n up through t h e MSRE core, which w i l l be r e q u i r e d
for t h e s o l u t i o n of t h e e q u a t i o n s d e s c r i b i n g t h e n e u t r o n i c model.
*The
r e a c t o r a c t u a l l y c o n s i s t s of h y d r a u l i c a l l y d i f f e r e n t p a r a l l e l channels, b u t t o d a t e no a t t e m p t has been made t o model them.
W
Y
CHAPTER 2
DEVELOPMENT OF THE EXTENDED MODEL P h y s i c a l C h a r a c t e r i s t i c s of t h e F u e l Pump and Fuel-Fump Bowl7 I n d i s c u s s i n g t h e extended h y d r a u l i c model of t h e MSRE c i r c u l a t i n g f u e l system, it may be h e l p f u l t o begin w i t h a b r i e f d e s c r i p t i o n of t h e
p h y s i c a l c h a r a c t e r i s t i c s of t h e f u e l pump and fuel-pump bowl which a r e shown i n F i g . 3. The f u e l - s a l t c i r c u l a t i o n pump i s a c e n t r i f u g a l sump-type pump w i t h
an overhung i m p e l l e r .
It i s d r i v e n by a n e l e c t r i c motor a t
-
1160 rpm and
has a c a p a c i t y of a b o u t 1200 gpm when o p e r a t i n g a t a head of 48.5 f t .
The
36-in.-diameter pump bowl, o r tank, i n which t h e pump v o l u t e and i m p e l l e r are l o c a t e d , s e r v e s as t h e s u r g e volume and expansion t a n k i n t h e primary c i r c u l a t i o n system. The pump bowl, o r tank, i s a b o u t a t the centerline.
36 i n . i n diameter and 17 i n . h i g h
The normal f u e l s a l t l e v e l i n t h e bowl i s a b o u t 11 i n .
from t h e bottom, measured a t t h e c e n t e r l i n e . A bypass f l o w of a b o u t
60 gpm i s t a k e n from t h e pump-bowl d i s c h a r g e
n o z z l e i n t o a r i n g of 2-in.-diameter t h e pump bowl.
p i p e e n c i r c l i n g t h e vapor space i n s i d e
T h i s d i s t r i b u t o r has r e g u l a r l y spaced h o l e s p o i n t i n g down-
ward toward t h e c e n t e r of t h e pump bowl.
The bypass f l o w i s sprayed from
t h e s e h o l e s i n t o t h e bowl t o promote t h e release of f i s s i o n p r o d u c t g a s e s t o t h e vapor space.
The bypass f l o w c i r c u l a t e s downward through t h e pump bowl and ree n t e r s t h e impeller.
The s p r a y c o n t a i n s a c o n s i d e r a b l e volume of cover
9
OANL-
DWG
69-
10172Ai
SHAFT
OFFGA
S BUBBLER
Bb
SPRAY RING
60
0 LEVEL SCALE _-. .-W)
VOLUTE’
“MPE‘LE’ ~PUMP
FIGURE 3.
Molten
Salt Reactor Experiment
BOWL
Fuel Pump and Fuel-Pump Bowl
11 U
gas i n t h e l i q u i d , and t h e tendency f o r t h i s e n t r a i n m e n t t o e n t e r t h e pump
i s c o n t r o l l e d by a b a f f l e on t h e v o l u t e .
Tests i n d i c a t e d t h a t t h e l i q u i d
r e t u r n i n g t o t h e i m p e l l e r w i l l c o n t a i n 1 t o 2 volume p e r c e n t of g a s . A purge f l o w of a b o u t 4.2 s t d l i t e r s / m i n of helium i s maintained
through t h e pump bowl t o a c t as a c a r r i e r for removing f i s s i o n - p r o d u c t g a s e s and t o c o n t r o l t h e p r e s s u r e i n t h e system. ,&sumptions Made i n t h e Development of t h e Model To extend t h e h y d r a u l i c model t o i n c l u d e t h e f u e l pump and f u e l pump bowl, it was n e c e s s a r y t o make some assumptions concerning t h e v a r i o u s r e g i o n s involved as shown i n Fig. 2, page 3. I n c l u d i n g t h e f u e l pump and fuel-pump bowl i n t h e model a c t u a l l y i n t r o d u c e d f o u r components i n t o t h e system: (2) t h e bypass f l o w connections,
supply and removal system.
(3)
(1) t h e f u e l pump i t s e l f ,
t h e fuel-pump bowl, and
(4) t h e helium
The assumptions made f o r e a c h of t h e s e com-
ponents are l i s t e d below.
1. F u e l Pump Assume t h a t t h e r e l a t i v e c o n c e n t r a t i o n of gas t o f u e l s a l t does n o t change as t h e f l u i d p a s s e s through t h e f u e l pump.
T h i s assumption was made
because t h e r e s i d e n c e time of material i n t h e pump is v e r y s h o r t , i.e., t h e r e i s p r a c t i c a l l y no holdup of c i r c u l a t i n g f l u i d i n t h e pump. F u r t h e r assume t h a t t h e 1150-rpm f u e l pump h e a d - c a p a c i t y curve (Fig.
4)
can be r e p r e s e n t e d by a n e q u a t i o n such as
60
IMPELLER MOLTEN-SALT
DIAM:
I1 !+ in. TEMPERATURE:
1200°F
\-
CONSTANT RESISTANCE
/
\
50
40
z 2 W
I J 2 2
30
i 20
10 PUMP HYDRAULIC BALANCE LINE-
‘/
I 0
0
200
400
I
I
600
600 Q , FLOW
FIGURE
4.
Hydraulic Fuel
‘i-
Performance Pump
of
I 1000
RATE
(gpm)
the
Molten
I200
Salt
Reactor
I 1400
Experiment
1600
2.
.
Bypass Flow Connections
For t h e 60-gpm bypass flow t h a t i s d i v e r t e d from t h e pump dizcl-mrge and sprayed i n t o t h e fuel-pump bowl, assume t h a t t h i s connection ( r e g i o n 11) can b e c o n s i d e r e d as a n o r i f i c e and t h e flow r a t e r e p r e s e n t e d by an e q u a t i o n of t h e form'
mll
=
AllCllJ2P(P9
-
p13)
*
(1-9)
A t s t e a d y - s t a t e c o n d i t i o n s ( n o change of l e v e l i n t h e fuel-pump bowl) t h e amount of f l u i d r e t u r n i n g t o t h e main loop f l o w a c r o s s t h e b a f f l e on t h e v o l u t e a t t h e pump s u c t i o n i s e q u a l t o t h e amount of bypass f l u i d l e a v i n g t h e main l o o p a t t h e pump d i s c h a r g e through r e g i o n 11. T h e r e f o r e , assume t h a t t h e bypass flow connection i d e n t i f i e d as r e g i o n 12 can also be c o n s i d e r e d as a n o r i f i c e and t h e f l o w r a t e r e p r e s e n t e d by an e q u a t i o n of t h e form
It i s f u r t h e r assumed t h a t E q s . (19) and (20) can each b e w r i t t e n as two separate equations:
f o r t h e gas mass f l o w r a t e .
one for the fuel s a l t mass flow rate, and one
The o r i f i c e c o e f f i c i e n t s w i l l b e d i f f e r e n t
depending on which material i s b e i n g c o n s i d e r e d .
Thus we have
14 V
and
3.
