One-dimensional drift-flux correlations for two-phase flow in medium-size channels

The drift-flux parameters such as distribution parameter and drift velocity are critical parameters in the one-dimensional two-fluid model used in nuclear thermal-hydraulic system analysis codes. These parameters affect the drag force acting on the gas phase. The accurate prediction of the drift-flux parameters is indispensable to the accurate prediction of the void fraction. Because of this, the current paper conducted a state-of-the-art review on one-dimensional drift-flux correlations for various flow channel geometries and flow orientations. The essential conclusions were: (1) a channel geometry affected the distribution parameter, (2) a boundary condition (adiabatic or diabatic) affected the distribution parameter in a bubbly flow, (3) the drift velocity for a horizontal channel could be approximated to be zero, and (4) the distribution parameter developed for a circular channel was not a good approximation to calculate the distribution parameter for a sub-channel of the rod bundle. In addition to the above, the review covered a newly proposed concept of the two-group drift-flux model to provide the constitutive equation to close the modified gas mixture momentum equation of the two-fluid model mathematically. The review was also extended to the existing drift-flux correlations applicable to a full range of void fraction (Chexel-Lellouche correlation and Bhagwat-Ghajar correlation).


Introduction
Nuclear thermal-hydraulic safety analysis codes are used for safety analyses of nuclear power plants. Since the nuclear power plant is a large complex system, full three-dimensional computational fluid dynamic simulation is currently unrealistic due to an immense amount of computational time and unavailability of reliable local constitutive equations Hibiki, 2015, 2017;Vaidheeswaran and Hibiki, 2017). The current practice for simulating thermal-hydraulic behavior in the nuclear power plant system heavily depends on one-dimensional system analysis codes such as TRACE (Bajorek, 2008), RELAP5 (ISL, 2001), and TRAC (Borkowski et al., 1992) codes. These one-dimensional system analysis codes are architected through rigorous one-dimensional two-phase flow formulation such as the two-fluid model (Ishii and Hibiki, 2010). The two-fluid model is composed of mass, momentum, and energy balance equations for each phase. Kinematic and thermal non-equilibrium two-phase flow dynamics can be simulated through interfacial transfer terms in the two-fluid model. The kinematic non-equilibrium between two phases appears as a relative velocity between two phases, whereas the thermal non-equilibrium between two phases causes heat transfer through the interface between two phases. The interfacial drag force and heat transfer coefficient are two of the most critical terms which govern the kinematic and thermal non-equilibrium, respectively.
The interfacial drag force includes various terms such as Basset force, virtual mass force, drag force, lift force, wall lubrication force, turbulence dispersion force, bubble-bubble collision force, and wall-collision force (Chuang and Hibiki, 2017). In a one-dimensional simulation, the virtual mass force is vital for stabilizing the numerical simulation, whereas robust formulation of the drag force is a key to predict void fraction accurately. In the one-dimensional simulation, it is common to neglect the terms other than virtual mass force and drag force terms. The rigorous formulation of the drag force based on a drag coefficient includes an interfacial area concentration, which is one of the weakest links in constitutive equations to close the two-fluid model mathematically. Unavailability of reliable constitutive equations for the prediction of the interfacial area concentration (Chuang and Hibiki, 2015) motivated developing an alternative method to formulate the onedimensional drag force without the interfacial area concentration (Andersen and Chu, 1982). Andersen and Chu (1982) expressed the one-dimensional drag force through a drift velocity. This simplified approach has been extended for three-dimensional drag force formulation .
The one-dimensional drift-flux model plays a crucial role in the drag force formulation regardless of the formulation methodology (drag-coefficient-based formulation or driftvelocity-based formulation). The one-dimensional or areaaveraged relative velocity appearing in the drag force is represented by void-fraction-weighted mean gas and liquid velocities, area-averaged void fraction, and distribution parameter . It has been pointed out that the area-averaged relative velocity formulation derived by Ishii and Mishima (1984) misses a void fraction or relative velocity covariance term (Brooks et al., 2012a). The constitutive equations of void fraction and relative velocity covariance terms have been developed for circular and sub-channel geometries (Hibiki and Ozaki, 2017;Ozaki and Hibiki, 2018). In the drift-velocity-based formulation, the overall drag coefficient is formulated through the drift velocity. The above brief discussion suggests the importance of constitutive equations for two drift-flux parameters such as distribution parameter and drift velocity.
The significance of the distribution parameter is the covariance of void fraction and mixture volumetric flux. The distribution parameter depending on flow channel geometry has been formulated empirically. The significance of the drift velocity is the difference between gas velocity and mixture volumetric flux. The drift velocity depending on the drag coefficient or bubble shape regime has been formulated with the aid of the drag law. Because of the importance of the constitutive equations for the drift-flux parameters in one-dimensional drag force formulation, a state-of-the-art review on one-dimensional drift-flux correlations for various flow channel geometries and flow orientations is conducted in this paper. Section 2 briefly describes the formulation of the one-dimensional drift-flux model and interfacial drag force using two drift-flux parameters such as distribution parameter and drift velocity. Section 3 reviews the state-of-the-art constitutive equations of the one-dimensional drift-flux model for two-phase flow in various medium-size channels. Section 4 covers a newly proposed concept of the two-group drift-flux model to provide the constitutive equation to close the modified gas mixture momentum equation mathematically. Section 5 describes existing drift-flux correlations applicable to a full range of void fraction. Zuber and Findlay (1965) proposed the drift velocity, gj v , to consider the effect of the relative velocity between two phases, r v , on the void fraction, α , as

