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Liquid film behavior in annular two-phase flow under flow oscillation conditions
International Journal of Heat and Mass Transfer 53 (2010) 962–971
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Liquid film behavior in annular two-phase flow under flow oscillation conditions Tomio Okawa *, Taisuke Goto, Yosuke Yamagoe Department of Mechanical Engineering, Osaka University 2-1, Yamadaoka, Suita-shi, Osaka 565-0871, Japan
a r t i c l e
i n f o
Article history: Received 3 July 2009 Received in revised form 8 October 2009 Accepted 30 October 2009 Available online 1 December 2009 Keywords: Annular two-phase flow Flow oscillation Liquid film Disturbance wave Dryout
a b s t r a c t Experiments were carried out to investigate the effects of sinusoidal forced oscillation of the inlet flow rate on the time variations of local liquid film thickness and the frequencies of large wave’s passing in steam–water annular two-phase flows. The liquid film thickness oscillated with the same period as the inlet flow rate. The mean film thickness in the thin film regions decreased and approached to an asymptotic value with an increase in the oscillation period of the inlet flow rate. This result was consistent with the experimental results of the occurrence of liquid film dryout under flow oscillation conditions reported in the literature. It was hence considered that the axial liquid transport from the thick to thin film regions mitigates the reduction of the critical heat flux caused by the flow oscillation. It was also found that the wave frequency in the thin film region increased with a decrease in the oscillation period. This observation suggested that the disturbance waves contribute to the enhancements of the liquid transport and consequently the critical heat flux associated with the liquid film dryout under flow oscillation conditions. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The use of high power density core enabling reductions of capital and maintenance costs is one of the most profitable ways to develop innovative boiling water reactors (BWRs) of higher economic efficiency. Various high power density BWR designs were hence proposed by many research and development teams [1–4]. It is, however, known that increased power density generally leads to the reduction of the margin to the onset of unanticipated flow instability in boiling two-phase flow systems [5]. Therefore, to ensure the safety of high power density BWRs, it is necessary to predict the occurrence of boiling transition with a high degree of accuracy even when the coolant flow rate becomes oscillatory. Extensive experimental studies were performed to clarify the effects of flow instability on critical heat flux in a boiling channel as summarized in the review article by Ozawa et al. [6]. These studies indicated that destabilization of the flow rate results in a noticeable decrease in the critical heat flux. However, in the experiments performed under unstable conditions, it is difficult to systematically investigate the effects of the amplitude and frequency of flow oscillation since these parameters are largely dependent on the geometries of the experimental apparatus. Ozawa et al. [7] and Umekawa et al. [8] therefore imposed forced oscillation on the inlet flow rate, enabling almost-arbitrary control of the amplitude and frequency. It was shown clearly in these exper-
* Corresponding author. Tel.: +81 6 6879 7257; fax: +81 6 6879 7247. E-mail address: [email protected] (T. Okawa). 0017-9310/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2009.11.024
iments that an increase in the amplitude and a decrease in the frequency lead to a reduction in the critical heat flux. Based on a one-dimensional three-fluid model [9], Okawa et al. carried out numerical simulations to calculate the critical heat flux associated with the liquid film dryout in annular two-phase flow regime [10]. In the simulations, the following sinusoidal oscillation was applied to the inlet mass flux GIN:
GIN ¼ GAVE þ DG sin
2p t T OSC
ð1Þ
where GAVE is the time-averaged inlet mass flux, DG is the oscillation amplitude, TOSC is the oscillation period and t is the time. They introduced the following dimensionless critical heat flux q* and the dimensionless heated length L* to investigate the influence of the flow oscillation on the critical heat flux:
q ¼
qOSC qMIN qAVE qMIN
ð2Þ
L ¼
2L uf0 T OSC
ð3Þ
where qOSC is the critical heat flux under the flow oscillation condition, qAVE and qMIN are the critical heat fluxes in the steady states when GIN is equal to GAVE and GMIN (= GAVE DG), respectively; L is the heated length, and uf0 is the characteristic film velocity. It was shown that the calculated values of q* were correlated in terms of L* well, and q* increased from 0 to 1 with increased value of L*. This result was interpreted that qOSC should be equal to qMIN (q* = 0) for small values of L* since the axial liquid transport to
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963
Nomenclature D dk E Eo Fr f fi fw G g j L M q q+ P Re T t tf uf0 u* x
tube diameter (m) image darkness entrainment fraction Eötvös number Froude number frequency (Hz) interfacial friction factor wall friction factor mass flux (kg/m2 s) gravitational acceleration (m/s2) superficial velocity (m/s) heated length (m) Morton number heat flux (W/m2) ratio of wall heat flux to CHF pressure (Pa) Reynolds number period (s) or temperature (K) time (s) liquid film thickness (m) characteristic film velocity (m/s) friction velocity (m/s) vapor quality
q r si
density (kg/m3) surface tension coefficient (N/m) interfacial shear stress (N/m2)
Superscripts dimensionless + dimensionless Subscripts AVE average DW disturbance wave EX exit f liquid film g gas phase IN inlet LW large wave l liquid phase MAX maximum MIN minimum OSC oscillation ST short time STA short-time average m vapor w wall
Greek symbols DG oscillation amplitude of mass flux (kg/m2 s) l viscosity (Pas)
the thinnest film region corresponding to GMIN was negligible, and it increased asymptotically to qAVE (q* ? 1) with increased value of L* since the liquid supply from other regions within a liquid film to the thinnest film region became noticeable and consequently the influence of the oscillation of the inlet mass flux was mitigated at the outlet of the heated section. Okawa et al. [10] then measured the critical heat flux under flow oscillation conditions in order to confirm the validity of the numerical results. It was found that q* could be correlated by a monotonically increasing function of L* as in the numerical simulations, but the experimental critical heat fluxes were generally higher than those calculated using the three-fluid model. This would indicate that available numerical models are satisfactory for conservative estimation of the critical heat flux, but the present understanding of the dynamic behavior of a liquid film is not sufficient for accurate prediction of the critical heat flux under flow oscillation conditions. Since the main objective of the experiments performed by Okawa et al. [10] was to elucidate the effect of the flow oscillation on the critical heat flux, particular attention was not paid to the dynamic behavior of a liquid film. Therefore, in the first half of this work, time variation of the local liquid film thickness is measured to confirm that the liquid transport to the thin film regions is enhanced with an increase in the dimensionless heated length L*. It is known that the surface of a liquid film in annular two-phase flow is commonly covered with complex interfacial waves such as ripple and disturbance waves. In particular, the disturbance waves that occur at high liquid flow rates have a height several times the mean film thickness, move at a velocity greater than the mean film velocity, and retain their identity throughout the flow channel [11–13]. It may hence be expected that the disturbance waves have a major role in transporting the liquid to the thin film regions. In view of this, in the second half of this work, wave frequency is measured to investigate the possibility for the disturbance waves to contribute to the enhancements of the liquid transport and the critical heat flux at large values of L*.
2. Experimental description 2.1. Experimental apparatus and measurement method Experiments were carried out with the experimental apparatus that is shown schematically in Fig. 1 and was described in detail in our previous study [10]. Filtrated and deionized tap water was used as a working fluid, and was circulated using a multi-stage pump. Two needle valves and a turbine flow meter were used to control and measure the inlet mass flux GIN; the flow meter was accurate to within ±0.5%. The inlet flow rate was oscillated by changing the electric power applied to the circulation pump sinusoidally. A 15-kW plug heater was used as a pre-heater to raise and then maintain the water temperature at the inlet of the heated section TIN. After exiting the pre-heating section, the water entered a vertical round tube of 12 mm in internal diameter and 0.8 mm in thickness made of SUS316 stainless-steel. A 60-kW (60 V 1000 A) DC power supply was used to heat the tube ohmically and generate steam–water annular flow inside it. The heated length L was 1360 mm. The transparent section was equipped downstream of the heated section for flow visualization and film thickness measurement. A silica glass cube 40 mm on a side, through which a 12 mm diameter circular hole was punctured as a flow channel, was used for this section. To minimize the effect of the connection between the stainless-steel and glass sections on the behavior of a liquid film, these components were manufactured with an accuracy of 10 lm. The step height at the connection was estimated around 5 lm under the temperature condition during the experiment. The distance from the top of the heated section to the center of the transparent section was 300 mm (25 tube diameters). The top of the vertical test section was connected with the upper tank in order to separate the vapor phase from the liquid phase. The liquid phase was then returned to the circulation pump, while the vapor phase was condensed in the separator or released into the atmosphere.
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shutter speed set to 1 ms was also performed at the transparent section to measure the frequency of large wave’s passing. Each image consisted of 256 512 pixels and the spatial resolution was approximately 50 lm/pixel. Since the displacement meter and the high-speed camera could not be placed on a measurement stage simultaneously, the flow visualization was performed separately from the film thickness measurement. The transparent section was backlit using a metal halide lamp in the visualization experiments. 2.2. Experimental conditions Main experimental conditions in the steady states and flow oscillation conditions are listed in Tables 1 and 2, respectively. The liquid film thickness and the wave frequency were measured under 10 steady states and 10 flow oscillation conditions. In all the experiments, the outlet pressure PEX and the inlet fluid temperature TIN were set to 129 ± 8 kPa and 374 ± 2 K, respectively. In the steadystate experiments, the inlet mass flux GIN and the wall heat flux qw were used as the parameters and changed within 54–162 kg/m2 s and 114–179 kW/m2, respectively. The resulting range of the equilibrium vapor quality at the channel exit (xEX) was 0.13–0.66. Under the flow oscillation conditions, the time-averaged inlet mass flux GAVE and the oscillation amplitude DG were set to 104 ± 3 kg/m2 s and 49 ± 7 kg/m2 s, respectively, and the oscillation period TOSC and qw were used as the experimental parameters. The ranges of TOSC and qw were 2–20 s and 114–178 kW/m2, respectively. The ratios of the wall heat flux to the critical heat flux (q+) are also shown in Tables 1 and 2. In the calculation of q+, the critical heat flux in each experimental condition was estimated from the CHF values measured using the same experimental facility in our previous work [10]. Table 2 indicates that the maximum heat flux of 178 kW/m2 tested in the
Table 1 Main experimental conditions in steady states (the values on the left and right sides of the slash marks are the experimental conditions in film thickness measurements and wave frequency measurements, respectively). Fig. 1. Schematic diagram of experimental apparatus.