Fuel-Pump Bowl The fuel-pump bowl ( r e g i o n 13) i s a two-component ( g a s and f l u i d )
r e g i o n , and it i s assumed t h a t each r e g i o n i s a s e p a r a t e , well-mixed volume because t h e r e s i d e n c e time of helium i n t h e gas space i s on t h e o r d e r of f i v e minutes, and t h e f l u i d i s a g i t a t e d by t h e s p r a y .
4.
Helium System The helium i n l e t and o u t l e t p r e s s u r e s a r e e i t h e r known o r assumed
t o be known, and t h e assumption i s made t h a t t h e flow of helium i n t o and o u t of t h e fuel-pump bowl gas space can be r e p r e s e n t e d by o r i f i c e - t y p e e q u a t i o n s such as
f o r the i n l e t (region
14),
and
for t h e o u t l e t ( r e g i o n 1 5 ) . Procedures Used i n Deriving t h e Ecluetions f o r t h e Model Using t h e assumptions d i s c u s s e d above, and w r i t i n g mass b a l a n c e e q u a t i o n s f o r t h e gas and f u e l s a l t a c r o s s t h e v a r i o u s r e g i o n s , a s e t of
1-3 simultaneous e q u a t i o n s w a s o b t a i n e d . 1. F u e l s a l t mass b a l a n c e f o r t h e fuel-pump bowl: mf13 -
dt
-
m
fll
- m
f12
,
(27)
or expressing the mass flow rates in terms of fuel salt density, v o i d
.
fraction, fuel salt velocity and f l o w areas, this becomes
2. Gas mass balance for the fuel-pump bowl:
- m
Q12
+ m
g14
- mQ15
7
or
3. Pressure-head relationship for the fuel pump: From Equation
(7)
But
m Q = fs P
so
Pg-P,
or
=
@ - o m
f8
’
(33)
16 4.
F u e l s a l t mass b a l a n c e f o r t h e f u e l pump: m
=
fa
m
f9
(35)
’
or
Assuming t h a t p
5.
f8
=
pfg,
and s i n c e A,
= Ag,
t h i s reduces t o
Gas mass b a l a n c e f o r t h e f u e l pump:
m
= E58
m E59
or
which reduces t o
6.
Fuel s a l t mass b a l a n c e a t t h e j u n c t i o n of r e g i o n s 1,
9, and 11:
17 W
C(l
.
7.
- a) VfAI,
r(1
=
- a)
V f A l l + [(l-
- a ) VfAl,,
Gas mass b a l a n c e a t t h e j u n c t i o n of r e g i o n s 1,
(43)
9, and
11:
or (Pg a V g A I 9
8.
avgql
(pg
=
.
+ ( pg a VQA)11
(45)
Fuel s a l t mass b a l a n c e a t t h e j u n c t i o n of r e g i o n s 6, 8, and 12:
m
fa
m
=
+ m
f6
'
f12
(46)
or CP&
- a> VfAI, =
Assuming t h a t
=
Pf6
[Pf(l =
f8
-
a ) VfAI, +
'
Of12
- 4 ~ $ 1= ~ r ( 1 9.
[P&
-
r(1 -
a) vfq12
Gas mass b a l a n c e a t t h e j u n c t i o n of r e g i o n s =
m
=
(Pg
a
g6
(47)
t h i s be c ome s
a ) v f ~ i 6+
mga
.
a) VfAI1,
e mg l 2
.
(48)
6, 8, and 12:
,
(49)
or
(P, a VgAIB 10.
",A),
+ (Pg
a Vgql2
Volume b a l a n c e f o r t h e fuel-pump bowl:
.
(50)
18 W
or
11. Void f r a c t i o n r e l a t i o n between r e g i o n s 1 and 9: A t t h e j u n c t i o n of r e g i o n s 1 and
a,
=
9 it i s
assumed t h a t
.
a,
(53)
12. Void f r a c t i o n r e l a t i o n between r e g i o n s 9 and 11:
9 and
A t t h e j u n c t i o n of r e g i o n s
a9 13.
=
11 it i s assumed t h a t
.
a,,
(54)
Void f r a c t i o n r e l a t i o n between r e g i o n s 6, 8, and 12:
The v o i d f r a c t i o n i n r e g i o n
8 may b e w r i t t e n
But
and
m
fE
-
r.
f6
+ m
f12
so
m
or
m
'
as
Y
The n e x t s t e p w a s t o w r i t e t h e s e 13 e q u a t i o n s i n l i n e a r i z e d , p e r t u r b e d form, as shown by t h e f o l l o w i n g examples. R e f e r r i n g t o Eq. (40), page 16, t h e e q u a t i o n r e p r e s e n t i n g t h e gas mass b a l a n c e f o r t h e f u e l pump w r i t t e n i n p e r t u r b e d form i s
Carrying out t h e indicated m u l t i p l i c a t i o n , t h e following r e s u l t
is obtained.
( S i n c e b o t h s i d e s are i d e n t i c a l e x c e p t f o r t h e r e g i o n s in-
volved, only t h e g e n e r a l procedure w i l l be d e s c r i b e d . )
The l e f t - h a n d
s i d e t h u s becomes
Because i n t e r e s t i s r e s t r i c t e d t o d e v i a t i o n s a b o u t t h e mean, t h e f i r s t term, which c o n t a i n s o n l y s t e a d y - s t a t e q u a n t i t i e s , c a n c e l s o u t . Furthermore, t h e l a s t two terms which c o n t a i n t h e p r o d u c t of two p e r t u r b e d q u a n t i t i e s , and are assumed small compared t o t h e remaining terms, were dropped.
I
This g i v e s t h e l i n e a r i z e d form of Eq. (40) as
20
4,
The l i n e a r i z e d v e r s i o n of Eq. (l),page
is
and t h e l i n e a r i z e d form of t h e r e l a t i o n s h i p between V
Eq.
g
and Vf
,
( 5 ) , Page 5, is 1AV
g
=
a.
K - a
AVf
+
0
Vfo0
(K -
-
K)
Ax.
"J2
Now r e f e r r i n g t o F i g . 2, page 3, it i s noted t h a t a t t h e j u n c t i o n
of r e g i o n s 1, 9, and 11 t h e p r e s s u r e s P1 = j u n c t i o n of r e g i o n s
6, 8, and
=
=
AP9
=
Pll, =
and at t h e
P8
=
P12.
APll and m6
=
APl,, r e s p e c t i v e l y . S u b s t i t u t i n g Eqs. (62) and
Eq.
=
12, t h e p r e s s u r e s Ps
L i n e a r i z e d , t h e s e e q u a l i t i e s become AP1
AI?,
P9
(63) and t h e p r e s s u r e e q u a l i t i e s i n t o
(61) g i v e s t h e f i n a l l i n e a r i z e d p e r t u r b e d form of t h e gas mass b a l a n c e
f o r t h e f u e l pump:
W
21 W
For a n o t h e r example, c o n s i d e r Eq. ( 2 - 0 , t h e f u e l s a l t mass b a l a n c e f o r t h e fuel-pump bowl, page
14.
Linenrixed, t h i s becomes
L i n e a r i z i n g t h e o r i f i c e r e l a t i o n s h i p s for t h e f u e l s a l t mass flow r a t e s for r e g i o n s 11 and 12 as g i v e n by E q s . (21) and (23), r e s p e c t i v e l y , page 13, g i v e s
and
S u b s t i t u t i n g E q s . (66) and (67) i n t o Eq.
