AE-28G UDC 535.248.2
CD 00 CN UJ < Calculation of Steam Volume Fraction in Subcooled Boiling
S. Z. Rouhani
AKTIEBOLAGET ATOMENERGI
STOCKHOLM, SWEDEN 1967
AE -286
CALCULATION OF STEAM VOLUME FRACTION IN SUBCOOLED BOILING
by
S. Z . Rouhani
SUMMARY
An analysis of subcooled boiling is presented. It is assumed that he at is removed by vapor generation, heating of the liquid that replaces the detached bubbles, and to some extent by single phase heat transfer. Two regions of subcooled boiling are considered and a criterion is provided for obtaining the limiting value of subcooling between the two regions. Condensation of vapor in the subcooled liquid is analysed and the relative velocity of vapor with respect to the liquid is neglected in these regions. The theoretical arguments result in some equations for the calculation of steam volume frac tion and true liquid subcooling.
Printed and distributed in June 1 967. LIST OF CONTENTS Page Introduction 3
Literature Survey 3
Theory 4 Assumptions and the Derivation of the Basic Equations 4 Remarks on the Liquid Subcooling 6
The integrated Equation of Subcooled Void 7
Estimation of the Rate of Bubble Condensation 9 Effects of Void and Channel Geometry on cp and cp^ 1 0 Effect of Subcooling on cp^ and cp^ 1 0 Effects of Mass Velocity, Heat Flux, and Physical Properties on cpj and cp,, 12
The Limiting Value of Subcooling in the First Region 1 3
Resume of Calculation Procedure 14 Calculations of a in the First Region 14 Calculation of O' in the Second Region 1 4 Steam Quality in Subcooled Boiling 1 5
Comparison with Experimental Data 1 5
Conclusion 15
Nomenclature 18 Greek Letters 20
References 22
Figure Captions 23 - 3 -
INTRODUCTION
The intensity of subcooled boiling depends on many factors, among which the degree of subcooling, surface heat flux, mass veloc ity, and physical properties of the liquid and its vapor. This report is intended to give a new method of calculating the steam volume fraction in the region of subcooled boiling. The method is based on a new way of considering the different mechanisms of heat removal in this region.
LITERATURE SURVEY
The subject of subcooled boiling has been treated by many inves tigators. However, the works have mostly been in connection with studies of heat transfer and surface temperature and only a few authors have concentrated on the question of determining the steam volume fractions in the region of subcooled boiling. Among the published works one may refer to the following. Griffith, Clark, and Rohsenow [l ] made photographic studies of subcooled boiling at high pressures on a rather short strip of metal in a rectangular channel. A fundamental result of their work was the observation of two separate regions in subcooled boiling. They sug gested a calculation method which was based on neglecting the steam generation in the first region but including it in the heat balance in the second region. For the first region they suggested that heat was re moved partly by the single phase heat transfer mechanism and partly by condensation. For the second region they assumed a constant con densing film coefficient. In order to find the transition point they drew upon some observation by Rohsenow [2 ] and assumed that at that point 80 per cent of the surface heat flux would go to steam generation. Maurer [3 ] made use of data from different sources and gave a method of void calculation which seems to be only suitable in cases of very high liquid velocities (several meters per second), and high pres sures (over 60 ata). For the subcooled void he used the arguments and derivations of Griffith et al. with a modification in the calculation of wall superheat. - 4 -
Bowring [4] gave a detailed method of calculation of void which was partly based on the forementioned works and included the assump tion that the bubble condensation was negligible in the second region. But, in calculating subcooled void according to his model it was neces sary to use an assumed and relatively high value of slip ratio. Costello [5] treated the problem of determining the time-averaged vapor bubble coating on the heater surface only for the case of high subcoolings. Delayre and Lavigne [6] gave a very brief treatment of the prob lem of calculating voids in local boiling with the assumption of a conden sation coefficient without further specifications.
THEORY
Assumptions and the Derivation of the Basic Equations
In the process of subcooled boiling the supplied heat through the surface is removed mainly by the following three mechanisms
1) single-phase heat transfer which will be effective as long as the heated surface is not covered with bubbles.
2) Steam generation.
3) Heating of that mass of water which replaces the detached bubbles.
Vapor condensation is a major factor in the heat exchange between the generated steam and subcooled liquid. This effect will be considered later. These mechanisms have been previously listed by some authors, among the possible forms of heat transfer in subcooled boiling [7 ]. An observation of the final results will show that the relative effective ness of each of the suggested mechanisms vary as mass velocity, pres sure, and subcooling are varied. One may assume that in a heated channel local boiling starts at a point where q/A - h(t -1^) > o. In local boiling at high subcoolings the single phase heat transfer will be still effective but augmented by the other mechanisms. In this region a part of the heated surface will be covered with bubbles and hence be unaccesible to the liquid flow. - 5 -
This is equivalent to a reduction in the single phase heat transfer co efficient. In the absence of an accurate method of estimating the ef fective h , we assume sb
h valid for t > t (0 sb w s
Based on this argument the non-boiling fraction of heat flux will be
. = hsn r-r-T- (tw-V = hRn- At (2) nb w o
As subcooling further decreases vapor clotting [5] takes place viz. , the bubbles cover the heated surface and eliminate single phase heat transfer. At this subcooling the steam volume fraction begins to rise rapidly with decreasing subcoolings. The heat balance on the heated surface is m qj; = h • At + m • X + —— C p. At (3) A s p p t ' ‘ g \ The amount of heat which goes to steam generation per unit time within an element of height of the channel, dz, will be
(^;) - hAt dQ, = m \ P, dz Pg'^SP^AtPg" Ph^z b s h (4 ) where mg is substituted from eq. (3).