Type-K thermocouples accurate to within ±1.5 K were used to measure the fluid temperatures at the inlet and outlet of the preheating section and the outer wall temperatures of the heated section. The pressures at the inlet and the outlet of the heated section were measured with pressure transducers accurate to within ±7.5 kPa. The heat flux applied to the fluid qw was calculated from the electric current passing through the heated section and the electrical resistance of the tube material. The heated section was not thermally insulated, but heat loss to the ambient air was estimated as less than 1 kW/m2 from calculation using the heat transfer equation proposed by Churchill and Chu [14]. At the transparent section, the local film thickness was measured with a laser focus displacement meter, in which the method to focus on a target adopted in an automatically focusing camera was applied to the displacement measurement [15]; the diameter of the laser beam spot was 7 lm, the spatial resolution was 0.1 lm and the linearity error was within ±3 lm. The effect of the non-uniformity of refractive index within the liquid film was neglected in the film thickness measurement since a heat conduction calculation showed that the temperature distribution within a liquid film was fairly uniform at the transparent section (300 mm downstream from the top of the heated section). The experimental data were recorded every 1 ms using a data acquisition system; the measurement time was 30 s in each experimental condition. Flow visualization using a high-speed camera at 1000 frames/s with
No.
PEX (kPa)
TIN (K)
GIN (kg/m2 s)
qw (kW/m2)
xEX (%)
q+ (%)
111 112 113 114 115 121 122 123 124 125
121/124 123/126 125/127 126/129 128/130 124/127 127/130 130/133 133/135 135/137
375/374 375/374 375/373 374/373 375/374 375/374 375/373 375/373 375/374 373/375
54/54 81/82 107/109 135/135 161/162 54/55 80/81 108/108 134/134 161/162
114/114 114/114 114/114 114/114 114/114 178/178 178/178 178/178 178/178 178/179
42/41 28/27 21/20 16/16 13/13 66/65 44/43 33/32 26/26 21/21
49/49 31/31 24/24 20/20 18/17 76/74 50/49 37/37 31/31 27/27
Table 2 Main experimental conditions under flow oscillation conditions (the values on the left and right sides of the slash marks are the experimental conditions in film thickness measurements and wave frequency measurements, respectively). No.
PEX (kPa)
TIN (K)
GAVE (kg/m2 s)
DG (kg/m2 s)
TOSC (s)
qw (kW/m2)
q+ (%)
211 212 213 214 215 221 222 223 224 225
124/127 124/127 124/127 124/127 124/127 129/131 130/132 130/132 129/132 129/132
375/374 376/375 375/374 375/373 375/374 372/374 374/374 375/375 374/374 374/374
106/105 107/107 105/107 104/105 106/106 102/101 106/106 106/106 103/106 105/105
43/42 54/54 54/56 53/53 55/55 45/44 54/54 55/55 52/56 56/54
2/2 3/3 5/5 10/10 20/20 2/2 3/3 5/5 10/10 20/20
114/114 114/114 114/114 114/114 114/114 178/178 178/178 178/178 178/178 178/178
29/29 33/33 39/38 44/43 47/47 46/46 52/52 60/60 69/70 77/74
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2
(b) 1.0
2
qw=114kW/m , GIN=54kg/m s
tf (mm)
tf (mm)
(a) 1.0 0.5
0.0 0 1.0
0.5
1 2
1.5
2
0.5
0.0 0 1.0
2
2
0.5
0.5
1 2
1.5
0.0 0 1.0
2
2
tf (mm)
tf (mm)
1 t (s)
1.5
2
1.5
2
1.5
2
2
0.5
1 2
2
qw=178kW/m , GIN=161kg/m s
0.5
0.5
1 2
0.5
qw=114kW/m , GIN=161kg/m s
0.0 0
0.5
qw=178kW/m , GIN=108kg/m s
tf (mm)
tf (mm)
qw=114kW/m , GIN=107kg/m s
0.0 0 1.0
2
qw=178kW/m , GIN=54kg/m s
1.5
2
0.5
0.0 0
0.5
1 t (s)
Fig. 2. Time variations of local film thickness measured in the steady states: (a) qw = 114 kW/m2 and (b) qw = 178 kW/m2.
3.1. Liquid film thickness Typical time variations of the local film thickness tf measured in the steady states are shown Fig. 2a and b. It can be seen that the fluctuation of tf becomes significant with an increase in GIN and a decrease in qw since the mean film thickness increases [16]. The data of average, minimum, and maximum film thicknesses for different values of qw are plotted against GIN in Fig. 3a and b; here, as done in the previous studies [15,17], the film thicknesses at which the cumulative distribution function was equal to 0.01 and 0.99 were defined as the minimum and maximum film thicknesses, respectively. It can be seen that the maximum thickness tf,MAX and the average thickness tf,AVE tend to increase with an increase in GIN and a decrease in qw, while the dependences of the minimum thickness tf,MIN on these parameters are less noticeable. These tendencies are consistent with those for air–water annular flows reported by Hazuku et al. [15]. The mean film thicknesses calculated by Sekoguchi’s correlation [18] and Okawa’s method [9] and the minimum film thickness calculated by Hazuku’s correlation [15] are displayed in the same figures (for details of the correlations, see Appendix A). Here, the droplet entrainment was neglected in Okawa’s method and the interfacial shear stress acting on a liquid film required in Hazuku’s correlation was evaluated using the method adopted by Okawa et al. [9]. Although it is considered that the present measurements were conducted for the developing region of the steam–water annular flow (the distance of the measuring section from the top of the heated section was 25 tube diameters), the calculated film thicknesses are in fairly good agreement with the present experimental data. This would suggest that the thermal–hydraulic field formed in the heated section was not deviated from the equilibrium state significantly in the present experiments since the values of mass flux and heat flux were rather low.
tf (mm)
3. Results and discussion
(a) 10 10
1 2
qw = 114 kW/m 0
10
–1
10
–2
tf,AVE by Sekogichi tf,AVE by Okawa tf,MIN by Hazuku
tf,MAX tf,AVE tf,MIN
0
50
(b) 101
100 2 GIN (kg/m s)
2
qw = 178 kW/m
tf (mm)
flow oscillation experiments corresponded to 46–77% of the critical heat flux, depending mainly on the oscillation period.
10
0
10
–1
10
–2
0
200
tf,AVE by Sekogichi tf,AVE by Okawa tf,MIN by Hazuku
tf,MAX tf,AVE tf,MIN
50
150
100
150
200
2
GIN (kg/m s) Fig. 3. Average, minimum and maximum film thicknesses in the steady states measured at (a) qw = 114 kW/m2 and (b) qw = 178 kW/m2.
Depicted in Figs. 4 and 5 are the typical time variations of GIN and tf measured under the flow oscillation conditions. Here, the moving average of 0.2 s was applied to the film thickness data to highlight long-term variations corresponding to the flow oscillation. It is found that high-frequency waves associated with the ripple and disturbance waves are superposed to the low-frequency wave in each experimental condition. Eq. (3) indicates that L* is inversely proportional to TOSC. Therefore, if the liquid transport to the
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(a)
250 200
150 100
150 100 50
50 0 0 0.3
5
10
15
0 0 0.3
20
2
tf (mm)
tf (mm)
0.2 0.1 0.0 0
5
10
15
20
15
20
15
20
15
20
15
20
15
20
2
(b)
2
qw=114kW/m ; TOSC=5s
0.2 0.1 0.0 0
20
5
250 200
10 t (s)
2
qw=178kW/m ; TOSC=5s
2
GIN (kg/m s)
200
15
2
GIN (kg/m s)
250
10 t (s)
5
qw=178kW/m ; TOSC=3s
qw=114kW/m ; TOSC=3s
(b)
2
qw=178kW/m ; TOSC=3s
2
GIN (kg/m s)
200
(a)
2
qw=114kW/m ; TOSC=3s
2
GIN (kg/m s)
250
150 100 50 0 0 0.3
5
10
15
150 100 50 0 0 0.3
20
2
tf (mm)
0.2
tf (mm)
5
10 2
qw=178kW/m ; TOSC=5s
qw=114kW/m ; TOSC=5s
0.1
0.2 0.1
Periods for shorter-time average
0.0 0
(c)
5
0.0 0
20
(c)
2
qw=114kW/m ; TOSC=10s
5
250 200
10 t (s) 2
qw=178kW/m ; TOSC=10s
2
GIN (kg/m s)
200
15
2
GIN (kg/m s)
250
10 t (s)
150 100
150 100
50 0 0 0.3
5
10
15
50 0 0 0.3
20
2
tf (mm)
tf (mm)
0.2 0.1
5
10 t (s)
10 2
qw=178kW/m ; TOSC=10s
qw=114kW/m ; TOSC=10s
0.0 0
5
15
0.2 0.1 0.0 0
20
5
10 t (s)
Fig. 4. Time variations of inlet mass flux and local film thickness under oscillatory conditions at qw = 114 kW/m2: (a) TOSC = 3 s and (b) TOSC = 5 s, and (c) TOSC = 10 s.