(65) and
using t h e line-
a r i z e d p r e s s u r e e q u a l i t i e s mentioned above, t h e l i n e a r i z e d f u e l s a l t mass b a l a n c e e q u a t i o n for t h e fuel-pump bowl becomes
The n e x t s t e p w a s t o Laplace t r a n s f o r m t h e time v a r i a b l e t o t h e frequency domain. v e r s i o n of Eq.
Thus, t h e l i n e a r i z e d , p e r t u r b e d , Laplace-transformed
(68) i s
22
As a r e s u l t of performing t h e manipulations d e s c r i b e d i n t h e above
examples on t h e remaining 11 e q u a t i o n s , a s e t of 13 l i n e a r i z e d , p e r t u r b e d , Laplace-transformed e q u a t i o n s c o n t a i n i n g
16 unknowns was o b t a i n e d .
Method Used t o Solve t h e Equations Derived f o r t h e Model I n o r d e r t o s o l v e t h e s e t of 13 e q u a t i o n s c o n t a i n i n g
j u s t d e r i v e d , t h r e e more e q u a t i o n s were needed.
16 unknowns
These t h r e e a d d i t i o n a l
e q u a t i o n s were o b t a i n e d from t h e o r i g i n a l h y d r a u l i c model. I n t h e o r i g i n a l model, t h e v a r i a b l e s i n r e g i o n i n r e g i o n 1 by means of a t r a n s f e r f u n c t i o n .
6 a r e r e l a t e d t o those
I n m a t r i x form, t h i s re-
l a t i o n s h i p can be e x p r e s s e d as
where X ( z , s ) i s t h e column m a t r i x
and B(s)
i s t h e 3 x 3 s q u a r e t r a n s f e r f u n c t i o n m a t r i x which r e l a t e s t h e
s t a t e v e c t o r X ( Z , S ) a t t h e o u t l e t of r e g i o n 6 t o t h e s t a t e v e c t o r X ( Z , S ) a t t h e i n l e t t o r e g i o n 1.
23 Expanding t h e matrix form of Eq. (70) gave t h e n e c e s s a r y t h r e e a d d i t i o n a l e q u a t i o n s t o complete t h e s e t of
16 e q u a t i o n s c’ontaining 16
unknowns. The
16 e q u a t i o n s c o n s t i t u t i n g t h e extended h y d r a u l i c model a r e a s
follows :
1. F u e l s a l t mass b a l a n c e f o r t h e fuel-pump bowl:
2. f
G a s mass b a l a n c e f o r t h e fuel-pump bowl:
24
3.
Pressure-head r e l a t i o n s h i p f o r t h e f u e l pump:
H 8-
H
+
4.
5 (9-
pgo)9
+
.0624 ( P ~ A AV, ) ~,
=
F
.
(74)
F u e l s a l t mass b a l a n c e f o r t h e f u e l pump:
(75)
5.
Gass mass b a l a n c e f o r t h e f u e l pump:
0 fo RTo K
-
a .
25
6.' F u e l
s a l t mass b a l a n c e a t t h e j u n c t i o n of r e g i o n s 1, 9, and 11: AVf9
7.
('go
8.
K
-
:),
Avfi
=
(77)
0
-
['go
'fo
(K
l &,
- ao> 2
9, and
=
F u e l salt mass b a l a n c e a t t h e j u n c t i o n of r e g i o n s
AV,@
9.
.0061 AVfl
Gas mass b a l a n c e a t t h e j u n c t i o n of r e g i o n s 1,
1-
-
- %,,-
-
AVf6
-
.127 AVf
12
=
11:
(78)
0
6, 8, and 12:
(79)
0
G a s m a s s b a l a n c e at t h e j u n c t i o n of r e g i o n s
6, 8, and 12:
1 - K 8
26
Volume b a l a n c e f o r t h e fuel-pump bowl:
10.
-
(k)
m f 1 3 + ’fl3
13
[
10,541
(RT ) 11
0 1 3
il. Void f r a c t i o n r e l a t i o n between r e g i o n s 1 and 9: La9 - L a 1
=
0
12
12.
Void f r a c t i o n r e l a t i o n between r e g i o n s 9 and 11: ciu9
-
f a l
=
(83)
0
13.
Void f r a c t i o n r e l a t i o n between r e g i o n s 6, 8, and 12:
14.
F u e l s a l t v e l o c i t y r e l a t i o n s h i p between r e g i o n s 1 and 6 (from
o r i g i n a l h y d r a u l i c model t r a n s f e r f u n c t i o n ) :
15. Void f r a c t i o n r e l a t i o n s h i p between r e g i o n s
1 and
6 (from
o r i g i n a l h y d r a u l i c model t r a n s f e r f u n c t i o n ) :
16. P r e s s u r e r e l a t i o n s h i p between r e g i o n s h y d r a u l i c model t r a n s f e r f u n c t i o n ) :
1 and
6 (from o r i g i n a l
28 The s o l u t i o n of t h e r e s u l t i n g
16 e q u a t i o n s w a s c a r r i e d o u t by
f i r s t w r i t i n g them i n m a t r i x form as f o l l o w s : AY
=
*
C ,
where
Y
=
t h e column v e c t o r c o n s i s t i n g of t h e
16 p e r t u r b e d v a r i a b l e s ,
o r d e r e d as f o l l o w s
where T i s t h e t r a n s p o s e ; A
=
t h e c o e f f i c i e n t m a t r i x c o n s i s t i n g of t h e c o n s t a n t s i n t h e
16 e q u a t i o n s , and o r d e r e d i d e n t i c a l l y t o Y, and C
=
t h e f o r c i n g f u n c t i o n column v e c t o r c o n s i s t i n g of t h e elements making up t h e r i g h t - h a n d sides of t h e
16 e q u a t i o n s .
Before t h e s o l u t i o n of t h e s e e q u a t i o n s c o u l d b e f o r m a l l y c a r r i e d o u t however, it w a s n e c e s s a r y t o c o n s i d e r t h e f o r c i n g f u n c t i o n v e c t o r C i n E q . (88).
r e s p e c t i v e l y , it i s noted t h a t t h e v a r i a b l e s involved i n v e c t o r C a r e t h e p e r t u r b a t i o n s i n t h e helium i n l e t p r e s s u r e , AP14; p e r t u r b a t i o n s i n t h e helium o u t l e t p r e s s u r e ,
and t h e fuel-pump i n p u t (head d e l i v e r e d )
f l u c t u a t i o n , F.
*Examples
showing how some of t h e c o n s t a n t s i n t h e A and C m a t r i c e s were o b t a i n e d a r e given i n t h e Appendix. S p e c i f i c e q u a t i o n s of t h e extended model are d i s c u s s e d , and a t a b l e of t h e c o n s t a n t s i s p r e s e n t e d .
29 V
From t h i s it i s seen t h a t t h e extended h y d r a u l i c model p o s t u l a t e s t h a t t h e helium system i s b u t one n a t u r a l s o u r c e o r cause of p r e s s u r e changes i n t h e c i r c u l a t i n g f u e l system; f l u c t u a t i o n s i n i n p u t (head del i v e r e d ) from t h e f u e l pump a l s o c o n t r i b u t e t o t h e n a t u r a l l y o c c u r r i n g p r e s s u r e p e r t u r b a t i o n s around t h e loop. The s o l u t i o n o b t a i n e d i n t h i s work was arrived a t by s p e c i f y i n g these variables individually.
