ORNL-TM-3007 by Jon Morrow

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MASTER

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HELP

O A K RIDGE N A T I O N A L LABORATORY operated by

UNION CARBIDE CORPORATION NUCLEAR DIVISION

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for the

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-

- -~

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:

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,

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.

TABU OF CONTENTS PAGE

CHAPTER

........................... 1. DESCRIFTION OF THE OPEN-LOOP MODEL . . . . . . . . . . . . .

II!I"RODUCTION

DESCRIFTION OF THE EXTENDED MODEL

..............

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

.....................

..... i n Deriving t h e Equations f o r t h e Model . .

Assumptions Made i n t h e Development of t h e Model

Procedures Used

RESULTS OF CALCULATIONS MADE WITH THE EXTENDED MODEL

............... 4. CONCWSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . . . LIST OF REFEFd3NCES . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . COMPARED TO EXPERIMENTAL DATA

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

..................... ..................

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,

F = O . O . . . . . . . . . . . . . . . . . . . . . . . . .

Hydraulic Performance o f t h e Molten S a l t Reactor Experiment F u e l Pump

................

Molten S a l t Reactor Experiment F u e l Pump and Fuel-Pump Bowl.

Extended Hydraulic Model of t h e Molten S a l t Reactor Experiment Fuel S a l t Loop

................

Modulus of t h e Neutron Flux-to-Pressure Frequency-Response Function for t h e Molten S a l t Reactor Experiment

.....

LIST OF SYMBOLS

Description

Symbol A

area

‘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

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

neutron d i f f u s i o n c o e f f i c i e n t

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 )

a c c e l e r a t i o n due t o g r a v i t y

gravitational constant heat transfer coefficient

e n t h a l p y of t h e f u e l s a l t

e n t h a l p y of t h e gas

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

thermal c o n d u c t i v i t y of t h e moderator

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

mass f l o w rate

f u e l s a l t mass flow rate

M Mf

gas mass flow r a t e mass mass of f u e l s a l t

v ii

viii

Symbol

PI1

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

f u e l pump v o l u m e t r i c flow r a t e

power d e n s i t y

u n i v e r s a l gas c o n s t a n t

Laplace-transformed time v a r i a b l e

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)

tine variable

a b s o l u t e temperature

f u e l s a l t temperature

gas temperature

...

TM Tw Uf U

v n'

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.

ORNL- DWG 68 - 8 4 1 7 OUTLET

UPPER PLENUM

-if

Y-

t t CHANNELED REGION

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

UPPER PLENUM

-f

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 ,

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

a Ipfvf2(1 pW

- 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)

and V

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 :

Power dens i t y

h(T

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

- B)cf

+),

cal@

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.

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)

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;

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:

t h e upper plenum, L5: and

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

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.

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

OANL-

DWG

69-

10172Ai

SHAFT

OFFGA

S BUBBLER

SPRAY RING

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.

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

IMPELLER MOLTEN-SALT

DIAM:

I1 !+ in. TEMPERATURE:

1200°F

CONSTANT RESISTANCE

z 2 W

I J 2 2

i 20

10 PUMP HYDRAULIC BALANCE LINE-

‘/

I 0

200

400

600

600 Q , FLOW

FIGURE

Hydraulic Fuel

‘i-

Performance Pump

I 1000

RATE

(gpm)

the

Molten

I200

Salt

Reactor

I 1400

Experiment

1600

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

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 .

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 -

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

3. Pressure-head relationship for the fuel pump: From Equation

(7)

But

m Q = fs P

Pg-P,

@ - o m

â&#x20AC;&#x2122;

(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

(35)

â&#x20AC;&#x2122;

Assuming t h a t p

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:

= E58

m E59

which reduces t o

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

- 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

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

f12

(46)

or CP&

- a> VfAI, =

Assuming t h a t

Pf6

[Pf(l =

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 =

(Pg

(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)

(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

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

9 it i s

assumed t h a t

(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

f12

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.

This g i v e s t h e l i n e a r i z e d form of Eq. (40) as

The l i n e a r i z e d v e r s i o n of Eq. (l),page

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.

and Vf

( 5 ) , Page 5, is 1AV

K - a

AVf

Vfo0

(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

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?,

(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:

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

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 câ&#x20AC;&#x2122;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:

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-

5 (9-

pgo)9

.0624 ( P ~ A AV, ) ~,

(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)

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 .

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

('go

:),

Avfi

(77)

['go

'fo

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,@

.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,

- %,,-

AVf6

.127 AVf

11:

(78)

6, 8, and 12:

(79)

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

Volume b a l a n c e f o r t h e fuel-pump bowl:

10.

(k)

m f 1 3 + â&#x20AC;&#x2122;fl3

[

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

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)

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

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

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, =

, 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,

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.

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

102

ORNL-DWG 70-1374

, LEGEND

CASE t

EXPERIMENTAL DATA

C 3

-. A L

-I

0 3

I w [r

3 v) v)

a -I

5a ,

-I

W U 3

0 W

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:

z 0 IV

z 3

LL W

8 v,

10-2

W [L

CY w

W [L

cn W

X 3 -I

10-3

z 0

t3 W

I ILL

cn 3 -I

n 0

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

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.

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.

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 .

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.

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.

21.

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

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).

Neal and Z i v i , STL 372-14 (l), p.

S . J . Ball and R. K. Adams,

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.

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.)

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;

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

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.

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

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

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 .

(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.

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 :

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,

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

Thus Eq. (A25) becomes

70.0 1670

( P ~ o ) -~ ~4635.4

Using t h e values of

(P )

27.7

psia (specified),

(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

g9 -

= m g9

?E9

-+-

Fgg

Of9

49 W

But from Eqs solving f o r m

g9â&#x20AC;&#x2122;

. (53)

and

(54), page 18, a

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

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 .

gll

8.12 x 0.40J2

5.61 x

x 49.4

x 9.24 x

10â&#x20AC;&#x2122;~

sec

Similarly,

0.12 x

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.

â&#x20AC;&#x2DC;

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

8.1x10-~

8.1x10+

4.8x10-~

4.8x10-'

lg.6

70.0

lg.6

20.6

27.7

14.7

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

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

0.087

g.2x1o-6

g.2x10w6

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

j,,

where cu i s s p e c i f i e d ;

582 i n . ,

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

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