Void fraction measurement and calculation model of vertical upward co-current air-water slug flow

Teng Wang ,Miao Gui,2 ,Jinle Zhao ,Qincheng Bi,*,Tao Zhang

1 State Key Laboratory of Multiphase Flow in Power Engineering,Xi’an Jiaotong University,Xi’an 710049,China

2 School of Nuclear Science and Technology,Xi’an Jiaotong University,Xi’an 710049,China

Keywords:Slug flow Gas-liquid flow Void fraction Measurement Optical probe Model

ABSTRACT The focus of this paper is on the measurement and calculation model of void fraction for the vertical upward co-current air-water slug flow in a circular tube of 15 mm inner diameter.High-speed photography and optical probes were utilized,with water superficial velocity ranging from 0.089 to 0.65 m·s-1 and gas superficial velocity ranging from 0.049 to 0.65 m·s-1.A new void fraction model based on the local parameters was proposed,disposing the slug flow as a combination of Taylor bubbles and liquid slugs.In the Taylor bubble region,correction factors of liquid film thickness Cδ and nose shape CZ* were proposed to calculate αTB.In the liquid slug region,the radial void fraction distribution profiles were obtained to calculate αLS,by employing the image processing technique based on superevised machine learning.Results showed that the void fraction proportion in Taylor bubbles occupied crucial contribution to the overall void fraction.Multiple types of void fraction predictive correlations were assessed using the present data.The performance of the Schmidt model was optimal,while some models for slug flow performed not outstanding.Additionally,a predictive correlation was correlated between the central local void fraction and the cross-sectional averaged void fraction,as a straightforward form of the void fraction calculation model.The predictive correlation showed a good agreement with the present experimental data,as well as the data of Olerni et al.,indicating that the new model was effective and applicable under the slug flow conditions.

1.Introduction

Gas-liquid two-phase flow exists widely in various industrial facilities.Given its importance to the secure operation of equipment and the complexity of its internal flow mechanism,comprehensive and in-depth studies have been necessary.As one of the most fundamental parameters to characterize two-phase flows,void fraction is crucial for determining other two-phase parameters such as pressure drop,gas velocity and average density [1].Thus,it is of great significance to accurately measure (or predict)the void fraction,simultaneously,due to the interaction between phases and multifarious two-phase flow patterns,it is full of difficulties.

In the method of obtaining the void fraction,the experimental investigations and the development of void fraction models are complementary.On one hand,benefitted from the advancement of measurement technology,more experimental methods have been developed,including differential pressure method [2],Gamma-ray densitometry [3],electrical capacitance tomography[4],neutron radiography technique [5],conductivity and optical probe [6,7].Through intervention or non-intervention,from local to the overall average,these advanced methods cover omnidirectional and multidimensional void fraction measurement,which provide a large amount of reliable data for the establishment of the void fraction model.On the other hand,prediction models and correlations have been developed based on a variety of experimental databases obtained by these techniques under various experimental conditions [8].The development of experimental studies also benefits from the more accurate models.For instance,variable density single-fluid model proposed by Bankoff [9] facilitates the studies of gas-phase distribution in the tube;Drift-flux model [10] brings more accurate two-phase velocity measurement;Slip ratio model [11] also deepens the understanding of the gas-liquid two-phase interaction.

Slug flow is a highly intermittent and irregular two-phase flow,with the alternation of Taylor bubbles and liquid slugs in the transportation of fluids.From the perspective of void fraction,Taylor bubbles and liquid slugs present discriminative time-domain characteristics.In a vertical upward slug flow,the Taylor bubble is composed of hollow gas and liquid film close to the inner wall surface,similar to an annular flow,while the liquid slug is more like a discrete bubbly flow[12].Thus,the undifferentiated general measurement implemented on slug flow has great limitations.The optical probeis an intrusive measurement method,based on the optical reflection principle and photoelectric signal conversion,which is known to demonstrate strong anti-interference,fast signal transmission speed and high measurement precision[13].More importantly,it has been verified that the optical probe coincides with the slug flow in time-domain characteristics.However,the optical probe can only perform local measurements,which is far from being to represent the averaged void fraction of the section.

As briefly reviewed above,an experimental investigation on vertical upward co-current air-water slug flow was carried out at atmospheric pressure,employing optical probes and high-speed photography.In this present study,a new void fraction model based on the local parameters was proposed,applicable to only slug flow,which was disposed as a combination of Taylor bubbles and liquid slugs.Multiple types of predictive correlation were employed to compare with the calculation results,including the general model,the modified-homogeneous model,the slip ratio model,the drift flux models,and the slug flow models.Moreover,the effectiveness and applicability of the newly proposed correlation were evaluated by comparing with some existing experimental data.

2.Literature Review of Void Fraction Models

2.1.Homogeneous model

The homogeneous model is the simplest model for calculating the void fraction,which is assumed that the actual velocities of the gas-liquid two phases are equal.Based on this assumption,the void fraction can be expressed in terms of volumetric void fraction:

where α is the cross-sectional averaged void fraction,β is the volumetric void fraction,UgsandUlsis the gas superficial velocity and water superficial velocity,respectively.The applicable condition of the homogeneous model has been extremely limited,owing to the complexity of two-phase flow.Previous studies [14] have shown that homogeneous model can only be applied to bubbly or mist flow with large mass flux,high pressure and relatively uniform phase distribution.

2.2.Separated-fluid model

The relative movement between gas and liquid phase and the non-uniform distribution in the cross-section has been considered in the separated-fluid model.There are two types of correlations based on separated-fluid model,one of which is proposed in the modified form of homogeneous model.These correlations are a constant or some functional multiple of the homogeneous void fraction [15],expressed as:

Armand[16]first proposed this model and believed thatKcould take a constant of 0.833.The variable density model of Bankoff[9]can also be expressed in the form ofK,which was shown to be a function of the system pressure.The expression ofKhas been improved by subsequent studies [17,18],in where more flow parameters and fluid properties have been taken into account.However,owing to the lack of theoretical basis,the modifiedhomogeneous correlations have limited accuracy in a wide range of conditions.The other type of void fraction prediction correlations based on separated-fluid model is slip ratio correlations,where the slip ratioSis defined as the ratio of gas-phase velocity to liquid-phase velocity.The void fraction can be calculated as:

The slip ratio model is usually applicable to steam-water twophase fluids.A general expression for this type of correlation was put forward by Butterworth [11] as a function of the ratios between wetness fraction(1-x)and dryness fractionx;the ratios of densities of the gas and liquid phase(ρgand ρl);and the ratios of the viscosities of the liquid and gas phase(μland μg).Premoliet al.[19] expressed the slip ratio as a function of dimensionless numbers (ReandWe),which extended the applicable range of the model.In recent decades,Schmidtet al.[20] proposed a void fraction prediction correlation based on the slip ratio for the highviscosity two-phase flow.

2.3.Drift-flux model

The drift-flux model,envisaged by Zuber and Findlay [10],is a simple and practical two-phase flow model,owing to its flexibility and modular structure [21].Due to the radial heterogeneities of void fraction and the relative velocity between the phases being taken into account,the drift-flux model has been recommended to predict void fraction over a broad range of two phase flow conditions by incorporating certain empirical parameters [22].The physical form of drift-flux model to determine the void fraction is defined as:

where α,Ugs,C0,UmandUgmare the void fraction,gas superficial velocity,distribution parameter,two-phase mixture superficial velocity and drift velocity,respectively.The notations of ＜＞present the cross-sectional averaged flow properties.The definitions of distribution parameterC0and drift velocityUgmare mathematically expressed as:

Thus,acquiring the reliableC0andUgmbecomes a crucial step towards to calculate the void fraction using drift-flux model.They are difficult to determine experimentally and inorder to establish a closure correlation form,both distribution parameter and the drift velocity must be modeled as a function of several two phase flow variables[21].Wallis[23],Ishii[24],and Hibiki and Ishii [25]promoted the development of drift-flux model by proposing appropriate correlations ofC0andUgm,which are affected by flow pattern,flow channel geometry,flow orientation,system pressure,gravitational acceleration and physical properties of the two-phase fluids.A summarized list of representative drift-flux model correlations is given in Table A1 in the Appendix.

