Application of the image analysis on the investigation of disturbance waves in vertical upward annular two-phase flow

Journal Pre-proofs Application of the image analysis on the investigation of disturbance waves in vertical upward annular two-phase flow Ruinan Lin, Ke Wang, Li Liu, Yongxue Zhang, Shaohua Dong PII: DOI: Reference:

S0894-1777(19)31353-6 https://doi.org/10.1016/j.expthermflusci.2020.110062 ETF 110062

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Experimental Thermal and Fluid Science

Received Date: Revised Date: Accepted Date:

20 August 2019 24 December 2019 26 January 2020

Please cite this article as: R. Lin, K. Wang, L. Liu, Y. Zhang, S. Dong, Application of the image analysis on the investigation of disturbance waves in vertical upward annular two-phase flow, Experimental Thermal and Fluid Science (2020), doi: https://doi.org/10.1016/j.expthermflusci.2020.110062

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Application of the image analysis on the investigation of disturbance waves in vertical upward annular two-phase flow Ruinan Lin1, Ke Wang1*, Li Liu2, Yongxue Zhang1, Shaohua Dong3** 1. Beijing Key Laboratory of Process Fluid Filtration and Separation, China University of Petroleum, Beijing 102249, China 2. School of Nuclear Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China 3. Pipeline Research Center, China University of Petroleum-Beijing, China * 1st Corresponding author: [email protected] (Ke Wang) ** 2nd Corresponding author: [email protected] (Shaohua Dong)

Abstract: Annular two-phase flow commonly exists in industries process, and the disturbance wave at the gas-liquid interface plays an essential role in the mass, momentum and energy exchange. Although extensive experimental and analytical approaches to properties of disturbance waves have been implemented, most of the empirical correlations constructed to predict the wave properties have defects in prediction, especially under the high-pressure conditions. In the present paper, an image analysis method by acquiring the spatiotemporal distribution of the liquid film is employed to investigate the wave properties in a 20 mm I.D. tube in the upward annular regime. Because the interfacial waves should first penetrate through the gas boundary layer before full development, the effect of the gas boundary layer cannot be ignored and is undoubtedly enhanced when pressure increases. Accordingly, new empirical correlations for wave velocity and frequency are proposed by introducing the excess liquid Reynolds number. Compared with the experimental data from the literature, the proposed equations agree well with the experimental data within the averaged relative deviation of ±35%, especially under high-pressure conditions. Keywords: Disturbance wave, Wave properties, Annular flow; Image analysis

1.

Introduction Annular flow is one of the most important two-phase flow regimes, which is often encountered

in many industrial applications. It is characterized as a liquid film at the pipe wall and a continuous 1

gas core with entrained droplets in the tube center. Remarkably, various scales of interfacial waves in wavelength and amplitude are usually present at the gas-liquid interface, playing an essential role in the mass, momentum and energy exchange. Essentially, the generation and development of interfacial waves mainly depend on the competition between Kelvin-Helmholtz instability and surface tension[1]. This instability, due to the unequal dynamic pressure on either side of the interface, occurs when there is sufficient velocity difference across the interface between two fluids. Generally, Weber number is considered as a measure of the relative strength of the KelvinHelmholtz instability relative to the dispersive stabilization effect associated with surface tension, which can be defined as[2]:

 g  ug  ul  D 2

We 



(1)

where D, ρg, ug, ul, σ are pipe diameter, gas density, gas velocity, liquid velocity and surface tension, respectively. When the Weber number is relatively small, the surface tension is the dominant force which makes the interface stable at equilibrium; in contrast, waves will generate when KelvinHelmholtz instability dominates. There have been enduring efforts to investigate the interfacial waves in annular two-phase flow. Generally, three main types of interfacial waves are distinguished as ripples, disturbance waves and huge waves, depending on their properties[3–6]. Ripples have small amplitudes compared with the liquid film thickness, move with low velocity and short lifetime, and they usually do not occupy the whole tube circumference[3,4,7]. Disturbance waves usually occupy the whole tube circumference and play an essential role in determining the typical characteristics of annular flow. They have a longer lifespan and serval times higher than the liquid film thickness, but smaller momentum compared with the huge waves[6–8]. Huge waves, usually accompanied by a large number of bubbles, have a larger amplitude and velocity than disturbance wave[9,10]. Owing to the effect of the interfacial shear stress, the wave crest is “stretched” to a thin ligament and sheared off into the gas core during wave movement, and immediately broken down into droplets[11]. This socalled ligament entrainment is of great concern for the design of industrial processes, which is of utmost importance for the calculation of pressure drop, the determination of film flow rate and the prediction of heat transfer [12–14]. Excessive liquid entrainment, particularly in nuclear power 2