Formulations of the drift-flux model and interfacial drag force
where g v and j are the gas velocity and mixture volumetric flux, respectively. Multiplying Eq. (1) by the void fraction and area-averaging the equation yields the one-dimensional drift-flux model as where and are the area-averaged and void-fractionweighted mean quantities, respectively. The distribution parameter, 0 C , and void-fraction-weighted mean drift velocity, , are defined as given by Eqs. (3) and (4), respectively.
The one-dimensional form of the interfacial drag force, D ig M , used in nuclear thermal-hydraulic safety analysis codes is expressed as When the interfacial drag force is given by Andersen approach (Andersen and Chu, 1982) or drift-velocity-based formulation, the area-averaged drag coefficient, i C , is represented by where ρ D and g are the density difference between the two phases and gravitational acceleration, respectively. The area-averaged relative velocity, r v , is formulated as where f v is the liquid velocity. The relative velocity covariance, α C¢, is defined by where the void fraction covariance, α C , is defined by It should be noted here that current nuclear thermalhydraulic safety analysis codes based on Ishii-Mishima formulation (1984) ignore the relative velocity covariance in Eq. (7). When the interfacial drag force is formulated through the drag coefficient, D C (drag-coefficient-based formulation), the area-averaged drag coefficient, i C , is represented by where i a and f ρ are the interfacial area concentration and liquid density, respectively.
As briefly described above, two drift-flux parameters such as the distribution parameter and void-fraction-weighted mean drift velocity (hereafter drift velocity for simplicity) are required for formulating the one-dimensional drag force. In what follows, the state-of-the-art review on constitutive equations of the drift-flux model for medium-size channels is presented.