2.4.Typical models applicable to slug flow

The aforementioned correlations for void fraction prediction are general models,with the application range independent of flow patterns.Several exclusive void fraction prediction models for gasliquid slug flow are summarized in this section.

Guetet al.[26]developed a physically based model for computing the mean void fraction and the liquid slug void fraction in vertical upward gas-liquid slug flow.Three parts,namely Taylor bubble region (TB),Taylor bubble wake region (W) and developed liquid slug region (LS) were separated;and the flow was assumed in a fully developed and equilibrium state.Based on the mean flux conservation of gas and liquid phase at each boundaries and hydrodynamics or drift-flux models for the three regions,13 independent relations were combined to calculate the void fraction.Relying on the study of Brauner and Ullmann [27],Guetet al.[26]introduced an additional relation for the rate of gas entrained at the rear of the Taylor bubble.Thus,the entrainment characteristics of the Taylor bubble trail were also described in the model.Table A2 in the Appendix lists the integrated closure formulation of this model.For the oil-gas two-phase flow,especially heavy oil high viscosity,Liuet al.[28] proposed a model for predicting slug flow liquid holdup (1 -α) in vertical pipes.Assuming that the thickness of the liquid film was constant and the liquid and gas phases were incompressible,a set of equations for calculating liquid holdup was constructed based on the mass conservation principle,which is also listed in Table A2 in the Appendix.The required inputs included Taylor bubble velocity,dispersed bubbles velocity in liquid slug,void fraction of slug liquid and two-phase superficial parameters.

As aforementioned in Section 2.3,the drift-flux model with flexible and modular structure also has the potential to be applied to slug flow.By analyzing the effect of a frictional pressure gradient due to a liquid flow in multi-bubble system,Hibiki and Ishii [25]developed the drift constitutive equation for dispersed two-phase flow.Regarding the slug flow,the diameter of the Taylor bubble becomes approximately that of the tube with a thin liquid film separating the bubbles from the wall.Thus,by balancing the drag force(given by Ishii[24])with the pressure force and the assumed bubble radius to beD/2,the drift velocityUgmfor slug flow was obtained.Further,the distribution parameterC0was considered to depend on the density ratio and the Reynolds number.Based on the experimental data,Hibiki and Ishii[25]proposed the empirical correlation ofC0for slug flow.Subsequently,Adekomaya [29]proposed an improved version of drift-flux model for predicting pressure gradient and void fraction in vertical slug flow.The drift velocityUgmwas determined according to the Taylor bubble rising velocity;and the distribution parameterC0adopted an empirically derived correlation containing the Reynolds number based on the mixture velocity.The drift-flux models of Hibiki and Ishii [25]and Adekomaya [29] for slug flow are both summarized in Table A2 in the Appendix.

It can be observed from the above review that the optimal methodology for establishing a slug flow model is to distinguish between Taylor bubbles and liquid slugs,due to their discriminative time-domain characteristics.The void fraction prediction models of Guetet al.[26]and Liuet al.[28]are following this thought.However,although these two models are inherently closed,the main inputs are overly dependent on some empirical correlations.Besides,several essential assumptions for establishing the models are too rough,ignoring some major influencing factors,for instance,the developing liquid film thickness and the length proportion of Taylor bubbles and liquid slugs in the slug flow.Therefore,to obtain an accurate void fraction calculation model,it is crucial to involve the local parameters of the slug flow that affect the characteristics of Taylor bubbles and liquid slugs.In this present study,a comprehensive experimental investigation on vertical upward air-water slug flow has been carried out,employing optical probes and high-speed photography.A new void fraction model based on the local parameters was proposed and evaluated.

3.Experimental Apparatus and Methods

3.1.Test loop

The void fraction measurement experiments for slug flow were carried out at the State Key Laboratory of Multiphase Flow in Power Engineering,Xi’an Jiaotong University.The air-water twophase flow test loop is an open system,mainly composed of a liquid flow control system,a gas flow control system,and a data acquisition system,schematically shown in Fig.1.A detailed introduction of the test loop can be referred to [30].

3.2.Test section geometry

The experiments of vertical upward co-current air-water slug flow were carried out in the polymethyl methacrylate test section with an inner diameter of 15 mm,as illustrated in Fig.2.Fig.2(a)shows the layout of the test section and measuring point arrangements;Fig.2(b) displays the location of the optical probes and the sealing method;Fig.2(c)exhibits the mixing method of air and water fluids.The total length of the test section was 1800 mm,and the measurement positions of high speed camera,double optical probe,and single optical probe were placed 1350,1550,and 1650 mm downstream of the air-water mixing unit,respectively.According to the results of Paranjapeet al.[31],a distance ofz/D=31 was enough for the formation of fully-developed two-phase flow,whereDwas the inner diameter andzwas the axial distance from the air-water mixing unit.In the present study,z/D=90,103.3,110,for the high speed camera,double optical probe,and single optical probe,respectively.These measurement positions were far enough downstream of the mixing unit so that the entrance effect could be neglected and all the measurements were considered to be conducted under fully-developed flow conditions.The optical probes were placed in the center of the tube section and sealed by a set of cutting ferrule components.A modified type-T air-water mixing unit was designed to acquire stable two-phase flow.

3.3.Measurement methods and instruments

High-speed photography and optical probes were utilized to study the local parameters and void fraction of the slug flow.The experiments were performed at steady states with pressures at the inlet of the testsection(P)ranging from 0.04 to 0.15 MPa,fluid temperature (Tin) ranging from 14.5 to 19.6 °C,water superficial velocity (Uls) ranging from 0.089 to 0.65 m·s-1,gas superficial velocity(Ugs)ranging from 0.049 to 0.65 m·s-1.The water flow rate was measured by an electromagnetic flow meter(KAIFENG,E-mag C type) with a measurement range of 0-15 m·s-1and an accuracy of 0.3%.The air flow rate was measured by aset of volume flow meters (AALBORG,TD9411M) with the measurement range of 0-16,0-100,and 0-1000 L·min-1,respectively,depending on the actual flow rate.Further,the accuracy of the AALBORG volume flow meters was 1%at the full scale.The two-phase fluids temperatures were determined using a set of 1.5 mm K-type (Omega) sheathed thermocouples.

3.4.Uncertainty analysis

Uncertainties in the parameters directly measured by instruments with clear factory precision,including inlet pressureP,differential pressure dP,water flow rateul,air volume flow fluxVg,and fluid temperatureT,were determined by the accuracy of measurement instruments and the testing range of the experiment.

Fig.1.Schematic diagram of the air-water two-phase flow test loop.

Fig.2.(a) Test section and measuring points arrangement;(b) Location of the probes and method of sealing;(c) Modified type-T air-water mixing unit.

Uncertainties of parameters measured by optical probes,including local void fraction αloc,interface frequencyfintand bubble velocityU,were determined by a set of elaborately designed visual calibrations [30,32].When a vertically rising bubble came into contact with the probe tip,a downward force exerted by the probe (which can be regarded as a fixed rigid body) decelerated and deformed the bubble.Thus,the contour of the bubble was reshaped,and the movement trajectory was shifted,resulting in corresponding measurement errors.Fig.3(a) shows the schematic diagram of the evaluation parameters caused by the deformation effect.When the bubble was approaching the probe,the bubble velocity and profile remained stable.Assuming that the probe was a virtual one,the chord length to be measured wasl,which can be defined as initial chord length.While the actual probe interacted with the bubble,the chord length to be measured became smaller under the action of squeezing,which can be determined asli.The ratio ofliandlwas used to quantitatively evaluate the measurement error of deformation effect.Fig.3(b) displays the schematic diagram of the evaluation parameters caused by the deceleration and drift effects,represented by the trajectory of centroids.In this case,the decrease in the distance between adjacent centroids indicated a decrease in the bubble velocity.Simultaneously,the lateral force applied to the bubble caused its trajectory to drift a way from the probe.Assuming that the vertical base line remained unchanged,the distance from the bubble centroid to the base line was defined as the drift semi-major axisai.The ratio ofaiandawas used to quantitatively evaluate the measurement error of drift effect.Synthetically,the uncertainties of local void fraction and bubble velocity measured by optical probes were mainly attributed to these three effects.

Fig.3.Schematic diagram of the evaluation parameters caused by (a) deformation effect;(b) deceleration and drift effect.