plants, will cause the complete removal of the liquid film from the wall (dryout), resulting in disastrous accidents [15,16]. To avoid those catastrophic events, more experimental data and indepth analysis are required to investigate the properties of disturbance waves in annular flow. So far, extensive research has been carried out in properties of disturbance waves in the annular flow[5,17–25]. A preliminary review of disturbance waves in annular flow has been published by Azzopardi [20], and an updated summary of wave properties can be found in his later work [26] and work of Berna et al. [27]. Various studies have verified that both the wave velocity and frequency increase with increasing gas and liquid flow rates [24,28–31]. Remarkably, wave velocity is more affected by gas flow than liquid flow. There is a dispute on the pipe diameter effect on wave characteristics, while Martin [32] proposed that pipe diameter have no substantial effects on wave velocity and observed an inverse relationship between diameter and wave frequency, Kaji et al. [33] showed that the wave frequency and velocity decrease with the increase of pipe diameter. Besides, the effect of pressure has not been given sufficient attention, and only a small amount of studies are directed at this effect [33,34], indicating that frequency and velocity of wave increase with the increasing pressure, while the amplitude and wavelength are on the contrary. Empirical correlations have been constructed to predict the velocity and frequency of disturbance waves. However, they are obviously of some certain limitations at different flow conditions. Numerous measurement methods are employed to investigate disturbance waves in annular flow based on the varying liquid film. Conductance probes, as a contact measurement method, are widely used[9,17,20,22,25,31,33,35]; however, it may disturb the downstream flow and has a relatively low spatial resolution. Therefore, the demand of such non-contact measurement methods (optical measurement techniques, for instance) is put forward. The laser-based method and highspeed photography method are two main non-contact measurement methods. Among so many laserbased methods, laser-induced fluorescence method (LIF) and its improvements such as planar laserinduced fluorescence (PLIF) and brightness-based laser-induced fluorescence (BBLIF) are the best representatives with the high spatial and temporal resolution [36–41]. There can be no doubt that such laser-based methods are more accurate than high-speed photography; however, its defects such as expensive, complex and sophisticated limit its application. The high-speed camera offers an economical optical method to capture a clear process of the generation and development of the 3

disturbance waves. Usually, high-speed photography is employed, more likely, to qualitatively investigate the behaviour of the interfacial waves in annular flow. Although quantitative measurements based on the direct high-speed photography for the thickness of the liquid film, wave frequency and velocity are available in the literature[21,23,42–44], accuracy is an inevitable weakness. Therefore, it is particularly valuable to develop a more accurate measurement with the help of direct photography. This paper presents the development of the velocity and frequency predictions of disturbance waves in vertical upward flow. The experiments of gas-liquid annular flow have been conducted in a 20-mm-inside-diameter pipe under room temperature and atmospheric pressure conditions, and the velocity and frequency of disturbance waves are obtained by the image analysis method. By analyzing and verifying the experiment data from literature and our experiments, we present the limitations of existing empirical correlations. To address this deficiency, we take the effect of the gas boundary layer into consideration and develop more applicable models for both wave velocity and frequency, albeit empirical, those incorporate three essential features: (1) the effect of diameter, (2) the effect of flow condition, (3) the effect of pressure on phase properties.

2.

Experimental facility and method

2.1 Experimental facility

Fig. 1 shows the schematic of the test facility which mainly consists of water and air supply systems, a test section as well as a measurement system. Air-water upward annular flow experiments are conducted in a 5.5 m long acrylic pipe of 20 mm internal diameter. In order to minimize the entrance effect, the porous wall mixer which features 22 rows of 88 holes of 1 mm in diameter is carefully designed, as illustrated in Fig. 2. In the present study, a FR-625 high-speed camera with Nikon 60 mm f/2.8 Micro lens is employed for the flow visualization. The frame setup is 1280*1024 pixels, and the spatial resolution is 0.04 mm/pixel. Also, the sampling frequency and shooting time are, respectively, set to be 500 frames/s and 5 s. To minimize the effect of refraction, the pipe at the shooting area is equipped with a transparent acrylic square box which is filled with water.

4

Fig.1 Schematic diagram of the experimental facility

Fig. 2 Design of porous walls mixer

2.2 Experimental procedure Note that all the experiments are carried out under atmospheric pressure. The temperature signals can be monitored through the liquid Coriolis mass flowmeter in the present study, and we conduct our experiment under the room temperature (25℃). Air is injected with a compressor unit at the bottom of the porous wall mixer, while water is fed into the pipeline from the centrifugal pump through small holes of porous wall mixer. Meanwhile, to ensure a fully developed annular flow, all measurements are located after the developing length of L/D =150 (the ratio of measuring 5

position from the inlet to pipe internal diameter). In the present study, the superficial liquid velocity ranges from 0.07 to 0.71 m/s while the superficial gas velocity from 10.41 to 30.82 m/s, i.e. all experiments are conducted corresponding to the annular flow regime (see Fig. 3).

gu2sg (kg·s-2·m-1)

104

Annular

103

Wispy-Annular

102

Bubble 101

Churn

Slug

0

10

10-1 100

101

102

103

104

105

106

lusl2 (kg·s-2·m-1) Fig.3 The flow operating point in flow pattern map by Hewitt and Roberts [45]

2.3 Data reduction By acquiring the spatiotemporal distribution of the liquid film, an image processing program is developed to identify and characterize the disturbance waves. As illustrated in Fig.4, The rectangular portion of the liquid film thickness area is extracted, and an image processing program is developed by MATLAB to convert the true colour image RGB to a binary image and minimize the effect of bubbles and droplets around the liquid film. Additionally, the function hysteresis3d inspired by Peter Kovesi’s 2D hysteresis function is adopted, providing two thresholding at the black shadow of the tube wall and gas core, respectively. Accordingly, the liquid film thickness of one-pixel-wide portion at any time can be acquired by the white area of the binary image. Consequently, the spatiotemporal distribution of liquid film thickness in the complete view at different times and locations is possible to acquire. The detailed data reduction can be referred to our previous paper[10].