That is, one v a r i a b l e w a s g i v e n a v a l u e and
t h e o t h e r two were made i d e n t i c a l l y e q u a l t o zero.
I n t h i s way t h e e f f e c t
of one v a r i a b l e on p r e s s u r e p e r t u r b a t i o n s i n t h e c i r c u l a t i n g f u e l system w a s s e p a r a t e d from t h e e f f e c t s of t h e o t h e r s .
Furthermore, t h i s method p e r m i t t e d t h e u s e of t h e concept of a purposely p e r t u r b e d system.
For example, e x p e r i m e n t a l data were o b t a i n e d
by i n t r o d u c i n g a t r a i n of sawtooth p r e s s u r e p u l s e s i n t o t h e fuel-pump bowl and a n a l y z i n g t h e r e s u l t i n g changes i n t h e n e u t r o n flux.''
The
c a p a b i l i t y of i n t r o d u c i n g a similar kind of p r e s s u r e p e r t u r b a t i o n i n t o t h e extended h y d r a u l i c model and t h e n u s i n g t h e n e u t r o n i c model t o c a l c u -
l a t e t h e e f f e c t s on t h e neutron f l u x e x i s t s .
Thus, a comparison of
t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s i s p o s s i b l e . Once t h e elements of t h e A and C m a t r i c e s were determined, s t a n d a r d computer t e c h n i q u e s g for s o l v i n g complex m a t r i x e q u a t i o n s were a p p l i e d t o Eq. (88) t o o b t a i n t h e d e s i r e d v a l u e s of t h e 16 v a r i a b l e s i n v e c t o r Y.
CHAF'I'ER
3
RESULTS OF CALCULATIONS MADE WITH THE EXTENDED MODEL COMPARED TO EXPERIMEIITAL DATA A s s t a t e d p r e v i o u s l y i n Chapter 2, page 29, t h e s o l u t i o n t o t h e
m a t r i x e q u a t i o n r e p r e s e n t i n g t h e extended h y d r a u l i c model, Eq.
(88),
page 28, was accomplished by a s s i g n i n g v a l u e s t o t h e f o r c i n g f u n c t i o n v a r i a b l e s one a t a time. Two c a s e s were c o n s i d e r e d : Case 1 The f u e l pump f o r c i n g function v a r i a b l e , F,
i n Eq.
(74),
page
24,
w a s s e t e q u a l t o u n i t y ; t h e fuel-pump bowl helium i n l e t and o u t l e t p r e s -
s u r e f l u c t u a t i o n s , aP14 and AP15, r e s p e c t i v e l y , i n Eq. (73), p g e 23, and Eq.
(81), page 26, were s e t e q u a l t o zero, i . e . , F
=
1.0
,
aP14
=
0.0
, and
m,,
=
0.0
Case 2 The fuel-pump bowl helium i n l e t p r e s s u r e f l u c t u a t i o n , AP14, w a s made e q u a l t o u n i t y ; t h e fuel-pump k,owl helium o u t l e t p r e s s u r e f l u c t u a t i o n ,
AP15, and t h e fuel-pump f o r c i n g f u n c t i o n v a r i a b l e , F, were s e t e q u a l to zero, i . e . ,
m,4
=
D1, =
F
=
, 0.0 , and 1.0
0.0
.
(92)
(93)
(94)
The fuel-pump bowl helium o u t l e t p r e s s u r e f l u c t u a t i o n ,
.
APlS,
is
z e r o i n b o t h c a s e s because of t h e assumption t h a t t h e helium system d i s charge p r e s s u r e , P15, i s c o n s t a n t (atmospheric)
.
The r e a s o n F and API4 a r e u n i t y i n Case 1 and Case 2, r e s p e c t i v e l y ,
i s t h e r e s u l t of t h e way i n which t h e frequency r e s p o n s e of a system i s computed.
The frequency response of a stable system can be d i r e c t l y ex-
p r e s s e d by an e q u a t i o n r e l a t i n g the F o u r i e r transform, F, of t h e system output t o the input:ll
4 F {Out u t
=
Frequency response.
(95)
To o b t a i n t h e neutron f l u x - t o - p r e s s u r e frequency response of t h e system d e s c r i b e d by t h e combined h y d r a u l i c - n e u t r o n i c model, it w a s necess a r y t o d i v i d e t h e o u t p u t s ( l e f t - h a n d s i d e s of t h e e q u a t i o n s ) by t h e inp u t s ( f o r c i n g f u n c t i o n v a r i a b l e s ) for each c a s e where t h e v a r i a b l e s were non- zero. I n Case 1, for example, d i v i d i n g Eq.
(74), page
24, by t h e f u e l -
pump f o r c i n g f u n c t i o n v a r i a b l e , F, makes t h e r i g h t - h a n d s i d e u n i t y . t h e element of t h e f o r c i n g f u n c t i o n v e c t o r C i n Eq.
Thus
(88), page 28, c o r r e -
sponding t o F becomes u n i t y , and a l l o t h e r elements of C are zero.
In
Case 2, t h e element of v e c t o r C corresponding t o CQl4 becomes a c o n s t a n t , and a l l o t h e r elements of v e c t o r C are zero. I n t h i s way it w a s p o s s i b l e t o o b t a i n i n f o r m a t i o n a b o u t t h e f r e quency response even though t h e a b s o l u t e values of t h e f o r c i n g f u n c t i o n variables were n o t known.
I n a d d i t i o n t o i n f o r m a t i o n a b o u t t h e frequency
response, r e s u l t s concerning t h e modulus of t h e p r e s s u r e i n t h e fuel-pump bowl, AP13, were a l s o determined for Cases 1 and 2.
32 Even though t h i s c a l c u l a t i o n a l procedure d i d n o t produce a b s o l u t e v a l u e s f o r t h e frequency response and fuel-pump bowl p r e s s u r e modulus, t h e r e s u l t s can s t i l l be compared t o t h e e x p e r i m e n t a l d a t a . The p e r t i n e n t e x p e r i m e n t a l data and t h e b a s e s f o r comparison w i t h c a l c u l a t e d r e s u l t s a r e as f o l l o w s : 1.
P r e s s u r e Power S p e c t r a l Density Datal2 f o r t h e E R E
These d a t a r e p r e s e n t measurements of t h e n a t u r a l l y o c c u r r i n g p r e s s u r e n o i s e i n t h e fuel-pump bowl d u r i n g normal r e a c t o r o p e r a t i o n . The p r e s s u r e power s p e c t r a l d e n s i t y i s p r o p o r t i o n a l t o t h e f u e l pump bowl p r e s s u r e frequency response f u n c t i o n modulus squared i f t h e d r i v i n g f u n c t i o n i s "white,
"
e.g.,
random and Gaussian.
Therefore, by p o s t u l a t i n g c e r t a i n white d r i v i n g f u n c t i o n s , it i s p o s s i b l e t o s e e i f t h e r e s u l t a n t c a l c u l a t e d fuel-pump bowl p r e s s u r e power s p e c t r a l d e n s i t y curve is similar i n shape compared w i t h t h a t which has been observed. 2.
The Modulus of t h e Neutron F l u x - t o - P r e s s u r e Frequency Response Function f o r t h e MSRE
These data r e p r e s e n t measurements t a k e n d u r i n g t h e saw-tooth p r e s s u r e p u l s e t e s t s mentioned i n Chapter 2. I n F i g . 5, t h e s q u a r e of t h e modulus of t h e p r e s s u r e i n t h e f u e l pump bowl c a l c u l a t e d f o r Cases 1 and 2 i s compared w i t h t h e e x p e r i m e n t a l p r e s s u r e power s p e c t r a l d e n s i t y data.
