A new model for predicting liquid holdup in two-phase ow ...

Scientia Iranica C (2019) 26(3), 1529{1539

Sharif University of TechnologyScientia Iranica

Transactions C: Chemistry and Chemical Engineeringhttp://scientiairanica.sharif.edu

A new model for predicting liquid holdup in two-phase ow under high gas and liquid velocities

Z. Liua,b,c, R. Liaoa,b,c;�, W. Luoa,b,c, and J.X.F. Ribeiroa,b,c

a. Petroleum Engineering College, Yangtze University, 430100, Wuhan Campus, No. 111, Caidian Street, Caidian District, HubeiProvince, China.

b. The Multiphase Flow Laboratory of Gas Lift Innovation Centre, CNPC, 430100, Wuhan Campus, No. 111, Caidian Street,Caidian District, Hubei Province, China.

c. The Branch of Key Laboratory of CNPC for Oil and Gas Production, Yangtze University, Jingzhou 434023, China.

Received 18 December 2017; received in revised form 30 July 2018; accepted 15 October 2018

KEYWORDSGas-liquid two-phase ow;Annular ow;Higher gas and liquidvelocities;Low liquid holdup.

Abstract. Existing liquid holdup models are generally based on low gas and liquidvelocities. To extend the applicable range of existing liquid holdup prediction modelsand improve their prediction accuracy, a gas-liquid two-phase ow experiment was carriedout using a pipe with an inner diameter of 60 mm and a length of 11.5 m. The super�cialgas and liquid velocity ranges were 14:07 � 56:50 m/s and 0:205 � 1:426 m/s, respectively.The results indicate that the liquid holdup decreases with an increase in the super�cial gasvelocity and increases with an increase in the super�cial liquid velocity. A new annular owmodel for calculating low liquid holdup in a horizontal pipe was developed and presented,considering the relationships between the friction factor ratio of the liquid phase and thegas-liquid interface as well as the super�cial Reynolds number of the gas and liquid. Thepredictions of the model were found to be accurate with an average absolute error of 4.8%.Further, by combining the approach of the Beggs-Brill model with that of the horizontalpipe model established in this paper, a new liquid holdup model that accounts for di�erentangles was presented. It was observed that the resultant model was also accurate and hadan average absolute error of 10%.© 2019 Sharif University of Technology. All rights reserved.

1. Introduction

Gas-liquid two-phase ow exists widely in petroleum,chemistry, nuclear power, aerospace, and other indus-tries. In the petroleum industry, gas-liquid two-phase ow often occurs in oil and gas production wells andgathering pipelines, and its ow characteristics havesigni�cant in uence on the production of reservoirs andthe equipment on the ground. Liquid holdup is notonly an important parameter for the classi�cation of

*. Corresponding author. Tel.: +86 13507212378;Fax: +86 02769110039E-mail address: [email protected] (R. Liao)

doi: 10.24200/sci.2018.50040.1480

ow patterns, but also related to the calculation of theexisting models of pressure drop.

Gas-liquid two-phase ow is more complicatedthan single-phase ow. Similarly, the determinationof liquid holdup is also complex, because there aremany factors that in uence liquid holdup. Due to itscomplexity, most researchers have initially used empir-ical models to predict liquid holdup [1-3]. However,empirical models are based on experiments and areapplicable to experimental data in a certain range ofconditions. It is di�cult, therefore, to generalize andapply the empirical models to other experimental andoperational conditions. With the development of gas-liquid two-phase ow mechanisms and ComputationalFluid Dynamics (CFD), several investigators havestudied mechanistic models that can predict the liquid

1530 Z. Liu et al./Scientia Iranica, Transactions C: Chemistry and ... 26 (2019) 1529{1539

holdup and subsequent pressure drop for di�erentangles [4-11]. One of the most signi�cant mechanisticmodels, which is based on the dynamics of slug ow,was carried out by Zhang [8]. By using the entire �lmzone as the control volume, the momentum exchangebetween the slug body and the �lm zone was introducedinto the momentum equations for slug ow.

