Drift-flux correlation for upward two-phase flow in inclined pipes

Journal Pre-proofs Drift-flux correlation for upward two-phase flow in inclined pipes Chuanshuai Dong, Somboon Rassame, Lizhi Zhang, Takashi Hibiki PII: DOI: Reference:

S0009-2509(19)30885-1 https://doi.org/10.1016/j.ces.2019.115395 CES 115395

To appear in:

Chemical Engineering Science

Received Date: Revised Date: Accepted Date:

2 July 2019 14 October 2019 27 November 2019

Please cite this article as: C. Dong, S. Rassame, L. Zhang, T. Hibiki, Drift-flux correlation for upward two-phase flow in inclined pipes, Chemical Engineering Science (2019), doi: https://doi.org/10.1016/j.ces.2019.115395

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Drift-flux correlation for upward two-phase flow in inclined pipes Chuanshuai Donga, Somboon Rassameb,*, Lizhi Zhang a, Takashi Hibikic, a

Key Laboratory of Enhanced Heat Transfer and Energy Conservation of Education Ministry,

School of Chemistry and Chemical Engineering, South China University of Technology, Guangzhou 510640, China b

Department of Nuclear Engineering, Faculty of Engineering, Chulalongkorn University, Wangmai, Patumwan, Bangkok 10330, Thailand

c

School of Nuclear Engineering, Purdue University, 400 Central Drive, West Lafayette, IN 47907-2017, USA * Corresponding author

Highlight 

Void fraction data and correlations of upward two-phase flow were reviewed extensively.



None of existing correlations could predict collected databases successfully.



The dependence of the drift-flux parameter on pipe inclination angles was analysed.



A new drift-flux correlation was developed for upward two-phase flow in inclined pipes.

Abstract In view of the practical importance of the drift-flux model to predict void fraction for gas-liquid flow in various engineering fields including chemical engineering, this study aims at developing a drift-flux correlation with a wide range of applicability for upward two-phase flows in inclined pipes. Firstly, over 3000 experimental data of void fraction were collected from 14 sources. Then, the correlations of distribution parameter and void-fraction-weighted-mean drift velocity were established. The comparison between the collected database and correlations indicated none of the existing correlations could predict the whole database satisfactorily. Therefore, the dependence of drift-flux parameters on inclination angles was comprehensively analysed, and a new drift-flux correlation was developed. The newly-developed drift-flux correlation demonstrated superior performance to existing

1

correlations. More than 95 % of the predicted void fraction were predicted within ±20 % error. In summary, the newly-developed drift-flux correlation is useful to predict void fraction in various engineering fields including chemical engineering.

Keywords: Drift-flux model, void fraction, upward two-phase flows, inclined pipe

Nomenclature C0

Distribution parameter

[-]



Gas

volumetric

flow [-]

fraction C

Asymptotic

distribution [-]



Surface roughness

[m]



Inclination angle

[°]

parameter DH

Hydraulic

equivalent [m]

diameter Fr

Flow inclination parameter

[-]



Dynamic viscosity

[ Pa  s ]

ftp

Two-phase friction factor

[-]



Density

[kg/m3]

j

Mixture volumetric flux

[m/s]



Density difference

[kg/m3]

jf

Superficial liquid velocity

[m/s]



Surface tension

[N/m]

jg

Superficial gas velocity

[m/s]

Subscripts

md

Mean error

[-]

Cal

Calculated

[-]

mrel

Mean relative deviation

[-]

f

Liquid

[-]

mrel,ab

Mean

relative [-]

g

Gas

[-]

absolute

deviation N

Total number of data points

[-]

h

Horizontal

[-]

Re

Reynolds number

[-]

Mea

Measured

[-]

sd

Standard deviation

[-]

v

Vertical

[-]

vg

Gas velocity

[m/s]

Superscripts

vgj

Drift velocity of gas phase

[m/s]

+

x

Two-phase flow quality

[-]

Mathematical symbols Area-averaged quantities

Greek symbols 

Void fraction

Non-dimensional value

[-]

Void

[-]

fraction-weighted [-]

properties

2

[-]

1. Introduction Upward two-phase flows in inclined pipes are often encountered in many engineering applications, such as oil/gas pipelines and nuclear reactors. The accurate prediction of void fraction for upward two-phase flow in inclined pipes is crucial. Two-fluid model and drift-flux model are widely used to predict the void fraction of two-phase flow (Yeoh, 2019; Hibiki, 2019; Ishii, 1977). In the drift-flux model, two drift-flux parameters, namely, distribution parameter and drift velocity, are introduced. The distribution parameter identifies the effect of local phase distribution and velocity on area-averaged void fraction, and the drift velocity characterizes the relative velocity between gas and liquid phases (Hibiki and Ishii, 2002; Zhao and Hibiki, 2019). Several one-dimensional (or area-averaged) void fraction correlations were developed based on the drift-flux model (Hibiki, 2019). Zuber and Findley (1965) gave the constitutive equations for the drift-flux model to estimate the void fraction in vertical gas-liquid two-phase flow. Ishii (1977) also proposed constitutive equations of distribution parameter and void fraction-weighted-mean drift velocity (drift velocity hereafter) for different flow regimes, such as bubbly flow, slug flow, churn flow and annular flow, etc. Due to the practical importance of drift-flux correlations, extensive researches on the constitutive equations for various inclination angles and pipe geometries have been conducted (Hibiki, 2019). The existing drift-flux correlations mainly focus on horizontal or vertical two-phase flows other than inclined flows. Franca and Lahey (1992) developed a drift-flux correlation for gas-liquid two-phase flow in horizontal pipes depending on various flow regimes. Rassame and Hibiki (2018) collected an abundant of experimental database and developed a drift-flux correlation for gas-liquid two-phase flow in horizontal pipes. Hibiki and Ishii (2003) proposed a one-dimensional drift-flux correlation for upward two-phase flow in vertical large diameter pipes and the comparison between experimental and calculated results indicated that their correlation could predict the void faction of upward two-phase flow in vertical large diameter pipes accurately. Although many drift-flux correlations of two-phase flows in horizontal and vertical pipes have been proposed, the drift-flux correlation of upward gas-liquid two-phase flow in inclined pipes is very limited. Chexal et al. (1991) developed a flow-regime-independent correlation using the concept of the drift-flux model, known as Chexal-Lellouche correlation or EPRI correlation. Many cascading 3

constitutive relationships with many empirical parameters were involved in Chexal-Lellouche correlation, which resulted in the complexity and inconvenience of the correlation. Bhagwat and Ghajar (2014) also proposed a flow-regime-independent void fraction correlation for two-phase flow in inclined pipes based on the concept of the drift-flux model. The comparison between the calculated and measured void fraction indicated that Bhagwat and Ghajar correlation (2014) could achieve good predictive accuracy. However, as void fraction was embedded in the constitutive equations of distribution parameter and drift velocity, it is difficult to calculate void fraction using Bhagwat and Ghajar correlation (2014) explicitly. Although the drift-flux correlations for upward two-phase flows in inclined pipes are significant, very few correlations are available. In view of this, this study aims at developing a new flow-regimeindependent drift-flux correlation, to serve as a comprehensive correlation predicting the void fraction of upward gas-liquid two-phase flow in inclined pipes. An extensive review is conducted to collect the experimental void fraction data and existing correlations for upward two-phase flow in inclined pipes. Based on the existing experimental data and correlations and the state-of-the-art knowledge of flow behaviours for upward two-phase flow in inclined pipes, a new void fraction correlation is developed based on the concept of the drift-flux model. The predictive capacity of the newlydeveloped drift-flux correlation is validated by the collected database.

