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Thermo-hydraulic behavior of inverted annular flow
Nuclear Engineering and Design 120 (1990) 281-291 North-Holland
THERMO-HYDRAULIC
281
BEHAVIOR OF INVERTED ANNULAR FLOW
M a s a n o r i A R I T O M I 1, A k i r a I N O U E 1 S h i g e b u m i A O K I 1 a n d K e i j i H A N A W A 2 I Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology, 2-12-10hokayama Meguro-ku Tokyo, 152, Japan 2 Toshiba Corp., Isogo Engineering Center, Shinsugita 8, Isogo, Yokohama, Kanagawa Prf., 235, Japan
The thermo-hydraulic behavior of inverted annular flow was investigated experimentally under various heat flux, inlet velocity, inlet subcooling and heating length conditions using freon 113. Empirical correlations are proposed concerning the net vaporization rate from the interface, heat flux from the interface to the liquid phase, interfacial shear stress and heat transfer coefficient from the wall to fluid. It is found that the roughness of the interface between the vapor film and liquid jet increases with the thickness of the vapor film and that an increase in the vapor film thickness causes the average Nusselt number and interracial friction factor to rise linearly. On the other hand, the wall shear stress in inverted annular flow is lower than that in liquid single-phase flow due to the existence of a vapor film at the wall.
1. Introduction
During the reflooding phase of a postulated loss-ofcoolant accident in LWRs, the clad temperature may exceed the minimum film boiling temperature. When subcooled water refloods into the PWR reactor core under bottom quench conditions, superheated vapor flows up around the hot clads and a subcooled liquid jet flows in the core space region between the rods. Since the positions of liquid and vapor phases are exactly opposite to the ones in normal annular flow, this flow pattern is called 'inverted annular flow'. Murao [1] ascertained from his visual experiment that this flow pattern was formed over a certain distance from the quench front under the bottom flooding condition. It may also occur in evaporators for cryogenic fluids with dual cycles for a natural liquid gas system [2]. Most works on inverted annular flow have been performed at velocities lower than 0.1 m / s to simulate the reflooding phase of LWRs. Under these flow conditions, heat transfer is usually analyzed as film pool boiling. For example, Berenson's correlation [3] is used in the RELAP-4 code [4] and Bailey's correlation, which is a revision of Bromley's one [5], is adopted in the TRAC code [6]. In addition, Murao [7] modified Ellison's correlation based on the FLECHT data, Sudo [8] took Kalinin's empirical correlation for subcooling into Bromley's one and Leonard et al. [9] modified Bromley's correlation for BWR LOCA. Regarding the heat transfer of film boiling, Kalinin et al. [10] have reviewed many works and tabulated an abundance of data for
various fluids. Regarding post-CHF heat transfer, Collier [11] has summarized a large number of works systematically. Ishii [12] has reviewed many works on the heat transfer and the hydrodynamics of the post-dryout region and modeled flow regime transitions of inverted annular flow in terms of jet break-up length, droplet size and so on. Elias and Chamble [13] concluded that void fractions at the quench front could be correlated with the equilibrium quality. This fact was also pointed out over a wider flow range by Fung and Groeneveld [14]. On the other hand, Fung and Groeneveld [15] clarified that the heat transfer of inverted annular flow almost coincided with EUison's correlation for flow rates less than 0.1 m / s but not for the flow rate higher than 0.5 m / s . Akagawa et al. [16] experimentally investigated the flow pattern and found that the heat transfer coefficient was correlated with the equilibrium quality in the quality region above -0.04. The liquid-core jet in inverted annular flow is an unstable flow, because it is surrounded by a compressible vapor film whose thickness fluctuates considerably. Since shear stresses in this flow pattern become zero at two radial points due to a vapor velocity higher than that of liquid (see fig. 1), the flow is not fully developed but also transient. Consequently, it is expected that the heat transfer in inverted annular flow is different from that in film pool boiling and also that the hydraulic behavior is different from normal two-phase flow. Although it is important in the safety analysis of LWRs under various accident conditions to understand non-equilibrium thermodynamics in two-phase flow, the
0 0 2 9 - 5 4 9 3 / 9 0 / $ 0 3 . 5 0 © 1990 - Elsevier Science P u b l i s h e r s B.V. ( N o r t h - H o l l a n d )
282
M. Aritomi et a L / Thermo-hydraulic behavior of inverted annular flow wall interface
vapor
film
liquid
core jet
q~i
q"
Tw
Tw
,I,
5
~o
Fig. 1. Flow characteristics of inverted annular flow. vaporization rate under these conditions is one of the facts which have never been adequately confirmed in the constitutive equations in safety analysis codes. The most interesting aspects of this flow pattern are that a thermal non-equilibrium state, with both a superheated vapor film and a subcooled liquid jet is readily formed in a cross section and that the shape of the interface is simple as compared with that in other flow patterns such as bubbly flow, slug flow and dispersed flow. Therefore, the investigation on this flow may offer clues to the modeling of the thermohydraulics of two-phase flow in a non-equilibrium thermal state. In this study, the heat transfer, pressure drop and vapor film thickness were measured during inverted annular flow using freon 113 to offer a fundamental data base. Furthermore, the heat, mass and momentum transfers at the interface between the two phases are discussed.
2. Experimental apparatus
adopted to shorten this line. The vapor generated in the test section was separated in a separator connected with the test section, passed into upper tanks and was liquefied in a condenser attached to the tanks. The liquid separated from vapor fell into a lower tank and was returned to the upper tanks by another pump. The flow rate was measured with an orifice flowmeter and was regulated with a control valve. A preheater and a precooler were provided to maintain the required fluid temperature at the entrance of the test section, and another precooler had the function of protecting the circulation pump from cavitation. A bypass was installed parallel to the test section to facilitate the formation of inverted annular flow and the control of flow conditions. The test section, whose geometry and dimensions are shown in fig. 3, consisted of a thin heat transfer tube made of stainless steel (O.D. 10 mm, I.D. 9.8 ram), with both upstream and downstream hot patches to fix the quench front. Heat loss from the heat transfer tube to atmosphere was minimized by adequate thermal insulation. These hot patches with controlled heaters were brazed to the heat transfer tube and had the function of electrodes for heating the tube electrically. Pairs of stainless steel wire (O.D. 0.1 mm) at about 2 mm spacings were welded to the tube at several points to measure the wall temperature by the resistance thermometer method. An eccentric rotary plug, in which static and impact pressure tubes (O.D. 0.88 mm) were installed, was mounted in the separator to measure the radial distribution of liquid velocity. The tubes projected as far as the exit of the heated section and led to a differential pressure transducer. The pitot-type measurement was calibrated with liquid single phase flow at the same inlet velocity. The pressure drop in the test section was measured between the static pressure tube
®
~) Test section (~) Separator (~) Orifice Preheater
Fig. 2 is a schematic diagram of the forced-convection freon 113 loop operated at atmospheric pressure. The loop was devised to suppress flow fluctuation by the adoption of a circulation pump with sufficient head and adequate throttling at the entrance to the test section to create stable inverted annular flow. Furthermore, since it was clear from our preliminary tests that slug bubbles were generated and pressure fluctuation occurred if the adiabatic line between the heated section and a separator was too long, dual circulating lines were
(~ (~ ~) ~)
Fig. 2. Freon 113 flow loop.
Cooler Pump Tank Condenser
283
M. Aritomi et al. / Thermo-hydraulic behavior of inverted annular flow Impact & stagnant pressure tubes
Condenser
('Too,.,looo,og
Tank~3 E/'ectroda ImII
,o i~ u
._g
transfer and a lower temperature resulted in moving the quench front. The experimental conditions are summarized in table 1.
3. Experimental results Aml me~
Iq
;,
healer
L~IIII~J
~
tap
9
Inlet
Fig. 3. Test section.
in the eccentric rotary plug and a pressure tap at the entrance of the test section using the differential pressure transducer. At the be~nning of an experiment, the fluid at regulated temperature and velocity flowed through the bypass, the heat transfer tube between both hot patches was heated at the required temperature with a DC power supply and the hot patches to 300 ° C using AC. When the temperatures stabilized, the bypass was closed and fluid flowed into the test section. A power controller in the DC power supply kept the heat transfer wall temperature constant. As stable inverted annular flow was formed, the temperature of the upstream hot patch was reduced to 150 ° C which was the most suitable condition found in our preliminary tests, because a higher temperature reduced downstream wall heat
3.1. Observation
Inverted annular flows were observed using a fused quartz tube (O.D. 10 mm, I.D. 8 mm and 300 mm long) wrapped with a nickel-chromium ribbon wire which was heated electrically. Typical observational results are sketched in fig. 4. In the case of inlet velocities lower than 0.2 m / s as shown in fig. 4(a), the liquid jet broke into liquid slugs at distances of 100-200 mm from the quench front and a fine filament of liquid formed between the two liquid slugs. On the other hand, the liquid core jet streamed up as it approached the heat transfer wall and a fine wave with a wavelength of about 1 turn was observed at the interface as drawn in fig. 4(c), in the case of inlet velocities higher than 0.5 m / s . As heat input increased or inlet velocity and subcooling decreased, a longer wavelength of several ta)
tb)
250
q
200
N
150
100
;4
Table 1 Experimental conditions Test fluid System pressure Inlet velocity Heat flux Wall temperature Inlet subcooling Heated length
(c)
12 Freon 113 atmospheric pressure 0.5 ~ 2.0 m/s 20 - 50 kW/m2 150 - 300 °C 10.7, 15.7, 20.7 K 100, 300, 500 mm
50
"s]
Quench front gin: 0.17 m/e
Uin: 0 , 5 7 m / s
Uin: 1 . 4 3 m l s
Fig. 4. Sketches of inverted annular flow.