The u n i t s of t h e s q u a r e of t h e
p r e s s u r e modulus a r e i n d i c a t e d as b e i n g a r b i t r a r y because t h e r e s u l t s computed by t h e extended model were s c a l e d f o r comparison purposes. The c a l c u l a t e d s q u a r e of t h e fuel-pump bowl p r e s s u r e modulus f o r Case 2 (AP14 =
1.0, F
=
0.0,
APl5
=
0.0) over t h e range of f r e q u e n c i e s
I
33
102
ORNL-DWG 70-1374
, LEGEND
A
CASE t
V
EXPERIMENTAL DATA
e
2
-
I
c
-
v)
c
C 3
-. A L
?
._
0
0
v)
3
-I
0 3
0
I w [r
3 v) v)
W
n
a -I
z
m
a
5a ,
-I
W U 3
Lc
0 W
LL
a
3
z
0.5
1.o
1.5
2 .O
2.5
3.0
3.5
FREQUENCY (cps)
FIGURE 5.
Square of t h e Modulus of t h e P r e s s u r e i n t h e Molten S a l t R e a c t o r Experiment Fuel-Pump Bowl Case 1: F = 1.0, aP14 = 0.0, aPl5 = 0.0; Case 2: aP14 = 1.0, AP15 = 0.0, F = 0.0.
34 shown i n F i g . 5 a g r e e s r e a s o n a b l y w e l l w i t h t h e g e n e r a l t r e n d of t h e exp e r i m e n t a l data, a l s o g i v e n i n F i g . 5, f o r t h e same range of f r e q u e n c i e s . The computation of t h e s q u a r e of t h e fuel-pump bowl p r e s s u r e modulus f o r Case 1 ( F
=
1.0, AP14 =
t h a t are a l s o shown i n Fig. 5.
0.0, AP15
=
0.0) gave r e s u l t s
T h i s curve showed f l u c t u a t i o n s more simi-
l a r t o t h o s e a c t u a l l y observed, e x c e p t i n t h e frequency range of a b o u t
0.5 t o 2.0 c p s .
Here it a p p e a r s t h a t t h e extended h y d r a u l i c model ex-
h i b i t s extreme under-damping o r resonance e f f e c t s . From t h e comparison of r e s u l t s shown i n F i g . 5, it seems r e a s o n a b l e t o p o s t u l a t e t h a t t h e fuel-pump bowl p r e s s u r e modulus i s b a s i c a l l y r e l a t e d t o t h e fuel-pump bowl h e l i u m i n l e t p r e s s u r e p e r t u r b a t i o n s .
Superimposed
on t h i s r e l a t i o n s h i p , however, are t h e random e f f e c t s of t h e f u e l pump input fluctuations. C o n s i d e r a t i o n of t h e i n f o r m a t i o n p r e s e n t e d i n F i g . 5 thus leads t o t h e expectation t h a t pressure f l u c t u a t i o n s i n the c i r c u l a t i n g f u e l system o f t h e MSRE are a l s o a r e s u l t of t h e combination of these two perturbations.
It a p p e a r s t h a t t h e r e s o n a n t s t r u c t u r e due t o t h e fuel-pump
i n p u t p e r t u r b a t i o n dominates t h e e f f e c t of t h e fuel-pump bowl helium i n l e t p r e s s u r e p e r t u r b a t i o n , however. Other e x p e r i m e n t a l evidence t o s u p p o r t t h i s c o n c l u s i o n i s a v a i l a b l e : It w a s noted t h a t when t h e fuel-pump bowl o f f g a s l i n e became plugged and t h e fxel-pump bowl p r e s s u r e remained e s s e n t i a l l y c o n s t a n t o v e r a l o n g
p e r i o d of t i m e , t h e p r e s s u r e n o i s e i n t h e fuel-pump bowl i n c r e a s e d app r e c i a b l y , b u t t h e n e u t r o n n o i s e remained e s s e n t i a l l y c o n s t a n t . l3 The modulus of t h e n e u t r o n f l u x - t o - p r e s s u r e frequency response
f u n c t i o n f o r Case 2 (AP14 =
1.0, F
=
0.0,
D'15
=
0.0) i s shown i n
35 v
Fig.
6 f o r two d i f f e r e n t fuel-pump bowl p r e s s u r e s .
Also shown i s t h e
modulus computed by t h e o r i g i n a l open-loop mathematical model.
The
modulus of t h e neutron f l u x - t o - p r e s s u r e frequency response f u n c t i o n determined from experimental r e s u l t s 1 4 obtained from a s e r i e s of saw- t o o t h p r e s s u r e p u l s e t e s t s i s included f o r comparison. Units of t h e modulus of t h e neutron f l u x - t o - p r e s s u r e frequency response f u n c t i o n a r e shown as being a r b i t r a r y because t h e r e s u l t s c a l c u l a t e d by t h e mathematical models Mere s c a l e d f o r t h e purpose of comparison. The open-loop model r e s u l t s more n e a r l y approximate t h e experi-
mental data, b u t both mathematical models appear t o be a t t e n u a t e d t o o r a p i d l y a t f r e q u e n c i e s above about 0.1 cps.
Unfortunately t h e r e a r e no
experimental d a t a below about 0.02 cps o r above about 0.5 cps with which t o compare t h e two models. The e f f e c t of d i f f e r e n t fuel-pump bowl p r e s s u r e s on t h e modulus of t h e neutron f l u x - t o - p r e s s u r e frequency response f u n c t i o n c a l c u l a t e d by t h e extended model i s apparent i n Fig. 6.
It i s noted t h a t t h e h i g h e r
fuel-pump bowl p r e s s u r e gave a g r e a t e r amplication of t h e p r e s s u r e perturbation.
ORNL-DWG
70-1375
lo-'
$1:
-
5
0
a
z 0 IV
2
z 3
LL W
cn
z
8 v,
10-2
W [L
>
V
Z
W
5
3
CY w
[L
LL
W [L
3
m
cn W
2
e
[L
a
X 3 -I
10-3
LL
z 0
[L
t3 W
z
5
W
I ILL
0
cn 3 -I
3
2
n 0
5
10-4
0.01
0.02
0 05
FREQUENCY
FIGURE
0.2
0.1
0.5
1 .o
(CPS)
6. Modulus of the Neutron Flux-to-Pressure Frequency-Response Function for the Molten Salt Reactor Experiment
CHAPTER
.
4
C O N C u l S I O N S AND RECOMMENDATIONS
The h y d r a u l i c p o r t i o n of t h e mathematical model o r i g i n a l l y formul a t e d by Robinson and Fry w a s extended by e x p l i c i t l y i n c l u d i n g t h e f u e l
pump and fuel-pump bowl t o c l o s e t h e loop. The extended model was t h e n combined w i t h t h e i r o r i g i n a l n e u t r o n i c model and used t o perform c a l c u l a t i o n s y i e l d i n g r e s u l t s such as t h e s q u a r e
of t h e modulus of t h e p r e s s u r e i n t h e fuel-pump bowl, and t h e modulus of t h e neutron-flux-to-pressure
frequency response f u n c t i o n for t h e MSRE.
