Application of a boiling water loop model to boiling sodium

(1)

Application of a boiling water loop model to boiling sodium

Citation for published version (APA):

Bogaardt, M., & Spigt, C. L. (1967). Application of a boiling water loop model to boiling sodium. Nuclear

Engineering and Design, 5(4), 465-476. https://doi.org/10.1016/0029-5493(67)90104-5

DOI:

10.1016/0029-5493(67)90104-5

Document status and date:

Published: 01/01/1967

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be

important differences between the submitted version and the official published version of record. People

interested in the research are advised to contact the author for the final version of the publication, or visit the

DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page

numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

(2)

NUCLEAR ENGINEERING AND DESIGN 5 (1967) 465-476. NORTH-HOLLAND PUBLISHING COMP.. AMSTERDAM

A P P L I C A T I O N OF A B O I L I N G W A T E R L O O P M O D E L

T O B O I L I N G S O D I U M

M. BOGAARDT and C. L. SPIGT

Technological University of Eindhoven. Eindhoven, The Netherlands

Received 13May1967

In this report results are given of an analytical study on the applicability to a boiling sodium system of existing physical-mathematical formulations for the performance characteristics of a boiling water system under steady or transient conditions. Furthermore, the effect of liquid superheat on the calcu- lated performance characteristics has been evaluated. It is sho~m that the application of theoretical models developed for water systems to liquid metal systems does not lead to difficulties but that the effect of liquid superheat should be taken into account in case appreciable superheats are found to occur.

I. INTRODUCTION In the T e c h n o l o g i c a l L a b o r a t o r y of R e a c t o r C e n t r u m N e d e r l a n d at P e t t e n a r e s e a r c h p r o - g r a m m e h a s b e e n s t a r t e d to i n v e s t i g a t e the b e - h a v i o u r of s o d i u m u n d e r a c c i d e n t c o n d i t i o n s . F o r t h i s p u r p o s e a 350 kW e x p e r i m e n t a l liquid s o d i - u m loop i s u n d e r c o n s t r u c t i o n . The e x p e r i m e n t a l p r o g r a m m e i s to be hacked up by t h e o r e t i c a l s t u d i e s . The f i r s t s t e p h a s b e e n to d e t e r m i n e i f the m o d e l s u s e d for d e s c r i b i n g the d y n a m i c s of b o i l i n g w a t e r s y s t e m s can be applied in the c a s e of b o i l i n g s o d i u m , b e a r i n g in m i n d the l a r g e d i f f e r e n c e in the p h y s i c a l p r o p e r - t i e s of the two l i q u i d s . It i s a l s o of i n t e r e s t to know if liquid s u p e r h e a t w i l l a f f e c t the d y n a m i c b e h a v i o u r of the sod~ium s y s t e m to any g r e a t e x - tent. T h e s a m e g e o m e t r y h a s been adopted for the b o i l i n g s o d i u m s y s t e m a s was u s e d for the b o i l i n g w a t e r s y s t e m a t the T e c h n o l o g i c a l U n i - v e r s i t y a t E i n d h o v e n , t h e r e b y f a c i l i t a t i n g c o m - p a r i s o n of r e s u l t s . In the a n a l y s i s a t t e n t i o n i s g i v e n to the d y n a m i c c h a r a c t e r i s t i c s of s o d i u m s y s t e m s with n a t u r a l a s w e l l a s f o r c e d c i r c u l a - tion. 2. THEORY Many p h y s i c a l - m a t h e m a t i c a l f o r m u l a t i o n s h a v e b e e n r e p o r t e d in l i t e r a t u r e a t t e m p t i n g to d e s c r i b e and to c a l c u l a t e the p e r f o r m a n c e c h a r - a c t e r i s t i c s of a t w o - p h a s e flow s y s t e m u n d e r s t a t i o n a r y o r t r a n s i e n t c o n d i t i o n s o r to a n a l y z e p a r t i c u l a r p h e n o m e n a o b s e r v e d such a s , f o r i n - s t a n c e , the o c c u r r e n c e of s e v e r e h y d r a u l i c o s - c i l l a t i o n s . In the p a s t few y e a r s a new a p p r o a c h to t h e s e p r o b l e m s h a s b e e n d e v e l o p e d in the L a b - o r a t o r y for Heat T r a n s f e r and R e a c t o r E n g i - n e e r i n g of the T e c h n o l o g i c a l U n i v e r s i t y of E i n d - hoven. T h i s t h e o r y w a s r e c e n t l y d e s c r i b e d by Spigi [1] and h a s b e e n u s e d for the p r e s e n t study. The a p p r o a c h s t a r t s by f o r m u l a t i n g the four b a s i c e q u a t i o n s of m o t i o n for the t w o - p h a s e flow. T h e s e e q u a t i o n s a r e f o r m e d by the l a w s of the c o n s e r v a t i o n of m a s s , m o m e n t u m and e n e r g y of the m i x t u r e and an e q u a t i o n for the n u m b e r d e n - s i t y d i s t r i b u t i o n of b u b b l e s d e s c r i b i n g the t r a n s p o r t of s t e a m in the liquid p h a s e . The n u m - b e r d e n s i t y d i s t r i b u t i o n of b o b b l e s i s g o v e r n e d by the p r o c e s s of bobble f o r m a t i o n and d i f f u s i o n , d e m i x i n g e f f e c t s n e a r the w a l l , c o n v e c t i o n of b u b b l e s , i n t e r a c t i o n e f f e c t s b e t w e e n a d j a c e n t b o b b l e s , and growth of the bubbles in the m i x - t u r e . T h e s e four b a s i c e q u a t i o n s h a v e b e e n f o r - m u l a t e d for a c o o l a n t c h a n n e l of c o n c e n t r i c a n - n u l a r s h a p e as a f u n c t i o n of the t h r e e i n d e p e n d e n t v a r i a b l e s , i.e. the t i m e , t , the c o o r d i n a t e a l o n g the c o o l a n t channel, z, and the r a d i a l c o o r d i n a t e , r . To t h e s e four e q u a t i o n s , the e q u a t i o n s of s t a t e h a v e b e e n added. T h e y a r e e x p r e s s i o n s for the p h y s i c a l q u a n t i t i e s of the fluid a s a function of the l o c a l t e m p e r a t u r e s . The q u a n t i t i e s of i m p o r - t a n c e a r e the d e n s i t y of the v a p o u r and liquid p h a s e s , the s p e c i f i c h e a t , the h e a t of e v a p o r a t i o n and the change in p r e s s u r e with t e m p e r a t u r e u n - d e r s a t u r a t e d c o n d i t i o n s . In the a p p r o a c h u n d e r d i s c u s s i o n t h e s e p h y s i c a l q u a n t i t i e s v a r y a l o n g

