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Void fraction measurement of stratified gas-liquid flow based on multi-wire capacitance probe
Accepted Manuscript Void fraction measurement of stratified gas-liquid flow based on multi-wire capacitance probe Denghui He, Senlin Chen, Bofeng Bai PII: DOI: Reference:
S0894-1777(18)30448-5 https://doi.org/10.1016/j.expthermflusci.2018.11.005 ETF 9655
To appear in:
Experimental Thermal and Fluid Science
Received Date: Revised Date: Accepted Date:
22 March 2018 20 July 2018 12 November 2018
Please cite this article as: D. He, S. Chen, B. Bai, Void fraction measurement of stratified gas-liquid flow based on multi-wire capacitance probe, Experimental Thermal and Fluid Science (2018), doi: https://doi.org/10.1016/ j.expthermflusci.2018.11.005
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Void fraction measurement of stratified gas-liquid flow based on multi-wire capacitance probe Denghui Hea,b, Senlin Chena, Bofeng Baib* a
State Key Laboratory of Eco-hydraulic in Northwest Arid Region, Xi’an University of Technology, Xi’an 710048,
Shaanxi, China b
State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, Shaanxi,
China
*Corresponding author E-mail: [email protected]
Abstract The void fraction of stratified gas-liquid flow is crucial to determine mixture density, calculate pressure gradient, obtain phase flow rate, analyze flow structure and monitor flow processes. The measurement and prediction of void fraction is of particular interest in many industrial processes. The objective of this paper is to develop a new method to accurately measure the void fraction in the stratified gas-liquid flow. A measurement device named multi-wire capacitance probe which is based on the single-wire capacitance probe is developed. The device can obtain the average void fraction as well as the local void fraction by measuring the water layer height at different circumferential positions of the pipe. We analyze the water layer height and its variations with time at different circumferential positions under different superficial gas and liquid velocity. We also discuss the local void fraction at different circumferential position and the average void fraction of the circular cross section by measuring the water layer height. Comparisons of ten existing prediction correlations using the measured void fraction are made. The correlation of Armand–Massina is recommended to predict the void fraction of stratified gas-liquid flow. The relative error of the void fraction predicted by this correlation is within ±3.0% under the 95.65% confidence level.
Key words: stratified gas-liquid flow, water layer height, void fraction, capacitance probe
1. Introduction The void-fraction in stratified gas-liquid flow is of significant importance in the heat exchanger applications [1, 2], the oil-gas transmission pipelines [3-5] and the nuclear reactors [6, 7]. It is one of the key
parameters to determine mixture density, calculate pressure gradient, obtain phase flow rate, analyze flow structure and monitor flow processes [7-10]. How to accurately measure and predict the void fraction remains a challenge. There are many typical methods to measure the void fraction [10], including the quick-closing valve (QCV) method [11, 12] (QCV is also used to calibrate other measurement techniques), the electrical methods (conductance and capacitance method) [13-18], the irradiation methods [19-20], the optical method [21], and ultrasonic method [22].The classical electrical methods are more popular owing to the characteristics of safety, economy and efficiency. However, they are greatly limited by the water salinity variation, the electrical properties of the liquid and the liquid phase distributions owing to their great dependence on the permittivity or the conductivity. Hence the temperature and pressure compensations for the flow are usually required. To address these concerns, Nettoet al. [23] proposed a new wire capacitance probe and employed it to characterize the wavy surface of the stratified flows and measure the liquid height in the horizontal slug flow. Guo et al [24] and Huang et al [25, 26] improved the structure of the wire capacitance probe and successfully use it to measure the void fraction and water holdup. They also concluded that the wire capacitance probe was independent of water salinity, temperature and less affected by the conductivity. Furthermore, the output capacitance is proportional to the water film height covered on the probe. So essentially it is a linear system. However, only one water layer height was measured in one cross section in previous studies, and the water layer shape cannot be obtained. Therefore, the measurement accuracy of the void fraction and holdup based on the water layer height is not satisfactory. In the past seven decades, extensive theoretical models and experimental correlations were proposed to predict the void fraction, which have been reviewed and compared by Dukler et al. [27], Marcano [28], Mandhane et al. [29], Papathanassiou [30], Abdulmajeed [31], Friedel and Diener [32], Melkamu and Ghajar [33], Xu and Fang [2], Rassame and Hibiki [34], etc. On the basis of the results, ten correlations proposed byLockhart–Martinelli (1949), Fauske (1961), Smith(1969), Rouhani (1970), Chisholm(1973), Minami–Brill (1987), Abdulmajeed (1996), Armand–Massina, Melkamu (2007), and Rassame–Hibiki (2018) are evaluated in the present study. The summary of these correlations is tabulated in Appendix A Table A1. This paper aims to develop a new device to accurately measure the void fraction in the stratified gas-liquid flow. A measurement device named multi-wire capacitance probe which is based on the wire capacitance probe is developed. The device can obtain the average void fraction as well as the local void fraction by measuring the water layer height at different circumferential positions. Comparisons of the existing
correlations using the measured void fraction are also made. Finally, the best performing correlation predicting the void fraction in the stratified flow is recommended. 2. Multi-wire capacitance probe 2.1 Measurement principle As is shown in Fig. 1, the capacitance probe 1 is consisting of the metal wire (1a), the insulating film (1b) and the water layer covered on it, which is formed a cylindrical capacitor. For the cylindrical capacitor, the insulating film is the dielectric of the capacitor, the metal wire with the insulating film (wire 1) and the water layer covering it are regarded as two electrodes; the wire without the insulating film(wire 2) is used to connect one of the electrodes with the measuring circuit. A plate capacitor will be formed between wire 1 and wire 2. The dielectric of the capacitor is the media (air, water and insulating film) between two wires, which composed the two pole plates. Compared with the cylindrical capacitance, the plate capacitance is very small. The plate capacitance is less than 0.25% of the cylindrical capacitance in the present study. Hence the plate capacitance between the two wires and the fringe effect are ignored. The capacitance, C, can be expressed as Eq. (1).
C
2 r 0 h kh ln d 2 d1
(1)
where εr is the relative dielectric constant, ε0 is the dielectric constant of vacuum (ε0= 8.8542×10-12F/m), d1, d2are the diameters of the metal wire without and with insulating film, respectively, h is the water layer height covering on the wire with insulating film, and k is the sensitivity coefficient, which is written as Eq. (2).
k
2 r 0 ln d 2 d1
(2)
For a given capacitance probe, εr, d1 andd2are constant, which results ink is a constant. Thus, according to Eq. (1), the capacitance, C, is linearly proportional to the water layer height, h, around the wire. Studies also showed that C is independent of the water salinity, the temperature (17℃ and 30℃), the working frequency and distance between two wires [26].The water layer height, h, can be determined by measuring the capacitance, C.
1a 1b
1
2
C
Air
Water
d1 d2
h
C
Fig. 1. Capacitance probe and Measurement principle.1-Wire with insulating film, 2- wire without insulating film, 1a-metal core, 1b- insulating film.
cylindrical
2.2 Structural design To obtain more accurate void fraction in the circular cross section, five capacitance probes (No.1 to No.5) were employed in one cross section, as shown in Fig. 2 (a). The diameter of the pipe, D, is 50mm. The distance of adjacent probes is 8mm and the No. 3 probe through the center of the circle. To detect the discontinuous water dispersed in the air, a parallel wire without insulating film (wire 2) is set downstream of each capacitance probe (wire 1). The distance between wire 1 and wire 2 is 5mm [25]. The average value of the void fractions measured by the five probes is taken as the void fraction in the cross section. The photo of
measurement device is shown in Fig. 2(c). In addition, to reduce effect of the joint between the pipe and measurement device on the flow, special emphases were put on the joint, including reduce the gap at the joint and keep concentricity between the pipe and measurement device. Y 8mm 8mm 8mm 8mm
(a)
5mm
(b) wire without insulating film
(c) wire with insulating film
Flow O
Z
2
1
No.1 No.2 No.3 No.4 No.5
Fig.2. Multi-wire capacitance probe device (a) side view (b) front view (c) device photo.
2.3 Circuit design The capacitance is measured by a circuit based on Integrated Circuit CAV444 from Analog Microelectronics, Germany. CAV444 is a capacitance-to-voltage (C/V) transmitter whose output voltage, V, is linearly proportional to the input capacitance, C, as shown in Eq. (3).
V k C b
(3)
where k′ is the proportionality coefficient and b is the intercept. The circuit diagram of the PCB (Printed Circuit Board) and the photos are shown in Fig. 3. To reduce the possible cross-talk between adjacent capacitance probes, five independent circuits are used and each water layer height is measured by one circuit. The working frequency of the circuit is1000 Hz and the measurement frequency of the void fraction is 500 Hz. The measuring circuit is calibrated by using different capacitance sensors to obtain the linear relationship between the capacitance and the output voltage. Figure 4 shows a typical calibration results between the capacitance and the voltage, which suggests that the linear relationship is excellent. Note that some stray capacitance will be introduced to the circuits and the parameters of the external element are not exactly the same, which results in the performance of the circuits are different with each other. Hence, to obtain the perfect relationship between the capacitance and the voltage, every circuit should be calibrated before void fraction measurement.
Fig. 3.Circuit diagram of printed circuit board and circuit photos.
4.0 Voltage Linear Fit of voltage
Voltage (V)
3.5 3.0 2.5 2.0
V=0.73598+0.01755C 2 Adj. R =0.99995
1.5 1.0
0 20 40 60 80 100 120 140 160 180 200 Capacitance (pF)
Fig.4.Typical calibration result between input capacitance and output voltage.
2.4 Calibration of multi-wire capacitance probe According to the above analyses, the capacitance is linearly proportional to the water layer height (Eq. (1)), and the output voltage is linearly proportional to the capacitance signal (Eq. (3)), so we get
V k C b k kh b ah b
(4)
where a and b are constant coefficients, a=k′k. Equation (4) demonstrates that the output voltage is linearly proportional to the water layer height, thus the multi-wire capacitance probe device can be calibrated by the system shown in Fig. 5. The method is as follows: 1)
Use two pieces of flat plate to clamp on both sides of the device, a gap of 2mm is left in one side to inject water into the device (Fig. 5).
2)
Inject water whose volume is initially known into the device, and then the water layer height covered on each probe can be calculated.
3)
Record the corresponding voltage, so the relationship between the water layer height and the output voltage is obtained.
4)
The repetitive experiments are conducted and the average values are used to determine the
coefficients a and b shown in Eq. (4). The coefficients a and b for the five probes are tabulated in Table 1. In the practical application, the
water layer height can be calculated by measuring the output voltage for knowing a and b. electrode injection water
circuit Fig.5. Calibration system of multi-wire capacitance probe device. Table 1 Parameters of relationship between liquid layer height and output voltage. Probe
a
b
R2
No. 1
0.049
1.408
0.99902
No. 2
0.049
1.532
0.99895
No. 3
0.047
1.547
0.99958
No. 4
0.049
1.559
0.99930
No. 5
0.049
1.549
0.99810
We divide the cross section into five control bands, with an area of Wi×Hi, where Wi and Hi are the width and height of the ith band, respectively (Fig. 6).The centre line of the band is coinciding with the probe and the width is 8mm (Wi=8mm). The void fraction of the ith band, αi, is approximately calculated by
i 1
Wi hi h 1 i Wi H i Hi
(5)
where hi is the water layer height of the ith probe. Thus the void fraction of the circular cross section, α, is calculated by
1 5 i 5 i 1
(6)
Y 8mm 8mm 8mm 8mm
Wi 8mm
Hi
O
Z
No.1 No.2 No.3 No.4 No.5
Fig.6.Control band of circular cross section.