Fig. 5. Time variations of inlet mass flux and local film thickness under oscillatory conditions at qw = 178 kW/m2: (a) TOSC = 3 s, (b) TOSC = 5 s, and (c) TOSC = 10 s.
thin film region is enhanced with an increase in L* as supposed in our previous study [10], it is expected that the mean film thickness in the thin film regions increases with a decrease in TOSC. In the present study, 10% of each oscillation period for which the shorter-time average of the film thickness was minimized was regarded as corresponding to the thin film region, as indicated schematically in Fig. 4b. The data of the mean film thickness in the thin film regions tf,STA are presented in Fig. 6a. It can be seen that tf,STA decreases with an increase in TOSC and approaches to asymptotic values. This result could be interpreted to represent that the axial
liquid transport to the thin film regions is negligibly small when the oscillation period is long, while the mean film thickness in the thin film regions increases due to the liquid transport from the thicker film regions for small values of TOSC. In order to quantify the dependence on L*, the same data of tf,STA are plotted against L* in Fig. 6b. Here, uf0 in Eq. (3) was estimated using the following equation [10]:
GAVE uf0 ¼ pffiffiffiffiffiffiffiffiffiffi
qf qv
DG GAVE
0:2 ð4Þ
T. Okawa et al. / International Journal of Heat and Mass Transfer 53 (2010) 962–971
(a) 0.14
frequency fLW from the time variation of the image brightness. It is noted that, although the film thickness data shown in the previous section can also be used for the measurement of fLW, it was decided to measure fLW from the brightness data in this work since the arrival of large waves could be verified in the video images. The size and position of the measuring window are shown schematically in the first image in Fig. 7. Since the distribution of brightness in the wave region was not uniform, the window height was set to the value in the same order with the typical wave length. The time variations of the space-averaged darkness within the measuring window dk measured in the steady states are presented in Fig. 8a and b. Since the range of brightness (gray level) was 0 (black)–255 (white), dk was calculated by
2
tf,STA (mm)
qw = 114 kW/m 2 qw = 178 kW/m
0.12
0.1
0.08 0
5
(b)
20
25
dk ¼ 1
2
qw = 114 kW/m 2 qw = 178 kW/m
1
tf,STA*
10 15 TOSC (s)
0.5
0 10
–2
–1
10 * L
10
0
Fig. 6. Mean film thicknesses in the thin film regions under flow oscillation conditions: (a) dependences on TOSC and (b) dependences of tf;STA on L*.
where qf and qv are the densities of liquid film and vapor phase, respectively, and tf,STA was scaled by
t f;STA ¼
tf;STA t f;AVE ðGMIN Þ t f;AVE ðGAVE Þ tf;AVE ðGMIN Þ
967
ð5Þ
where tf,AVE(G) is the mean film thickness in the steady state when the inlet mass flux is set to G. It is found that t f;STA is correlated fairly well as a monotonically increasing function of L* within the experimental conditions tested in this work. This result indicates that L* is a promising dimensionless parameter to characterize the liquid transport to the thin film regions under flow oscillation conditions. 3.2. Wave frequency Typical sequential images of large waves passing through the transparent section are depicted in Fig. 7. It is found that the brightness level is low in the wave region, reflecting that the surface of a large wave is highly perturbed. Hence, the rectangular measuring window was located in the images to measure the wave
N 1 X bi 255N i¼1
ð6Þ
where bi is the brightness of the ith pixel and N is the number of pixels contained in the measuring window (the value of dk is equal to 0 and 1 for totally white and black windows, respectively). In Fig. 8a and b, the circular symbols are used to indicate the waves that were classified as large waves. Since the film thickness increases sharply downstream of a large wave [17], it was assumed that large waves arrived at the measuring section when the time derivative of the mean darkness exceeded the threshold value. The maximum time derivative was calculated in the same manner as tf,MAX in each experimental condition and the half of the maximum value was used for the threshold. The frequencies of large waves fLW measured using the above-described method were compared with the frequencies of disturbance waves fDW calculated by Sekoguchi’s correlation [18] in Fig. 9 (details of the correlation is found in Appendix A). It can be seen that fLW measured in the present experiments increases with increased values of GIN and qw and agrees with the values of fDW calculated by Sekoguchi’s correlation fairly well. It should, however, be noted that Sekoguchi et al. [18] used more precise criteria for determining the disturbance wave. The waves indicated with circular symbols in Fig. 8 are hence identified just as large waves in this paper. The time variations of dk under the oscillatory conditions are depicted in Fig. 10a and b; here, the moving average of 0.2 s was applied to the raw data. It can be confirmed that the high-frequency variations of dk associated with the interfacial waves such as disturbance and ripple waves are superposed to the low-frequency variation. Since dk tended to increase with increased value of GIN in the steady state experiments, 10% of each oscillation period for which the shorter-time average of dk was minimized was regarded as corresponding to the thin film region. The time variations of dk in the thin film regions are displayed in Fig. 11a and b. At the bottom of these figures, the data for the steady states when GIN was set to GMIN were also presented for comparisons. It can be seen
Fig. 7. Typical time-elapsed images of large waves passing through a transparent section (qw = 178 kW/m2, GIN = 81 kg/m2 s).
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(a) 1.5
GIN=54kg/m
(b) 1.5
2
classified as large wave
2
GIN=55kg/m
classified as large wave
1.0 dk
dk
1.0
0.5
0.5 0.0 0 1.5
0.5
0.0 0 1.5
1
2
GIN=109kg/m
0.5
1
0.5
1
0.5 t (s)
1
2
GIN=108kg/m
1.0 dk
dk
1.0
0.5
0.5 0.0 0 1.5
0.5
0.0 0 1.5
1
2
GIN=162kg/m
2
GIN=162kg/m
1.0 dk
dk
1.0
0.5
0.5 0.0 0
0.5 t (s)
1
0.0 0
Fig. 8. Time variations of relative darkness and the arrivals of large waves in the steady states: (a) qw = 114 kW/m2 and (b) qw = 178 kW/m2.
100
2
qw = 114 kW/m 2 qw = 178 kW/m
fLW (Hz)
fDW by Sekogichi
50
0 0
50
100 2 GIN (kg/m s)
150
200
Fig. 9. Comparisons of large wave frequencies with disturbance wave frequencies calculated by Sekoguchi’s correlation.
that the time variations of dk in the thin film regions are similar to those in the steady states when the oscillation period is long (TOSC = 10 s). However, the film surface is covered with larger number of interfacial waves when TOSC is short. The effect of TOSC on the wave frequency in the thin film regions fLW,ST is shown in Fig. 12; here, the values used in the steady state experiments of GIN = GMIN were used for the thresholds to determine the arrivals of large waves. It can be confirmed that the frequency of large wave’s passing is fairly constant for sufficiently large values of TOSC while increases with a decrease in TOSC when the oscillation period is short. The dependence of the wave frequency under the flow oscillation conditions on the oscillation period is hence consistent with the results of the CHF measurement reported in the previous work [10] and the film thickness measurement. This result may support the hypothesis that the disturbance waves contribute to transport the liquid from the thick to the thin film regions and to mitigate the reduction of critical heat flux caused by the flow oscillation.
4. Summary and conclusions Time variations of local liquid film thickness and frequencies of large wave’s passing were measured for steam–water annular
two-phase flows near the atmospheric pressure. The diameter of the test section tube was 12 mm and the heated length was 1360 mm; the measurements were conducted 300 mm downstream from the top of the heated section. In order to investigate the role of liquid transport within a liquid film in the occurrence of liquid film dryout under the flow oscillation conditions, the experiments were performed under oscillation conditions as well as in steady states. Although it was considered that the measurement section was located in the developing region of annular two-phase flow, the average and minimum film thicknesses and the wave frequencies measured in the steady states were in qualitative agreements with those calculated by available correlations. This might reflect that the thermal–hydraulic condition in the heated section was not deviated from the equilibrium state significantly since the heat flux applied to the channel wall was rather low. In the experiments in which sinusoidal oscillation was applied to the inlet mass flux, the liquid film thickness oscillated in the same period as the inlet mass flux, although higher-frequency fluctuations associated with the interfacial waves such as ripple and disturbance waves were superposed to the base oscillation. It is expected that, if the heat flux is gradually increased, the film dryout first occurs at the thin film region corresponding to the minimum inlet mass flux. In view of this, the liquid film behavior within the thin film region was explored. It was shown that the mean film thickness in the thin film regions tends to increase with a decrease in the oscillation period. This result was consistent with the dependence of the critical heat flux on the oscillation period reported in the literature. Hence, if the variation of the film thickness with the oscillation period was assumed to be the consequence of the liquid transport from the thick to the thin film regions, an increase in the critical heat flux with a decrease in the oscillation period could be attributed to enhancement of the liquid transport within a liquid film. A strong correlation of the dimensionless heated length defined in our previous study with the mean film thickness in the thin film region indicated that the dimensionless heated length is a promising parameter to characterize the liquid transport to the thin film region. In the present experiments, the wave frequency in the thin film region also increased with a decrease in the oscillation period.