(3)

466 M. BOGAARDT and C. L. SPIGT the c o o l a n t channel so any p r e s s u r e and t e m p e r -

a t u r e e f f e c t s along the c o o l a n t channel and c o m - p r e s s i b i l i t y e f f e c t s a r e taken into account.

F o r d e s i g n c a l c u l a t i o n s the b a s i c e q u a t i o n s have been i n t e g r a t e d with r e s p e c t to r b e t w e e n the l i m i t s of the i n n e r and o u t e r r a d i u s of the annular coolant channel. It was shown by Spigt [1] that t h i s i n t e g r a t i o n can be p e r f o r m e d by i n - t r o d u c i n g e m p i r i c a l or t h e o r e t i c a l c o r r e l a t i o n s taking into account the r a d i a l v a r i a t i o n s of the flow q u a n t i t i e s . The t h r e e m o s t i m p o r t a n t c o r r e - l a t i o n s a r e those (a) for the s l i p r a t i o , (b) for the t w o - p h a s e f r i c t i o n f o r c e and (c) for the h e a t d i v i s i o n p a r a m e t e r K which d e n o t e s the f r a c t i o n of the l o c a l heat input that i s supplied to the s t e a m phase. B e s i d e s , o t h e r c o r r e l a t i o n s a r e needed for an a c c u r a t e e v a l u a t i o n of the p e r - f o r m a n c e c h a r a c t e r i s t i c s of a b o i l e r . K has to be defined along the length of the coolant channel. By m e a n s of K any s u p e r h e a t i n g in the bulk b o i l - ing r e g i o n or any s u b c o o l e d boiling can be i n t r o - duced.

The a s s u m p t i o n n o r m a l l y m a d e that the s u b - cooled r e g i o n i s not at t h e r m o d y n a m i c e q u i l i b r i - um and that the bulk boiling r e g i o n i s in c o m - p l e t e t h e r m o d y n a m i c e q u i l i b r i u m has b e e n a b a n - doned in this study. T h i s a s s u m p t i o n a p p e a r s to have been m a d e f a i r l y a r b i t r a r i l y . The m a i n a d - v a n t a g e of the p r o p o s e d c h o i c e for the p a r a m e t e r i s that only a s i n g l e s e t of e q u a t i o n s is u s e d for the e n t i r e channel and that the a p p r o a c h can be used for any t w o - p h a s e flow s y s t e m .

In the study s p e c i a l a t t e n t i o n was g i v e n to the f o r m u l a t i o n of the boundary c o n d i t i o n s for the boiling channel and thus to the d e s c r i b i n g e q u a - tions, which i n t r o d u c e the c h a r a c t e r i s t i c s of the o t h e r p a r t s of the s y s t e m , e.g. the c o n d e n s o r , d o w n c o m e r , pump, etc. A d e t a i l e d f o r m u l a t i o n of the boundary c o n d i t i o n s was obtained by s e t t i n g up the c o n s e r v a t i o n l a w s for the e x t e r n a l p a r t s of the t e s t c i r c u i t .

F o r studying the d y n a m i c b e h a v i o u r of a t w o - p h a s e flow s y s t e m the d e s c r i b i n g e q u a t i o n s have been m a d e d i m e n s i o n l e s s . T h i s p r o c e d u r e holds c e r t a i n a d v a n t a g e s , such as a m o r e r e a d i l y a p - p a r e n t c o m p a r i s o n b e t w e e n v a r i o u s o p e r a t i n g c o n d i t i o n s , g e o m e t r i e s and w o r k i n g fluids. A f t e r that the equations have b e e n l i n e a r i s e d with r e - s p e c t to s m a l l d e v i a t i o n s f r o m the s t e a d y state. Then a study i s m a d e of the r e s p o n s e to a s i n u - s o i d a l m o d u l a t i o n in the c o n t r o l l i n g v a r i a b l e s , e.g. h e a t input, h e a t r e m o v a l or m a s s flow. In this way t r a n s f e r functions m a y be obtained f r o m t h e s e v a r i a b l e s to dependent v a r i a b l e s such a s m a s s flow, void f r a c t i o n , etc. It should be noted that the equations d e s c r i b i n g the s t e a d y - s t a t e

p e r f o r m a n c e c h a r a c t e r i s t i c s a r e not l i n e a r i s e d . The s y s t e m of e q u a t i o n s i s i n t e g r a t e d n u m e r i - c a l l y with r e s p e c t to z and has b e e n p r o g r a m m e d for a d i g i t a l c o m p u t e r as d e s c r i b e d by Van d e r W a l l e [2]. The c o r r e l a t i o n functions a r e i n t r o - duced in the p r o g r a m m e in the f o r m of s u b - r o u - t i n e s , which r e s u l t in a l a r g e f l e x i b i l i t y .