6

Fig.4 Extraction of liquid film thickness by image analysis method

Fig. 5 depicts a typical spatiotemporal distribution of liquid film thickness. Note that the flow direction is from right to left. The disturbance waves are distinctly identified by higher velocity (compared with the slow ripples nearby), longer existence time than ripples (exists in the whole axial shooting area), and at least twice larger than the amplitude of ripples nearby. Meanwhile, ripples around disturbance wave are absorbed and released with lower velocity. Additionally, the trajectories of disturbance waves are identified manually by GetData Graph Digitizer. Thus, the velocity of disturbance wave can be obtained by the slope of each spatiotemporal trajectory, and the frequency of disturbance wave can be estimated relating to the number of waves.

7

(a)

(b)

Fig.5 Example of spatiotemporal distribution of liquid film thickness, usg=19.36 m/s, usl=0.12 m/s (a. Spatiotemporal records of the liquid film; b. Evolution of the interfacial waves)

2.4 Error analysis The water flow is measured by liquid Coriolis mass flowmeter with the uncertainty of 0.5%, and the gas flow is measured by gas thermal mass flowmeter with the uncertainty of 0.5%. In the present study, the identification of interfacial waves is based on the extraction of liquid film thickness. Thus, the precise measurement of the thickness of the liquid film directly affects the accuracy of wave properties obtained. As illustrated in Fig.6, due to the pixels and measured liquid film thickness values are discrete, the limit of the ideal error is within one pixel (about 0.04 mm); therefore, the actual error of the film thickness measurement might be larger than one pixel (about 0.04 mm) due to the limitation of the camera resolution, camera focus and optical distortion. Meanwhile, the different refractive indices of acrylic pipe, water and air lead the imaging value of the liquid film thickness slightly higher than the actual value, which detailed quantification of these errors can be referred to our previous work[10,46]. Note that only two-dimensional (plane) characteristics of the gas-liquid interface are extracted in the present study. Inevitably, there is a slight difference between the measurement and the real value because the film thickness in the vicinity of the investigated section of the pipe is non-uniform along the circumferential coordinate. According to the work of Alekseenko et al.[38], the two-dimensional approach is relatively reliable for the wave characteristics in annular flow. Since there are many factors that affect the measurement of liquid film thickness, it is worthwhile discussing the reliability of the measured wave properties in the present study. According to the spatiotemporal records of liquid film thickness, the amplitude, velocity and existence time of different interfacial waves are obtained. For ripples, the errors in liquid film thickness can significantly lead the inaccurate recognition, yielding a greater uncertainty in wave properties analysis. With regard to disturbance waves or huge waves, we believe that the effect of these errors is limited. The disturbance waves, for instance, have a height several times the mean film thickness and travel at a velocity greater than that of the film. Accordingly, they are relatively easier to be distinguished from the ripples nearby by analyzing the 8

differences in amplitude, velocity and existence time. Therefore, we are unlikely to see any significant impact on the accuracy of the measured wave properties.

Fig.6 Conversion of pixel values and real values

In addition, the existence of a large number of bubbles in the chaotic flow field would affect the precision of the liquid film thickness measurement (see Fig. 7a). The similar chaotic flow field was also observed by Sekoguchi and Takeishi[9] where they defined them as huge waves. As depicted in Fig.7b, huge waves have many discontinuity points (bubbles effect) and are obviously faster than normal disturbance waves. Therefore, it is easy to remove theses huge waves in the chaotic flow field when calculating the wave velocity, and only the velocity of identifiable disturbance wave in the stable region is measured. To improve the accuracy of wave frequency measurement, the waves in both stable region and chaotic flow field are taken into account. In order to minimize human errors, the sample size for each set of flow conditions is over 30 to obtain the average value of wave velocity and frequency.

(a)

(b)

9

Fig. 7 Huge wave in the chaotic flow field (a. huge wave captured by high-speed camera, usg=14.14 m/s, usl = 0.22 m/s; b. Spatiotemporal records of liquid film thickness, usg=25 m/s, usl = 0.07 m/s)

3.

Results and discussion

3.1 Wave velocity Fig. 8 depicts the variation of the wave velocity under different flow conditions. Obviously, the wave velocity increases with the increase in both superficial gas and liquid velocities, which is in agreement with the work from the literature[20,24–27]. It is also worthwhile to mention that the effect of the gas velocity on the wave velocity apparently becomes less pronounced at lower liquid mass flowrate but more prominent at higher liquid mass flowrate.