M. Aritomi et al. / Thermo-hydraulic behavior of inverted annular flow
284
0.08
millimeters appeared at the interface, the vapor film became thicker and some constriction of the liquid jet was seen as shown in fig. 4(b). The latter flow pattern, where a continuous liquid core-jet is formed, is investigated in this paper.
Uin
0.06
00•
1.0 m/s
~,
1.~.
/
[]
2.0 m/s
/
o' d /
/
3.2. Flow characteristics a I,~
3.2.1. Vapor film thickness The radial distribution of liquid jet velocity was measured at 84, 284 and 484 mm from the upstream hot patch, and typical results are shown in fig. 5. The distribution retains the convex profile of single-phase flow at 84 mm, but the profile becomes concave at 284 mm. That is, the vapor velocity is higher than that of the liquid in the fully developed region of inverted annular flow. This transformation in the distribution occurs at a greater distance from the quench front as the inlet velocity increases. The change in liquid mass flow rate due to vaporization can be neglected in comparison with the accuracy of the measurements, since it is only about 1% even if the total heat input contributes to evaporation. The radial distributions of liquid velocity obtained over several measurements were expressed in a cubic equation by using the least squares method. The vapor film thickness was obtained by integration. The thickness increased linearly along the channel and the average was taken over the test section between 84 mm and 484 mm. The average vapor film thickness is illustrated in fig. 6 by making it dimensionless on the basis of the channel diameter (D). From the figure, it is clear that an increase in heat flux (q") increases the thickness ( ~ / D ) linearly but that increases in inlet
0.04
o / , / ,/
0.02
#I
ill
I I
, /
/ A Tsub oAn
q"
10.7 K
OA
15.7 K
•
20.7 K
I I 40 (kW/m2)
20
60
i
80
Fig. 6. Experimental results of average vapor film thickness.
The value of heat flux obtained by extrapolating the thickness to zero is a minimum, depends only on the inlet subcooling and represents the lower limit required to form stable inverted annular flow. Fig. 7 presents the axial profile of vapor film thickness as compared with Bromley's correlation. The vapor film of inverted annular flow is thicker than that of the saturated film pool boiling despite the existence of subcooling. The ex-
0.3 u : 1.5m/s q" : 4 0 k W / m =
velocity (Ui.) and inlet subcooling (ATe.b) reduce it. "o
,2l
t2i i 01o11 1.0~-0 ~
i
0¢//
i i / // /i/ / i / i I / / I / / // // / ill O/ I /// / / / • i1/i/ i // // i/I/ I / / /// //// / I// / / o, // /
0.2
ATsub : 10.7K
\o
~-0--0
~
0
h (experiment) 0 0
1.0 0-
J
f z - 484mm 0.8J
o.g I
~'~ 1.0 L _ 0 ~ 0 ~0 ~0"~
I z- 284mrn 1.2
i,.oi o-o-o 0.8
z- 284mml
t 1.2 i
Teub:10.7K
Uln:t.Omla q,:3OkWlm 2 Teub: I0,7K i
1.0~ O ~
0 ~0"~
015
r/R
I
0.8 L 1.2
0
i
0.5 r/R
= Uin:2.0m/B
i.OLO~o~ I
0.5 [ 0
~..~ i~
z - 84ram 0.5
rlR
Fig. 5. Radial distribution of liquid core jet velocity.
O0 "Ol
°~ - ~ ~ ~ .~
,,; .......
q°:5OkW/m 2 Tsub:lO.7K
i z - 84rnm 0.8
0.1
t
Uin : 1,5m/s q,:4OkWlm 2
1.0 ~ o------.._o ~o.
z -84mm 0.8
:
I
L
z=4841mm i
St.2
1.2 ~
0.8'
0,8l
' 0.1
~
0.5
'°~
h (Bromley)
i);- &;o-m~e;i- - ~ ' 0.2
'. 0 3 Z
-
-
' 0.4
0 0.5
(m)
Fig. 7. Comparison between inverted annular flow and film pool boiling for vapor film thickness and heat transfer coefficient.
M. Aritomi et aL / Thermo-hydraulic behavior of inverted annular flow
0.4, o O•
0.3
Neglecting the change of liquid mass flow rate due to vaporization as aforementioned, and integrating eq. (1) from 84 mm to 484 mm,
/O
U in 1.0 m/s
A&
1.5 m / s
D
2.0 m/s
285
&
/
nA
®
4~iDi
[
1 ~0.484r
0.4 ]0.084
D2 -
+ ~z { p l ( a - a ) u +Pl(1
://
~ 0.2
•
"o
AAÜ
®
OAn
0 -0.15
i -0.10
2)
0 /
10.7 K
®&
15.7 K
•
20.7 K
= -0.05
0
Equilibriumquality (-) Fig. 8. Comparison between void fraction and equilibrium quality.
- or)g] dz.
(2)
Assuming that the void fraction can be approximated by a linear function, the average interfacial shear stress (~i) is obtained by eq. (2) after substituting measured values of pressure drop, void fraction and liquid velocity. Though pressure drop is measured to an accuracy of 10 Pa, the error including experimental reproducibility and the above stated evaluation method is thought to be about 15%. The results are shown in fig. 9, and indicate a similar tendency to that of the average vapor film thickness, that is, a rise in the heat flux increases the average interracial shear stress linearly and the reductions in the inlet velocity and subcooling also increases it.
10 Uin
istence of compressible vapor flow between the wall and liquid core-jet makes the jet unstable due to the mobility and brings about a wavy interface. Consequently, the vaporization rate is increased and the vapor flow is suppressed by the roughness of the interface, increasing the thickness of the vapor film. Although Ellias and Chamble [13] and Fung and Groeneveld [14] indicated that void fractions could be correlated only with equilibrium quality, our experimental results also depend on the subcooling as shown in fig. 8. It is considered that this difference is caused by the different range of equilibrium quality which was measured, that is, the range investigated by them is higher than -0.02.
000
1.0 rnls
&&
1.5 m / s
r~
2.0 m / s O Z~
6
//
®
3.2.2. ln terfacial shear stress The steady-state momentum equation for liquid is
~---[p,(1-a)u 2] + ( 1 - a)-~-zP- 4~'iDi az D2 + pl(1 - a ) g = --Fgu i.
zl Tsu b OZ~ m
'
20
' q"
(1)
40
10.7 K
O&
15.7 K
•
20.7 K
'
610
( k W l m 2)
Fig. 9. Experimental results of average interfacial shear stress.
M. A ritomi et aL / Thermo-hydraulic behavior of inverted annular flow
286
0.25
3.2.3. Wall shear stress
gin = 1.0 m / s ( a ) 4Tsu b = 10.7 K
The steady-state mixture momentum equation is 0
~ [psau~ +
p](1 - o,,u,,',.21 +
~
o
4%, P + 19 -
40.2
v
35.5
0.2018
-
+ [p,a+p|(l-a)]g=O.
v
(3)
o
8
, i
q" = 40 k W / m 2 ATsub = 10,7 K
Uin ( m / s )
o
0.4 v0.084 L ~2 P - ~2 ( pl(] - Ol)t/~ } 0.30
-p,(1 - a)g] dz
25.5
D
0.35
1 f0.484[ _
30.4
V v
i
0.15 (b)
0
v
~
Neglecting the mass of the vapor phase and using the same procedure as for average interracial shear stress, 47w O-
q° ( k W / m = )
1.0
~,
1.5
[]
2.0
(4) 0.25
~
J=
The average wall shear stress (~w) is obtained by eq. (4) using the same procedure for the average interfacial shear stress. The ratio of the wall shear stress in inverted annular flow to that of liquid single-phase flow is shown in fig. 10. Although the experimental data are scattered because the values are not very large compared with the accuracy of measurement, it is clear that the frictional loss in inverted annular flow is less than 50% of the value in liquid single-phase flow. This is a result of the existence of the thin vapor film on the wall which eases the liquid phase flow.