2. Existing drift-flux correlations and database of upward two-phase flow in inclined pipes 2.1 One dimensional drift-flux model Zuber and Findley (1965) first gave the expression of the one-dimensional drift-flux model as: jg





vg

 C0 j 

(1)

vgj

where j g ,  , vg and j represent the superficial gas velocity, void fraction, gas velocity and mixture volumetric flux, respectively.

and

mean the area-averaged and void fraction-weighted area-

averaged quantities over the cross-sectional flow area, respectively. C0 and distribution parameter and drift velocity, defined by: 4

vgj

are the

C0 

j  j

vgj



(2)

 vgj

(3)



The drift velocity, vgj , expresses the difference between the gas velocity, vg , and the mixture volumetric flux, j . vgj  vg  j

(4)

The drift-flux model is non-dimensionalized using a velocity scale of   g /  2f  , proposed 1/4

by Kataoka and Ishii (1987). Therefore, the non-dimensionalized drift-flux model is expressed as: jg

vg





 C0 j   Vgj

(5)

The non-dimensionalized parameters are defined as follows. 1/ 4

j

 g

 jg

  g   /   2f   

 j

  g   /   2f   

(6)

1/ 4

j



(7)

1/ 4

v

 g



vg

  g   /   2f   

(8)

1/ 4

Vgj 

vgj

  g   /   2f   

(9)

2.2 Existing drift-flux correlations for upward two-phase flows in inclined pipes The drift-flux correlations for upward gas-liquid two-phase flows in inclined pipes are very limited. Only a few void fraction correlations are available. This section briefly reviews the existing drift-flux correlations of upward gas-liquid two-phase flows in inclined pipes. Chexal et al. (1991) developed a flow-regime-independent drift-flux correlation, known as Chexal-Lellouche correlation, based on a large amount of experimental data of two-phase flows in

5

inclined pipes (-90° to +90°). Chexal-Lellouche correlation (1991) is a continuous correlation in void fraction and covers the full range of void fractions (0-to-1). The experimental data of air-water, steamwater and refrigerants are collected to develop the correlation. Chexal-Lellouche correlation (1991) can be applied to pipes as large as 450 mm in diameter. To develop a flow-regime-independent void fraction correlation, the internal physical mechanisms between void fraction and drift-flux parameters in Chexal-Lellouche correlation (1991) were sacrificed, and several cascading constitutive equations, as well as empirical parameters, were involved. The expression of Chexal-Lellouche correlation (1991) for upward two-phase flows in inclined pipes is given by: C0  FrC0v  (1  Fr )C0 h

(10)

where C0v and C0h are the concentration parameters evaluated for vertical and horizontal flows, respectively. Fr is the flow inclination parameter, defined as follows. Fr   / 90o for 0o    90o

(11)

where  is the pipe inclination angle measured from horizontal orientation. The concentration parameter for vertical flow is given by: C0v  L / [ K0  (1  K0 )  ] r

(12)

where L is Chexal-Lellouche fluid parameter. For steam-water,

L

1  exp(C1  )

(13)

1  exp(C1 )

For air-water,

L  min[1.15 

0.45

,1.0]

(14)

For refrigerant,

L 

0.025(110  )

exp[0.5(1   )]

(15)

where C1 , K 0 and r are the parameters, referenced in Chexal et al. (1991). The concentration parameter for horizontal flow is given as follows.



C0 h  1  

0.05

1-    C 2

0v

6

(16)

where C0v is defined by Eq. (10) and the Chexal-Lellouche fluid parameter, L , used in C0v is given as follows. For steam-water,

L

1  exp(C1  )

(17)

1  exp(C1 )

For air-water,

L  min[1.25 

0.60

,1.0]

(18)

For refrigerant,

L   [1.375  1.5    0.5 ] 2

The drift velocity,

vgj

(19)

, for upward two-phase flow is given as follows. vgj

=FrVgjv  (1  Fr )Vgjh

(20)

where Vgjv and Vgjh are the drift velocities of vertical and horizontal flows, respectively. Vgjv =Vgj0 C9

Vgj0 =1.41[

 g

 2f

]0.25C2C3C4

(21)

(22)

where g , C2 , C3 , C4 and C9 are the parameters, referenced in Chexal et al. (1991). The drift velocity of horizontal flow, Vgjh , is evaluated by Eq. (21) as used for vertical flows, using positive values of the superficial velocity. Bhagwat and Ghajar (2014) proposed another flow-regime-independent void fraction correlation based on the concept of the drift-flux model. The collected database contains many fluid systems, such as air-water, natural gas-water, air-kerosene, air-glycerine, steam-water, and air-oil, etc. Circular, annular and rectangular pipe geometries are considered in the database, and the hydraulic equivalent diameters range from 0.5 mm to 305 mm. The correlation considered the pipe inclination. However, the embedded void fraction in the constitutive equations of distribution parameter and drift velocity leads to the inconvenience of the Bhagwat and Ghajar correlation (2014). The detailed expression of Bhagwat and Ghajar correlation (2014) is given as follows. 7

C0 

 2 g  f 

  

2

1   Retp / 1000 

2

  [1    g /  f  



2

1 

 cos  ] / (1  cos  )  

1  1000 / Retp 

  

2/5

 C0,1

2

(23)

where Retp is the two-phase mixture Reynolds number, defined by: Retp =

 f j DH f

(24)

C0,1 is determined as a function of the ratio of gas to liquid density,  g /  f , gas volumetric flow

fraction,  , two-phase friction factor, f tp , and two-phase flow quality, x .





C0,1 = C1 -C1  g /  f (2.6   )0.15 

1.5 ftp  1  x 

(25)

where C1 is assumed 0.2 for circular and annular pipes and 0.4 for rectangular pipes. The two-phase Fanning friction factor, f tp , is calculated by Colebrook correlation (1939) as:

 / D 1 1.256 H =-4.0log10    3.7 ftp Retp ftp 

   

(26)

where  is the surface roughness and DH is the hydraulic equivalent diameter. The drift velocity,

vgj

vgj

, is given by:

  0.35sin   0.45cos 

 gDH

f

1 





0.5

C2C3C4

(27)

where C2 , C3 , C4 are the parameters, referenced in Bhagwat and Ghajar (2014).