2.8

wave velocity v (m/s)

2.4

usg= 10.41 m/s usg = 19.36 m/s usg = 30.82 m/s

usg = 14.14 m/s usg = 25.00 m/s

2.0 1.6 1.2 0.8 0.4 0.0

0.2

0.4

0.6

0.8

Superficial liquid velocity usl (m/s) Fig. 8 Variation of wave velocity under different flow conditions

As illustrated in Table 1, empirical correlations have been widely constructed to predict disturbance wave velocity v in literature. Fig.9 shows the performance of the listed empirical correlations against the present experimental data and the experimental data existing in literature[17,21–23,25,33,34,47,48]. Obviously, the present experimental data fit well with the Marmottant’s correlation[49] and Al-Sarkhi’s correlation[17], but Kumar’s correlation[50] and Schubring’s correlation[23] apparently overpredict the data. It is suspected that they did not strictly distinguish huge waves from disturbance waves under annular flow conditions, but the influence of 10

such huge waves are ignored in the present study. Experimental data from similar non-contact highspeed photography method by Dasgupta et al. [21] are also smaller in these two correlations due to the removal of short-lived waves. Although the empirical correlation proposed by Schubring et al.[23] provides a fair degree of accuracy in experiments by direct measurement; it underestimates the wave velocity at higher-pressure conditions, and the deviation increases with the increasing pressure. Meanwhile, the correlation proposed by Schubring et al. [23] overestimates the wave velocity in some experiments by non-contact measurement [21,24] and present experiment. It is evident that the empirical correlation proposed by Kumar et al. [50] provides relatively good accuracy in atmospheric pressure condition, except for a small amount of experimental data has a large error. However, the correlation proposed by Kumar et al. [50] also underestimates the wave velocity at higher-pressure conditions. Since the existing correlations fail to predict the velocity of disturbance waves, especially under high-pressure conditions, it is necessary to find a more reasonable empirical correlation according to the physical mechanism during the generation and development of disturbance waves.

Table 1 Summary of empirical correlations for the wave velocities References Swanson[51]

Empirical correlations v  vfric,g 

w g

Where τw is wall shear stress. Kumar et al.[50]

v

Pearce[52]

v

0.25

g l

 Rel     usg  usl  Re g  0.25   Re  1  5.5 g  l   l  Reg 

5.5

Kulf  usg g / l

u lf 

K   g / l D m lf 4  l

Where K,  , D and m lf are Pearce coefficient, average film thickness, pipe diameter and liquid film flow rate, respectively. Schubring and Shedd[8]

v  vfric,g 

w g

w 0.0109Gusg Reg0.15 Where G is mass flux.

11

Marmottant and Villermaux[49]

 g usg   l usl

v Al-Sarkhi et al.[17]

g  l

v  1.942 usl X 0.91

Where X is Lockhart-Martinelli number:

Wang et al.[34]

l g

X=

usl usg

v

 u gc  u lf 1   g  E   l  

 = 0 .0 0 1 7 

 0 .5

 Rel   R e  g  

 0 .2 4

 Psyste m     Pa tm 

0 .4 8

Where u gc and u lf are the gas core velocity and mean velocity of the liquid film surface, respectively.

Atmospheric pressure

Higher pressure 10

8

+35%

+35%

8

6

Calculated v (m/s)

Calculated v (m/s)

Kumar’s correlation[50]

7

5 4

-35%

3 2

6 -35%

4 2

1 0

0

1

2

3

4

5

6

7

0

8

0

2

Experimental v (m/s)

6

8

10

Experimental v (m/s) +35%

10

12 10 -35%

8 6 4

Calculated v (m/s)

+35%

14

Calculated v (m/s)

Schubring’s correlation[23]

16

4

8 6

-35%

4 2

2 0

0 0

2

4

6

8

10

12

14

16

0

2

4

6

8

Experimental v (m/s)

Experimental v (m/s)

12

10

+35%

Calculated v (m/s)

8

5 4

-35%

3 2

6

-35%

4 2

1 0

0

1

2

3

4

5

6

0

7

0

2

Experimental v (m/s)

8

10

+35%

+35%

-35%

-35%

260 40 1 20 1

0

2

3

4

5

20 Experimental 40 60 v (m/s) 80

6

7

100

8 6 4

8

Calculated v (m/s)

10

+35%

140 5 120 4 100 380

Calculated v (m/s)

Calculated v (m/s)v (m/s) Calculated

6

Experimental v (m/s)

6

00 0 -20

4

10

7

Al-Sarkhi’s correlation[17]

10

+35%

6

Calculated v (m/s)

Marmottant’s correlation[49]

7

+35%

6

-35%

4 -35%

2

2

0 0

0

0

2 2

4

6

8

10

4 Experimental 6 v8 (m/s) 10

Experimental v (m/s)

Experimental v (m/s)