To quantify the uncertainties,two time-domain signals were defined in this study.One was the acquired signal τopby the optical probe,which represented the actual measurement value.The other was the ideally standard signal τs,which assumed that the bubble did not interact with the probe tip,but maintained its original contour,velocity,and trajectory before touching the probe.Based on the relationship between the two signals,the corresponding measurement uncertainty can be obtained.By analyzing the signals from a series of bubbles,with various semi-major axisaand piercing depthsx,the uncertainties of local void fraction and bubble velocitywere 8.4%and 12.6%,respectively.The uncertainty analysis results are summarized in Table 1.

Table 1Uncertainties of the main experimental parameters

4.Development of a Void Fraction Model for Slug Flow

4.1.Derivation of the void fraction model

Regarding the vertical upward air-water slug flow,in an infinitesimal control volume along the axial direction,the crosssectional averaged void fraction α can be determined as:

where dVis the volume of the infinitesimal control volume and dVgis the volume occupied by the gas phase.It is supposed that the control volume contains the entire cross section in the radial direction,thus,dVand dVgcan be calculated,respectively,as:

where dLis the length of the control volume along the axial direction,Ais the cross-sectional area,andAgis the averaged area occupied by the gas phase.For a series of representative slug units,both Taylor bubbles and liquid slugs may exist in the control volume.Further,Eq.(7)can be expressed as a combination of Taylor bubbles and liquid slugs,as:

whereirepresents an independent Taylor bubble segment andjrepresents an independent liquid slug segment.Considering the addition in axial direction and average in cross section for Taylor bubbles and liquid slugs,respectively,Eq.(10) can be expressed as following:

where dLTBand dLLSare the total length of Taylor bubbles and liquid slugs in the control volume,Ag,TBandAg,LSare the averaged area occupied by the gas phase of Taylor bubbles and liquid slugs,respectively.Particularly,Ag,TB/AandAg,LS/Acan be expressed as void fraction in Taylor bubbles αTBand liquid slugs αLS,respectively.Simultaneously,in co-current air-water slug flow,the aforementioned infinitesimal length is determined as product of phase velocity and infinitesimal time.

whereUBis the mean velocity of slug units,UTBandULSare the mean velocity of Taylor bubbles and liquid slugs,dtTBand dtLSare the total time of Taylor bubbles and liquid slugs in the control volume,respectively.Therefore,the cross-sectional averaged void fraction α in continuous slug flow of durationtis determined as:

where ΦTBand ΦLSare the dimensionless velocity ratio of Taylor bubbles and liquid slugs,respectively.

4.2.Void fraction in Taylor bubbles

Fig.4 presents the distribution characteristics of air-water two phases in the axial direction and cross section of a representative slug flow unit.Due to non-uniform distribution,the central local void fraction αlocis far from replacing the cross-sectional averaged void fraction α.Thus,a reasonable derivation from αlocto α becomes a crucial step towards a reliable establishment of the void fraction model.

As shown in Fig.4,the center of a Taylor bubble has been always maintained in the gas phase,with the local void fraction αloc,TBbeing constant 1.On one hand,the cross-sectional averaged void fraction is extremely overestimated on account of the falling liquid film around the Taylor bubble.Therefore,it is necessary to introduce a corresponding correction factor in allusion to the fully developed liquid film thickness,noted asCδ.On the other hand,in nose region,as the radius of the cross-section of the bubble increases further from the Taylor bubble tip,the liquid film thickness will change greatly.Considering the nose shape of Taylor bubble,the corresponding correction factor is introduced,noted asCZ*.Thus,void fraction in Taylor bubbles is determined as:

where αloc,TBis constant 1.

4.2.1.Optical corrections

The image processing method was employed to extract the gasliquid two-phase interface to obtain the fully developed liquid film thickness and the nose shape of Taylor bubble.However,the light rays are scattered near the phase interface during the propagation process due to changes in refractive index [33].Therefore,a series of optical corrections on the distorted images were performed.The focus of the corrections fell on the true distance between the airwater two-phase interface and the tube center [34].Fig.5 graphically illustrates the pathway of light propagating inside and outside the tube.

It can be seen from Fig.5 that the light is refracted between the liquid film and the inner wall of the tube,the outer wall of the tube and the air,respectively.The two-dimensional coordinate system is established as shown in Fig.5,and the intersection points of the light pathway and the three phase interfaces areA(xA,yA),B(xB,yB),andC(xC,yC) respectively,whereyAis the true distance from the air-water interface to the center of the tube,andyCis the imaging value from the air-water interface to the center of the tube.At intersection pointC,the following relationship can be deduced according to Snell’s law and triangle law:

whereRoutis the outer diameter of the tube,nairandntubeare the refractive indices of air and tube (polymethyl methacrylate),respectively.

The refraction angles θ1and θ2can be solved from the above equations.Further,at intersection pointB,the geometric relations of coordinates are determined as:

The relationship between intersection pointCand intersection pointBis established by the refraction angles:

Besides,use of Snell’s law allows the refracted angle,θ4,to be solved.

wherenwateris the refractive index of water.Further,at intersection pointA,yAis determined from the geometric relationship:

Meanwhile,the true liquid film thickness δ can be calculated by the following formula:

Fig.4.Schematic diagram of the main hydrodynamic features of a slug flow.

Fig.5.Schematic illustration of the light pathway propagation across different media.

4.2.2.Correction factor of liquid film thickness Cδ

Between the Taylor bubble and the tube wall,a thin liquid film flows downward and gradually develops to a stable thickness δ.Accordingly,the Taylor bubble shape stabilizes as a cylinder with a radius of (R-δ) [12].The developed liquid film is known to be relevant when the slug flow hydrodynamic model is used for a better understanding of the processes of heat and mass transfer [35]and of the mechanisms of interaction between two consecutive Taylor bubbles [36].In the early decades,Brown [37] proposed an alternative model to the film flow around the Taylor bubble.The thickness of the annular fully developed film for laminar regime as derived by Brown becomes:

The liquid film assumption is applicable to relatively low viscosity liquids.In some subsequent experimental investigations to predict the thickness of the liquid film,the dimensionless inverse viscosity numberNfhas been crucial.Nfeliminates the effect of surface tension,also can be obtained by combining the Eotvos number (Eo=ρlgD2/σl) and the Morton number

where μlis the liquid viscosity,ρlis the liquid density,gis the gravitational acceleration,andDis the tube inner diameter.

Llewellinet al.[38] conducted a dimensionless analysis of Brown’s theoretical work,the liquid film δ was estimated in dimensionless form,written as:

where δ’is the dimensionless thickness,δ’=δ/R.Eq.(25)is based on the assumption of potential flow proposed by Dumitrescu [39],which is applied when viscosity is not important,and correspondingly,the applicable conditions can be couched in terms of inverse viscosity number for validity:Nf＞120.

Further,synthesizing existing theoretical models and experimental data,Llewellinet al.[38] proposed a single empirical correlation:

Eq.(26) provides an excellent fit to experimental conditions in the range 0.1 ＜Nf＜100000,and verified by the data of Nogueiraet al.[40].

Azevedoet al.[41] employed non-intrusive pulse-echo ultrasonic technique to measure the falling liquid film around single Taylor bubbles,proposed an empirical correlation to estimate dimensionless equilibrium film thickness forNfranging from 15 to 12900:

In this present study,the stable liquid film thickness of fullydeveloped slug flow was obtained employed image processing method.During the raw image acquisition process,a unilateral focused light source was adopted,resulting in a distinct outline of air-water two-phase interface on the backlight side,as shown by the vertical red line in Fig.6.The dimensionless liquid film thickness δ’was obtained by calculating the pixel ratio of the liquid film in the horizontal direction,which had high identification accuracy.From the aforementioned analysis,the pivotal dimensionless number that determined the thickness of the liquid film was the inverse viscosity numberNf.Thus,under stable slug flow conditions,the inlet temperature was gradually increased from 15 to 55 °C,correspondinglyNfranged from 5050 to 11248.Further,the comparison between several sets of typical experimental data and three prediction models is presented in Fig.7.It was observed that the prediction divergence of these three models was mainly reflected whenNfwas greater than approximately 3000.The experimental results in this study indicated that in a higherNfrange,the liquid film thickness was almost unchanged,and there was no obvious evidence to support the trend that the thickness of liquid film decreased as the inverse viscosity number increased.Therefore,the prediction model of liquid film thickness proposed by Llewellinet al.[38] was recommended in a wide range ofNf.

