Calculation of Void Volume Fraction in the Subcooled and Quality … · 2015. 3. 30. · m • X This is the differential change in the true vapour weight fraction with dz regardless

AE-336 UDC 536.248.2

Calculation of Void Volume Fraction

in the Subcooled and Quality Boiling

Regions

S. Z. Rouhani and E. Axelsson

AKTIEBOLAGET ATOMENERGI

STOCKHOLM, SWEDEN 1968

AE-336

CALCULATION OF VOID VOLUME FRACTION IN THE

SUBCOOLED AND QUALITY BOILING REGIONS

S. Z. Rouhani and E. Axelsson

SUMMARY

The complex problem of void calculation in the different

regions of flow boiling is divided in two pa r t s .

The f i rs t par t includes only the descr ip t ion of the mechan i sms

and the calculation of the r a t e s of heat t ransfe r for vapour and liquid.

It is assumed that heat is removed by vapour generat ion, heating of

the liquid that replaces the detached bubbles, and in some p a r t s , by

single phase heat t r ans fe r . By considering the ra te of vapour conden­

sation in liquid, an equation for the differential changes in the t rue

s team quality throughout the boiling regions is obtained. Integrat ion

of this equation yields the vapour weight fraction at any position.

The second par t of the problem concerns the determinat ion

of the void fractions corresponding to the calculated s team qual i t ies .

For this purpose we use the derivat ions of Zuber and Findlay [ 9 ] .

This model is compared with data from different geomet r i e s

including small rectangular channels and large rod bundles. The data

covered p r e s s u r e s from 19 to 138 b a r s , heat fluxes from 18 to 120

W/c m with many different subcoolings and mass veloci t ies . The

agreement is general ly very good.

P r in ted and distr ibuted in October 1968.

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C O N T E N T S

Page No.

Introduction 3

L i te ra tu re survey 3

Theory 4

1 Basic Assumptions 4

2 The Separate Regions of Subcooled Boiling 5

3 Derivation of the Basic Equations 6

Est imat ion of the Rate of Bubble Condensation 8

1 Selection of the Distr ibution P a r a m e t e r 8

2 Effect of Void and Channel Geometry on k 9 1 c

3 Effect of Mass Velocity and Heat Flux 9 4 Effect of Physica l P rope r t i e s on k 10 5 The Proposed Corre la t ion for the Condensation

P a r a m e t e r 10

Comparison with Data under Various Conditions 10

Conclusions 12

Table of Data used in Compar ison with this Model 13

Nomenclature 14

References 16

gures

- 3 -

1. INTRODUCTION

The calculation of void in subcooled flow boiling is particularly-

complex because of the absence of the rmal equi l ibr ium between the

two phases . For this reason the problem of subcooled void calculation

must be divided in two p a r t s :

1. a close est imation of the t rue liquid subcooling and the vapour

weight fraction at any position,

2. the determinat ion of the vapour volume fraction for the calcu­

lated s team quali t ies in the given conditions.

A par t icu lar approach to the solution of these problems was de­

scribed in a previous repor t [ 1 ] .

The method of void calculation suggested in [1] was only appli­

cable to the regions of subcooled boiling and was approximative r e ­

garding the influence of slip velocity.

The main improvement in the present model over that of [1 ] is

the admiss ion of slip velocity between vapour and liquid even in the

region of subcooled boiling. The inclusion of a slip ra t io other than

unity in this model improves it considerably and makes it applicable

to a wider range of conditions. The present model is valid even in

the net boiling region.

Another improvement is the inclusion of a unified corre la t ion

for the condensation constant throughout the regions of subcooled

boiling. This e l iminates any discontinuity in t rans i t ions between the

different boiling regions .

2. LITERATURE SURVEY

A l i t e ra tu re survey on the papers dealing with the calculation

of void fraction in subcooled boiling was given in ref. [1]. The survey

included works of Griffith, Clark, and Rohsenow [2] , Maurer [3] ,

Bowring [ 4 ] , Costello [ 5 ] , and. Delayre and Lavmge [ 6 ] ,

Several additional r e p o r t s on this subject have appeared in the

l i t e ra tu re recent ly . Among these a re the works of Levy [ 7 ] , and

Zuber, Staub, and Bijwaard [ 8 ] .

- 4 -

The main points of the paper by Levy [7] is a new method of

calculating the liquid subcooling at the point of bubble depa r tu re . This

is different from Bowring 's method [ 4 ] , Levy suggests a lso a cer ta in

re la t ionship between the t rue local vapour weight fraction and the c o r r e ­

sponding the rma l equi l ibr ium value. Finally, by applying an accepted

slip cor re la t ion he calculates the void fraction in subcooled boiling.

Zuber , Staub, and Bijwaard [8] emphasize the influence of flow

reg ime upon the re la t ive vapour velocity throughout the boiling regions .

With the inclusion of a concentration constant and the drift velocity of

the bubbles as presented in [9 ] they give a be t ter descr ipt ion of the

average slip velocity. F o r the par t icu la r region of subcooled boiling

they a s sume a mathemat ica l ly feasible function for liquid t empera tu re

dis tr ibut ion along the heated channels . They apply Bowring 's c o r r e ­

lat ions [4 ] to de termine the location of bubble detachment and finally,

in the absence of a method of predicting the l imits of var ious flow

r e g i m e s , they make use of a fixed value of concentration constant for

al l conditions.

3. THEORY

3.1 Basic Assumptions

As explained in the introduction one should f i r s t es t imate the

t rue vapour weight fraction at any position along the channel. This

may be done through proper heat balance equations for each phase in

axial and t r a n s v e r s e d i rec t ions in the channel.

We consider f i rs t the t r a n s v e r s e heat flow from the heated

surface to the boiling flow. The assumed mechanisms of heat removal

in this model a r e the same as given in [ 1 ] . These a re briefly repeated

below:

1. single phase heat t ransfe r which will be par t ia l ly effective as

long as the heated surface is not covered with bubbles

2. s t eam generat ion

3. heating of that m a s s of water which rep laces the detached

bubbles.