[] 0.20 .o
n
o
o
~
(el
u~ = 1.0 m / s " = 40 k W m 2
t
0.15
a
o
~
g
o~ I
i
(K)
aTsub
0.30 .o
o
10.7
o
20.7
15.7
0.25 • & o o
o
o
o
o
0.20
o 0.15
~
8
8
6 L
011
022 z
0.3
0.4
0.5
(m)
Fig. 11. Typical axial distribution of heat transfer coefficient. 1.0 Ui n
0.8
ATsub
OO•
1.0 m / s
O&D
10.7 K
z~&
1.5 m / s
O&
15.7 K
[]
2.0 m/s
•
20.7 K
3.3. Heat transfer
The wall temperature was measured using the electrical resistance method. The calibration curve was obtained by using a tube identical to the heat transfer tube. The error, including the experimental reproducibility, could be controlled to less than 3 K. The heat transfer coefficient is defined by q " / ( T w - T~at). Fig. 7 shows a typical axial profile of heat transfer coefficient compared with Bromley's correlation [5] and Sudo's version [8]. The heat transfer coefficient in inverted annular flow is larger than that in film pool boiling despite the thicker vapor film. Fig. 11 illustrates the axial profiles of the heat transfer coefficient obtained. Increases in inlet velocity and in subcooling enhance the
~. 0.6 F.-
I~"
0.4
&
o
&
•
0.2
J
20 q"
i
L
40
i
60
(kw/m =)
Fig. 10. Experimental results of average frictional multiplier.
heat transfer coefficient, but a decrease in heat flux increases it only in the region less than 0.25 mm from the quench front. From figs. 7 and 11, it can be seen that the heat transfer coefficient along the channel decreases remarkably in the region less than 0.1 m from the quench front but scarcely varies in the region more than 0.1 m away. Therefore, the average heat transfer
M. A ritomi et aL / Thermo-hydraulic behaoior of inverted annular flow
8
coefficient in the fully developed region is considered hereinafter. The average film Nusselt number (Nuf) is defined by
Nuf
=
h~g/,/Xg,
287
//" Uin 1.0 m/s
/
zxA
1.5 m/s
+10% //O /
El
2.0 m/s
0®0
(5)
6
//~
IZ
Experimental results are shown in fig. 12, and indicate a similar tendency to the average vapor film thickness and the average interracial shear stress as regards the effects of heat flux, inlet velocity and inlet subcooling. The results are illustrated in fig. 13 of plotting the average film Nusselt number against ~g/D. They are well correlated, within 10% error, and the following correlation, shown on the diagram by a solid line, is obtained.
4
//~/0
A 0
/ /
El ~
where ~g is determined by the average vapor film temperature ( ~ )
Tg=½(Twq- Tsat).
//
,/ /
""
//
"
/ GO'/'~'0" 0
S. 2
z~"
.1Tsub
/
OLeO 10.7 K
I
00
I
I
0.02
I
0.04
®A
15.7 K
•
20.7 K I
I
0.06
bg/D
Nut = 1 + 9 0 . 9 ~ .
(7)
Heat flux, inlet velocity and inlet subcooling have an influence only through the thickness. Consequently, the heat transfer coefficient scarcely varies axially in spite of changes in the vapor film thickness as shown in figs. 7 and 11. From fig. 13 and eq. (7), the average film Nusselt number is seen to converge to unity when the
Uin o
^
/ ,/ / u
txA
1.5m/s
•
13
2.0 m/s
O
Z~ El
o//° / / -~ I~0 i r"
-
/ 4
O
/
e/
•
/ •
2
ATsub OZXEI 10.7 K
00
'
vapor film thickness approaches zero. That illustrates the fact that a decrease in the vapor film thickness reduces turbulence in the vapor film and allows thermal conductivity to govern the heat transfer.
4. Discussion
O®O 1.0 m/s
6
Fig. 13. Average film Nusselt number in reference to the dimensionless average vapor film thickness.
2'o
i qR
®&
15.7 K
•
20.7 K
, 40
,
6'o
(kW/m2)
Fig. 12. Experimental results of average film Nusselt number.
In the case of inverted annular flow, the liquid thermal boundary layer near the interface and the vapor film are very thin and the interface fluctuates. Therefore, the accuracy obtained in measurements of radial distributions of liquid jet temperature, vapor temperature and vapor velocity cannot be guaranteed. However, it is necessary for the further consideration of the experimental results to evaluate the average vapor film velocity and temperature. The average vapor film temperature is assumed to be estimated by eq. (6), and the procedure for evaluating the average vapor film velocity is described below. Replotting the values of the interfacial shear stress against the vapor film thickness, a correlation is seen as shown in fig. 14, though it decreases slightly with increasing inlet velocity. In cases where the interface is considered to be smooth, it has been made clear from our experiments on stratified flow
288
M. A ritomi et aL / Thermo-hydraulic behaoior of inoerted annular flow 100
10 0@•
1.0 m / s
0 e •
1.0 m / s
t~ Z~
1.5 m / s
z~ &
1.5 r n / s
[]
2.0 m/s
O
2.0 m/s
50
/ITsub
//
OZ~D 10.7 K
2
~6
De = 4bg ( 1 - b g / D )
Uin
gin
®&
15.7 K
•
20.7 K
A
...... Eq.(7)
/ O
DAO" [:]i,"
ie"
~lTsub OZXD 10.7 K
20
O/®/~7o/~o/° / /•/~
I
De
i
i
0.04
i
i
0.06
bg/D
Fig. 14. Average interracial shear stress with reference to the average dimensionlessvapor film thickness.
[17] that the interracial shear stress could be evaluated by 0.0791
- u,) 2,
,
,
i
,
10
(mm)
is examined to evaluate the average vapor velocity. The experimental values are replotted in fig. 15, which makes it clear that the average Nusselt number is proportional to the 0.8th power of the average equivalent diameter. Supposing that vapor film flow is turbulent, the heat transfer coefficient can be evaluated through DittusBoelter's correlation. Therefore, evaluating the average vapor film velocity on the basis of Dittus-Boelter's correlation for Nusselt number by using the equivalent diameter defined by eq. (11), the calculated results are presented in fig. 16.
(S) 15
where (Ug -- Ul)O
Re i -
,
5
2
Fig. 15. Average film Nusselt number with reference to the equilibrium diameter.
0.02
~'i = - Rei0.2 5 ½Pg(Ug
1
/
I
20.7 K
i
0.5
/
0
15.7 K
•
/
/® ®
®A
i
us
(9) Q
We consider that the average vapor film velocity scarcely varies under the conditions shown in the figure, that an increase in vapor film thickness increases interfacial roughness and interracial shear stress, and also that an increase in inlet velocity slightly decreases the slip velocity between the phases and decreases the interfacial shear stress. On the other hand, although an empirical correlation of average film Nusselt number is obtained by eq. (7), the average film Nusselt number, N,u = hDe/?~g,
(10)
10
o2
U in
,t Tsu b
O®•
1.0 m / s
OZXO
10.7 K
Ad~
1.5 m / s
®&
15.7 K
13
2.0 m/s
•
20.7 K
~ 20
which is commonly used, and is defined by the equivalent diameter, D~ = 4~g(1 - ~ J D )
(11)
~ q"
4 "O
'
60
( k W / m =)
Fig. 16. Average vapor velocity obtained by Dittus-Boelter's correlation.
289
M. Aritomi et aL / Thermo-hydraulic behavior of inverted annular flow
The void fraction (a) is given by the relation
Q"
// Uin
OO•
The average net vaporization rates (Pg) are computed from the exit void fraction and vapor film velocity and illustrated in fig. 17. The average net vaporization rate in the heated section depends only on ~ / D and the following empirical correlation is obtained from the figure.
.i ®
1.0 m / s
Z~A
1.5 m / s
[]
2.0 m / s
i
2 ///®
O
A
/
(13)
/
/
////
dTsub
/~//
--ttt
The phasic average volumetric heat generation rates (qg and Eh'" ) are
= r,{ L +
r.,)},
(14)
. . . . = ~4q . . .-. . .q~ • q~
--
tit
O®•
?
0
I
0
l/
1.0 m / s 1.5 m / s
[3
2.0 m / s
i
15,7 K
•
20.7 K
i
i
2 li (obtained by Eq.(16))
3
(kW/m=K)
Fig. 18. Heat transfer coefficient from the interface to the liquid core jet.
substituted into eq. (15) and E/l'" is obtained. Evaluating and adjusting the results, the following empirical co~telations concerning the average volumetric heat generations of the fiquid phase (Yh") and the heat transfer coefficients from the interface to the liquid core jet (hli) are obtained.
,'Q
zx A
i
1
10.7 K
®&
--
U in
40
OZ~D
/ "
(15)
The average vaporization rates are calculated by Dittus-Boelter's correlation on the basis of the experimental results of the heat transfer coefficient. Substituting it/ these values into eq. (14), qg is obtained. The qg is
/
/
/
I J::
1
_Fg= 6 5 0 ~ / D .
A/ +io~. / A J,...