3. S T A B I L I T Y C R I T E R I A

As no g e o m e t r i c a l data of e x i s t i n g s o d i u m s y s t e m s w e r e e a s i l y a v a i l a b l e the c a l c u l a t i o n s w e r e p e r f o r m e d for the g e o m e t r y of the b o i l i n g w a t e r loop of the T e c h n o l o g i c a l U n i v e r s i t y of Eindhoven. T h i s m a k e s it a l s o p o s s i b l e to c o m - p a r e the r e s u l t s with t h o s e a l r e a d y obtained for b o i l i n g w a t e r in the s a m e s y s t e m . While, up t i l l now, t h e s e r e s u l t s w e r e only a v a i l a b l e for n a t u r a l l y c i r c u l a t i n g c o n d i t i o n s , the c a l c u l a t i o n s have b e e n p e r f o r m e d without c o n s i d e r i n g the c h a r a c t e r i s t i c s of any pump. A flow s h e e t of the s y s t e m i s g i v e n in fig. 1. The m a i n p a r t c o n s i s t s of a p r e s s u r e v e s s e l with a d i a m e t e r of 150 m m and a length of about 3 m, an i n t e r n a l s h r o u d with a d i a m e t e r of 50 m m and a length of 2.7 m and, in the c e n t e r , an e l e c t r i c a l l y h e a t e d s t a i n - l e s s s t e e l tube with a d i a m e t e r of 33.84 m m and a length of 2.4 m. By the h e a t t r a n s f e r r e d to the liquid n a t u r a l c o n v e c t i o n o c c u r s , in upward d i - r e c t i o n in the s h r o u d , and d o w n w a r d s in the v e s - sel. B e f o r e e n t e r i n g the s h r o u d the liquid p a s s e s through a s u b c o o l e r . The h y d r a u l i c i n s t a b i l i t i e s that a r e of i n t e r e s t h e r e a r e t h o s e that a r e t y p i c a l for a n a t u r a l c i r - c u l a t i o n s y s t e m . In a f o r c e d c i r c u l a t i o n b o i l e r with v e r y s t e e p h e a d - f l o w c h a r a c t e r i s t i c s of the pump, no h y d r a u l i c i n s t a b i l i t i e s , such as c o n s i d - e r e d h e r e , h a v e b e e n found. T h i s s u g g e s t s c o n - s i d e r i n g a f o r c e d c i r c u l a t i o n b o i l e r , as shown in figs. 2 and 3. A pump i s p r e s e n t in the down- c o m e r , which pump g e n e r a t e s a p r e s s u r e r i s e , c o r r e s p o n d i n g with a r i s e in s a t u r a t i o n t e m p e r a - t u r e AT s. The pump m e a s u r e s the f l u c t u a t i o n s in m a s s flow

V1, i

and t r a n s l a t e s t h e s e v a r i a t i o n s into a r i s e in s a t u r a t i o n t e m p e r a t u r e of

ATs,i.

The index i d e n o t e s v a r i a t i o n s f r o m the s t e a d y - state. Now it i s a s s u m e d that the s y s t e m i s brought into e x c i t a t i o n by c o n t r o l l i n g the pump with a s i n u s o i d a l s i g n a l c o r r e s p o n d i n g to a d e - s i r e d f l u c t u a t i o n in m a s s flow V 1 i" The m a g n i - tude of

ATs, i

( s e e fig. 3) i s dependent upon the d i f f e r e n c e b e t w e e n

V1, i

and l~l, i. The d e p e n d - e n c e m a y be s i m p l y e x p r e s s e d by:

(4)

APPLICATION OF BOILING WATER LOOP MODEL 467

r--- ---1

I-________,

multi-manometer

remote level gauge 1 heating clement

w

-electrode

T -thermocouple

8 p -pressure gauge

H-

sewage

(5)

468 M. BOGAARDT and C. L. SPIGT

]

_~X

2x4

12x/,

differentiat

boundary

equations for

J conditions

riser

:ontrotter

ve

( Ts.i)do

increase

iding to

. . . .~se in

saturation temp. hTs

Fig. 2. Forced circulation boiler.

w h e r e Hp i s a m e a s u r e of the pump c h a r a c t e r i s - t i c s . The l a r g e r Hp b e c o m e s , the h i g h e r i s the p r e s s u r e r i s e upon a c e r t a i n change in m a s s flow. A c t u a l l y , Hp i s a t r a n s f e r function and thus a c o m p l e x quantity. The following t r a n s f e r f u n c - t i o n s for the o p e n - l o o p without a pump and with a cut, a s i n d i c a t e d by the b r o k e n line in fig. 3, a r e defined a s :

l ( T s ' i ) d ° f , = t

Vl'i I

. (2) G1 =~(Ts, )i---n)Hp=0

G2 ~(Ts,i)in~Up=O

H e r e G 1 and G 2 d e t e r m i n e the change in s a t u r a - tion t e m p e r a t u r e ( p r e s s u r e ) and m a s s flow r a t e at the outlet of the d o w n c o m e r upon a v a r i a t i o n in s a t u r a t i o n t e m p e r a t u r e ( p r e s s u r e ) at the i n l e t of the heated channel.

The t r a n s f e r function for the c l o s e d - l o o p f r o m i m p o s e d m o d u l a t i o n in m a s s flow to the s a t u r a - tion t e m p e r a t u r e at the i n l e t of the channel can then be w r i t t e n a s :

Ts,i _

Hp

V1, i

1 -GI + HpG 2

(3)

_ ~

~ A T s ump-

characterist ics

-'*

/cut. open toop

**(Ts, i)in :(Ts, i)do *ATs. i**

Fig. 3. Block diagram of forced circulation boiler. An i n s t a b i l i t y condition i s obtained when the t r a n s f e r function defined in eq. (3) a p p r o a c h e s infinity. T h e r e a r e two c o n d i t i o n s w h e r e t h i s i s indeed the c a s e :

a) F o r l a r g e v a l u e s of Hp, the s y s t e m w i l l a l - ways be s t a b l e , as long a s G 2 d o e s not a p p r o a c h z e r o . F o r l a r g e v a l u e s of Hp, the t r a n s f e r f u n c - tion for the c l o s e d loop (eq. (3)) a p p r o a c h e s the v a l u e of

1/G 2.