Slug ow shares transition boundaries with allthe other ow patterns. The equations of slug oware used not only to calculate the slug characteristicsbut also to predict transitions from slug ow to other ow patterns. Slug liquid holdup is a critical slug ow characteristic for predicting average liquid holdup.Wang et al. [12] developed a complex mechanisticmodel for slug liquid holdup by using an empiricalproportionality parameter generated from their databank as a function of the Froude number and theinclination angle. Al-Safran [13] postulated severalmechanisms of gas entrainment and evolution withinthe slug body and proposed a new simple empirical slugliquid holdup closure relationship valid for di�erentliquid viscosities. Hassanlouei et al. [14] presented amethodology for calculating a slug liquid holdup in ahorizontal pipe. The advantage of this method is thatthe slug unit holdup can be calculated directly based onthe solutions of the ow �eld equations with no need touse correlations. Zuber and Findlay [15] �rst proposedthe drift- ux model to predict the liquid holdup forslug ow, which has been well-researched and appliedto gas-liquid two-phase pipe ows [16-21].

Some researchers used the two- uid model tocalculate liquid holdup of strati�ed and annular owregimes [22-24]. The two- uid model was developedby establishing and solving a momentum equation foreach phase with corresponding closure relationships,which are often empirical in nature. Khaledi et al. [25]performed experiments with high density gas phase andoils of two di�erent viscosities, and the obtained ex-perimental results were compared with the predictionsfrom a two- uid gas-liquid model, which was presentedby Unander et al. [26]. Dabirian et al. [27] studied thee�ects of phase velocities and uid properties on thecharacteristics of strati�ed ow and proposed a closurerelationship for the interfacial friction factor to predictthe liquid holdup.

The mechanistic model is more accurate for liquidholdup prediction; however, it usually incorporatesa system of nonlinear equations. In addition, themechanistic model also relies on some assumptions andempirical constants obtained by experiments. Dueto the limitation of such experimental conditions, thesuper�cial gas and liquid velocities for most gas-liquidtwo-phase ow experiments reported in the literatureare generally not very high. Most of the experimentaldata for liquid holdup based on gas-liquid two-phase ow are derived from the Tulsa University Fluid Flow

Projects (TUFFP), where the super�cial gas velocityrange of the experimental loops in the laboratory is0�44.8 m/s, while the super�cial liquid velocity rangeis 0�1.83 m/s.

Therefore, it is necessary to develop a predictionmodel for liquid holdup and pressure drop based onexperimental data and take advantage of di�erentmodels. To obtain an accurate model for predictingliquid holdup at higher gas and super�cial liquidvelocities applicable to di�erent angles, a two-phase ow experiment with higher gas and liquid velocitieswas carried out on a multi-phase ow experimentalplatform at Yangtze University where the super�cialliquid velocity range was from 0.205 to 1.426 m/s, andthe super�cial gas velocity range was from 14.07 to56.50 m/s. The ow patterns in the experiment wereobserved, and the in uence of di�erent factors on theliquid holdup under the experimental conditions wasobtained by analyzing the experimental data. Finally,a new model for predicting low liquid holdup (HL <0:1) in horizontal pipes was presented. In addition,based on the Beggs-Brill model, a model for predictingthe holdup at di�erent angles was obtained under thecondition of higher super�cial gas and liquid velocities.

2. Experimental facility and measurementmethods

2.1. Experimental facilityThe experiment was carried out on the multi-phase ow experimental platform, which is the platformof the gas lift test base of CNPC located at theSchool of Petroleum Engineering, Yangtze University.The diagram of the experimental device is shown inFigure 1.

Liquid from the mixing tank is pressurized by apump. After pressure stabilization and measurement,the liquid is mixed with measured compressed gas,and the mixed media are introduced together into thetest section. The liquid returns to the mixing tankafter separation from the gas as they pass throughthe gas-liquid separator. The inner diameter of thetest section used in the experiment is 60 mm. Thelength of the plexiglass pipe used to observe the owpattern is 7 m. The test section can be adjusted atan angle of 0 � 90�. Pressure, temperature, andpressure di�erential sensors, as well as quick closingvalves and other devices, are installed on the pipesection. The distance between the two quickly closingvalves is 9.5 m, including the 7-m Plexiglas tube and2.5-m-long stainless steel tube. Control of the devicesand extraction of data are done directly online at thecontrol center. The test section is shown in Figure 2.