3. Experimental setup and method 3.1 Experimental setup The experiments were carried out in the Gas–Liquid Two-Phase Flow Loop of Xi’an Jiaotong University as is shown in Fig. 7. The air and tap water are used as the test fluid. The reference air flow rate is measured by a gas Coriolis mass flow meter, the reference water flow rate is measured by an electromagnetic flow meter or a Coriolis mass flow meter depending on the flow rate. The pressure and the temperature are also measured by corresponding sensors. The quick-closing valves (QCV) method is used to obtain the void fraction under different test conditions. To record the flow pattern flowing through the V-Cone flow meter, a high-speed camera is employed. The parameters of the measurement devices are shown in Table 2. The data are collected by the NIUSB-6229 data acquisition module and the LabVIEW based software. The sampling frequency and sampling time are 500Hz and 60s, respectively, in present test. To air
gas–liquid mixer
Test section
Valve 2 Gas–liquid separator P
Electromagnetic flow meter
T
Valve 1 Valve 3 Water tank
Centrifugal pump
Water mass flow meter
Bypass
Pressure gauge Thermometer P
T
Valve 4 Air Air storage compressor tank
Air freezing dryer
Air mass flow meter
Fig. 7. Flow diagram of experimental setup.
Table 2 Parameters of the measurement devices used in present experiments. Device
Measurement range
Uncertainty
Air mass flow meter
0-700 kg/h
±0.5%
Siemens
Electromagnetic flowmeter
0.0076-0.76 m3/h
± 0.2%
YOKOGAWA
Water mass flow meter
0-10 000 kg/h
± 0.1%
Siemens
Temperature transmitter
0-60℃
±0.15℃
Xi'an Instruments Factory
Pressure transmitter
0-1.0 MPa
±0.0750%
Emerson Process Management
High-speed camera
0-10000 fps
i-SPEED TR
Olympus
16 bits
National Instrumentation
Data acquisition board
48input channel, 80 ks/s
Manufacturer
3.2 Experimental scheme The test conditions in the present experiments are listed in Table 3, where Usg and Usl are the superficial gas and liquid velocity, respectively, and are defined by Eqs. (7) and (8), β is the gas volume fraction, which is higher than 97.9% in present cases.
U sg
U sl
4 mg
(7)
D2 g 4ml D 2 l
(8)
where mg and ml are the gas and liquid mass flow rate, respectively, ρg and ρl are the gas density and liquid density, respectively.
Table 3 Test conditions in present experiment. D (mm)
Usg(m/s)
Usl(m/s)
β (%)
0.10
6.73–20.23
0.00654–0.113
98.64–99.96
0.15
6.07–18.82
0.00642–0.114
98.34–99.96
0.20
5.56–21.34
0.00587–0.119
98.30–99.93
0.30
4.99–17.82
0.00767–0.108
97.93–99.91
Pressure (MPa)
50
The flow pattern displayed on the classical Mandhane flow pattern map [35] is shown in Fig. 8.We found that the wave stratified flow is the primary flow pattern observed in the present experiment. Note that
although some data lie in the transition regions according to the Mandhane flow pattern map, these data have been observed as the stratified gas-liquid flow by recording the flow pattern using the high-speed camera. The detailed experimental data are available in Appendix B Table B1. 10
Usl (m/s)
1
Fine bubbly
Plug
Slug
Annular
0.1
0.01 1E-3 0.01
Smooth stratified P = 0.10MPa P = 0.15MPa P = 0.20MPa P = 0.30MPa
0.1
Wave stratified
1 Usg (m/s)
10
100
Fig. 8.Test condition in present study in the Mandhane flow pattern map [36] (the data have been validated by recording the flow pattern using the high-speed camera).
3.3 Error analysis To further validate the measurement accuracy of the proposed multi-wire capacitance probe device, the quick-closing valves (QCV) method is employed. The QCV is located downstream of the capacitance probe device, as shown in Fig. 9. When the system reaches a steady state, the two ball valves simultaneously close to
trap the gas and liquid in the test section. At the same time, the bypass valve opens to force the fluid to flow so the system could continue to work normally. A water box is added to reduce the effect of refraction of tube wall. The high-speed camera captures the side view of the test section (Fig. 9) to illustrate clear separation between gas and liquid phases. We estimate the height of gas from the interface to the top of the inner tube for void fraction calculation. To eliminate the system error, the averaged void fraction for eight times by the QCV system was regarded as the actual time-averaged void fraction under the flow condition.
Fig. 9. Quick closing valves system(D=50mm).
Al
Ag
(9)
Ag Al
D2 sin 8
(10)
Ag A Al π
2 cos 1 1
(11) θ
D hg D / 2
(12)
± ≤
Relative error (%)
5 4 3 2 1 0 -1 -2 -3 -4 -5 3
Relative error
+2.5%
-2.5%
6
9
12 15 Usg(m/s)
18
21
24
Fig. 10. Relative error of void fraction. Table 4 Maximum relative uncertainty of the measurement Parameters in present experiments. Measurement Parameter Gas flow rate
Measurement device
Maximum relative uncertainty
Gas Coriolis mass flow meter
3.29%
Electromagnetic flow meter
4.42%
Water Coriolis mass flow meter
0.521%
Pressure
Pressure transmitter
0.225%
Temperature
Temperature transmitter
1.06%
Void fraction
Multi-wire capacitance probe
3.71%
Liquid flow rate
4. Results and analysis 4.1. Water layer height To compare the water layer height at different circumferential positions, the equivalent water layer height, h*, is used and defined by Eq. (13)
hi*
hi Hi
(13)
As shown in Fig. 11, the time within 3s is employed to compare the different experimental results. It can be seen that the equivalent water layer heights at different circumferential positions are different, thus it is unreasonable to measure the void fraction of the circular pipe by one probe. The equivalent water layer height varies with time and is characterized by irregular "disturbance" waves under lower superficial gas velocity while the disturbance waves will more regular under higher superficial gas velocity. We also find that the
variation frequency of the water layer at different circumferential position is similar with each other (Figs. 12 and 13), and all the probes can detect the disturbance waves of the gas-liquid interface. The equivalent water layer height and the amplitude of the disturbance waves increase with the superficial liquid velocity increasing (Figs. 11-13). Figures 12 and 13 also show that the increase of superficial liquid velocity has little effect on
(a) 1.0
Y
Usg=6.73m/s, Usl=0.0214m/s
0.8 0.6
h1*
h2*
h3*
h4*
h5*
O
h*
Z
0.4 h1 h2 h3 h4 h5
0.2 0.0 0.0
(b) 1.0
0.5
1.0
1.5 2.0 Time (s)
h1*
h2*
h3*
h4*
h5*
O
Z
h*
0.6
3.0
Y
Usg=6.73m/s, Usl=0.0706m/s
0.8
2.5
0.4
h1 h2 h3 h4 h5
0.2 0.0 0.0
(c) 1.0
0.5
1.0
1.5 2.0 Time (s)
h1*
h2*
h3*
h4*
h5*
O
Z
h*
0.6
3.0
Y
Usg=14.06m/s, Usl=0.0288m/s
0.8
2.5
0.4 h1 h2 h3 h4 h5
0.2 0.0 0.0
(d) 1.0
0.5
1.0
1.5 2.0 Time (s)
h2*
h3*
h4*
h5*
O
Z
h*
0.6
h1*
3.0
Y
Usg=14.17m/s, Usl=0.113m/s
0.8
2.5
0.4 h1 h2 h3 h4 h5
0.2 0.0 0.0
0.5
1.0
1.5 2.0 Time (s)
2.5
3.0
Fig. 11. Variation of equivalent water layer height with time at different circumferential position (P=0.1MPa).
the disturbance wave frequency, whereas the disturbance wave frequency increases slightly as the superficial gas velocity is increased. The multi-wire capacitance probe method exhibits great potential to investigate the
wave structures and the wave oscillations from a two-dimensional perspective just as the WMS did [3].
h4 *
h5 *
0
5
10
15
20
25
0.006
Amplitude
h3 *
Amplitude
h2 *
0.006
Amplitude
(b)
h1 *
Amplitude
0.0027 0.0018 0.0009 0.0000 0.0027 0.0018 0.0009 0.0000 0.0027 0.0018 0.0009 0.0000 0.0027 0.0018 0.0009 0.0000 0.0027 0.0018 0.0009 0.0000
Amplitude
Amplitude
Amplitude
Amplitude
Amplitude
Amplitude
(a)
30
h1 *
0.003 0.000 h2 *
0.003 0.000 0.006
h3 *
0.003 0.000 0.006
h4 *
0.003 0.000 0.006
h5 *
0.003 0.000
0
5
Frequency (Hz)
10
15
20
25
30
Frequency (Hz)
Fig. 12. Frequency spectrogram of equivalent water layer height corresponding to Fig. 9(a) and (b). (a) Usg=6.73m/s,
(b)
h1 *
0.002 0.000 0.004
Amplitude
h2 *
0.002 0.000 0.004
Amplitude
h3 *
0.002 0.000 0.004
Amplitude
h4 *
0.002 0.000 0.004
h5 *
0.002 0.000 0
5
10
15
20
Frequency (Hz)
25
Amplitude
0.004
Amplitude
Amplitude
Amplitude
Amplitude
Amplitude
(a)
Amplitude
Usl=0.0214 m/s (b) Usg=6.73m/s, Usl=0.070 m/s (P=0.1MPa).
30
0.009
h1 *
0.006 0.003 0.000 0.009
h2 *
0.006 0.003 0.000 0.009
h3 *
0.006 0.003 0.000 0.009
h4 *
0.006 0.003 0.000 0.009
h5 *
0.006 0.003 0.000
0
5
10
15
20
25
30
Frequency (Hz)
Fig. 13. Frequency spectrogram of equivalent water layer height corresponding to Fig. 9(c) and (d). (a) Usg=14.06m/s, Usl=0.0288 m/s (b) Usg=14.17m/s, Usl=0.113 m/s (P=0.1MPa).
The average water layer heights at different circumferential positions and flow conditions are obtained by numerically time-averaging the water layer height signal, as shown in Fig. 14. It can be seen that the water
layer is nonuniform at different circumferential position due to the influence of gravity. The distribution curves of the layer are of the shape of downwardly concave particularly at lower superficial liquid velocity (Fig. 14(a) and (f)). Fukano and Ousaka [38], Flores et al. [39] pointed that this was caused by the pumping action of disturbance waves and the secondary flow effect of the gas phase. The water layer is higher in the middle of the pipe cross section (measured by No. 3 probe) than the other circumferential positions. The non-uniformity of the water layer will increase as the superficial liquid velocity is increased. It seems that the non-uniformity of the water layer will decrease as the superficial gas velocity is increased.
(a)
(b)
(c)
Usg=6.74m/s, Usl=0.00654m/s
Usg=6.73m/s, Usl=0.0214m/s
Usg=6.77m/s, Usl=0.0424m/s
-25-20-15-10 -5 0 5 10 15 20 25 Z (mm)
-25-20-15-10 -5 0 5 10 15 20 25 Z (mm)
-25-20-15-10 -5 0 5 10 15 20 25 Z (mm)
(e)
(d)
(f)
Usg=6.73m/s, Usl=0.0706m/s
Usg=7.07m/s, Usl=0.0972m/s
Usg=13.83m/s, Usl=0.00823m/s
-25-20-15-10 -5 0 5 10 15 20 25
-25-20-15-10 -5 0 5 10 15 20 25
-25-20-15-10 -5 0 5 10 15 20 25
Z (mm)
Z (mm)
Z (mm)
(g)
(h)
(i)
Usg=14.06m/s, Usl=0.0288m/s
Usg=14.17m/s, Usl=0.0564m/s
Usg=14.17m/s, Usl=0.113m/s
-25-20-15-10 -5 0 5 10 15 20 25
-25-20-15-10 -5 0 5 10 15 20 25
-25-20-15-10 -5 0 5 10 15 20 25
Z (mm)
Z (mm)
Z (mm)
Fig.14. Average water layer height at different circumferential positions (P=0.1MPa).