Fig.6.Liquid film thicknesses of typical slug units,by image processing.

It can be seen from the section A-A in Fig.4,in the region of fully-developed liquid film,the cross-sectional void fractionis a constant,which is only related to the tube inner diameter and the thickness of liquid film.Thus,the correction factor of liquid film thicknessCδis determined as:

Further,when using the dimensionless liquid film thickness δ’,the above formula can be expressed as:

4.2.3.Correction factor of nose shape CZ*

In front of the Taylor bubble,the velocity field,determined by the rising bubble and falling liquid film,drives the gas-liquid interface to form the shape of a nose,which is known to be an important feature to be assessed while studying the motion of Taylor bubbles [12].

In the reported literature,a number of experimental and numerical studies onthe individual Taylor bubble shape of the nose region have been carried out.Dumitrescu[39]derived,from potential flow,a mathematical expression of Taylor bubble shape rising trough stagnant water contained in a vertical tube.Mao and Dukler[42,43]verified the applicability of Dumitrescu model under specific flow conditions,based on experimental data and numerical simulation.The investigation results of Nogueiraet al.[40] also showed an excellent agreement with Dumitrescu’s profile at highRenumber.Recently,the effects of gas and liquid superficial velocity [44],flow direction [45] and the presence of dispersed bubbles around the Taylor bubble [46] on the shape of nose region have been analyzed in detail.Numerous evidences demonstrated the feasibility that employing the Dumitrescu model to describe the bubble shape for slug flow with low gas-liquid superficial velocity and low viscosity working fluid.Dumitrescu suggested that the shape profile could be expressed in the nose and film regions,respectively,as:

Fig.7.Comparison between experimental data and three prediction models of liquid film thickness.

wherez/Dis the dimensionless axial distance pointing downwards from Taylor bubble tip,andr/Dis the dimensionless radial distance from the tube center line.

The comparison between prediction results of Eq.(30) and experimental data under a typical co-current slug flow withUlsof 0.1 m·s-1,Ugsof 0.1 m·s-1,andReUTBof 4483,is shown in Fig.8.The experimental data was obtained by image processing,and a brief illustration of the processing method and steps are shown in Fig.9.The initial image taken by a high-speed camera,had 480 (V) × 640 (H) pixels.Cut off the outside view of the flow channel,the cropped image size was 480 (V) × 233 (H) pixels,as shown in Fig.9(a).After the operation of image enhancement,the outline of a Taylor bubble was drawn using the tracing method to obtain Fig.9(b),where the red scattered dots were the outline of the Taylor bubble.Due to the symmetry,only the right half of the Taylor bubble was considered in this study.By setting the threshold of the R channel in the RGB image,the binary image shown in Fig.9(c) was generated.Further,the relative position curve of the Taylor bubble outline was obtained through the pixel coordinates of the white scatters.It was observed from Fig.8 that the Eq.(30) perfectly coincided with the outline of the experimental Taylor bubble under the specific flow condition.

To further verify the applicability of Eq.(30) in a wide range of conditions,Fig.10 presents the effects of water superficial velocityUlsand gas superficial velocityUgson the shape of the Taylor bubble.Fig.10(a) displaysthe shape of the Taylor bubble with a fixedUgsof 0.1 m·s-1and a varyingUlsfrom 0.1 to 0.5 m·s-1.Fig.10(b)shows the shape of the Taylor bubble with a fixedUlsof 0.1 m·s-1and a varyingUgsfrom 0.1 to 0.5 m·s-1.It was found both the variations ofUlsandUgshad certain influence on the bubble shape in the nose region,among which the influence ofUgswas more significant,while that ofUlswas slightly weaker.This effect can be better explained when usingReUTB(=ρlUTBD/μl),which was based on the rising velocity of an individual Taylor bubble.In general,the nose shape was found to shrink towards the interior of the bubble asReUTBincreased.

Fig.8.Comparison between experimental image data and Dumitrescu model of Taylor bubble shape.

Fig.9.Image processing steps of Taylor bubble outline.

Two reasons are considered to contribute to the above phenomena.On one hand,the increase inReUTBstrengthens the turbulence intensity of Taylor bubble wake,so that more dispersed bubbles are entrained into the liquid slug [47].The presence of small bubbles will disequilibrate the phase interface at the Taylor bubble nose,generating an additional force inward the Taylor bubble[46].On the other hand,an increasingReUTBleads to an increase in the relative velocity between the falling liquid film and the rising Taylor bubble,which reshapes the velocity field around the Taylor bubble,resulting in a sharper nose shape.Simultaneously,from this perspective,the effect of gas superficialvelocity is stronger than that of liquid superficial velocity.

Further,the Taylor bubble was visually elongated in the axial direction at higherReUTB,and the Dumitrescu model was no longer applicable at corresponding flow conditions.Thus,a correction factor was introduced to modify the Dumitrescu model along the axial direction.Obviously,the correction factor was related toReUTB,and the modified Dumitrescu model can be expressed as:

The validity of the modified Dumitrescu model is recommended to involveUlsfrom 0.089 to 0.65 m·s-1,Ugsfrom 0.1 to 0.5 m·s-1,and accordingly,ReUTBfrom 4483 to 14429.Fig.11 indicates the nose outline of Taylor bubble at varyingReUTBusing the modified Dumitrescu model and a good agreement of Eq.(31) with the experimental data can be observed.

From the schematic diagram of the main hydrodynamic features of a slug flow in Fig.4,another important parameter related with Taylor bubble nose and falling liquid film is the length needed to have fully developed annular liquid filmZ*,for which the boundary layer near the wall reaches the free streamline.Z*is the axial range that needs to be considered in the nose correction factorCZ*.

Nicklinet al.[48]found the velocity inthe liquid film relative to the nose applying Bernoulli’s equation along the free streamline.Further,Campos and Guedes de Carvalho [49] proposed the calculation correlation of the film development lengthZ*,determined as:

Fig.10.Effects of water superficial velocity Uls and gas superficial velocity Ugs on the shape of the Taylor bubble.

Azevedoet al.[41]compared Eq.(32)through pulse-echo ultrasonic experiments and found thatZ*also depended on the dimensionless inverse viscosity numberNf.Based on their experimental results,an empirical correlation to estimate dimensionless film development length was proposed forNfranging from 15 to 12900:

Eq.(33) has also been verified by the experimental data of Nogueiraet al.[40].Meanwhile,the above results,where performed in the studies of Azevedoet al.[41] and Nogueiraet al.[40],indicated that whenNfincreased,a longer bubble length was required to allow the liquid film reach the fully-developed condition.Regarding the experimental conditions in this present work,inverse viscosity numberNfis not less than 5000,accordingly,the film development lengthZ*exceeds 10D.

Fig.11.Comparison of modified Dumitrescu model with the experimental nose outline of Taylor bubble.

To obtain a reasonable calculation result of void fraction,the stable liquid film thickness must be the maximum value of the development of the liquid film.However,the modified Dumitrescu model cannot converge satisfactorily to the stable liquid film thickness in the axial direction.Therefore,it is necessary to set a correction length to limit the development of the liquid film.Theoretically,the intersection of the profile calculated by the modified Dumitrescu model and the stable liquid film thickness can be regarded as the correction length.In this study,the correction length is 2.03D-3.12Dfor the flow conditions.

Fig.12 displays the local curvature variation of the Taylor bubble outline and the difference between local liquid film thickness and the equilibrium liquid film thickness,respectively.Results showed that in the nose region,the local curvature dropped sharply with the development of liquid film.Whenz＞2D,the local curvature was approximately 0,where the local liquid film thickness has been extremely close to the equilibrium liquid film thickness.For simplicity,from the tip of Taylor bubble nose,the correction range of 0-2Dwas adopted under all working conditions.To evaluate the error caused by this simplification,a set of calibrations were performed based on the image areas of Taylor bubble,and the results showed that the maximum relative deviation was only 0.25%.Thus,the correction length of 0-2Dcan ensure sufficient accuracy.