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(a) 0.5 0.3
0.2
0.1
0.1 10
0 0 0.5
15
2
qw=114kW/m ; TOSC=5s
0.4
0.3
dk
dk
0.4
5
0.2
0.1
0.1
0.4
5
10
2
qw=114kW/m ; TOSC=10s
0.4 dk
0.2
0.1
0.1 5
10
5
10
15
10
15
2
qw=178kW/m ; TOSC=10s
0 0
15
15
0.3
0.2
0 0
10
qw=178kW/m ; TOSC=5s
0 0 0.5
15
0.3
5 2
0.3
0.2
0 0 0.5
2
qw=178kW/m ; TOSC=3s
0.3
0.2
0 0 0.5
dk
0.4 dk
dk
0.4
(b) 0.5
2
qw=114kW/m ; TOSC=3s
5
t (s)
t (s)
Fig. 10. Time variations of relative darkness measured under the oscillatory conditions (moving average of 0.2 s was applied): (a) qw = 114 kW/m2 and (b) qw = 178 kW/m2.
(b) 1.0
TOSC=3s
dk
dk
(a) 1.0 0.5
0.0 0 1.0
0.5
1
0.5
0.0 0 1.0
1.5
0.5
0.5
1
0.0 0 1.0
1.5
dk
dk
0.5
1
1.5
0.5
1
0.5
1
1.5
1
1.5
0.5
0.0 0 1.0
1.5
Steady (GIN=GMIN)
Steady (GIN=GMIN)
dk
dk
1.5
TOSC=10s
0.5
0.5
0.0 0
1
0.5
TOSC=10s
0.0 0 1.0
0.5 TOSC=5s
dk
dk
TOSC=5s
0.0 0 1.0
TOSC=3s
0.5
1 t (s)
1.5
0.5
0.0 0
0.5 t (s)
Fig. 11. Time variations of relative darkness in the thin film regions measured under the oscillatory conditions: (a) qw = 114 kW/m2 and (b) qw = 178 kW/m2.
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T. Okawa et al. / International Journal of Heat and Mass Transfer 53 (2010) 962–971
100
tf;AVE
2
fLW,ST (Hz)
qw = 114 kW/m 2 qw = 178 kW/m
sffiffiffiffiffiffiffiffiffi fw ql ð1 EÞjl D fi qg jg
ðA:14Þ
where the entrainment fraction E was assumed to be zero in this work, neglecting the droplet entrainment. The interfacial friction factor fi and the wall friction factor fw are evaluated by the following correlations by Wallis [16]:
50
0 0
1 ¼ 4
5
10 15 TOSC (s)
20
25
fi ¼ 0:005 1 þ 300t f;AVE =D
ðA:15Þ
fw ¼ maxð16=Ref ; 0:005Þ
ðA:16Þ
Hazuku et al. [15] correlated the minimum film thickness tf,MIN by
Fig. 12. Dependences of the wave frequencies in the thin film regions fLW,ST on qw and TOSC.
tf;MIN ¼
l2l =g q2l
1=3
t f;MIN
ðA:17Þ
where It was considered that this observation reflected the contribution of the disturbance waves in transporting the liquid to the thin film regions.
si ¼
Acknowledgement
Appendix A. Correlations for liquid film thickness and wave frequency In Sekoguchi’s correlation [18], the mean film thickness tf,AVE and the wave frequency fDW are calculated by
fDW ¼
si g q2l ql g l2l
ðA:18Þ
1=3 ðA:19Þ
In this work, the interfacial shear stress si in Eq. (A.19) was calculated by
This work was supported by KAKENHI (No. 20360419).
ll þ t u ql f;AVE
ðA:1Þ
jg f1 ðEoÞg 1 ðnÞ D
ðA:2Þ
t f;AVE ¼
tf;MIN ¼ 0:977Re0:143 ðsi Þ0:117 l
where
. 0:1 f1 ðEoÞ t þf;AVE ¼ 0:046fgðnÞ þ 5:5gRe0:4 l M
ðA:3Þ
gðnÞ ¼ 100fg 2 ðnÞ g 1 ðnÞg
ðA:4Þ
g 1 ðnÞ ¼ 0:0076 ln n 0:051
ðA:5Þ
g 2 ðnÞ ¼ 0:0142n0:1
ðA:6Þ
f1 ðEoÞ ¼ Eo0:5 ð0:5 ln Eo 0:47Þ
ðA:7Þ
n ¼ Re2:5 l =Fr g
ðA:8Þ
Rel ¼ ql jl D=ll
ðA:9Þ
M ¼ g l4l =ql r
ðA:10Þ
Eo ¼ gD2 ðql qg Þ=r
ðA:11Þ
pffiffiffiffiffiffi Fr g ¼ jg = gD
ðA:12Þ
u ¼ 0:2ðll =ql Þ1=8 D1=8 ðjg þ jl Þ7=8
ðA:13Þ
In Okawa’s method [9], tf,AVE is evaluated assuming the force balance between the interfacial shear force and the wall friction force acting on a liquid film. The resulting equation is
1 2
si ¼ fi qg j2g
ðA:20Þ
As in Okawa’s method, Eqs. (A.14)–(A.16) were used to estimate fi. References [1] K. Arai, A. Murase, R. Hamazaki, M. Kuroki, AB1600 – progress of ABWR technology toward next generation ABWR, Nucl. Eng. Des. 238 (8) (2008) 1902–1908. [2] M. Aoyama, K. Haikawa, L.E. Fennern, R. Yoshioka, T. Ohta, T. Anegawa, Optimization of core design for the next generation BWR, in: Proceedings of the Fifth International Conference on Nuclear Engineering, Nice, France, 1997, Paper No. ICONE5-2636. [3] T. Okamoto, A. Hotta, T. Ama, K. Mishima, A. Tomiyama, T. Okawa, Y. Kudo, Y. Yamamoto, Y. Takeuchi, M. Chaki, Research on instability design method without occurring boiling transition for hyper ABWR plants of extended core power density, in: Proceedings of the Sixth Japan–Korea Symposium on Nuclear Thermal Hydraulics and Safety, Okinawa, Japan, 2008, Paper No. N6P1122. [4] P. Ferroni, C.S. Handwerk, N.E. Todreas, Steady state thermal–hydraulic analysis of hydride-fueled grid-supported BWRs, Nucl. Eng. Des. 239 (2009) 1544–1559. [5] M. Ozawa, Two-phase flow instabilities, in: S.G. Kandlikar, M. Shoji, V.K. Dhir (Eds.), Handbook of Phase Change: Boiling and Condensation, Taylor & Francis, Philadelphia, PA, 1999, pp. 261–278. [6] M. Ozawa, M. Hirayama, H. Umekawa, Critical heat flux condition induced by flow instabilities in boiling channels, Chem. Eng. Technol. 25 (2002) 1197– 1201. [7] M. Ozawa, H. Umekawa, Y. Yoshioka, A. Tomiyama, Dryout under oscillatory flow condition in vertical and horizontal tubes – experiments at low velocity and pressure conditions, Int. J. Heat Mass Transfer 36 (1993) 4076–4078. [8] H. Umekawa, M. Ozawa, T. Mitsunaga, K. Mishima, T. Hibiki, Y. Saito, Scaling parameters of CHF under oscillatory flow conditions, Heat Transfer – Asian Res. 28 (1999) 541–550. [9] T. Okawa, T. Kitahara, K. Yoshida, T. Matsumoto, I. Kataoka, New entrainment rate correlation in annular two-phase flow applicable to wide range of flow condition, Int. J. Heat Mass Transfer 45 (2002) 87–98. [10] T. Okawa, T. Goto, J. Minamitani, Y. Yamagoe, Liquid film dryout in a boiling channel under flow oscillation conditions, Int. J. Heat Mass Transfer 52 (2009) 3665–3675. [11] N.S. Hall-Taylor, R.M. Nedderman, The coalescence of disturbance waves in annular two phase flow, Chem. Eng. Sci. 23 (1968) 551–564. [12] G.F. Hewitt, N.S. Hall-Taylor, Annular Two-Phase Flow, Pergamon Press, Oxford, 1970. pp. 98–126. [13] B.J. Azzopardi, Drops in annular two-phase flow, Int. J. Multiph. Flow 23 (1997) 1–53. [14] S.W. Churchill, H.H.S. Chu, Correlating equations for laminar and turbulent free convection from a vertical plate, Int. J. Heat Mass Transfer 18 (1975) 1323– 1329.