R e s u l t s c a l c u l a t e d by t h e extended and open-loop models were compared t o t h o s e o b t a i n e d from e x p e r i m e n t a l measurements t o determine t h e v a l i d i t y of t h e models.
It w a s found t h a t t h e extended model i n i t s p r e s e n t form cannot c a l c u l a t e t h e modulus of t h e p r e s s u r e i n t h e fuel-pump bowl i n a n a b s o l u t e sense.
On t h e o t h e r hand, examination of t h i s parameter as c a l c u l a t e d by
t h e extended model l e a d t o some i n t e r e s t i c g q u a l i t a t i v e c o n c l u s i o n s which
are d i s c u s s e d below. F l u c t u a t i o n s i n p r e s s u r e , and t h u s i n n e u t r o n f l u x , i n t h e c i r c u l a t i n g f u e l system of t h e MSRE r e s u l t from p e r t u r b a t i o n s i n s e r t e d by t h e
f u e l pump and t h e helium cover-gas system. The e f f e c t s of t h o s e p e r t u r b a t i o n s i n t r o d u c e d by t h e f u e l pump are dominant, however, and are r e s p o n s i b l e f o r t h e r e s o n a n t s t r u c t u r e s e e n i n t h e s q u a r e of t h e modulus of t h e p r e s s u r e i n t h e fuel-pump bowl shown i n Fig.
5 , Page 33.
37
v
The modulus of t h e n e u t r o n - f l u x - t o - p r e s s u r e frequency r e s p o n s e f u n c t i o n c a l c u l a t e d by t h e o r i g i n a l open-loop model a g r e e s much b e t t e r w i t h t h e e x p e r i m e n t a l r e s u l t s t h a n does t h a t c a l c u l a t e d by t h e extended
model. The l a r g e resonances i n t h e modulus of t h e n e u t r o n f l u x - t o - p r e s s u r e frequency response f u n c t i o n noted i n F i g . 6, page 36, i n d i c a t e t h a t t h e extended model i s n o t s u f f i c i e n t l y damped.
It w a s f u r t h e r found t h a t t h e extended model a l s o cannot c a l c u l a t e t h e frequency response of t h e MSRE a b s o l u t e l y .
It was s t a t e d e a r l i e r (page 35) t h a t a s c a l e f a c t o r was a p p l i e d t o t h e modulus o f t h e n e u t r o n f l u x - t o - p r e s s u r e frequency response f u n c t i o n c a l c u l a t e d by t h e extended model i n o r d e r t o compare it t o t h e e x p e r i mental data i n F i g . 6, page 36.
The u s e of t h i s scale f a c t o r w a s n e c e s s a r y
because t h e c a l c u l a t e d modulus w a s much s m a l l e r t h a n t h e e x p e r i m e n t a l l y determined modulus.
T h i s s u g g e s t s t h a t t h e r e i s t o o much a t t e n u a t i o n
( p r e s s u r e drop) i n t h e extended model between t h e p r e s s u r e i n t h e f u e l pump bowl and t h e c o r e . S i m i l a r l y , t h e s q u a r e of t h e modulus of t h e p r e s s u r e i n t h e f u e l pump bow1 w a s much smaller than t h e e x p e r i m e n t a l r e s u l t s shown i n F i g . 5, page 33.
T h i s i n d i c a t e s t h a t t h e r e w a s a l s o t o o much a t t e n u a t i o n i n t h e
p o r t i o n of t h e l o o p ( f u e l pump and fuel-pump bowl) added by t h e extended model. The f o l l o w i n g recommendations, based on t h e above c o n c l u s i o n s , a r e made w i t h t h e hope t h a t t h e y might prove h e l p f u l i n improving t h e extended model, or i n f u t u r e modeling of similar h y d r a u l i c systems. Y
39 Although t h e extended model i s n o t a b l e t o g e n e r a t e a b s o l u t e v a l u e s f o r t h e modulus of t h e p r e s s u r e i n t h e fuel-pump bowl and t h e modulus of t h e neutron f l u x - t o - p r e s s u r e frequency response f u n c t i o n , it could be p u t t o u s e i n t h e following way.
By normalizing t h e r e s u l t s t o g i v e c o r r e c t
v a l u e s based on experimental data, t h e normalized model could then be a p p l i e d i n p e r f o m i n g parameter s t u d i e s on t h e MSRE c i r c u l a t i n g f u e l system. The a t t e n u a t i o n i n t h e p o r t i o n of t h e loop between t h e f u e l pump and fuel-pump bowl must be reduced.
I n t h i s case, it appears t h a t t h e
choice of o r i f i c e - t y p e equations t o r e p r e s e n t t h e bypass flow r e s t r i c t i o n s
a t t h e pump discharge and pump s u c t i o n w a s n o t a sound assumption. Furthermore, t h e use of s e p a r a t e o r i f i c e - t y p e equations f o r t h e gas and f u e l s a l t w a s a l s o unfortunate.
The 1 t o 2 volume p e r c e n t of gas i n t h e
l i q u i d observed r e t u r n i n g t o t h e i m p e l l e r during pump t e s t s could be a n important e f f e c t . These o b s e r v a t i o n s a r e pointed up by t h e good agreement of t h e r e s u l t s obtained from t h e o r i g i n a l model with t h e experimental data.
In
p l a c e of o r i f i c e - t y p e equations t o c l o s e t h e loop, t h e o r i g i n a l model used as boundary c o n d i t i o n s (1) p r e s s u r e f l u c t u a t i o n s a t t h e e x i t of r e g i o n 6, F i g , 1, page 2, a r e t h e same as i n l e t p r e s s u r e f l u c t u a t i o n s t o r e g i o n 1, and ( 2 ) t h e f l u c t u a t i o n s i n t h e t o t a l mass flow r a t e t o r e g i o n 1 w a s zero.
A b e t t e r s e l e c t i o n f o r t h e extended model would have been t o u s e
a momentum balance i n t h e fuel-pump bowl combined w i t h o r i f i c e - t y p e equations which accounted f o r momentum t r a n s f e r between t h e gas and f u e l
s a l t i n t h e bypass loop.
These equations would r e p l a c e t h e s e p a r a t e gas
and f u e l s a l t o r i f i c e r e l a t i o n s h i p s t h a t were a p p l i e d .
W
Another recommendation t h a t may b e b e n e f i c i a l i s t o remove t h e assumption t h a t t h e r e i s no holdup volume i n t h e f u e l pump, :.e., t h e pump t o r e p r e s e n t a well-mixed volume.
consider
T h i s would h e l p a c c o u n t f o r
t h e f a c t t h a t mass can be s t o r e d i n t h e f u e l pump as it i s i n t h e f u e l pump bowl. F i n a l l y , t h e n e u t r o n i c p o r t i o n of t h e model, a l t h o u g h n o t d i r e c t l y a d d r e s s e d by t h i s t h e s i s , could be improved by s i m u l a t i n g t h e e f f e c t s of t h e h e a t exchanger on t h e c i r c u l a t i n g f u e l system.
V
.
LIST OF REF'EFGNCES
LIST OF REFERENCES 1. J. C . Robinson and D. N. Fry, ?Determination of t h e Void F r a c t i o n i n t h e MSRE Using Small Induced P r e s s u r e P e r t u r b a t i o n s , " USAEC Report ORNL-TM-2318, Oak Ridge N a t i o n a l Laboratory (February 6,
1969) ~ , 18 2. Robinson and Fry, O R N L - T M - ~ ~ ~pp.