(2 ZX
16,
//'
0 i
+20%
v
[]
//
/
30
= 1.287
/ tA
/ / -20%
&®
d~ o e~
// /
20
/
,/
®
/
//I !/
/ /o
0~ O) >
//
10 //
//
///
ZlTsub oZXrl
// // /~///
0
I
0
I
I
0.02
0.04
ub + 1.03 aL b I" ,
(17)
//
10.7 K
®A
15.7 K
[]
20.7 K I
i
0.06
~m
Fig. 17. Average net vaporization rate.
where A~u b = Ts, t - T l and Tl is the average liquid temperature calculated by heat balance. Fig. 18 compares the heat transfer coefficients obtained from fig. 16 with the calculated ones from eq. (16). In eq. (17), the first term corresponds to the minimum heat flux required to create stable inverted annular flow as shown in fig. 6. The second term in the equation is interpreted as follows: (1) The effect of liquid velocity on heat convection is larger than that of channel flow because of the existence of the compressible vapor film between the wall. (2) Both vaporization and condensation simultaneously occur at the interface and the fluctuations at the interface renew the surface, so that the effect of subcooling on the heat convection is the square and
290
M. Aritomi et al. / Thermo-hydraulic behavior of inverted annularflow
the vapor film thickness influences the heat convection linearly. To verify these empirical correlations, the usual Nusselt number defined by hD/Xg is evaluated. The calculated results including the condition Ui~ = 0.5 m/s, where the vapor film thickness could not be measured due to the limitations of measurement accuracy, are illustrated in fig. 19 and compared with the experimental ones. On the basis of eq. (8), the average interfacial friction factors are examined in terms of the experimental average interfacial shear stress, the average vapor film velocity and the average liquid velocity calculated from the average void fraction and are shown in fig. 20. The empirical correlation for the average interfacial friction factor (fi) is obtained by
0.4
0.3
1.0 m / s
A ~,
1.5 m / s
0
2.0 m / s
// /// // +15% / /
I
,g
O
O'
~ /
/
o.-
/O /
ig:
165-
/ /
&• /
/////0// / O / // • / // //
d Tsu b
/ /e
o~0
z/~////
200
/////
j10,115, 20,
i
A E
•
0.5
•r. 150 v
1.0
0
1.5
C,
2.0
O
v r
•
0
/
ioo
/ • ~ ® / ///
-15%
/ / /J/// /
/
//
/
50
i
0 0
50 Nusselt
i
0.02
i
0.04
I
15.7 K
O
20.7 K |
0.06
i _ _
0.08
bg/Di
Fig. 20. Comparison of interfacial friction coefficient with dimensionless vapor film thickness.
cause of the scatter in the experimental values, the results do coincide within an order of magnitude. The experimental results can be systematically arranged using a model in which the heat and momentum transfers are similar to those of the usual singlephase channel flow. In this model the heat, mass and momentum transfers depend on the interfacial roughness which is proportional to the vapor film thickness.
5. C o n c l u s i o n s
///////
Z
L
00
10.7 K
®A
+15%// //
///~
E
El/ -15%
~o.2
(18)
It increases linearly with the ]g/Di, because increasing the vapor film thickness increases the surge and roughness of the interface. The experimental results of average wall shear stress under the conditions shown in table 1 (except Uin = 0.5 m / s ) have values between 0.9 and 4.5 Pa. The results calculated by the average vapor film velocity in fig. 15 and Blasius's correlation are from 1.6 to 2.1 Pa. Although a correlation between the experimental results and those calculated cannot be obtained exactly be-
/
//
0.1
Rei°'25 Di"
/1
U in O®¢
L 100
number
i 150
200
(calculation)
Fig. 19. Comparison between experimental results and calculated results using empirical correlations proposed in the present paper for the Nusselt number.
The thermo-hydraulic behavior of inverted annular flow was investigated experimentally and the measured results were rearranged as average values in the fully developed region. From consideration of the results, the following points are clarified: (1) The heat transfer coefficient decreases along the channel only in the inlet region where z / D is less than 10, but scarcely changes in the fully developed region where z / D is larger than 10. (2) An increase in the vapor film thickness causes a linear increase in the interfacial shear stress and the film Nusselt number because the interface becomes rougher with increasing thickness. The roughness
M. Aritomi et al. / Thermo-hydraulic behavior of inverted annularflow leads to a thicker v a p o r film as c o m p a r e d with saturated film pool boiling due to the increase in flow resistance at the interface in spite of the existence of subcooling. (3) A s s u m i n g that the heat transfer coefficient is given b y D i t t u s - B o e l t e r ' s correlation o n the basis of the experimental results, the results correlate well in terms of heat transfer coefficient from the interface to the liquid jet, interfacial friction factor a n d net vaporization rate. (4) T h e existence of the v a p o r film at the wall decreases the flow resistance of the liquid phase, so t h a t the frictional loss in inverted a n n u l a r flow is less t h a n that in single p h a s e flow. Since the p r o p o s e d empirical correlations c a n b e applied only to freon 113 u n d e r a t m o s p h e r i c pressure, it is necessary to m a k e further efforts to e x p a n d their applicable range in future work.
Nomenclature
D g h L
Nu P Pr q" q Re T
= = = = = =
= =
= = =
= ATsub = U = Z = ~t = = = h = = # = 'r =
specific heat ( k J / k g K) diameter (m) acceleration due to gravity ( m / s 2) heat transfer coefficient ( k W / m 2 K) latent heat of vaporization ( k J / k g ) Nusselt n u m b e r ( - ) pressure (Pa) Prandtl number (-) heat flux ( k W / m 2) volumetric heat generation rate ( k W / m 3) Reynolds n u m b e r ( - ) t e m p e r a t u r e ( ° C) subcooling (K) velocity ( m / s ) position along c h a n n e l (m) void fraction ( - ) vaporization rate ( k g / m 3 s) v a p o r film thickness (m) t h e r m a l conductivity ( k W / m K ) kinematic viscosity ( m 2 / s ) density ( k g / m 3) shear stress (Pa)
Subscripts and superscripts 1 = liquid, g = gas, f = film, i = interface, w = wall, sub = subcooled, sat = saturated, - = averaged.
291
References
[1] Y. Murao, Analytical study of the thermo-hydrodynamic behavior of the reflood-phase during LOCA, J. Nucl. Sci. Tecnol. 16 [11] (1979) 802-817. [2] K. Akagawa and T. Fuji, Power generation by utilization of LNG-cold, (in Japanese), J. JSME 83 (1980) 650-656. [3] P.J. Berenson, Film-boiling heat transfer from a horizontal surface, Trans. ASME, J. Heat Transf. 83 (1961) 351-357. [4] K.V. Moore and W.H. Rettig, RELAP4: A computer program for transient thermal-hydraulic analysis, ANCR1127 (1973). [5] L. Bromley, Heat transfer in stable f i l l boiling, Chem. Eng. Prog. 46 (1950) 221-227. [6] TRAC-P1A: An advanced best-estimate computer program for PWR LOCA analysis, NUREG/CR-0665 (1979). [7] Y. Murao and J. Sugimoto, Correlation of heat transfer coefficient for saturated film boiling during reflood phase prior to quenching, J. Nucl. Sei. Tecllol. 18 (1981) 275284. [8] Y. Sudo, Film boiling heat transfer during reflood phase in postulated PWR loss-of-coolant accident, J. Nucl. Sci. Tcchnol. 17 (1980) 516-530. [9] J.E. Leonald, K.H. Sun and G.E. Dix, Low flow f i l l boiling heat transfer on vertical surface, Part II: Empirical formulations and application to BWR-LOCA analysis, Solar and Nucl. Heat Transl., AICliE Syrup. Series, No. 164, Vol. 73 (1977) 7-13. [10] E.K. Kalinin, I.I. Berlin and V.V. Kostyuk, Film-boiling heat transfer, Advances in Heat Transfer Vol. 11 (Academic Press, New York, 1975) pp. 51-198. [11] J.G. Collier, Post dryout heat transfer, Two-Phase Flow and Heat Transfer in the Power and Process Industries (McGraw-Hill, New York, (1981) pp. 282-329. [12] M. Ishii and G.De JarlaJs, Flow regime transition and interfacial characteristics of inverted annular flow, Proc. Japan-US Seminar on Two-Phase Flow Dynamics (1984) pp.B.7.1-B.7.9. [13] E. Ellias and P. Cambte, Inverted-annular film boiling heat transfer from vertical surface, Nucl. Engrg. Des. 64 (1981) 249-257. [14] K.K. Fung and D.C. Groeneveld, Measurement of void fraction in steady state subcooled and low quality f i l l boiling, Int. J. Multiphase Flow 6 (1980) 357-361. [15] K.K. Fung and D.C. Groeneveld, Subcooled and low quality flow f i l l boiling of water at atmospheric pressure, Nucl. Engrg. Des. 55 (1979) 51-57. [16] K. Akagawa, T. Fujii, N. Takenaka, K. Nishida and M. Iseki, Flow pattern and heat transfer of inverted annular flow, (in Japanese), 22nd Nat. Heat Transl. Symp. Japan (1985) pp. 130-132. [17] M. Takahashi, M. Aritomi, A. Inoue and S. Aoki, Interface turbulent heat transfer of horizontal co-current liquid-liquid stratified flow (3rd. Boiling in Water), (in Japanese), 17th Nat. Heat Transf. Symp. Japan (1980) pp. 220-222.