Wolf et al.(2001), 0.10 MPa Kaji et al.(2010), 0.10 MPa Belt et al.(2010), 0.10 MPa Schubring(2010), 0.10 MPa Al-Sarkhi et al.(2011), 0.10 MPa Dasgupta et al.(2017), 0.10 MPa Matsuyamaetal.(2017), 0.10 MPa Vasquesa et al.(2018), 0.10 MPa Present study, 0.10 MPa Sawant et al.(2008), 0.12 MPa Sawant et al.(2008), 0.40 MPa Sawant et al.(2008), 0.58 MPa Wang et al.(2018), 0.30 MPa Wang et al.(2018), 0.70 MPa Wang et al.(2018), 0.90 MPa

Fig. 9. Performances of empirical correlations for wave velocity proposed by Kumar et al.[50], Schubring et al.[23], Al-Sarkhi et al.[17] and Marmottant et al.[49]

With the development of the disturbance waves, entrained droplets are sheared off from the wave crests by the gas flow. Ishii and Grolmes [53] elaborately investigated the experimental data of entrainment inception in co-current two-phase flow. By plotting the data on the liquid film Reynolds number Relf versus the critical gas velocity ug, they found that no entrainment occurs at any gas velocity below the critical Reynolds number of the liquid film, which is related to the submergence of the liquid film in the gas turbulent boundary layer. Due to the waves is the prerequisites of droplets entrainment, we consider this critical Reynolds number to be related to the generation and development of disturbance wave. To investigate the dynamic interaction between the liquid film and gas core, the effect of the gas boundary layer should be taken into consideration, i.e., waves should first penetrate through the gas boundary layer before full development. The energy transfer from the gas turbulence to the 13

liquid can be characterized by the ratio of the gas boundary layer thickness l to the film thickness δ as follows: l  y

  0.347 Re l2/3

g g

g i

(2)

 l l  i l

(3)

where τi is the interfacial shear force, y+ is the dimensionless distance from the wall based on the shear velocity, μg is gas viscosity, μl is liquid viscosity, Rel is liquid Reynolds number, ρg is gas density, and ρl is liquid density. Obviously, the gas boundary layer thickness is much different in different pressure conditions, and this is probably the fact for the limitation of existing correlations at high-pressure conditions. By analyzing the available experimental data, Ishii and Grolmes[53] suggested that y+=10 and proposed a correlation for the critical Reynolds number of the liquid film R e on (the onset of l entrainment): 3/2

 y  Relon     0.347 

3/4

3/2

 l   g       g   l 

(4)

Azzopardi[26] reviewed Eq. (3) and found that the prediction was smaller than the experimental data. Therefore, he suggested that y+ was more appropriate to be modified as 30. In this paper, y+ is set as 30. Moreover, Re lon is more sensitive with the density ratio when pressure varies. It is deducible that Re lon decreases with the increasing pressure, which can be explained as the gas boundary layer becomes much thinner at higher pressure. Accordingly, the gas boundary layer effect and extra energy from the gas turbulence to the liquid should be carefully considered to describe wave velocity. With this in mind, it is more reasonable to take excess Reynolds number Re lEX ( Re lEX = Re l - Re on ) to analyze the properties of l disturbance waves in the present study. The same excess Reynolds number has been reported by Azzopardi[26], he replaced Reynolds number by Shearer [54] to analyze the properties of disturbance waves; however, it is only for the convenience of plotting. Due to the wave velocity is more affected by the gas phase and physical properties vary at different pressures, the newly developed wave velocity correlation is related to gas superficial 14

velocity usg, gas Reynolds number Reg, excess liquid Reynolds number RelEX and density ratio ρl/ρg. By analyzing the experimental data from Sawant et al. [22] at 0.58 MPa, Schubring et al. [23], Matsuyama et al. [48] and the present study near atmosphere, a new correlation for wave velocity is proposed as follow: v  23.52usg Re

0.25 g

(Re

EX 0.15 l

)

 l   g

  

0.60

(5)

where u sg and Reg are superficial gas velocity and gas-phase Reynolds number, respectively. The inaccuracy of the above coefficients shown in Table 2, while the F-value and R2 value of the nonlinear curve fit are 592.51 and 0.79, respectively. The regression result is also illustrated in Fig.10. Table 2 The inaccuracy of the coefficients in new wave velocity correlation by nonlinear curve fit Coefficients value

Standard Error

Dependency

1

23.52

15.82

0.999

2

-0.25

0.046

0.998

3

0.15

0.029

0.992

4

-0.60

0.029

0.988

10 9 +35%

Calculated v (m/s)

8 7 6 5

-35%

4 3

Sawant et al.(2008) at 0.58MPa Schubring et al.(2010) Matsuyama et al.(2017) Present experimemtal data

2 1 0

0

1

2

3

4

5

6

7

8

9

10

Experimental v (m/s)

Fig. 10. Regression result of the newly developed wave velocity correlation

15

Fig. 11 depicts the validation of the newly developed correlation against the experimental data in literature and present study. Obviously, the newly developed correlation is consistent with the variation of experiment data in the research, especially at higher pressure conditions, and it may be anticipated to be used in more general cases. Due to the data in Kaji et al.[33] did not distinguish the huge wave and disturbance wave, the experiment data are higher than the calculation value. Also, the experimental value in the present study and Dasgupta et al.[21] by non-contact highspeed photography method is smaller than the calculated velocity value; this is probably due to the removal of short-lived waves [21] and waves in the chaotic flow field (present study). Instead, other results indicate that the new correlation gives a relatively good prediction at various conditions within ±35% error.