The flow structure from the front view and the average water layer distribution in cross section are compared in Fig. 15. It is found that the water layer height flow recorded by the high-speed camera increases
with the superficial liquid velocity increasing, which agrees well with the measurement by the capacitance probes. The disturbance of the gas-liquid interface increases as the superficial gas velocity is increased, which is also in accordance with the above analysis in Figs. 11-13.
(b)
(a) Usg=6.74m/s, Usl=0.00654m/s
Usg=6.73m/s, Usl=0.0214m/s
(d)
(c)
Usg=6.73m/s, Usl=0.0706m/s
Usg=6.77m/s, Usl=0.0424m/s
(f)
(e)
Usg=13.83m/s, Usl=0.00823m/s
Usg=7.07m/s, Usl=0.0972m/s
(g)
Usg=14.06m/s, Usl=0.0288m/s
(h)
Usg=14.17m/s, Usl=0.0564m/s
(i) Usg=14.17m/s, Usl=0.113m/s
Fig. 15. Flow structure from the front view and the average water layer distribution in cross section (P=0.1MPa).
4.2. Void fraction The local void fraction at different circumferential positions calculated by Eq. (5) is shown in Fig. 16. It can be seen that the local void fraction decreases from the wall to the center line of the pipe. We also find that the local void fraction decreases as the superficial liquid velocity is increased (Fig. 16 (a)-(e)). The distribution curves of the local void fraction are not flat and have a valley at the centerline. When further increasing the superficial gas velocity, the distribution curves are relatively flat, as shown in Fig. 16 (f) and (g). The local void fraction at different circumferential positions is consistent with the flow structure (Fig. 15). The average void fraction of the circular cross section calculated by Eq. (6) is shown in Figs. 17 and 18. The void fraction increases with the operating pressure when the gas volume fraction keeps constant. Figure 18 also shows that the void fraction increases with the gas volume fraction and decreases as the superficial gas velocity is increased. These results are consistent with the existing studies.
0.9
0.9
0.9
0.8
Usg=6.74m/s, Usl=0.00654m/s
0.7 0.6
0.8
Void fraction
(c) 1.0
Void fraction
(b) 1.0
Void fraction
(a) 1.0
Usg=6.73m/s, Usl=0.0214m/s
0.7 0.6
0.8
Usg=6.77m/s, Usl=0.0424m/s
0.7 0.6
0.5 0.5 0.5 -25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 Z (mm) Z (mm) Z (mm) (d) 1.0 (e) 1.0 (f) 1.0
0.8 0.7
Usg=6.73m/s, Usl=0.0706m/s
0.6
0.9
0.8 0.7
Usg=7.07m/s, Usl=0.0972m/s
0.6
Void fraction
0.9 Void fraction
Void fraction
0.9
0.8 Usg=13.83m/s, Usl=0.00823m/s
0.7 0.6
0.5 0.5 0.5 -25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 Z (mm) Z (mm) Z (mm) (i) 1.0 (g) 1.0 (h) 1.0
Usg=14.06m/s, Usl=0.0288m/s
0.7 0.6
0.5 -25 -20 -15 -10 -5 0 5 10 15 20 25 Z (mm)
0.8
0.9 Usg=14.17m/s, Usl=0.0564m/s
0.7 0.6
Void fraction
Void fraction
0.8
0.9
0.5 -25 -20 -15 -10 -5 0 5 10 15 20 25 Z (mm)
0.8 0.7
Usg=14.17m/s, Usl=0.113m/s
0.6 0.5 -25 -20 -15 -10 -5 0 5 10 15 20 25 Z (mm)
Fig. 16. Local void fraction at different circumferential positions (P=0.1MPa).
1.00 P=0.10MPa, Usg=6.85m/s
Measured void fraction
Void fraction
0.9
P=0.10MPa, Usg=13.85m/s
0.96
P=0.15MPa, Usg=6.25m/s P=0.15MPa, Usg=12.58m/s P=0.20MPa, Usg=5.85m/s
0.92
P=0.20MPa, Usg=11.35m/s P=0.30MPa, Usg=5.08m/s P=0.30MPa, Usg=10.19m/s
0.88 0.84 0.80 0.975
0.980 0.985 0.990 0.995 Gas volume fraction
Fig. 17. Effect of pressure on void fraction.
1.000
Measured void fraction
(a)
1.00 P =0.15 MPa Usg=6.25m/s
0.95
Usg=12.58m/s Usg=18.74m/s
0.90 0.85 0.80 0.985
0.990 0.995 Gas volume fraction
1.000
Measured void fraction
(b) 1.00 P =0.30 MPa Usg=5.08m/s
0.95
Usg=10.19m/s Usg=15.16m/s
0.90 0.85 0.80 0.975
0.980 0.985 0.990 0.995 Gas volume fraction
1.000
Fig. 18. Effect of superficial gas velocity on void fraction.
4.3. Comparison of existing correlations with experimental data Ten correlations are those of Lockhart–Martinelli (1949), Fauske (1961), Smith(1969), Rouhani (1970), Chisholm(1973), Minami–Brill (1987), Abdulmajeed (1996), Armand–Massina, Melkamu (2007), and Rassame–Hibiki (2018) are compared to predict the void fraction of the stratified flow. The relative error (RE) the average relative error (ARE), the average absolute relative error (AARE), and the standard deviation (SD) are employed to evaluate these correlations. These parameters are defined by Eqs. (14)-(17).
predicted measured 100 measured
RE
SD
(14)
ARE
1 N REi N i 1
(15)
AARE
1 N REi N i 1
(16)
1 N2
2 N N 2 N RE RE i i i 1 i 1
(17)
where αpredicted and αmeasured are the predicted void fraction and measured void fraction, respectively, N is the
total test number. Studies [40-42] showed that the void fraction correlation directly influences the two-phase mixture density and this influence is of different magnitude for different ranges of the void fraction. Ghajar and Bhagwat [40] reported that an error less than ±10 % is considered acceptable for the range of the void fraction between 0.75 and 1 (0.75＜α＜1). As shown in Fig. 19 and Table 5, all of the correlations predict the void fraction with the relative error less than ±10%. The correlations of Rouhani (1970), Abdulmajeed (1996), Armand–Massina and Melkamu (2007) are able to predict more than 90% of the data points within the RE of ±5.0%, among which the correlations of Rouhani (1970) and Armand–Massina predict all the data points with the RE of ±5.0%. For the more restrictive ±3.0% error band, the correlation of Armand–Massina performs best with 95.65% of the data points within the RE of ±3.0%. the next best prediction comes from Rouhani(1970) with 84.06% of the data points within the RE of ±3.0%. The other correlations predict less than 80% of the data points within the RE of ±3.0%, which is not so satisfactory. From the statistical results in Table 6, we also find that the correlations of Rouhani (1970), Abdulmajeed (1996) and Armand–Massina tends to underestimate the void fraction (ARE＜0), whereas the others overestimate the void fraction data (ARE＞0). The AARE of correlation by Armand–Massina is minimum followed by the correlations of Rouhani (1970) and Melkamu (2007). According to the above comparisons and analysis, the correlation of Armand–Massina shows the best prediction capability, and has more simpler equation form, thus is recommended to predict the void fraction of stratified gas-liquid flow. Lockhart -Martinelli(1949) Fauske (1961) Smith (1969) Rouhani (1970) Chisholm (1973)
fraction Predicted Void fraction PredictedVoid
1.00
Minami and Brill (1987) Abdulmajeed (1996) Armand-Massina Melkamu (2007) Rassame-Hibiki(2018)
0.95
1.00 0.95
+10.0%
0.90
0.90
+10.0%
0.85
+5.0%
0.85
0.80
+5.0%
+3.0% -3.0%
0.75 0.80 0.75
+3.0%
0.80
-3.0%
0.75 0.75
-5.0%
-10.0%
-10.0% -5.0%0.85
0.90 Measured Void fraction
0.80 0.85 0.90 0.95 Measured Void fraction
0.95
1.00
1.00
Fig. 19. Comparisons of existing correlations with measured experimental data.
Table 5 Confidence level of relative error predicted by correlations under different error band. Correlation
Confidence level (%)
Error band
±3.0%
±5.0%
±10.0%
Lockhart–Martinell (1949)
47.83
81.16
100
Fauske (1961)
66.67
85.51
100
Smith(1969)
15.94
78.26
100
Rouhani(1970)
84.06
100
100
Chisholm(1973)
33.33
68.12
100
Minami–Brill (1987)
39.13
65.22
100
Abdulmajeed (1996)
66.67
91.3
100
Armand–Massina
95.65
100
100
Melkamu (2007)
71.01
92.75
100
Rassame–Hibiki (2018)
26.09
75.36
100
Table 6 Statistical results using present measured data. Correlation
RE (%)
ARE (%)
AARE (%)
SD (%)
Lockhart–Martinell (1949)
1.12–7.09
3.37
3.37
1.57
Fauske (1961)
-8.61–5.03
0.13
2.67
3.29
Smith(1969)
2.28–7.13
4.04
4.04
1.23
Rouhani(1970)
-3.87–3.10
-1.07
1.66
1.70
Chisholm(1973)
1.82–7.87
4.16
4.16
1.68
Minami–Brill (1987)
0.59–10.20
4.19
4.19
2.38
Abdulmajeed (1996)
-5.54–5.89
-0.92
2.50
2.79
Armand–Massina
-3.09–3.54
-0.44
1.41
1.62
Melkamu (2007)
-3.39–6.81
1.19
2.28
2.57
Rassame–Hibiki (2018)
0.69–8.06
4.02
4.02
1.45
5. Conclusions A measurement device of the void fraction based on the wire capacitance probe was developed. The water layer height and its variations with time were analyzed. Based on which the local void fraction at different circumferential positions and the average void fraction of the circular cross section were obtained. Ten existing correlations were compared using the measured void fraction data and the best performing correlation was recommended. The main conclusions are as follows:
1)
The equivalent water layer heights at different circumferential positions are different and vary with time. The variation frequency at different circumferential positions is similar with each other and less affected by the increasing of superficial liquid velocity, whereas it increases slightly as the superficial gas velocity is increased. The distribution curves of the layer are of the shape of downwardly concave particularly at lower superficial liquid velocity. The water layer in the middle of the pipe cross section is higher than that of at the other circumferential positions. The non-uniformity of the water layer will increase as the superficial liquid velocity is increased.
2)
The distribution curves of the local void fraction are not flat and have a valley at the centerline; the curves are relatively flat when further increasing the superficial gas velocity. The local void fraction decreases from the wall to the centerline of the pipe and decreases as the superficial liquid velocity is increased. The average void fraction decreases with the superficial gas velocity and increases with the operating pressure.
3)
Comparisons of the existing correlations using the measured void fraction demonstrate that all the ten correlations predict the void fraction with the relative error less than ±10%. The correlation of Armand–Massina is recommended to predict the void fraction of stratified gas-liquid flow owing to it has the best prediction capability and simpler equation form. The relative error of the void fraction predicted by this correlation is within ±3.0% under the 95.65% confidence level.