T. Okawa et al. / International Journal of Heat and Mass Transfer 53 (2010) 962–971 [15] T. Hazuku, T. Takamasa, Y. Matsumoto, Experimental study on axial development of liquid film in vertical upward annular two-phase flow, Int. J. Multiph. Flow 34 (2008) 111–127. [16] G.B. Wallis, One-Dimensional Two-Phase Flow, McGraw-Hill, New York, 1969. pp. 315–374. [17] K. Sekoguchi, O. Tanaka, T. Ueno, T. Furukawa, S. Esaki, M. Nakasatomi, An investigation of the flow characteristics in the disturbance wave region of
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annular flow: 1st report, effect of tube diameter, Bull. JSME 26 (1983) 1719– 1726. [18] K. Sekoguchi, T. Ueno, O. Tanaka, An investigation of the flow characteristics in the disturbance wave region of annular flow: 2nd report, on correlation of principal flow parameters, Trans. JSME B 51 (1985) 1798–1806. in Japanese.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Liquid film behavior in annular two-phase flow under flow oscillation conditions Tomio Okawa *, Taisuke Goto, Yosuke Yamagoe Department of Mechanical Engineering, Osaka University 2-1, Yamadaoka, Suita-shi, Osaka 565-0871, Japan
a r t i c l e
i n f o
Article history: Received 3 July 2009 Received in revised form 8 October 2009 Accepted 30 October 2009 Available online 1 December 2009 Keywords: Annular two-phase flow Flow oscillation Liquid film Disturbance wave Dryout
a b s t r a c t Experiments were carried out to investigate the effects of sinusoidal forced oscillation of the inlet flow rate on the time variations of local liquid film thickness and the frequencies of large wave’s passing in steam–water annular two-phase flows. The liquid film thickness oscillated with the same period as the inlet flow rate. The mean film thickness in the thin film regions decreased and approached to an asymptotic value with an increase in the oscillation period of the inlet flow rate. This result was consistent with the experimental results of the occurrence of liquid film dryout under flow oscillation conditions reported in the literature. It was hence considered that the axial liquid transport from the thick to thin film regions mitigates the reduction of the critical heat flux caused by the flow oscillation. It was also found that the wave frequency in the thin film region increased with a decrease in the oscillation period. This observation suggested that the disturbance waves contribute to the enhancements of the liquid transport and consequently the critical heat flux associated with the liquid film dryout under flow oscillation conditions. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The use of high power density core enabling reductions of capital and maintenance costs is one of the most profitable ways to develop innovative boiling water reactors (BWRs) of higher economic efficiency. Various high power density BWR designs were hence proposed by many research and development teams [1–4]. It is, however, known that increased power density generally leads to the reduction of the margin to the onset of unanticipated flow instability in boiling two-phase flow systems [5]. Therefore, to ensure the safety of high power density BWRs, it is necessary to predict the occurrence of boiling transition with a high degree of accuracy even when the coolant flow rate becomes oscillatory. Extensive experimental studies were performed to clarify the effects of flow instability on critical heat flux in a boiling channel as summarized in the review article by Ozawa et al. [6]. These studies indicated that destabilization of the flow rate results in a noticeable decrease in the critical heat flux. However, in the experiments performed under unstable conditions, it is difficult to systematically investigate the effects of the amplitude and frequency of flow oscillation since these parameters are largely dependent on the geometries of the experimental apparatus. Ozawa et al. [7] and Umekawa et al. [8] therefore imposed forced oscillation on the inlet flow rate, enabling almost-arbitrary control of the amplitude and frequency. It was shown clearly in these exper-
* Corresponding author. Tel.: +81 6 6879 7257; fax: +81 6 6879 7247. E-mail address: [email protected] (T. Okawa). 0017-9310/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2009.11.024
iments that an increase in the amplitude and a decrease in the frequency lead to a reduction in the critical heat flux. Based on a one-dimensional three-fluid model [9], Okawa et al. carried out numerical simulations to calculate the critical heat flux associated with the liquid film dryout in annular two-phase flow regime [10]. In the simulations, the following sinusoidal oscillation was applied to the inlet mass flux GIN:
GIN ¼ GAVE þ DG sin
2p t T OSC
ð1Þ
where GAVE is the time-averaged inlet mass flux, DG is the oscillation amplitude, TOSC is the oscillation period and t is the time. They introduced the following dimensionless critical heat flux q* and the dimensionless heated length L* to investigate the influence of the flow oscillation on the critical heat flux:
q ¼
qOSC qMIN qAVE qMIN
ð2Þ
L ¼
2L uf0 T OSC
ð3Þ
where qOSC is the critical heat flux under the flow oscillation condition, qAVE and qMIN are the critical heat fluxes in the steady states when GIN is equal to GAVE and GMIN (= GAVE DG), respectively; L is the heated length, and uf0 is the characteristic film velocity. It was shown that the calculated values of q* were correlated in terms of L* well, and q* increased from 0 to 1 with increased value of L*. This result was interpreted that qOSC should be equal to qMIN (q* = 0) for small values of L* since the axial liquid transport to
T. Okawa et al. / International Journal of Heat and Mass Transfer 53 (2010) 962–971
963
Nomenclature D dk E Eo Fr f fi fw G g j L M q q+ P Re T t tf uf0 u* x
tube diameter (m) image darkness entrainment fraction Eötvös number Froude number frequency (Hz) interfacial friction factor wall friction factor mass flux (kg/m2 s) gravitational acceleration (m/s2) superficial velocity (m/s) heated length (m) Morton number heat flux (W/m2) ratio of wall heat flux to CHF pressure (Pa) Reynolds number period (s) or temperature (K) time (s) liquid film thickness (m) characteristic film velocity (m/s) friction velocity (m/s) vapor quality
q r si
density (kg/m3) surface tension coefficient (N/m) interfacial shear stress (N/m2)
Superscripts dimensionless + dimensionless Subscripts AVE average DW disturbance wave EX exit f liquid film g gas phase IN inlet LW large wave l liquid phase MAX maximum MIN minimum OSC oscillation ST short time STA short-time average m vapor w wall
Greek symbols DG oscillation amplitude of mass flux (kg/m2 s) l viscosity (Pas)
the thinnest film region corresponding to GMIN was negligible, and it increased asymptotically to qAVE (q* ? 1) with increased value of L* since the liquid supply from other regions within a liquid film to the thinnest film region became noticeable and consequently the influence of the oscillation of the inlet mass flux was mitigated at the outlet of the heated section. Okawa et al. [10] then measured the critical heat flux under flow oscillation conditions in order to confirm the validity of the numerical results. It was found that q* could be correlated by a monotonically increasing function of L* as in the numerical simulations, but the experimental critical heat fluxes were generally higher than those calculated using the three-fluid model. This would indicate that available numerical models are satisfactory for conservative estimation of the critical heat flux, but the present understanding of the dynamic behavior of a liquid film is not sufficient for accurate prediction of the critical heat flux under flow oscillation conditions. Since the main objective of the experiments performed by Okawa et al. [10] was to elucidate the effect of the flow oscillation on the critical heat flux, particular attention was not paid to the dynamic behavior of a liquid film. Therefore, in the first half of this work, time variation of the local liquid film thickness is measured to confirm that the liquid transport to the thin film regions is enhanced with an increase in the dimensionless heated length L*. It is known that the surface of a liquid film in annular two-phase flow is commonly covered with complex interfacial waves such as ripple and disturbance waves. In particular, the disturbance waves that occur at high liquid flow rates have a height several times the mean film thickness, move at a velocity greater than the mean film velocity, and retain their identity throughout the flow channel [11–13]. It may hence be expected that the disturbance waves have a major role in transporting the liquid to the thin film regions. In view of this, in the second half of this work, wave frequency is measured to investigate the possibility for the disturbance waves to contribute to the enhancements of the liquid transport and the critical heat flux at large values of L*.
2. Experimental description 2.1. Experimental apparatus and measurement method Experiments were carried out with the experimental apparatus that is shown schematically in Fig. 1 and was described in detail in our previous study [10]. Filtrated and deionized tap water was used as a working fluid, and was circulated using a multi-stage pump. Two needle valves and a turbine flow meter were used to control and measure the inlet mass flux GIN; the flow meter was accurate to within ±0.5%. The inlet flow rate was oscillated by changing the electric power applied to the circulation pump sinusoidally. A 15-kW plug heater was used as a pre-heater to raise and then maintain the water temperature at the inlet of the heated section TIN. After exiting the pre-heating section, the water entered a vertical round tube of 12 mm in internal diameter and 0.8 mm in thickness made of SUS316 stainless-steel. A 60-kW (60 V 1000 A) DC power supply was used to heat the tube ohmically and generate steam–water annular flow inside it. The heated length L was 1360 mm. The transparent section was equipped downstream of the heated section for flow visualization and film thickness measurement. A silica glass cube 40 mm on a side, through which a 12 mm diameter circular hole was punctured as a flow channel, was used for this section. To minimize the effect of the connection between the stainless-steel and glass sections on the behavior of a liquid film, these components were manufactured with an accuracy of 10 lm. The step height at the connection was estimated around 5 lm under the temperature condition during the experiment. The distance from the top of the heated section to the center of the transparent section was 300 mm (25 tube diameters). The top of the vertical test section was connected with the upper tank in order to separate the vapor phase from the liquid phase. The liquid phase was then returned to the circulation pump, while the vapor phase was condensed in the separator or released into the atmosphere.
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T. Okawa et al. / International Journal of Heat and Mass Transfer 53 (2010) 962–971
shutter speed set to 1 ms was also performed at the transparent section to measure the frequency of large wave’s passing. Each image consisted of 256 512 pixels and the spatial resolution was approximately 50 lm/pixel. Since the displacement meter and the high-speed camera could not be placed on a measurement stage simultaneously, the flow visualization was performed separately from the film thickness measurement. The transparent section was backlit using a metal halide lamp in the visualization experiments. 2.2. Experimental conditions Main experimental conditions in the steady states and flow oscillation conditions are listed in Tables 1 and 2, respectively. The liquid film thickness and the wave frequency were measured under 10 steady states and 10 flow oscillation conditions. In all the experiments, the outlet pressure PEX and the inlet fluid temperature TIN were set to 129 ± 8 kPa and 374 ± 2 K, respectively. In the steadystate experiments, the inlet mass flux GIN and the wall heat flux qw were used as the parameters and changed within 54–162 kg/m2 s and 114–179 kW/m2, respectively. The resulting range of the equilibrium vapor quality at the channel exit (xEX) was 0.13–0.66. Under the flow oscillation conditions, the time-averaged inlet mass flux GAVE and the oscillation amplitude DG were set to 104 ± 3 kg/m2 s and 49 ± 7 kg/m2 s, respectively, and the oscillation period TOSC and qw were used as the experimental parameters. The ranges of TOSC and qw were 2–20 s and 114–178 kW/m2, respectively. The ratios of the wall heat flux to the critical heat flux (q+) are also shown in Tables 1 and 2. In the calculation of q+, the critical heat flux in each experimental condition was estimated from the CHF values measured using the same experimental facility in our previous work [10]. Table 2 indicates that the maximum heat flux of 178 kW/m2 tested in the
Table 1 Main experimental conditions in steady states (the values on the left and right sides of the slash marks are the experimental conditions in film thickness measurements and wave frequency measurements, respectively). Fig. 1. Schematic diagram of experimental apparatus.