Atmospheric pressure

Higher pressure

10

+35%

Calculated v (m/s)

Calculated v (m/s) v (m/s) Calculated

Newly developed

6 140 5 120

-35%

4 100 -35%

1 0

2 20

3

4 60

7

6

5

Experimental v (m/s) 40

Calculated v (m/s)

+35%

7

380 60 2 40 1 20 0 00 -20

+35%

10

8

80

8 6 4

100

+35%

6

-35%

4 -35%

2

2

0

8 0

8

0

0 2

2 4

4

6

8

10

6 8 v (m/s) 10 Experimental

Experimental v (m/s)

Experimental v (m/s)

Wolf et al.(2001), 0.10 MPa Kaji et al.(2010), 0.10 MPa Belt et al.(2010), 0.10 MPa Schubring(2010), 0.10 MPa Al-Sarkhi et al.(2011), 0.10 MPa Dasgupta et al.(2017), 0.10 MPa Matsuyamaetal.(2017), 0.10 MPa Vasquesa et al.(2018), 0.10 MPa Present study, 0.10 MPa Sawant et al.(2008), 0.12 MPa Sawant et al.(2008), 0.40 MPa Sawant et al.(2008), 0.58 MPa Wang et al.(2018), 0.30 MPa Wang et al.(2018), 0.70 MPa Wang et al.(2018), 0.90 MPa

Fig. 11. Validation of newly developed correlation against experiment data in the literature and present study.

3.2 Wave frequency The variation of the wave frequency under different flow conditions are illustrated in Fig. 12. Evidently, the wave frequency almost increases linearly with both the increasing superficial gas and liquid velocity.

16

wave frequency f (Hz)

35 30

usl= 0.07 m/s usl = 0.22 m/s

usl = 0.12 m/s usl = 0.39 m/s

12

20

25 20 15 10 5

8

16

24

28

32

Superficial gas velocity usg (m/s) Fig. 12. Variation of wave frequency under different flow conditions

Generally, the wave frequency could be correlated through a dimensionless number namely Strouhal number Sr (gas-based Srg or liquid-based Srl), which can be defined as[55]: Srg(l) 

fD usg(sl)

(6)

where Sr is Strouhal number, f is wave frequency, D is pipe diameter, usg is superficial gas velocity, usl is superficial liquid velocity. Empirical correlations for both the liquid-based and gas-based Strouhal number have been developed by many researchers, as illustrated in Table 3.

Table 3 Summary of empirical correlations for the wave frequencies References

Empirical correlations

Sekoguchi et al.[56]

Srg  E00.5(0.5ln(E0) 0.47)(0.0076ln 0.051) E0 



gD 2 (  l   g )

 2.5 l

Re Frg

Frg 

usg gD

Where E0, Frg and ξ are Eötvös number, Froude number and non-dimensional parameter, respectively.

Schubring Shedd[4]

and

u sg

f  0.005

f  0.035 x

D

x

usg Frmod D

m g

m g  m l

17

Frmod 

 g usg  l gD

Azzopardi[57]

Srl  0.25X  1.2

Alamu[55]

Srl  0.4292X 0.908

Al-Sarkhi et al.[17]

Srl  1.1X 0.93

Sawant et al.[22]

Srg  0.086(Rel )0.27 (

 l 0.64 ) g

Majority of the experimental data available in the literature about disturbance wave frequency are obtained at the near atmospheric conditions; however, there is a dearth of wave frequency data on the pressure effect. Fig.13 shows the comparison of existing liquid-based Strouhal number Srl correlations at atmospheric and higher-pressure conditions. The result implies that only Alamu [55]’s correlation gives a relatively good prediction for some experimental data; however, the wave frequency is strongly influenced by the superficial gas velocity, and the gas-based Strouhal Srg is more reasonable to describe the wave frequency property. 120

120

80

80

60

Calculated Srl

100

-35%

40

+35%

60

-35%

40 20

20

Alamu's correlation(2010)

Azzopardi's correlation(2006)

0

0

20

40

60

80

100

0

120

0

50

Experimental Srl

Experimental Srl

(a) Azzopardi’s correlation[57]

(a) Alamu’s correlation[55]

300 250

Calculated Srl

Calculated Srl

+35%

100

+35%

200 -35%

150 100 50

Al-Sarkhi's correlation(2012)

0

0

50

100

150

200

250

300

Experimental Srl Wolf et al.(2001) Schubring et al.(2008) Sawant et al.(2008) Belt et al.(2010) Al-Sarkhi et al.(2011) Zhao et al.(2013) Matsuyama et al.(2017) Dasgupta et al.(2017) Vasquesa et al.(2018) Present study

(c) Al-Sarkhi’s correlation[17] 18

100

Fig. 13. Comparison of liquid based Strouhal number correlations (a) Azzopardi’s correlation[57] (b) Alamu’s correlation[55] (c) Al-Sarkhi’s correlation[17]