To ensure the reliability of the experimentaldata, high-precision instruments are used. The liquid owmeters installed include Endress+Hauser/80E25

Z. Liu et al./Scientia Iranica, Transactions C: Chemistry and ... 26 (2019) 1529{1539 1531

Figure 1. Multiphase ow test device.

Figure 2. Liquid holdup test section.

and Endress+Hauser/80E50, while the gas owmeteremployed is Endress+Hauser/65F1H. All owmetersmeasure the volume ow rate during the experiment.The response time of the quick-closing valve is 0:3 �0:5 s. The pressure and temperature sensors installedbehind the quick-closing valve are used to measure thetemperature and pressure in the test section. Themeasurement parameters and errors of each device areshown in Table 1.

Air is the gas phase, while water constitutes theliquid phase. The surface tension of the water is0.071 Nm�1 (23:58�C). The densities of water andair are 1000 kg/m3 and 1.205 kg/m3 (0.101 MPa),respectively, while their respective viscosities are 1mPa.s and 0.0181 mPa.s (20�C).

2.2. Experimental measurements

Numerous researchers have reported various develop-ment lengths (L=D ratios) for fully developed ow.Omebere et al. [28] demonstrated that L=D = 40was su�cient for a reasonably well-developed churn-turbulent ow using probability density functions ofthe void fraction that showed the same shapes for thisvalue and higher. In addition, some of the reportedexperimental observations are temporary or developing ow patterns, as described by Brennen [29]. Aliyu etal. [30] observed fully developed ow in a pipe positionwith L=D = 46 for annular ow in upward verticalpipes. In the literature, no previous study providesconclusive experimental evidence or estimates of theminimum required length for the ow developmentsection. This study provides approximately 2.4 m forthe ow to develop, L=D = 40, and our observationsshow that ow almost reaches development in thisstage; therefore, the liquid holdup measurements couldproceed.

In the experiment, a constant liquid ow ratewas maintained, while the gas ow rate was adjusted.

Table 1. Measurement parameters and errors of the experimental devices.

Equipment Measuring range Measurement error

Pressure 0�3.5 MPa �0:1%Temperature 0-90�C �0:5%

Liquid ow rate 2 � 20 m3/h �0:3%Gas ow rate 160 � 2000 m3/h �1%


Table 2. Range of parameters under the experimental conditions.

Experimental conditions

Super�cial liquid velocity (m/s) 0:205 � 1:426Super�cial gas velocity (m/s) 14:07 � 56:50

Pressure (MPa) 0:03 � 0:46Temperature (�C) 4:0 � 20:83

Liquid holdup 0:007 � 0:329

Angle of upward inclination (�) 0 15 30 45 60 75 90 TotalNumber of data points 92 91 85 68 72 69 71 548

When the system was steady, the data were recorded,and the experimental ow pattern was observed. Theexperimental data were recorded every 5 seconds for 3minutes; �nally, the average value of each measurementparameter was obtained. After completing the datarecording, the quick-closing valve was closed, andthe liquid holdup was measured. The measurementmethod of liquid holdup was used to test the volumeof residual liquid in the test section, and the ratio ofthe volume of the residual liquid to the volume of thewhole pipe was used as the average liquid holdup of theexperiment. The range of measurement parameters inthe experiment is shown in Table 2.

3. Experimental results and analysis

3.1. Experimental ow patternUnder the experimental conditions, the ow patternsobserved in the horizontal pipe were slug and annular ow, attributed to the high super�cial gas and liquidvelocities. Experimental data points, indicating theobserved ow patterns, plotted on the ow patternmaps presented by Mandhane et al. [31], Taitel andDukler [32] and Zhang are shown in Figure 3. It canbe observed that when the super�cial gas velocity isapproximately 12 m/s, the slug ow is transited toannular ow. With the increase of the super�cial liquidvelocity, the super�cial gas velocity is required whenthe slug ow changing into annular ow increases grad-ually. It can also be observed that the experimentaldata fall near the transition boundary between slug ow and annular ow in the three ow pattern maps.It is found that the Mandhane ow pattern map is inexcellent agreement with the experimental data.