F r o m the d e f i n i t i o n of G2, it may be concluded that G 2 w i l l n o r m a l l y n e v e r a p - p r o a c h z e r o , u n l e s s r e s o n a n c e c o n d i t i o n s a p p e a r within the boiling channel.

b) F o r s u f f i c i e n t l y low v a l u e s of H , the t r a n s f e r function for the c l o s e d loop (eq. (~)) a p - p r o a c h e s the v a l u e of 1/(1 - G 1 ) , and an i n s t a b i l i - ty condition i s obtained when G 1 b e c o m e s +1 for s o m e f r e q u e n c y .

It a p p e a r s to be a d v a n t a g e o u s , t h e r e f o r e , to c a l c u l a t e the open loop t r a n s f e r functions G 1 and G 2 and s e e w h e t h e r the m o d u l u s of G 1 a p p r o a c h - e s a condition of +1, and at the s a m e t i m e w h e t h - e r the p h a s e a n g l e h a s r e a c h e d a v a l u e of about 0 ° or 360 ° or w h e t h e r the modulus of G 2 has b e - c o m e s u f f i c i e n t l y s m a l l .

The c a l c u l a t e d s t e a d y - s t a t e c h a r a c t e r i s t i c s and the t h r e s h o l d channel p o w e r s for the o n s e t of

(6)

APPLICATION OF BOILING WATER LOOP MODEL 469 f l o w o s c i l l a t i o n s i n t h e b o i l i n g w a t e r s y s t e m , c a l c u l a t e d b y m e a n s of G1 a n d G 2 , c o m p a r e d q u i t e w e l l w i t h e x p e r i m e n t a l l y d e t e r m i n e d v a l - u e s , s e e S p i g i [1]. F u r t h e r m o r e , i t w a s s h o w n t h a t i n a b o i l i n g w a t e r s y s t e m t h r e e d i f f e r e n t t y p e s of f l o w o s c i l l a t i o n s m a y o c c u r , w i t h f r e - q u e n c i e s of r o u g h l y 0 . 0 3 , 1 a n d 15 c p s . 4. D E S C R I P T I O N O F T H E C O N D I T I O N S T h e m o s t i m p o r t a n t g e o m e t r i c a l a n d h y d r o - d y n a m i c d a t a o f t h e b o i l i n g l o o p a r e g i v e n i n t a - b l e 1. Table 1

G e o m e t r i c and hydrodynamic data of the boiling loop of the Technological U n i v e r s i t y of Eindhoven.

Quantity Units Value

length of channel

leng~th of heated p a r t of channel length of downcomer c r o s s - s e c t i o n a l a r e a of channel c r o s s - s e c t i o n a l a r e a of d o w n c o m e r a r e a of w a t e r s u r f a c e in c o n d e n s e r volume of s u b c o o l e r c o n d e n s e r volume p a r t of the c o n d e n s e r occupied by s t e a m

height of w a t e r level above channel inlet o u t e r d i a m e t e r of heating e l e m e n t h y d r a u l i c d i a m e t e r of channel i n n e r d i a m e t e r of shroud p r e s s u r e l o s s coefficient at channel inlet p r e s s u r e l o s s coefficient at channel exit m 2.70 m 2.4 m 4.42 m 2 0.00106 m 2 0.00958 m 2 0.495 m 3 0.0330 m 3 0.1454 - 0.75 m 2.74 m 0.03384 m 0.01616 m 0.05 - 1 . 4 - 0 T h e c a l c u l a t i o n s h a v e b e e n p e r f o r m e d a t t h r e e p r e s s u r e s , 1, 8.7 a n d 16.4 a t m , w i t h n o s u b c o o l i n g a t t h e i n l e t . T h e p h y s i c a l c o n s t a n t s f o r b o i l i n g s o d i u m w e r e d e r i v e d f r o m S p i l l e r [3]. T h e d a t a u s e d a r e g i v e n i n t a b l e 2. A s n o c o r r e l a t i o n f o r t h e s l i p r a t i o a n d two p h a s e f r i c t i o n w e r e a v a i l a b l e f o r b o i l i n g s o d i u m , t h e c o r r e l a t i o n s f o r b o i l i n g w a t e r a t t h e s a m e s y s t e m p r e s s u r e w e r e u s e d . F o r t h e s l i p r a t i o t h e c o r r e l a t i o n of Z u b e r a n d F i n d l a y [4], b a s e d o n e x p e r i m e n t a l d a t a of t h e T e c h n o l o g i c a l U n i - v e r s i t y of E i n d h o v e n , h a v e b e e n a p p l i e d . T h e c o r r e l a t i o n of M a r t i n e l l i - N e l s o n a s f o r m u l a t e d b y J o n e s [5], h a s b e e n u s e d f o r t h e c a l c u l a t i o n of t h e two p h a s e f r i c t i o n l o s s e s . It i s k n o w n f r o m e x p e r i m e n t a l d a t a t h a t a p - p r e c i a b l e s u p e r h e a t s of t h e l i q u i d p h a s e c a n o c - c u r i n b o i l i n g s o d i u m . A s a l r e a d y m e n t i o n e d i n t h e t h e o r y i t i s p o s s i b l e to s i m u l a t e t h i s p h e - n o m e n o n by t h e c h o i c e of K. T h i s c a n e a s i l y b e d o n e b y v a r y i n g a c o n s t a n t F 1 , t h e s u p e r h e a t p a - r a m e t e r , i n t h e c o r r e l a t i o n f o r K. T h e i n f l u e n c e of t h e a m o u n t of s u p e r h e a t o n t h e r e s u l t s h a s b e e n c a l c u l a t e d a t a s y s t e m p r e s s u r e of 16.4 a t m . A t t h i s p r e s s u r e t h e s u p e r h e a t p a r a m e t e r F 1 h a s b e e n v a r i e d f r o m 500 to 50 w h i c h c o r r e - s p o n d s w i t h a v a r i a t i o n i n t h e s u p e r h e a t a t t h e e x i t of t h e c h a n n e l of 2 t o 1 5 ° C . In a l l t h e o t h e r c a l c u l a t i o n s F 1 w a s 500. T h e s t e a d y - s t a t e r e s u l t s h a v e b e e n c a l c u l a t e d a s a f u n c t i o n of c h a n n e l p o w e r . In t h e f o l l o w i n g r e s u l t s of t h e c a l c u l a t i o n s of t h e r e c i r c u l a t i o n r a t e , t h e v o i d f r a c t i o n a n d t h e t e m p e r a t u r e s o f t h e l i q u i d a n d v a p o u r p h a s e a l o n g t h e c h a n n e l a r e g i v e n . T h e h a r m o n i c a n a l y s i s h a s b e e n c a r r i e d o u t f o r a l i m i t e d n u m b e r of c h a n n e l p o w e r s i n t h e f r e q u e n c y r a n g e of 0 . 0 3 to 17 c p s . R e s u l t s a r e Table 2 P h y s i c a l c o n s t a n t s for sodium [3]. Quantity Units p r e s s u r e s a t u r a t i o n t e m p e r a t u r e liquid density vapour density specific heat heat of e v a p o r a t i o n heat conduction coefficient dynamic v i s c o s i t y