The amount of heat removed from the steam bubbles by con densation is an unknown factor. The available information on the col lapse of the individual bubbles [8,9, 10] is not easily applicable to an unknown population of bubbles in a boiling system. Therefore, one may try to express the rate of condensation in terms of some macro scopic parameters of the system. Here we assume that the amount of heat which will be removed from steam bubbles by condensation per unit time within an element of height, dz, will be
dQ = k (t - t.) dz (5 ) c cx s -v - 6 -
where k is a condensation constant with the same dimensions as c that of the thermal conductivity. This constant will be analysed in more detail later. Considering the flow of steam into and out of a small section of the channel, ' dz, one should get
steam heat flow out - steam heat flow in = dQ^ - dQ^ (6)
Neglecting the relative velocity of vapor with respect to the liquid in the subcooled region:
m V ~ Yl ~ = V (7) g A [cvp + (1 - of) p . ] C g ' -V and eq. (6) becomes
A [ (V + dV) (a + da) - Va ] p X = dQ. - dQ (8) c ' g be
But from eq. (7)
(° l - P g) da m dV A~ (9) c [apg+ (1 - a) p^]2
Elimination of dQ^, dQ^, and dV between cqs. (4), (5), (8) and (9) yields :
mX dof - (J) - h At p X P - k At dz (10) ~T Li - (i-i/p)of32 g h c .
Remarks on the Liquid Subcooling
In subcoolcd boiling a part of the supplied heat goes with the 1« iin bubbles. Tt is clear th.il the average liquid subcooling is dif
b rent 11 •»i n I hr average bulk subcooling which is normal Iv obtained from a simple heat balance, viz.
z
o At, a At. (H) b in m • C P - 7 -
This would represent the liquid subcooling only if the steam bub bles had zero velocity. The assumption of equal velocities of vapor and liquid yields the steam quality in the region of subcooled boiling;
a Xs “ a + (l - a) p (12)
The heat carried by steam bubbles will be
m X a x m X = (13) Q. a + 0 -<*) P
And hence a closer approximation of the average liquid subcooling will be . z
At. (1 4 - a) Att in rh- C P or a • X At (14- b) % AV [ar+ (1 -a) p] C
The last term becomes increasingly significant as pressure and steam fraction are increased.
THE INTEGRATED EQUATION OF SUBCOOLED VOID
The more correct value of At^ from eq. (l 4-a) yields m C dO dz =------2----- [d(At^) - -—-] (15) (q/A) mC
But, the second term in the brackets may be neglected and hence m C dz = - d(At„) (16) ^ (q/A)
Rewriting eq. (10) with dz substituted from eq. (l 6) we get
Cp^ q/A-hAt^ dQf 2 ” q/A ' X • p ^ + C^p^A t^ Ll -a(l-l/p)] ‘ g P
k At, p • x • p ) d (At-t) (17) h g _ 8 -
In the case of constant values of heat flux and physic ai prope 1 uc s this equation may be integrated between the inlet (a = 0) and any other point within the first region (this will be defined later).
a C p. q/A-h At. da P -v f1 t * -v _ .... ^ Jo[l - a(]-]/(3)] q/Z A^^Pg+V^^
-) d(At^) (18) Ph ' P
Carrying on the integrations and neglecting ^ one obtains
, r h- X p Xp+Cp.At., t m,. 2 l m + —ln ' 1 - a cPpt Xpg+CpP^Att
91 C p. -> h(Atin"AV - ] (M)
In which At p. (qA) V, = I 4 AW = - S dQ = At. rh C z~o in
Ph (q/A) Qc (z) (20) rh C P where (z) is a factor which must be determined from some ex perimental data. Eqs. (l 8) and (l 9) are valid only between the point of initiation of local boiling (roughly at At. = ff/A) and a limiting value of At. = At -v n.gp b c at which the heated surface becomes clotted with vapor. At subcoolings less than At^ (At^ > At^ >o) the single phase heat transfer is non existent and the corresponding equations for the second region will be obtained by setting h = o in all the previous equations. Particularly eqs. (l 8) and (l 9) will become (neglecting |3) - 9 -
a r n At da pp<- r (-----ilA------S 0 -<*)2 a-a q/A &tc X-pg+CpPtAtt c
kc &tt —) d(&t„) (21) PhX'p g
or Xp + C p . At Jjg P..^—c- ^2 Gp^L - In (22) a 1 - a Tp +C p. At.' TTT^-q/A g p l l
In which htl = I - d(AtJ (23) At
For very low inlet subcoolings At^ = At. . in A method of estimation of At will be given later on in this report. c
ESTIMATION OF THE RATE OF BUBBLE CONDENSATION
The integral terms in cp^ and cp^ of eqs. (20) and (23) indicate a dependence on the liquid subcoolings at some initially fixed points (At. or At ) as well as on the local values of At. in each region. In addition one should also expect some influence of other parameters such as liquid and vapor properties, heat flux, mass flow rate and, finally, the average contact area between vapor and liquid which is, to some extent, dependent on the steam volume fraction and the chan nel geometry. These factors are collectively expressed as k , that is, in cp^
kcl " 6l(pV A’ G) Abl (24)
and in cp^
kc2 = e2
Effects of Void and Channel Geometry on cp^ and cp-.