In addition, slug ow, churn ow, and annular ow were observed in the inclined and vertical pipesections. The experimental data points plotted on the ow pattern map of Kaya [33] for vertical pipes arepresented in Figure 4. It can be seen from Figure 4that the area of churn ow decreases with increasingsuper�cial liquid velocity. It can also be observed thatthe ow pattern map of Kaya is consistent with theexperimental data.

Figure 3. Test points on di�erent horizontal ow patternmaps (black represents the Mandhane ow pattern map;blue color represents the Taitel-Dukler ow pattern map;and red represents the Zhang ow pattern map).

Figure 4. Test points on the Kaya ow pattern map.

3.2. Liquid holdup versus super�cial velocityand inclination

The variation in liquid holdup with the super�cial gasand liquid velocities is shown in Figure 5. From thehorizontal to vertical orientation, the liquid holdupdecreases with the increase of super�cial gas veloc-ity. When the super�cial gas velocity is greater than


Figure 5. Liquid holdup versus super�cial gas velocitywith di�erent inclinations (from horizontal to vertical).

Figure 6. Liquid holdup versus angle of inclination.

35 m/s, with an increase in super�cial gas velocity, theliquid holdup decreases a little. When the super�cialgas velocity is high enough, the ow pattern changesinto annular ow, the liquid holdup is very low, andthe in uence of increasing gas velocity on liquid holdupis not obvious. The �gure also shows that the liquidholdup increases with the increase of the super�cialliquid velocity at the same super�cial gas velocity.

The variation in liquid holdup with the inclinationangle is shown in Figure 6. The liquid holdup increases

with the increase of the angle at the same super�cialgas and liquid velocity and reaches the maximum valuewhen the inclination angle is 45�. The liquid holdupdecreases slightly; however, the change is small withthe increase of the angle. This e�ect may result fromthe in uence of gravity and viscosity, which can beseen in the literature study of Beggs [1]. When theinclination angle is less than 45�, the in uence of angleon the liquid holdup decreases with the increase of thesuper�cial gas velocity.

3.3. Prediction of liquid holdup by empiricaland mechanistic models

The Beggs-Brill (B-B) empirical model and the Kayamechanistic model were used to predict the experimen-tal liquid holdup data. The Kaya model was applicableto deviated pipes and not suitable for horizontal pipes.Therefore, the Kaya model was used to predict theliquid holdup at the angle from 15� to 90�, whilethe B-B model was used to predict the liquid holdupat the angle from 0� to 90�. The results of thecomparison between the predicted values of the twodi�erent models and the experimental data of the liquidholdup are shown in Figure 7. When the liquid holdupis low, the prediction of the liquid holdup from thesetwo models is not accurate.

The prediction average relative error and theaverage absolute error are determined as follows:

Average relative error:

E1=

1n

nX1

HLcal �HL exp

HL exp

!� 100: (1)

Average absolute error:

E2=

1n

nX1

��HLcal �HL exp

HL exp

��!� 100; (2)

where HLcal is the liquid holdup predicted by themodel, and HLexp is the liquid holdup measured in theexperiment.

Figure 7. Comparison of di�erent models predicting liquid holdup with experimental data.


Table 3. Errors of the B-B model and Kaya model.

HL < 0:1 HL > 0:1 All dataModel E1 (%) E2 (%) E1 (%) E2 (%) E1 (%) E2 (%)B-B 43.6 46.7 -5.2 10.2 15.6 25.8Kaya 66.4 67.1 -3.9 8.9 19.7 28.5

Table 4. The errors of B-B and Kaya models at di�erent angles.

Model Angle (�) 0 15 30 45 60 70 90

B-B E1 (%) 38.5 35.2 33.2 -4.0 -0.8 -5.5 -4.5E2 (%) 41.9 41.7 40.1 11.6 13.4 10.0 8.7

Kaya E1 (%) { 50.1 31.6 6.0 13.1 5.5 6.9E2 (%) { 60.0 40.2 21.7 17.6 13.4 13.2

The liquid holdup errors of HL < 0:1 and HL >0:1 are shown in Table 3. Accordingly, when HL < 0:1,the prediction values of the two models are greater thanthe experimental value. The average absolute errorshows that the errors of the two models are high andmore than 40%. When HL > 0:1, the errors of the twomodels are lower, and the predicted values are lowerthan the experimental value. Based on the error ofall the data, it can be seen that the errors of the twomodels are relatively high. However, the performanceof the B-B model is better than that of the Kaya model;the errors of the B-B model are 15.6% and 25.8%.