Acknowledgements The National Natural Science Foundation of China under Grant No. 51709227, the China National Funds for Distinguished Young Scientists under Grant No. 51425603 are acknowledged for supporting this study.
Appendixes Appendix A:Void fraction correlation Table A1 Void fraction correlations considered for this study. Author/source
Void fraction correlation
Lockhart–Martinelli (1949)
0.36 0.64 1 x g l 1 0.28 x l g
Fauske (1961)
1 x g 0.5 1 x l
Smith(1969)
1 g l
Rouhani (1970)
x x 1 x U gm C g 0 g l G C0 1 0.2 1 x
0.07 1
1
l g 0.4 1 x 1 1 x 0.4 0.6 1 0.4 1 x 1 x
0.5
1
1
0.25 1.18 U gm 0.5 g l g l
Chisholm(1973)
1 1 x 1 l g
Minami–Brill (1987)
1 x g x l
1
ln Z 9.21 4.3374 8.7115
exp Z Abdulmajeed (1996)
1.84U sl0.575 l0.5804 U sg D 0.0277 g 0.3696 0.1804
1 0.528 U sgU sl
0.25
0.216121
P 101325
0.05
l0.1
El theo
For turbulent flow
El theo exp 0.9304919 0.5285852R 9.219634 102 R2 9.02418 104 R4 For laminar flow
1.1 0.6788496 R 0.1232191 102 R 2 1.778653 103 R 3 E exp l theo 3 4 1.626819 10 R
U sg g l U sl l g
where R ln X and X
L
lU sl2 2 gU sg
where L=0.2 for turbulent flow and L=1 for laminar flow Armand–Massina
1 x g 0.833 0.167 x 1 x l
Melkamu (2007)
1
U sg U U sg 1 sl U sg
g l
0.1
0.25 gD 1 cos l g P 1.22 1.22sin atm 2.9 2 l Psystem
Rassame–Hibiki (2018)
U sg C0U gm U gm
For 0 ≤ β < 0.9 0.5 1.50 1.50 g C0 0.800exp 0.815 0.800exp 0.815 1 0.900 0.900 l
For 0.9 ≤ β ≤1
C0 8.08 9.08 8.08 1 g l
0.5
U gm 0
Appendix B: Experimental data Table B1 Experimental data in present experiment. P (MPa)
T (℃)
Usg (m/s)
Usl (m/s)
h1(mm)
h2(mm)
h3(mm)
h4(mm)
h5(mm)
α
β
x
0.104489
15.4
6.741
0.00654
0.82
1.1
2.59
1.82
1.34
0.9661
0.999
0.7194
0.106279
17.2
6.792
0.01334
1.26
2.13
3.65
2.76
2.1
0.9473
0.998
0.5596
0.103083
18.6
6.733
0.02142
1.62
3.18
4.55
3.59
2.55
0.9315
0.9968
0.4346
0.106448
19.3
7.034
0.02754
1.89
3.81
5.15
4.24
2.82
0.9209
0.9961
0.3877
0.104707
20.6
6.772
0.04241
2.47
4.78
6.26
5.3
3.43
0.9016
0.9938
0.281
0.104838
21.3
6.883
0.05618
3.04
5.57
7.1
6.12
4
0.8856
0.9919
0.2304
0.10339
22
6.73
0.0706
3.65
6.34
8.01
6.89
4.61
0.8691
0.9896
0.1874
0.107765
22.7
6.893
0.08217
4.21
6.9
8.54
7.83
5.16
0.8549
0.9882
0.1714
0.104997
22.9
7.067
0.09721
5.23
7.54
9.24
8.19
5.75
0.8394
0.9864
0.1502
0.104918
15.2
13.83
0.00823
1.07
1.4
1.71
1.4
1.46
0.9682
0.9994
0.8074
0.105668
16.3
13.47
0.01448
2.25
2.14
2.43
1.94
1.95
0.9512
0.9989
0.6989
0.105577
18.3
14.06
0.02884
2.73
2.4
3.66
3.11
2.56
0.9346
0.998
0.5471
0.105411
20.4
14.17
0.05636
3.27
3.62
5.53
4.5
3.67
0.9075
0.996
0.3819
0.107609
21.6
13.42
0.08624
4.23
5.06
7.02
5.8
4.71
0.8795
0.9936
0.2778
0.105855
21.9
14.17
0.11254
4.91
5.8
8.15
6.78
5.41
0.8606
0.9921
0.2357
0.104298
15.4
20.2
0.0073
1.52
1.02
1.28
1.16
1.46
0.9701
0.9996
0.8731
0.104677
16.9
20.23
0.01443
2.43
1.76
1.71
1.6
2.3
0.9544
0.9993
0.7765
0.154078
15.1
6.328
0.00642
0.88
1.42
2.49
1.74
1.37
0.965
0.999
0.7528
0.154
17.1
6.271
0.01447
1.32
2.41
3.78
2.79
2.07
0.9452
0.9977
0.5708
0.15151
18.5
6.251
0.0225
1.67
3.09
4.67
3.59
2.55
0.9311
0.9964
0.4566
0.157276
19.2
6.074
0.02785
1.96
3.55
5.22
4.13
2.83
0.9217
0.9954
0.4024
0.152107
20.4
6.264
0.04361
2.52
4.36
6.32
5.25
3.45
0.9031
0.9931
0.302
0.154156
21.2
6.216
0.05688
3.1
5.28
7.19
6.13
4.02
0.886
0.9909
0.2486
0.150947
21.8
6.418
0.07091
3.28
6.02
7.99
6.88
4.65
0.8723
0.9891
0.2127
0.152969
22.3
6.151
0.0851
3.92
6.77
8.98
8
5.37
0.8534
0.9864
0.1783
0.150098
22.4
6.263
0.10568
4.99
7.41
9.77
8.61
6.11
0.8355
0.9834
0.1496
0.152063
14.8
12.68
0.00778
1.05
1.16
1.7
1.32
1.48
0.9696
0.9994
0.8334
0.152601
16.2
12.52
0.01596
1.4
1.79
2.53
1.92
1.91
0.957
0.9987
0.706
0.152841
18.2
12.59
0.03021
2.75
2.69
3.56
2.95
2.61
0.934
0.9976
0.5593
0.151648
20.3
12.82
0.05625
3.32
3.5
5.48
4.3
3.56
0.9093
0.9956
0.4068
0.151757
21.9
12.53
0.08468
4.3
4.77
6.95
5.54
4.61
0.8822
0.9933
0.307
0.152651
22
12.34
0.11405
4.92
5.78
8.24
6.16
5.47
0.8626
0.9908
0.2452
0.156765
15.6
18.65
0.00722
1.02
1.89
1.37
1.17
2.2
0.9648
0.9996
0.8896
0.15457
17
18.82
0.01417
2.5
1.57
1.91
1.7
3.35
0.9481
0.9992
0.8034
0.201385
15
5.679
0.00587
0.9
1.37
2.16
1.56
1.3
0.9675
0.999
0.7799
0.202457
16.8
5.558
0.01419
1.31
2.14
3.44
2.61
1.94
0.9493
0.9975
0.5888
0.199876
18.1
5.853
0.0222
1.6
2.76
4.31
3.39
2.35
0.9362
0.9962
0.4875
0.20229
18.9
5.97
0.02859
1.91
3.28
5
4.04
2.69
0.9252
0.9952
0.431
0.202171
20.1
5.715
0.04299
2.47
4.2
6.14
5.14
3.36
0.9057
0.9925
0.3243
0.20246
21
5.863
0.0573
3.1
5.02
7.09
6.14
4.01
0.8875
0.9903
0.2694
0.199712
21.6
6.115
0.0715
3.69
5.68
7.88
6.91
4.65
0.8719
0.9884
0.2336
0.204241
22.2
5.811
0.08692
4.43
6.48
8.91
8.08
5.41
0.8517
0.9853
0.1944
0.201229
22.2
6.098
0.10573
5.04
7.65
9.67
8.88
6.18
0.8331
0.983
0.1709
0.203854
14.1
11.24
0.0076
1.08
1.27
1.78
1.45
1.44
0.9683
0.9993
0.8456
0.204686
15.8
11.41
0.01506
1.4
1.66
2.53
2.12
1.96
0.9564
0.9987
0.7367
0.201159
18.6
11.37
0.03646
2.23
3.04
4.2
3.73
2.93
0.9277
0.9968
0.53
0.202884
20.1
11.41
0.05721
3.63
3.86
5.72
4.89
3.62
0.9024
0.995
0.4192
0.204459
21.6
11.41
0.08897
4.52
5.06
7.46
6.38
4.88
0.8729
0.9923
0.317
0.202868
21.8
11.3
0.11884
5.27
6.14
8.63
7.36
5.72
0.8513
0.9896
0.2549
0.203437
16.5
17.32
0.01415
2.48
1.55
2.17
1.76
2.66
0.9506
0.9992
0.8179
0.203327
17.9
21.34
0.01518
2.6
2.26
2.79
2.18
2.76
0.9422
0.9993
0.837
0.301425
15.3
5.032
0.01043
1.09
1.9
3.09
2.26
1.74
0.9553
0.9979
0.7016
0.300847
16.8
5.109
0.01465
1.35
2.31
3.78
2.84
2.12
0.9451
0.9971
0.6279
0.302543
17.8
5.102
0.02188
1.69
2.92
4.65
3.67
2.56
0.9314
0.9957
0.5304
0.30246
18.4
4.989
0.0284
2.02
3.45
5.31
4.32
2.87
0.9205
0.9943
0.4591
0.300592
19.2
5.225
0.0361
2.36
3.95
5.9
4.94
3.16
0.9101
0.9931
0.4098
0.303191
19.8
5.094
0.0437
2.7
4.49
6.54
5.65
3.59
0.8983
0.9915
0.3597
0.30417
21.3
5.044
0.07204
3.11
6.17
8.54
7.6
5.31
0.8639
0.9859
0.2518
0.300521
22.1
5.104
0.08825
4.79
6.95
9.23
8.25
5.06
0.8476
0.983
0.2155
0.30387
22
5.033
0.10647
5.89
7.73
10.2
8.5
6.78
0.8249
0.9793
0.1847
0.305137
14.6
9.814
0.00767
1.09
1.23
1.9
1.43
1.52
0.9676
0.9992
0.8631
0.303139
16.5
9.822
0.01591
1.4
1.96
2.75
2.13
1.98
0.9542
0.9984
0.7504
0.303611
17.5
10.08
0.02225
2.03
2.27
3.31
2.69
2.23
0.9437
0.9978
0.6876
0.301632
19
10.31
0.03696
2.34
3.13
4.44
3.63
4.19
0.9197
0.9964
0.5731
0.29899
21
10.46
0.07374
3.99
4.55
6.71
3.86
5.76
0.8869
0.993
0.4023
0.302721
21.7
10.66
0.10799
4.89
5.93
8.2
4.75
6.84
0.8611
0.99
0.3207
0.303369
17.4
15.34
0.01577
2.67
1.52
2.06
1.81
2.47
0.9509
0.999
0.8254
0.29956
19.7
14.97
0.04462
3.32
3.3
4.27
3.46
3.62
0.9182
0.997
0.6158
0.302467
16.8
17.82
0.0156
2.66
2.29
2.76
2.26
2.99
0.9403
0.9991
0.8474
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Highlights
1.
A multi-wire capacitance probe measurement device was developed.
2.
Water layer height and void fraction at different circumferential positions was measured.
3.