In an ideal case of uniform and spherical bubbles one could cal culate
A = n tt d, b b (26) in which n would be the number of bubbles per unit length of the chan nel and d^ the bubble diameter. Then, from the relation
6 A (27) c
one could find
AhM=62/3.(„.r,)'/3 a V3 „2/3 (28)
Now, in the actual cases of cp^ and cp^ one may assume
A (a) = e A2/3u 2/3 bl ' 3 c (29) and 2/3 2/3 ^b2^=^4 c (3 0)
“ 1 I3 whore s and e ^ are some multipliers (in m ' )
Effect of Subcooling on and
Plots of the measured values of a vs At^ which are obtained with constant heat flux show that at very large subcoolings the steam volume fraction increases slightly and linearly with decreasing At^. This cor responds to the first region of subcooled boiling. Then, at some value of subcooling around At^, at which vapor clotting appears in the channel, the curve of a starts to rise rapidly with At^. A typical curve of this type is shown in Fig. (1). For any point in the first region
At. -At. = mhatr (31) in 'L in which At, refers to any value of At, between At. and At I 7 v in ( - 11
In the second region a nearly linear relation is found between 2/3 the changes in or' and At^, viz,
2/3 ^c"^2 2/3 a a (JZ) W = At -At. c u
Hence we employ these relations in eqs. (29) and (3 0). The discrepancies between these simplifying approximations and the true variations of a with At^ will only have a secondary effect on the final results since cp^ and cp^ are like correction terms in the integrated equations of void in the two regions. Based on these arguments the detailed form of the condensation equations will be: At Z e ] (P» i- cd/V,)^At ] d(At]) (33) At. in for the first region in which At. >-^At >At„ >/At , and b m 1 -t- c 5^ y /o o f-x At -At- ^(P' ^'G)A/ ^VAt^Atr^^'^y At c b c for the second region in which At^>^At^> At^^ 0.
Assuming that the involved properties of vapor and liquid (yet to be defined) do not change considerably between the limits of sub coolings in the integrals, one may carry out the integrations with respect to At^ and get:
T, = "A, ' *c^ ^ (O' 6 Alqn+Ay (At^ -At&) (35) and • ?2 = ^2 ’ A2/3 v2/3 (At^) (A t c -h 2 At-j) (Atc -At^) (36)
in which T) and T|^ represent the products of the previous constants and those which resulted from the integrations. 12
Effects of Mass Velocity, Heat Flux, and Physical Properties on ^ and q>2 The physical properties which are expected to influence cp^ and cp^ are:
k^, X, C^, Pgj Pf,’ P1^• and o.
The effect of mass velocity may be included in a nondimensional form by using Reynolds number
G- D e Re (37) and in order to include the effect of heat flux also in a nondimensional form we consider the following dimensionless number:
(q/A) |R (3 8) N q = TX------o - g
In an attempt to find proper expressions for 1^ in cpj and in cp^ this analysis was applied to some experimental data [l 1 ] and after many trials it was concluded that the following expressions would give the best fit to most of the measured data.
(39)
with =3.5* 1 03 m 4^3 and
SRe0.5„0.5__l TU = (40) q (Atc-Att) with az a 7. 0 • 105 m'4/3
Hence, the final expressions for the condensation terms in eqs. (19) and (22) will be:
0. 25 . cp] =a] Re • N 9 A2/3 a2/3 (0. 6At. + At.) (At. -At.) c ' m V x m V (41) - 13
and L = a2K r P"' R”'5 N”'5 Ac/3 0,2/3 (PV2-^) (42) e q c with = 3. 5 • 1 0^ m ^ and a^ = 7.0" 1 0^ m 4 /^
The dimensional constants a^ and a^ may probably depend on the number of nucleation sites per unit area of the heated surface as well as on the bubbling frequency and its relation to the mass velocity. But no attempt is made to analyse this matter here.
THE LIMITING VALUE OF SUBCOOLING IN THE FIRST REGION
An estimation of the limiting value of subcooling in the first re gion, At^, may be made by using a known value of which corresponds to the limiting value of void at the point where the heated surface be comes fully covered with tiny bubbles. For an estimation of o^ we draw upon an argument presented in the published works by Maurer [3], Bowring [4 ] and Costello [5]. That is, when the heated surface becomes fully covered with one layer of small bubbles the average vapor thickness on the wall will be about:
8 = 0.67 R , d
in which R^ is the average radius of vapor bubbles when they leave the heated surface. Values of R^ obtained experimentally at lower pressures [ 1 2 ] and extrapolated to higher pressures with the guidance of theoretical arguments [ 1 3 ] are presented as a curve in ref. [4 ]. In the pressure range between 1 to 1 00 bar one may use the fol lowing approximation:
Rd = 0. 237 ‘ 1 ° m (44) P
Once R^ and 8 are determined one may easily calculate
a = ~ 6 s 1. 59 • 1 0“4 • p"°* 237 P /A (45) ' c A h c c - 14
The particular value of subcooling At^ may now be calculated by using this in eqs. (l 9) and (41) and solving for At^ by iteration. Since at the end of first region (clotting point) or = a?c and At^ = At^ are known, the corresponding value of the heated length may be easily determined:
m Z -Atc) Cp + u + (1 -a )(3_ (46) c c x c (q/A)
RESUME OF CALCULATION PROCEDURE
Once u , At are determined as described in section 6 the local c c values of of may be calculated as follows.
Calculations of a in the First Region
Usually the values of of in the first region are quite negligible unless P^/Ac is very large and the system is operated at very low (near atmospheric) pressures. But, in any case, for heated lengths less than the values of Of may be approximated by
At. - At, (47) “-siT^r “c in c
Since At^ is dependent on a, one has to use a method of trial and error to solve eqs. (l 4-b) and (47). For a more detailed calculation of of in this region one may solve eqs. (l 1 ), (l 4-b), (41), and (l 9) simultaneously (by trial and error) to obtain values of At „ and of.