Type-K thermocouples accurate to within ±1.5 K were used to measure the fluid temperatures at the inlet and outlet of the preheating section and the outer wall temperatures of the heated section. The pressures at the inlet and the outlet of the heated section were measured with pressure transducers accurate to within ±7.5 kPa. The heat flux applied to the fluid qw was calculated from the electric current passing through the heated section and the electrical resistance of the tube material. The heated section was not thermally insulated, but heat loss to the ambient air was estimated as less than 1 kW/m2 from calculation using the heat transfer equation proposed by Churchill and Chu [14]. At the transparent section, the local film thickness was measured with a laser focus displacement meter, in which the method to focus on a target adopted in an automatically focusing camera was applied to the displacement measurement [15]; the diameter of the laser beam spot was 7 lm, the spatial resolution was 0.1 lm and the linearity error was within ±3 lm. The effect of the non-uniformity of refractive index within the liquid film was neglected in the film thickness measurement since a heat conduction calculation showed that the temperature distribution within a liquid film was fairly uniform at the transparent section (300 mm downstream from the top of the heated section). The experimental data were recorded every 1 ms using a data acquisition system; the measurement time was 30 s in each experimental condition. Flow visualization using a high-speed camera at 1000 frames/s with
No.
PEX (kPa)
TIN (K)
GIN (kg/m2 s)
qw (kW/m2)
xEX (%)
q+ (%)
111 112 113 114 115 121 122 123 124 125
121/124 123/126 125/127 126/129 128/130 124/127 127/130 130/133 133/135 135/137
375/374 375/374 375/373 374/373 375/374 375/374 375/373 375/373 375/374 373/375
54/54 81/82 107/109 135/135 161/162 54/55 80/81 108/108 134/134 161/162
114/114 114/114 114/114 114/114 114/114 178/178 178/178 178/178 178/178 178/179
42/41 28/27 21/20 16/16 13/13 66/65 44/43 33/32 26/26 21/21
49/49 31/31 24/24 20/20 18/17 76/74 50/49 37/37 31/31 27/27
Table 2 Main experimental conditions under flow oscillation conditions (the values on the left and right sides of the slash marks are the experimental conditions in film thickness measurements and wave frequency measurements, respectively). No.
PEX (kPa)
TIN (K)
GAVE (kg/m2 s)
DG (kg/m2 s)
TOSC (s)
qw (kW/m2)
q+ (%)
211 212 213 214 215 221 222 223 224 225
124/127 124/127 124/127 124/127 124/127 129/131 130/132 130/132 129/132 129/132
375/374 376/375 375/374 375/373 375/374 372/374 374/374 375/375 374/374 374/374
106/105 107/107 105/107 104/105 106/106 102/101 106/106 106/106 103/106 105/105
43/42 54/54 54/56 53/53 55/55 45/44 54/54 55/55 52/56 56/54
2/2 3/3 5/5 10/10 20/20 2/2 3/3 5/5 10/10 20/20
114/114 114/114 114/114 114/114 114/114 178/178 178/178 178/178 178/178 178/178
29/29 33/33 39/38 44/43 47/47 46/46 52/52 60/60 69/70 77/74
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T. Okawa et al. / International Journal of Heat and Mass Transfer 53 (2010) 962–971
2
(b) 1.0
2
qw=114kW/m , GIN=54kg/m s
tf (mm)
tf (mm)
(a) 1.0 0.5
0.0 0 1.0
0.5
1 2
1.5
2
0.5
0.0 0 1.0
2
2
0.5
0.5
1 2
1.5
0.0 0 1.0
2
2
tf (mm)
tf (mm)
1 t (s)
1.5
2
1.5
2
1.5
2
2
0.5
1 2
2
qw=178kW/m , GIN=161kg/m s
0.5
0.5
1 2
0.5
qw=114kW/m , GIN=161kg/m s
0.0 0
0.5
qw=178kW/m , GIN=108kg/m s
tf (mm)
tf (mm)
qw=114kW/m , GIN=107kg/m s
0.0 0 1.0
2
qw=178kW/m , GIN=54kg/m s
1.5
2
0.5
0.0 0
0.5
1 t (s)
Fig. 2. Time variations of local film thickness measured in the steady states: (a) qw = 114 kW/m2 and (b) qw = 178 kW/m2.
3.1. Liquid film thickness Typical time variations of the local film thickness tf measured in the steady states are shown Fig. 2a and b. It can be seen that the fluctuation of tf becomes significant with an increase in GIN and a decrease in qw since the mean film thickness increases [16]. The data of average, minimum, and maximum film thicknesses for different values of qw are plotted against GIN in Fig. 3a and b; here, as done in the previous studies [15,17], the film thicknesses at which the cumulative distribution function was equal to 0.01 and 0.99 were defined as the minimum and maximum film thicknesses, respectively. It can be seen that the maximum thickness tf,MAX and the average thickness tf,AVE tend to increase with an increase in GIN and a decrease in qw, while the dependences of the minimum thickness tf,MIN on these parameters are less noticeable. These tendencies are consistent with those for air–water annular flows reported by Hazuku et al. [15]. The mean film thicknesses calculated by Sekoguchi’s correlation [18] and Okawa’s method [9] and the minimum film thickness calculated by Hazuku’s correlation [15] are displayed in the same figures (for details of the correlations, see Appendix A). Here, the droplet entrainment was neglected in Okawa’s method and the interfacial shear stress acting on a liquid film required in Hazuku’s correlation was evaluated using the method adopted by Okawa et al. [9]. Although it is considered that the present measurements were conducted for the developing region of the steam–water annular flow (the distance of the measuring section from the top of the heated section was 25 tube diameters), the calculated film thicknesses are in fairly good agreement with the present experimental data. This would suggest that the thermal–hydraulic field formed in the heated section was not deviated from the equilibrium state significantly in the present experiments since the values of mass flux and heat flux were rather low.
tf (mm)
3. Results and discussion
(a) 10 10
1 2
qw = 114 kW/m 0
10
–1
10
–2
tf,AVE by Sekogichi tf,AVE by Okawa tf,MIN by Hazuku
tf,MAX tf,AVE tf,MIN
0
50
(b) 101
100 2 GIN (kg/m s)
2
qw = 178 kW/m
tf (mm)
flow oscillation experiments corresponded to 46–77% of the critical heat flux, depending mainly on the oscillation period.
10
0
10
–1
10
–2
0
200
tf,AVE by Sekogichi tf,AVE by Okawa tf,MIN by Hazuku
tf,MAX tf,AVE tf,MIN
50
150
100
150
200
2
GIN (kg/m s) Fig. 3. Average, minimum and maximum film thicknesses in the steady states measured at (a) qw = 114 kW/m2 and (b) qw = 178 kW/m2.
Depicted in Figs. 4 and 5 are the typical time variations of GIN and tf measured under the flow oscillation conditions. Here, the moving average of 0.2 s was applied to the film thickness data to highlight long-term variations corresponding to the flow oscillation. It is found that high-frequency waves associated with the ripple and disturbance waves are superposed to the low-frequency wave in each experimental condition. Eq. (3) indicates that L* is inversely proportional to TOSC. Therefore, if the liquid transport to the
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T. Okawa et al. / International Journal of Heat and Mass Transfer 53 (2010) 962–971
(a)
250 200
150 100
150 100 50
50 0 0 0.3
5
10
15
0 0 0.3
20
2
tf (mm)
tf (mm)
0.2 0.1 0.0 0
5
10
15
20
15
20
15
20
15
20
15
20
15
20
2
(b)
2
qw=114kW/m ; TOSC=5s
0.2 0.1 0.0 0
20
5
250 200
10 t (s)
2
qw=178kW/m ; TOSC=5s
2
GIN (kg/m s)
200
15
2
GIN (kg/m s)
250
10 t (s)
5
qw=178kW/m ; TOSC=3s
qw=114kW/m ; TOSC=3s
(b)
2
qw=178kW/m ; TOSC=3s
2
GIN (kg/m s)
200
(a)
2
qw=114kW/m ; TOSC=3s
2
GIN (kg/m s)
250
150 100 50 0 0 0.3
5
10
15
150 100 50 0 0 0.3
20
2
tf (mm)
0.2
tf (mm)
5
10 2
qw=178kW/m ; TOSC=5s
qw=114kW/m ; TOSC=5s
0.1
0.2 0.1
Periods for shorter-time average
0.0 0
(c)
5
0.0 0
20
(c)
2
qw=114kW/m ; TOSC=10s
5
250 200
10 t (s) 2
qw=178kW/m ; TOSC=10s
2
GIN (kg/m s)
200
15
2
GIN (kg/m s)
250
10 t (s)
150 100
150 100
50 0 0 0.3
5
10
15
50 0 0 0.3
20
2
tf (mm)
tf (mm)
0.2 0.1
5
10 t (s)
10 2
qw=178kW/m ; TOSC=10s
qw=114kW/m ; TOSC=10s
0.0 0
5
15
0.2 0.1 0.0 0
20
5
10 t (s)
Fig. 4. Time variations of inlet mass flux and local film thickness under oscillatory conditions at qw = 114 kW/m2: (a) TOSC = 3 s and (b) TOSC = 5 s, and (c) TOSC = 10 s.