Water layer height and its variations with time at different circumferential positions were analyzed.
4.
Void fraction prediction correlations were compared and best recommended.
performing correlation was
S0894-1777(18)30448-5 https://doi.org/10.1016/j.expthermflusci.2018.11.005 ETF 9655
To appear in:
Experimental Thermal and Fluid Science
Received Date: Revised Date: Accepted Date:
22 March 2018 20 July 2018 12 November 2018
Please cite this article as: D. He, S. Chen, B. Bai, Void fraction measurement of stratified gas-liquid flow based on multi-wire capacitance probe, Experimental Thermal and Fluid Science (2018), doi: https://doi.org/10.1016/ j.expthermflusci.2018.11.005
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Void fraction measurement of stratified gas-liquid flow based on multi-wire capacitance probe Denghui Hea,b, Senlin Chena, Bofeng Baib* a
State Key Laboratory of Eco-hydraulic in Northwest Arid Region, Xi’an University of Technology, Xi’an 710048,
Shaanxi, China b
State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, Shaanxi,
China
*Corresponding author E-mail: [email protected]
Abstract The void fraction of stratified gas-liquid flow is crucial to determine mixture density, calculate pressure gradient, obtain phase flow rate, analyze flow structure and monitor flow processes. The measurement and prediction of void fraction is of particular interest in many industrial processes. The objective of this paper is to develop a new method to accurately measure the void fraction in the stratified gas-liquid flow. A measurement device named multi-wire capacitance probe which is based on the single-wire capacitance probe is developed. The device can obtain the average void fraction as well as the local void fraction by measuring the water layer height at different circumferential positions of the pipe. We analyze the water layer height and its variations with time at different circumferential positions under different superficial gas and liquid velocity. We also discuss the local void fraction at different circumferential position and the average void fraction of the circular cross section by measuring the water layer height. Comparisons of ten existing prediction correlations using the measured void fraction are made. The correlation of Armand–Massina is recommended to predict the void fraction of stratified gas-liquid flow. The relative error of the void fraction predicted by this correlation is within ±3.0% under the 95.65% confidence level.
Key words: stratified gas-liquid flow, water layer height, void fraction, capacitance probe
1. Introduction The void-fraction in stratified gas-liquid flow is of significant importance in the heat exchanger applications [1, 2], the oil-gas transmission pipelines [3-5] and the nuclear reactors [6, 7]. It is one of the key
parameters to determine mixture density, calculate pressure gradient, obtain phase flow rate, analyze flow structure and monitor flow processes [7-10]. How to accurately measure and predict the void fraction remains a challenge. There are many typical methods to measure the void fraction [10], including the quick-closing valve (QCV) method [11, 12] (QCV is also used to calibrate other measurement techniques), the electrical methods (conductance and capacitance method) [13-18], the irradiation methods [19-20], the optical method [21], and ultrasonic method [22].The classical electrical methods are more popular owing to the characteristics of safety, economy and efficiency. However, they are greatly limited by the water salinity variation, the electrical properties of the liquid and the liquid phase distributions owing to their great dependence on the permittivity or the conductivity. Hence the temperature and pressure compensations for the flow are usually required. To address these concerns, Nettoet al. [23] proposed a new wire capacitance probe and employed it to characterize the wavy surface of the stratified flows and measure the liquid height in the horizontal slug flow. Guo et al [24] and Huang et al [25, 26] improved the structure of the wire capacitance probe and successfully use it to measure the void fraction and water holdup. They also concluded that the wire capacitance probe was independent of water salinity, temperature and less affected by the conductivity. Furthermore, the output capacitance is proportional to the water film height covered on the probe. So essentially it is a linear system. However, only one water layer height was measured in one cross section in previous studies, and the water layer shape cannot be obtained. Therefore, the measurement accuracy of the void fraction and holdup based on the water layer height is not satisfactory. In the past seven decades, extensive theoretical models and experimental correlations were proposed to predict the void fraction, which have been reviewed and compared by Dukler et al. [27], Marcano [28], Mandhane et al. [29], Papathanassiou [30], Abdulmajeed [31], Friedel and Diener [32], Melkamu and Ghajar [33], Xu and Fang [2], Rassame and Hibiki [34], etc. On the basis of the results, ten correlations proposed byLockhart–Martinelli (1949), Fauske (1961), Smith(1969), Rouhani (1970), Chisholm(1973), Minami–Brill (1987), Abdulmajeed (1996), Armand–Massina, Melkamu (2007), and Rassame–Hibiki (2018) are evaluated in the present study. The summary of these correlations is tabulated in Appendix A Table A1. This paper aims to develop a new device to accurately measure the void fraction in the stratified gas-liquid flow. A measurement device named multi-wire capacitance probe which is based on the wire capacitance probe is developed. The device can obtain the average void fraction as well as the local void fraction by measuring the water layer height at different circumferential positions. Comparisons of the existing
correlations using the measured void fraction are also made. Finally, the best performing correlation predicting the void fraction in the stratified flow is recommended. 2. Multi-wire capacitance probe 2.1 Measurement principle As is shown in Fig. 1, the capacitance probe 1 is consisting of the metal wire (1a), the insulating film (1b) and the water layer covered on it, which is formed a cylindrical capacitor. For the cylindrical capacitor, the insulating film is the dielectric of the capacitor, the metal wire with the insulating film (wire 1) and the water layer covering it are regarded as two electrodes; the wire without the insulating film(wire 2) is used to connect one of the electrodes with the measuring circuit. A plate capacitor will be formed between wire 1 and wire 2. The dielectric of the capacitor is the media (air, water and insulating film) between two wires, which composed the two pole plates. Compared with the cylindrical capacitance, the plate capacitance is very small. The plate capacitance is less than 0.25% of the cylindrical capacitance in the present study. Hence the plate capacitance between the two wires and the fringe effect are ignored. The capacitance, C, can be expressed as Eq. (1).
C
2 r 0 h kh ln d 2 d1
(1)
where εr is the relative dielectric constant, ε0 is the dielectric constant of vacuum (ε0= 8.8542×10-12F/m), d1, d2are the diameters of the metal wire without and with insulating film, respectively, h is the water layer height covering on the wire with insulating film, and k is the sensitivity coefficient, which is written as Eq. (2).
k
2 r 0 ln d 2 d1
(2)
For a given capacitance probe, εr, d1 andd2are constant, which results ink is a constant. Thus, according to Eq. (1), the capacitance, C, is linearly proportional to the water layer height, h, around the wire. Studies also showed that C is independent of the water salinity, the temperature (17℃ and 30℃), the working frequency and distance between two wires [26].The water layer height, h, can be determined by measuring the capacitance, C.
1a 1b
1
2
C
Air
Water
d1 d2
h
C
Fig. 1. Capacitance probe and Measurement principle.1-Wire with insulating film, 2- wire without insulating film, 1a-metal core, 1b- insulating film.
cylindrical
2.2 Structural design To obtain more accurate void fraction in the circular cross section, five capacitance probes (No.1 to No.5) were employed in one cross section, as shown in Fig. 2 (a). The diameter of the pipe, D, is 50mm. The distance of adjacent probes is 8mm and the No. 3 probe through the center of the circle. To detect the discontinuous water dispersed in the air, a parallel wire without insulating film (wire 2) is set downstream of each capacitance probe (wire 1). The distance between wire 1 and wire 2 is 5mm [25]. The average value of the void fractions measured by the five probes is taken as the void fraction in the cross section. The photo of
measurement device is shown in Fig. 2(c). In addition, to reduce effect of the joint between the pipe and measurement device on the flow, special emphases were put on the joint, including reduce the gap at the joint and keep concentricity between the pipe and measurement device. Y 8mm 8mm 8mm 8mm
(a)
5mm
(b) wire without insulating film
(c) wire with insulating film
Flow O
Z
2
1
No.1 No.2 No.3 No.4 No.5
Fig.2. Multi-wire capacitance probe device (a) side view (b) front view (c) device photo.
2.3 Circuit design The capacitance is measured by a circuit based on Integrated Circuit CAV444 from Analog Microelectronics, Germany. CAV444 is a capacitance-to-voltage (C/V) transmitter whose output voltage, V, is linearly proportional to the input capacitance, C, as shown in Eq. (3).
V k C b
(3)
where k′ is the proportionality coefficient and b is the intercept. The circuit diagram of the PCB (Printed Circuit Board) and the photos are shown in Fig. 3. To reduce the possible cross-talk between adjacent capacitance probes, five independent circuits are used and each water layer height is measured by one circuit. The working frequency of the circuit is1000 Hz and the measurement frequency of the void fraction is 500 Hz. The measuring circuit is calibrated by using different capacitance sensors to obtain the linear relationship between the capacitance and the output voltage. Figure 4 shows a typical calibration results between the capacitance and the voltage, which suggests that the linear relationship is excellent. Note that some stray capacitance will be introduced to the circuits and the parameters of the external element are not exactly the same, which results in the performance of the circuits are different with each other. Hence, to obtain the perfect relationship between the capacitance and the voltage, every circuit should be calibrated before void fraction measurement.
Fig. 3.Circuit diagram of printed circuit board and circuit photos.
4.0 Voltage Linear Fit of voltage
Voltage (V)
3.5 3.0 2.5 2.0
V=0.73598+0.01755C 2 Adj. R =0.99995
1.5 1.0
0 20 40 60 80 100 120 140 160 180 200 Capacitance (pF)
Fig.4.Typical calibration result between input capacitance and output voltage.
2.4 Calibration of multi-wire capacitance probe According to the above analyses, the capacitance is linearly proportional to the water layer height (Eq. (1)), and the output voltage is linearly proportional to the capacitance signal (Eq. (3)), so we get
V k C b k kh b ah b
(4)
where a and b are constant coefficients, a=k′k. Equation (4) demonstrates that the output voltage is linearly proportional to the water layer height, thus the multi-wire capacitance probe device can be calibrated by the system shown in Fig. 5. The method is as follows: 1)
Use two pieces of flat plate to clamp on both sides of the device, a gap of 2mm is left in one side to inject water into the device (Fig. 5).
2)
Inject water whose volume is initially known into the device, and then the water layer height covered on each probe can be calculated.
3)
Record the corresponding voltage, so the relationship between the water layer height and the output voltage is obtained.
4)
The repetitive experiments are conducted and the average values are used to determine the
coefficients a and b shown in Eq. (4). The coefficients a and b for the five probes are tabulated in Table 1. In the practical application, the
water layer height can be calculated by measuring the output voltage for knowing a and b. electrode injection water
circuit Fig.5. Calibration system of multi-wire capacitance probe device. Table 1 Parameters of relationship between liquid layer height and output voltage. Probe
a
b
R2
No. 1
0.049
1.408
0.99902
No. 2
0.049
1.532
0.99895
No. 3
0.047
1.547
0.99958
No. 4
0.049
1.559
0.99930
No. 5
0.049
1.549
0.99810
We divide the cross section into five control bands, with an area of Wi×Hi, where Wi and Hi are the width and height of the ith band, respectively (Fig. 6).The centre line of the band is coinciding with the probe and the width is 8mm (Wi=8mm). The void fraction of the ith band, αi, is approximately calculated by
i 1
Wi hi h 1 i Wi H i Hi
(5)
where hi is the water layer height of the ith probe. Thus the void fraction of the circular cross section, α, is calculated by
1 5 i 5 i 1
(6)
Y 8mm 8mm 8mm 8mm
Wi 8mm
Hi
O
Z
No.1 No.2 No.3 No.4 No.5
Fig.6.Control band of circular cross section.