2.3 Existing database for upward two-phase flows in inclined pipes Many researchers performed experimental studies on void fraction of upward gas-liquid twophase flows in inclined pipes. This section identifies more than 3000 experimental data of upward gas-liquid two-phase flows in inclined pipes from 14 sources. The fluid systems, geometrical information of the test section, inclination angle, flow conditions, number of data for each reference, void fraction measurement methods and measurement accuracy are summarized in Table 1. As the experimental data in Beggs (1972), Mukherjee. (1979), Payne et al. (1979), Kokal (1987) and Ottens

8

et al. (2001) were presented numerically in tables, these experimental data were collected from the references directly. However, as the experimental data in Spedding et al. (1998), Perez (2008), Oyewole (2013), Yan et al. (2014), Ghajar and Bhagwat (2014), Wiesche and Kapitz (2015), Bhagwat and Ghajar (2016), Luo et al. (2016) and Wen et al. (2017) were presented in figures, these experimental data were collected by the image processing software using positioning technology. As shown in Table 1, five fluid systems, including air-water, air-kerosene, air-lube oil, natural gas-water, and air-light oil, are involved. Table 2 summarizes the physical properties of fluid systems in collected database. The inclination angle,  , ranges from 0° to 90°, measured from horizontal orientation and the pipe diameter ranges from 1.27 cm to 7.63 cm. The measurement methods of void fraction include quick-closing valve, parallel-plate capacitance, capacitance probes, and high-speed camera, etc., with the measurement accuracy ranging from ±1.3 % to ±16.5 %. The ranges of the superficial gas velocity, jg , and the superficial liquid velocity, j f , are from 0.0366 m/s to 204 m/s and from 0.0152 m/s to 5.54 m/s, respectively. Figure 1 presents part of the collected experimental conditions in the flow regime map proposed by Bhagwat and Ghajar (2016). The red dashed line, blue dotted line, pink short dashed line and black solid line indicate the flow regime transition boundaries of the upward gas-liquid two-phase flows in inclined pipes with the inclination angles of 0° (horizontal), 30°, 60° and 90° (vertical), respectively. The symbols represent the experimental conditions from different sources. The figure indicates that the collected database covers the whole flow regimes. In summary, the collected database includes upward gas-liquid two-phase flows of different fluid systems (such as air-water, air-kerosene, air-lube oil, natural gas-water, and air-light oil) in inclined pipes with different inclination angles (ranging from 0° to 90°).

3. Comparison of existing correlations with collected database As discussed in Section 2.2, the existing drift-flux correlations of void fraction in upward twophase flows in inclined pipes are very limited. Chexal-Lellouche correlation (1991) and Bhagwat and

9

Ghajar correlation (2014) are the most known correlations. This section conducts a performance evaluation of the existing correlations with the collected database. Figure 2 shows the comparison of Chexal-Lellouche correlation (1991) with the collected database from 14 sources in different inclination angles. The abscissa and ordinate in the figure represent the measured and calculated void fraction, respectively. The solid and dotted lines indicate 0 % and ±20 % error, respectively. As shown in Figure 2, Chexal-Lellouche correlation (1991) can predict the void fraction of two-phase flows in vertical and near vertical pipes (larger than 70°) well but tends to overestimate the void fraction. The degree of scattering becomes relatively high when the void fraction increases. As the inclination angle decreases, Chexal-Lellouche correlation (1991) tends to further overestimate the void fraction, especially in horizontal and slightly inclined pipes. Although most of the predicted void fraction fall within ±20 % error band of the measured values, the predicted void fractions are monotonously higher than the measured values, especially in horizontal or slightly inclined pipes. Figure 3 shows the comparison of Bhagwat and Ghajar correlation (2014) with the collected database. Bhagwat and Ghajar correlation (2014) tends to overestimate the void fraction of upward inclined two-phase flows. When the void fraction is lower than 0.4, the predicted void fractions agree well with the measured values in horizontal and slightly inclined pipes (lower than 10°). The prediction accuracy turns unacceptable with the increase of void fraction. As the inclination angle increases, the degree of scattering is rather high, most of the predicted void fractions fall outside of ±20 % error band of the measured values. In vertical pipes (θ=90°), the calculated void fractions fall within 0% to +20 % error band of the measured values. To quantitatively evaluate the prediction performance of the existing drift-flux correlation, four statistical parameters, namely mean error, md, defined by Eq. (28), standard deviation, sd, defined by Eq. (29), mean relative deviation, mrel, defined by Eq. (30), and mean absolute relative deviation, mrel, ab,

defined by Eq. (31) are introduced (Hibiki et al., 2017). The mean error, md, represents the mean

difference between the predicted and measured void fractions and standard deviation, sd, represents the random uncertainty of a correlation or scatter of measured data around a correlation. The mean

10

relative deviation, mrel, represents the relative bias of a correlation, whereas the mean absolute relative, mrel, ab, is often used as a prediction error. md 

sd 

1 N

N

[ i 1

cal

  mea ]

(28)

1 N [cal   mea  md ]2 N  1 i 1

(29)

1 N  cal   mea 100  N i 1  mea

(30)

1 N  cal   mea 100   N i 1 mea

(31)

mrel 

mrel , ab 

where N is the number of samples. Subscripts of cal and mea indicate the calculated and measured values, respectively. Table 3 summarizes the prediction performance of the existing drift-flux correlations for upward two-phase flows in inclined pipes with different inclination angles. The mean relative deviations of Chexal-Lellouche correlation (1991) ranges from 4.91 % to 14.4 % with the mean value of 8.40 %, whereas the mean absolute relative deviations range from 7.97 % to 14.4 % with the mean value of 11.3 % in different inclination angles. Most of the predicted void fractions by Chexal-Lellouche correlation (1991) fall within ±20 % error. However, as shown in Table 3, all the mean relative deviations of Chexal-Lellouche are positive, indicating that Chexal-Lellouche correlation (1991) tends to overestimate the void fraction of upward inclined two-phase flows. As for Bhagwat and Ghajar correlation (2014), the mean relative deviations range from 8.29 % to 23.8 % with the mean value of 16.0 %, whereas the mean absolute relative deviations range from 11.7 % to 24.3 % with the mean value of 18.2 % in different inclination angles. The degree of scattering for Bhagwat and Ghajar correlation (2014) is much higher than that for Chexal-Lellouche correlation (1991), especially when the void fraction is high. The above performance evaluation reveals that the existing drift-flux correlations cannot predict the void fraction of upward two-phase flows in inclined pipes well. The effect of pipe inclination angles on the void fraction is not clearly elaborated. In view of this, it is necessary to develop an accurate and robust void fraction correlation based on the drift-flux model for upward gas-liquid two11

phase flows in inclined pipes, which can be applicable to a wide range of conditions as well as conditions even beyond the conditions validated for the correlation.

4. Development of drift-flux correlation for upward two-phase flow in inclined pipes The goal of this study is to develop an accurate and simple drift-flux correlation for upward twophase flows in inclined pipes. The effect of the inclination angle on the drift-flux parameters such as distribution parameter and drift velocity should be incorporated into the correlation to be developed. The following steps are taken in the development process.

Step 1: The limiting cases of an inclined pipe at

=90° and 0° are vertical and horizontal pipes,

respectively. The drift-flux correlation to be developed for upward two-phase flow in inclined pipes should be reduced to drift-flux correlations for vertical upward and horizontal flows at

=90° and 0°,

respectively. As the first step of the correlation development, reliable and accurate drift-flux correlations for vertical and horizontal flows will be selected. Step 2: The difference in the distribution parameter and drift velocity between vertical upward and horizontal flow correlations reflects the effect of the inclination angle on the distribution parameter and drift velocity between

=90° and 0°. A general tendency on how the drift-flux

parameters vary with the changing inclination angles will be discussed. Step 3: Key parameters to correlate the effect of the inclination angle on the drift-flux parameters will be identified. A new drift-flux correlation for upward two-phase flows in inclined pipes will be developed.

Step 1: Selection of reliable and accurate drift-flux correlations for vertical upward and horizontal flows Vertical pipe A most well-received drift-flux correlation for vertical upward two-phase flow is considered to be Ishii correlation (1977). This correlation works well for upward slug and churn flows in vertical medium-size pipes (Hibiki, 2019), but the prediction accuracy of the correlation is deteriorated for 12

bubbly flow and large diameter pipes.