v a r i a t i o n of liquid density with s a t u r a t i o n t e m p e r a t u r e

v a r i a t i o n of vapour .density with s a t u r a t i o n t e m p e r a t u r e v a r i a t i o n of v a p o u r :pressure with s a t u r a t i o n t e m p e r a t u r e atm, abs o c k g / m 3 k g / m 3 J/kgOC J / k g W / m °C N m / s e c 2 k g / m 3 o c k g / m 3 o c N / m 2 o c 1 880 739.5 0.28 1285 3.415 x 106 47.9 0.000169 -0.256 0.0025 950 8.7 1180 661.5 2.22 1392 3.106 x 106 33.4 0.000145 -0.2625 0.012 4800 16.4 1300 629 3.86 1445 2.9455 x 106 27.75 0.000140 -0.281 0.015 7780

(7)

470 M. BOGAARDT and C. L. SPIGT g i v e n of t h e c a l c u l a t i o n of G 1 a n d G 2 a n d t h e c l o s e d l o o p t r a n s f e r f u n c t i o n f r o m c h a n n e l p o w e r to t h e l i q u i d v e l o c i t y a t t h e i n l e t . F r o m t h e s e c a l c u l a t i o n s t h e i n s t a b i l i t y t h r e s h o l d c h a n n e l p o w e r a n d t h e f r e q u e n c y of t h e r e s u l t i n g f l o w o s - c i l l a t i o n s a r e d e d u c e d . 5. R E S U L T S

5.1. Results o f / h e s/eady-state calculations

T h e c a l c u l a t e d r e c i r c u l a t i o n r a t e Vci ( i . e . t h e l i q u i d v e l o c i t y V a t t h e c h a n n e l i n l e t ) i n m e t e r s p e r s e c o n d a n d t h e c a l c u l a t e d v o i d f r a c t i o n a t t h e Vci ! 0,9 0,8 0,7 0,6 \ " ~ . ]lS, Sa m \ o INSTABILITY \ ~ THRESHOLD ~, FI. = soo \ 0,5 50 100 150

Fig. 4. R e c i r c u l a t i o n r a t e as a function of channel power.

---m- GjKW

0,7

= = 2,02; t m.~. ,~ l a t i n ~ .--- " " " &7 a t m _- _/ 15,8 at m , . . - - ~ 1 ~ a t ~ _ . . ~ ----'--" - " - - / " " ~ ~ ~ S O D I U M _ _ _ W A T E R O I N S T A B I L I T Y T H R E S H O L D F| =S00 50 100 150 - - - ~ - %1< w

(8)

APPLICATION OF BOILING WATER LOOP MODEL 471 e x i t o f t h e c h a n n e l a e x a r e s h o w n i n f i g s . 4 a n d 5 a s a f u n c t i o n o f t h e c h a n n e l p o w e r Q f o r t h r e e s y s t e m p r e s s u r e s . A s i s s h o w n , t h e r e c i r c u l a - t i o n r a t e d e c r e a s e s w i t h i n c r e a s i n g c h a n n e l p o w - e r . T h i s m e a n s tha~: a n i n c r e m e n t a l i n c r e a s e i n c h a n n e l p o w e r c a u s e s t h e f r i c t i o n a l a n d a c c e l e r - a t i o n l o s s e s t o i n c r e a s e b y a l a r g e r a m o u n t t h a n t h e d r i v i n g h e a d . T h e i n c r e a s e i n v o i d f r a c t i o n w i t h i n c r e a s i n g c h a n n e l p o w e r i s c a u s e d by t h e i n c r e a s e d v a p o u r p r o d u c t i o n . I t c a n b e s h o w n t h a t a t f i r s t a p p r o x i m a t i o n a / ( 1 - ~ ) i s p r o p o r - t i o n a l to t h e c h a n n e l p o w e r w h i c h e x p l a i n s t h e 1320 1 ,3,8 0,9 Tt, Tsar =~. °C I 1316 0,8 <~(F L=SO) ~ - ~ ,3,, 0,7 ~ ' ~ I ~ ,31z o,6 ~

~ / ~

't(Ft:S°)

13,o o., .,m 1308 0,4 1306 0j3 V Tsat( Ft:500 en SO)

13o4

o,z

/ /

T t (F L : 500) =

o,i 0,2 0,3 0/. o,5 0,5 0,7 o,e o,g , 1

Fig. 6. Void f r a c t i o n and t e m p e r a t u r e s along the channel for two values of the s u p e r h e a t p a r a m e t e r F 1.