Fig.12.Local curvature variation of the Taylor bubble outline and the difference between local liquid film thickness and the equilibrium liquid film thickness.

In summary,correction factor of nose shapeCZ*can be expressed by the volume ratio:

whereVZ* is the nose volume of the Taylor bubble,ranging from 0 to 2Din axial direction.Based on the two-dimensional distribution curve of nose outline represented in Eq.(31),the volume within the closed interface can be calculated by integration rotated around the center line.Vδis the volume of the equilibrium liquid film,ranging from 2Dto the tail of Taylor bubble.VLTBis the volume of the entire Taylor bubble regardless of the nose shape and developing liquid film.BothVδandVLTBcan be calculated by the product of equilibrium sectional area of the gas-phase and corresponding bubble length.The expressions of these three volumes are as follows:

4.3.Void fraction in liquid slugs

The local void fraction incontinuous slug flow units(αloc)and in the liquid slugs(αloc,LS)were measured by the single optical probe.In the liquid slug region as illustrated in Fig.4,the gas-liquid twophase flow regime composed of discrete small bubbles can be regarded as a bubbly flow.Regarding the bubbly flow,the radial void fraction distribution profile is crucial for determining the cross-sectional averaged void fraction.Therefore,based on the mathematical description ofradial void fraction distribution curve αloc,LS(r),the cross-sectional averaged void fraction in the liquid slugs αLSis calculated as:

4.3.1.Review of the radial void fraction distribution

Due to the high similarity of bubble performances between bubbly flow and the liquid slugs in slug flow,a research review of the radial bubble phase distribution was carried out in this section.A preliminary distribution form of the radial local void fraction in the liquid slugs was obtained.

Two types of radial void fraction distribution forms have been observed in extensive numerical and experimental studies [50-55].The bubbles may accumulate in the center of the tube (corepeak) or near the walls (wall-peak),depending on the flow conditions and channel geometries [56].For the gas-liquid bubbly flow in a circular tube,the determinant factors affecting the distribution characteristics include the inner diameter (D),bubble size (d) and liquid turbulence structure around the bubbles.

Per the observation of Shawkatet al.[50],the bubbles tended to migrate toward the centerline forming a core-peak distribution in a large diameter vertical tube(typically,the diameter of a large tube is greater than approximately 100 mm).Similar trends were also reported by Jinet al.[51]and Babaeiet al.[52],in their studies with corresponding diameter of 160 and 248 mm,respectively.On the contrary,the wall-peak distribution was generally observed in a relatively small tube (Liuet al.[53],D=38 mm;Marfainget al.[54],D=40 mm).However,the core-peak distribution may also occur in small diameter tubes,which mainly depends on the bubble diameter.Nakoryakovet al.[55]conducted a contrastive experimental investigation using two types of gas-liquid mixers in a 14.8 mm inner diameter vertical tube.The N-injector with 6 hypodermic needles of 0.4 mm diameter was easier to produce large bubbles than P-injector with 18 holes of 0.15 mm diameter,and resulting in a core-peak distribution of the void fraction.In addition,the liquid turbulence structure around the bubbles was also observed to affect the void fraction distribution trend[50].According to the study of Lopez de Bertodanoet al.[57],the turbulent dispersion force has a stabilizing effect on the two-phase flow.

Moreover,by analyzing a vast experimental database,a criterion to identify the radial void fraction distribution (core-peak or wall-peak)was proposed by Mendez-Diazet al.[56].The fluid properties and bubble size and shape,which were expressed in dimensionless forms as Reynolds number and Weber number,were considered to evaluate the identification criterion.

whereUris the bubble relative velocity,Dsmis the bubble Sauter diameter,ρl,μl,and σlare the density,viscosity,and interfacial tension of liquid phase,respectively.The distribution transition from wall peak to core peak occurs when both Reynolds and Weber numbers reach critical values (Rer,crit=1500 andWer,crit=8).

The radial void fraction distribution of the discrete bubbles in liquid slugs of this study was verified using the identification criterion of Mendez-Diazet al.[56],as shown in Fig.13.The independent variable was the dimensionless Sauter diameter of bubbles(determined by the ratio of bubble Sauter diameterDsmto the tube inner diameterD) under specific flow conditions.The bubble relative velocityUrwas the equilibrium rise velocity measured by the optical probe in the stagnant water visual calibration test.Results indicated that the radial void fraction distribution of the discrete bubbles in liquid slugs conformed to the wall-peak distribution.

Fig.13.Identification of the radial void fraction distribution of the discrete bubbles in liquid slugs,employing the criterion proposed by Mendez-Diaz et al. [56].

4.3.2.Image processing technique based on machine learning

To further obtain the actual radial void fraction distribution profile of the liquid slugs,the acquired high-speed images were processed using image processing technique based on machine learning.The principle and procedure of this image processing method are described in detail in the present section.

The focus of high-speed photography to study gas-liquid twophase flow is on the interface recognition.Fig.14 illustrates three typical bubble forms in the liquid slugs,which are separated bubbly flow (a),interacting bubbly flow (b) and irregular bubbly flow(c),respectively.The traditional image processing algorithms[58],including boundary recognition,morphological operation,opening and closing operation,have a better performance in the gas-liquid interface extraction for separated bubbly flow.However,regarding the interacting bubbly flow,as shown in Fig.14(b),the bubble number density becomes so large that the bubbles begin to interact with each other directly or indirectly because of collisions or the effects of wakes caused by other bubbles[59].The interaction phenomenon makes the bubbles overlap in the visual direction,aggravates the extraction difficulty of the gas-liquid interface.Additionally,in the Taylor bubble wake region,as shown in Fig.14(c),the falling annular film enters the liquid slug as an expanding jet,creating a recirculation region with irregular bubbles.The interface extraction of irregular bubbles is also challenging.

In the field of artificial intelligence image processing technology,interface recognition and extraction are classified into the category of image segmentation.Therefore,a modified image segmentation technology based on U-shaped network,proposed by Ronnebergeret al.[60],is employed to conduct superevised learning for the extraction of gas-liquid interface.The U-shaped network has inherent advantages in the interface extraction,and the simple principle is: Regarding a gas-liquid two-phase raw image,first reduce the image size through multiple rounds of convolution and down-sampling,and extract some shallow quantified features in each convolution.Then in the up-sampling processing,some deep features of the image are obtained.Simultaneously,a fusion operation is performed on the same image scale corresponding to the feature extraction part for each round of up-sampling step.The final output is the binary segmentation result corresponding to the raw image.After that,iterative back propagation is conducted to train the image segmentation network according to the difference between the intermediate results and the real segmentation result previously processed.Fig.15 illustrates the image processing steps,mainly consisted of two stages.

(1)Training stage

The establishment of image training database and the optimization of image segmentation network were the emphases in this stage.The raw image was segmented with high precision through ITK-SNAP,which is a software application for 3D medical image segmentation.In this study,the gas phase was filled manually to obtain the gas-liquid segmentation interface.Since the image can be enlarged infinitely during the filling process,the accuracy of the recognition interface can be guaranteed,and the error will not exceed one pixel.

For superevised machine learning,theoretically,the larger the image training database,the higher the accuracy of the final segmentation result.In this study,the original image training database contained 50,75,and 100 pairs of images (each pair of images was composed of a raw image and a corresponding manually filled image)respectively,and three training results were output accordingly.Use these three training results to perform batch image processing and compare the radial void fraction distribution profiles.Result showed that there was no significant difference between the distribution profiles of the three image databases.Thus,the image training database containing 100 pairs of images was finally adopted.

Fig.14.Sample high-speed images in liquid slugs with (a) separated bubbles;(b) interacting bubbles;(c) irregular bubbles affected by the Taylor bubble wake.

Fig.15.Image processing steps based on machine learning.(a) Training stage;(b) processing stage.

When adhesion or overlap occurs between adjacent bubbles,the connection part of the bubbles is the most prone to segmentation error.Therefore,when training the network to optimize,the proportion of quantified feature extraction at these locations was increased.Meanwhile,combined with morphological operations such as threshold segmentation,image erosion and image expansion,the adhesion bubbles were divided into separate bubbles,with an extremely narrow gap as the boundary.Regarding the bubbles that overlapped with each other,the separate bubbles with boundary lines were first processed by the aforementioned operation,and then they were connected according to their characteristic features.These optimization operations ensured that the hardto-recognize boundary can be better identified at the positions where the bubbles overlapped.