Upward two-phase flow in a vertical circular channel
The distribution parameter and drift velocity are in general dependent on flow regime. Before a drift-flux correlation is applied to upward two-phase flow in a vertical circular channel, the flow regime should be identified. There are several reliable flow regime transition criteria for upward two-phase flow in a vertical circular channel Mishima and Ishii, 1984).
The drift velocity of dispersed two-phase flow can be modeled by the drag law. Ishii (1977) summarized the drift velocity correlations for upward two-phase flow in a vertical circular channel as follows.
Bubbly flow: where σ is the surface tension. Slug flow: where D is the channel diameter. Churn flow: The distribution parameter correlation for upward adiabatic two-phase flow in a vertical circular channel is given by where g ρ is the gas density. Although Eq. (14) is applicable to slug and churn flow regimes, it does not consider the complicated phase distribution mechanism observed in bubbly flow regime. The phase distribution is primarily governed by local non-drag forces such as lift, wall-lubrication, turbulence dispersion, bubble-bubble collision, and bubble-wall collision forces (Chuang and Hibiki, 2017). Hibiki and Ishii (2002a) calculated the distribution parameter from measured profiles of local void fraction and local gas and liquid velocities, and found the dependence of the distribution parameter on the bubble Sauter mean diameter. They developed the distribution parameter correlation for upward adiabatic bubbly flow in a vertical circular channel as ( ) where Sm D is the bubble Sauter mean diameter, which can be calculated by Hibiki-Ishii correaltion (2002b). Equation (15) has been validated by the atmospheric bubbly flow data taken in circular channels with the inner diameter ranging from 25.4 to 60.0 mm. The distribution parameter correlation for subcooled bubbly flow is given by ( ) [ ] The drift velocity of separated two-phase flow such as pure annular flow cannot be defined locally because the local flow is either gas single-phase or liquid single-phase. In this case, the drift-flux model is formulated through the force balance between liquid film and gas core. For annular flow, the drift-flux correlation is given in the form of mean drift velocity, gj V , as which is further simplified for g f 1 ρ ρ  as In a strict sense, the distribution parameter and drift velocity should be experimentally obtained from Eqs. (3) and (4) with measured profiles of local void fraction and gas and liquid velocities to validate the above correlations. Limited availability of the measured profiles only allows for the rigorous validation of Eqs. (11) and (15) as shown in Fig. 1 (Hibiki and Ishii, 2002a). The validation for other cases relies on a drift-flux plot ( j vs. g v plot). The prediction error is estimated for a set of distribution parameter and drift velocity correlations. Ishii (1977) showed the validity of the drift-flux correlation for air-water, boiling Freon-22, boiling water, and boiling heavy water systems under the conditions of g f 0.16 ρ ρ  and the pipe diameter up to 168 mm. The constitutive equations described in Section 3.1 have been widely accepted and implemented into nuclear thermal-hydraulic safety analysis codes.

Upward two-phase flow in a vertical rectangular channel
The distribution parameter and drift velocity are in general dependent on flow regime. Before a drift-flux correlation is applied to upward two-phase flow in a vertical rectangular channel, the flow regime should be identified. A reliable model of flow regime transition criteria for upward two-phase flow in a vertical rectangular channel is available (Hibiki and Mishima, 2001).
The applicability of Eqs. (19) and (20) to a rectangular channel with the aspect ratio of the order of unity, namely nearly a square channel has not been confirmed. When the aspect ratio of a rectangular channel is unity, a round channel may be an assumption better than a rectangular channel with a certain aspect ratio. The distribution parameters for a circular channel, Eqs. (14) and (16), are lower than those for a rectangular channel, Eqs. (19) and (20). It is expected that the distribution parameter for a rectangular channel may decrease as the aspect ratio approaches unity. Due to the lack of detailed local flow parameters such as void fraction and gas and liquid velocities in a rectangular channel, the distribution parameter determined through Eq. (3) does not exist. The distribution parameter correlation for an adiabatic bubbly flow in a rectangular channel is not available. The drift velocity correlations for dispersed flow regimes, Eqs. (11), (12), and (13), were developed based on local drag law depending on a bubble shape regime, similarity hypothesis between a single-particle system and a multi-particle system, and simplification to replace local void fraction with an area-averaged void fraction (Ishii, 1977). These assumptions are not susceptible to a medium-size channel geometry. The drift velocity correlations recommended for a circular channel, Eqs. (11), (12), (13), and (17), are applied to a rectangular channel. Figure 2 compares the drift-flux correlation, Eqs. (13) and (19), with the data taken in a rectangular channel under the pressure ranging from 0.789 to 4.23 MPa (Abbs and Hibiki, 2019). Ishii (1977) showed the validity of the drift-flux correlation for boiling water, boiling R-11, and nitrogen-NaK systems under the conditions of g f 0.04 ρ ρ  and the aspect ratio ranging from 2.8 to 9.4.