Calculation of a in the Second Region
For heated lengths larger than Z^ steam volume fraction is ob tained by the following steps.
1) Obtain a and At as described in section 6. ' c c 2) Calculate At from eq. (ll).
3) Assume an Qf (a > or ^) and calculate A from eq. (I 4-b). - 15
4) Use At^ along with the assumed a in cq. (42).
5) With that value o£ c and the same At^ solve the following cq. for ca (and iterate)
X p + C p,At Cp ^2 J = In—s___Eli—£ 1 - a 1 -a (id) (q/A)P^)vp^
Steam Quality in Subcooled Boiling
For a A tf >"• o x (49) real oi + (1 - O') (3 and fo r Q(Z) - m (isat - ,in) At, "< o x = x, (50) u real b m
COMPARISON WITH EXPERIMENTAL DATA
The described methods of calculating At^ and a in the second region have been applied to some experimental data from ref. [l 1 ]. The results are shown in Table 1 . The calculated values of At^ seem to be almost independent of AL^ for large values of this parameter. But, as At. decreases the difference between At. and At becomes in in c smaller and, below certain values, At = At. . c in Figures 2 to 5 show the comparison of the calculated steam fractions with some of the experimental data from the annular geom etry of ref. [l 1 ]. Figure 6 shows the comparison of this analysis with some data obtained with a six-rod cluster [l 4].
CONC FUSION
It appears that the agreement between the calculated and meas ured data is quite good in most cases. This analysis and the suggested method of calculation do dis play the effects of various parameters on subcooled void as found by experiments. - 16 -
The present method of calculation will not yield satisfactory re sults in the case of very low mass velocities. The limiting values of liquid velocity (single phase) is about 0.5 m/sec. At lower liquid ve locities this analysis yields relatively large values of steam volume fraction. Another limitation may be the range of heat flux in which cp^ and correlations are valid. The constants which are given here were ob tained from the data in the range of 60 < q/A < 120 W’/cm'A Perhaps it will be necessary to find more suitable constants for other ranges of this parameter. Finally, it may be expected that the nature of the heated surface influence the form and the empirical constants of cp^ and cp^. - 17
Table 1 - The subcoolings at the end of first region calculated for different inlet subcoolings
p = 50 b p = 50 b 2 G = 665 kg/m * s G = 1 065 kg/m'G s ^ = 90 W/cm2 = 90 W/cm2
At. At At. At in c in c °C °C °C °C
58. 70 32.85 48.95 21. 80 53. 70 36. 07 43. 95 24.98 48. 70 38.97 38.95 28.16 43. 70 39.87 33.95 30.08 38.70 37. 12 28.95 27. 73 33.70 32.83 23.95 23.36 27. 70 27. 19 17. 95 17. 63
p = 39.4 b P = 29. 5 b G = 665 kg/m • s G = 1450 kg/m ■ s ^ = 120 w/cm2 — = 90 W/cm2
At. At At. At in c om o c °C °c C °C
71. 60 45. 34 44. 90 17. 52 66. 60 48.46 39.90 20. 56 61.60 51.06 34.90 23. 74 56. 60 51.70 29.90 26. 00 51. 60 49.26 24.90 23.82 46. 60 45. 24 19. 90 19. 41 40. 60 39.76 13. 90 1 3. 66 - 18
NOMENC LATURE
Dimensions
Abl average condensation area per unit m length of channel in the first region
Ab2 average condensation area per unit m length of channel in the second region 2 A channel area m 3. 5 • 1 03 -4/3 al m
Ln -4/3 a2 o o m '
C specific heat of liquid at constant kj/kg °C P pressure d differential sign -
bubble diameter m
D equivalent^(hydraulic) diameter of m e channel ~ p - tot kg/m2- sec G mass velocity = ~ c (kgmm/sec^) / gc conversion factor equivalent to 9. 81 kW/m2- °C h single phase heat transfer coefficient calculated according to Collburn ''s correlation kw/m2- °C hsb single phase heat transfer coefficient in subcooled boiling h the same as h kW/m2- °C sp i. enthalpy of liquid at inlet to the heated kj/kg m channel
1sat enthalpy of liquid at saturation tem- kj/kg pe rature k condensation constant defined by eq. kj/m« °C‘ sec c (5) k condensation constant in the first re kj/m- °C- sec C1 gion defined by eq. (24) -19-
k . condensation constant in the second kj/m, °C sec c2 region defined by eq. (25) k-L thermal conductivity of liquid kj/m- °C- sec m mass flow rate kg/sec kg/m^' sec ™s mass of liquid which is converted in to steam per unit time per unit area of the heated surface - 1 n number of bubbles per unit length of m channel
19 (q/A) p/(X-CT. gc-p-t) - q p pressure bar ph heated periphery m
Pr Prandtl number = p/k^ -
P total periphery m tot kW / m^ q/A heat flux
Qb the amount of heat which is absorbed k W/m by boiling per unit time per unit length of channel
Qc the amount of heat which is exchanged kW/m by condensation of bubbles per unit time per unit length of channel
Qs heat carried by steam defined by eq. kW (13)
Rd bubble radius at the instant of detach m ment
R Reynolds number = G-De/p - e \ bulk temperature °C t saturation temperature °c s t wall temperature °c w . V average velocity of the mixture m/sc c
V vapor velocity m/ sec 5 V-L liquid velocity m/sec - 20
x steam quality calculated from heat - b balance according to eq. (50)
^real true steam quality - x steam quality in the region of sub- - s cooled boiling defined by equation
Z distance along the heated channel m measured from the inlet point
Z the heated length at which the first m c region ends
Greek Letters O' steam volume fraction (void) - o' steam volume fraction at the end of the C first region, defined by eq. (46)
At temperature difference in general °C
Mb t -t =ave rage bulk sub cooling defined °C by (11)
At liquid subcooling at which the first re °c c gion ends and the second region begins
At rO °c o e j and e dimensional constants in eqs. (24) and kj/m* °C* sec (25)
T1 ^ and T\^ dimensional constants in eqs. (39) and kj/m2- 25 . °C see (40)
X latent heat of vaporization kj/kg
M- dynamic viscosity kg /m* sec m ksmAn3 Pg vapor density kgm/m3 P-t liquid density - 21
cr surface tension of liquid against its kg^/m vapor
condensation function in the first region kW• °C/m *1 defined by eqs. (20) and (41)
condensation function in the second re- kW• °C/m ^2 gion defined by eqs. (23) and (42) - 22