Sawant et al.[22] systematically studied the properties of disturbance waves at three pressure conditions (0.12, 0.4 and 0.58 MPa) in a vertical tube with an inside diameter 9.4 mm. He developed the gas-based Strouhal number Srg as a function of liquid phase Reynolds number Rel and pressure ratio. Indeed, Sawant’s correlation is in good agreement with the experimental data under different pressure conditions. However, from the previous analysis, the disturbance wave should first overcome the influence of the gas boundary layer before full development. It is more reasonable to alternative the Reynolds number Rel in Sawant’s correlation to the excess Reynolds number RelEX . By analyzing the experimental data from Sawant et al.[22] and Al-Sarkhi et al.[17], a new modified correlation can be described as follow:   Srg  0.047(RelEX )0.177  l     g

0.37

(7)

The inaccuracy of the coefficients in modified wave frequency correlation by nonlinear curve fit is shown in Table 4, while the F-value and R2 value of nonlinear curve fit are 524.96 and 0.40, respectively. Due to the large measurement errors in different experiments, although the R2 value is not as good as that of the wave velocity correlation, our fitting effect is more accurate than many existing empirical equations. The regression result is also illustrated in Fig.14.

Table 4 The inaccuracy of the coefficients in modified wave frequency correlation by nonlinear curve fit Coefficients value

Standard Error

Dependency

1

0.047

0.016

0.995

2

0.177

0.038

0.992

3

-0.37

0.039

0.987

19

0.04

Sawant et al.(2008) Al-Sarkhi et al.(2011)

+35%

Calculated Srg

0.03

0.02

-35%

0.01

0.00 0.00

0.01

0.02

0.03

0.04

Experimental Srg Fig.14. Regression result of the modified wave frequency correlation

Fig.15 shows the comparison of Srg at atmospheric pressure condition among Sekoguchi et al.[56], Sawant et al.[22] and the new modified correlation. The experimental data from references[4,17,21,24,25,35,47,48] and the present experiment are taken into account. It can be referred that the correlation proposed by Sekoguchi et al.[56] is unsatisfactory to predict the most data and even fails to predict the wave frequency with the data from Dasgupta et al.[21] and Vasquesa et al.[24], which may be ascribed to the small Reynolds number and pipe diameter. Both the accuracy of the newly presented correlation and Sawant’s correlation seems acceptable to a certain extent, but they overestimate the experimental data from Al-Sarkhi et al.[17] and Zhao et al.[18] and underestimate the data from Dasgupta et al.[21] and Vasquesa et al.[24]. This is because that the disturbance waves and huge waves were not strictly differentiated in the work of[17,35], and the number of disturbance waves possibly has not been fully counted due to the different definition and uncertainty of the tracking algorithm in the work of[21,24]. It is worthwhile to mention that the newly presented correlation overestimates the data from [25,48]. This is probably because the result may be erroneous for blurred wave visualization especially at lower water flowrate in Wolf’s experiment and decreasing frequency by the reduction of surface tension in Matsuyama’s experiment. It might be argued that the improvement proposed in the present paper seems little contributions to the accurate prediction of wave frequency. However, inspired by the work of [53], the meaning of the optimization is that the reasonable consideration of the effect of 20

the gas boundary layer.

0.05

0.05

Sekoguchi's correlation(1985)

-0.05 -35%

-0.10

-0.15 -0.15

+35%

0.04

+35%

Calculated Srg

Calculated Srg

0.00

Sawant's correlation(2008)

0.03 -35%

0.02 0.01

-0.10

-0.05

0.00

0.05

0.00 0.00

0.01

0.02

0.03

0.04

0.05

Experimental Srg

Experimental Srg

(a) Sekoguchi’s correlation[56]

(a) Sawant’s correlation[22]

0.05 Modified correlation +35%

Calculated Srg

0.04 0.03

-35%

0.02 0.01 0.00 0.00

0.01

0.02

0.03

Experimental Srg

0.04

0.05

Wolf et al.(2001) Schubring et al.(2008) Belt et al.(2010) Al-Sarkhi et al.(2011) Zhao et al.(2013) Matsuyama et al.(2017) Dasgupta et al.(2017) Vasquesa et al.(2018) Present experimental data

(c) Modified correlation Fig.15. Comparison of the gas-based Strouhal number correlations at atmospheric pressure condition (a) Sekoguchi’s correlation[56] (b) Sawant’s correlation[22] (c) Modified correlation

Fig.16 shows the comparison of Srg at high-pressure conditions among the newly developed correlation, Sawant’s correlation and Sekoguchi’s correlation. Although Sekoguchi’s correlation gives good prediction under atmospheric pressure conditions, it fails to predict the wave frequency at high-pressure conditions. The results also imply that the new correlation prediction is higher than Sawant’s correlation. Another issue that should be pointed is the effect of gas boundary layer has an inverse relationship with pressure, and it can be observed that the higher the pressure, the less difference between the new correlation and Sawant’s correlation; therefore, the effect of the gas boundary is less prominent at higher pressure conditions. Although the correlation by Sawant et al.[22] has a better prediction in wave frequency in some low-pressure conditions, they ignored the 21

physical mechanism on wave generation and development that the disturbance waves should first overcome the influence of gas boundary layer; therefore, adopting the new modified correlation is more reasonable.