3. Experimental setup and method 3.1 Experimental setup The experiments were carried out in the Gas–Liquid Two-Phase Flow Loop of Xi’an Jiaotong University as is shown in Fig. 7. The air and tap water are used as the test fluid. The reference air flow rate is measured by a gas Coriolis mass flow meter, the reference water flow rate is measured by an electromagnetic flow meter or a Coriolis mass flow meter depending on the flow rate. The pressure and the temperature are also measured by corresponding sensors. The quick-closing valves (QCV) method is used to obtain the void fraction under different test conditions. To record the flow pattern flowing through the V-Cone flow meter, a high-speed camera is employed. The parameters of the measurement devices are shown in Table 2. The data are collected by the NIUSB-6229 data acquisition module and the LabVIEW based software. The sampling frequency and sampling time are 500Hz and 60s, respectively, in present test. To air
gas–liquid mixer
Test section
Valve 2 Gas–liquid separator P
Electromagnetic flow meter
T
Valve 1 Valve 3 Water tank
Centrifugal pump
Water mass flow meter
Bypass
Pressure gauge Thermometer P
T
Valve 4 Air Air storage compressor tank
Air freezing dryer
Air mass flow meter
Fig. 7. Flow diagram of experimental setup.
Table 2 Parameters of the measurement devices used in present experiments. Device
Measurement range
Uncertainty
Air mass flow meter
0-700 kg/h
±0.5%
Siemens
Electromagnetic flowmeter
0.0076-0.76 m3/h
± 0.2%
YOKOGAWA
Water mass flow meter
0-10 000 kg/h
± 0.1%
Siemens
Temperature transmitter
0-60℃
±0.15℃
Xi'an Instruments Factory
Pressure transmitter
0-1.0 MPa
±0.0750%
Emerson Process Management
High-speed camera
0-10000 fps
i-SPEED TR
Olympus
16 bits
National Instrumentation
Data acquisition board
48input channel, 80 ks/s
Manufacturer
3.2 Experimental scheme The test conditions in the present experiments are listed in Table 3, where Usg and Usl are the superficial gas and liquid velocity, respectively, and are defined by Eqs. (7) and (8), β is the gas volume fraction, which is higher than 97.9% in present cases.
U sg
U sl
4 mg
(7)
D2 g 4ml D 2 l
(8)
where mg and ml are the gas and liquid mass flow rate, respectively, ρg and ρl are the gas density and liquid density, respectively.
Table 3 Test conditions in present experiment. D (mm)
Usg(m/s)
Usl(m/s)
β (%)
0.10
6.73–20.23
0.00654–0.113
98.64–99.96
0.15
6.07–18.82
0.00642–0.114
98.34–99.96
0.20
5.56–21.34
0.00587–0.119
98.30–99.93
0.30
4.99–17.82
0.00767–0.108
97.93–99.91
Pressure (MPa)
50
The flow pattern displayed on the classical Mandhane flow pattern map [35] is shown in Fig. 8.We found that the wave stratified flow is the primary flow pattern observed in the present experiment. Note that
although some data lie in the transition regions according to the Mandhane flow pattern map, these data have been observed as the stratified gas-liquid flow by recording the flow pattern using the high-speed camera. The detailed experimental data are available in Appendix B Table B1. 10
Usl (m/s)
1
Fine bubbly
Plug
Slug
Annular
0.1
0.01 1E-3 0.01
Smooth stratified P = 0.10MPa P = 0.15MPa P = 0.20MPa P = 0.30MPa
0.1
Wave stratified
1 Usg (m/s)
10
100
Fig. 8.Test condition in present study in the Mandhane flow pattern map [36] (the data have been validated by recording the flow pattern using the high-speed camera).
3.3 Error analysis To further validate the measurement accuracy of the proposed multi-wire capacitance probe device, the quick-closing valves (QCV) method is employed. The QCV is located downstream of the capacitance probe device, as shown in Fig. 9. When the system reaches a steady state, the two ball valves simultaneously close to
trap the gas and liquid in the test section. At the same time, the bypass valve opens to force the fluid to flow so the system could continue to work normally. A water box is added to reduce the effect of refraction of tube wall. The high-speed camera captures the side view of the test section (Fig. 9) to illustrate clear separation between gas and liquid phases. We estimate the height of gas from the interface to the top of the inner tube for void fraction calculation. To eliminate the system error, the averaged void fraction for eight times by the QCV system was regarded as the actual time-averaged void fraction under the flow condition.
Fig. 9. Quick closing valves system(D=50mm).
Al
Ag
(9)
Ag Al
D2 sin 8
(10)
Ag A Al π
2 cos 1 1
(11) θ
D hg D / 2
(12)
± ≤
Relative error (%)
5 4 3 2 1 0 -1 -2 -3 -4 -5 3
Relative error
+2.5%
-2.5%
6
9
12 15 Usg(m/s)
18
21
24
Fig. 10. Relative error of void fraction. Table 4 Maximum relative uncertainty of the measurement Parameters in present experiments. Measurement Parameter Gas flow rate
Measurement device
Maximum relative uncertainty
Gas Coriolis mass flow meter
3.29%
Electromagnetic flow meter
4.42%
Water Coriolis mass flow meter
0.521%
Pressure
Pressure transmitter
0.225%
Temperature
Temperature transmitter
1.06%
Void fraction
Multi-wire capacitance probe
3.71%
Liquid flow rate
4. Results and analysis 4.1. Water layer height To compare the water layer height at different circumferential positions, the equivalent water layer height, h*, is used and defined by Eq. (13)
hi*
hi Hi
(13)
As shown in Fig. 11, the time within 3s is employed to compare the different experimental results. It can be seen that the equivalent water layer heights at different circumferential positions are different, thus it is unreasonable to measure the void fraction of the circular pipe by one probe. The equivalent water layer height varies with time and is characterized by irregular "disturbance" waves under lower superficial gas velocity while the disturbance waves will more regular under higher superficial gas velocity. We also find that the
variation frequency of the water layer at different circumferential position is similar with each other (Figs. 12 and 13), and all the probes can detect the disturbance waves of the gas-liquid interface. The equivalent water layer height and the amplitude of the disturbance waves increase with the superficial liquid velocity increasing (Figs. 11-13). Figures 12 and 13 also show that the increase of superficial liquid velocity has little effect on
(a) 1.0
Y
Usg=6.73m/s, Usl=0.0214m/s
0.8 0.6
h1*
h2*
h3*
h4*
h5*
O
h*
Z
0.4 h1 h2 h3 h4 h5
0.2 0.0 0.0
(b) 1.0
0.5
1.0
1.5 2.0 Time (s)
h1*
h2*
h3*
h4*
h5*
O
Z
h*
0.6
3.0
Y
Usg=6.73m/s, Usl=0.0706m/s
0.8
2.5
0.4
h1 h2 h3 h4 h5
0.2 0.0 0.0
(c) 1.0
0.5
1.0
1.5 2.0 Time (s)
h1*
h2*
h3*
h4*
h5*
O
Z
h*
0.6
3.0
Y
Usg=14.06m/s, Usl=0.0288m/s
0.8
2.5
0.4 h1 h2 h3 h4 h5
0.2 0.0 0.0
(d) 1.0
0.5
1.0
1.5 2.0 Time (s)
h2*
h3*
h4*
h5*
O
Z
h*
0.6
h1*
3.0
Y
Usg=14.17m/s, Usl=0.113m/s
0.8
2.5
0.4 h1 h2 h3 h4 h5
0.2 0.0 0.0
0.5
1.0
1.5 2.0 Time (s)
2.5
3.0
Fig. 11. Variation of equivalent water layer height with time at different circumferential position (P=0.1MPa).
the disturbance wave frequency, whereas the disturbance wave frequency increases slightly as the superficial gas velocity is increased. The multi-wire capacitance probe method exhibits great potential to investigate the
wave structures and the wave oscillations from a two-dimensional perspective just as the WMS did [3].
h4 *
h5 *
0
5
10
15
20
25
0.006
Amplitude
h3 *
Amplitude
h2 *
0.006
Amplitude
(b)
h1 *
Amplitude
0.0027 0.0018 0.0009 0.0000 0.0027 0.0018 0.0009 0.0000 0.0027 0.0018 0.0009 0.0000 0.0027 0.0018 0.0009 0.0000 0.0027 0.0018 0.0009 0.0000
Amplitude
Amplitude
Amplitude
Amplitude
Amplitude
Amplitude
(a)
30
h1 *
0.003 0.000 h2 *
0.003 0.000 0.006
h3 *
0.003 0.000 0.006
h4 *
0.003 0.000 0.006
h5 *
0.003 0.000
0
5
Frequency (Hz)
10
15
20
25
30
Frequency (Hz)
Fig. 12. Frequency spectrogram of equivalent water layer height corresponding to Fig. 9(a) and (b). (a) Usg=6.73m/s,
(b)
h1 *
0.002 0.000 0.004
Amplitude
h2 *
0.002 0.000 0.004
Amplitude
h3 *
0.002 0.000 0.004
Amplitude
h4 *
0.002 0.000 0.004
h5 *
0.002 0.000 0
5
10
15
20
Frequency (Hz)
25
Amplitude
0.004
Amplitude
Amplitude
Amplitude
Amplitude
Amplitude
(a)
Amplitude
Usl=0.0214 m/s (b) Usg=6.73m/s, Usl=0.070 m/s (P=0.1MPa).
30
0.009
h1 *
0.006 0.003 0.000 0.009
h2 *
0.006 0.003 0.000 0.009
h3 *
0.006 0.003 0.000 0.009
h4 *
0.006 0.003 0.000 0.009
h5 *
0.006 0.003 0.000
0
5
10
15
20
25
30
Frequency (Hz)
Fig. 13. Frequency spectrogram of equivalent water layer height corresponding to Fig. 9(c) and (d). (a) Usg=14.06m/s, Usl=0.0288 m/s (b) Usg=14.17m/s, Usl=0.113 m/s (P=0.1MPa).
The average water layer heights at different circumferential positions and flow conditions are obtained by numerically time-averaging the water layer height signal, as shown in Fig. 14. It can be seen that the water
layer is nonuniform at different circumferential position due to the influence of gravity. The distribution curves of the layer are of the shape of downwardly concave particularly at lower superficial liquid velocity (Fig. 14(a) and (f)). Fukano and Ousaka [38], Flores et al. [39] pointed that this was caused by the pumping action of disturbance waves and the secondary flow effect of the gas phase. The water layer is higher in the middle of the pipe cross section (measured by No. 3 probe) than the other circumferential positions. The non-uniformity of the water layer will increase as the superficial liquid velocity is increased. It seems that the non-uniformity of the water layer will decrease as the superficial gas velocity is increased.
(a)
(b)
(c)
Usg=6.74m/s, Usl=0.00654m/s
Usg=6.73m/s, Usl=0.0214m/s
Usg=6.77m/s, Usl=0.0424m/s
-25-20-15-10 -5 0 5 10 15 20 25 Z (mm)
-25-20-15-10 -5 0 5 10 15 20 25 Z (mm)
-25-20-15-10 -5 0 5 10 15 20 25 Z (mm)
(e)
(d)
(f)
Usg=6.73m/s, Usl=0.0706m/s
Usg=7.07m/s, Usl=0.0972m/s
Usg=13.83m/s, Usl=0.00823m/s
-25-20-15-10 -5 0 5 10 15 20 25
-25-20-15-10 -5 0 5 10 15 20 25
-25-20-15-10 -5 0 5 10 15 20 25
Z (mm)
Z (mm)
Z (mm)
(g)
(h)
(i)
Usg=14.06m/s, Usl=0.0288m/s
Usg=14.17m/s, Usl=0.0564m/s
Usg=14.17m/s, Usl=0.113m/s
-25-20-15-10 -5 0 5 10 15 20 25
-25-20-15-10 -5 0 5 10 15 20 25
-25-20-15-10 -5 0 5 10 15 20 25
Z (mm)
Z (mm)
Z (mm)
Fig.14. Average water layer height at different circumferential positions (P=0.1MPa).