- 5 -

At l e a s t two of t h e s e m e c h a n i s m s a r e e f fec t ive in p a r a l l e l on

the hea t ed s u r f a c e whi le s o m e . h e a t exchange b e t w e e n s t e a m bubb le s and

the subcoo led l iquid wi l l t ake p l ace t h r o u g h c o n d e n s a t i o n .

3. 2 The s e p a r a t e R e g i o n s of Subcooled Boi l ing

As poin ted out in m a n y r e p o r t s on the sub jec t of subcoo led boi l ing

[ 1 - 8 ] t h e r e e x i s t s a c e r t a i n l i m i t of subcool ing at which the bubb l e s b e ­

gin to d e t a c h f r o m the h e a t e d w a l l . It i s a s s u m e d tha t the bubb l e s g e n e ­

r a t e d a t subcoo l ings l a r g e r than that of the point of d e t a c h m e n t a r e m o s t ­

ly s t a t i o n a r y and c o l l a p s e b e f o r e moving away f r o m the wa l l . The void

f r ac t i on due to the s t a t i o n a r y bubb les i s t e r m e d wa l l vo idage and it h a s

an upper l i m i t which d e p e n d s on p r e s s u r e , hea t ed p e r i m e t e r and the

flow a r e a .

A c c o r d i n g to the w o r k s of M a u r e r [3*1, Bowr ing [ 4 ] , and C o s t e l l o

[ 5 ] , and a s exp la ined in [ 1 ] , we c o n s i d e r two r e g i o n s of subcoo led

bo i l ing :

1. l oca l boi l ing with s t a t i o n a r y bubb le s on the s u r f a c e and high

subcoo l ing ,

2. l o c a l boi l ing wi th low enough subcool ing to a l low bubble d e t a c h ­

men t and flow of vapou r bubb les wi th l iquid .

The m a x i m u m va lue of wal l vo idage o c c u r s at the end of the f i r s t

r e g i o n . We r e f e r to the void f r a c t i o n at the end of the f i r s t r e g i o n by <% .

The b a s i s of ca l cu l a t ion of Q; w a s exp la ined in [ 1 ] and it w a s conc luded

tha t for -water a s the boi l ing m e d i u m

p -> A -> r A /\~3 - 0 . 2 3 7 h /, \

<y = 2 . 4 3 5 • 10 p — (1) *c - —- — r A

c 2 In th i s equat ion p is in N / m , P , in m and A in m

T h e second r e g i o n s t a r t s at the point of d e t a c h m e n t and ends

a t a pos i t ion w h e r e the l iquid subcool ing b e c o m e s n e g l i g i b l e .

- 6 -

3.3 Derivation of the Basic Equations

We assume that local boiling starts at a point where

i - h ( t s - t l ) > 0 (2)

h is the single phase heat transfer coefficient for only liquid flow.

We use Collburn's correlation which gives

h = 0 ^ 2 | G . c . p - 2 / 3 }

„ 0. 2 p r w

Re r

At high subcoolings the single phase heat transfer will still be

effective but accompanied by the other mechanisms. As the subcool-

ing decreases the heated surface will become more and more covered

with bubbles and hence less accessible to the bulk liquid flow. For

this reason we assume that the non-boiling fraction of heat flux will be

©nb^-f^^s-^ f°r-^c <4> c

in which <y is the local void fraction and <* is the void fraction at the ^ c

point of vapour clotting.

The non-boiling fraction of heat flux will gradually decrease

with increasing wall voidage and it vanishes when the wall voidage reaches jy . c

The heat balance oh the heated surface is

m i = h e ( 1 _ J 3 L ) + ^ X + ̂ C • 0 l . 9 l (5)

For values of & larger than a , the first term on the right hand side

of this equation should be eliminated.

The amount of heat which goes to steam generation per unit time

within dz along the channel will be

(f) ~ h9x(l -%) dQ, = m \ P • dz = ———— T— o I P, dz (6)

b s h Dg p °1 1 g

- 7 -

Again for w >Q/ the t e r m conta in ing h m u s t be e l i m i n a t e d .

T h e a m o u n t of h e a t which goes to the subcoo led l iquid t h r o u g h

condensa t i on of vapour b u b b l e s p e r unit t i m e wi th in dz m a y be expressed

a s

dQ = k • 9, dz (7) c c 1

in which k is a condensa t i on coeff ic ient with the s a m e d i m e n s i o n s a s c

that of the t h e r m a l conduc t iv i ty . It wi l l be shown tha t t h i s c o n s t a n t is

a c t u a l l y p r o p o r t i o n a l to the t h e r m a l conduc t iv i ty of the l iquid p h a s e

d iv ided by the P r a n d t l n u m b e r .

C o n s i d e r i n g the hea t b a l a n c e in the a x i a l d i r e c t i o n we u s e two

s e p a r a t e e q u a t i o n s for the two p h a s e s . The connec t ion b e t w e e n the

two hea t b a l a n c e e q u a t i o n s i s found in eq. (7) which g i v e s the r a t e of

hea t exchange b e t w e e n v a p o u r and l iquid .

'Wi thou t mak ing any d i s t i n c t i o n b e t w e e n the d i f fe ren t r e g i o n s of

bo i l ing , one m a y w r i t e the hea t b a l a n c e for the v a p o u r p h a s e wi th in dz

a s

dQ - dQ dx = 2 £ (8)

m • X

T h i s is the d i f f e r en t i a l change in the t r u e v a p o u r weight f r a c t i o n with

dz r e g a r d l e s s of the flow r e g i m e or s l i p r a t i o .

The t o t a l hea t b a l a n c e a c r o s s dz g i v e s the d i f f e r en t i a l change

in the t r u e l iquid subcool ing a s

(A> • P • dz - (dQ - dQ ) d9l = A K— ± c- (9)

m - C p

Now, a s s u m i n g tha t the v a r i a t i o n s of dQ, and dQ with z a r e 0 b e

known, one m a y i n t e g r a t e equa t ion (8) to obtain the t r u e s t e a m qua l i t y

at any he igh t .