THERMO-HYDRAULIC
281
BEHAVIOR OF INVERTED ANNULAR FLOW
M a s a n o r i A R I T O M I 1, A k i r a I N O U E 1 S h i g e b u m i A O K I 1 a n d K e i j i H A N A W A 2 I Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology, 2-12-10hokayama Meguro-ku Tokyo, 152, Japan 2 Toshiba Corp., Isogo Engineering Center, Shinsugita 8, Isogo, Yokohama, Kanagawa Prf., 235, Japan
The thermo-hydraulic behavior of inverted annular flow was investigated experimentally under various heat flux, inlet velocity, inlet subcooling and heating length conditions using freon 113. Empirical correlations are proposed concerning the net vaporization rate from the interface, heat flux from the interface to the liquid phase, interfacial shear stress and heat transfer coefficient from the wall to fluid. It is found that the roughness of the interface between the vapor film and liquid jet increases with the thickness of the vapor film and that an increase in the vapor film thickness causes the average Nusselt number and interracial friction factor to rise linearly. On the other hand, the wall shear stress in inverted annular flow is lower than that in liquid single-phase flow due to the existence of a vapor film at the wall.
1. Introduction
During the reflooding phase of a postulated loss-ofcoolant accident in LWRs, the clad temperature may exceed the minimum film boiling temperature. When subcooled water refloods into the PWR reactor core under bottom quench conditions, superheated vapor flows up around the hot clads and a subcooled liquid jet flows in the core space region between the rods. Since the positions of liquid and vapor phases are exactly opposite to the ones in normal annular flow, this flow pattern is called 'inverted annular flow'. Murao [1] ascertained from his visual experiment that this flow pattern was formed over a certain distance from the quench front under the bottom flooding condition. It may also occur in evaporators for cryogenic fluids with dual cycles for a natural liquid gas system [2]. Most works on inverted annular flow have been performed at velocities lower than 0.1 m / s to simulate the reflooding phase of LWRs. Under these flow conditions, heat transfer is usually analyzed as film pool boiling. For example, Berenson's correlation [3] is used in the RELAP-4 code [4] and Bailey's correlation, which is a revision of Bromley's one [5], is adopted in the TRAC code [6]. In addition, Murao [7] modified Ellison's correlation based on the FLECHT data, Sudo [8] took Kalinin's empirical correlation for subcooling into Bromley's one and Leonard et al. [9] modified Bromley's correlation for BWR LOCA. Regarding the heat transfer of film boiling, Kalinin et al. [10] have reviewed many works and tabulated an abundance of data for
various fluids. Regarding post-CHF heat transfer, Collier [11] has summarized a large number of works systematically. Ishii [12] has reviewed many works on the heat transfer and the hydrodynamics of the post-dryout region and modeled flow regime transitions of inverted annular flow in terms of jet break-up length, droplet size and so on. Elias and Chamble [13] concluded that void fractions at the quench front could be correlated with the equilibrium quality. This fact was also pointed out over a wider flow range by Fung and Groeneveld [14]. On the other hand, Fung and Groeneveld [15] clarified that the heat transfer of inverted annular flow almost coincided with EUison's correlation for flow rates less than 0.1 m / s but not for the flow rate higher than 0.5 m / s . Akagawa et al. [16] experimentally investigated the flow pattern and found that the heat transfer coefficient was correlated with the equilibrium quality in the quality region above -0.04. The liquid-core jet in inverted annular flow is an unstable flow, because it is surrounded by a compressible vapor film whose thickness fluctuates considerably. Since shear stresses in this flow pattern become zero at two radial points due to a vapor velocity higher than that of liquid (see fig. 1), the flow is not fully developed but also transient. Consequently, it is expected that the heat transfer in inverted annular flow is different from that in film pool boiling and also that the hydraulic behavior is different from normal two-phase flow. Although it is important in the safety analysis of LWRs under various accident conditions to understand non-equilibrium thermodynamics in two-phase flow, the
0 0 2 9 - 5 4 9 3 / 9 0 / $ 0 3 . 5 0 © 1990 - Elsevier Science P u b l i s h e r s B.V. ( N o r t h - H o l l a n d )
282
M. Aritomi et a L / Thermo-hydraulic behavior of inverted annular flow wall interface
vapor
film
liquid
core jet
q~i
q"
Tw
Tw
,I,
5
~o
Fig. 1. Flow characteristics of inverted annular flow. vaporization rate under these conditions is one of the facts which have never been adequately confirmed in the constitutive equations in safety analysis codes. The most interesting aspects of this flow pattern are that a thermal non-equilibrium state, with both a superheated vapor film and a subcooled liquid jet is readily formed in a cross section and that the shape of the interface is simple as compared with that in other flow patterns such as bubbly flow, slug flow and dispersed flow. Therefore, the investigation on this flow may offer clues to the modeling of the thermohydraulics of two-phase flow in a non-equilibrium thermal state. In this study, the heat transfer, pressure drop and vapor film thickness were measured during inverted annular flow using freon 113 to offer a fundamental data base. Furthermore, the heat, mass and momentum transfers at the interface between the two phases are discussed.
2. Experimental apparatus
adopted to shorten this line. The vapor generated in the test section was separated in a separator connected with the test section, passed into upper tanks and was liquefied in a condenser attached to the tanks. The liquid separated from vapor fell into a lower tank and was returned to the upper tanks by another pump. The flow rate was measured with an orifice flowmeter and was regulated with a control valve. A preheater and a precooler were provided to maintain the required fluid temperature at the entrance of the test section, and another precooler had the function of protecting the circulation pump from cavitation. A bypass was installed parallel to the test section to facilitate the formation of inverted annular flow and the control of flow conditions. The test section, whose geometry and dimensions are shown in fig. 3, consisted of a thin heat transfer tube made of stainless steel (O.D. 10 mm, I.D. 9.8 ram), with both upstream and downstream hot patches to fix the quench front. Heat loss from the heat transfer tube to atmosphere was minimized by adequate thermal insulation. These hot patches with controlled heaters were brazed to the heat transfer tube and had the function of electrodes for heating the tube electrically. Pairs of stainless steel wire (O.D. 0.1 mm) at about 2 mm spacings were welded to the tube at several points to measure the wall temperature by the resistance thermometer method. An eccentric rotary plug, in which static and impact pressure tubes (O.D. 0.88 mm) were installed, was mounted in the separator to measure the radial distribution of liquid velocity. The tubes projected as far as the exit of the heated section and led to a differential pressure transducer. The pitot-type measurement was calibrated with liquid single phase flow at the same inlet velocity. The pressure drop in the test section was measured between the static pressure tube
®
~) Test section (~) Separator (~) Orifice Preheater
Fig. 2 is a schematic diagram of the forced-convection freon 113 loop operated at atmospheric pressure. The loop was devised to suppress flow fluctuation by the adoption of a circulation pump with sufficient head and adequate throttling at the entrance to the test section to create stable inverted annular flow. Furthermore, since it was clear from our preliminary tests that slug bubbles were generated and pressure fluctuation occurred if the adiabatic line between the heated section and a separator was too long, dual circulating lines were
(~ (~ ~) ~)
Fig. 2. Freon 113 flow loop.
Cooler Pump Tank Condenser
283
M. Aritomi et al. / Thermo-hydraulic behavior of inverted annular flow Impact & stagnant pressure tubes
Condenser
('Too,.,looo,og
Tank~3 E/'ectroda ImII
,o i~ u
._g
transfer and a lower temperature resulted in moving the quench front. The experimental conditions are summarized in table 1.
3. Experimental results Aml me~
Iq
;,
healer
L~IIII~J
~
tap
9
Inlet
Fig. 3. Test section.
in the eccentric rotary plug and a pressure tap at the entrance of the test section using the differential pressure transducer. At the be~nning of an experiment, the fluid at regulated temperature and velocity flowed through the bypass, the heat transfer tube between both hot patches was heated at the required temperature with a DC power supply and the hot patches to 300 ° C using AC. When the temperatures stabilized, the bypass was closed and fluid flowed into the test section. A power controller in the DC power supply kept the heat transfer wall temperature constant. As stable inverted annular flow was formed, the temperature of the upstream hot patch was reduced to 150 ° C which was the most suitable condition found in our preliminary tests, because a higher temperature reduced downstream wall heat
3.1. Observation
Inverted annular flows were observed using a fused quartz tube (O.D. 10 mm, I.D. 8 mm and 300 mm long) wrapped with a nickel-chromium ribbon wire which was heated electrically. Typical observational results are sketched in fig. 4. In the case of inlet velocities lower than 0.2 m / s as shown in fig. 4(a), the liquid jet broke into liquid slugs at distances of 100-200 mm from the quench front and a fine filament of liquid formed between the two liquid slugs. On the other hand, the liquid core jet streamed up as it approached the heat transfer wall and a fine wave with a wavelength of about 1 turn was observed at the interface as drawn in fig. 4(c), in the case of inlet velocities higher than 0.5 m / s . As heat input increased or inlet velocity and subcooling decreased, a longer wavelength of several ta)
tb)
250
q
200
N
150
100
;4
Table 1 Experimental conditions Test fluid System pressure Inlet velocity Heat flux Wall temperature Inlet subcooling Heated length
(c)
12 Freon 113 atmospheric pressure 0.5 ~ 2.0 m/s 20 - 50 kW/m2 150 - 300 °C 10.7, 15.7, 20.7 K 100, 300, 500 mm
50
"s]
Quench front gin: 0.17 m/e
Uin: 0 , 5 7 m / s
Uin: 1 . 4 3 m l s
Fig. 4. Sketches of inverted annular flow.