Vci

~ o

~9

o, Te

0,77

oTe

0,75

1,1

lp

v¢i

I~._ L

, Ft= 50

I.

,1"

/ :50 ,6,4 A T .

O INSTABILITY

/ % FI:I(O THRESHOLD U

lOO 11o 12o 13o 14o

O,,KW

(9)

472 M. BOGAARDT and C. L. SPIGT 1 0 5.0

2,0

1,0 0,5 0,2 0)1 30 - ( 90 12C 16C 2t, 30( 361

Fig. 8. Open-loop t r a n s f e r f u n c t i o n s , Fig. 9. C l o s e d - l o o p t r a n s f e r f u n c t i o n s ,

(10)

APPLICATION OF BOILING WATER LOOP MODEL 473 d e c r e a s i n g s l o p e of the v o i d f r a c t i o n v e r s u s

p o w e r c u r v e s with i n c r e a s i n g c h a n n e l p o w e r . An i n c r e a s e in s y s t e m p r e s s u r e r e s u l t s in a d e - c r e a s e in void f r a c t i o n , which i s l a r g e l y due to the i n c r e a s e in s p e c i f i c d e n s i t y with p r e s s u r e . T h i s d e c r e a s e in v o i d f r a c t i o n r e s u l t s in a d e - c r e a s e in the d r i v i n g head and c o r r e s p o n d s , t h e r e f o r e , to an i n c r e a s i n g c i r c u l a t i o n r a t e with a d e c r e a s e in s y s t e m p r e s s u r e . F o r c o m p a r i s o n s o m e r e s u l t s for the b o i l i n g w a t e r s y s t e m a r e a l s o given. Owing to the l o w e r v a l u e of the heat of e v a p o r a t i o n for w a t e r , the void f r a c t i o n for the b o i l i n g w a t e r s y s t e m i s h i g h e r and the r e c i r - c u l a t i o n r a t e l o w e r than for the boiling s o d i u m s y s t e m .

In fig. 6 the dist:cibution along the channel of the void f r a c t i o n a, the liquid t e m p e r a t u r e T 1 and the s a t u r a t i o n t e m p e r a t u r e T s a t a r e g i v e n f o r a s y s t e m p r e s s u r e of 16.4 a t m , a c h a n n e l p o w e r of 110 kW and two v a l u e s of the s u p e r h e a t p a r a m e t e r F 1. As i s shown, the liquid s u p e r h e a t c h a n g e s f r o m 1.8 to 14.4°C for a change in F 1 f r o m 500 to 50. At F 1 = 50 t h e r e i s m o r e h e a t s t o r e d in the s o d i u m than at F 1 = 500, which r e - s u l t s in l o w e r v a l u e s of the void f r a c t i o n for a l o w e r F 1. The v a r i a t i o n in p r e s s u r e ( s a t u r a t i o n t e m p e r a t u r e ) along the c h a n n e l i s , p r a c t i c a l l y spoken, independent of the liquid s u p e r h e a t .

In fig. 7 f i n a l l y , the i n f l u e n c e i s shown of F 1 on the r e c i r c u l a t i o ; a r a t e and on the e x i t void f r a c t i o n v e r s u s ch~mnel p o w e r c u r v e s . The i n -

c r e a s e in void f r a c t i o n with d e c r e a s i n g liquid s u p e r h e a t c o r r e s p o n d s with the d e c r e a s e in r e - c i r c u l a t i o n , a s was the c a s e with the i n f l u e n c e of s y s t e m p r e s s u r e ( s e e figs. 4 and 5).

5.2.

Results o f / h e stability calculations

The open loop t r a n s f e r functions G 1 and G 2 (i.e. the p h a s e a n g l e and the modulus) as w e l l as the t r a n s f e r function for the c l o s e d loop f r o m the channel p o w e r to the r e c i r c u l a t i o n r a t e have been c a l c u l a t e d for the t h r e e s y s t e m p r e s s u r e s m e n t i o n e d b e f o r e and a n u m b e r of channel p o w - e r s . The r e s u l t s for a s y s t e m p r e s s u r e of 16.4 a t m a r e p r e s e n t e d in figs. 8-11.

In figs. 8 and 9 the open and c l o s e d loop c h a r - a c t e r i s t i c s have been plotted for a f r e q u e n c y r a n g e of 0.3 to 2 cps for d i f f e r e n t channel p o w - e r s . In the u p p e r p a r t s of the d i a g r a m s the m o d - ulus has been plotted and in the l o w e r p a r t s the p h a s e shift of the r e s p o n d i n g s i g n a l with r e s p e c t to the o s c i l l a t i o n of the e x c i t a t i o n . In the plot of the open loop c h a r a c t e r i s t i c s , G1, it is shown that an u n s t a b l e condition i s p a s s e d when p r o g - r e s s i n g f r o m 130 to 135 kW c h a n n e l power. At 130 kW the modulus of G 1 b e c o m e s l a r g e r than unity but the p h a s e a n g l e d o e s not a p p r o a c h the v a l u e of 0 ° or 360 °. At 135 kW the m o d u l u s of G 1 i s l a r g e r than unity and the p h a s e a n g l e b e c o m e s z e r o at a f r e q u e n c y of 1.32 c p s , which i n d i c a t e s that the s y s t e m is u n s t a b l e .