With the known v a l u e s of s t e a m qua l i ty x , one m a y u s e a su i t ab l e

r e l a t i o n s h i p for s l ip r a t i o and c a l c u l a t e the l o c a l v a l u e s of the void

vo lume f r a c t i o n .

It wi l l be shown tha t k i s dependen t on the l o c a l void f r a c t i o n

in a n o n - l i n e a r m a n n e r . F o r t h i s r e a s o n dx in e q . (8) b e c o m e s a n o n -

- 8 -

l inear function of z and therefore it may only be integrated by numer i ­

cal methods.

F o r the calculation of the average local void from s team quality

the authors draw upon the derivat ions of Zuber and Findlay (9) and use

the following relat ion:

g u p g 1 p l

In this equation C is a dis tr ibut ion pa rame te r which is dependent on

the velocity profile and void distr ibution over flow a r ea .

Although one would expect that C should vary with channel geo­

m e t r y and flow r e g i m e s , we have found that an average value of about

1. 1 would be adequate to match the data from a large var ie ty of tes t

geome t r i e s . However, a r a the r strong dependence of C upon the mass

velocity was observed for the lower values of the la t te r pa rame te r

(G < 200). Fo r low velocit ies C was found to be much l a rge r than 1 .1 .

4. ESTIMATION OF THE RATE OF BUBBLE CONDENSATION

The condensation coefficient, k , in equation (7) is dependent

on many p a r a m e t e r s . Physical ly , this coefficient must depend on the

t h e r m a l conductivity of the liquid and some other p roper t i e s of the two

phases . It must be a function of the local values of contact a r ea be ­

tween vapour and liquid which is to some extent dependent on the void

fraction and the channel geometry . Final ly, m a s s velocity and heat flux

must have some influence upon the condensation p a r a m e t e r .

The individual effects of these p a r a m e t e r s were determined by

a sys temat ic comparison of this model with the data of references[10] .

The genera l validity of the model and the related dependencies upon

var ious p a r a m e t e r s were then verified by comparing this calculation

procedure with a large number of data from different sources [ 1 1 - 1 4 ] .

4. 1 Selection of the Distr ibution P a r a m e t e r

Before any studies on the effects of different p a r a m e t e r s upon

k the data of net boiling regions were used in equation (10) to obtain

- 9 -

a su i t ab l e va lue of C. In m o s t c a s e s the a v e r a g e va lue of t h i s p a r a ­

m e t e r t u r n e d out to be

C = 1 .12 (11 a)

T h i s va lue of the d i s t r i b u t i o n p a r a m e t e r was t hen u sed in equa t ion (10)

for the r e g i o n of subcoo led bo i l ing .

F o r the c a s e of low m a s s v e l o c i t i e s of ref. [ 1 0 ] it w a s found

tha t

C = 1 .54 (11 b)

4 . 2 Effect of Void and C h a n n e l G e o m e t r y on k 1 c

A s d e s c r i b e d in [1"] the a v e r a g e con tac t a r e a b e t w e e n t h e b u b b l e s

and l iquid m a y be e x p r e s s e d a s

A b = & • A c2 / 3 • a 2 ' 3 (pe r unit l ength) (12)

in which a i s a p r o p o r t i o n a l i t y cons tan t which m a y depend on t h e h e a t

f lux (bubble g e n e r a t i o n f r e q u e n c y ) .

4 . 3 Effect of M a s s Ve loc i ty and Hea t F l u x

T h e effect of m a s s v e l o c i t y i s inc luded in a n o n - d i m e n s i o n a l

f o r m by us ing R e y n o l d s n u m b e r c a l c u l a t e d for the l o c a l l iquid v e l o c i t y

G_- De ( 1 3 ) ( R e ) = , " ' "*

A s y s t e m a t i c c o m p a r i s o n with the e x p e r i m e n t a l da t a of [ 1 0 ]

showed that the condensa t ion coeff ic ient v a r i e d l i n e a r l y with (Re) , .

T h i s equa t ion i l l u s t r a t e s a n o t h e r d e p e n d e n c e of k upon # . The effect

of hea t flux i s a l s o inc luded in a n o n - d i m e n s i o n a l m a n n e r by us ing

the following d i m e n s i o n l e s s n u m b e r :

N = - £ (14)

The d e p e n d e n c e of k on N was found to be as l /< /N . r c q V q

- 10 -

4. 4 Effect of Physical P rope r t i e s on k

Apar t from the effect of physical proper t ies through (Re), and

N , it was seen that k var ied with p r e s s u r e considerably with all the

other p a r a m e t e r s being the same . The effect of p r e s s u r e could be r e ­

presented with the following group of p a r a m e t e r s

e ( p ) = a2p^(-°-S) for p ^ 19 b (15) r q l

in which a ? is a dimensionless constant.

4. 5 The Proposed Corre la t ion for the Condensation P a r a m e t e r

Based on the above mentioned resu l t s it was found that

- 4 / 3 in which a = a. • a_ = 30. 0 m ' is a dimensional constant and may

probably depend on the number of nucleation si tes per unit a r e a of the

heated surface as well as on the bubbling frequency and other fac tors .

Equation (16) is applicable in both regions of subcooled boiling.

5. COMPARISON WITH DATA UNDER VARIOUS CONDITIONS

The genera l validity of the model and the related dependencies

upon var ious p a r a m e t e r s were verified by comparing this calculation

procedure with a large number of data from different geometr ies ob­

tained over a wide range of p a r a m e t e r s .

Table 1 gives a br ief descr ipt ion of the test geomet r ies and

the range of p a r a m e t e r s covered.

The void volume fractions corresponding to the exper imenta l

conditions were computed by numer ica l integration of eqs. (6), (7), (8)

and (9) and using eqs. (1), (3), (10) and (16).

Graphical compar isons of the resu l t s of computations with the

exper imenta l data a r e shown in F igs . 1 to 7.