Fig. 5. Time variations of inlet mass flux and local film thickness under oscillatory conditions at qw = 178 kW/m2: (a) TOSC = 3 s, (b) TOSC = 5 s, and (c) TOSC = 10 s.
thin film region is enhanced with an increase in L* as supposed in our previous study [10], it is expected that the mean film thickness in the thin film regions increases with a decrease in TOSC. In the present study, 10% of each oscillation period for which the shorter-time average of the film thickness was minimized was regarded as corresponding to the thin film region, as indicated schematically in Fig. 4b. The data of the mean film thickness in the thin film regions tf,STA are presented in Fig. 6a. It can be seen that tf,STA decreases with an increase in TOSC and approaches to asymptotic values. This result could be interpreted to represent that the axial
liquid transport to the thin film regions is negligibly small when the oscillation period is long, while the mean film thickness in the thin film regions increases due to the liquid transport from the thicker film regions for small values of TOSC. In order to quantify the dependence on L*, the same data of tf,STA are plotted against L* in Fig. 6b. Here, uf0 in Eq. (3) was estimated using the following equation [10]:
GAVE uf0 ¼ pffiffiffiffiffiffiffiffiffiffi
qf qv
DG GAVE
0:2 ð4Þ
T. Okawa et al. / International Journal of Heat and Mass Transfer 53 (2010) 962–971
(a) 0.14
frequency fLW from the time variation of the image brightness. It is noted that, although the film thickness data shown in the previous section can also be used for the measurement of fLW, it was decided to measure fLW from the brightness data in this work since the arrival of large waves could be verified in the video images. The size and position of the measuring window are shown schematically in the first image in Fig. 7. Since the distribution of brightness in the wave region was not uniform, the window height was set to the value in the same order with the typical wave length. The time variations of the space-averaged darkness within the measuring window dk measured in the steady states are presented in Fig. 8a and b. Since the range of brightness (gray level) was 0 (black)–255 (white), dk was calculated by
2
tf,STA (mm)
qw = 114 kW/m 2 qw = 178 kW/m
0.12
0.1
0.08 0
5
(b)
20
25
dk ¼ 1
2
qw = 114 kW/m 2 qw = 178 kW/m
1
tf,STA*
10 15 TOSC (s)
0.5
0 10
–2
–1
10 * L
10
0
Fig. 6. Mean film thicknesses in the thin film regions under flow oscillation conditions: (a) dependences on TOSC and (b) dependences of tf;STA on L*.
where qf and qv are the densities of liquid film and vapor phase, respectively, and tf,STA was scaled by
t f;STA ¼
tf;STA t f;AVE ðGMIN Þ t f;AVE ðGAVE Þ tf;AVE ðGMIN Þ
967
ð5Þ
where tf,AVE(G) is the mean film thickness in the steady state when the inlet mass flux is set to G. It is found that t f;STA is correlated fairly well as a monotonically increasing function of L* within the experimental conditions tested in this work. This result indicates that L* is a promising dimensionless parameter to characterize the liquid transport to the thin film regions under flow oscillation conditions. 3.2. Wave frequency Typical sequential images of large waves passing through the transparent section are depicted in Fig. 7. It is found that the brightness level is low in the wave region, reflecting that the surface of a large wave is highly perturbed. Hence, the rectangular measuring window was located in the images to measure the wave
N 1 X bi 255N i¼1
ð6Þ
where bi is the brightness of the ith pixel and N is the number of pixels contained in the measuring window (the value of dk is equal to 0 and 1 for totally white and black windows, respectively). In Fig. 8a and b, the circular symbols are used to indicate the waves that were classified as large waves. Since the film thickness increases sharply downstream of a large wave [17], it was assumed that large waves arrived at the measuring section when the time derivative of the mean darkness exceeded the threshold value. The maximum time derivative was calculated in the same manner as tf,MAX in each experimental condition and the half of the maximum value was used for the threshold. The frequencies of large waves fLW measured using the above-described method were compared with the frequencies of disturbance waves fDW calculated by Sekoguchi’s correlation [18] in Fig. 9 (details of the correlation is found in Appendix A). It can be seen that fLW measured in the present experiments increases with increased values of GIN and qw and agrees with the values of fDW calculated by Sekoguchi’s correlation fairly well. It should, however, be noted that Sekoguchi et al. [18] used more precise criteria for determining the disturbance wave. The waves indicated with circular symbols in Fig. 8 are hence identified just as large waves in this paper. The time variations of dk under the oscillatory conditions are depicted in Fig. 10a and b; here, the moving average of 0.2 s was applied to the raw data. It can be confirmed that the high-frequency variations of dk associated with the interfacial waves such as disturbance and ripple waves are superposed to the low-frequency variation. Since dk tended to increase with increased value of GIN in the steady state experiments, 10% of each oscillation period for which the shorter-time average of dk was minimized was regarded as corresponding to the thin film region. The time variations of dk in the thin film regions are displayed in Fig. 11a and b. At the bottom of these figures, the data for the steady states when GIN was set to GMIN were also presented for comparisons. It can be seen
Fig. 7. Typical time-elapsed images of large waves passing through a transparent section (qw = 178 kW/m2, GIN = 81 kg/m2 s).
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T. Okawa et al. / International Journal of Heat and Mass Transfer 53 (2010) 962–971
(a) 1.5
GIN=54kg/m
(b) 1.5
2
classified as large wave
2
GIN=55kg/m
classified as large wave
1.0 dk
dk
1.0
0.5
0.5 0.0 0 1.5
0.5
0.0 0 1.5
1
2
GIN=109kg/m
0.5
1
0.5
1
0.5 t (s)
1
2
GIN=108kg/m
1.0 dk
dk
1.0
0.5
0.5 0.0 0 1.5
0.5
0.0 0 1.5
1
2
GIN=162kg/m
2
GIN=162kg/m
1.0 dk
dk
1.0
0.5
0.5 0.0 0
0.5 t (s)
1
0.0 0
Fig. 8. Time variations of relative darkness and the arrivals of large waves in the steady states: (a) qw = 114 kW/m2 and (b) qw = 178 kW/m2.
100
2
qw = 114 kW/m 2 qw = 178 kW/m
fLW (Hz)
fDW by Sekogichi
50
0 0
50
100 2 GIN (kg/m s)
150
200
Fig. 9. Comparisons of large wave frequencies with disturbance wave frequencies calculated by Sekoguchi’s correlation.
that the time variations of dk in the thin film regions are similar to those in the steady states when the oscillation period is long (TOSC = 10 s). However, the film surface is covered with larger number of interfacial waves when TOSC is short. The effect of TOSC on the wave frequency in the thin film regions fLW,ST is shown in Fig. 12; here, the values used in the steady state experiments of GIN = GMIN were used for the thresholds to determine the arrivals of large waves. It can be confirmed that the frequency of large wave’s passing is fairly constant for sufficiently large values of TOSC while increases with a decrease in TOSC when the oscillation period is short. The dependence of the wave frequency under the flow oscillation conditions on the oscillation period is hence consistent with the results of the CHF measurement reported in the previous work [10] and the film thickness measurement. This result may support the hypothesis that the disturbance waves contribute to transport the liquid from the thick to the thin film regions and to mitigate the reduction of critical heat flux caused by the flow oscillation.
4. Summary and conclusions Time variations of local liquid film thickness and frequencies of large wave’s passing were measured for steam–water annular
two-phase flows near the atmospheric pressure. The diameter of the test section tube was 12 mm and the heated length was 1360 mm; the measurements were conducted 300 mm downstream from the top of the heated section. In order to investigate the role of liquid transport within a liquid film in the occurrence of liquid film dryout under the flow oscillation conditions, the experiments were performed under oscillation conditions as well as in steady states. Although it was considered that the measurement section was located in the developing region of annular two-phase flow, the average and minimum film thicknesses and the wave frequencies measured in the steady states were in qualitative agreements with those calculated by available correlations. This might reflect that the thermal–hydraulic condition in the heated section was not deviated from the equilibrium state significantly since the heat flux applied to the channel wall was rather low. In the experiments in which sinusoidal oscillation was applied to the inlet mass flux, the liquid film thickness oscillated in the same period as the inlet mass flux, although higher-frequency fluctuations associated with the interfacial waves such as ripple and disturbance waves were superposed to the base oscillation. It is expected that, if the heat flux is gradually increased, the film dryout first occurs at the thin film region corresponding to the minimum inlet mass flux. In view of this, the liquid film behavior within the thin film region was explored. It was shown that the mean film thickness in the thin film regions tends to increase with a decrease in the oscillation period. This result was consistent with the dependence of the critical heat flux on the oscillation period reported in the literature. Hence, if the variation of the film thickness with the oscillation period was assumed to be the consequence of the liquid transport from the thick to the thin film regions, an increase in the critical heat flux with a decrease in the oscillation period could be attributed to enhancement of the liquid transport within a liquid film. A strong correlation of the dimensionless heated length defined in our previous study with the mean film thickness in the thin film region indicated that the dimensionless heated length is a promising parameter to characterize the liquid transport to the thin film region. In the present experiments, the wave frequency in the thin film region also increased with a decrease in the oscillation period.