0.02

Calculated Srg

+35%

0.01

-35%

data from Sawant et al.(2008) Sekoguchi's(1985) correlation Sawant's(2008) correlation Modified correlation p=0.12 MPa

0.00 0.00

0.01

0.02

Expreimental Srg

(a) 0.12 MPa 0.04

0.04 +35%

0.03

Calculated Srg

Calculated Srg

0.03

data from Sawant et al.(2008) Sekoguchi's(1985) correlation Sawant's(2008) correlation Modified correlation p=0.40 MPa

-35%

0.02

data from Sawant et al.(2008) Sekoguchi's(1985) correlation Sawant's(2008) correlation Modified correlation p=0.58 MPa

+35%

-35%

0.02

0.01 0.01 0.00 0.00

0.01

0.02

0.03

0.04

Expreimental Srg

0.01

0.02

0.03

0.04

Expreimental Srg

(b) 0.40 MPa

(c) 0.58 MPa

Fig. 16. Comparison of the modified correlations with other gas-based Strouhal number correlations at highpressure conditions

4.

Conclusions The nature and causes of interfacial waves have an essential dependence on structure and flow

condition. To study the properties of disturbance waves in vertical upward annular flow, this paper mainly carries out the investigation of experiments in a 20-mm-inner-diameter pipe near atmospheric pressure and predictive correlation for wave velocity and frequency. The following main conclusions are drawn: 22

1. The image analysis method by acquiring the spatiotemporal distribution of the liquid film is applicable for the extraction of the information of the disturbance waves. 2. The velocity and frequency of disturbance wave increase with the increase in both superficial gas and liquid velocities. 3. Comparing the experimental data with the existing empirical equations, most of them are unsatisfactory to evaluate the wave properties, especially at high-pressure conditions. The effect of the gas boundary layer is taken into consideration to describe the wave properties, and this effect is more significant in relatively high-pressure conditions. Taking this influencing factor into account, the excess Reynolds number is introduced in newly developed correlations. Comparing with other empirical correlations, newly developed correlations have a relatively satisfactory precision within ±35% deviation, especially under high-pressure conditions.

Nomenclature D

pipe diameter (m)

x

flow quality (-)

E0

Eötvös number (-)

X

Lockhart-Martinelli number (-)

f

wave frequency (Hz)

y+

dimensionless distance from the wall (-)

Fr

Froude number (-)

Greek symbols

G

mass flux (kg m-2 s-1)

K

Pearce coefficient (-)

l

gas boundary layer thickness (m)

L

measuring distance from the inlet (m)

m g

gas mass flow rate (kg

m l

liquid mass flow rate (kg m-2 s-1)

m lf

liquid film flow rate (kg m-2 s-1)

Patm

atmospheric pressure (Pa)

m-2 s-1)

Psystem system pressure (Pa) Re

Reynolds number (-)

RelEX excess liquid Reynolds number (-)

23

ρE

density resulting from liquid entrainment (kg m-3)

ρg

gas density (kg m-3)

ρl

liquid density (kg m-3)

μ

viscosity (Pa s)

ξ

non-dimensional parameter (-)

τ

shear stress (kg m−1 s−2)

τi

interfacial shear stress (kg m−1 s−2)

τw

wall shear stress (kg m−1 s−2)

λ

wavelength (m)

δ

the film thickness (m)



average film thickness (m)

Relon

critical Reynolds number of liquid film (-)

σ

Sr

Strouhal number (-)

Ψ

Srg

gas-based Strouhal number (-)

Subscripts

Srl

liquid-based Strouhal number (-)

frc

friction

u gc

gas core velocity (m s-1)

g

gas phase

u lf

mean velocity of the liquid film surface (m s-1)

gc

gas core

usg

superficial gas velocity (m s-1)

i

interface

usl

superficial liquid velocity (m s-1)

l

liquid phase

lf

liquid film

v

wave velocity (m s-1)

vfric,g gas friction velocity (m s-1) We Weber number (-)

surface tension (N m-1) parameter used in Wang et al. (2018) correlation (-)

mod

modified

w

wall

Acknowledgements Authors acknowledge the financial support from National Natural Science Foundation of China under grant No. 51706245 and Science Foundation of China University of Petroleum, Beijing No. 2462016YJRC029.

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27

CRediT author statement Ruinan Lin: Data curation, Writing Original draft preparation, Software Ke Wang: Conceptualization, Methodology. Li Liu: Visualization, Investigation. Yongxue Zhang: Supervision, Validation. Shaohua Dong: Reviewing, Editing

28

Highlights 

The image analysis method is applicable for disturbance waves detecting.



Wave velocity and frequency are carefully investigated.



The effect of the gas boundary layer is considered.



The excess Reynolds number is introduced to describe the liquid phase.



Newly developed correlations for wave properties are developed.

29