Upward two-phase flow in vertical annulus channel
The distribution parameter and drift velocity are in general dependent on flow regime. Before a drift-flux correlation is applied to upward two-phase flow in a vertical annulus  channel, the flow regime should be identified. A reliable model of flow regime transition criteria for upward two-phase flow in a vertical annulus channel is available (Julia and Hibiki, 2011). Equation (14) has been utilized for predicting the distribution parameter for upward adiabatic two-phase flow in a vertical annulus channel. Ozar et al. (2008) calculated the distribution parameter with the profiles of measured local void fraction, assumed local mixture volumetric flux, and proposed the following distribution parameter correlation for upward adiabatic bubbly flow in a vertical annulus channel.
where H D is the hydraulic equivalent diameter. For a higher void fraction, the distribution parameter is given by Equation (21) indicates that the distribution parameters for an annulus channel, Eqs. (21) and (22), are smaller than those for a circular channel, Eqs. (14) and (15). This has been explained through Eq. (3) with assumed power-law profiles of local void fraction and mixture volumetric flux (Ozar et al., 2008).  developed the bubble-layer thickness model to predict the distribution parameter for subcooled boiling flow in a vertical annulus channel. The distribution parameter correlation is analytically derived with the assumed distribution parameter for saturated boiling flow, Eq. (14), as Figures 3(a) and 3(b) compare Eqs. (21) and (24)  data taken for upward adiabatic and boiling flows in vertical annuli, respectively. Equations (21) and (24) have been respectively validated by the atmospheric and boiling bubbly flow data taken in annulus channels with the rod diameter of 19.1 mm and the inner diameter of the outer pipe being 38.1 mm. As discussed in Section 3.2, the drift velocity correlations recommended for a circular channel, Eqs. (11), (12), (13), and (17), are applied to an annulus channel.

Upward two-phase flow in a vertical sub-channel of rod bundle geometry
The distribution parameter and drift velocity are in general dependent on flow regime. Before a drift-flux correlation is applied to upward two-phase flow in a vertical sub-channel of the rod bundle geometry, the flow regime should be identified. A reliable model of flow regime transition criteria for upward two-phase flow in a vertical sub-channel of the rod bundle geometry is available (Liu and Hibiki, 2017). Julia et al. (2009) analytically derived the distribution parameter for upward subcooled boiling flow in a vertical sub-channel of the rod bundle geometry using the bubble layer thickness model  as where 0 D and 0 P are the rod diameter and rod pitch, respectively. Julia et al. (2009) recommended the following drift velocity correlation modified for a confined channel to be used together with Eq. (25) in bubbly flow regime.
where sf B is the bubble size factor to consider the rod wall effect on the bubble rising velocity. Ozaki and Hibiki (2015) developed the distribution parameter correlation for upward boiling two-phase flow in a vertical 8×8 rod bundle under prototypic pressure and temperature conditions of a nuclear reactor as Equation (28) is applicable to the void fraction of 0.87. Equation (28) is optimized with the use of Hibiki-Ishii's drift velocity correlation  given by where gj V + and + g j are, respectively, the non-dimensional drift velocity and superficial gas velocity defined by The subscripts of B and P are the drift velocity for bubbly flow calculated by Eq. (11) and pool condition calculated by Kataoka-Ishii correlation (1987), respectively. Figure 4 compares the drift-flux correlation, Eqs. (28) and (29), with the data taken in a vertical 8×8 rod bundle under the pressure of 7.2 MPa (Ozaki and Hibiki, 2015). Equations (28) and (29) have been validated by NUPEC BTBF data (Morooka et al., 1991), JAERI TPTF data (Kondo et al., 1993), and Purdue adiabatic data (Yang et al., 2012). The applicable range covers the pressure from 0.1 to 11.8 MPa, mass velocity from 5 to 2000 kg/(m 2 ·s), hydraulic equivalent diameter from 9.8 to 21.7 mm, and casing size from 79 to 140 mm (Ozaki et al., 2013). Ozaki and Hibiki (2018) modified Eq. (25) to match with Eq. (28) for a high void fraction and analytically derived the distribution parameter for upward subcooled boiling flow in a vertical sub-channel of the rod bundle geometry as Liu et al. (2018) empirically developed the distribution parameter correlation for upward adiabatic bubbly flow in a vertical 5×5 rod bundle as Equation (32) has been validated by the data taken for upward adiabatic bubbly flow in a vertical 5×5 rod bundle under the atmospheric condition.