REFERENCES
1. GRIFFITH P. , CLARK I. A., and ROHSENOW W.M. , Void Volumes in Subcooled Boiling Systems. 1 958. (ASME 58 -HT -19.)
2. ROHSENOW W. M. , A method of correlating heat transfer data for surface boiling of liquids . Trans. AS ME, Vol. 74 (1 952) Aug.
3. MAURER G. W. , A Method of Predicting Steady -State Boiling Vapor Fractions in Reactor Coolant Channels . I960. (WAPD-BT-1 9- )
4. BOWRING R. W. , Physical Model, Based on Bubble Detachment and Calculation of Steam Voidage in the Subcooled Region of a Heated Channel. 1 962. (HRP 10.)
5. COSTELLO C.P. , Aspects of Local Boiling Effects on Density and Pressure Drop. 1959. ' (ASME 59-HT-l 8. )
6. DELAY RE R. and LAVIGNE P. , Friction Pressure Drops for Flow of Boiling Water at High Pressure. Appendix-Model of Void Fraction. (In French.) E AES Symposium on Two -Phase Flow, Steady State Burn-out and Hydrodynamic Instability. AB Atomenergi, Studsvik, Sweden, 1 -3 Oct. , 1 963. Vol. 1 . 1 963.
7. FORSTER K.E. and GREIF R. , Heat Transfer to a Boiling Liquid - Mechanism and Correlation. J. Heat Transfer, 81 (1959) pp. 43-53.
8. FLORSCHUETZ L.W. and CHAO B. T. , On the Mechanics of Vapor Bubble Collapse. J. Heat Transfer, 87 (l 965) pp. 209-220.
9. NISHIWAKI I. , Brief Survey on Growth and Collapse of Steam Bubbles. 1 960. (TID-1 1 060. )
10. LEVENSPIEL O. , Collapse of Steam Bubbles in Water. Ind. Eng. and Chem. 51 (1959) p. 787.
1 1 . ROUHANI S. Z . , Void Measurements in the Regions of Subcooled and Low Quality Boiling. Part 2. 1 966. (AE -239. ) - 23
12. ZYSINA-MOLOSHEN L.M. and KUTATELADZE S.S., The Effect of Pressure on the Mechanism of Steam Forma tion in a Boiling Liquid. Zhur. Tekh. Fiz. 20 (l 950) p. 1 10.
13. JAKOB M. , Heat Transfer. Vol. 1, Chap. 29. J. Wiley, N.Y. 1 949.
14. EKLUND G. and NY LUND O. , FROJA - Preliminary Results of Measurements During Feb. - March 1965. Personal Communication 1 965. - 24 -
FIGURE CAPTIONS 2/3 Figure 1 Typical trends of variations of a and a ' with sub cooling along the heated length of a channel.
2 Comparison of the present analysis with the experi mental data of ref. [ 1 1 ] at 9. 9 bars.
3 Comparison of the present analysis with the experi mental data of ref. [ 1 1 ] at 29. 5 bars.
4 Comparison of the present analysis with the experi mental data of ref. [ 1 1 ] at 39. 0 bars.
5 Comparison of the present analysis with the experi mental data of ref. [ 11 ] at 50. 0 bars.
6 Comparison of the suggested method of calculation with data from void measurements in a 6-rod cluster [ ref. 14]. Of AND Of
FIG. 1 - TYPICAL TRENDS OF VARIATIONS OF Oi AND Qf 2/3 WITH SUB COOLING ALONG THE HEATED LENGTH OF A CHANNEL. P * 9.9 1 0. 1 BAR _ 60
EXPERIMENTAL DATA 122. 5 CALCULATED VALUES
EXPERIMENTAL DATA *61.5 _____ CALCULATED VALUES
— 40
- 30
— 20
DEGREE OF BULK SUB COOLING, A t
FIG. 2 COMPARISON OF THE PRESENT ANALYSIS WITH THE EXPERIMENTAL DATA OF REF. ill j AT 9.9 BARS. P = 29. 5 to. 5 BAR Of EXPERIMENTAL DATA = 91.5 CALCULATED VALUES - 50 EXPERIMENTAL DATA 1455 - CALCULATED VALUES
- 40
- 20
- 1 0
DEGREE OF BULK SUBCOOLING, A t, FIG. 3 - COMPARISON OF THE PRESENT ANALYSIS WITH THE EXPERIMENTAL DATA OF REF. Ill] AT 29.5 BARS. EXPERIMENTAL DATA G = 665 CALCULATED VALUES m -s - 50 EXPERIMENTAL DATA
CALCULATED VALUES
- 20
DEGREE OF BULK SUB COOLING, A t0
ITG. 4 — COMPARISON OF THE PRESENT ANALYSIS WITH THE EXPERIMENTAL DATA OF REF. Ill] AT 39. 0 BARS. B 50.0 : 1.3 BAR
EXPERIMENTAL DATA G = 665 CALCULATED VALUES
EXPERIMENTAL DATA
CALCULATED VALUES - 40
DEGREE OF BULK SUB COOLING, A t
FIG. 5 — COMPARISON OF THE PRESENT ANALYSIS WITH THE EXPERIMENTAL DATA OF REF. [ 11] AT 50. 0 BARS. *31.6 BAR EXPERIMENTAL DATA = 1345 kg/ r 46. 7 w/cm -----CALCULATED VALUES
= 51.4 BAR EXPERIMENTAL DATA
CALCULATED VALUES 64. 5 w/cm
DEGREES OF SUBCOOLING, A t
FIG. 6 — COMPARISON OF THE SUGGESTED METHOD OF CALCULATION WITH DATA FROM VOID MEAS UREMENTS IN A 6-ROD CLUSTER [REF. 14].