6
-
-
21.
3.
Robinson and Fry, ORNL-TM-2318, pp.
4.
L. G. Neal and S. M. Zivi, "Hydrodynamic S t a b i l i t y of N a t u r a l Circul a t i o n B o i l i n g Systems," Volume 1, "A Comparative Study of Anal y t i c a l Models and Experimental Data," USAEC Research and Development Report, STL 372-14 (l), Thomson Ram0 Woolridge, I n c . , (June 30, 1965).
5.
Neal and Z i v i , STL 372-14 (l), p.
6.
S . J . Ball and R. K. Adams,
7.
R. C . Robertson, '%ISRE Design and Operations Report, P a r t I, D e s c r i p t i o n of Reactor Design," USAEC Report ORNL-TM-728, pp. 133 - 142, 325, ( J a n u a r y 1965)
10.
85.
''MATEXP, A General Purpose D i g i t a l Computer Program for S o l v i n g Ordinary D i f f e r e n t i a l Equations by t h e M a t r i x E x p o n e n t i a l Method, USAEC Report ORNL-TM-1933, Oak Ridge N a t i o n a l Laboratory, (August 1967).
.
8. R. B. Bird,
W. E. S t e w a r t , and E . N. L i g h t f o o t , T r a n s p o r t Phenomena, P. 226, John Wiley and Sons, Inc., New York, 1960.
9.
R. E. Funderlich, E d i t o r , Programmer's Handbook, USAEC Report K-1729, "CMATEQ, Program ORD-9067, Gaussian E l i m i n a t i o n for Real and Complex Systems, (February 9, 1968).
10. Robinson and Fry, ORNL-TM-2318, p . 22.
11. M. A . S c h u l t z , C o n t r o l of Nuclear R e a c t o r s and Power P l a n t s , Second E d i t i o n , p. 70, McGraw-Hill Book Company, New York, 1961. 12.
D. N. Fry, p e r s o n a l communication t o W. C . U l r i c h , November 26,
1969.
13.
D. N. Fry, p e r s o n a l communication t o W. C . U l r i c h , November 26,
1969.
14.
J. C . Robinson, p e r s o n a l communication t o W. C . U l r i c h , November 26, 1969. (To b e p u b l i s h e d i n Nuclear S c i e n c e and Engineering.)
42
APPENDIX
APPENDIX
The v a r i o u s c o n s t a n t s i n t h e A and C matrices i n Eq
.
(88), page 28,
were g e n e r a t e d u s i n g i n f o r m a t i o n o b t a i n e d from t h r e e s o u r c e s : 1.
o b s e r v a b l e q u a n t i t i e s such as system t e m p e r a t u r e s , p r e s s u r e s , a r e a s , and volumes;
2.
p h y s i c a l p r o p e r t i e s , such as f u e l s a l t d e n s i t y , based on cond i t i o n s d e s c r i b e d by t h e o b s e r v a b l e q u a n t i t i e s ; and
3.
c a l c u l a t i o n s of o t h e r c o n s t a n t s u s i n g f l o w rates, p r e s s u r e s , v e l o c i t i e s , and v o i d f r a c t i o n s o b t a i n e d from t h e s o l u t i o n of t h e o r i g i n a l h y d r a u l i c model i n c o n j u n c t i o n w i t h items 1 and 2.
The f o l l o w i n g examples show how t h e i n f o r m a t i o n d e s c r i b e d above w a s a p p l i e d t o two s p e c i f i c e q u a t i o n s of t h e extended h y d r a u l i c model.
Example 1 -
Consider Eq. ( T Z ) , page 23, t h e f u e l s a l t mass b a l a n c e
f o r t h e fuel-pump bowl: a.
An o b s e r v a b l e q u a n t i t y i s t h e t e m p e r a t u r e of t h e f u e l s a l t
i n r e g i o n s 11 and 12, (Tfo)
The a r e a s All drawings.
11
and A12
and (Tfo)
12,
respectively:
were o b t a i n e d from d e t a i l pump d e s i g n
The a r e a f o r r e g i o n 11 w a s t a k e n as t h e s i z e of t h e connection
between t h e pump d i s c h a r g e l i n e and t h e 2 - i n . - d i a .
pipe spray ring; the
a r e a of t h e opening a t t h e pump s u c t i o n w a s c a l c u l a t e d . t h e s e are
44
Specifically,
.
45 Y
A s s t a t e d on page
p a s s r e g i o n i s 60 gpm.
A11
=
0.48 i n ?
,
(A3 1
A12
=
9.98 i n ?
,
(A41
9, t h e v o l u m e t r i c r a t e of f l o w through t h e by-
S i n c e t h e v o i d f r a c t i o n of gas i s l e s s t h a n
1 p e r c e n t a t normal o p e r a t i n g c o n d i t i o n s ,
f l o w c o n s i s t e d e n t i r e l y of f u e l s a l t .
=
20.1 l b / s e c
it w a s assumed t h a t t h e bypass
Thus
.
(A5 1
Furthermore, w i t h no change of mass of f u e l s a l t i n t h e pump bowl, t h e mass r a t e of f l o w through r e g i o n 1 2 e q u a l s t h a t through r e g i o n 11, o r
(mfo)l, b.
-
(mfo)12
=
20.1 l b / s e c .
The t e m p e r a t u r e i n r e g i o n s 11 and 12 determines t h e f u e l s a l t
density; thus
Pfll
pf 12
c.
=
0.087 l b / i n ?
=
0.087 l b / i n ?
The o r i f i c e c o e f f i c i e n t s C f l l
,
and C
. f12
were c a l c u l a t e d by
a p p l y i n g some o f t h e above i n f o r m a t i o n t o E q s . (21) and (23), page 13, as f o l l o w s :
46 From Eq
. (21), .
where (Po)9 and (Po)l3 a r e given i n t h e s o l u t i o n o b t a i n e d 'by t h e o r i g i n a l h y d r a u l i c model, namely
(p ) 0
Therefore, Eq.
=
20.6 p s i a .
13
(Ag) becomes 20.1
-
?
Cfll
0.48 J2 x 0.087 x 49.4
-
14.6 in,4 - s e e - 1
Cfll -
.
S i m i l a r l y , Eq. (23) g i v e s (mf0 ) 1 2
i s a l s o o b t a i n e d from t h e o r i g i n a l h y d r a u l i c model s o l u t i o n :
where (Po) 8
(PO) 8
Thus Eq.
=
19.6 p s i a .
(A14) i s now 20.1 cf 1 2
-
?
9.98,/2
x 0.087 x 1.0
47 Now a l l t h e numbers r e q u i r e d t o determine t h e c o n s t a n t s i n Eq. (72) a r e a v a i l a b l e ; s u b s t i t u t i n g them i n t o t h e equation and c a r r y i n g o u t t h e indicated operations w i l l give t h e desired values. Example 2
- Now
consider Eq. (73), pages 23 and
24, t h e gas mass
balance f o r t h e fuel-pump bowl.
a.
b.
The observable q u a n t i t i e s f o r t h i s example i n c l u d e :
R
=
4635.4 in./"R
All
=
0.48 i n ?
A,,
=
9.98 i n .2
,
(A221
,
.
There a r e no p h y s i c a l p r o p e r t i e s such as t h e f u e l s a l t d e n s i t y
which depend only on t h e observable q u a n t i t i e s .
That is, t h e gas d e n s i t y
i s dependent a l s o on t h e system p r e s s u r e which must be obtained from t h e o r i g i n a l model s o l u t i o n .