However, if the mixture volumetric flux is large, the

deterioration of the prediction accuracy becomes insignificant. For a simple correlation development, Ishii correlation (1977) is selected as the drift-flux correlation for upward two-phase flow in vertical pipes. A brief explanation of Ishii correlation is given below. Ishii (1977) proposed a simple correlation of the distribution parameter for gas-liquid two-phase flow in upward vertical pipes. The distribution parameter, C0 , was assumed depending on the ratio of gas density to liquid density,  g  f , and Reynolds number, Re . As the ratio of gas density to liquid density,  g  f , approaches unity, the distribution parameter, C0 , should approach unity. Based on this assumption, Ishii (1977) proposed a simple functional form of distribution parameter to account for the effect of inertia force on the distribution parameter as: C0  C ( Re)  C ( Re)  1

g f

(32)

where C is the asymptotic value of the distribution parameter when  g /  f approaches zero. The functional form in Eq. (32) ensures that the distribution parameter approaches unity when gas density approaches liquid density. Ishii correlation (1977) indicated that the lighter phase tended to migrate into the higher velocity region and resulted in high void fraction region in the central region. After a comprehensive investigation on the effect of Reynolds number on the distribution parameter, Ishii (1977) found that C was approximated to be 1.2 for upward gas-liquid two-phase in vertical pipes. Therefore, Eq. (32) is re-expressed as follows. C0  1.2  0.2

g f

(33)

Ishii (1977) also gave the drift velocity correlations depending on the flow regime. It should be noted here that C is dependent on channel geometry (Hibiki, 2019).

Horizontal pipe Several empirical drift-flux correlations have been proposed to predict gas-liquid two-phase flow in horizontal pipes. Rassame and Hibiki (2018) performed extensive reviews of existing data and 13

drift-flux correlations and evaluated the applicability of the collected correlations. They found that no correlations could predict the void fraction in a wide range of the collected test conditions. Due to no accurate correlations available for horizontal flow, Rassame and Hibiki (2018) developed a drift-flux correlation to predict the void fraction of gas-liquid two-phase flow in horizontal pipes. A large amount of measured void fractions in different flow regimes were collected from different sources to verify the correlation. In this study, Rassame and Hibiki correlation (2018) is selected as the driftflux correlation for two-phase flows in horizontal pipes. A brief explanation of Rassame and Hibiki correlation (2018) is given below. Rassame and Hibiki (2018) approximated the drift velocity being zero for horizontal flows and back-calculated the distribution parameters from measured superficial gas and liquid velocities and void fraction through Eq. (1). They found that the distribution parameter depended on the ratio of superficial gas velocity to mixture volumetric flux and proposed the following correlation of the distribution parameter.   For 0  jg / j  0.9 ,

  jg / j  C0  0.800exp 0.815    0.900  

1.50

   

    j / j   0.800exp 0.815  g     0.900    

1.50

   

     1 g   f  

(34)

  For 0.9  jg / j  1 ,





C0  8.08 jg / j   9.08  8.08





jg / j   1

g f

(35)

Step 2: Effect of inclination angles on drift-flux parameters Figure 4 shows the variation of the asymptotic value of the distribution parameter, C , against the ratio of superficial gas velocity to mixture volumetric flux,

jg / j  . Black solid and red

broken lines indicate the calculated values by Rassame and Hibiki correlation (2018) (horizontal flow) and Ishii correlation (1977) (vertical upward flow). The ratio of superficial gas velocity to mixture volumetric flux approximately represents the void fraction. 14

For asymptotic distribution parameter of horizontal flows indicated by the black solid line, C   gradually increases with jg / j

  until it reaches 0.9, and C decreases with jg / j beyond

jg / j  =0.9. The asymptotic value of C at jg / j  =0 is 0.80, whereas the peak value of C     at jg / j =0 is 1.80. The asymptotic value of C at jg / j =1 is 1.0.

For asymptotic distribution parameter of vertical upward flows indicated by the broken red line, C   1.20  is constant up to the transition between churn and annular flow regimes. The value of

jg / j  at the transition is tentatively set at 0.9, which will make the correlation development process more straightforward. Since Ishii's distribution parameter given by Eq. (33) is only applicable   up to churn flow, a straight interpolation line between jg / j =0.9 and 1.0 is introduced.

In the following discussion, the values of C at

jg / j  =0, 0.9 and 1 are represented by

C , A , C , B , and C , C , respectively. As can be seen in Figure 4, the value of C , A increases from 0.80 to 1.20 as the inclination angle increases from 0° (horizontal pipe) to 90° (vertical pipe). The value of C , B decreases from 1.80 to 1.20 as the inclination angle increases from 0° (horizontal pipe) to 90° (vertical pipe). The above observation indicates that the exponential increase trend of C with

jg / j  in the region of 0  jg / j   0.9 and the linear decrease trend of C with jg / j    in the region of 0.9  jg / j  1.0 observed in horizontal flows should also appear in upward

two-phase flow in inclined pipes. In order to validate this hypothesis, further analysis with an assumed drift velocity correlation will be performed as follows. Firstly, the correlation of the drift velocity for upward two-phase flow in inclined pipes will be proposed. As discussed above, the drift velocity for horizontal flows is approximated to be zero,

vgj

 0 m/s , in Rassame and Hibiki correlation (2018). The drift velocity for vertical upward

flows depends on flow regime (Ishii, 1977). For the development purpose of the drift-velocity

15

correlation applicable to a whole range of void fraction, the following drift velocity correlation   recommended for vertical upward churn flow regime is adopted in the region of 0  jg / j  0.9 .

vgj

  g  2   2f 

1/4

    for 0  jg / j  0.9 

(36)

where  and  are the density difference between gas and liquid phases and surface tension, respectively. It should be noted here that the drift velocity at

jg / j   1.0 should be zero. Thus,

a linear interpolation scheme should be applied to calculate the drift velocity in the region of

0.9  jg / j   1.0 as:

vgj

  g  2   2f 

 1  jg / j    0.1 

1/4

  

  for 0.9  jg / j   1.0  

(37)

As a correlation to match the above drift velocities at the limiting cases such as at

=0° and 90°,

the following correlation for the drift velocity of upward two-phase flows in inclined pipes is proposed as:

vgj

   g sin   2  2f       g sin   2   2f  

1/4

    for 0  jg / j  0.9  1/4

  

 1  jg / j    0.1 

(38)

  for 0.9  jg / j   1.0  

where g is the gravitational acceleration (=9.8 m/s2). Eq. (38) is reduced to

vgj

 0 m/s for

horizontal flows and Eqs. (36) and (37) for vertical upward flows. Using Eq. (38), the asymptotic distribution parameter for upward two-phase flows in inclined pipes can be estimated from measured superficial gas and liquid velocities and void fraction through Eq. (39) which can be derived from Eqs. (1) and (32).

vg C 

 vgj j

g 1 f

16



g f

(39)