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T. Okawa et al. / International Journal of Heat and Mass Transfer 53 (2010) 962–971
(a) 0.5 0.3
0.2
0.1
0.1 10
0 0 0.5
15
2
qw=114kW/m ; TOSC=5s
0.4
0.3
dk
dk
0.4
5
0.2
0.1
0.1
0.4
5
10
2
qw=114kW/m ; TOSC=10s
0.4 dk
0.2
0.1
0.1 5
10
5
10
15
10
15
2
qw=178kW/m ; TOSC=10s
0 0
15
15
0.3
0.2
0 0
10
qw=178kW/m ; TOSC=5s
0 0 0.5
15
0.3
5 2
0.3
0.2
0 0 0.5
2
qw=178kW/m ; TOSC=3s
0.3
0.2
0 0 0.5
dk
0.4 dk
dk
0.4
(b) 0.5
2
qw=114kW/m ; TOSC=3s
5
t (s)
t (s)
Fig. 10. Time variations of relative darkness measured under the oscillatory conditions (moving average of 0.2 s was applied): (a) qw = 114 kW/m2 and (b) qw = 178 kW/m2.
(b) 1.0
TOSC=3s
dk
dk
(a) 1.0 0.5
0.0 0 1.0
0.5
1
0.5
0.0 0 1.0
1.5
0.5
0.5
1
0.0 0 1.0
1.5
dk
dk
0.5
1
1.5
0.5
1
0.5
1
1.5
1
1.5
0.5
0.0 0 1.0
1.5
Steady (GIN=GMIN)
Steady (GIN=GMIN)
dk
dk
1.5
TOSC=10s
0.5
0.5
0.0 0
1
0.5
TOSC=10s
0.0 0 1.0
0.5 TOSC=5s
dk
dk
TOSC=5s
0.0 0 1.0
TOSC=3s
0.5
1 t (s)
1.5
0.5
0.0 0
0.5 t (s)
Fig. 11. Time variations of relative darkness in the thin film regions measured under the oscillatory conditions: (a) qw = 114 kW/m2 and (b) qw = 178 kW/m2.
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T. Okawa et al. / International Journal of Heat and Mass Transfer 53 (2010) 962–971
100
tf;AVE
2
fLW,ST (Hz)
qw = 114 kW/m 2 qw = 178 kW/m
sffiffiffiffiffiffiffiffiffi fw ql ð1 EÞjl D fi qg jg
ðA:14Þ
where the entrainment fraction E was assumed to be zero in this work, neglecting the droplet entrainment. The interfacial friction factor fi and the wall friction factor fw are evaluated by the following correlations by Wallis [16]:
50
0 0
1 ¼ 4
5
10 15 TOSC (s)
20
25
fi ¼ 0:005 1 þ 300t f;AVE =D
ðA:15Þ
fw ¼ maxð16=Ref ; 0:005Þ
ðA:16Þ
Hazuku et al. [15] correlated the minimum film thickness tf,MIN by
Fig. 12. Dependences of the wave frequencies in the thin film regions fLW,ST on qw and TOSC.
tf;MIN ¼
l2l =g q2l
1=3
t f;MIN
ðA:17Þ
where It was considered that this observation reflected the contribution of the disturbance waves in transporting the liquid to the thin film regions.
si ¼
Acknowledgement
Appendix A. Correlations for liquid film thickness and wave frequency In Sekoguchi’s correlation [18], the mean film thickness tf,AVE and the wave frequency fDW are calculated by
fDW ¼
si g q2l ql g l2l
ðA:18Þ
1=3 ðA:19Þ
In this work, the interfacial shear stress si in Eq. (A.19) was calculated by
This work was supported by KAKENHI (No. 20360419).
ll þ t u ql f;AVE
ðA:1Þ
jg f1 ðEoÞg 1 ðnÞ D
ðA:2Þ
t f;AVE ¼
tf;MIN ¼ 0:977Re0:143 ðsi Þ0:117 l
where
. 0:1 f1 ðEoÞ t þf;AVE ¼ 0:046fgðnÞ þ 5:5gRe0:4 l M
ðA:3Þ
gðnÞ ¼ 100fg 2 ðnÞ g 1 ðnÞg
ðA:4Þ
g 1 ðnÞ ¼ 0:0076 ln n 0:051
ðA:5Þ
g 2 ðnÞ ¼ 0:0142n0:1
ðA:6Þ
f1 ðEoÞ ¼ Eo0:5 ð0:5 ln Eo 0:47Þ
ðA:7Þ
n ¼ Re2:5 l =Fr g
ðA:8Þ
Rel ¼ ql jl D=ll
ðA:9Þ
M ¼ g l4l =ql r
ðA:10Þ
Eo ¼ gD2 ðql qg Þ=r
ðA:11Þ
pffiffiffiffiffiffi Fr g ¼ jg = gD
ðA:12Þ
u ¼ 0:2ðll =ql Þ1=8 D1=8 ðjg þ jl Þ7=8
ðA:13Þ
In Okawa’s method [9], tf,AVE is evaluated assuming the force balance between the interfacial shear force and the wall friction force acting on a liquid film. The resulting equation is
1 2
si ¼ fi qg j2g
ðA:20Þ
As in Okawa’s method, Eqs. (A.14)–(A.16) were used to estimate fi. References [1] K. Arai, A. Murase, R. Hamazaki, M. Kuroki, AB1600 – progress of ABWR technology toward next generation ABWR, Nucl. Eng. Des. 238 (8) (2008) 1902–1908. [2] M. Aoyama, K. Haikawa, L.E. Fennern, R. Yoshioka, T. Ohta, T. Anegawa, Optimization of core design for the next generation BWR, in: Proceedings of the Fifth International Conference on Nuclear Engineering, Nice, France, 1997, Paper No. ICONE5-2636. [3] T. Okamoto, A. Hotta, T. Ama, K. Mishima, A. Tomiyama, T. Okawa, Y. Kudo, Y. Yamamoto, Y. Takeuchi, M. Chaki, Research on instability design method without occurring boiling transition for hyper ABWR plants of extended core power density, in: Proceedings of the Sixth Japan–Korea Symposium on Nuclear Thermal Hydraulics and Safety, Okinawa, Japan, 2008, Paper No. N6P1122. [4] P. Ferroni, C.S. Handwerk, N.E. Todreas, Steady state thermal–hydraulic analysis of hydride-fueled grid-supported BWRs, Nucl. Eng. Des. 239 (2009) 1544–1559. [5] M. Ozawa, Two-phase flow instabilities, in: S.G. Kandlikar, M. Shoji, V.K. Dhir (Eds.), Handbook of Phase Change: Boiling and Condensation, Taylor & Francis, Philadelphia, PA, 1999, pp. 261–278. [6] M. Ozawa, M. Hirayama, H. Umekawa, Critical heat flux condition induced by flow instabilities in boiling channels, Chem. Eng. Technol. 25 (2002) 1197– 1201. [7] M. Ozawa, H. Umekawa, Y. Yoshioka, A. Tomiyama, Dryout under oscillatory flow condition in vertical and horizontal tubes – experiments at low velocity and pressure conditions, Int. J. Heat Mass Transfer 36 (1993) 4076–4078. [8] H. Umekawa, M. Ozawa, T. Mitsunaga, K. Mishima, T. Hibiki, Y. Saito, Scaling parameters of CHF under oscillatory flow conditions, Heat Transfer – Asian Res. 28 (1999) 541–550. [9] T. Okawa, T. Kitahara, K. Yoshida, T. Matsumoto, I. Kataoka, New entrainment rate correlation in annular two-phase flow applicable to wide range of flow condition, Int. J. Heat Mass Transfer 45 (2002) 87–98. [10] T. Okawa, T. Goto, J. Minamitani, Y. Yamagoe, Liquid film dryout in a boiling channel under flow oscillation conditions, Int. J. Heat Mass Transfer 52 (2009) 3665–3675. [11] N.S. Hall-Taylor, R.M. Nedderman, The coalescence of disturbance waves in annular two phase flow, Chem. Eng. Sci. 23 (1968) 551–564. [12] G.F. Hewitt, N.S. Hall-Taylor, Annular Two-Phase Flow, Pergamon Press, Oxford, 1970. pp. 98–126. [13] B.J. Azzopardi, Drops in annular two-phase flow, Int. J. Multiph. Flow 23 (1997) 1–53. [14] S.W. Churchill, H.H.S. Chu, Correlating equations for laminar and turbulent free convection from a vertical plate, Int. J. Heat Mass Transfer 18 (1975) 1323– 1329.
T. Okawa et al. / International Journal of Heat and Mass Transfer 53 (2010) 962–971 [15] T. Hazuku, T. Takamasa, Y. Matsumoto, Experimental study on axial development of liquid film in vertical upward annular two-phase flow, Int. J. Multiph. Flow 34 (2008) 111–127. [16] G.B. Wallis, One-Dimensional Two-Phase Flow, McGraw-Hill, New York, 1969. pp. 315–374. [17] K. Sekoguchi, O. Tanaka, T. Ueno, T. Furukawa, S. Esaki, M. Nakasatomi, An investigation of the flow characteristics in the disturbance wave region of
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annular flow: 1st report, effect of tube diameter, Bull. JSME 26 (1983) 1719– 1726. [18] K. Sekoguchi, T. Ueno, O. Tanaka, An investigation of the flow characteristics in the disturbance wave region of annular flow: 2nd report, on correlation of principal flow parameters, Trans. JSME B 51 (1985) 1798–1806. in Japanese.