Horizontal two-phase flow in a circular channel
The distribution parameter and drift velocity are in general dependent on flow regime. Before a drift-flux correlation is applied to horizontal two-phase flow, the flow regime should be identified. The flow regime transition criteria for horizontal two-phase flow are rather complicated because the flow regime for horizontal gas-liquid two-phase flow is susceptible to inlet conditions and developing length. Because the flow regime transition criteria for horizontal two-phase flow have not been well-developed (Barnea et al., 1980), a general drift-flux correlation applicable to the full-range of the void fraction has been developed by Rassame and Hibiki (2018). The buoyancy force acting on the gas phase is perpendicular to the flow direction resulting in no relative velocity or drift velocity along the flow direction.
The distribution parameter correlation for horizontal twophase flow is given by ( ) ( )   Figure 5 compares the drift-flux correlation, Eqs. (33) and (34), with the data taken in horizontal channels . Equations (33) and (34) have been validated by the data taken for adiabatic air-water and air-kerosene flows in horizontal channels. The applicable range covers the superficial gas velocity from 0.0253 to 47.5 m/s, superficial liquid velocity from 5.7×10 -5 to 5.97 m/s, and the channel diameter from 19 to 77.9 mm.
where o v is the oil-fraction-weighted mean oil velocity. Equation (36) has been validated by the data taken for adiabatic oil-water flows in horizontal channels. The applicable range covers the superficial oil velocity from 0.01 to 2.69 m/s, superficial water velocity from 0.01 to 2.7 m/s, and the channel diameter from 20 to 106 mm. The physical properties of the fluids in the database are the oil density from 787 to 998 kg/m 3 , water density from 980 to 1000 kg/m 3 , oil viscosity from 1.2 to 280 mPa·s, water viscosity from 0.84 to 1 mPa·s, and surface tension from 28 to 45 mN/m.

Downward two-phase flow in a vertical circular channel
The distribution parameter and drift velocity are in general dependent on flow regime. Before a drift-flux correlation is applied to downward two-phase flow in a vertical circular channel, the flow regime should be identified. The flow regime transition criteria for downward two-phase flow are rather complicated because the flow regime for downward two-phase flow is susceptible to buoyancy force acting in the opposite direction of the liquid flow. Because the flow regime transition criteria for downward two-phase flow in a vertical circular channel have not been well-developed (Lokanathan and Hibiki, 2018), a general drift-flux correlation applicable to the full-range of the void fraction has been developed by Goda et al. (2003).
To develop a general drift-flux correlation independent on the flow regime, the drift velocity is approximated by 0.25 The distribution parameter correlation for downward  Figure 6 compares the drift-flux correlation, Eqs. (37) and (38), with the data taken for downward two-phase flow in vertical circular channels (Goda et al., 2003). Equations (37) and (38) have been validated by the data taken for downward air-water and steam-water flows in vertical circular channels. The applicable range covers the mixture volumetric flux from -0.45 to -24.6 m/s, pressure from 0.1 to 1.5 MPa, and the channel diameter from 16 to 102.3 mm.