Therefore it i s necessary t o u s e t h e o r i g i n a l
model t o c a l c u l a t e t h e remainder of t h e c o n s t a n t s . c.
The gas d e n s i t i e s may be c a l c u l a t e d by u s i n g Eq. (l), page
For example,
4.
48 I I
where (P )
i s obtained from t h e o r i g i n a l h y d r a u l i c model s o l u t i o n ,
0 11
i.e.,
70.0 p s i a
=
11
.
.
Thus Eq. (A25) becomes
1
-
70.0 1670
( P ~ o ) -~ ~4635.4
'
Using t h e values of
(P )
=
27.7
psia (specified),
14
=
=
(Po),
(p )
=
0 12
19.6 p s i a
.
from t h e s o l u t i o n by t h e o r i g i n a l hydraulic model, t h e remaining gas d e n s i t y terms can be c a l c u l a t e d . The gas mass flow r a t e f o r region 11 w a s c a l c u l a t e d as follows: The v o i d f r a c t i o n i n region 9 may be expressed as
m
g9 -
a
9
= m g9
?E9
m
-+-
f9
Fgg
Of9
49 W
But from Eqs solving f o r m
g9’
. (53)
and
(54), page 18, a
z
1
a
9
.
Therefore,
we g e t
Now t h e f u e l pump v o l u m e t r i c flow r a t e i s 1200 gpm, a11 of which p a s s e s through r e g i o n
9; t h e bypass flow r a t e i s 60 gpm, or 5 p e r c e n t of t h i s .
Because t h e p r e s s u r e s and temperatures of t h e gas i n r e g i o n s 9 and 11 a r e assumed t o be equal, t h e d e n s i t y o f t h e gas i n t h e s e two r e g i o n s i s a l s o t h e same.
Therefore, it w a s assumed t h a t
5 p e r c e n t of t h e t o t a l gas
mass flow r a t e also flowed through t h e bypass, i . e . ,
(A34) i n t o Eq. (A35) g i v e s
S u b s t i t u t i n g Eq.
where
(a )
i s o b t a i n e d from t h e s o l u t i o n p r e s e n t e d by t h e o r i g i n a l hy-
0 1
d r a u l i c model.
Numerically, Eq.
( ~ 3 6 )i s
0.004
(m go
11
=
Ooo5
(mgo ) 11
=
8.12 x
1 - 0.004 lb/sec.
0.087
x 9.24 x
lod6
, (A38)
Furthermore, when t h e r e i s no mass accumulation i n t h e fuel-pump bowl,
50 1
The o r i f i c e c o e f f i c i e n t s f o r t h e gas a r e determined by u s i n g E q s .
-
C
gll
C
gll
8.12 x 0.40J2
=
5.61 x
x 49.4
x 9.24 x
10’~
sec
.
Similarly,
0.12 x
-
C
g12
9.98J2
x 2.6 x
x 1.0
S i m i l a r c a l c u l a t i o n s were made t o o b t a i n t h e remaining coeff i c i e n t s f o r t h e 16 e q u a t i o n s ( E q s . (72) through
(87)) which make up t h e
extended h y d r a u l i c model: t h e r e s u l t s a r e p r e s e n t e d i n Table A 1 and t h e f o l l o w i n g list.
.
.
,
‘
f
TABLF: Al Constants Used to Calculate
A, in?
--
Cf, in. -4 set-l c g7
the Coefficients
9.98
--
14.6
2.8
--
--
--
--
5.6x10-~
3.6~10'~
--
1.5x10-2
1.9x10-;
--
--
20.1
20.1
--
--
--
--
--
--
8.1x10-~
8.1x10+
--
4.8x10-~
4.8x10-'
lg.6
lg.6
70.0
70.0
lg.6
20.6
27.7
14.7
78.4
78.4
--
78.4 em
--
--
--
--
--
mfo7 lb/set
--
--
m lb/set go7 Po, psia
--
70.0
T o7 vfoJ
- -4 see-l In*
"R in./sec
aO
pf,
lb/in3
pgo, lb/in?
1670 55.5
1670 55.9
4.0x10-3
1.2x1o-z
--
--
9.2x1@
for Eq. (88)
Used in the A and C Matrices
2.5~10~~
1670 55.9 1.2x10-*
0.087 2.5~10~~
1670 55.5
0.48
1670 481
4.ox1o-3
4.ox1o-3
0.087
0.087
g.2x1o-6
g.2x10w6
1670
1670
0.304
23.2
--
--
530 --
1.2x1o-2
--
--
--
--
--
1.1~10~~
7.2x10-'
0.087 0.087 2.5~10~~
--
530
0.30'
wl P
The remaining c o n s t a n t s a r e :
and
v
s
=
j,,
where cu i s s p e c i f i e d ;
H
=
582 i n . ,
K
=
s p e c i f i e d v a l u e between 0.8 and 1.0,
f13
-
5184
in?
.
53 v
ORNL-TM-3007
.
INTERNAL DISTRIBUTION
1. N. J . Ackerman 2. S. J . B a l l 3. H. F. Bauman 4. S. E. B e a l l 5. E. S. B e t t i s 6. R. Blumberg 7. E . G. Bohlmann 8. R . B. Briggs 9. R . A. Buhl 10. W. B. C o t t r e l l 11. J . L. Crowley 12. F. L. C u l l e r 13. J . R . Engel 14. D . N . Fry 1 5 . C . H . Gabbard 16. W. R. G r i m e s 17. A. G . G r i n d e l l 18. R . H. Guymon 1 9 . P. N . Haubenreich 20. A . Houtzeel 21. T. L. Hudson 22. P. R . Kasten 23. T. W. K e r l i n 24. R . J . K e d l 53-54.
55-56. 57-59. 60.
61.
25 26. 27 28. 29 30.
31. 32.
33. 34. 35. 36. 37
38. 39 40.
41. 42.
43.
44. 45. 46. 47-51.
52 *
A. I . Krakoviak T . S. Kress M. I . Lundin R . N . Lyon H. G. MacPherson H . E . MacPherson L . C . Oakes A. M. P e r r y H. B. P i p e r B. E. P r i n c e J . L. Redford J . C . Robinson M . W. Rosenthal H. M. Roth, AEC-OR0 A. W. Savolainen Dunlap S c o t t M. J . Skinner I. Spiewak D. A. Sundberg J . R. Tallackson R . E. Thoma D. B. Trauger W. C . U l r i c h Gale Young
C e n t r a l Research L i b r a r y ( C R L ) Y-12 Document Reference S e c t i o n (DRS) Laboratory Records Department (LRD) Laboratory Records Department Record Copy (LRD-RC ) ORNL P a t e n t W f i c e
-
EXTERNAL DISTRIBUTION 62. 63.
64.
65.
66. 67-81.
S. J . Hanauer, U n i v e r s i t y o f Tennessee, K n o x v i l l e , Tennessee 37916 J. E. Mott, U n i v e r s i t y of Tennessee, Knoxville, Tennessee 37916 P. F. Pasqua, U n i v e r s i t y o f Tennessee, Knoxville, Tennessee 37916 R . C. Steff‘y, Jr., Tennessee Valley A u t h o r i t y , 303 Power B u i l d i n g , Chattanooga, Tennessee 37401 Laboratory and U n i v e r s i t y Division, QRO D i v i s i o n of‘ Technical Information Extension, OR