For the case of data of ref. [10] in which voids a re given at

a fixed position in the channel at 109 cm from the inlet, the comparisons

- 11 -

a r e made by plotting voids as a function of the local average subcooling

or local average s team quality. These average values a re calculated

by assuming t he rma l equi l ibr ium in the channel and neglecting the t rue

vapour flow. These a r e good only as some reference points for com­

par ison. The true liquid subcoolings calculated from equation (9) a r e

considerably different from average subcoolings, 0 . Likewise, the

lower values of the average s team quality a re considerably different

from the t rue s team qualit ies obtained from equation (8).

As can be seen in F ig . 3 b the agreement between the model and

data with very low m a s s veloci t ies (G = 130 kg /m • sec) is ra ther poor

for very large inlet subcoolings (9. > 150 C). It has not been possible

to find out whether this d iscrepancy depends on the effect of m a s s

velocity, on the ra te of condensation, or on the effect of var ia t ions of

physical p roper t i es because of the large t empera tu re var ia t ions . Al­

though the effect of na tura l convection on the s ingle-phase heat t ransfe r

has been considered, it is plausible that in the presence of steam bubb­

les near the inner surface of the annular geometry there has been some

sort of intensified convection at very low m a s s veloci t ies . Heat removal

through such mechanisms has not been accounted for in these calcula­

t ions .

F igs , l a through 3a show very good agreement between the cal­

culation and the data under different conditions. These a r e only samp­

les of many s imilar t es t s of this model against the data of ref. [ 1 0 ] .

F igs . 4 and 5 show the comparisons with data from rod bundles

with six and th i r tys ix rods respect ively ( refs . 11 and 12). The calcu­

lated voids match the data quite well.

Comparison with the data from two rectangular geomet r i e s a r e

shown in F igs . 6 and 7. These include runs at p r e s s u r e s up to about

140 a tm. The la rges t deviation (in the case of Chr i s t ensen ' s data at

56 atm) seems to be about 6 % void. The admitted l imits of exper i ­

mental e r r o r s for these data are_+5 %.

The computed voids for the conditions of data from Battelle

Memoria l Institute (14) a r e shown in F igs . 7a and b . On the average

the computed resu l t s show good agreement even with the data from

these exper iments .

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6. CONCLUSIONS

Based on the r e su l t s of comparison with the exper imenta l data

it may be concluded that the present model gives a very close approxi­

mation of the t rue physical phenomena involved in the changes of s team

volume fraction in flow boiling throughout the boiling regions .

In the absence of data from subcooled boiling of other l iquids,

nothing can be said on the applicability of the corre la t ion for the con­

densation factor in genera l . However, it seems to agree quite well

with the p roper t i e s of water and heavy water for p r e s s u r e s f rom 1 9 to

140 a tmospheres .

7. Table 1 - Range of data used in comparison with this model

Source of Data

Ref 10

Ref 11

Ref 12

Ref 13

Ref 14

Test section

geometry-

Annular

6-rod d u s t e r

36-rod d u s t e r

Rectangular

1. 11 x 4. 44 cm

Rectangular 0.261 x 2.54

flow area

2 cm

3.78

30.5

142. 7

4 .93

0.665

heated per imeter

cm

3.77

26.2

156.

11. 1

5 .6

P r e s s u r e

b a r

19 - 50

31.6 -51.4

50.

51. - 68.9

137.9

•

q / A

W / c m 2

60-120

46 .7 -64 .5

22-64

49.6

18.9-0

126. 1

G

m . sec

130-1450

1345-1607

1110-1159

877-906

9080-1165

0 in

°C

0-130

5 .7-27 .2 (corrected)

11-22.4

12.5

5 - 7 3 . 4

•

%

0-12

0-6

2 -9

6-7

8-18

- 14 -

8. N O M E N C L A T U R E

S y m b o l D i m e n s i o n s

- 4 / 3 a = a d i m e n s i o n a l cons t an t in eq . (16) m

a 4 = " " " " " (12) m " 4 / 3

a ? = a p r o p o r t i o n a l i t y c o n s t a n t in eq . (15)

2 A, = con tac t a r e a b e t w e e n the bubble and m

*b

s

c

the l iquid p e r unit l eng th of the channe l

2 A = flow a r e a in the c h a n n e l m

c C = d i s t r i b u t i o n p a r a m e t e r in eq . (8)

c = spec i f i c hea t of l iquid a t c o n s t a n t p r e s s u r e j / k g C ir

4 A D = equ iva l en t h y d r a u l i c d i a m e t e r = — m e ^ t

2 G = m a s s v e l o c i t y k g / m s e c

h = hea t t r a n s f e r coeff ic ient ( C o l l b u r n ' s c o r r e - W / m C la t ion)

k = c o n d e n s a t i o n f a c t o r W / m C c

k , = t h e r m a l conduc t iv i ty of l iquid W / m C

m = t o t a l m a s s flow r a t e k g / s e c

m = m a s s of l iquid which i s c o n v e r t e d into s t e a m k g / m • s e c p e r unit t i m e p e r unit a r e a of t h e h e a t e d s u r f a c e

P = P r a n d t l n u m b e r = C U,/k, r P 1' 1

c o n d e n s a t i o n of bubble p e r unit t i m e p e r unit l eng th of channe l

N = a d i m e n s i o n l e s s n u m b e r def ined by eq. (14)

p = p r e s s u r e N / m

P, = h e a t e d p e r i m e t e r m h

P 2

q / A = hea t flux W / m Q, = the a m o u n t of hea t which i s a b s o r b e d by W / m

bo i l ing p e r unit t i m e p e r uni t l eng th of channe l

Q = the a m o u n t of hea t which i s exchanged by W/: m

- 15 -

Re = Reynolds number = G • De/p,

o

0?

c

o. ti = liquid t empera tu re in the presence C of s team flow

t = saturat ion t empera tu re C s

x = t rue vapour weight fraction (s team quality)

x = average s team quality

= distance along the heated channel m

measured from the inlet

= vapour volume fraction (void)

= upper limit of wall voidage given by eq. (1) o. 9-, = liquid suhcooling as an in tegral of C

eq. (9) = tg - tx

0 = average liquid subcooling obtained from C a heat balance for the whole flow

e(p) = a p ressure-dependent non-dimensional pa ramete r defined in eq. (15)

\ = latent heat of vapourization j / k

H = dynamic viscosi ty of liquid kg/:

p = vapour density k g / m

p, = liquid density kg/:

a = surface tension of liquid N / m

m

m

- 16 -

9. REFERENCES

ROUHANI, S Z, Calculation of Steam Volume Frac t ion in Subcooled Boiling. Journal of Heat Transfe r 90 (1968) 158-164.