The flow structure from the front view and the average water layer distribution in cross section are compared in Fig. 15. It is found that the water layer height flow recorded by the high-speed camera increases
with the superficial liquid velocity increasing, which agrees well with the measurement by the capacitance probes. The disturbance of the gas-liquid interface increases as the superficial gas velocity is increased, which is also in accordance with the above analysis in Figs. 11-13.
(b)
(a) Usg=6.74m/s, Usl=0.00654m/s
Usg=6.73m/s, Usl=0.0214m/s
(d)
(c)
Usg=6.73m/s, Usl=0.0706m/s
Usg=6.77m/s, Usl=0.0424m/s
(f)
(e)
Usg=13.83m/s, Usl=0.00823m/s
Usg=7.07m/s, Usl=0.0972m/s
(g)
Usg=14.06m/s, Usl=0.0288m/s
(h)
Usg=14.17m/s, Usl=0.0564m/s
(i) Usg=14.17m/s, Usl=0.113m/s
Fig. 15. Flow structure from the front view and the average water layer distribution in cross section (P=0.1MPa).
4.2. Void fraction The local void fraction at different circumferential positions calculated by Eq. (5) is shown in Fig. 16. It can be seen that the local void fraction decreases from the wall to the center line of the pipe. We also find that the local void fraction decreases as the superficial liquid velocity is increased (Fig. 16 (a)-(e)). The distribution curves of the local void fraction are not flat and have a valley at the centerline. When further increasing the superficial gas velocity, the distribution curves are relatively flat, as shown in Fig. 16 (f) and (g). The local void fraction at different circumferential positions is consistent with the flow structure (Fig. 15). The average void fraction of the circular cross section calculated by Eq. (6) is shown in Figs. 17 and 18. The void fraction increases with the operating pressure when the gas volume fraction keeps constant. Figure 18 also shows that the void fraction increases with the gas volume fraction and decreases as the superficial gas velocity is increased. These results are consistent with the existing studies.
0.9
0.9
0.9
0.8
Usg=6.74m/s, Usl=0.00654m/s
0.7 0.6
0.8
Void fraction
(c) 1.0
Void fraction
(b) 1.0
Void fraction
(a) 1.0
Usg=6.73m/s, Usl=0.0214m/s
0.7 0.6
0.8
Usg=6.77m/s, Usl=0.0424m/s
0.7 0.6
0.5 0.5 0.5 -25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 Z (mm) Z (mm) Z (mm) (d) 1.0 (e) 1.0 (f) 1.0
0.8 0.7
Usg=6.73m/s, Usl=0.0706m/s
0.6
0.9
0.8 0.7
Usg=7.07m/s, Usl=0.0972m/s
0.6
Void fraction
0.9 Void fraction
Void fraction
0.9
0.8 Usg=13.83m/s, Usl=0.00823m/s
0.7 0.6
0.5 0.5 0.5 -25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 Z (mm) Z (mm) Z (mm) (i) 1.0 (g) 1.0 (h) 1.0
Usg=14.06m/s, Usl=0.0288m/s
0.7 0.6
0.5 -25 -20 -15 -10 -5 0 5 10 15 20 25 Z (mm)
0.8
0.9 Usg=14.17m/s, Usl=0.0564m/s
0.7 0.6
Void fraction
Void fraction
0.8
0.9
0.5 -25 -20 -15 -10 -5 0 5 10 15 20 25 Z (mm)
0.8 0.7
Usg=14.17m/s, Usl=0.113m/s
0.6 0.5 -25 -20 -15 -10 -5 0 5 10 15 20 25 Z (mm)
Fig. 16. Local void fraction at different circumferential positions (P=0.1MPa).
1.00 P=0.10MPa, Usg=6.85m/s
Measured void fraction
Void fraction
0.9
P=0.10MPa, Usg=13.85m/s
0.96
P=0.15MPa, Usg=6.25m/s P=0.15MPa, Usg=12.58m/s P=0.20MPa, Usg=5.85m/s
0.92
P=0.20MPa, Usg=11.35m/s P=0.30MPa, Usg=5.08m/s P=0.30MPa, Usg=10.19m/s
0.88 0.84 0.80 0.975
0.980 0.985 0.990 0.995 Gas volume fraction
Fig. 17. Effect of pressure on void fraction.
1.000
Measured void fraction
(a)
1.00 P =0.15 MPa Usg=6.25m/s
0.95
Usg=12.58m/s Usg=18.74m/s
0.90 0.85 0.80 0.985
0.990 0.995 Gas volume fraction
1.000
Measured void fraction
(b) 1.00 P =0.30 MPa Usg=5.08m/s
0.95
Usg=10.19m/s Usg=15.16m/s
0.90 0.85 0.80 0.975
0.980 0.985 0.990 0.995 Gas volume fraction
1.000
Fig. 18. Effect of superficial gas velocity on void fraction.
4.3. Comparison of existing correlations with experimental data Ten correlations are those of Lockhart–Martinelli (1949), Fauske (1961), Smith(1969), Rouhani (1970), Chisholm(1973), Minami–Brill (1987), Abdulmajeed (1996), Armand–Massina, Melkamu (2007), and Rassame–Hibiki (2018) are compared to predict the void fraction of the stratified flow. The relative error (RE) the average relative error (ARE), the average absolute relative error (AARE), and the standard deviation (SD) are employed to evaluate these correlations. These parameters are defined by Eqs. (14)-(17).
predicted measured 100 measured
RE
SD
(14)
ARE
1 N REi N i 1
(15)
AARE
1 N REi N i 1
(16)
1 N2
2 N N 2 N RE RE i i i 1 i 1
(17)
where αpredicted and αmeasured are the predicted void fraction and measured void fraction, respectively, N is the
total test number. Studies [40-42] showed that the void fraction correlation directly influences the two-phase mixture density and this influence is of different magnitude for different ranges of the void fraction. Ghajar and Bhagwat [40] reported that an error less than ±10 % is considered acceptable for the range of the void fraction between 0.75 and 1 (0.75＜α＜1). As shown in Fig. 19 and Table 5, all of the correlations predict the void fraction with the relative error less than ±10%. The correlations of Rouhani (1970), Abdulmajeed (1996), Armand–Massina and Melkamu (2007) are able to predict more than 90% of the data points within the RE of ±5.0%, among which the correlations of Rouhani (1970) and Armand–Massina predict all the data points with the RE of ±5.0%. For the more restrictive ±3.0% error band, the correlation of Armand–Massina performs best with 95.65% of the data points within the RE of ±3.0%. the next best prediction comes from Rouhani(1970) with 84.06% of the data points within the RE of ±3.0%. The other correlations predict less than 80% of the data points within the RE of ±3.0%, which is not so satisfactory. From the statistical results in Table 6, we also find that the correlations of Rouhani (1970), Abdulmajeed (1996) and Armand–Massina tends to underestimate the void fraction (ARE＜0), whereas the others overestimate the void fraction data (ARE＞0). The AARE of correlation by Armand–Massina is minimum followed by the correlations of Rouhani (1970) and Melkamu (2007). According to the above comparisons and analysis, the correlation of Armand–Massina shows the best prediction capability, and has more simpler equation form, thus is recommended to predict the void fraction of stratified gas-liquid flow. Lockhart -Martinelli(1949) Fauske (1961) Smith (1969) Rouhani (1970) Chisholm (1973)
fraction Predicted Void fraction PredictedVoid
1.00
Minami and Brill (1987) Abdulmajeed (1996) Armand-Massina Melkamu (2007) Rassame-Hibiki(2018)
0.95
1.00 0.95
+10.0%
0.90
0.90
+10.0%
0.85
+5.0%
0.85
0.80
+5.0%
+3.0% -3.0%
0.75 0.80 0.75
+3.0%
0.80
-3.0%
0.75 0.75
-5.0%
-10.0%
-10.0% -5.0%0.85
0.90 Measured Void fraction
0.80 0.85 0.90 0.95 Measured Void fraction
0.95
1.00
1.00
Fig. 19. Comparisons of existing correlations with measured experimental data.
Table 5 Confidence level of relative error predicted by correlations under different error band. Correlation
Confidence level (%)
Error band
±3.0%
±5.0%
±10.0%
Lockhart–Martinell (1949)
47.83
81.16
100
Fauske (1961)
66.67
85.51
100
Smith(1969)
15.94
78.26
100
Rouhani(1970)
84.06
100
100
Chisholm(1973)
33.33
68.12
100
Minami–Brill (1987)
39.13
65.22
100
Abdulmajeed (1996)
66.67
91.3
100
Armand–Massina
95.65
100
100
Melkamu (2007)
71.01
92.75
100
Rassame–Hibiki (2018)
26.09
75.36
100
Table 6 Statistical results using present measured data. Correlation
RE (%)
ARE (%)
AARE (%)
SD (%)
Lockhart–Martinell (1949)
1.12–7.09
3.37
3.37
1.57
Fauske (1961)
-8.61–5.03
0.13
2.67
3.29
Smith(1969)
2.28–7.13
4.04
4.04
1.23
Rouhani(1970)
-3.87–3.10
-1.07
1.66
1.70
Chisholm(1973)
1.82–7.87
4.16
4.16
1.68
Minami–Brill (1987)
0.59–10.20
4.19
4.19
2.38
Abdulmajeed (1996)
-5.54–5.89
-0.92
2.50
2.79
Armand–Massina
-3.09–3.54
-0.44
1.41
1.62
Melkamu (2007)
-3.39–6.81
1.19
2.28
2.57
Rassame–Hibiki (2018)
0.69–8.06
4.02
4.02
1.45
5. Conclusions A measurement device of the void fraction based on the wire capacitance probe was developed. The water layer height and its variations with time were analyzed. Based on which the local void fraction at different circumferential positions and the average void fraction of the circular cross section were obtained. Ten existing correlations were compared using the measured void fraction data and the best performing correlation was recommended. The main conclusions are as follows:
1)
The equivalent water layer heights at different circumferential positions are different and vary with time. The variation frequency at different circumferential positions is similar with each other and less affected by the increasing of superficial liquid velocity, whereas it increases slightly as the superficial gas velocity is increased. The distribution curves of the layer are of the shape of downwardly concave particularly at lower superficial liquid velocity. The water layer in the middle of the pipe cross section is higher than that of at the other circumferential positions. The non-uniformity of the water layer will increase as the superficial liquid velocity is increased.
2)
The distribution curves of the local void fraction are not flat and have a valley at the centerline; the curves are relatively flat when further increasing the superficial gas velocity. The local void fraction decreases from the wall to the centerline of the pipe and decreases as the superficial liquid velocity is increased. The average void fraction decreases with the superficial gas velocity and increases with the operating pressure.
3)
Comparisons of the existing correlations using the measured void fraction demonstrate that all the ten correlations predict the void fraction with the relative error less than ±10%. The correlation of Armand–Massina is recommended to predict the void fraction of stratified gas-liquid flow owing to it has the best prediction capability and simpler equation form. The relative error of the void fraction predicted by this correlation is within ±3.0% under the 95.65% confidence level.