The errors of the two models at di�erent anglesare shown in Table 4. It can be observed that the errorsof the two models decrease with the increase of theangle. The performance of the B-B model is relativelybetter. When the angle is greater than 45�, the B-Bmodel is more accurate and the error is approximately10%. However, the error is high; when the angle is lessthan 45�, all the errors are greater than 40%.

4. Model development and evaluation

According to the error analysis, the error of the B-B model is high when the liquid holdup is less than0.1. However, the liquid holdup prediction at di�erentangles of the B-B model is based on the calculation ofthe liquid holdup of the horizontal pipe. Therefore, the

development of a low liquid holdup correlation capableof predicting values less than 0.1 in the horizontalpipe is necessary. When the liquid holdup is low, the ow pattern of the horizontal pipe is either strati�edor annular. However, no strati�ed ow was observedin this experiment. Therefore, the low liquid holdup(HL < 0:1) model was developed based on annular ow.

Hart et al. [34] proposed the \Apparent RoughSurface" (ARS) model. Badie et al. [35] validatedthe two models by using the experimental data andattached the importance of liquid holdup predictionto the accuracy of pressure drop prediction. Meng etal. [22] added the concept of droplet entrainment on thebasis of the \double-circle" model that was developedby Chen [36] and presented a new model. Fan [23]presented a two- uid model that requires the initialvalue calculated by the ARS model proposed by Hart.Xu [24] combined the research results of Meng andFan and presented a new model for calculating the lowholdup and verifying it with the laboratory data fromthe experiments of Meng and Fan.

4.1. Development of a liquid holdup model forthe horizontal pipe

The liquid phase in annular ow exists in two forms: aliquid �lm owing along the pipe wall and the dropletsentrained in the gas core. The shape of the liquid inannular ow is shown in Figure 8.

Figure 8. The morphology of gas-liquid two-phase ow in the pipe.


The study follows the approach of Xiao [5], whopresented an annular ow model with an average liquid�lm thickness around the pipe wall. The model wasdeveloped based on the momentum equation in theliquid �lm and the gas core.

Momentum equation of gas core:

�AG�dpdL

�G� �iSi � �GAGgsin� = 0; (3)

where:�dpdL

�G

The pressure gradient of the gas phase,Pa;

AG The cross-sectional area inside the pipeof the gas phase, m2;

�i The shear stress of the interface, N/m;Si The perimeter of the interface, m;�G Density of the gas, kg/m3;g The gravity acceleration, m/s2;� The inclination angle, �.

The momentum equation of the liquid �lm:

�AL�dpdL

�L� �LSL + �iSi � �LALgsin� = 0; (4)

where:�dpdL

�L

The pressure gradient of the liquidphase, Pa;

AL The cross-sectional area inside the pipeof the liquid phase, m2;

�L The shear stress of liquid �lm and wall,N/m;

�L The density of the liquid, kg/m3.

If the pressure drop gradient in the gas is equal tothat in the liquid, Eqs. (3) and (4) can be combined toobtain the following combined momentum equation:

�iSiAp � �LSLAG �ALAG(�L � �G)gsin� = 0; (5)

where AL (m2) is the cross-sectional area inside thepipe of the liquid phase.

The shear stresses are:

�L = fL�LvL2

2; (6)

�i = fi�G(vG � vi)2

2; (7)

where:

fL; fi The liquid phase friction factor andthe interfacial friction factor, m2;

vL The average liquid velocity, m/s;vL The average gas velocity, m/s;vi The gas-liquid interface velocity, m/s.

The area of each phase can be expressed approx-imately as functions of pipe cross-sectional area andliquid holdup:

AL = ApHL; (8)

AG = Ap(1�HL); (9)

where HL is the liquid holdup.Then, we can obtain that:

fL�LvL2

2SLAp(1�HL)� fi �G(vG � vL)2

2SiAp

+ApHLAp(1�HL)(�L � �G)gsin� = 0: (10)

When the pipe is horizontal, Eq. (10) can be simpli�edas:�

vLvG

�2

� 11�HL

fi�GSifL�LSL

(1� vLvG

)2 = 0: (11)

The average gas and liquid velocities are:

vG =vsg

1�HL; (12)

vL =vslHL

; (13)

where vsg (m/s) is the super�cial gas velocity, and vsl(m/s) is the super�cial liquid velocity.