GRIFFITH, P , CLARK, J A and ROHSENOW, W M, Void Volumes in Subcooled Boiling Sys tems . 1958. (ASME 58 - HT-19 . )

MAURER, G W, A Method of Predict ing Steady-State Boiling Vapor F rac t ions in Reactor Coolant Channels . I960. (WAPD-BT-19, p. 59.)

BOWRING, R W, Physica l Model, Based on Bubble Detachment, and Calculation of Steam Voidage in Subcooled Region of a Heated Channel. 1962. (HPR-10.)

COSTELLO, C P , Aspect s of Local Boiling Effects on Density and P r e s s u r e Drop. 1959. (ASME 5 9 - H T - 1 8 . )

DELA YRE, R, and LAVIGNE, P , Fr ic t ion P r e s s u r e Drop for Flow of Boiling Water at High P r e s s u r e (Appendix-Model of Void Frac t ion) . EAES Symposium on Two-Phase Flow, Steady State Burnout and Hydrodynamic Instabil i ty, AB Atomenergi , Stud svik, Sweden, October l s t - 3 rd , ' 1963, Vol. 1 (1963).

LEVY, S, Forced Convection Subcooled Boiling r Predic t ion of Vapor Volumetr ic Frac t ion . 1966. (GEAP-5157.)

ZUBER, N, STAUB, F W and BIJWAARD, G, Vapour Void F rac t ion in Subcooled Boiling and in Saturated Boiling Sys tems . Int. Heat Trans fe r Conf. 3. Chicago, 111., Aug. 7-12, 1966. Vol. 5, p. 24-38, New York, Amer ican Inst, of Chem. Eng. , 1966.

ZUBER, N and FINDLAY, J A, Average Volumetr ic Concentration in Two-Phase Flow Sys tems . Journal of Heat T rans fe r , Vol. 87 (1965) 453-468.

ROUHANI, S Z, Void Measurements in the Regions of Sub-Cooled and Low-Quality Boiling. P a r t 2. Higher Mass Veloci t ies . 1966. (AE-239. )

- 17 -

EKLUND, R, GELIUS, O and NYLUND, O, ASEA-PM-KAB 65-8 . 1965. ( internal Report from ASEA, V ä s t e r å s , Sweden.)

NYLUND, O et a l . , FRIGG Loop Pro jec t . 1968. FRIGG-2. AB Atomenergi , Stockholm, and ASEA, V ä s t e r å s , Sweden.

CHRISTENSEN, H, Power-To-Void Transfe r Funct ions . 1961. (ANL-6385. )

EGEN, R A, DINGEE, D A and CHASTAIN,, J W, Vapor Format ion and Behavior in Boiling Heat Trans fe r . 1957. (BMI-1163.)

p s 50 bar

q/A g 88 W / c m 2

G = 665 k g / m • s

58 > 6. > 1 °C i n

i 1 1

-t ' 6 i

. 5 .

. 4 -

. 3 .

. 1 .

• y®

— \ —

Gyt

—\ v—

®/T

-\ » 1

£)***^

y^o

i — i — i — i — i

30 °C

20 10

Average »ubcooling, 9

1 2 3 4 5 6 7 8 9 10

Average s team quality, x

F ig . 1 a - Comparison of this model with data of ref. (10).

Annular tes t sect ion, low m a s s velocity.

%

p a 50 bar

q/A s 121 W/cm

G s 665 k g / m 2 • s

> 1 °C

30 °C

20 10

Average subcooling, 9

1 2 3 4 5 6 7 8 9 10 11 12 %

Average s team quality, x o " o

F ig . 1 b - Comparison with data of ref. (10). High heat flux, low m a s s

veloci ty .

£ 19.8 bar

q/A s 60 .7 W/cm

G =1055 k g / m 2 • s

62 > e. > 1 ° C

C 30 20 10 1 2 3 4 5 6 7 Average subcooling, 9 Average s team quality, x

F ig . 2 a - Comparison with data of ref. (10). Low heat flux.

'C 30 20 10 1 2 3 4 5 6 7

Average subcooling, 8 Average steam, quality, x

F ig . 2 b - Comparison with data of ref. (10). High heat flux.

8 %

p s 19.8 ba r

q/A s 118 W/cm 2

G = 1 0 6 0 k g / m 2 -

62 > 9. > 1 °C i n

. — . 1 1

s

^

1

. t . 8 .

. 7

• 6 .

. 5 .

. 3 .

. 2 .

t->

1

© ^

1

s*

-i

© ^ —

-I 1 -— » 1 1 i

8 %

p £ 29. 5 bar

q/A = 92 W / c m 2

G s 1430 k g / m 2 .

38 > 9. > 1 °C m

• - """"©

— ?

s

H

4 • 6 .

• 5 .

. 4 .

. i .

— » —

e / ^

— J « 1 1 i 1

C 20 10 Average subcooling, 0

1 2 3 4 5 6 %

Average s team quality, x o ° x ' o

F ig . 3 a - Comparison with data from annular tes t section (ref. 10).

p = 29. 2 bar

q/A s 91.1 W / c m 2

G = 1 3 0 . 5 k g / m 2 -

177 > 9. > 157 °C i n

— 1 \r

S

a 1 . 6 .

• t».

. 4 .

• 3 .

• 2 .

. 1 .

j o

_ J

o ©

-1 1—

o

—i

o

H 1 O

C 20 10 Average subcooling, 8

1 2 3 4 5 6 % Average s team quality, x

o " • o

F ig . 3 b - Comparison with data from annular tes t section (ref. 10)

Very low m a s s velocity.

4 <

3 .

2 .

1 .

P

q /A

G

e. m A c

D e

= 51.4 b

= 64.5 W/cm'

= 1607 kg/m2

= 27.2 °C

= 30.5 cm2

= 26. 2 cm

= 2. 51 cm

—I P

Distance from inlet - I 1

1.