LIST OF PUBLISHED AE-REPORTS 247. Neutron-activation analysis of biological material with high radiation levels. By K. Samsahl. 1966. 15 p. Sw. cr. 8:—. 1-200. (See the back cover earlier reports.) 248. One-group perturbation theory applied to measurements with void. By R Persson. 1966. 19 p. Sw. cr. 8:-. 201. Heat transfer analogies. By A. Bhattacharyya. 1965. 55 p. Sw. cr. 8:-. 249. Optimal linear filters. 2. Pulse time measurements in the presence of 202. A study of the "384" KeV complex gamma emission from plutonium-239. noise. By K. Nygaard. 1966 9 p. Sw. cr. 8:—. By R. S. Forsyth and N. Ronqvist. 1965. 14 p. Sw. cr. 8:—. 250. The interaction between control rods as estimated by second-order one- 203. A scintillometer assembly for geological survey. By E. Dissing and O. group perturbation theory. By R. Persson. 1966. 42 p. Sw. cr. 8:—. Landstrom. 1965. 16 p. Sw. cr. 8:—. 251. Absolute transition probabilities from the 453.1 keV level in 183W. By S. G 204. Neutron-activation analysis of natural water applied to hydrogeology. By Malmskog. 1966. 12 p. Sw. cr. 8:-. O. Landstrom and C. G. Wenner. 1965. 28 p. Sw. cr. 8:—. 252. Nomogram for determining shield thickness for point and line sources of 205. Systematics of absolute gamma ray transition probabilities in deformed gamma rays. By C. Jonemalm and K. Malen. 1966. 33 p. Sw. cr. 8:— odd-A nuclei. By S. G. Malmskog. 1965. 60 p. Sw. cr. 8:-. 253. Report on the personnel dosimetry at AB Atomenergi during 1965. By K. A 206. Radiation induced removal of stacking faults in quenched aluminium. By Edwardsson. 1966. 13 p. Sw. cr. 8:—. U. Bergeniid. 1965. 11 p. Sw. cr. 8:—. 254 Buckling measurements up to 250°C on lattices of Agesta clusters and on 207. Experimental studies on assemblies 1 and 2 of the fast reactor FRO. Part 2. DiO alone in the pressurized exponential assembly TZ. By R. Persson, By E. Hellstrand, T. Andersson, B. Brunfelter, J. Kockum, S-O. Londen A. J. W. Andersson and C.-E. Wikdahl. 1966. 56 p. Sw. cr. 8:—. and L. I. Tiren. 1965. 50 p. Sw. cr. 8:—. 255 Decontamination experiments on intact pig skin contaminated with beta- 208. Measurement of the neutron slowing-down time distribution at 1.46 eV gamma-emitting nuclides. By K. A. Edwardsson, S. HagsgSrd and A Swens- and its space dependence in water. By E. Moller. 1965. 29 p. Sw. cr. 8:—. son. 1966. 35 p. Sw. cr. 8:—. 209. Incompressible steady flow with tensor conductivity leaving a transverse 255. Perturbation method of analysis applied to substitution measurements of magnetic field. By E. A. Witalis. 1965. 17 p. Sw. cr. 8:-. buckling. By R. Persson. 1966. 57 p. Sw. cr. 8:-. 210. Methods for the determination of currents and fields in steady two 257. The Dancoff correction in square and hexagonal lattices. By I. Carlvik. 1966. dimensional MHD flow with tensor conductivity. By E. A. Witalis. 1965. 35 p. Sw. cr. 8:-. 13 p. Sw. cr. 8:—. 258. Hall effect influence on a highly conducting fluid. By E. A. Witalis. 1966 211. Report on the personnel dosimetry at AB Atomenergi during 1964. By 13 p. Sw. cr. 8:—. K. A. Edvardsson. 1966. 15 p. Sw. cr. 8:—. 259. Analysis of the quasi-elastic scattering of neutrons in hydrogenous liquids. 212. Central reactivity measurements on assemblies 1 and 3 of the fast reactor By S. N. Purohit. 1966. 26 p. Sw. cr. 8:-. FRO. By S-O. Londen. 1966. 58 p. Sw. cr. 8:—. 260. High temperature tensile properties of unirradiated and neutron irradiated 213. Low temperature irradiation applied to neutron activation analysis of 20Cr—35Ni austenitic steel By R B Roy and B Solly. 1966. 25 p. Sw mercury in human whole blood. By D. Brune. 1966. 7 p. Sw. cr. 8:—. cr. 8:-. 214. Characteristics of linear MHD generators with one or a tew loads. By 261. On the attenuation of neutrons and photons in a duct filled with a helical E. A. Witalis. 1966. 16 p. Sw. cr. 8:-. plug. By E. Aalto and A. Krell. 1966. 24 p. Sw. cr. 8:—. 215. An automated anion-exchange method for the selective sorption of five 262. Design and analysis of the power control system of the fast zero energy groups of trace elements in neutron-irradiated biological material. By reactor FR-O. By N. J. H. Schuch. 