The gas-liquid interface velocity is di�cult toobtain. Many researchers assume that the gas-liquidinterface velocity equals the average liquid velocity,vi � vL.

We de�ne the parameter a = HL1�HL , where HL =

a1+a . Eq. (11) can be transformed into:�

vslvsg

�2

� (1 + a)fi�GSifL�LSL

(a� vslvsg

)2 = 0: (14)

According to the geometry of the ow pattern, thefollowing equations can be obtained:

SL = �D; (15)

Si = �(D � 2�); (16)

where D (m) is the internal diameter, and � (m) is theliquid �lm thickness.

According to the assumption that the liquid �lmthickness on the wall of the annular ow is the same,


the following relation between the liquid �lm thicknessand the liquid holdup is given by Fan [23]:

� =D(1�p1�HL)

2: (17)

By substituting Eq. (15) to (17) into Eq. (14), a canbe calculated as follows (Eq. 18):

a =vslvsg

(

sfL�Lfi�G

(1�HL)0:25 + 1): (18)

Finally, the implicit equations of liquid holdup areobtained as follows:

HL =vslvsg

(

sfL�Lfi�G

(1�HL)0:25 + 1)(1�HL): (19)

Eq. (19) shows that the key to calculating the liquidholdup is to calculate the friction coe�cient ratiobetween the liquid phase and the gas-liquid interfacefLfi . Hart et al. (1989) [34] reported the relationshipbetween fL

fi and the super�cial liquid Reynolds number.According to the experimental results,

fLfi

= 108Resl�0:726: (20)

According to the experimental data, fLfi can be calcu-

lated by Eq. (19); then, the corresponding super�cialgas and liquid Reynolds number can also be calculated.The ratio of the friction coe�cient between the liquidphase and the gas-liquid interface fL

fi of di�erentsuper�cial gas and liquid Reynolds number is shown inFigure 9. It can be seen from the �gure that fL

fi changes

Figure 9. The ratio of friction coe�cient between theliquid phase and the gas-liquid interface ( fLfi ) versussuper�cial gas Reynold number (Resg) and super�cialliquid Reynold number (Resl).

not only with the super�cial liquid Reynolds number,but also with the super�cial gas Reynolds number.

Therefore, in the reference form of Eq. (20), thefollowing equations are presented:

fLfi

= xReslyResgz; (21)

where:

Resl =�LvslD�L

The super�cial liquid Reynoldsnumber;

Resg =�gvsgD�G

The super�cial gas Reynolds number;

�G The gas viscosity, mPa.s;�L The liquid viscosity, mPa.s.

By �tting the experimental data, the obtainedparameters are as follows: x = 0:5756, y = 0:5372;and z = �0:7734.

4.2. Evaluation of the new horizontal lowliquid holdup model

The calculation procedure for the new horizontal lowliquid holdup model is as follows:

1. Select an initial value for the liquid holdup (HL =0:1);

2. Use Eq. (19) to calculate a new liquid holdupHLnew;

3. If jHLnew � HLj > 0:0001, then HL = HLnew.Repeat step (2), until jHLnew �HLj < 0:0001.

There are 49 groups of data for which the liquidholdup is HL < 0:1 in the horizontal pipe. The resultsof a comparison between the predictions of 6 existingmodels (presented by B-B, M-B, Hart, Xiao, Chen, andFan) and those of the present model and experimentaldata are shown in Figure 10. The �gure shows that theB-B, Xiao, Fan, Chen and Hart models over-predict theexperimental values. The average relative error and theaverage absolute error are all greater than 70%. Theperformance of the M-B model is good, and the averagerelative error is 17.8%. The average absolute error is19.3%; however, the error is relatively high. On theother hand, the present model gives better predictionsthan the existing models. The absolute error of thepresent model is within 20%, the average absolute erroris 4.8%, and the average relative error is 0.62%.

