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
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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 ] .
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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.
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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 .
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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
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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 -
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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
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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
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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
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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
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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
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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