1966. 70 p. Sw. cr. 8:-. K. Samsahl. 1966. 14 p. Sw. cr. 8:—. 263. Possible deformed states in115 ln and mln. By A. Backlin, B. Fogelberg and 216. Measurement of the time dependence of neutron slowing-down and therma- S. G. Malmskog. 1967. 39 p. Sw. cr. 10: —. Hzation in heavy water. By E. Moller. 1966. 34 p. Sw. cr. 8:—. 264. Decay of the 16.3 min. 182 Ta isomer. By M. Hojeberg and S. G. Malmskog. 217. Electrodeposition of actinide and lanthanide elements. By N-E. Barring. 1967. 13 p. Sw. cr. 10: —. 1966. 21 p. Sw. cr. 8:—. 265. Decay properties of ,47Nd. By A. Backlin and S. G. Malmskog. 1967. 15 p. 218. Measurement of the electrical conductivity of He3 plasma induced by Sw. cr. 10: -. neutron irradiation. By J. Braun and K. Nygaard. 1966. 37 p. Sw. cr. 8:-. 266. The half life of the 53 keV level in 197Pt. By S. G. Malmskog. 1967. 10 p. 219. Phytoplankton from Lake Magelungen, Central Sweden 1960-1963. By T. Sw. cr. 10: —. Willen. 1966. 44 p. Sw. cr. 8:—. 267. Burn-up determination by hight resolution gamma spectrometry: Axial and 220. Measured and predicted neutron flux distributions in a material surround diametral scanning experiments. By R. S. Forsyth, W. H. Blackadder and ing av cylindrical duct. By J. Nilsson and R. Sandlin. 1966. 37 p. Sw. N. Ronqvist. 1967. 18 p. Sw. cr. 10:-. cr. 8:—. 268. On the properties of the s]j2 ----- >- d3^2 transition in 199 Au. By A. Backlin 221. Swedish work on brittle-fracture problems in nuclear reactor pressure and S. G. Malmskog. 1967. 23 p, Sw. cr. 10: —. vessels. By M. Grounes. 1966 34 p. Sw. cr. 8:—. 222. Total cross-sections of U, UOi and ThOz for thermal and subthermal 269. Experimental equipment for physics studies in the Agesta reactor. By G. neutrons. By S. F. Beshai. 1966. 14 p. Sw. cr. 8:—. Bernander, P. E. Blomberg and P.-O. Dubois. 1967. 35 p. Sw. cr. 10: —. 223. Neutron scattering in hydrogenous moderators, studied by the time de 270. An optical model study of neutrons elastically scattered by iron, nickel, pendent reaction rate method. By L. G. Larsson, E, Moller and S. N. cobalt, copper, and indium in the energy region 1.5 to 7.0 MeV. By B. Purohit. 1966, 26 p. Sw. cr. 8:—. Holmqvist and T. Wiedling. 1967. 20 p. Sw. cr. 10: —. 271. Improvement of reactor fuel element heat transfer by surface roughness. 224. Calcium and strontium in Swedish waters and fish, and accumulation of By B. Kjellstrom and A. E. Larsson. 1967. 94 p. Sw. cr. 10: —. strontium-90. By P-O. Agnedal. 1966. 34 p. Sw. cr. 8:—. 272. Burn-up determination by high resolution gamma spectormetry Fission pro 225. The radioactive waste management at Studsvik, By R. Hedlund and A. duct migration studies. By R, S. Forsyth, W. H. Blackadder and N. Ron Lindskog. 1966. 14 p. Sw cr. 8:—. qvist. 1967. 19 p. Sw. cr. 10: —. 226. Theoretical time dependent thermal neutron spectra and reaction rates 273. Monoenergetic critical parameters and decay constants for small spheres in HiO and DzO. S. N. Purohit. 1966. 62 p. Sw. cr. 8:-. and thin slabs. By I. Carlvik. 24 p. Sw. cr. 10:-. 227. Integral transport theory in one-dimensional geometries. By I. Carlvik. 274. Scattering of neutrons by an enharmonic crystal. By T. Hogberg, L. BohHn 1966. 65 p. Sw. cr. 8:—. and I. Ebbsjo. 1967. 38 p. Sw. cr. 10:-. 228. Integral parameters of the generalized frequency spectra of moderators. 275. The IAKI = 1, E1 transitions in odd-A isotopes of Tb and Eu. By S. G. Malm By S. N. Purohit. 1966. 27 p. Sw. cr. 8:—. skog, A. Marelius and S. Wahlborn. 1967. 24 p. Sw. cr. 10: —. 229. Reaction rate distributions and ratios in FRO assemblies 1, 2 and 3. By 276. A burnout correlation for flow of boiling water in vertical rod bundles. By T. L. Andersson. 1966. 50 p. Sw. cr. 8:-. Kurt M. Becker. 1967. 102 p. Sw. cr. 10: —. 230. Different activation techniques for the study of epithermal spectra, app 277. Epithermal and thermal spectrum indices in heavy water lattices. By E. K. lied to heavy water lattices of varying fuel-to-moderator ratio. By E. K Sokolowski and A. Jonsson. 1967. 44 p. Sw, cr. 10:-. Sokolowski. 1966. 34 p. Sw. cr. 8:—. 