Fig. 4 a - Comparison with measurements in a 6-rod cluster

(Run No. 13028 of ref. 11)

m

5 .

4

3 .

2 .

1

p = 51.6 b

q/A = 64.5 W/cm2

G =1597 kg/m2 • s

9. = 13.2 °C i n

° X © v " ^

Fig. 4 b - Comparison with measurements in a 6-rod cluster

(Run No. 13027 of ref. 11)

7 .

6 .

5 .

4 .

3 .

2 .

1 .

o

p = 31.6 b

q/A = 46.7 W/ c m

G =1345 kg/m 2 . s

9. = 5.7 °C i n .

>̂ a

y ^

>*©

1 «

<^Q

— i -

o * * ©

Distance from inlet

1 1 1. 4 . m

Fig. 4 c - Comparison with measurements in a 6-rod cluster

(Run No. 13026 of ref. 11)

,4

3

2

1

O

= 50.0 b q/A = 22.8 W/c m '

0 F ig . 5

6J.

.5

4

3±

2

1 4.

O -measu red data

1 2 ' 3 4 5 m a - Comparison with m e a s u r e m e n t s in a 36-rod cluster

(Run No. 313007 of ref. 12)

p = 49 .7 b

q/A = 4 2 . 7 W / c m ?

G = l l l 6 k g / m 2 - s ^ " " ^ "calculated o

O -measu red data

Fig.

.7

.6

,5

4 . .

3

2

1

1 ' 2 3 4 5 m 5 b - Comparison with m e a s u r e m e n t s in a 36-rod c lus te r

(Run No. 313015 of ref. 12)

p = 49 .7 b q/A = 64.6 W/c m

= 1159 kg /m

= 22.4 calculated

O - m e a s u r e d data

1 " 2 3 4 5 m

Fig . 5 c - Comparison -with m e a s u r e m e n t s in a 36-rod cluster

(Run No. 313020 of ref. 12)

o • r-i

O n) u

O >

0.6

0.5

0.4

0 . 3 .

0 . 2

0 . 1

0.0 0

rr \f t

p = 55.12 b

q/A = 49 .65 W/ cm

G = 906 k g / m 2 • s

9. = 12.5 °C i n ?

Test geometry ; 4 . 4 x 1 . 1 1 cm

© > © ^^

© ^r

© ^ X ^ © _^^

© ^/^

© ^^^^^

—1 1 1 1 1 1 1 1—

°>^^

Distance from inlet

1 1 1 1 0 0.1 0.2 0.3 0.4 0 .5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 m

F ig . 6 a - Comparison of this model with Chr i s tensen ' s data (13).

Rectangular t es t sect ion.

0.5 ..

0 • »H -t->

u u

<*-{

V

fcj 1—1

0 >

T3 • r-* O >

0

0

0.

0

0

4

3

2

1

0

p = 68.9 b

q/A = 4 9 . 6 5 W / c m 2

G = 877.5 k g / m 2 • s

= 12.1 UC

0.5

Fig. 6 b - Comparison of this model with Chr i s tensen ' s data (13).

Rectangular tes t sect ion.

.54.

.4

at (q /A) = 1 8 . 9 3 W / c m

G = 908 k g / m 2 • s

i n = 5 .0 UC

.7 m

F i g . 7 a - C o m p a r i s o n wi th the da ta of BMI a t 138 b a r s .

(Condi t ion N o . 11 of ref . 14)

• 5 . .

.4

3

2

QC q /A s 4 7 . 3 W / c m 2 , G ™=" "867~kg/m2 T" s"ec

e . = 3 6 . 6 7 ° C

0 . 1 0 .2 0 .3 0 .4

| X Q = 0

i D i s t a n c e f r o m in l e t

-H-0 .5

—i— 0 . 6

—i— 0 . 7 m

F i g . 7 b - C o m p a r i s o n wi th the da ta of BMI a t 138 b a r s ,

(Condi t ion N o . 6 of ref . 14)

LIST OF PUBLISHED AE-REPORTS

1-260. (See the back cover earlier reports.) 261. On the attenuation of neutrons and photons in a duct filled with a helical

plug. By E. Aalto and A. Krell. 1966. 24 p. Sw. cr. 8 : - . 262. Design and analysis of the power control system of the fast zero energy

reactor FR-0. By N. J. H. Schuch. 1966. 70 p. Sw. cr. 8 : - . 263. Possible deformed states in '" In and " ' I n . By A. Bäcklin, B. Fogelberg and

S. G. Malmskog. 1967. 39 p. Sw. cr. 10:- .

264. Decay of the 16.3 min. m T a isomer. By M. Höjeberg and S. G. Malmskog. 1967. 13 p. Sw. cr. 10: - .

265. Decay properties of ' "Nd . By A. Bäcklin and S. G. Malmskog. 1967. 15 p. Sw. cr. 10:- .

266. The half life of the 53 keV level in " 'Pt . By S. G. Malmskog. 1967. 10 p. Sw. cr. 10: - .

267. Burn-up determination by high resolution gamma spectrometry: Axial and diametral scanning experiments. By R. S. Forsyth, W. H. Blackladder and N. Ronqvist. 1967. 18 p. Sw. cr. 10:- .

268. On the properties of thes , i 2 >• d 3 / 2 transition in "<Au. By A. Bäcklin and S. G. Malmskog. 1967. 23 p. Sw. cr. 10: - .

269. Experimental equipment for physics studies in the Agesta reactor. By G. Bernander, P. E. Blomberg and P.-O. Dubois. 1967. 35 p. Sw. cr. 10:- .

270. An optical model study of neutrons elasticaliy scattered by iron, nickel, cobalt, copper, and indium in the energy region 1.5 to 7.0 MeV. By B. Holmqvist and T. Wiedling. 1967. 20 p. Sw. cr. 10:- .

271. Improvement of reactor fuel element heat transfer by surface roughness. By B. Kjellström and A. E. Larsson. 1967. 94 p. Sw. cr. 10:- .