H e r e i t should be pointed out that c o n d i t i o n s

150 1,9

t/,0

1,8

O, inst

fmst

KW I 130 r"l~S~ 1,7

110

I,S

I O0

1,4

90 1,3

1,2

)~;0

0,9

0 (~inst

/ /

/

J

/

/ /

/

SODIUM

WATER

/

80

t J ' ~ ~

~. f|nst

5O

2 /.

6

8 1 0

12 I/4

1 $

lS

20 P, atrn

Fig. 10. Influence of the system pressure on Qinst and finst'

, /

/

(11)

474 M. B O G A A R D T and C. L . S P I G T SO0 100

°'I

5O

k / /

r

( ~ 135KW

//

I ? S K W i i 16+4 A T " I0 I

V

I

V

0~3 2 ] ~ 5 6 7 8 9 1 0

Fig. 11. Open-loop transfer function G2, high frequency range.

20

of G 1 in which the a r g u m e n t is z e r o but the m o d - u l u s i s in e x c e s s of unity a r e not u n s t a b l e in the l i n e a r i s e d a p p r o x i m a t i o n s . H o w e v e r , t h e s e c o n - d i t i o n s a r e g e n e r a l l y t e r m e d " c o n d i t i o n a l l y s t a - b l e " c o n d i t i o n s in c o n t r o l t h e o r y , as l a r g e a m - plitude f l u c t u a t i o n s tend to d e c r e a s e e f f e c t i v e l y the m o d u l u s with only s m a l l c h a n g e s in the p h a s e angle b e c a u s e of n o n - l i n e a r s a t u r a t i o n e f f e c t s . In p r a c t i c e t h e s e c o n d i t i o n s a r e t h e r e f o r e only s t a - ble f o r v e r y s m a l l o s c i l l a t i o n s and tend to d e v e l - op s e l f - s u s t a i n e d finite a m p l i t u d e o s c i l l a t i o n s . F u r t h e r m o r e , by c a l c u l a t i n g the c o n d i t i o n s b e - tween 130 and 135 kW c h a n n e l p o w e r a condition w i l l be found w h e r e the condition G 1 = 1 i s e x a c t - ly fulfilled.

The c l o s e d loop r e s p o n s e of the n a t u r a l c i r - c u l a t i o n loop to a m o d u l a t i o n of the channel p o w - e r i s shown in fig. 9. V e r t i c a l l y the f l u c t u a t i o n

in i n l e t v e l o c i t y p e r kW p o w e r m o d u l a t i o n i s plotted. A c o m p a r i s o n b e t w e e n the open and c l o s e d loop r e s u l t s shows a s t r o n g l y p e a k e d b e - h a v i o u r in

Vci

at the f r e q u e n c y at which the p h a s e a n g l e for G 1 p a s s e s 0 °. A l s o in the c l o s e d loop, a condition w i l l be found c o r r e s p o n d i n g to G 1 = 1 w h e r e the a m p l i t u d e r a t i o b e c o m e s i n f i - nite. Also for the o t h e r s y s t e m p r e s s u r e s the i n - s t a b i l i t y t h r e s h o l d was c l e a r l y i n d i c a t e d by the t r a n s i t i o n of the p h a s e a n g l e c u r v e of G 1 at a c e r t a i n c h a n n e l p o w e r f r o m a c l o s e d c u r v e to an open one. The n a t u r a l c i r c u l a t i o n t h r e s h o l d data a r e s u m m a r i s e d in fig. 10. F o r c o m p a r i s o n the data for boiling w a t e r a r e a l s o given. The g e n e r - al b e h a v i o u r f o r the boiling sodium s y s t e m i s r o u g h l y the s a m e a s for the b o i l i n g w a t e r s y s - t e m . T h e r e s o n a n c e f r e q u e n c y i s a p p r e c i a b l y h i g h e r than in the c a s e of w a t e r , i.e. around 1.4 cps i n s t e a d of a r o u n d 1.0 cps. T h i s i s p r o b a b l y due to the l o w e r v a p o u r d e n s i t y of sodium. F u r - t h e r m o r e , the r e s o n a n c e f r e q u e n c y s h i f t s a p p r e - ciably with s y s t e m p r e s s u r e w h e r e a s in b o i l i n g w a t e r this f r e q u e n c y change i s m u c h l e s s p r o - nounced. F i n a l l y , in the b o i l i n g s o d i u m c a s e a p r o n o u n c e d m i n i m u m in the r e s o n a n c e f r e q u e n c y e x i s t s w h e r e a s in the boiling w a t e r c a s e this m i n i m u m i s m u c h l e s s p r o n o u n c e d .

In fig. 11 the m o d u l u s of the open loop t r a n s - f e r function G 2 i s p r e s e n t e d for the f r e q u e n c y r a n g e f r o m 2 to 17 cps and a s y s t e m p r e s s u r e of 16.4 atm. The m o d u l u s of the t r a n s f e r function G 2 shows a s h a r p dip at the h i g h e r f r e q u e n c i e s . T h i s i n d i c a t e s the a p p r o a c h to a f o r c e d c i r c u l a - tion i n s t a b i l i t y at a f r e q u e n c y of about 15 cps and at an a p p r e c i a b l y h i g h e r channel p o w e r than c o r - r e s p o n d s to the h y d r a u l i c i n s t a b i l i t y of the n a t - u r a l c i r c u l a t i o n s y s t e m . The m i n i m u m v a l u e s a t t a i n e d for G 2 a r e 3.12, 1.56 and 0.40 for a c h a n n e l p o w e r of 135, 150 and 175 kW, r e s p e c - t i v e l y . As the f o r c e d c i r c u l a t i o n loop b e c o m e s u n s t a b l e for G 2 = 0 ( s e e s e c t i o n 3), it i s e v i d e n t that the i n s t a b i l i t y i s a t t a i n e d at a channel p o w e r s l i g h t l y a b o v e 175 kW. The c u r v e s for G 1 in t h i s f r e q u e n c y r a n g e (fig. 12) show that the n a t u r a l c i r c u l a t i o n loop e x h i b i t s a l s o a tendency to i n - s t a b i l i t y although at a s o m e w h a t l o w e r f r e q u e n c y . The n a t u r e of this i n s t a b i l i t y i s , h o w e v e r , u n - doubtedly of the s a m e o r i g i n a s that of the f o r c e d c i r c u l a t i o n loop. In [1] it i s shown that this in- s t a b i l i t y i s c h a r a c t e r i s e d by a standing wave o s - c i l l a t i o n with z e r o v e l o c i t y f l u c t u a t i o n at the inlet. The i n f l u e n c e of the liquid s u p e r h e a t on the i n s t a b i l i t y t h r e s h o l d for a n a t u r a l c i r c u l a t i o n b o i l i n g s o d i u m s y s t e m has been e v a l u a t e d at a s y s t e m p r e s s u r e of 16.4 a t m . The r e s u l t s of t h e s e c a l c u l a t i o n s a r e s u m m a r i s e d in fig. 13.