After multiple rounds of iterative training,an optimized Ushaped network for image segmentation was obtained.

(2)Processing stage

In the processing stage,based on the U-shaped network for image segmentation from the above training,batch of raw images were converted into binary images.For the binary image,by reading the pixel values at the specified positions,the transient distribution of void fraction was obtained.77 pixels symmetrical about the tube center were involved to describe the radial distribution of void fraction.Further,the time-average radial distribution was obtained by processing thousands of images under specific flow condition.In this study,three 5 s duration high-speed videos were collected during each test,and correspondingly,more than 20 liquid slugs were involved in the statistical analysis.

The trained U-shaped network for image segmentation was used to analyze and predict the void fraction of liquid slug in the experiments,and its recognition accuracy was very important.In the field of image recognition and segmentation,the manually processed image is generally regarded as the standard.To evaluate the uncertainty of the image processing program,eight different shapes of images were involved,as shown in Fig.16.These eight images covered multiple types of outer boundary contours.The pixel areas of the standard images were obtained by manual filling using the ITK-SNAP program and program processing using the trained U-shaped network for image segmentation respectively.By comparing the results of the two methods,it was found the maximum relative deviation was 0.58%,and the average relative deviation was 0.3%.Therefore,the image processing program in this study has relatively high accuracy.

4.3.3.Radial void fraction distribution profile of the liquid slugs

Fig.16.Eight graph shapes used to evaluate the uncertainty of image processing programs.

Employing the image processing method based on the Ushaped network,the radial void fraction distribution profiles of the liquid slugs were obtained.Fig.17 indicates typical distributions in varying gas superficial velocity and varying water superficial velocity,respectively.In the radial direction,a total of 77 local void fraction data points were included.Generally,the distribution form was found to conform the wall-peak trend,which was consistent with the aforementioned theoretical prediction results in Section 4.3.1.It was observed that the overall void fraction increased with the increase of gas superficial velocity,and the peak position became closer to the inner wall surface.While under the same gas superficial velocity,results showed that the effects of water superficial velocity on both overall void fraction and peak position were not significant.

Further,the peak position of the radial void fraction distribution profile is a crucial parameter.The position of the wall peak in the liquid slug was found to be farther away from the wall,compared with that in bubbly flow.The results of the present study showed that the wall peak occurred between 0.25Dand 0.4D,while the wall peak positions in several typical studies of bubbly flow[53-55] were at least 0.38D,and most of them were not less than 0.42D.In addition to the factors of channel geometry,bubble size,and turbulent intensity of the gas-liquid flow,the effect of Taylor bubble wake was also the main reason for the above phenomenon.In the weak region of slug flow,the liquids of the near-wall and the tube center were flowing downwards and upwards,respectively.The violently stirred liquids created a local recirculation,causing the gas-phase at the tail of Taylor bubble to be torn,forming trailing irregular bubbles.According to the study of flow field in the wake of a Taylor bubble by Nogueiraet al.[40],the central location of the recirculation was between 0.25Dand 0.3D.Accompanied by the effect of horizontal lift(driving the bubbles with a small diameter toward the wall)and turbulent diffusion(making the distribution of the gas-phase more uniform) [56],the wall peak of the radial void fraction distribution profile occurred between 0.25Dand 0.4D.A higher gas superficial velocity will increase the gasphase content in the main liquid slug region,weakening the influence of the Taylor bubble weak,and resulting in a shift of the wall peak to the wall direction.

Fig.17.Radial void fraction distribution profile of the liquid slugs in(a)varying gas superficial velocity and (b) varying water superficial velocity.

To quantify the peak position,a predictive correlation of (r/D)peakin the liquid slug was proposed,based on the fitting of experimental data:

whereRetis the Reynolds number of Taylor bubble tail,determined as ρl(Umain-Uft)D/μl,which reflects the influence of the turbulence intensity and the Taylor bubble weak;andWeris the Weber number of bubble relative motion,which reflects the influence of bubble size and horizontal lift [56].

Correspondingly,regarding the mathematical description of the radial void fraction distribution profile,we divided it into two parts: before the wall peak and after the wall peak.From the tube center(r/D=0) to the peak location,the void fraction rose slowly.Based on the Reynolds number of Taylor bubble tailRet,the relationship between the local void fraction αloc,LSand the dimensionless radial distancer/Dis as follows:

From the peak location to the tube inner wall (r/D=0.5),the void fraction dropped rapidly,and the expression of the void fraction distribution profile is as:

To verify the applicability of the predictive correlations,a comparison between the prediction curves and the image data was performed,as shown in Fig.18.The prediction curves were observed to be in good agreement with the experimental image data in the distribution form,the peak location,and the variation on superficial parameters.By calculation,more than 95% of the image data can be captured within an accuracy of ±10%.Thus,the predictive correlations composed of Eqs.(41),(42)and(43)can well describe the radial void fraction distribution profile in liquid slugs.

4.4.Closure formulation

In the present approach,the correlation structure and calculation process of the proposed void fraction model for vertical upward co-current air-water slug flow based on the local parameters are as follows:

●Eq.(13) is derived from the definition of cross-sectional averaged void fraction,disposing the slug flow as a combination of Taylor bubbles and liquid slugs.

●Void fraction in Taylor bubbles is determined as Eq.(14),including the correction factor of liquid film thicknessCδand the correction factor of nose shapeCZ*.

Fig.18.Comparison of predictive correlation against the experimental image data for the radial void fraction distribution in liquid slugs.

●Eq.(29)gives the expression ofCδ,where the dimensionless liquid film thickness is calculated by Eq.(26).

●Eq.(34)gives the expression ofCZ*,which involves three volume calculations expressed in Eqs.(35)-(37);Regarding the Taylor bubble nose shape profile,the modified Dumitrescu model by Eq.(31) is used.

●Void fraction in liquid slugs is determined as Eq.(38),based on the radial void fraction distribution.

●The mathematical description of the radial void fraction distribution profile in the liquid slug is determined as Eqs.(41)-(43).

5.Results and Discussion

5.1.Results of the newly proposed model

5.1.1.Void fraction in Taylor bubbles and in liquid slugs

The slug flow has been disposed as a combination of Taylor bubbles and liquid slugs,in which the void fraction is determined as Eqs.(14) and (38),respectively.Fig.19 displays the crosssectional averaged void fraction in Taylor bubbles and in liquid slugs with a varying gas superficial velocity and a varying water superficial velocity.Regarding the fully developed Taylor bubble with a regular interface outline,the local void fraction αloc,TBis constant 1,and the cross-sectional averaged void fraction αTBmainly depends on the equilibrium liquid film thickness and the bubble nose shape.Researches [38,41] showed that both equilibrium liquid film thickness and bubble nose shape were greatly affected by the physical properties of the flowing fluid while less affected by the working conditions.Therefore,it was found that the void fraction of Taylor bubbles was almost unchanged with the varying gas superficial velocity,and slightly decreased with the increase of water superficial velocity.These slight differences were mainly related to the length and nose shape of Taylor bubbles.When the water superficial velocity increased,the average Taylor bubble length decreased and the shape of nose became more sharp,leading to a gradual decrease of the cross-sectional averaged void fraction in the Taylor bubble region.The void fraction in the liquid slug region primarily depended on the influence of the turbulence intensity and the radial distribution of the gas phase.It was observed from Fig.19 that with the increase of gas superficial velocity,the void fraction in the liquid slug region increased significantly,while for a given gas superficial velocity,an increase in water superficial velocity was not coupled with an increase in the void fraction.

Fig.19.Cross-sectional averaged void fraction in Taylor bubbles and in liquid slugs for a given(a)water superficial velocity of 0.1 m·s-1 and(b)gas superficial velocity of 0.2 m·s-1.