To investigate the effect of pipe inclination angles on distribution parameter, C0 , the dependence of the asymptotic distribution parameter on the ratio of superficial gas velocity to mixture volumetric   flux in different inclination angles is analysed. Figure 5 presents the variation of C with jg / j

in different inclination angles (0°, 5°, 10°, 15°, 20°, 30°, 45°, 50°, 60°, 70°, 80° and 90°). As can be   seen in Figure 5, C gradually increases with jg / j

until it reaches 0.9, and C decreases with

jg / j  beyond jg / j  =0.9. This trend is more pronounced when the inclination angle is smaller. When the inclination angle approaches 90°, the asymptotic distribution parameter is almost   constant in the region of 0  jg / j  0.9 . In order to elucidate the effect of inclination angle on

C , A and C , B , the data obtained in each inclination angle is correlated with the following functions used in the development of Rassame and Hibiki correlation (2018) for horizontal flows.    C, B   jg / j   C  C, A exp  ln   C , A   0.900  

C  10(1  C, B )

jg j

1.5

   

 10C, B  9

   

for 0  jg / j   0.9

for 0.9  jg / j   1

(40)

(41)

Red solid lines in Figure 5 show the asymptotic distribution parameter calculated by Eqs. (40) and (41) with C , A and C , B determined by the data for each inclination angle. As can be seen in Figure 5, the calculated asymptotic distribution parameters agree with the data well. The value of the asymptotic distribution parameter, C , A , increases from 0.80 to 1.20 and the peak value, C , B , decreases from 1.80 to 1.20 as the pipe inclination angle increases from 0° to 90°. The obtained results in Figure 5 are expected as shown in Figure 4. The correlations of asymptotic distribution parameter for each inclination angle are summarized in Table 4.

Step 3: Development of drift-flux correlation for upward two-phase flow in inclined pipes In step 3, the drift-flux correlation for upward two-phase flows in inclined pipes will be developed. To quantitatively investigate the effect of pipe inclination on the asymptotic distribution 17

parameter, the dependence of C , A and C , B on the sine of inclination angles is analysed. Figure 6 presents the variation of C , A and C , B with the sine of inclination angles. It is observed that C , A increases linearly with the sine of inclination angles, while C , B decreases linearly. Then, the correlations of C , A and C , B in terms of the sine of inclination angles are expressed as: C, A =0.400sin   0.800

(42)

C, B =1.80-0.700sin 

(43)

When the pipe inclination angle,  , is 0° (horizontal flow), C , A and C , B , are 0.800 and 1.80, respectively, which agree well with Rassame and Hibiki correlation (2018). As the pipe inclination angle,  , approaches 90° (vertical flow), C , A and C , B approach 1.20 and 1.20, respectively, which agree with Ishii correlation (1977). Therefore, the newly-developed correlation of distribution parameter, Eqs. (40) and (41) together with Eqs. (42) and (43) achieves the built-in accordance with Rassame and Hibiki correlation (2018) and Ishii correlation (1977). Finally, the correlation of asymptotic distribution parameter for upward gas-liquid two-phase flow in different inclined pipes is given as follows.   For 0  jg / j  0.9     1.80-0.700sin    jg / j C   0.400sin   0.800  exp ln    0.400sin   0.800   0.900  

1.5

   

   

(44)

  For 0.9  jg / j  1

C   -8.00+7.00sin  

jg j

 9.0-7.00sin 

(45)

The newly-developed drift-flux correlation for upward two-phase flows in inclined pipes is summarized in Table 5.

5. Performance evaluation of the newly-developed drift-flux correlation

18

As discussed above, a new drift-flux correlation for upward gas-liquid two-phase flows in inclined pipes is developed based on a large amount of experimental data. Figure 7 presents the comparison between measured and calculated void fractions by the newly-developed drift-flux correlation in different pipe inclination angles. Table 6 summarizes the performance evaluation of the drift-flux correlation quantitatively. More than 95 % of the calculated void fraction fall within ±20 % of the measured values. The predicted void fraction values are concentrated evenly around the measured values, and no systematic bias in the prediction is observed. The mean relative deviations range from -1.52 % to +3.38 % with the mean value of 1.86 %, whereas the mean absolute deviations range from 2.71 % to 8.98 % with the mean value of 6.83 % in different inclination angles. The statistical indices of the newly-developed drift-flux correlation are much lower than that of ChexelLellouche correlation (1991) and Bhagwat-Ghajar correlation (2016), indicating that the newlydeveloped correlation demonstrates superior predictive performance. As the effect of pipe inclination angles on void fraction is considered in the newly-developed drift-flux correlation well, the new driftflux correlation can be applied in the whole ranges of pipe inclination angles, ranging from 0° to 90° as well as in the whole ranges of void fraction, ranging from 0 % to 100 %. The distribution of 5 % of the calculated void fraction out of ±20 % error might be attributed to several aspects. Firstly, the measurement accuracy ranges from ±1.3 % to ±16.5 %, as shown in Table 1. The high measurement error could result in high uncertainty of the measured values. Secondly, linear assumption is made in the determination of asymptotic distribution parameter to simply the newly-developed drift-flux correlation. Therefore, linear formulas are adopted to correlate C , A and

C , B with the sine of inclination angles instead of polynomial or other complicated formulas which might improve the predictive accuracy a bit but increase the complexity dramatically of the newlydeveloped correlation. The good agreement between the measured and calculated void fraction by the newly-developed drift-flux correlation proves the correctness of this assumption. To examine the applicability of the newly-developed drift-flux correlation to different fluid systems, the comparison between measured and calculated void fraction of air-water, air-kerosene, air-lube oil, natural gas-water and air-light oil systems are conducted separately, as shown in Figure 8. 19

Most of the calculated void fraction are predicted within ±20 % of the measured values. Table 7 summarizes the performance evaluation of the drift-flux correlation for upward two-phase flows with different fluid systems quantitatively. The mean relative deviations for air-water, air-kerosene, airlube oil, natural gas-water and air-light oil systems are 2.98 %, 1.51 %, 1.94 %, -6.84 % and 1.88 %, respectively, while the mean absolute relative deviations for air-water, air-kerosene, air-lube oil, natural gas-water and air-light oil systems are 6.45 %, 5.80 %, 5.15 %, 10.8 % and 7.61 %, respectively. The comparison results indicate that the newly-developed drift-flux correlation could be well applied to predict the void fraction of upward two-phase flows with various fluid systems in inclined pipes. To simplify the model development process, the measured void fraction data in the inclination angles of 1°, 2°, 4.23°, 7.02° and 8.24° are not used in the determination of drift-flux parameters. Figure 9 presents the comparison between the measured and calculated void fraction by the newlydeveloped drift-flux correlation in the pipe inclination angles of 1°, 2°, 4.23°, 7.02° and 8.24°. More than 92 % of the calculated void fraction fall within ±20 % of the measured values with the mean relative deviation and mean absolute relative deviation of -3.55% and 10.7 %, respectively. As the experimental data in the inclination angles of 1°, 2°, 4.23°, 7.02° and 8.24° are not included in the database which was used to correlate the drift-flux correlation, the good agreement between the measured and calculated void fraction proves the robust applicability of the newly-developed driftflux correlation. Figure 10 presents the comparison between the newly-developed drift-flux correlation, ChexelLellouche correlation (1991) and Bhagwat-Ghajar correlation (2016) with measured upward gas velocities in different inclined pipes. Symbols and lines represent the measured and calculated gas velocities by different drift-flux correlations, respectively. In this comparison, the measured data by Perez (2008) with a diameter of 0.038 m are utilized. The pipe inclination angles are 30° and 50°. The comparison results indicate that Chexel-Lellouche correlation (1991) and Bhagwat-Ghajar correlation (2014) tend to underestimate the gas velocity with remarkable prediction error. The newly-developed correlation agrees with the data well.