M. Aritomi et al. / Thermo-hydraulic behavior of inverted annular flow
284
0.08
millimeters appeared at the interface, the vapor film became thicker and some constriction of the liquid jet was seen as shown in fig. 4(b). The latter flow pattern, where a continuous liquid core-jet is formed, is investigated in this paper.
Uin
0.06
00•
1.0 m/s
~,
1.~.
/
[]
2.0 m/s
/
o' d /
/
3.2. Flow characteristics a I,~
3.2.1. Vapor film thickness The radial distribution of liquid jet velocity was measured at 84, 284 and 484 mm from the upstream hot patch, and typical results are shown in fig. 5. The distribution retains the convex profile of single-phase flow at 84 mm, but the profile becomes concave at 284 mm. That is, the vapor velocity is higher than that of the liquid in the fully developed region of inverted annular flow. This transformation in the distribution occurs at a greater distance from the quench front as the inlet velocity increases. The change in liquid mass flow rate due to vaporization can be neglected in comparison with the accuracy of the measurements, since it is only about 1% even if the total heat input contributes to evaporation. The radial distributions of liquid velocity obtained over several measurements were expressed in a cubic equation by using the least squares method. The vapor film thickness was obtained by integration. The thickness increased linearly along the channel and the average was taken over the test section between 84 mm and 484 mm. The average vapor film thickness is illustrated in fig. 6 by making it dimensionless on the basis of the channel diameter (D). From the figure, it is clear that an increase in heat flux (q") increases the thickness ( ~ / D ) linearly but that increases in inlet
0.04
o / , / ,/
0.02
#I
ill
I I
, /
/ A Tsub oAn
q"
10.7 K
OA
15.7 K
•
20.7 K
I I 40 (kW/m2)
20
60
i
80
Fig. 6. Experimental results of average vapor film thickness.
The value of heat flux obtained by extrapolating the thickness to zero is a minimum, depends only on the inlet subcooling and represents the lower limit required to form stable inverted annular flow. Fig. 7 presents the axial profile of vapor film thickness as compared with Bromley's correlation. The vapor film of inverted annular flow is thicker than that of the saturated film pool boiling despite the existence of subcooling. The ex-
0.3 u : 1.5m/s q" : 4 0 k W / m =
velocity (Ui.) and inlet subcooling (ATe.b) reduce it. "o
,2l
t2i i 01o11 1.0~-0 ~
i
0¢//
i i / // /i/ / i / i I / / I / / // // / ill O/ I /// / / / • i1/i/ i // // i/I/ I / / /// //// / I// / / o, // /
0.2
ATsub : 10.7K
\o
~-0--0
~
0
h (experiment) 0 0
1.0 0-
J
f z - 484mm 0.8J
o.g I
~'~ 1.0 L _ 0 ~ 0 ~0 ~0"~
I z- 284mrn 1.2
i,.oi o-o-o 0.8
z- 284mml
t 1.2 i
Teub:10.7K
Uln:t.Omla q,:3OkWlm 2 Teub: I0,7K i
1.0~ O ~
0 ~0"~
015
r/R
I
0.8 L 1.2
0
i
0.5 r/R
= Uin:2.0m/B
i.OLO~o~ I
0.5 [ 0
~..~ i~
z - 84ram 0.5
rlR
Fig. 5. Radial distribution of liquid core jet velocity.
O0 "Ol
°~ - ~ ~ ~ .~
,,; .......
q°:5OkW/m 2 Tsub:lO.7K
i z - 84rnm 0.8
0.1
t
Uin : 1,5m/s q,:4OkWlm 2
1.0 ~ o------.._o ~o.
z -84mm 0.8
:
I
L
z=4841mm i
St.2
1.2 ~
0.8'
0,8l
' 0.1
~
0.5
'°~
h (Bromley)
i);- &;o-m~e;i- - ~ ' 0.2
'. 0 3 Z
-
-
' 0.4
0 0.5
(m)
Fig. 7. Comparison between inverted annular flow and film pool boiling for vapor film thickness and heat transfer coefficient.
M. Aritomi et aL / Thermo-hydraulic behavior of inverted annular flow
0.4, o O•
0.3
Neglecting the change of liquid mass flow rate due to vaporization as aforementioned, and integrating eq. (1) from 84 mm to 484 mm,
/O
U in 1.0 m/s
A&
1.5 m / s
D
2.0 m/s
285
&
/
nA
®
4~iDi
[
1 ~0.484r
0.4 ]0.084
D2 -
+ ~z { p l ( a - a ) u +Pl(1
://
~ 0.2
•
"o
AAÜ
®
OAn
0 -0.15
i -0.10
2)
0 /
10.7 K
®&
15.7 K
•
20.7 K
= -0.05
0
Equilibriumquality (-) Fig. 8. Comparison between void fraction and equilibrium quality.
- or)g] dz.
(2)
Assuming that the void fraction can be approximated by a linear function, the average interfacial shear stress (~i) is obtained by eq. (2) after substituting measured values of pressure drop, void fraction and liquid velocity. Though pressure drop is measured to an accuracy of 10 Pa, the error including experimental reproducibility and the above stated evaluation method is thought to be about 15%. The results are shown in fig. 9, and indicate a similar tendency to that of the average vapor film thickness, that is, a rise in the heat flux increases the average interracial shear stress linearly and the reductions in the inlet velocity and subcooling also increases it.
10 Uin
istence of compressible vapor flow between the wall and liquid core-jet makes the jet unstable due to the mobility and brings about a wavy interface. Consequently, the vaporization rate is increased and the vapor flow is suppressed by the roughness of the interface, increasing the thickness of the vapor film. Although Ellias and Chamble [13] and Fung and Groeneveld [14] indicated that void fractions could be correlated only with equilibrium quality, our experimental results also depend on the subcooling as shown in fig. 8. It is considered that this difference is caused by the different range of equilibrium quality which was measured, that is, the range investigated by them is higher than -0.02.
000
1.0 rnls
&&
1.5 m / s
r~
2.0 m / s O Z~
6
//
®
3.2.2. ln terfacial shear stress The steady-state momentum equation for liquid is
~---[p,(1-a)u 2] + ( 1 - a)-~-zP- 4~'iDi az D2 + pl(1 - a ) g = --Fgu i.
zl Tsu b OZ~ m
'
20
' q"
(1)
40
10.7 K
O&
15.7 K
•
20.7 K
'
610
( k W l m 2)
Fig. 9. Experimental results of average interfacial shear stress.
M. A ritomi et aL / Thermo-hydraulic behavior of inverted annular flow
286
0.25
3.2.3. Wall shear stress
gin = 1.0 m / s ( a ) 4Tsu b = 10.7 K
The steady-state mixture momentum equation is 0
~ [psau~ +
p](1 - o,,u,,',.21 +
~
o
4%, P + 19 -
40.2
v
35.5
0.2018
-
+ [p,a+p|(l-a)]g=O.
v
(3)
o
8
, i
q" = 40 k W / m 2 ATsub = 10,7 K
Uin ( m / s )
o
0.4 v0.084 L ~2 P - ~2 ( pl(] - Ol)t/~ } 0.30
-p,(1 - a)g] dz
25.5
D
0.35
1 f0.484[ _
30.4
V v
i
0.15 (b)
0
v
~
Neglecting the mass of the vapor phase and using the same procedure as for average interracial shear stress, 47w O-
q° ( k W / m = )
1.0
~,
1.5
[]
2.0
(4) 0.25
~
J=
The average wall shear stress (~w) is obtained by eq. (4) using the same procedure for the average interfacial shear stress. The ratio of the wall shear stress in inverted annular flow to that of liquid single-phase flow is shown in fig. 10. Although the experimental data are scattered because the values are not very large compared with the accuracy of measurement, it is clear that the frictional loss in inverted annular flow is less than 50% of the value in liquid single-phase flow. This is a result of the existence of the thin vapor film on the wall which eases the liquid phase flow.