4.3. Prediction and evaluation of the newliquid holdup model with di�erentinclination angles at high velocity

The presented model modi�es the B-B model when theangle is less than 45� and the liquid holdup is less than0.1. The procedure for predicting the liquid holdupwith the modi�ed B-B model is presented in Figure 11.


Figure 10. Comparison of predictions of liquid holdup in di�erent models and experimental values: (a) B-B, M-B, Fan,Xiao, Chen, and Hart models, and (b) the present model.

Figure 11. Calculation ow chart for the new model.

Table 5. B-B model errors for di�erent angles.

Angle of inclination (�) 0 15 30 45 60 75 90 0 � 90

E1 (%) -2.2 -0.7 -0.5 -0.4 -0.8 -5.5 -4.5 -2.0

E2 (%) 7.2 9.3 11.2 11.6 13.3 10.0 8.7 10.0

The average relative error and average absoluteerror of the new model are shown in Table 5. Thetable shows that, compared with the B-B model, theerror of the new model in 0 � 30� is clearly reduced,and the average absolute error is approximately 10%.Overall, the predicted average relative error and aver-age absolute error of the new model are 2% and 10%,respectively. In addition, the prediction accuracy isimproved compared with those of the B-B model and

the Kaya model. The new model is useful for predictingthe liquid holdup of gas and liquid at high velocities.

5. Conclusions

To obtain an accurate two- uid model for the predic-tion of liquid holdup for all inclinations at high gasand liquid velocities, 548 liquid holdup data pointswere obtained from the multi-phase ow experimental


laboratory of the Yangtze University from the study.The following conclusions can be reached:

1. Under a certain super�cial liquid velocity, the liquidholdup decreased with the increase of super�cial gasvelocity, while the liquid holdup increased graduallywith the increase of super�cial liquid velocity. Theliquid holdup increased with the increase of theangle and reached the maximum value at the angleof 45�. Then, with the increase of the angle, the liq-uid holdup changed a little and slightly decreased.With the insigni�cant increase of gas velocity, thee�ect of angle on liquid holdup decreased when theinclination of angle was less than 45�.

2. A new two- uid model for predicting liquid holdupfor annular ow in horizontal pipes was presented.Predictions of the model were found to be accurateand in agreement with experimental data. Theaverage absolute error was 4.8%. In addition,a modi�ed B-B model for calculating the liquidholdup at di�erent angles was also presented. Theerror of the new model was lower, and the averageabsolute error was 10%. Both models were applica-ble to high gas and liquid velocities.

Acknowledgments

The authors are grateful to everyone at the Branch ofKey Laboratory of CNPC for Oil and Gas Productionand Key Laboratory of Exploration Technologies forOil and Gas Resources for their assistance. Thiswork was supported by the National Natural ScienceFound Project (No. 61572084) and National Key Scien-ti�c and Technological Project (2016ZX05056004-002,2017ZX05030-005).

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Biographies

Zilong Liu obtained both his BSc degree (2007-2011)and MSc degree (2013-2016) in Petroleum EngineeringCollege at Yangtze University, China. He is currentlypursuing his PhD in Oil and Gas Transportation andStorage at Yangtze University, Wuhan, China. Hismain research interest is multiphase ow and gas lift.

Ruiquan Liao is a Professor of Petroleum Engineeringat Yangtze University, China. He received his PhDdegree (2002) from Huazhong University of Science andTechnology, China. He has published over 100 researchpapers. His research interests include multiphase ow,oil and gas �eld development, gas lift, and controlscience and engineering.

Wei Luo is a Lecturer of Petroleum Engineering atYangtze University, China. He has worked as anengineer in Gas lift Technology Center in PetroChinaTuha Oil�eld Company from 2009 to 2012. He receivedhis PhD degree (2012-2015) from Yangtze University,China. He has published over 10 research papers.His research interests include gas lift design method,multiphase ow, and oil well in ow performance.

Joseph Xavier Francisco Ribeiro obtained both hisBSc degree (1999-2003) and MSc degrees (2009-2011)in Mechanical Engineering at the Kwame NkrumahUniversity of Science and Technology, Ghana. He iscurrently pursuing his PhD in Oil and Gas Transporta-tion and Storage at Yangtze University, China. Hismain research interest is multiphase ow.

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