20

In summary, the newly-developed drift-flux correlation, much simpler than existing correlations, demonstrates superior predictive performance in void fraction of upward gas-liquid two-phase flows in inclined pipes. The mean error, md, and the standard deviation, sd, are 0.0220 and 0.0706, respectively. The mean relative deviation and the mean absolute relative deviations are 1.86 % and 6.83 %, respectively. The newly-developed drift-flux correlation was well validated by a large amount of experimental data (more than 3000 experimental data), and its applicable range is 0.0152 m/s   j f   5.54 m/s , 0.0366 m/s   jg   205 m/s , 1.25 cm  D  7.63 cm and 0°   

90°. The newly-developed drift-flux correlation can be well applied in predicting the void fraction of upward two-phase flows in inclined pipes.

6. Conclusions In view of the practical importance of the drift-flux model for two-phase flow analysis, this study develops a new drift-flux correlation to predict the void fraction for upward gas-liquid two-phase flows in inclined pipes. The essential achievements are summarized below. 

An extensive literature survey of existing experimental data (more than 3000 data from 14 sources with various fluid systems) of void fraction and existing drift-flux correlations in upward inclined pipes has been performed.



The comparison between the collected data and existing correlations is conducted. The comparison results indicate that none of the existing correlations can predict the void fraction of upward gas-liquid two-phase flows in inclined pipes with acceptable accuracy.



The effect of pipe inclination angles on distribution parameter and drift velocity is comprehensively analysed, and a new drift-flux correlation for upward two-phase flows in inclined pipes is developed.



The performance evaluation of the newly-developed drift-flux correlation proves that the newly-developed drift-flux correlation demonstrated superior predictive performance to the existing correlations. More than 95 % of the predicted void fraction fall within ±20 % error

21

of the measured values. The mean relative deviation and the mean absolute deviations are 1.86 % and 6.83 %, respectively. 

The newly-developed drift-flux correlation for upward two-phase flows in inclined pipes is well validated by a large amount of experimental data (more than 3000 experimental data), and its applicable range is 0.0152 m/s   j f   5.54 m/s , 0.0366 m/s   jg   205 m/s , 1.25 cm  D  7.63 cm and 0°    90°.

Acknowledge This project is supported by the China MOST-Japan JICA International Cooperation Key Project of the National Key Research and Development Program (No. 2017YFE0116100). It is also supported by the National Science Fund for Distinguished Young Scholars of China (No. 51425601). This project is also supported by the National Natural Science Foundation of China (No. 51906071), the Postdoctoral Science Foundation of China (No. 2019M652889) and Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education (No. ares-2019-09).

References Beggs, H.D., 1972. An experimental study of two-phase flow in inclined pipes. Dissertation, The University of Tulsa, Tulsa, USA. Bhagwat, S.M., Ghajar, A.J., 2014. A flow pattern independent drift flux model based void fraction correlation for a wide range of gas-liquid two phase flow. Int. J. Multiphase Flow 59, 186-205. Bhagwat, S.M., Ghajar, A.J., 2016. Experimental investigation of non-boiling gas-liquid two phase flow in upward inclined pipes. Exp. Therm. Fluid. Sci. 79, 301-318. Colebrook, C.F., 1939. Turbulent flow in pipes, with particular reference to the transition between the smooth and rough pipe laws. Journal of the Institution of Civil Engineers (London). Chexal, B., Lellouche, G., Horowitz, J., Healzer, J., Oh, S., 1991. The Chexal-Lellouche void fraction correlation for generalized applications. EPRI, NSAC-139. Dong, C.S., Hibiki, T., 2018. Correlation of heat transfer coefficient for two-component two-phase slug flow in a vertical pipe. Int. J. Multiphase Flow 108, 124-139.

22

Dong, C.S., Hibiki, T., 2018. Heat transfer correlation for two-component two-phase slug flow in horizontal pipes. Appl. Therm. Eng. 141, 866-876. Dong, C.S., Lu, L., Wang, X., 2019. Experimental investigation on non-boiling heat transfer of twocomponent air-oil and air-water slug flow in horizontal pipes. Int. J. Multiphase Flow 119, 28-41. Franca, H., Lahey, R.T., 1992. The use of drift flux techniques for the analysis of phase flows. Int. J. Multiphase Flow 18, 787-801. Hibiki, T., Ishii, M., 2002. Distribution parameter and drift velocity of drift-flux model in bubbly flow. Int. J. Heat Mass Transfer 45(4), 707-721. Hibiki, T., Ishii, M., 2003. One-dimensional drift–flux model for two-phase flow in a large diameter pipe. Int. J. Heat Mass Transfer 46, 1773-1790. Hibiki, T., Mao, K., Ozaki, T., 2017. Development of void fraction-quality correlation for two-phase flow in horizontal and vertical tube bundles. Prog. Nuc. Energy 97, 38-52. Hibiki, T., 2019. One-dimensional drift-flux correlations for two-phase flow in medium-size channels, Exp. Comp. Multiphase Flow 1(2) (in print). Ishii, M., 1977. One-dimensional drift–flux model and constitutive equations for relative motion between phases in various two-phase flow regimes. ANL-77-47, USA. Ishii, M., Hibiki, T., 2010. Thermo-Fluid Dynamics of Two-Phase Flow. Springer Science & Business Media. Kataoka, I., Ishii, M., 1987. Drift flux model for large diameter pipe and new correlation for pool void fraction. Int. J. Heat Mass Transfer 30, 1927-1939. Kokal, S.L., 1987. An experimental study of two phase flow in inclined pipes. Dissertation, University of Calgary, Calgary, Canada. Luo, W., Li, Y., Wang, Q.H., Li, J.L., Liao, R.Q., Liu, Z.L., 2016. Experimental study of gas-liquid two-phase flow for high velocity in inclined medium size tube and verification of pressure calculation methods. Int. J. Heat Technology 34(3), 455-464. Mukherjee, H. 1979. An experimental study of inclined two-phase flow. Dissertation, The University of Tulsa, Tulsa, USA.