[] 0.20 .o
n
o
o
~
(el
u~ = 1.0 m / s " = 40 k W m 2
t
0.15
a
o
~
g
o~ I
i
(K)
aTsub
0.30 .o
o
10.7
o
20.7
15.7
0.25 • & o o
o
o
o
o
0.20
o 0.15
~
8
8
6 L
011
022 z
0.3
0.4
0.5
(m)
Fig. 11. Typical axial distribution of heat transfer coefficient. 1.0 Ui n
0.8
ATsub
OO•
1.0 m / s
O&D
10.7 K
z~&
1.5 m / s
O&
15.7 K
[]
2.0 m/s
•
20.7 K
3.3. Heat transfer
The wall temperature was measured using the electrical resistance method. The calibration curve was obtained by using a tube identical to the heat transfer tube. The error, including the experimental reproducibility, could be controlled to less than 3 K. The heat transfer coefficient is defined by q " / ( T w - T~at). Fig. 7 shows a typical axial profile of heat transfer coefficient compared with Bromley's correlation [5] and Sudo's version [8]. The heat transfer coefficient in inverted annular flow is larger than that in film pool boiling despite the thicker vapor film. Fig. 11 illustrates the axial profiles of the heat transfer coefficient obtained. Increases in inlet velocity and in subcooling enhance the
~. 0.6 F.-
I~"
0.4
&
o
&
•
0.2
J
20 q"
i
L
40
i
60
(kw/m =)
Fig. 10. Experimental results of average frictional multiplier.
heat transfer coefficient, but a decrease in heat flux increases it only in the region less than 0.25 mm from the quench front. From figs. 7 and 11, it can be seen that the heat transfer coefficient along the channel decreases remarkably in the region less than 0.1 m from the quench front but scarcely varies in the region more than 0.1 m away. Therefore, the average heat transfer
M. A ritomi et aL / Thermo-hydraulic behaoior of inverted annular flow
8
coefficient in the fully developed region is considered hereinafter. The average film Nusselt number (Nuf) is defined by
Nuf
=
h~g/,/Xg,
287
//" Uin 1.0 m/s
/
zxA
1.5 m/s
+10% //O /
El
2.0 m/s
0®0
(5)
6
//~
IZ
Experimental results are shown in fig. 12, and indicate a similar tendency to the average vapor film thickness and the average interracial shear stress as regards the effects of heat flux, inlet velocity and inlet subcooling. The results are illustrated in fig. 13 of plotting the average film Nusselt number against ~g/D. They are well correlated, within 10% error, and the following correlation, shown on the diagram by a solid line, is obtained.
4
//~/0
A 0
/ /
El ~
where ~g is determined by the average vapor film temperature ( ~ )
Tg=½(Twq- Tsat).
//
,/ /
""
//
"
/ GO'/'~'0" 0
S. 2
z~"
.1Tsub
/
OLeO 10.7 K
I
00
I
I
0.02
I
0.04
®A
15.7 K
•
20.7 K I
I
0.06
bg/D
Nut = 1 + 9 0 . 9 ~ .
(7)
Heat flux, inlet velocity and inlet subcooling have an influence only through the thickness. Consequently, the heat transfer coefficient scarcely varies axially in spite of changes in the vapor film thickness as shown in figs. 7 and 11. From fig. 13 and eq. (7), the average film Nusselt number is seen to converge to unity when the
Uin o
^
/ ,/ / u
txA
1.5m/s
•
13
2.0 m/s
O
Z~ El
o//° / / -~ I~0 i r"
-
/ 4
O
/
e/
•
/ •
2
ATsub OZXEI 10.7 K
00
'
vapor film thickness approaches zero. That illustrates the fact that a decrease in the vapor film thickness reduces turbulence in the vapor film and allows thermal conductivity to govern the heat transfer.
4. Discussion
O®O 1.0 m/s
6
Fig. 13. Average film Nusselt number in reference to the dimensionless average vapor film thickness.
2'o
i qR
®&
15.7 K
•
20.7 K
, 40
,
6'o
(kW/m2)
Fig. 12. Experimental results of average film Nusselt number.
In the case of inverted annular flow, the liquid thermal boundary layer near the interface and the vapor film are very thin and the interface fluctuates. Therefore, the accuracy obtained in measurements of radial distributions of liquid jet temperature, vapor temperature and vapor velocity cannot be guaranteed. However, it is necessary for the further consideration of the experimental results to evaluate the average vapor film velocity and temperature. The average vapor film temperature is assumed to be estimated by eq. (6), and the procedure for evaluating the average vapor film velocity is described below. Replotting the values of the interfacial shear stress against the vapor film thickness, a correlation is seen as shown in fig. 14, though it decreases slightly with increasing inlet velocity. In cases where the interface is considered to be smooth, it has been made clear from our experiments on stratified flow
288
M. A ritomi et aL / Thermo-hydraulic behaoior of inoerted annular flow 100
10 0@•
1.0 m / s
0 e •
1.0 m / s
t~ Z~
1.5 m / s
z~ &
1.5 r n / s
[]
2.0 m/s
O
2.0 m/s
50
/ITsub
//
OZ~D 10.7 K
2
~6
De = 4bg ( 1 - b g / D )
Uin
gin
®&
15.7 K
•
20.7 K
A
...... Eq.(7)
/ O
DAO" [:]i,"
ie"
~lTsub OZXD 10.7 K
20
O/®/~7o/~o/° / /•/~
I
De
i
i
0.04
i
i
0.06
bg/D
Fig. 14. Average interracial shear stress with reference to the average dimensionlessvapor film thickness.
[17] that the interracial shear stress could be evaluated by 0.0791
- u,) 2,
,
,
i
,
10
(mm)
is examined to evaluate the average vapor velocity. The experimental values are replotted in fig. 15, which makes it clear that the average Nusselt number is proportional to the 0.8th power of the average equivalent diameter. Supposing that vapor film flow is turbulent, the heat transfer coefficient can be evaluated through DittusBoelter's correlation. Therefore, evaluating the average vapor film velocity on the basis of Dittus-Boelter's correlation for Nusselt number by using the equivalent diameter defined by eq. (11), the calculated results are presented in fig. 16.
(S) 15
where (Ug -- Ul)O
Re i -
,
5
2
Fig. 15. Average film Nusselt number with reference to the equilibrium diameter.
0.02
~'i = - Rei0.2 5 ½Pg(Ug
1
/
I
20.7 K
i
0.5
/
0
15.7 K
•
/
/® ®
®A
i
us
(9) Q
We consider that the average vapor film velocity scarcely varies under the conditions shown in the figure, that an increase in vapor film thickness increases interfacial roughness and interracial shear stress, and also that an increase in inlet velocity slightly decreases the slip velocity between the phases and decreases the interfacial shear stress. On the other hand, although an empirical correlation of average film Nusselt number is obtained by eq. (7), the average film Nusselt number, N,u = hDe/?~g,
(10)
10
o2
U in
,t Tsu b
O®•
1.0 m / s
OZXO
10.7 K
Ad~
1.5 m / s
®&
15.7 K
13
2.0 m/s
•
20.7 K
~ 20
which is commonly used, and is defined by the equivalent diameter, D~ = 4~g(1 - ~ J D )
(11)
~ q"
4 "O
'
60
( k W / m =)
Fig. 16. Average vapor velocity obtained by Dittus-Boelter's correlation.
289
M. Aritomi et aL / Thermo-hydraulic behavior of inverted annular flow
The void fraction (a) is given by the relation
Q"
// Uin
OO•
The average net vaporization rates (Pg) are computed from the exit void fraction and vapor film velocity and illustrated in fig. 17. The average net vaporization rate in the heated section depends only on ~ / D and the following empirical correlation is obtained from the figure.
.i ®
1.0 m / s
Z~A
1.5 m / s
[]
2.0 m / s
i
2 ///®
O
A
/
(13)
/
/
////
dTsub
/~//
--ttt
The phasic average volumetric heat generation rates (qg and Eh'" ) are
= r,{ L +
r.,)},
(14)
. . . . = ~4q . . .-. . .q~ • q~
--
tit
O®•
?
0
I
0
l/
1.0 m / s 1.5 m / s
[3
2.0 m / s
i
15,7 K
•
20.7 K
i
i
2 li (obtained by Eq.(16))
3
(kW/m=K)
Fig. 18. Heat transfer coefficient from the interface to the liquid core jet.
substituted into eq. (15) and E/l'" is obtained. Evaluating and adjusting the results, the following empirical co~telations concerning the average volumetric heat generations of the fiquid phase (Yh") and the heat transfer coefficients from the interface to the liquid core jet (hli) are obtained.
,'Q
zx A
i
1
10.7 K
®&
--
U in
40
OZ~D
/ "
(15)
The average vaporization rates are calculated by Dittus-Boelter's correlation on the basis of the experimental results of the heat transfer coefficient. Substituting it/ these values into eq. (14), qg is obtained. The qg is
/
/
/
I J::
1
_Fg= 6 5 0 ~ / D .
A/ +io~. / A J,...