Acknowledgements The National Natural Science Foundation of China under Grant No. 51709227, the China National Funds for Distinguished Young Scientists under Grant No. 51425603 are acknowledged for supporting this study.
Appendixes Appendix A:Void fraction correlation Table A1 Void fraction correlations considered for this study. Author/source
Void fraction correlation
Lockhart–Martinelli (1949)
0.36 0.64 1 x g l 1 0.28 x l g
Fauske (1961)
1 x g 0.5 1 x l
Smith(1969)
1 g l
Rouhani (1970)
x x 1 x U gm C g 0 g l G C0 1 0.2 1 x
0.07 1
1
l g 0.4 1 x 1 1 x 0.4 0.6 1 0.4 1 x 1 x
0.5
1
1
0.25 1.18 U gm 0.5 g l g l
Chisholm(1973)
1 1 x 1 l g
Minami–Brill (1987)
1 x g x l
1
ln Z 9.21 4.3374 8.7115
exp Z Abdulmajeed (1996)
1.84U sl0.575 l0.5804 U sg D 0.0277 g 0.3696 0.1804
1 0.528 U sgU sl
0.25
0.216121
P 101325
0.05
l0.1
El theo
For turbulent flow
El theo exp 0.9304919 0.5285852R 9.219634 102 R2 9.02418 104 R4 For laminar flow
1.1 0.6788496 R 0.1232191 102 R 2 1.778653 103 R 3 E exp l theo 3 4 1.626819 10 R
U sg g l U sl l g
where R ln X and X
L
lU sl2 2 gU sg
where L=0.2 for turbulent flow and L=1 for laminar flow Armand–Massina
1 x g 0.833 0.167 x 1 x l
Melkamu (2007)
1
U sg U U sg 1 sl U sg
g l
0.1
0.25 gD 1 cos l g P 1.22 1.22sin atm 2.9 2 l Psystem
Rassame–Hibiki (2018)
U sg C0U gm U gm
For 0 ≤ β < 0.9 0.5 1.50 1.50 g C0 0.800exp 0.815 0.800exp 0.815 1 0.900 0.900 l
For 0.9 ≤ β ≤1
C0 8.08 9.08 8.08 1 g l
0.5
U gm 0
Appendix B: Experimental data Table B1 Experimental data in present experiment. P (MPa)
T (℃)
Usg (m/s)
Usl (m/s)
h1(mm)
h2(mm)
h3(mm)
h4(mm)
h5(mm)
α
β
x
0.104489
15.4
6.741
0.00654
0.82
1.1
2.59
1.82
1.34
0.9661
0.999
0.7194
0.106279
17.2
6.792
0.01334
1.26
2.13
3.65
2.76
2.1
0.9473
0.998
0.5596
0.103083
18.6
6.733
0.02142
1.62
3.18
4.55
3.59
2.55
0.9315
0.9968
0.4346
0.106448
19.3
7.034
0.02754
1.89
3.81
5.15
4.24
2.82
0.9209
0.9961
0.3877
0.104707
20.6
6.772
0.04241
2.47
4.78
6.26
5.3
3.43
0.9016
0.9938
0.281
0.104838
21.3
6.883
0.05618
3.04
5.57
7.1
6.12
4
0.8856
0.9919
0.2304
0.10339
22
6.73
0.0706
3.65
6.34
8.01
6.89
4.61
0.8691
0.9896
0.1874
0.107765
22.7
6.893
0.08217
4.21
6.9
8.54
7.83
5.16
0.8549
0.9882
0.1714
0.104997
22.9
7.067
0.09721
5.23
7.54
9.24
8.19
5.75
0.8394
0.9864
0.1502
0.104918
15.2
13.83
0.00823
1.07
1.4
1.71
1.4
1.46
0.9682
0.9994
0.8074
0.105668
16.3
13.47
0.01448
2.25
2.14
2.43
1.94
1.95
0.9512
0.9989
0.6989
0.105577
18.3
14.06
0.02884
2.73
2.4
3.66
3.11
2.56
0.9346
0.998
0.5471
0.105411
20.4
14.17
0.05636
3.27
3.62
5.53
4.5
3.67
0.9075
0.996
0.3819
0.107609
21.6
13.42
0.08624
4.23
5.06
7.02
5.8
4.71
0.8795
0.9936
0.2778
0.105855
21.9
14.17
0.11254
4.91
5.8
8.15
6.78
5.41
0.8606
0.9921
0.2357
0.104298
15.4
20.2
0.0073
1.52
1.02
1.28
1.16
1.46
0.9701
0.9996
0.8731
0.104677
16.9
20.23
0.01443
2.43
1.76
1.71
1.6
2.3
0.9544
0.9993
0.7765
0.154078
15.1
6.328
0.00642
0.88
1.42
2.49
1.74
1.37
0.965
0.999
0.7528
0.154
17.1
6.271
0.01447
1.32
2.41
3.78
2.79
2.07
0.9452
0.9977
0.5708
0.15151
18.5
6.251
0.0225
1.67
3.09
4.67
3.59
2.55
0.9311
0.9964
0.4566
0.157276
19.2
6.074
0.02785
1.96
3.55
5.22
4.13
2.83
0.9217
0.9954
0.4024
0.152107
20.4
6.264
0.04361
2.52
4.36
6.32
5.25
3.45
0.9031
0.9931
0.302
0.154156
21.2
6.216
0.05688
3.1
5.28
7.19
6.13
4.02
0.886
0.9909
0.2486
0.150947
21.8
6.418
0.07091
3.28
6.02
7.99
6.88
4.65
0.8723
0.9891
0.2127
0.152969
22.3
6.151
0.0851
3.92
6.77
8.98
8
5.37
0.8534
0.9864
0.1783
0.150098
22.4
6.263
0.10568
4.99
7.41
9.77
8.61
6.11
0.8355
0.9834
0.1496
0.152063
14.8
12.68
0.00778
1.05
1.16
1.7
1.32
1.48
0.9696
0.9994
0.8334
0.152601
16.2
12.52
0.01596
1.4
1.79
2.53
1.92
1.91
0.957
0.9987
0.706
0.152841
18.2
12.59
0.03021
2.75
2.69
3.56
2.95
2.61
0.934
0.9976
0.5593
0.151648
20.3
12.82
0.05625
3.32
3.5
5.48
4.3
3.56
0.9093
0.9956
0.4068
0.151757
21.9
12.53
0.08468
4.3
4.77
6.95
5.54
4.61
0.8822
0.9933
0.307
0.152651
22
12.34
0.11405
4.92
5.78
8.24
6.16
5.47
0.8626
0.9908
0.2452
0.156765
15.6
18.65
0.00722
1.02
1.89
1.37
1.17
2.2
0.9648
0.9996
0.8896
0.15457
17
18.82
0.01417
2.5
1.57
1.91
1.7
3.35
0.9481
0.9992
0.8034
0.201385
15
5.679
0.00587
0.9
1.37
2.16
1.56
1.3
0.9675
0.999
0.7799
0.202457
16.8
5.558
0.01419
1.31
2.14
3.44
2.61
1.94
0.9493
0.9975
0.5888
0.199876
18.1
5.853
0.0222
1.6
2.76
4.31
3.39
2.35
0.9362
0.9962
0.4875
0.20229
18.9
5.97
0.02859
1.91
3.28
5
4.04
2.69
0.9252
0.9952
0.431
0.202171
20.1
5.715
0.04299
2.47
4.2
6.14
5.14
3.36
0.9057
0.9925
0.3243
0.20246
21
5.863
0.0573
3.1
5.02
7.09
6.14
4.01
0.8875
0.9903
0.2694
0.199712
21.6
6.115
0.0715
3.69
5.68
7.88
6.91
4.65
0.8719
0.9884
0.2336
0.204241
22.2
5.811
0.08692
4.43
6.48
8.91
8.08
5.41
0.8517
0.9853
0.1944
0.201229
22.2
6.098
0.10573
5.04
7.65
9.67
8.88
6.18
0.8331
0.983
0.1709
0.203854
14.1
11.24
0.0076
1.08
1.27
1.78
1.45
1.44
0.9683
0.9993
0.8456
0.204686
15.8
11.41
0.01506
1.4
1.66
2.53
2.12
1.96
0.9564
0.9987
0.7367
0.201159
18.6
11.37
0.03646
2.23
3.04
4.2
3.73
2.93
0.9277
0.9968
0.53
0.202884
20.1
11.41
0.05721
3.63
3.86
5.72
4.89
3.62
0.9024
0.995
0.4192
0.204459
21.6
11.41
0.08897
4.52
5.06
7.46
6.38
4.88
0.8729
0.9923
0.317
0.202868
21.8
11.3
0.11884
5.27
6.14
8.63
7.36
5.72
0.8513
0.9896
0.2549
0.203437
16.5
17.32
0.01415
2.48
1.55
2.17
1.76
2.66
0.9506
0.9992
0.8179
0.203327
17.9
21.34
0.01518
2.6
2.26
2.79
2.18
2.76
0.9422
0.9993
0.837
0.301425
15.3
5.032
0.01043
1.09
1.9
3.09
2.26
1.74
0.9553
0.9979
0.7016
0.300847
16.8
5.109
0.01465
1.35
2.31
3.78
2.84
2.12
0.9451
0.9971
0.6279
0.302543
17.8
5.102
0.02188
1.69
2.92
4.65
3.67
2.56
0.9314
0.9957
0.5304
0.30246
18.4
4.989
0.0284
2.02
3.45
5.31
4.32
2.87
0.9205
0.9943
0.4591
0.300592
19.2
5.225
0.0361
2.36
3.95
5.9
4.94
3.16
0.9101
0.9931
0.4098
0.303191
19.8
5.094
0.0437
2.7
4.49
6.54
5.65
3.59
0.8983
0.9915
0.3597
0.30417
21.3
5.044
0.07204
3.11
6.17
8.54
7.6
5.31
0.8639
0.9859
0.2518
0.300521
22.1
5.104
0.08825
4.79
6.95
9.23
8.25
5.06
0.8476
0.983
0.2155
0.30387
22
5.033
0.10647
5.89
7.73
10.2
8.5
6.78
0.8249
0.9793
0.1847
0.305137
14.6
9.814
0.00767
1.09
1.23
1.9
1.43
1.52
0.9676
0.9992
0.8631
0.303139
16.5
9.822
0.01591
1.4
1.96
2.75
2.13
1.98
0.9542
0.9984
0.7504
0.303611
17.5
10.08
0.02225
2.03
2.27
3.31
2.69
2.23
0.9437
0.9978
0.6876
0.301632
19
10.31
0.03696
2.34
3.13
4.44
3.63
4.19
0.9197
0.9964
0.5731
0.29899
21
10.46
0.07374
3.99
4.55
6.71
3.86
5.76
0.8869
0.993
0.4023
0.302721
21.7
10.66
0.10799
4.89
5.93
8.2
4.75
6.84
0.8611
0.99
0.3207
0.303369
17.4
15.34
0.01577
2.67
1.52
2.06
1.81
2.47
0.9509
0.999
0.8254
0.29956
19.7
14.97
0.04462
3.32
3.3
4.27
3.46
3.62
0.9182
0.997
0.6158
0.302467
16.8
17.82
0.0156
2.66
2.29
2.76
2.26
2.99
0.9403
0.9991
0.8474
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Highlights
1.
A multi-wire capacitance probe measurement device was developed.
2.
Water layer height and void fraction at different circumferential positions was measured.
3.
Water layer height and its variations with time at different circumferential positions were analyzed.
4.
Void fraction prediction correlations were compared and best recommended.
performing correlation was