5.1.2.Void fraction in co-current slug flow

In addition to the void fraction in Taylor bubble and liquid slug region,the void fraction in co-current slug flow is also highly relevant to the duration ratio and gas-phase velocity ratio between the two regions.The duration ratio reflects the proportion of Taylor bubbles (mainly composed of gas phase) and liquid slugs (mainly composed of liquid phase)in the slug flow,being crucial for determining the cross-sectional averaged void fraction.Therefore,comprehensively considering the average void fraction,dimensionless velocity ratio and duration ratio in Taylor bubble and liquid slug region,the cross-sectional averaged void fraction based on the new model was obtained.

The measurement results are shown in Fig.20,containing 33 standard slug flow conditions,under water superficial velocity ranging from 0.1 to 0.65 m·s-1and gas superficial velocity ranging from 0.1 to 0.65 m·s-1.The gray bars and red bars represent the contribution of Taylor bubbles and liquid slugs to the overall void fraction,respectively.In general,the cross-sectional averaged void fraction of the slug flow was maintained at a moderate level between 0.25 and 0.7.With the increase of gas superficial velocity and the decrease of water superficial velocity,α presented an increasing trend.It was also observed that the void fraction proportion in Taylor bubbles was much higher than that in liquid slugs,which indicated the crucial contribution of Taylor bubbles to the overall void fraction.At a fixed water superficial velocity,the percentage of void fraction in liquid slugs exhibited unconspicuous difference,while as the water superficial velocity increased from 0.1 to 0.65 m·s-1,the corresponding percentage increased significantly from approximately 5% to more than 20%.

Fig.20.Cross-sectional averaged void fraction in co-current slug flow based on the new model.

The cross-sectional averaged void fraction of the slug flow was calculated from the void fraction model based on the measured local parameters,employed the optical probes.Since the calculation model relied on several measured parameters and predictive correlations,it was limited in practical applications.Thus,establishing a simple relationship between cross-sectional averaged void fraction and central local void fraction becomes a feasible strategy.The variation of α with αlocunder the slug flow conditions is illustrated in Fig.21.Obviously,α was generally lower than αloc.Moreover,as the void fraction increased,the difference between the two gradually decreased from 40% to 20%.It was also found that the larger difference between α and αlococcurred in the conditions of high water superficial velocity and low gas superficial velocity,in which the length of Taylor bubble was proved to be relatively small,and resulting a larger nose shape correction factorCZ*.On the contrary,as the length of Taylor bubble increased under low water superficial velocity and high gas superficial velocity,the flow pattern transformed to an annular flow,leading to a long and equilibrium falling liquid film.

Fig.21.Variations in cross-sectional averaged void fraction with the central local void fraction.

Further,a predictive correlation for calculating the crosssectional averaged void fraction from the central local void fraction was proposed,expressed as:

The predictive correlation is applicable to the air-water twophase slug flow in a vertical upward tube,with the local void fraction ranging from 0.4 to 0.9,and the water and gas superficial velocity ranging from 0.1 to 0.65 m·s-1.The good agreement of Eq.(44)with the experimental data was observed in Fig.22,where the maximum relative deviation did not exceed 7.5%.Additionally,more than 80% of the points were located between ±3%,which indicated that feasibility of extending from the central local void fraction to the cross-sectional averaged void fraction.

When using Eq.(44) to calculate the cross-sectional averaged void fraction,the central local void fraction can be acquired by experimental measurement.With the development of two-phase flow measurement technology,several advanced methods,including optical probe[32],electrical resistance tomograph(ERT),wiremesh sensor (WMS) [61],and conductivity probe [6] have been adopted to measure the local parameters.Besides,according to our previously published study[30],the central local void fraction was closely related to the volumetric void fraction β,which is calculated from the gas and liquid superficial velocity,as expressed in Eq.(1).Based on the variation trend as indicated in Fig.23,the central local void fraction can be correlated as:

It was observed that all data points can be captured within an accuracy of ±10%.Therefore,the combination of Eqs.(44) and(46)constitutes a simple model to predict the cross-sectional averaged void fraction of slug flow.

5.2.Comparison of experimental results with existing models

A comparison of the performance of 10 void fraction correlations was conducted against the experimental data.Multiple types of predictive correlation were employed,including the general model,the modified-homogeneous model,the slip ratio model,the drift flux models,and the slug flow models.The summarized lists of these representative correlations are given in Tables A1 and A2 in the Appendix.The prediction performances were quantitatively assessed using the mean average error (MAE) defined as [1]:

Fig.22.Prediction performance of the fitted curves Eq.(44) against the experimental α.

Fig.23.Prediction performance of the fitted curves Eq.(45) against the experimental αloc.

whereNis the number of experimental data,αcalis the calculated void fraction and αexpis the experimental void fraction.

Fig.24 presents the comparison of the calculated void fractions by the existing models with the experimental void fractions in the present study.The results of quantitative assessments for these correlations are listed in Table 2.The experimental void fraction data was divided into two different ranges of void fraction,namely 0.25 ＜α ＜0.5 and 0.5 ＜α ＜0.7.In addition,the percentage of experimental data within a certain accuracy range was also calculated.The correlations by Bonnecazeet al.[18],Speddinget al.[62] and Schmidtet al.[20] were the representatives of the modifiedhomogeneous model,general model and slip ratio model,respectively,as shown in Fig.24(a).It was observed that the correlations of Bonnecazeet al.and Schmidtet al.captured 100% of the experimental data with an accuracy of ±30%.Meanwhile,95.3% of the data canbe predicted within ±30% by the correlation of Speddinget al.More importantly,the Schmidt correlation presented the optimal assessment performance,with the MAE being 4.3%.It can be seen from Table A1 in the Appendix that the constant parameters of the Schmidt correlation were floating.In this study,the parametersa,b,andcwere set to be 1.95,0.218,and 0.789,respectively.Three typical predictive correlations based on the drift flux model,respectively by Zuber and Findlay [10],Woldesemayatet al.[15],and Bhagwatet al.[21],were contained in Fig.24(b).The drift-flux model was first envisaged by Zuber and Findlay,who proposed simple expressions of distribution parameterC0and drift velocityUgm(see Table A1).Therefore,the prediction performance of Zuber and Findlay correlation was extremely limited.Woldesemayatet al.collected a number of experimental void fraction data and proposed an improved void fraction correlation regardless of flow patterns and inclination angles.For the vertical upward slug flow,the predicted values and the experimental values were in good agreement at low void fraction.The MAE was 12.9% in the range of 0.25 ＜α ＜0.5.Further,Bhagwatet al.adequately considered multiple two-phase flow parameters(including the flow channel geometry,flow orientation,system pressure,gravitational acceleration and physical properties of the twophase fluids) and definedC0andUgmas complicated functions of these variables.The Bhagwat correlation was widely applicable and captured 87.0% of the experimental data with an accuracy of±20%.Especially for higher void fraction,more than 90%of the data can be predicted within±10%and the MAE was 5.8%in the range of 0.5 ＜α ＜0.7.

Fig.24.Comparisons of experimental void fraction with calculated void fraction by existing models: (a) general models;(b) drift flux models;(c) slug flow models.

The above correlations were independent of flow patterns and applicable to a wide range of parameters.These proposed correlations gave better performance over the entire range of the void fraction(0 ＜α ＜1)and without any reference to flow pattern maps.Nevertheless,these correlations cannot reveal the physical mechanism of a specified flow pattern,due to the essential differences(including the morphology and distribution of gas/liquid phase)between them.In certain industrial equipment,the two-phase flow pattern has been usually univariate.Thus,the void fraction predictive correlation that depends on a specified flow pattern becomes particularly critical.Fig.24(c) exhibits the comparison results of several void fraction prediction models in allusion to slug flow.The four prediction models can also be divided into two categories.One is under the framework of the drift flux model,to determine the distribution parameterC0and drift velocityUgmaccording to the characteristics of the slug flow.For instance,Hibiki and Ishii[25] developed the drift constitutive equation for dispersed twophase flow.Then he introduced the Taylor bubbles of slug flow into the constitutive equation by assuming bubble radius to beD/2 and obtained correspondingUgm.The assessment results showed that the correlation of Hibiki and Ishii [25] was superior to that of Adekomaya[29],with the overall MAEs of 13.6%and 20.9%,respectively.The other category is the mechanism model,which is based on the two-phase flow mechanism,and dividing the slug flow into Taylor bubbles and liquid slugs.These models employ the principle of mass conservation to analyze the relationship between gas/liquid fluxes at each boundaries and derive the corresponding formula for calculating the void fraction.The models by Guetet al.[26]and Liuet al.[28]were compared with the experimental data of this present study.The evaluation results of the latter were found to be better than that of the former,with the overall MAEs of 14.8% and 28.4%,respectively.Although these models considered the flow mechanism of slug flow,some concessive assumptions were inevitably involved to achieve the closure formulation.Moreover,it was found in Table A2 that the main inputs of these two models were overly dependent on empirical correlations,some of which were derived from drift-flux models.Thus,it was observed that the prediction results of Liu model[28] were similar to that of Hibiki and Ishii drift-flux model [25].