(2 ZX
16,
//'
0 i
+20%
v
[]
//
/
30
= 1.287
/ tA
/ / -20%
&®
d~ o e~
// /
20
/
,/
®
/
//I !/
/ /o
0~ O) >
//
10 //
//
///
ZlTsub oZXrl
// // /~///
0
I
0
I
I
0.02
0.04
ub + 1.03 aL b I" ,
(17)
//
10.7 K
®A
15.7 K
[]
20.7 K I
i
0.06
~m
Fig. 17. Average net vaporization rate.
where A~u b = Ts, t - T l and Tl is the average liquid temperature calculated by heat balance. Fig. 18 compares the heat transfer coefficients obtained from fig. 16 with the calculated ones from eq. (16). In eq. (17), the first term corresponds to the minimum heat flux required to create stable inverted annular flow as shown in fig. 6. The second term in the equation is interpreted as follows: (1) The effect of liquid velocity on heat convection is larger than that of channel flow because of the existence of the compressible vapor film between the wall. (2) Both vaporization and condensation simultaneously occur at the interface and the fluctuations at the interface renew the surface, so that the effect of subcooling on the heat convection is the square and
290
M. Aritomi et al. / Thermo-hydraulic behavior of inverted annularflow
the vapor film thickness influences the heat convection linearly. To verify these empirical correlations, the usual Nusselt number defined by hD/Xg is evaluated. The calculated results including the condition Ui~ = 0.5 m/s, where the vapor film thickness could not be measured due to the limitations of measurement accuracy, are illustrated in fig. 19 and compared with the experimental ones. On the basis of eq. (8), the average interfacial friction factors are examined in terms of the experimental average interfacial shear stress, the average vapor film velocity and the average liquid velocity calculated from the average void fraction and are shown in fig. 20. The empirical correlation for the average interfacial friction factor (fi) is obtained by
0.4
0.3
1.0 m / s
A ~,
1.5 m / s
0
2.0 m / s
// /// // +15% / /
I
,g
O
O'
~ /
/
o.-
/O /
ig:
165-
/ /
&• /
/////0// / O / // • / // //
d Tsu b
/ /e
o~0
z/~////
200
/////
j10,115, 20,
i
A E
•
0.5
•r. 150 v
1.0
0
1.5
C,
2.0
O
v r
•
0
/
ioo
/ • ~ ® / ///
-15%
/ / /J/// /
/
//
/
50
i
0 0
50 Nusselt
i
0.02
i
0.04
I
15.7 K
O
20.7 K |
0.06
i _ _
0.08
bg/Di
Fig. 20. Comparison of interfacial friction coefficient with dimensionless vapor film thickness.
cause of the scatter in the experimental values, the results do coincide within an order of magnitude. The experimental results can be systematically arranged using a model in which the heat and momentum transfers are similar to those of the usual singlephase channel flow. In this model the heat, mass and momentum transfers depend on the interfacial roughness which is proportional to the vapor film thickness.
5. C o n c l u s i o n s
///////
Z
L
00
10.7 K
®A
+15%// //
///~
E
El/ -15%
~o.2
(18)
It increases linearly with the ]g/Di, because increasing the vapor film thickness increases the surge and roughness of the interface. The experimental results of average wall shear stress under the conditions shown in table 1 (except Uin = 0.5 m / s ) have values between 0.9 and 4.5 Pa. The results calculated by the average vapor film velocity in fig. 15 and Blasius's correlation are from 1.6 to 2.1 Pa. Although a correlation between the experimental results and those calculated cannot be obtained exactly be-
/
//
0.1
Rei°'25 Di"
/1
U in O®¢
L 100
number
i 150
200
(calculation)
Fig. 19. Comparison between experimental results and calculated results using empirical correlations proposed in the present paper for the Nusselt number.
The thermo-hydraulic behavior of inverted annular flow was investigated experimentally and the measured results were rearranged as average values in the fully developed region. From consideration of the results, the following points are clarified: (1) The heat transfer coefficient decreases along the channel only in the inlet region where z / D is less than 10, but scarcely changes in the fully developed region where z / D is larger than 10. (2) An increase in the vapor film thickness causes a linear increase in the interfacial shear stress and the film Nusselt number because the interface becomes rougher with increasing thickness. The roughness
M. Aritomi et al. / Thermo-hydraulic behavior of inverted annularflow leads to a thicker v a p o r film as c o m p a r e d with saturated film pool boiling due to the increase in flow resistance at the interface in spite of the existence of subcooling. (3) A s s u m i n g that the heat transfer coefficient is given b y D i t t u s - B o e l t e r ' s correlation o n the basis of the experimental results, the results correlate well in terms of heat transfer coefficient from the interface to the liquid jet, interfacial friction factor a n d net vaporization rate. (4) T h e existence of the v a p o r film at the wall decreases the flow resistance of the liquid phase, so t h a t the frictional loss in inverted a n n u l a r flow is less t h a n that in single p h a s e flow. Since the p r o p o s e d empirical correlations c a n b e applied only to freon 113 u n d e r a t m o s p h e r i c pressure, it is necessary to m a k e further efforts to e x p a n d their applicable range in future work.
Nomenclature
D g h L
Nu P Pr q" q Re T
= = = = = =
= =
= = =
= ATsub = U = Z = ~t = = = h = = # = 'r =
specific heat ( k J / k g K) diameter (m) acceleration due to gravity ( m / s 2) heat transfer coefficient ( k W / m 2 K) latent heat of vaporization ( k J / k g ) Nusselt n u m b e r ( - ) pressure (Pa) Prandtl number (-) heat flux ( k W / m 2) volumetric heat generation rate ( k W / m 3) Reynolds n u m b e r ( - ) t e m p e r a t u r e ( ° C) subcooling (K) velocity ( m / s ) position along c h a n n e l (m) void fraction ( - ) vaporization rate ( k g / m 3 s) v a p o r film thickness (m) t h e r m a l conductivity ( k W / m K ) kinematic viscosity ( m 2 / s ) density ( k g / m 3) shear stress (Pa)
Subscripts and superscripts 1 = liquid, g = gas, f = film, i = interface, w = wall, sub = subcooled, sat = saturated, - = averaged.
291
References
[1] Y. Murao, Analytical study of the thermo-hydrodynamic behavior of the reflood-phase during LOCA, J. Nucl. Sci. Tecnol. 16 [11] (1979) 802-817. [2] K. Akagawa and T. Fuji, Power generation by utilization of LNG-cold, (in Japanese), J. JSME 83 (1980) 650-656. [3] P.J. Berenson, Film-boiling heat transfer from a horizontal surface, Trans. ASME, J. Heat Transf. 83 (1961) 351-357. [4] K.V. Moore and W.H. Rettig, RELAP4: A computer program for transient thermal-hydraulic analysis, ANCR1127 (1973). [5] L. Bromley, Heat transfer in stable f i l l boiling, Chem. Eng. Prog. 46 (1950) 221-227. [6] TRAC-P1A: An advanced best-estimate computer program for PWR LOCA analysis, NUREG/CR-0665 (1979). [7] Y. Murao and J. Sugimoto, Correlation of heat transfer coefficient for saturated film boiling during reflood phase prior to quenching, J. Nucl. Sei. Tecllol. 18 (1981) 275284. [8] Y. Sudo, Film boiling heat transfer during reflood phase in postulated PWR loss-of-coolant accident, J. Nucl. Sci. Tcchnol. 17 (1980) 516-530. [9] J.E. Leonald, K.H. Sun and G.E. Dix, Low flow f i l l boiling heat transfer on vertical surface, Part II: Empirical formulations and application to BWR-LOCA analysis, Solar and Nucl. Heat Transl., AICliE Syrup. Series, No. 164, Vol. 73 (1977) 7-13. [10] E.K. Kalinin, I.I. Berlin and V.V. Kostyuk, Film-boiling heat transfer, Advances in Heat Transfer Vol. 11 (Academic Press, New York, 1975) pp. 51-198. [11] J.G. Collier, Post dryout heat transfer, Two-Phase Flow and Heat Transfer in the Power and Process Industries (McGraw-Hill, New York, (1981) pp. 282-329. [12] M. Ishii and G.De JarlaJs, Flow regime transition and interfacial characteristics of inverted annular flow, Proc. Japan-US Seminar on Two-Phase Flow Dynamics (1984) pp.B.7.1-B.7.9. [13] E. Ellias and P. Cambte, Inverted-annular film boiling heat transfer from vertical surface, Nucl. Engrg. Des. 64 (1981) 249-257. [14] K.K. Fung and D.C. Groeneveld, Measurement of void fraction in steady state subcooled and low quality f i l l boiling, Int. J. Multiphase Flow 6 (1980) 357-361. [15] K.K. Fung and D.C. Groeneveld, Subcooled and low quality flow f i l l boiling of water at atmospheric pressure, Nucl. Engrg. Des. 55 (1979) 51-57. [16] K. Akagawa, T. Fujii, N. Takenaka, K. Nishida and M. Iseki, Flow pattern and heat transfer of inverted annular flow, (in Japanese), 22nd Nat. Heat Transl. Symp. Japan (1985) pp. 130-132. [17] M. Takahashi, M. Aritomi, A. Inoue and S. Aoki, Interface turbulent heat transfer of horizontal co-current liquid-liquid stratified flow (3rd. Boiling in Water), (in Japanese), 17th Nat. Heat Transf. Symp. Japan (1980) pp. 220-222.