272. Burn-up determination by high resolution gamma spectrometry: Fission pro­duct migration studies. By R. S. Forsyth, W. H. Blackadder and N. Ron­qvist. 1967. 19 p. Sw. cr. 10:- .

273. Monoenergetic critical parameters and decay constants for small spheres and thin slabs. By I. Carlvik. 1967. 24 p. Sw. cr. 10:- .

274. Scattering of neutrons by an anharmonic crystal. By T. Högberg, L. Bohlin and I. Ebbsjö. 1967. 38 p. Sw. cr. 10: - .

275. T h e l A K I = 1 , E1 transitions in odd-A isotopes of Tb and Eu. By S. G. Malm-skog, A. Mareiius and S. Wahlbom. 1967. 24 p. Sw. cr. 10:- .

276. A burnout correlation for flow of boiling water in vertical rod bundles. By Kurt M. Becker. 1967. 102 p. Sw. cr. 10: - .

277. Epithermal and thermal spectrum indices in heavy water lattices. By E. K. Sokolowski and A. Jonsson. 1967. 44 p. Sw. cr. 10: - .

278. On the istl^-^^n transitions in odd mass Pm nuclei. By A. Bäcklin and S. G. Malmskog. 1967. 14 p. Sw. cr. 10:- .

279. Calculations of neutron flux distributions by means of integral transport methods. By I. Carlvik. 1967. 94 p. Sw. cr. 10: - .

280. On the magnetic properties of the K = 1 rotational band in ' "Re. By S. G. Malmskog and M. Höjeberg. 1967. 18 p. Sw. cr. 10: - .

281. Collision probabilities for finite cylinders and cuboids. By I. Carlvik. 1967. 28 p. Sw. cr. 10:- .

282. Polarized elastic fast-neutron scattering of " C in the lower MeV-range. I. Experimental part. By O. Aspelund. 1967. 50 p. Sw. cr. 10:- .

283. Progress report 1966. Nuclear chemistry. 1967. 26 p. Sw. cr. 10:- . 284. Finite-geometry and polarized multiple-scattering corrections of experi­

mental fast-neutron polarization data by means of Monte Carlo methods. By O. Aspelund and B. Gustafsson. 1967. 60 p. Sw. cr. 10:- .

285. Power disturbances close to hydrodynamic instability in natural circulation two-phase flow. By R. P. Mathisen and O. Eklind. 1967. 34 p. Sw. cr. 10: - .

286. Calculation of steam volume fraction in subcooled boiling. By S. Z. Rou-hani. 1967. 26 p. Sw. cr. 10:- .

287. Absolute E1, AK = 0 transition rates in odd-mass Pm and Eu-isotopes. By S. G. Malmskog. 1967. 33 p. Sw. cr. 10:- .

288. Irradiation effects in Fortiweld steel containing different boron isotopes. By M. Grounes. 1967. 21 p. Sw. cr. 10:- .

289. Measurements of the reactivity properties of the Agesta nuclear power reactor at zero power. By G. Bernander. 1967. 43 p. Sw. cr. 10:- .

290. Determination of mercury in aqueous samples by means of neutron activa­tion analysis with an account of flux disturbances. By D. Brune and K. Jir-low. 1967. 15 p. Sw. cr. 10:- .

291. Separtaion of "Cr by means of the Szilard-Chalmers effect from potassium chromate irradiated at low temperature. By D. Brune. 1967. 15 p. Sw. cr. 10:- .

292. Total and differential efficiencies for a circular detector viewing a circu­lar radiator of finite thickness. By A. Lauber and B. Tollander. 1967. 45 p. Sw. cr. 10: - .

293. Absolute M l and E2 transition probabilities in U ! U . By S. G. Malmskog and M. Höjeberg. 1967. 37 p. Sw. cr. 10:- .

294. Cerenkov detectors for fission product monitoring in reactor coolant water. By O. Strindehag. 1967. 56 p. Sw. cr. 10:-.

295. RPC calculations for K-forbidden transitions in ' "W. Evidence for large inertial parameter connected with high-lying rotational bands. By S. G. Malmskog and S. Wahlbom. 1967. 25 p. Sw. cr. 10:- .

296. An investigation of trace elements in marine and lacustrine deposits by means of a neutron activation method. By O. Landström, K. Samsaht and C-G. Wenner. 1967. 40 p. Sw. cr. 10:- .

297. Natural circulation with boiling. By R. P. Mathisen. 1967. 58 p. Sw. cr. 10:- . 298. Irradiation effects at 160-240°C in some Swedish pressure vessel steels.

By M. Grounes, H. P. Myers and N-E. Hannerz. 1967. 36 p. Sw. cr. 10:- . 299. The measurement of epithermal-to-thermal U-238 neutron capture rate (P2a)

in Ågesta power reactor fuel. By G. Bernander. 1967. 42 p. Sw. cr. 10:- . 300. Levels and transition rates in <"Au. By S. G. Malmskog, A. Bäcklin and B.

Fogelberg. 1967. 48 p. Sw. cr. 10: - . 301. The present status of the half-life measuring equipment and technique at

Studsvik. By S. G. Malmskog. 1967. 26 p. Sw. cr. 10: - . 302. Determination of oxygen in aluminum by means of 14 MeV neutrons with

an account of flux attenuation in the sample. By D. Brune and K. Jirlow. 1967. 16 p. Sw. cr. 10: - .

303. Neutron elastic scattering cross sections of the elements Ni , Co, and Cu between 1.5 and 8.0 mev. By B. Holmqvist and T. Wiedling. 1967. 17 p. Sw. cr. 10:- .

304. A study of the energy dependence of the Th232 capture cross section in the energy region O. I to 3.4 eV. By G. Lundgren. 1967. 25 p. Sw. cr. 10:- .

305. Studies of the reactivity effect of polythene in the fast reactor FRO. By L. I. Tirén and R. Håkansson. 1967. 25 p. Sw. cr. 10:- .

306. Final report on IFA-10, the first Swedish instrumented fuel assembly irra­diated in HBWR, Norway. By J-A. Gyllander. 1967. 35 p. Sw. cr. 10:- .