23

Ottens, M., Hoefsloot, H.C.J., Hamersma, P.J., 2001. Correlations predicting liquid hold-up and pressure gradient in steady-state (nearly) horizontal co-current gas-liquid pipe flow. Chem Eng Res Des. 79(5), 581-592. Oyewole, A.L., 2009. Study of flow patterns and void fraction in inclined two phase flow. Dissertation, Oklahoma State University, Oklahoma, USA. Payne, G.A., Palmer, C.M., Brill, J.P., Beggs, H.D., 1979. Evaluation of inclined-pipe, two-phase liquid holdup and pressure-loss correlations using experimental data. Int. J. Pet. Technol. 1198-1208. Perez, H., 2008. Gas-liquid two-phase flow in inclined pipes. Dissertation, University of Nottingham, Nottingham, UK. Rassame, S., Hibiki, T., 2018. Drift-flux correlation for gas-liquid two-phase flow in a horizontal pipe. Int. J. Heat Fluid Flow 69, 33-42. Spedding, P.L., Watterson, J.K., Raghunathan, S.R., Ferguson, M.E.G., 1998. Two-phase co-current flow in inclined pipe. Int. J. Heat Mass Transfer 30, 3194-3117. Wen, Y., Wu, Z.H., Wang, J.L., Wu, J., Yin, Q.G., Luo W., 2017. Experimental study of liquid holdup of liquid-gas two-phase flow in horizontal and inclined pipes. Int. J. Heat Technology 35(4), 713-720. Wiesche, S.A.D., Kapitz, M., 2015. Effect of pipe inclination on void fraction of two-phase gas-liquid flow. Proceedings of the ASME-JSME-KSME 2015 Joint Fluids Engineering Conference, Seoul, Korea. Yan, C.X., Yan, C.Q., Shen, Y.H., Sun, L.C., Wang, Y., 2014. Evaluation analysis of correlations for predicting the void fraction and slug velocity of slug flow in an inclined narrow rectangular duct. Nuc Eng Des. 273, 155-164. Yeoh, G. H., 2019. Thermal hydraulic considerations of nuclear reactor systems: Past, present and future challenges, Exp. Comp. Multiphase Flow 1(1), 3-27. Zhao, Q., Hibiki, T., 2019. One-dimensional drift-flux correlation for vertical upward two-phase flow in large size concentric and eccentric annuli. Int. J. Multiphase Flow 113, 33-44. Zuber, N., Findlay, J.A., 1965. Average volumetric concentration in two-phase flow systems. J. Heat Transfer 87, 453-468. 24

Captions of Figures Figure 1 Flow conditions of collected database in Bhagwat and Ghajar (2016) flow regime map. Figure 2 Comparison between measured and calculated void fraction by Chexel-Lellouche correlation (1991) in different inclination angles. Figure 3 Comparison between measured and calculated void fraction by Bhagwat-Ghajar correlation (2014) in different inclination angles. Figure 4 Variation of asymptotic distribution parameter, C , with ratio of superficial gas velocity to   mixture volumetric flux, jg / j , by Rassame and Hibiki correlation (2018) and Ishii

correlation (1977). Figure 5 Dependence of asymptotic distribution parameter, C , on ratio of superficial gas velocity to   mixture volumetric flux, jg / j , in different pipe inclination angles.

Figure 6 Effect of pipe inclination angle on asymptotic distribution parameter constants, C , A and

C , B . Figure 7 Comparison between measured and calculated void fraction by the newly-developed driftflux correlation for upward two-phase flows in different inclined pipes. Figure 8 Comparison between measured and calculated void fraction by the newly-developed driftflux correlation for upward two-phase flows with various fluid systems. Figure 9 Comparison between the measured and calculated void fraction by the newly-developed drift-flux correlation in the pipe inclination angles of 1°, 2°, 4.23°, 7.02° and 8.24°.

25

Figure 10 Comparison of various drift-flux correlations with measured data taken by Perez (2008) of upward two-phase flows in inclined pipes.

Captions of Tables Table 1 Summary of two-phase void fraction data for upward two-phase flows in inclined pipes. Table 2 Physical properties of fluid systems in collected database Table 3 Performance evaluation of existing void fraction correlations based on collected database of void fraction for upward two-phase flows in inclined pipes. Table 4 Summary of newly-developed correlations of C for upward gas-liquid two-phase flows in different inclined pipes. Table 5 Newly-developed drift-flux correlation. Table 6 Performance evaluation of the newly-developed drift-flux correlation for upward two-phase flows in inclined pipes. Table 7 Performance evaluation of the newly-developed drift-flux correlation for upward two-phase flows with different fluid systems

26

Figure 1 Flow conditions of collected database in Bhagwat and Ghajar (2016) flow regime map.

27

28

Figure 2 Comparison between measured and calculated void fraction by Chexel-Lellouche correlation (1991) in different inclination angles.

29

30

Figure 3 Comparison between measured and calculated void fraction by Bhagwat-Ghajar correlation (2014) in different inclination angles.

31

Figure 4 Variation of asymptotic distribution parameter, C , with ratio of superficial gas velocity   to mixture volumetric flux, jg / j , by Rassame and Hibiki correlation (2018) and Ishii

correlation (1991).

32

33

Figure 5 Dependence of asymptotic distribution parameter, C , on ratio of superficial gas velocity   to mixture volumetric flux, jg / j , in different pipe inclination angles.

34

Figure 6 Effect of pipe inclination angle on asymptotic distribution parameter constants, C , A and

C , B .

35

36

Figure 7 Comparison between measured and calculated void fraction by the newly-developed driftflux correlation for upward two-phase flows in different inclined pipes.

37

Figure 8 Comparison between measured and calculated void fraction by the newly-developed driftflux correlation for upward two-phase flows with various fluid systems.

38

Figure 9 Comparison between the measured and calculated void fraction by the newly-developed drift-flux correlation in the pipe inclination angles of 1°, 2°, 4.23°, 7.02° and 8.24°.

39

Figure 10 Comparison of various drift-flux correlations with measured data taken by Perez (2008) of upward two-phase flows in inclined pipes.

40

Table 1 Summary of two-phase void fraction data for upward two-phase flows in inclined pipes. Sources

Beggs (1972)

Fluid Systems

Diameter,

Inclination Angle

 jf 

 jg 

Number of

Measurement

Measurement

[-]

D [cm]

[°]

[m/s]

[m/s]

Data [-]

Methods [-]

Accuracy [%]

Air-Water

3.84

5°, 10°, 15°, 35°,

0.0226-

0.250-24.1

263

Quick-closing

N/A

55°, 75°, 90°

5.54

0°, 5°, 20°, 30°,

0.0152-

50°, 60°, 70°,

4.36

Mukherjee.

Air-Kerosene,

(1979)

Air-Lube Oil

3.81

valve 0.0366-30.8

767

Parallel-plate

N/A

capacitance

80°, 90° Payne et al.

Natural Gas-

(1979)

Water

Kokal (1987)

Air-Light Oil

5.1

2.58, 5.12, 7.63

4.23°, 7.02°,

0.118-

8.24°

0.778

0°, 1°, 5°, 9°

0.0300-

0.325-18.8

87

N/A

valve 0.0824-27.4

558

3.05 Spedding et al.

Quick-closing

Capacitance-type

N/A

volume sensor

Air-Water

5.8

0°, 5°

0.0411

0.325-54.0

34

Borescope

N/A

Air-Water

5.2

0°, 1°, 2°

0.141-

7.53-14.5

53

Flow meter

N/A

0.106-12.8

427

Capacitance

±15.0

(1998) Ottens et al. (2001) Perez (2008)

Oyewole

1.48 Air-Water

Air-Water

3.8, 6.7

1.27

0°, 5°, 30°, 60°,

0.0400-

80°, 90°

0.730

5°, 10°, 20°, 45°

0.179-

(2013) Yan et al. (2014)

probes 0.169-24.3

228

0.896 Air-Water

4.3×0.325 (Rectangular)

60°, 70°, 80°, 90°

0.2051.41

41

Quick-closing

±1.3-±16.5

valve 0.124-2.19

44

High-speed camera

±10.2

Sources

Ghajar and

Fluid Systems

Diameter,

Inclination Angle

 jf 

 jg 

Number of

Measurement

Measurement

[-]

D [cm]

[°]

[m/s]

[m/s]

Data [-]

Methods [-]

Accuracy [%]

Air-Water

1.27

0°, 5°, 10°, 20°

0.142-

0.167-22.3

164

Quick-closing

±14.0

Bhagwat

0.992

valve

(2014) Wiesche and

Air-Water

1.3

Kapitz (2015) Bhagwat and

0°, 15°, 30°, 45°,

1.00-2.00

1.10-4.63

70

60°, 75°, 90 Air-Water

1.25

Ghajar (2016)

Quick-closing

±5.7

valve

0°, 5°, 10°, 15°,

0.150-

20°, 30°, 45°,

1.05

0.170-16.4

288

Quick-closing

±1.3-±10.0

valve

60°, 75°, 90 Luo et al.