Basic concept of two-group drift-flux model
The concept of the two-group drift-flux model has been proposed to close the one-dimensional modified two-fluid model (Ishii and Hibiki, 2010). The two-group approach treats a bubble in two groups, namely, group-1 bubbles (spherical or distorted bubbles) and group-2 bubbles (cap or slug bubbles). The drag coefficient and interfacial area concentration available to mass, momentum, and heat transfers are dependent on the bubble shape regime. Twogroup interfacial area transport equation has been proposed to enhance the prediction accuracy of the interfacial area concentration which affects the prediction accuracy of the interfacial drag force ). The introduction of the two-group interfacial area transport equation into the two-fluid model requires two gas continuity equation and two gas momentum equations resulting in 8 equation-based two-fluid model. To avoid the additional computational burden and numerical instability in the computation, the modified two-fluid model has been proposed to simplify the two gas momentum equations into a gas mixture momentum equation while still preserving the ability to treat the two bubble groups separately. The onedimensional form of the mixture gas momentum equation in the modified two-fluid model is expressed as where P , τ , T τ , Γ , and 12 m  D are the pressure, viscous shear stress, turbulent shear stress, mass generation rate per unit mass, and intergroup mass transfer rate per unit mass, respectively. The subscripts of w, i, 1, 2, and k are the wall, interface between two phases, group-1 bubble, group-2 bubble, and group-k bubble, respectively. The difference between the phase-fraction-weighted mean velocities, g21 V , is defined by To close the one-dimensional modified two-fluid model, the closure relationships for given though the two-group drift-flux model expressed by The distribution parameter and void-fraction-weighted mean drift velocity can be expressed by the distribution parameters and drift velocities for group-1 and group-2 as and g 02 f Figures 7(a) and 7(b) compare Eqs. (48) and (49) with the group-1 and group-2 bubble data taken for upward adiabatic two-phase flows in vertical annulus channels, respectively. Equations (48) and (49) have been validated by the data taken in upward adiabatic air-water flow in a vertical annulus channel under the pressure from 0.12 to 0.6 MPa.
The drift velocity for a group-1 bubble is given by The drift velocity for a group-2 bubble in a channel with a width, w , smaller than 52 times Laplace length is given as follows.
Churn flow: It should be noted here that the void fraction for group-1 bubble and Sauter mean diameter for group-2 bubble in a medium-size channel can be estimated by a constitutive correlation (Ozar et al., 2012). Brooks et al. (2012b) recommended Eqs. (54) and (55) for the distribution parameters of group-1 and group-2 bubbles of upward boiling two-phase flow in an annulus channel.

Upward boiling two-phase flow in a vertical annulus channel
and g 02 f Figures 8(a) and 8(b) compare Eqs. (54) and (55) with the group-1 and group-2 bubble data taken for upward boiling two-phase flows in a vertical annulus, respectively. Equations (54) and (55) have been validated by the data taken in upward boiling steam-water flow in a vertical annulus channel under the pressure from 0.18 to 0.95 MPa. The drift velocity correlations given in Section 4.2 can be used for upward boiling two-phase flow in a vertical annulus channel.

Chexel-Lellouche correlation
Chexel-Lellouche correlation also known as EPRI correlation has been implemented into RELAP5 code (ISL, 2001). The unique features of the Chexel-Lellouche correlation are (1) the independence on flow regime, (2) the applicability to the full range of thermodynamic conditions and geometries typical of PWR and BWR fuel assemblies as well as pipes up to 450 mm in diameter, (3) the applicability to various fluid types (steam-water, air-water, hydrocarbons and oxygen), and (4) the continuous function against the void fraction. The correlation set is given as follows.
The distribution parameter is expressed as where 0v C and 0h C are the concentration parameters evaluated for vertical and horizontal flows and r F is the flow orientation parameter defined by for  80  90 10 , for 80 90 where v θ is the pipe orientation angle measured from the vertical axis. The gas Reynolds number, g Re , is defined by where g μ is the gas viscosity. The concentration parameter for vertical flow is represented as follows: where L is the Chexal-Lellouche fluid parameter given as follows: for steam-water where f j and f μ are the area-averaged superficial liquid velocity and liquid viscosity, respectively.
The concentration parameter for horizontal flow is represented as follows: ( ) [ ] 2 0.05 0h 0v where 0v C is defined by Eq. (60) with the Chexal-Lellouche fluid parameter given as follows: for steam-water for air-water The drift velocity for co-current upward flow and pipe orientation angles where g v j V and g h j V are the drift velocities for vertical and horizontal flows, respectively. The drift velocity for cocurrent downward flow is expressed as The drift velocity for a vertical pipe covers co-current upward and downward flows and counter-current flow and is given by  Re For counter-current flow ( g j is positive and f j is negative) On the counter-current flooding limit (CCFL) line In the region of counter-current flow, there are two solutions for the void fraction, 1 α and 2 α , at every point. The desired void fraction, des α , known a priori from pressure drop or other information must be used in selecting the appropriate 3 C as follows:

Bhagwat-Ghajar correlation
The unique features of the Bhagwat-Ghajar correlation (2014) are (1) the independence on flow regime, (2) the applicability to the wide range of thermodynamic conditions and geometries (circular, annular, and rectangular channels) as well as pipes up to 305 mm in diameter, (3) the applicability to various fluid types (steam-water, gas-water, refrigerants and air-oil), and (4) the continuous function against the void fraction. The correlation set is given as follows.
The distribution parameter is expressed as where h θ is the pipe orientation angle measured from the horizontal axis. The two-phase mixture Reynolds number is defined by The parameter, 0,1 C , is defined by 0.15 1.5 0,1 where the constant, 1 C , is 0.2 for circular and annular channels and 0.4 for a rectangular channel and x is the two-phase flow quality. For horizontal and near horizontal channel orientations, Bhagwat and Ghajar recommended The non-dimensional Laplace length is defined by The applicable range of Bhagwat-Ghajar correlation covers the two-phase Reynolds number from 10 to 5×10 6 , pressure from 0.1 to 18.1 MPa, liquid viscosity from 0.0001 to 0.6 Pa·s, hydraulic equivalent diameter from 0.5 to 305 mm (circular, rectangular, and annular channels), and pipe orientation of h 90 90 θ   -£ £ . The fluid systems used for the validation are air-water, argon-water, natural gas-water, air-kerosene, air-glycerin, argon-acetone, argon-ethanol, argon-alcohol, refrigerants (R11, R12, R22, R134a, R114, R410A, R290, and R1234yf), steam-water, and air-oil fluid combinations.

Conclusions
The drift-flux parameters such as distribution parameter and drift velocity are critical parameters in the one-dimensional two-fluid model used in nuclear thermal-hydraulic system analysis codes. These parameters affect the drag force acting on the gas phase. The accurate prediction of the drift-flux parameters is indispensable to the accurate prediction of the void fraction. Because of this, the current paper conducted a state-of-the-art review on one-dimensional drift-flux correlations for various flow channel geometries and flow orientations. Section 2 described the formulation of the one-dimensional drift-flux model and interfacial drag force using the two drift-flux parameters. Section 3 reviewed the state-of-the-art constitutive equations of the one-dimensional drift-flux model for two-phase flow in various mediumsize channels. The covered flow conditions and channel geometries were upward two-phase flow in vertical circular, rectangular and annulus channels and vertical sub-channel of the rod bundle, horizontal two-phase flow in a circular channel, and downward two-phase flow in a circular channel. The essential conclusions in Section 3 were:  A channel geometry affected the distribution parameter.  A boundary condition (adiabatic or diabatic) affected the distribution parameter in a bubbly flow.


The drift velocity for a horizontal channel was approximated to be zero. The distribution parameter developed for a circular channel was not a good approximation to calculate the distribution parameter for a sub-channel of the rod bundle. It should be noted here that this approximation was often used in thermal-hydraulic system analysis codes.
Section 4 covered a newly proposed concept of the twogroup drift-flux model to provide the constitutive equation to close the modified gas mixture momentum equation mathematically. The covered flow conditions and channel geometry were upward adiabatic and boiling two-phase flow in annulus channels. Because of the similarity in the distribution parameter between annulus channel and subchannel of the rod bundle, the constitutive equations for the annulus channel could be used for calculating the distribution parameter of a sub-channel of the rod bundle as a first approximation.
Section 5 described the existing drift-flux correlations applicable to a full range of void fraction. They were Chexel-Lellouche correlation and Bhagwat-Ghajar correlation. These correlations used many empirical constants and branches. The correlations were not explicit correlations for the void fraction, and some iterations were required for obtaining the solution of the void fraction. Due to their empirical nature, the correlations should be used within the validated range.
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