278. On the ^Si2<~~^^7i2 transitions in odd mass Pm nuclei. By A. Backlin and 231. Calibration of the failed-fuel-element detection systems in the Agesta reactor. By O. Strindehag. 1966. 52 p. Sw. cr. 8:—. S. G. Malmskog. 1967. 14 p. Sw. cr. 10: —. 232. Progress report 1965. Nuclear chemistry. Ed. by G. Carleson. 1966. 26 p. 279. Calculations of neutron flux distributions by means of integral transport Sw. cr. 8:-. methods. By I. Carlvik. 1967. 94 p. Sw. cr. 10:-. 233. A summary report on assembly 3 of FRO. By T. L. Andersson, B. Brun 230. On the magnetic properties of the K = 1 rotational band in 188Re. By S. G. felter, P. F. Cecchi, E. Hellstrand, J. Kockum, S-O. Londen and L. 1. Malmskog and M. Hojeberg. 1967. 18 p. Sw. cr. 10:-. Tiren. 1966. 34 p. Sw. cr. 8:— 281. Collision probabilities for finite cylinders and cuboids. By I. Carlvik. 1967. 234. Recipient capacity of Tvaren, a Baltic Bay. By P.-O. Agnedal and S. O. W. 28 p. Sw. cr. 10:-. Bergstrom. 1966. 21 p. Sw. cr. 8:-. 282. Polarized elastic fast-neutron scattering off 12 C in the lower MeV-range. 235. Optimal linear filters for pulse height measurements in the presence of I. Experimental part. By O. Aspelund. 1967. 50 p. Sw. cr. 10:-. noise. By K. Nygaard. 1966. 16 p. Sw. cr. 8:—. 283. Progress report 1966 nuclear chemistry. 1967. 26 p. Sw. cr. 10:-. 236. DETEC, a subprogram for simulation of the fast-neutron detection pro 284. Finite-geometry and polarized multiple-scattering corrections of experi cess in a hydro-carbonous plastic scintillator. By B, Gustafsson and O. mental fast-neutron polarization data by means of Monte Carlo methods. Aspelund. 1966. 26 p. Sw. cr. 8:-. By 0. Aspelund and B. Gustafsson. 1967. 60 p. Sw. cr. 10:-. 237. Microanalys of fluorine contamination and its depth distribution in zircaloy 285. Power disturbances close to hydrodynamic instability in natural circulation by the use of a charged particle nuclear reaction. By E. Moller and N. two-phase flow. By R. P. Mathisen and O. Eklind. 1967. 34 p. Sw. cr. 10:-. Starfelt. 1966. 15 p. Sw. cr, 8:-. 286. Calculation of steam volume fraction in subcooled boiling. 1967. 26 p. 238. Void measurements in the regions of sub-cooled and low-quality boiling. Sw. cr. 10:-. P. 1. By S. Z. Rouhani. 1966. 47 p. Sw. cr. 8:-. 239. Void measurements in the regions of sub-cooled and low-quality boiling. P. 2. By S. Z. Rouhani. 1966. 60 p. Sw. cr. 8:—. 240. Possible odd parity in 1 ”Xe. By L. Broman and S. G. Malmskog. 1966. 10 p. Sw. cr. 8:-. Forteckning over publicerade AES-rapporter 241. Burn-up determination by high resolution gamma spectrometry: spectra from slightly-irradiated uranium and plutonium between 400-830 keV. By 1. Analys medelst gamma-spektrometri. Av D. Brune. 1961. 10 s. Kr 6:-. R. S. Forsyth and N. Ronqvist. 1966. 22 p. Sw. cr. 8:—. 2. BestrSIningsforandringar och neutronatmosfar i reaktortrycktankar — nSgra 242. Half life measurements in 15$ Gd. By S. G. Malmskog. 1966. 10 p. Sw. synpunkter. Av M. Grounes. 1962. 33 s. Kr 6:-. cr. 8:—. 3. Studium av strackgransen i mjukt st&l. Av G. Dstberg och R. Attermo 243. On shear stress distributions for flow in smooth or partially rough annuli. 1963. 17 s. Kr 6:-. By B. Kjellstrom and S. Hedberg. 1966. 66 p. Sw. cr. 8:—. 4. Teknisk upphandling inom reaktoromradet. Av Erik Jonson. 1963. 64 s. 244. Physics experiments at the Agesta power station. By G. Apelqvist, P.-A. Kr 8:—. Bliselius, P. E. Blomberg, E. Jonsson and F. Akerhielm. 1966. 30 p. Sw. 5. Agesta Kraftvarmeverk. Sammanstallning av tekniska data, beskrivningar cr. 8:-. m. m. for reaktordelen. Av B. Lilliehook. 1964. 336 s. Kr 15: —. 245. Intercrystalline stress corrosion cracking of inconel 600 inspection tubes in 6. Atomdagen 1965. Sammanstallning av foredrag och diskussioner. Av S the Agesta reactor. By B. Gronwall, L, Ljungberg, W. Hiibner and W. Sandstrom. 1966. 321 s. Kr 15:-. Stuart. 1966. 26 p. Sw. cr. 8:—. Additional copies available at the library of AB Atomenergi, Studsvik, Ny- 246. Operating experience at the Agesta nuclear power station. By S. Sand- koping, Sweden. Micronegatives of the reports are obtainable through Film- strom. 1966. 113 p. Sw. cr. 8:-. produkter, Gamla landsvagen 4, Ektorp, Sweden.
EOS-tryckerierna, Stockholm 1967