(12)

APPLICATION OF BOILING WATER LOOP MODEL 475 GI

//

tTSKW f v O jl

o,o'.

2 3 16/, ATM 135 KW _r 2/,0 150 KW

,70KWl

rl\l,

36O 4 5 6 7 6 9 10 20 2 3 /~ 5 G 7 8 9 - - ~ f.c.p.s. - - I D , - f.c.ps.

Fig. 12. Open-loop transfer function G1, high frequency range.

20 O.inst KW 170 150 !/,0 130 120 110 100 IJ. 18 16 f inst AT

i "i

c..si, 3 1 !. 12

~,2

~o

8 It1 6 /., 0 0;9 ~ n s t Q, ,nst 0 100 200 300 /, 00 500 - - - I m ~ FL

(13)

476 M. BOGAARDT and C. L. SPIGT H o r i z o n t a l l y the s u p e r h e a t p a r a m e t e r F 1 has

been plotted. V e r t i c a l l y the c o r r e s p o n d i n g liquid s u p e r h e a t A T , the i n s t a b i l i t y t h r e s h o l d channel p o w e r Qinst and the r e s o n a n c e f r e q u e n c y f i n s t h a v e been plotted. As can be concluded f r o m t h i s f i g u r e the channel p o w e r for i n s t a b i l i t y d e c r e a s e s with an i n c r e a s e in liquid s u p e r h e a t , although the t o t a l void v o l u m e p r e s e n t in the channel d e c r e a s e s ( s e e figs. 6 and 7). F u r t h e r - m o r e , t h e r e i s an a p p r e c i a b l e e f f e c t of liquid s u p e r h e a t on the r e s o n a n c e f r e q u e n c y . When the d e n s i t y of the fluid in the channel d e c r e a s e s , t h e r e w i l l be a d e c r e a s e in the r e s o n a n c e f r e - quency. T h i s i s in a g r e e m e n t with the r e s u l t s of e x p e r i m e n t s c a r r i e d out in b o i l i n g w a t e r t e s t s , s e e Spigt [1]. It i s e v i d e n t that the p h e n o m e n o n of liquid s u p e r h e a t should be taken into a c c o u n t in a s t a b i l i t y a n a l y s i s in c a s e a p p r e c i a b l e s u p e r h e a t s a r e found to o c c u r .

6. CONCLUDING REMARKS

The p r e s e n t study s e e m s to i n d i c a t e that the r e l a t i v e l y l a r g e d i f f e r e n c e s in p h y s i c a l p r o p e r - t i e s between liquid s o d i u m and w a t e r do not lead to d i f f i c u l t i e s in a p p l y i n g the t h e o r e t i c a l m o d e l s that have b e e n d e v e l o p e d f o r w a t e r s y s t e m s to liquid sodium. E x p e r i m e n t a l v e r i f i c a t i o n of the p r e d i c t e d r e s u l t s i s , h o w e v e r , r e q u i r e d .

In b o i l i n g s o d i u m s y s t e m s the s a m e type of

h y d r a u l i c i n s t a b i l i t i e s a r e found as in b o i l i n g w a t e r s y s t e m s , although the p o w e r at which the loop b e c o m e s u n s t a b l e i s l o w e r in the c a s e of liquid sodium and the r e s o n a n c e f r e q u e n c y h i g h - e r .

As could be e x p e c t e d , the i n f l u e n c e of liquid s u p e r h e a t on the d y n a m i c b e h a v i o u r of the loop i s r a t h e r l a r g e . T h i s should be w e l l kept in m i n d in p r e d i c t i n g loop b e h a v i o u r u n d e r t r a n s i e n t c o n - ditions.

R E F E R E N C E S

[1] C.L.Spigt, On the hydraulic characteristics of a boiling water channel with natural circulation, The- sis Technological University Eindhoven (1966). [2] F.Van der Walle and H.J. Lamein, A digital com-

puter prog]:amme for the non-linear steady-state and quasi-linear dynamic calculation of boiling hydraulic loops, Report Rescona Ltd., The Nether- lands, No. 64-10 (1966).

[3] K.H. Spiller, Physikalisch-thermische Eigenschaf- ten yon Na-, K- und NaK-Legierungen im Tempe- raturbereich zwischen Siedepunkt und etwa 1300oc. Euratom report Nr. EUR 357.d (1963).

[4] N. Zuber and J.A. Findlay, Average volumetric concentration in two-phase flow systems, Trans. ASME, J. Heat Transfer 87 (1965) 453.

[5] A. B. Jones and D. G. Dight, Hydrodynamic instability of a boiling channel, Reports of Knolls Atomic Power Laboratory, Schenectady, KAPL 2170 (1961),