307. Solution of large systems of linear equations with quadratic or non-qua­dratic matrices and deconvoiution of spectra. By K. Nygaard. 1967. 15 p. Sw. cr. 10:- .

308. Irradiation of superheater test fuel elements in the steam loop of the R2 reactor. By F. Ravndal. 1967. 94 p. Sw. cr. 10: - .

309. Measurement of the decay of thermal neutrons in water poisoned with the non-1/v neutron absorber cadmium. By. L. G. Larsson and E. Möller. 1967. 20 p. Sw. cr. 10: - .

310. Calculated absolute detection efficiencies of cylindrical Nal (Tl) scintill­ation crystals for aqueous spherical sources. By. O. Strindehag and B. Tollander. 1968. 18 p. Sw. cr. 10:- .

311. Spectroscopic study of recombination in the early afterglow of a helium plasma. By J. Stevefelt. 1968. 49 p. Sw. cr. 10: - .

312. Report on the personnel dosimetry at AB Atomenergi during 196S. By J . Carlsson and T. Wahlberg. 1968. 10 p. Sw. cr. 10: - .

313. The electron temperature of a partially ionized gas in an electric field. By F. Robben. 1968. 16 p. Sw. cr. 10:- .

314. Activation Doppler measurements on U238 and U235 in some fast reactor spectra. By L. I. Tirén and I. Gustafsson. 1968. 40 p. Sw. cr. 10: - .

315. Transient temperature distribution in a reactor core with cylindrical fuel rods and compressible coolant. By H. Vollmer. 1968. 38 p. Sw. cr. 10: - .

316. Linear dynamics model for steam cooled fast power reactors. By H. Voll­mer. 1968. 40 p. Sw. cr. 10:- .

317. A low level radioactivity monitor for aqueous waste. By E. J . M. Quirk. 1968. 35 p. Sw. cr. 10:- .

318. A study of the temperature distribution in UOi reactor fuel elements. By I. Devoid. 1968. 82 p. Sw. cr. 10:- .

319. An on-line water monitor for low level /^-radioactivity measurements. By E. J. M. Quirk. 1968. 26 p. Sw. cr. 10:- .

320. Special cryostats for lithium compensated germanium detectors. By A. Lauber, B. Malmsten and B. Rosencrantz. 1968. 14 p. Sw. cr. 10: - .

321. Stability of a steam cooled fast power reactor, its transients due to mode­rate perturbations and accidents. By H. Vollmer. 1968. 36 p. Sw. cr. 10: - .

322. Progress report 1967. Nuclear chemistry, 1968. 30 p. Sw. cr. 10: - . 323. Noise in the measurement of light with photomultipliers. By F. Robben.

1968. 74 p. Sw. cr. 10: - . 324. Theoretical investigation of an electrogasdynamic generator. By S. Palm­

gren. 1968. 36 p. Sw. cr. 10: - . 325. Some comparisons of measured and predicted primary radiation levels in

the Agesta power plant. By E. Aalto, R Sandlin and A. Krell. 1968. 44 p. Sw. cr. 10:- .

326. An investigation of an irradiated fuel pin by measurement of the production of fast neutrons in a thermal column and by pile oscillation technique. By Veine Gustavsson. 1968. 24 p. Sw. cr. 10:- .

327. Phytoplankton from Tvären, a bay of the Baltic, 1961-1963. By Torbjörn Willén. 1968. 76 p. Sw. 10:- .

328. Electronic contributions to the phonon damping in metals. By Rune Jonson. 1968. 38 p. Sw. cr. 10:- .

329. Calculation of resonance interaction effects using a rational approximation to the symmetric resonance line shape function. By H. Häggblom. 1968. 48 p. Sw. cr. 10:- .

330. Studies of the effect of heavy water in the fast reactor FRO. By L. I. Tirén, R. Håkansson and B. Karmhag. 1968. 26 p. Sw. cr. 10: - .

331. A comparison of theoretical and experimental values of the activation Dop­pler effect in some fast reactor spectra. By H. Häggblom and L. I. Tirén. 1968. 28 p. Sw. cr. 10:-.

332. Aspects of low temperature irradiation in neutron activation analysis. By D. Brune. 1968. 12 p. Sw. cr. 10:- .

333. Application of a betatron in photonuclear activation analysis. By D. Brune, S. Mattsson and K. Liden. 1968. 13 p. Sw. cr. 10:- .

334. Computation of resonance-screened cross section by the Dorix-Speng system. By H. Häggblom. 1968. 34 p. Sw. cr. 10:- .

335. Solution of large systems of linear equations in the presence of errors. A constructive criticism of the least squares method. By K. Nygaard. 1968. 28 p. Sw. cr. 10:-.

336. Calculation of void volume fraction in the subcooled and quality boiling regions. By S. Z. Rouhani and E. Axelsson. 1968. 26 p. Sw. cr. 10:- .

List of published AES-reports (In Swedish)

1 . Analysis be means of gamma spectrometry. By D. Brune. 1961. 10 p. Sw. cr. 6:- .

2. Irradiation changes and neutron atmosphere in reactor pressure vessels-some points of view. By M. Grounes. 1962. 33 p. Sw. cr. 6:- .

3. Study of the elongation limit in mild steel. By G. Östberg and R. After-mo. 1963. 17 p. Sw. cr. 6:- .

4. Technical purchasing in the reactor field. By Erik Jonson. 1963. 64 p. Sw. cr. 8 : - .

5. Agesta heat generating station. Summary of technical data, descriptions, etc. for the reactor. By B. Lilliehöök. 1964. 336 p. Sw. cr. 15:-.

6. Atom Day 1965. Summary of lectures and discussions. By S. Sandström. 1966. 321 p. Sw. cr. 15:- .

7. Building materials containing radium considered from the radiation pro­tection point of view. By Stig O. W. Bergström and Tor Wahlberg. 1967. 26 p. Sw. cr. 10:- .

Additional copies available from the library of AB Atomenergi, Fack, S-611 01 Nyköping, Sweden.

EOS-tryckerierna, Stockholm 1968