Table 2Comparison of the experimental data with the void fraction predictive correlations

Table 3Experimental data used in the comparison against the proposed model

5.3.Comparison of the prediction model with existing experimental data

In this section,some experimental data of void fraction measurements were selected to verify the relationship between the cross-sectional and local void fraction proposed in this study.The central local void fraction and two-phase flow parameters for calculating the average void fraction were the prerequisites to determine the selected experimental data.In addition to the optical probe,electrical resistance tomograph (ERT),wire-mesh sensor(WMS),and conductivity probe (CP) have also been important methods to measure the local parameters of two-phase flow.The experimental data to be evaluated include the void fraction measurement experiments conducted by Olerniet al.[61] and Wanget al.[63],whose measurement techniques and test conditions are shown in Table 3.The radial void fraction distribution profiles were obtained by employing the ERT,WMS and CP,respectively.Only the slug flows were intercepted,containing 10 and 14 data points,respectively.The cross-sectional averaged void fraction was calculated by the area-weighted processing method.

Table A1Summary of correlations considered in the present study (independent of flow pattern)

Table A1 (continued)

Table A2Summary of correlations considered in the present study (for slug flow)

The comparison results of the experimental data by Olerniet al.[61]against the present correlation(Eq.(44))are shown in Fig.25.It was observed that a good agreement of Eq.(44) with the Olerni data in the range of 0.28 ＜α ＜0.4.Most of the data were captured by the present correlation with an accuracy of ±5%,and the MAE was only 1.92%.Fig.25(b)exhibits the relative evaluation error distribution of the Olerni data under specific gas and water superficial velocity.The flowing parameters were found to be thoroughly covered in the range of the present test conditions,which became the decisive factor for the good agreement between the two.Moreover,two techniques were employed to measure the void fraction by Olerniet al.,which are separately marked in Fig.25(b).The results showed a slight difference between the two techniques.Olerniet al.reported that radial void fraction distribution profiles of the ERT remained the typical core peak while WMS presented larger local void fraction values than the corresponding ERT except one or two points close to the tube wall.This indicated that the cross-sectional averaged void fraction of WMS was larger than that of ERT under the same central localvoid fraction.

Additionally,the present correlation (Eq.(44)) was compared against the experimental data by Wanget al.[63],as shown in Fig.26.The prediction trend of Eq.(44) was found to capture the experimental data well over the range of 0.25 ＜α ＜0.65,while the prediction accuracy was barely satisfactory with the MAE of 10.9%.It was observed from Fig.26(b) that the maximum water and gas superficial velocity was 2.0 and 5.0 m·s-1,respectively,which were far beyond the test conditions of the present investigation.Based on the observation of the visualized two-phase flow,the flowing parameters of Wanget al.cannot fully guarantee the morphological characteristics of the standard slug flow.A variety of transitional flow patterns including bubbly-slug,churn-slug,and slug-annular flow were involved in the experiment of Wanget al.Therefore,the application of the proposed model in this study has certain limitations on the void fraction prediction in a wide range of flowing parameters.

Fig.25.Comparison of the calculated void fraction by the newly proposed model with the experimental data of Olerni et al. [61].

6.Conclusions

Comprehensive experimental investigations on vertical upward air-water slug flow were carried out at atmospheric pressure in this study.Per the experimental results,following conclusions were drawn.

(1) A new void fraction model based on the local parameters was proposed,applicable to only slug flow,which was disposed as a combination of Taylor bubbles and liquid slugs.The model indicated that in addition to the void fraction of each sub-region,the duration ratio and gas-phase velocity ratio were also crucial parameters for determining the cross-sectional averaged void fraction.

(2) In the Taylor bubble region,correction factors of liquid film thicknessCδand nose shapeCZ*were proposed to calculate αTB,which was mainly affected by the equilibrium liquid film thickness,the nose shape of Taylor bubble,and the film development length.In the liquid slug region,the radial void fraction distribution profiles were found to conform to the wall-peak trend.Based on the experimental data obtained by image processing method,the mathematical description of the radial void fraction distribution profile was established to calculate αLS.Correspondingly,an integrated calculation model containing 13 independent equations has been improved.

Fig.26.Comparison of the calculated void fraction by the newly proposed model with the experimental data of Wang et al. [63].

(3) The Taylor bubble region occupied a dominant contribution to the overall cross-sectional void fraction of the slug flow,and under different flow conditions,its proportion was between 80% and 95%.The cross-sectional averaged void fraction was found to be closely related to thecentral local void fraction.Hence,a predictive correlation was correlated between them,as a straightforward form of the void fraction calculation model.

(4) The experimental result in the present study was compared with some typical void fraction prediction models,including those independent of flow patterns and applicable for slug flow.Results showed that the performance of the Schmidt model was optimal.However,although Guet model and Liu model considered the flow mechanism of slug flow,some concessive assumptions were inevitably involved to achieve the closure formulation.Moreover,the main inputs of these two models were overly dependent on empirical correlations.Thus,the measured void fraction and calculated void fractions by these two models differed significantly.

(5) The proposed predictive correlation based on the central local void fraction was applicable to the air-water twophase slug flow in a vertical upward tube,with αlocranging from 0.4 to 0.9,UlsandUgsranging from 0.1 to 0.65 m·s-1.The correlation was observed to agree well with the experimental data of Olerniet al.,whose flowing parameters were covered in the range of the present test conditions.

Declaration of Competing Interest

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.

Acknowledgements

This work was supported by National Key Research and Development Program of China (2018YFE011061).The authors thank the staff at the Institute of High Pressure Steam-water Two-Phase Flow and Heat Transfer in the State Key Laboratory of Multiphase Flow in Power Engineering,XJTU,for their constructive discussions and suggestions.

Nomenclature

Across-sectional area,m2

Cz*correction factor of nose shape

Cδcorrection factor of liquid film thickness

C0distribution parameter

Dinner diameter,m

Dsmbubble Sauter diameter,m

dbubble diameter,m

EoEotvos number (=ρlgD2/σl)

ggravitational acceleration,m·s-2

Kmodified coefficient of homogeneous model

Llength,m

MMorton number (=gμl4/(ρlσl3))

Nnumber of experimental data

Nfinverse viscosity number (=ρl(gD3)1/2/μl)

Rtube radius,m

RerReynolds number of bubble relative motion (=UrDsmρl/μl)

RetReynolds number of Taylor bubble tail (=ρl(Umain-Uft)D/μl)

ReUTBReynolds number based on the Taylor bubble velocity(=ρl-UTBD/μl)

rradial distance from the center line,m

Sslip ratio

ttime,s

Ububble velocity,m·s-1

Ugsgas superficial velocity,m·s-1

Ugmdrift velocity,m·s-1

Ulswater superficial velocity,m·s-1

Umtwo-phase mixture superficial velocity,m·s-1

Urbubble relative velocity,m·s-1

Vvolume,m3

VLTBvolume of the entire Taylor bubble,m3

nose volume of the entire Taylor bubble,m3

Vδvolume of equilibrium liquid film region,m3

WerWeber number of bubble relative motion (=smρl/μl)

xdryness fraction

Z*length needed to have fully developed annular liquid film,m

zaxial distance from mixing unit,m

α void fraction

αloclocal void fraction

β volumetric local void fraction

δ film thickness,m

δ’dimensionless thickness (=δ/R)

μ dynamic velocity,Pa·s

ρ density,kg·m-3

σ surface tension,N·m

Φ dimensionless velocity ratio (=UTBorULS/UB)

Subscripts

B bubble

cal calculated

crit critical

exp experimental

g gas

LS liquid slug

lliquid

TB Taylor bubble