Air-Water

6

(2016) Wen et al.

0°, 15°, 30°, 45°,

1.01-1.95

20.4-159

322

60°, 75°, 90 Air-Water

6

0°, 15°, 30°

Quick-closing

N/A

valve 1.03-1.23

(2017)

40.9-204

36

Quick-closing valve

42

N/A

Table 2 Physical properties of fluid systems in collected database Fluid Systems

f

f

f

g

g

[-]

[mN/m]

[kg/m3]

[ Pa  s ]

[kg/m3]

[ Pa  s 106 ]

Air-Water

72.0

998.0

0.000894

1.19

18.3

Air-Kerosene

22.6-28.8

786.6-844.1

0.00118-0.00383

1.87-7.40

16.8-19.6

Air-Lube Oil

32.1-37.5

832.3-867.7

0.0108-0.0744

2.23-7.08

16.3-19.0

Natural Gas-Water

72.0

998.0

0.000894

28.4

11.0

Air-Light Oil

30.9

858.6

0.00695

1.19

18.3

43

Table 3 Performance evaluation of existing void fraction correlations based on collected database of void fraction for upward two-phase flows in inclined pipes. Correlations

Chexal-Lellouche correlation (1991)

Bhagwat and Ghajar correlation (2014)

Inclination

md

sd

mrel

mrel, ab

Angle [°]

[-]

[-]

[%]

[%]

0°

0.0591

0.0873

4.91

12.0

5°

0.0736

0.0954

9.01

13.2

10°

0.0771

0.0897

8.67

12.1

15°

0.0844

0.0957

9.67

10.6

20°

0.0622

0.0716

8.47

10.1

30°

0.0693

0.0624

9.63

10.5

45°

0.0567

0.0599

6.57

9.45

50°

0.0593

0.0745

8.62

9.98

60°

0.0872

0.0370

14.4

14.4

70°

0.0476

0.0407

8.89

9.66

80°

0.0501

0.0240

9.91

9.94

90°

0.0319

0.0332

7.01

7.97

Overall

0.0650

0.0755

8.40

11.3

0°

0.141

0.197

12.8

25.2

5°

0.165

0.150

19.9

22.7

10°

0.198

0.135

23.8

24.3

15°

0.0988

0.129

12.3

13.9

20°

0.151

0.123

19.9

21.2

30°

0.116

0.126

15.0

16.2

45°

0.137

0.106

17.1

17.5

50°

0.0986

0.119

13.2

15.5

60°

0.0882

0.0636

14.0

14.1

70°

0.161

0.0964

23.7

23.9

80°

0.0968

0.0137

16.5

18.3

90°

0.0423

0.0838

8.29

11.7

Overall

0.128

0.140

16.0

18.2

44

Table 4 Summary of newly-developed correlations of C for upward gas-liquid two-phase flows in different inclined pipes. Inclination

0  jg / j   0.9

0.9  jg / j   1

angles





 / 0.9



C  8.26



jg / j 

  9.26





 / 0.9



C  7.46



jg / j 

  8.46





 / 0.9



C  6.81 jg / j 

  7.81





 / 0.9



C  6.50



jg / j 

  7.50





 / 0.9



C  5.70



jg / j 

  6.70





 / 0.9



C  5.07



jg / j 

  6.07





 / 0.9



C  3.53 jg / j 

  4.53





 / 0.9



C  3.10



jg / j 

  4.10





 / 0.9



C  2.50



jg / j 

  3.50





 / 0.9



C  2.40



jg / j 

  3.40





 / 0.9



C  2.20



jg / j 

  3.20





 / 0.9



C  1.99



jg / j 

  2.99

0°

C  0.799exp 0.826  jg / j  

5°

C  0.844exp 0.727  jg / j  

10°

C  0.885exp 0.642  jg / j  

15°

C  0.895exp 0.612  jg / j  

20°

C  0.975exp 0.476  jg / j  

30°

C  0.991exp 0.419  jg / j  

45°

C  1.02exp 0.283  jg / j  

50°

C  1.00exp 0.270  jg / j  

60°

C  1.13exp 0.100  jg / j  

70°

C  1.16exp 0.667  jg / j  

80°

C  1.18exp 0.0333  jg / j  

90°

C  1.20exp 0.00251  jg / j  

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

45

1.5

1.5

1.5





Table 5 Newly-developed drift-flux correlation.

0  jg / j   0.9

C0

0.9  jg / j   1

0 j

 g

/ j



 0.9

      1.80-0.700sin    jg / j C0   0.400sin   0.800  exp ln    0.400sin   0.800   0.900    

   

      1.80-0.700sin    jg / j   0.400sin   0.800  exp ln    0.400sin   0.800   0.900    

   

  jg   C0   -8.00+7.00sin     9.0-7.00sin   j       g jg   -  -8.00+7.00sin     8.0-7.00sin   f j    

vgj

  

vgj

  g sin  = 2   2f 

  

vgj 0.9  j

 g

/ j



1

1/ 4

  g sin  = 2   2f 

46

1/ 4

 1  jg / j    0.1 

   

1.5

1.5

     

     1 g   f  

Table 6 Performance evaluation of newly-developed drift-flux correlation for upward two-phase flows in inclined pipes. Inclination Angle

md

sd

mrel

mrel, ab

[°]

[-]

[-]

[%]

[%]

0°

0.0178

0.0554

1.95

6.54

5°

0.0188

0.0590

-1.52

8.01

10°

0.0307

0.0546

3.38

7.48

15°

0.0170

0.0347

2.08

3.68

20°

0.0207

0.0505

2.56

6.19

30°

0.0253

0.0513

2.76

5.76

45°

0.0277

0.0593

-0.826

8.98

50°

0.0147

0.0710

-0.546

7.08

60°

0.0327

0.0454

3.13

5.87

70°

0.0159

0.0369

1.44

3.92

80°

0.0153

0.0281

1.70

3.25

90°

0.00520

0.0260

-0.517

2.71

Overall

0.0220

0.0706

1.86

6.83

47

Table 7 Performance evaluation of the newly-developed drift-flux correlation for upward twophase flows with different fluid systems Fluid System

md

sd

mrel

mrel, ab

[-]

[-]

[-]

[%]

[%]

Air-Water

0.0262

0.0522

2.98

6.45

Air-Kerosene

0.0140

0.0466

1.51

5.80

Air-Lube Oil

0.0190

0.0487

1.94

5.15

Natural Gas-Water

-0.0385

0.0840

-6.84

10.8

Air-Light Oil

0.0153

0.0531

1.88

7.61

48

Highlights: 

Void fraction data and correlations of upward two-phase flow were reviewed extensively.



None of existing correlations could predict collected databases successfully.



The dependence of the drift-flux parameter on pipe inclination angles was analysed.



A new drift-flux correlation was developed for upward two-phase flow in inclined pipes.

Declaration of interests

☐ √The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: