One-dimensional drift-flux model and a new approach to calculate drift velocity and gas holdup in bubble columns

Journal Pre-proofs One-dimensional drift-flux model and a new approach to calculate drift velocity and gas holdup in bubble columns Azadeh Bahramian, Siamak Elyasi PII: DOI: Reference:

S0009-2509(19)30792-4 https://doi.org/10.1016/j.ces.2019.115302 CES 115302

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Chemical Engineering Science

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Please cite this article as: A. Bahramian, S. Elyasi, One-dimensional drift-flux model and a new approach to calculate drift velocity and gas holdup in bubble columns, Chemical Engineering Science (2019), doi: https:// doi.org/10.1016/j.ces.2019.115302

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One-dimensional drift-flux model and a new approach to calculate drift velocity and gas holdup in bubble columns Azadeh Bahramian1, Siamak Elyasi Department of Chemical Engineering, Lakehead University, Thunder Bay, Ontario, Canada

Abstract In view of the practical importance of the drift-flux model and the variety of industrial applications of large-diameter columns, the model coefficients, including the distribution parameter (𝐶0) and drift velocity (𝐶1), have been investigated for the upward two-phase flow in large- and small-diameter columns. The distribution parameter and drift velocity reflect the nonuniformity in the flow and the local relative motion between two phases, respectively. As the formation of cap bubbles and occurrence of liquid recirculation are two of the most important characteristics in a large-diameter column, the distribution parameter and drift velocity in such a column are rather different from those in a small-diameter column. In this study, the local flow parameters measured by other researchers were collected, and 𝐶0 and 𝐶1 were calculated based on their definitions. The experimental values in small- and large-diameter columns were compared, and the performances of various existing constitutive equations for the drift-flux model were reevaluated. Two semi-empirical correlations for the drift velocity and gas holdup were developed, and their performances were verified against experimental data for various two-phase flow conditions. Keywords: Multiphase flow; Bubbly flow; Gas holdup; Distribution parameter; Drift velocity 1

Introduction Two-phase gas–liquid flow is the most commonly observed phenomenon in the chemical,

petrochemical, and biochemical industries. For instance, the distribution of gas through liquid is crucial for certain processes, such as oxidation, chlorination, polymerization, and fermentation (e.g. Debellefontaine et al., 1999; De Sousa et al., 2018; Doig et al., 2005; Kantarci et al., 2005; Lübbert et al., 1996; Maretto and Krishna, 2001; Smith et al., 1996; Woo et al., 2010). Furthermore, the presence of vapor–liquid in boiling and condensation phenomena is inevitable. In-depth Tel.:+1(807)6339762 E-mail address:[email protected] (A. Bahramian) 1

1

knowledge of the two-phase behavior, in terms of the flow patterns, fractional gas holdup, and gas and liquid velocities, is important for understanding the interactions between phases, and consequently for suitably designing or optimizing the related equipment. In particular, bubble columns are utilized to facilitate two-phase processes. However, despite the simplicity of their construction and operation, the hydrodynamics of bubble columns are complex. The fractional gas holdup (αg) is one of the most important design parameters, playing an important role in the performance of a bubble column. The gas holdup explicitly affects the reactor volume, because the fraction of the volume is occupied by gas. In addition, a spatial variation of αg changes the pressure, and eventually results in a change in the liquid phase momentum. This internal source of momentum in the liquid phase governs the rates of mixing, heat transfer, and mass transfer (Joshi et al., 1998). As a result, the ability to predict the gas holdup in a bubble column as a function of the geometry and operating parameters has attracted considerable research attention, and many experimental and theoretical methods have been proposed (e.g. Kawasa and Moo-Young, 1987; Kim et al., 2017; Jamilahmadi et al., 2000; Nedeltchev and Schumpe, 2008; Singh et al., 2017; Woldesemayat and Ghajar, 2007). As a two-phase flow consists of the relative motion between two phases, a general two-phase flow problem should be formulated using a two-phase flow model, or a drift-flux model. In the Eulerian–Eulerian two-phase approach, a two-fluid model treats each phase as a separate fluid with its own set of governing equations. In general, the solutions of the governing equations provide the velocity, temperature, and pressure of each phase individually. Although this approach is capable of accounting for the dynamic and non-equilibrium interactions between phases, considering separate sets of momentum and energy equations for each phase introduces a high degree of complexity, and for most applications only a numerical solution is available. In contrast to the two-fluid model, in which each phase is considered separately, the drift-flux model considers the mixture as a whole. Furthermore, important phenomena such as the influence of non-uniformity in the flow and phase holdup distribution profile, as well as the local relative motion between phases, are taken into account. This approach reduces the number of numerical equations and the resulting complexity of the problem. In addition, this model is a general one, which can be applied to any two-phase flow regime. The concept of drift-flux theory was first proposed by Zuber and Findlay (1965), and was later modified by Wallis (1969), Ishii (1977), and Joshi et al. (1990). The drift-flux model considers the effect of the non-uniformity of the holdup distribution across a cross-sectional area, as well as the local 2

relative velocity between phases. Owing to the flexibility and simplicity of the drift-flux model, many investigators have applied this method to predict the gas holdup over a broad range of operating conditions and column diameters (e.g. Bhagwat and Ghajar, 2014; Shen et al., 2010; Woldesemayat and Ghajar, 2007). Recently, experimental and modeling activities have mostly been limited to small-diameter columns, and the constitutive equations for the drift-flux model have been developed effectively for vertical upward two-phase flows with relatively small diameters ranging from 25 to 50 mm. Although these constitutive equations effectively predict the void fraction in small-diameter columns, mostly large-diameter columns are encountered in industrial plants. The hydrodynamics of two-phase flows in large-diameter columns are different from those in small-diameter columns. In large-diameter columns, slug bubbles are no longer sustained, owing to interfacial instability, which leads to the distortion and collapse of the upper surfaces of large bubbles, consequently resulting in the formation of cap bubbles. This phenomenon produces additional turbulence, which results in secondary recirculation in the flow. On the other hand, the bubbles in large-diameter columns tend to move toward the column centerline and form a core-peak void fraction profile, instead of the wall-peak profile that is commonly observed in smaller columns (e.g. Cheng et al., 1998; Hibiki and Ishii, 2003; Ohnuki and Akimoto, 2000; Serizawa et al. 1991). These differences lead to significant changes in the fractional gas holdup and velocity profiles, and consequently the distribution parameter and drift velocity. Therefore, the constitutive equations for the drift-flux model that are obtained in a smalldiameter column can no longer be guaranteed to accurately predict the two-phase behavior in a large-diameter column, and their applicability and accuracy must be verified. From this viewpoint, the main purpose of this study is to develop a model that can predict the fractional gas holdup for bubble columns over a wide range of operating conditions. The distribution parameter and drift velocity in large- and small-diameter columns are studied, and the existing constitutive equations of drift-flux theory are re-evaluated in comparison with the published experimental data. This approach results in new semi-empirical equations for the drift velocity and gas holdup. The performance and accuracy of the proposed correlations are tested and verified against a wide range of experimental data that are available in open literature for various experimental conditions, such as column diameters, pressures, liquid phase properties, and mixture volumetric fluxes.

3

2

Theory

2.1 One-dimensional drift-flux model 2.1.1

Basic drift-flux model

Drift-flux theory is one of the most practical and accurate two-phase models, taking into account the effect of the relative motion between phases as well as the non-uniformity of the void fraction profile. An advantage of this model is that it can be utilized for any two-phase (liquid– gas) flow analysis, and in particular it has been successfully applied to forced convection systems (Hibiki and Ishii, 2002). The one-dimensional drift-flux model proposed by Zuber and Findlay (1965), is formulated as follows:

〈𝑗𝑔〉 = 〈〈𝜐𝑔〉〉 = 𝐶0〈𝑗〉 + 𝐶1 , 〈𝛼𝑔〉

(1)

where 𝑗𝑔, 𝛼𝑔, 𝜐𝑔, and 𝑗 are the superficial gas velocity, fractional gas holdup, gas velocity, and mixture volumetric flux, respectively. Here, <> and ≪≫ denote the area-weighted average of a quantity over the column cross-section and the void fraction-weighted mean value, respectively. The distribution parameter (C0) and drift velocity (C1) are defined by Equations (2) and (3): 𝐶0 =

〈𝛼𝑔𝑗〉 , 〈𝛼𝑔〉

(2)

𝐶1 =

〈𝛼𝑔𝜐𝑔𝑗〉 , 〈𝛼𝑔〉

(3)

where 𝜐𝑔𝑗 is the local drift velocity of the gas phase, which is defined as 𝜐𝑔𝑗 = 𝜐𝑔 ― 𝑗 = (1 ― 𝛼𝑔)(𝜐𝑔 ― 𝜐𝑙) , 2.1.2

(4)

Existing equations for the distribution parameter and drift velocity in small-diameter columns

Ishii (1977) developed a comprehensive set of equations for the relative motions between phases for small-diameter columns and rectangular channels. Kawanishi et al. (1990) examined a steam–water two-phase flow under different flow conditions of a vertical upward or downward flow and a counter-current flow at different pressures. They studied the effect of the column diameter on the drift-flux parameters using vertical tubes with diameters of 19.7 and 102.3 mm. In another study, Mishima and Hibiki (1996) calculated the distribution parameter for capillary tubes with inner diameters between 1 and 4 mm, and derived an equation for 𝐶0 as a function of 4

the column diameter. Hibiki and Ishii (2002) measured the local flow parameters of adiabatic air– water bubbly flows in vertical columns with inner diameters of 25.4 and 50.8 mm, and developed a new model. The proposed correlations for the drift-flux model in small diameter vertical columns are summarized in Table 1. Table1: Distribution parameter and drift velocity correlations for small diameter columns. Distribution parameter (𝑪𝟎)

Researchers Ishii (1977)

Drift velocity (𝑪𝟏)

1.2 ― 0.2 𝜌𝑔 𝜌𝑓

( ) ( ) ( ) ( ) ( )

2

𝑔𝜎∆𝜌

2 1.2 ― 0.2 𝜌𝑔 𝜌𝑓

0.52 Mishima and Hibiki

𝑔𝐷𝐻∆𝜌

12

Application range Bubbly flow

Slug flow

𝜌𝑓

𝑔𝜎∆𝜌

0.35

(1990)

(1 ― 〈𝛼〉)1.75

𝜌2𝑓

0.35

Kawanishi et al.

14

14

Churn flow

𝜌2𝑓

𝑔𝐷𝐻∆𝜌

12

𝑗 ≥ 0.24, 𝑃 ≤ 1.5, 𝐷𝐻 ≤ 0.05m

12

𝑗 ≥ 0.24, 𝑃 ≤ 1.5, 𝐷𝐻 ≥ 0.05m

𝜌𝑓

𝑔𝐷𝐻∆𝜌 𝜌𝑓

1.2 + 0.51𝑒 ―0.691 𝐷𝐻

1.0𝑚𝑚 ≤ 𝐷𝐻 ≤ 4.0𝑚𝑚

(1996) Hibiki and Ishii

(1.2 ― 0.2

𝜌𝑔 𝜌 )(1 ― 𝑒 ―22〈𝐷𝑆𝑚〉 𝐷𝐻) 𝑓

Same as Ishii (1977)

25.4𝑚𝑚 ≤ 𝐷𝐻 ≤ 60𝑚𝑚

(2002)

2.1.3

Existing equations for the distribution parameter and drift velocity in large-diameter columns

Shen et al. (2014) proposed definitions for small and large pipe diameters based on the nondimensional hydraulic diameter: 𝐷𝐻∗ =

𝐷𝐻 𝜎 𝑔∆𝜌

,

(5)

They defined small pipes as 𝐷𝐻∗ < 18.5 and large pipes as 𝐷𝐻∗ > 40. The intervening range of 18.5 ≤ 𝐷𝐻∗ ≤ 40 represents a transition region between the two behaviors. For air–water systems under atmospheric pressure, pipes with a diameter larger than 102 mm can be considered as largediameter pipes. Hills (1976) derived a drift-flux correlation in an air-water bubble column with an inner diameter of 150 mm and a height of 10.5 m. Although his model confirms his dataset very well, it does not take into account the physical properties of the fluids. Shipley (1984) proposed a 5

model based on his own data in a pipe with a diameter of 457 mm and height of 5.64 m. Shipley considered the column diameter in the drift velocity term, which is the main drawback of the proposed model. This leads to an unrealistic increase in the value of the relative velocity for a column of very large diameter. The drift velocity for bubbling or pool boiling systems was examined in detail by Kataoka and Ishii (1987). They developed a correlation for the drift velocity based on a large amount of experimental data with different physical properties, column diameters, and pressures. Hibiki and Ishii (2003) later demonstrated that the flow pattern and drift-flux parameters are dependent on the physical properties and fluid velocities for bubbly flows in largediameter columns. The existing correlations for the distribution parameter and drift velocity for large-diameter pipes are presented in Table 2. It is worth noting that the distribution parameter and drift velocity used to develop the existing drift-flux models for small- and large-diameter columns have been obtained indirectly from the plot of 〈𝑗𝑔〉 〈𝛼〉 versus〈𝑗〉. When the drift velocity is independent of the fractional phase holdup, or, in other words, when the flow regime is fully developed, the data points form a straight line. The slope of this line represents the distribution parameter, and the drift velocity can be interpreted as the intercept of the graph. However, the majority of two-phase flows in large-diameter columns are not fully developed, and a recirculation flow pattern may develop at low flow rates. As a result, the parameters obtained by graphical methods may not predict the real flow behavior. Despite the practical importance of understanding two-phase flow characteristics in a large-diameter column, only a few analytical and experimental studies have been conducted.

6

Table2: Distribution parameter and drift velocity correlations for large diameter pipes. Distribution parameter (𝑪𝟎)

Drift velocity (𝑪𝟏)

Application range

1.0

(0.24 + 4.0〈𝛼〉

𝑗𝑓 ≤ 0.3

1.35𝑗 ―0.07

0.24

Shipley (1984)

1.2

0.24 + 0.35

Kataoka and Ishii

1.2 ― 0.2 𝜌𝑔 𝜌𝑓

Researchers Hills (1976)

𝑗𝑓 ≥ 0.3

et

al.

(1990)

Hibiki (2003)

and

1.95〈𝑗𝑔〉 + 0.93〈𝑗𝑓〉

𝜎𝑔∆𝜌

―0.157

1.69

1―

〈𝑗〉

𝜎𝑔∆𝜌

―2.88

〈𝑗〉

―0.562 𝑁𝜇𝑓

𝑁𝜇𝑓 ≤ 2.25 × 10 ―3 𝐷𝐻∗ ≤ 30 𝑁𝜇𝑓 ≤ 2.25 × 10 ―3

―0.562 𝑁𝜇𝑓

𝐷𝐻∗ ≥ 30

14

𝑁𝜇𝑓 ≥ 2.25 × 10 ―3

𝜌2𝑓

𝐷𝐻∗ ≥ 30

14

𝑔𝐷𝐻∆𝜌 𝜌𝑓

12

)

𝜌𝑔

(

𝜌𝑓

𝜌𝑔

+

](

𝜌𝑓

𝐷𝐻 ≥ 0.05m

( ) 𝜎𝑔∆𝜌 𝜌2𝑓

[ (

𝜌𝑓

+ 𝐶1,𝑃 1 ― 𝑒𝑥𝑝 ―1.39〈𝑗𝑔〉

)

𝜌𝑔

+ 4.08 1 ―

𝑗 ≥ 0.24, 𝑃 ≤ 1.5,

)

(

𝐶1,𝐵𝑒𝑥𝑝 ―1.39〈𝑗𝑔〉

[ ( )

―0.157

14

𝜌2𝑓

𝐶1,𝐵𝑒𝑥𝑝 ―1.39〈𝑗𝑔〉

{ ( ) }( 〈𝑗𝑔〉

𝜌𝑔 𝜌𝑓

𝜌2𝑓

𝜌𝑓

𝜎𝑔 𝜌𝑓

(

0.52

Ishii 𝑒𝑥𝑝 0.475

14

0.25

1.2 ― 0.2 𝜌𝑔 𝜌𝑓

〈𝑗𝑔〉

𝜎𝑔∆𝜌

𝜌𝑓

1.53

〈𝑗〉

Kawanishi

0.809

( )

0.934(1 + 1.42〈𝛼〉)

(1986)

𝑔𝐷𝐻〈𝛼〉

〈𝑗〉

―0.157

𝜌𝑔

𝜌𝑔

0.92

Clark and Flemmer

2

( )() () ( ) () ( )

0.0019𝐷𝐻∗

0.030

(1985)

〈𝑗𝑔〉

( )

(1987)

Clark and Flemmer

)(1 ― 〈𝛼〉)

1.72

+

𝜌𝑔

( ) 𝜎𝑔∆𝜌 𝜌2𝑓

[ (

𝜌𝑓

+ 𝐶1,𝑃 1 ― 𝑒𝑥𝑝 ―1.39〈𝑗𝑔〉

)

―1 4

( ) 𝜎𝑔∆𝜌 𝜌2𝑓

Bubbly flow 〈𝛼〉 ≤ 0.3

)]

―1 4

)

0≤

〈𝑗𝑔〉 〈𝑗〉

≤ 0.9

―1 4

( ) 𝜎𝑔∆𝜌 𝜌2𝑓

Bubbly flow 〈𝛼〉 ≤ 0.3

)]

―1 4

0.9 ≤

〈𝑗𝑔〉 〈𝑗〉

≤1

Cap Bubbly Flow

{ [(

1.2𝑒𝑥𝑝 0.11 〈𝑗〉

𝜎𝑔∆𝜌 𝜌2𝑓

)

] }(

―1 4

2.22

1―

)

𝜌𝑔 𝜌𝑓

+

𝜌𝑔

〈𝛼〉 > 0.3 𝐶1,𝑃

𝜌𝑓

0 ≤ 〈𝑗〉

7

( ) 𝜎𝑔∆𝜌 𝜌2𝑓

―1 4

≤ 1.8

{ { [ { { [( 0.6𝑒𝑥𝑝 ―1.2 〈𝑗〉

( ) 𝜎𝑔∆𝜌 𝜌2𝑓

― 0.6𝑒𝑥𝑝 ―1.2 〈𝑗〉

3

𝜎𝑔∆𝜌 𝜌2𝑓

]} } ) ]} }

―1 4

Cap Bubbly Flow

― 1.8 + 1.2

―1 4

― 1.8 + 0.2

〈𝛼〉 > 0.3 𝐶1,𝑃 𝜌𝑔

〈𝑗〉

𝜌𝑓

( ) 𝜎𝑔∆𝜌 𝜌2𝑓

―1 4

> 1.8

Results and discussion

3.1 Datasets utilized to develop new correlations With an emphasis on the above facts, a re-evaluation of existing models shows that although the tabulated correlations (Tables 1 and 2) predict the void fraction to a reasonable accuracy for the specific range of operating conditions and geometries from which they were obtained, their performance cannot be guaranteed over a wide range of operating and flow conditions. Therefore, many sets of experimental data taken over a broad range of superficial gas and liquid velocities, column diameters, system pressures, and fluid properties are essential to develop the drift-flux correlation for a two-phase flow. The comprehensive databases utilized in this study are listed in Table 3. A total of 1,611 datasets available from open literature are extracted to validate the void fraction correlation. These databases cover extensive experimental conditions, such as the column diameter (0.019–0.304 m), system pressure (0.1–0.28 MPa), superficial gas velocity (0.001–4.7 m/s), superficial liquid velocity (0.001–5.0 m/s), liquid viscosity (0.3–127 mPa.s), liquid density (767–1,199 m3/s), and surface tension (0.02–0.10 N/m). 3.2 Flow regime map In small-diameter columns, the flow regimes are typically classified into four phase distribution patterns: homogeneous (bubbly), heterogeneous (churn), slug, and annular flow. Zhang et al. (1997) investigated flow regimes experimentally using an air-water system under ambient conditions in a small-diameter column (0.0826 m). However, as discussed in the previous section, stable slug bubbles cannot exist in largediameter columns. This means that the flow regime maps developed for small-diameter columns cannot be applied to large-diameter columns. Schlegel et al. (2009) developed a flow regime map for large-diameter columns, and this was validated using a large database of flow regime identification data. The authors conducted the measurements in an upward air–water two-phase 8

flow along a column with an inner diameter of 0.15 m, and they classified the flow patterns into bubbly, cap-turbulent, churn-turbulent, and annular flow. In this study, we utilized these flow maps to identify the flow patterns of the experimental data in the small- and large-diameter columns listed in Table 4, and the results are illustrated in Figures 1(a) and 1(b). The dashed lines represents the flow regime transition boundaries developed by Zhang et al. (1997) and Schlegel et al. (2009).

9

Table 3: Databases utilized in this study. Researcher(s)

Fluid system

Column diameter (m)

Superficial liquid velocity (m/s)

Superficial gas velocity (m/s)

System pressure (MPa)

Liquid viscosity (mPa.s)

Number of data points

Hills (1976)

Air–Water

0.150

0.00–0.50

0.04–0.85

0.1

1.0

282

Shen (2010)

Air–Water

0.200

0.05–0.30

0.001–0.1

0.1

1.0

17

Schlegel et al. (2010)

Air–Water

0.152, 0.203

0.05–1.00

0.1–5.0

0.18, 0.28

1.0

265

Schlegel et al. (2012)

Air–Water

0.152, 0.203

0.40–0.63

0.15–3.0

0.18, 0.28

1.0

101

Schlegel et al. (2013)

Air–Water

0.152, 0.203, 0.304

0.25–1.00

0.04–4.0

0.18, 0.28

1.0

374

Hibiki & Ishii (2000)

N2–Water

0.102

0.01–0.40

0.05–0.28

0.1

1.0

58

Besagni & Inzoli (2017)

Air–Aqueous MEG solution

0.240

0.00

0.003–0.2

0.1

1.0–8.0

157

Rollbusch et al. (2015)

N2–Water

0.160

0.140

0.003–0.1

0.1

1.0

15

N2–Acetone

0.300

0.207

0.003–0.05

0.1

0.3

22

Das et al. (1992)

Air–Aqueous CMC solution

0.019

0.31–0.69

0.28–1.1

0.1

100–127

27

Abdulkadir et al. (2010)

Air–Silicon oil

0.067

0.05–0.38

0.027–4.7

0.1

5.3

78

Xing et al. (2013)

Air–Glycerol solution

0.190

0.00

0.01–0.3

0.1

1.0–39.6

67

Shawkat et al. (2008)

Air–Water

0.200

0.20–0.68

0.005–0.1

0.1

1.0

26

Esmaeili et al. (2015)

Glucose

0.292

0.00

0.02-0.22

0.1

185

16

Boger

117-122

16

CMC

34-66

16

Xanthan gum

13-66

16

1.0

58

Yang & Fan (2003)

Air–Water

0.051

0.00-0.02

10

0.015-0.20

0.1

Hills (1976) Hibiki & Ishii (2000) Das (1992) Abdulkadir (2010) Flow regime map by Zhang et al. (1997) 0.1

Dispersed bubble flow

0.01

Discrete bubble

Churn flow

Superficial liquid velocity, (m/s)

1

Slug flow

Annular flow

(a) 0.001 0.001

0.01

Superficial liquid velocity, (m/s)

10

0.1

Superficial gas velocity, (m/s)

1

10

Schlegel (2010) Schlegel (2012) Schlegel (2013) Shawkat (2008) Rollbusch (2015) Flow regime map by Schlegel (2009)

1

Annular 0.1

Bubbly

Cap- Turbulent

Churn- Turbulent

(b) 0.01 0.01

0.1

1 Superficial gas velocity, (m/s)

10

Fig. 1. Flow regime map (a) for small-diameter and (b) large-diameter columns.

11

100

3.3 Comparison of existing correlations for the distribution parameter and drift velocity with available experimental data It was emphasized above that the two-phase flow in large-diameter columns is multidimensional and the flow regime is not fully developed. As a result, the drift-flux parameters obtained by the graphical method may not correctly reflect the physics. Using the definitions of 𝐶0 and 𝐶1 in Eqs. (2) and (3), it is possible to calculate their local values from the radial profiles of the gas holdup, gas velocity, and liquid velocity. The experimental data of Shawkat et al. (2008) and Shen et al. (2010), which were obtained in a 200 mm column, were selected to re-evaluate the existing correlations. Based on the local data captured by their experiments, a database for 𝐶0 and 𝐶1 in a large-diameter pipe was established. Furthermore, a literature survey on the most commonly utilized constitutive equations of the drift-flux model for an upward two-phase flow in a vertical large-diameter column was conducted. These existing correlations are presented in Table 2. The existing correlations for the distribution parameter proposed by Clark and Flemmer (1985, 1986), Hibiki and Ishii (2003), and Ishii (1977) are compared with the experimental results in Figure 2. It can be observed that Ishii’s model and the two models of Clark and Flemmer do not satisfactorily predict the distribution parameter. The equation of Hibiki and Ishii agrees well with the experimental distribution parameters at lower values of 〈𝑗𝑔〉 〈𝑗〉, but deviations are noticeable when 〈𝑗𝑔〉 〈𝑗〉 is greater than 0.2.

12

1.40

Shawkat et al. (2008) Shen et al. (2010) Ishii (1977) Clark & Flemmer (1985) Clark & Flemmer (1986) Hibiki & Ishii (2003)

Distribution parameter, C0 (-)

1.35 1.30 1.25 1.20 1.15 1.10 1.05 1.00 0.95 0.90 0

0.1

0.2

0.3 / (-)

0.4

0.5

0.6

Fig. 2. Comparison of the distribution parameter determined in the present study with experimental results.

The existing correlations from Clark and Flemmer (1985), Hibiki and Ishii (2003), Hills (1976), Kataoka and Ishii (1987) and Shipley (1984) for the mean drift velocity are compared with the experimental results in Figure 3. The scatter of the data points appears to be relatively large, and falls between the model of Clark and Flemmer and that presented by Kataoka and Ishii. The high value of the drift velocity predicted by Kataoka and Ishii can be justified by the fact that their model is based on the measurements in a pool. As the wall effect is negligible in a pool, the bubbles can rise freely, and the bubble terminal velocity is higher than that of a bounded fluid (Clift et al., 19778). As a result, the drift velocity predicted by their equation is at the maximum value. On the other hand, the correlations developed by Clark and Flemmer are based on the experimental measurements in a 100-mm-diameter column, which is relatively small, and the wall effect significantly reduces the bubble terminal velocity. The equations of Hibiki and Ishii and of Shipley also do not predict the drift velocity very accurately. Although the drift-flux equations presented by Hibiki and Ishii satisfactorily predict the void fraction, they cannot be employed individually, because their correlations for the distribution parameter and drift velocity have not been validated separately by the local flow measurements. These exhibit a ±30% error for a relatively low velocity

(〈𝑗𝑔〉 < 2.0 𝑚/𝑠) and ±60% for a relatively high velocity (〈𝑗𝑔〉 ≥ 2.0 𝑚/𝑠). Additionally, the 13

correlations suggested by Kataoka and Ishii, Clark and Flemmer, Shipley, and Hills were derived by calculating the intercept of the plot of 〈𝑗𝑔〉 〈𝛼〉 versus〈𝑗〉. This leads to a constant value, and consequently, a significant deviation from the experimental data.

0.60

Shawkat et al. (2008) Shen et al. (2010) Hills (1976): jl=0.2 Shipely (1984) Clark & Flemmer (1985) Kataoka & Ishii (1987) Hibiki & Ishii (2003): jl=0.2 Hibiki & Ishii (2003): jl=0.35

Mean drift velocity, C1 (m/s)

0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0

0.1

0.2

0.3 (-)

0.4

0.5

0.6

Fig. 3. Comparison of the mean drift velocity determined in the present study with experimental results.

To demonstrate the difference between the values of the drift-flux parameters determined by the graphical method and those obtained using of their definitions, the graph of 〈𝑗𝑔〉 〈𝛼〉 versus〈𝑗〉 is presented in Figure 4.

14

1.20 1.00

/<α> (m/s)

0.80 0.60 ⟨�� ⟩/⟨�⟩ =0.867⟨�⟩+0.35

0.40 0.20 0.00 0

0.2

0.4

(m/s)

0.6

0.8

1

Fig. 4. Graphical method of calculating the distribution parameter and drift velocity.

Thus, values of 𝐶0 = 0.867 and 𝐶1 = 0.35 𝑚/𝑠 are obtained from the slope and the intercept of the fitted line, respectively. A comparison of these parameters with those in Figures 2 and 3 indicates that the graphical method produces constant values for 𝐶0 and 𝐶1, but these vary for different superficial gas and liquid velocities. Furthermore, the flows in large and small columns are different. Therefore, the distribution parameter and drift velocity were investigated for various column diameters. The databases utilized in this study and the detailed experimental conditions are summarized in Table 4. Figure 5 compares the distribution parameters calculated in columns with different diameters. The figure shows that the distribution parameter is higher in small-diameter columns. It can be concluded that slug bubbles cannot be formed in large-diameter columns because of the surface tension instability. As a result, the large bubbles break into smaller cap bubbles (Hibiki and Ishii, 2003). This phenomenon may generate secondary turbulent movement owing to liquid recirculation. The existence of turbulent movement and liquid recirculation along with small-cap bubbles gives rise to a relatively uniform radial profile for the gas holdup, whereas the void fraction profile in smalldiameter columns under the same operating conditions is sharper. As the distribution parameter reflects the non-uniformity of the gas holdup profile, the values for large-diameter columns are lower than those for small-diameter columns. 15

1.40

Shawkat et al. (2008), D=200 mm

Distribution parameter, C0 (-)

1.35

Hibiki & Ishii (2002), D=50.8 mm

1.30

Hibiki & Ishii (2002), D=25.4 mm

1.25 1.20 1.15 1.10 1.05 1.00 0.95 0.90 0.00

0.05

0.10

0.15 0.20 Gas holdup (-)

0.25

0.30

Fig. 5. Comparison of the distribution parameter in small and large diameter columns. Table 4: Databases utilized in this study to compare the distribution parameter and drift velocity. Researcher(s)

Fluid system

Column diameter (m)

Superficial liquid velocity (m/s)

Superficial gas velocity (m/s)

Pressure (MPa)

Number of data points

Shawkat et al. (2008)

Air–Water

0.200

0.2–0.68

0.005–0.1

0.1

26

Shen et al. (2010)

Air–Water

0.200

0.2–0.311

0.004–0.065

0.1

17

Serizawa at al. (1991)

Air–Water

0.060

0.442–1.03

0.0941–0.416

0.1

12

Liu (1989)

Air–Water

0.0381

0.376–1.39

0.0197–0.353

0.1

41

Hibiki & Ishii (2002)

Air–Water

0.0508

0.491–5.0

0.0257–4.88

0.1

18

Hibiki & Ishii (1999)

Air–Water

0.0254

0.262–3.49

0.0156–1.27

0.1

25

The drift velocity has been investigated for different column diameters, and the results are shown in Figure 6. The datasets exhibit higher values of 𝐶1 for large diameters. The higher drift velocity in large-diameter columns may be attributed to the formation of large cap bubbles at the centers of the columns, resulting from the enhanced bubble coalescence owing to liquid recirculation. These cap bubbles move faster than the dispersed bubbles, consequently leading to a greater drift velocity.

16

Mean drift velocity, C1 (m/s)

1.00

0.10 Shawkat et al. (2008), D=200 mm Serizawa et al. (1991), D=60 mm Hibiki & Ishii (2002), D=50.8 mm Liu (1989), D=38.1 mm Hibiki & Ishii (1999), D=25.4 mm

0.01 0.00

0.05

0.10

0.15

0.20 0.25 Gas holdup (-)

0.30

0.35

0.40

Fig. 6. Comparison of the drift velocity in small- and large-diameter columns.

3.4 Semi-empirical modeling and introduction of new correlations 3.4.1

Semi-empirical model development for the drift velocity

As discussed in the previous section, liquid recirculation may develop in large-diameter columns, particularly at low flow rates. The developed flow pattern affects the void fraction and velocity profiles, and consequently the drift-flux parameters. As the contribution of the drift velocity to the calculation of the gas velocity is relatively significant for a bubbly flow regime compared with other regimes, such as churn-turbulent and annular flow patterns, a new drift velocity correlation is developed in this section. Zuber and Findlay (1965) showed that the void-weighted average drift velocity is derived by the following equation: 𝐶1 =

〈𝛼𝑔𝜐𝑔𝑗〉 〈𝛼𝑔(1 ― 𝛼𝑔)𝜐𝑟〉 , = 〈𝛼𝑔〉 〈𝛼𝑔〉

(6)

where 𝜐𝑟 is the relative local velocity. Assuming that the relative velocity is uniform across the cross-section area, the drift velocity can be rewritten as in Equation (7):

〈𝛼𝑔2〉 〈𝛼𝑔𝜐𝑔𝑗〉 〈𝛼𝑔(1 ― 𝛼𝑔)〉 〈𝜐𝑟〉 = 1 ― 〈𝜐 〉 , = 𝐶1 = 〈𝛼𝑔〉 〈𝛼𝑔〉 〈𝛼𝑔〉 𝑟

(

)

17

(7)

The assumption of a uniform relative velocity can be verified using the detailed local experimental data of Hibiki and Ishii (1999) and Shawkat et al. (2008) as shown in Figure 7. This assumption is also supported by Brooks et al. (2014). The void fraction covariance can be defined using Equation (8). As a result, the drift velocity correlation can be simplified as in Equation (9).

〈𝛼𝑔2〉

,

(8)

𝐶1 = (1 ― 𝐶𝛼〈𝛼𝑔〉)〈𝜐𝑟〉 ,

(9)

𝐶𝛼 =

〈𝛼𝑔〉2

The void fraction is calculated from the local gas holdup measurements. Brooks et al. (2014) plotted 𝐶𝛼〈𝛼𝑔〉 (covariance parameter) versus 〈𝛼𝑔〉 based on experimental data taken for two flow directions (upward and downward flow) and two different flow geometries (pipe and annulus). For a two-phase upward flow, they showed that for a wide range of pipe sizes and two geometries, the data converge to a line with a slope of 1.18, which represents the magnitude of the void fraction covariance.

Local relative velocity, υr (m/s)

0.60 0.50

Shawkat et al. (2008), D=200 mm

jl=0.2 m/s,

jg=0.1 m/s

Hibiki & Ishii (1999), D=25.4 mm

jl=0.262 m/s,

jg=0.117 m/s

0.40 0.30 0.20 0.10 0.00 0

0.2

0.4

r/R (-)

0.6

0.8

1

Fig. 7. Local radial relative velocity profile in small- and large-diameter pipes.

In the present study, this value is considered to estimate the covariance parameter. 𝐶𝛼〈𝛼𝑔〉 =

〈𝛼𝑔2〉 〈𝛼𝑔〉

= 1.18〈𝛼𝑔〉 ,

(10)

18

For the next step, the area-weighted average relative velocity 〈𝜐𝑟〉, with the correlation as a function of the void fraction, must be estimated. Lapidus and Elgin (1957) found that in liquid fluidization systems, the relative velocity is related to the terminal velocity and gas holdup: 𝜐𝑟 = 𝑢∞𝑓(𝛼𝑔) ,

(11)

where 𝑢∞ is the terminal velocity of a bubble in an infinite medium. Several researchers have proposed different correlations for the dependency of the relative velocity on the terminal velocity and gas holdup. Some of these equations are summarized in Table 5. All existing correlations were tested, and it was found that the equation proposed by Wallis with n = 2 yields an accurate prediction compared with the other equations. Therefore, Equation (11) can be reformulated as follows: 𝜐𝑟 = 𝑢∞(1 ― 𝛼𝑔) ,

(12)

Table 5: Existing correlations for the relative velocity. Author(s)

Proposed correlation

Richardson and Zaki (1954)

𝜐𝑟 = 𝑢∞(1 ― 𝛼𝑔)2.39

Marrucci

5 ( ) 𝜐𝑟 = 𝑢∞ 1 ― 𝛼𝑔 (1 ― 𝛼𝑔 3)

Davidson and Harrison (1966)

𝜐𝑟 = 𝑢∞ (1 ― 𝛼𝑔)

Wallis (1969)

𝜐𝑟 = 𝑢∞(1 ― 𝛼𝑔)𝑛 ― 1

n=2 for small bubbles, n=0 for large bubbles

In view of the significant contribution of the terminal velocity in determining the overall hydrodynamics of the system, the bubble rise velocity in a system and possible bubble size need to be determined. As the terminal velocity and bubble diameter vary with the system properties, many investigators have developed empirical, semi-empirical, and theoretical correlations. Summaries of the available correlations for the terminal velocity and bubble diameter/volume are presented in Tables 6 and 7, respectively. We utilized different combinations of terminal velocity and bubble diameter correlations to develop a model that is capable of estimating the void fraction over a broad range of flow conditions. The model proposed by Krishna et al. (1999) was selected for the terminal velocity, because it provides accurate predictions in comparison with the other models. The authors developed the model to estimate the rise velocity of a swarm of bubbles in bubble columns: 19

𝑢∞ = 0.71 𝑔𝑑𝑏(𝑆𝐹)(𝐴𝐹) ,

(13)

where AF is the acceleration factor, which considers the effect of the interaction between a bubble and the wake of the bubble preceding it. The AF for both low- and high-viscosity liquids can be given by Equation (14): 𝐴𝐹 = 2.25 + 4.09(𝑗𝑔 ― 𝑈𝑡𝑟𝑎𝑛𝑠) ,

(14)

Furthermore, SF is the scale correction factor, which accounts for the influence of the column diameter. 𝑆𝐹 = 1

𝑓𝑜𝑟

𝑑𝑏 𝐷𝐻

< 0.125

( )

𝑆𝐹 = 1.13 𝑒𝑥𝑝 ―

𝑆𝐹 = 0.496

𝐷𝐻 𝑑𝑏

𝑑𝑏

(15)

𝑓𝑜𝑟 0.125 <

𝐷𝐻

𝑑𝑏 𝐷𝐻

< 0.6

𝑑𝑏 𝑓𝑜𝑟 > 0.6 𝐷𝐻

(16)

(17)

Here, 𝑑𝑏 is the bubble diameter, which can be estimated using Akita and Yoshida’s (1974) formula (Eq. 18) for regimes with a small bubble size, including bubbly, churn-turbulent (churn flow), and annular flow regimes. For slug/cap-turbulent regimes, which are characterized by large bubble diameters, the correlation proposed by Krishna et al. (1999) (Eq. 19) yields the best estimation: ―0.5

―0.12

( ) ( ) ( )

𝑑𝑏 = 26𝐷𝐻

𝐷2𝐻𝑔𝜌𝑙 𝜎

𝑔𝐷3𝐻 𝜗2𝑙

𝑗𝑔

𝑔𝐷𝐻

―0.12

,

𝑑𝑏 = 0.069(𝑗𝑔 ― 𝑈𝑡𝑟𝑎𝑛𝑠)0.376,

(18) (19)

20

Table 6: Existing correlations for bubble terminal velocity. Researchers

Rise velocity

Remarks

∆𝜌𝑔𝑑2𝑏

Pure gasses and clean liquids

Stokes (1851)

𝑢∞ =

Davies and Taylor (1950)

𝑢∞ = 0.707 𝑔𝑑𝑏

Haberman & Morton (1956)

𝑢∞ =

Mandelson (1967)

𝑢∞ =

𝑢∞ = 𝑢∞ = Peebles and Garber (1953)

18𝜇𝑙

𝑑𝑏 < 0.7 𝑚𝑚

∆𝜌𝑔𝑑2𝑏

viscosity and surface tension

3𝜇𝑙 + 3𝜇𝑔

2𝜎 𝑔𝑑𝑏 + 𝑑𝑏𝜌𝑙 2

1.4 𝑚𝑚 < 𝑑𝑏 < 6 𝑚𝑚 For intermediate–large bubbles in pure liquids

𝑅𝑒 < 2

𝑔𝜌𝑙𝑑2𝑏 18𝜇𝑙

3.1𝑀𝑜 ―0.25 < 𝑅𝑒

Lehrer(1976)

𝑢∞ = 𝑢∞ =

4.02𝑀𝑜 ―0.214 < 𝑅𝑒 < 3.1𝑀𝑜 ―0.25

1.18𝜎𝑔2 𝜌𝑙

𝑢∞ = 0.35 𝑔𝑑𝑏

Applicable for slugs with a clean interface

3𝜎 𝑔𝑑𝑏∆𝜌 + 𝑑𝑏𝜌𝑙 2𝜌𝑙

( ) 𝜇𝑙

𝜌𝑙𝑔

6 𝑚𝑚 < 𝑑𝑏

𝑀𝑜 ―0.149(𝐽 ― 0.857)

𝐽 = 0.94𝐻0.747

2 < 𝐻 < 59.3

0.441

59.3 < 𝐻

𝐽 = 3.42𝐻

()

𝜇𝑙 4 𝐻 = 𝐸𝑜𝑀𝑜 ―0.149 3 𝜇𝑤

(

Nickens et al. (1987)

𝑢∞ = 0.361 1 +

Jamilalahmadi et al. (1994)

𝑢∞ =

Rodrigu (2002)

𝑉=

―0.14

0.25

)

4.89 𝐸𝑜

𝑢∞,𝐻𝑢∞,𝑀

𝑢∞,𝐻 is Haberman and Morton’s correlation and

𝑢∞,𝐻2 + 𝑢∞,𝑀2

𝑢∞,𝑀 is Mandlson’s correlation

𝐹

V: Velocity number

12(1 + 0.018𝐹)0.75 ―0.273

0.03

( ) () ( ) ( ) ()

𝜎 𝜎3𝜌𝑙 𝑢∞ = 2.25 𝜇𝑙 𝑔𝜇4𝑙 Wilkinson et al. (1992)

+2.4

Krishna et al. (1999)

2 < 𝑅𝑒 < 4.02𝑀𝑜 ―0.214

0.136𝑔0.76𝜌0.52 𝑑1.28 𝑙 𝑏

Dumitrescu (1943)

Krishna & Ellenberger (1996)

For small bubbles considering the effect of the inherent circular flow within the bubble

)

(

18𝜇𝑙 2𝜇𝑙 + 3𝜇𝑔

10𝜇0.52 𝑙 2𝜎𝑔 𝑢∞ = 1.35 𝑑𝑏𝜌𝑙 𝑢∞ =

Clift et al. (1978)

For large bubbles without consideration of

𝑢∞ =

(𝑗𝑔 ― 𝑈𝑡𝑟𝑎𝑛𝑠)𝜇𝑙

𝜌𝑙

𝜌𝑔

0.757

𝜎

𝜎3𝜌𝑙

―0.077

𝜌𝑙

0.077

𝜌𝑔

𝑔𝜇4𝑙

1 𝐷0.18(𝑗𝑔 ― 𝑈𝑡𝑟𝑎𝑛𝑠)0.42 0.268 𝐻 𝑑𝑏

𝑆𝐹 = 1

𝑢∞ = 0.71 𝑔𝑑𝑏(𝑆𝐹)(𝐴𝐹)

𝐷𝐻

( )

𝐴𝐹 = 2.25 + 4.09(𝑗𝑔 ― 𝑈𝑡𝑟𝑎𝑛𝑠)

𝑆𝐹 = 1.13 𝑒𝑥𝑝 ―

21

𝑑𝑏

𝐷𝐻

< 0.125

0.125 <

𝑑𝑏 𝐷𝐻

< 0.6

𝑆𝐹 = 0.496

𝐷𝐻

𝑑𝑏

𝑑𝑏

𝐷𝐻

> 0.6

Table 7: Existing correlations for bubble diameter. Researchers

Bubble diameter/volume

Eversole et al. (1941)

𝜎 𝑑𝑏 = 81.18 𝑃0𝑔

Hughes et al. (1955)

𝜋𝑔𝑑ℎ𝜎 𝑉𝑏 = 1.82 𝑔∆𝜌 6

(1960)

𝑄 5 𝑉𝑏 = 1.772 3 𝑔 5

Kumar et al. (1970)

𝑉𝑏 =

Acharya et al. (1978)

𝑉𝑏 = 0.976

Tsuge et al. (1986)

𝑉𝑏 = 6.9

Akita & Yoshida (1974)

𝑑𝑏 = 26𝐷𝐻

Davidson

and

Schuler

Remarks

0.25

0.75

( ) ( ) 4𝜋 3

15𝜇𝑄 2𝜌𝑙𝑔 35

𝑄2 𝑔

( ) () ( ) ( ) ( ) 𝜎 𝜌𝑙

0.5

𝑗0.44 𝑔

𝐷2𝐻𝑔𝜌𝑙

―0.5

𝜎

𝑔𝐷3𝐻

―0.12

𝑗𝑔

𝜈2𝑙

―0.04

𝑔𝐷𝐻

―0.12

( ) ( ) ()

𝜎 𝑗𝑔𝜇𝑙 𝑔𝜌𝑙 𝜎

𝜎3𝜌𝑙

Wilkinson & Haring (1994)

𝑑𝑏 = 38.8

Krishna at al. (1999)

𝑑𝑏 = 0.069(𝑗𝑔 ― 𝑈𝑡𝑟𝑎𝑛𝑠)0.376

―0.12

𝜌𝑙

0.22

𝜌𝑔

𝑔𝜇4𝑙

Low- and high-viscous liquids

Many attempts have been made to predict the flow regime transition. A comprehensive study by Sheikh and al-Dahan (2007) reviewed existing efforts to understand the flow regime transition. By comparing different available correlations, the relations derived by Reilly et al. (1994) were utilized to calculate the transition velocity and gas holdup. The authors conducted experiments in 150-mm-diameter bubble columns using water and non-aqueous liquids as a continuous phase and different gases as a dispersed phase. They studied the effects of gas and liquid physical properties on the flow regime transition, and proposed the following correlations for the velocity and gas holdup at the transition point: 𝑈𝑡𝑟𝑎𝑛𝑠 =

1 2.84𝜌0.04 𝑔

𝜎0.12𝛼𝑡𝑟𝑎𝑛𝑠(1 ― 𝛼𝑡𝑟𝑎𝑛𝑠) ,

0.12 𝛼𝑡𝑟𝑎𝑛𝑠 = 0.59𝐵1.5(𝜌0.96 /𝜌𝑙) 𝑔 𝜎

0.5

(20)

,

(21)

where B=3.85, based on their experimental data. Substituting Equations (20) and (21) into Equation (13), the terminal velocity can be computed as follows: 22

(

(

𝑢∞ = 0.71 𝑔𝑑𝑏(𝑆𝐹) 2.25 + 4.09 𝑗𝑔 ―

1 2.84𝜌0.04 𝑔

))

𝜎0.12𝛼𝑡𝑟𝑎𝑛𝑠(1 ― 𝛼𝑡𝑟𝑎𝑛𝑠)

(22)

When Equations (10) and (11) are utilized, a new correlation for the drift velocity can be derived, where 𝑢∞ is obtained by Equation (22): 𝐶1 = (1 ― 1.18〈𝛼𝑔〉)(1 ― 〈𝛼𝑔〉)𝑢∞ , 3.4.2

(23)

New correlations for the drift-flux model and gas holdup

A new correlation for the drift velocity was developed in the previous section. To derive the drift flux model, a proper formula for the distribution parameter also needs to be selected. Shipley (1984) worked with a wide range of gas and liquid velocities (with a total flux of 0.1–10 m/s) in a 240-mm-diameter column, and reported a value of 𝐶0 equal to 1.2. This value is supported by several researchers working with different ranges of flow conditions (e.g. Clark and Flemmer, 1985; Joshi et al., 1998; Hills, 1976; Wallis, 1969; Zuber and Findlay, 1965). Because the distribution parameter for most two-phase flows is approximately 1.2, this value is selected hereafter to analyze the problem. Using the basic drift-flux model, the newly developed drift velocity obtained in the previous section, and the distribution parameter proposed by Shipley, a new correlation can be derived to predict the gas holdup for either large or small columns. Inserting 𝐶0 = 1.2 and 𝐶1 from Equation (23) into Equation (1) yields the following correlation:

〈𝑗𝑔〉 = 1.2〈𝑗〉 + (1 ― 1.18〈𝛼𝑔〉)(1 ― 〈𝛼𝑔〉)𝑢∞ , 〈𝛼𝑔〉

(24)

After some manipulation, the following equation is proposed for the gas holdup: 1.18

𝑢∞

〈𝑗𝑔〉

〈𝛼𝑔〉3 ― 2.18

𝑢∞

〈𝑗𝑔〉

(

〈𝛼𝑔〉2 + 1 +

〈𝑗𝑙〉 𝑢∞ 〈𝛼 〉 ― 1 = 0 , + 〈𝑗𝑔〉 〈𝑗𝑔〉 𝑔

)

(25)

The gas holdup can be calculated by solving the above equation under different flow conditions. 4 Comparison of the newly developed semi-empirical correlations with the experimental data The proposed models for the drift velocity and gas holdup in a large-diameter column are evaluated using the available experimental data. The performance of the developed correlation in Equation (23), is compared with the data captured by Hibiki and Ishii (2003), Shawkat et al. (2008), 23

and Shen et al. (2010) shown in Figure 8. The proposed correlation predicts the majority of the data points within ±20% error bands. 0.60 Predicted mean drift velocity, C1 (m/s)

Shen et al. (2010) Shawkat et al. (2008)

0.50

Hibiki & Ishii (2003) Hibiki & Ishii (1999)

0.40

Liu (1989) ±20% error bands

0.30 0.20 0.10 0.00 0.00

0.10 0.20 0.30 0.40 Experimental mean drift velocity, C1 (m/s)

0.50

0.60

Fig. 8. Performance of the proposed correlation for the drift velocity against experimental data.

To verify that the correlation proposed in Equation (25) is applicable for different flow patterns, column diameters, or fluid systems, its performance was examined against experimental void fraction data. The comparison between the predicted values and available experimental data listed in Table 3 is illustrated in Figure 9.

24

0.60

Hills (1976) Das et al. (1992) Hibiki & Ishii (2002) Shawkat (2008) Abdulkadir et al. (2010) Schlegel (2010) Schlegel (2012) Schlegel (2013) Rollbucsh et al. (2015) Besagni & Inzoli (2017) ±20% error bands

Predicted gas hodup, 〈�� 〉 (-)

0.50 0.40 0.30 0.20 0.10 0.00 0

0.1

0.2 0.3 0.4 Experimental gas hodup, 〈�� 〉 (-)

0.5

0.6

Fig. 9. Performance of the proposed correlation for the void fraction against available experimental data.

To investigate the quantitative performance of the proposed equation more comprehensively, the predictions of the void fraction for different column diameters, system pressures, and liquid phase properties are presented individually in Figures 10–12. It is observed that for a wide range of column diameters and system pressures the proposed correlation predicts the majority of the data within ±20% error bands, and within ±10% for different liquid phase dynamic viscosities. All the correlations developed for large and small columns in this study are summarized in Table 8. Table 8: Developed and recommended correlations. Proposed correlations

Remarks

𝐶1 = (1 ― 1.18〈𝛼𝑔〉)(1 ― 〈𝛼𝑔〉)𝑢∞

The mean drift velocity

1.18

𝑢∞

〈𝑗𝑔〉

〈𝛼𝑔〉3 ― 2.18

𝑢∞

〈𝑗𝑔〉

(

〈𝛼𝑔〉2 + 1 +

〈𝑗𝑙〉 𝑢∞ 〈𝛼 〉 ― 1 = 0 + 〈𝑗𝑔〉 〈𝑗𝑔〉 𝑔

)

Average gas holdup

〈𝑗𝑔〉 = 1.2〈𝑗〉 + (1 ― 1.18〈𝛼𝑔〉)(1 ― 〈𝛼𝑔〉)𝑢∞ 〈𝛼𝑔〉

The drift-flux model

𝑆𝐹 = 1

25

𝑓𝑜𝑟

𝑑𝑏 𝐷𝐻

< 0.125

(

(

𝑢∞ = 0.71 𝑔𝑑𝑏(𝑆𝐹) 2.25 + 4.09 𝑗𝑔 ―

1

𝜎0.12𝛼𝑡𝑟𝑎𝑛𝑠(1 ― 𝛼𝑡𝑟𝑎𝑛𝑠) 0.04

2.84𝜌𝑔

))

( )

𝑆𝐹 = 1.13 𝑒𝑥𝑝 ―

𝑆𝐹 = 0.496 ―0.5

―0.12

( ) ( ) ( )

𝑑𝑏 = 26𝐷𝐻

𝐷2𝐻𝑔𝜌𝑙 𝜎

𝑔𝐷3𝐻 𝜗2𝑙

𝑗𝑔

―0.12

1

𝐷𝐻

𝐷𝐻

𝑓𝑜𝑟 0.125 <

𝑑𝑏 𝐷𝐻

< 0.6

𝑑𝑏 𝑓𝑜𝑟 > 0.6 𝐷𝐻

𝑑𝑏

For bubbly flow, churn-turbulent, and annular flow

𝑔𝐷𝐻

(Proposed by Akita and Yoshida, 1974)

For Slug/ cap-turbulent flow

𝑑𝑏 = 0.069(𝑗𝑔 ― 𝑈𝑡𝑟𝑎𝑛𝑠)0.376

𝑈𝑡𝑟𝑎𝑛𝑠 =

𝑑𝑏

(Proposed by Krishna et al., 1999) Transition velocity and gas holdup

𝜎0.12𝛼𝑡𝑟𝑎𝑛𝑠(1 ― 𝛼𝑡𝑟𝑎𝑛𝑠) 0.04

(Proposed by Reilly et al., 1994)

2.84𝜌𝑔

0.5

0.8 0.6 0.4 0.2

Das et al. (1992) ±10% error bands

0 0

Predicted gas holdup, 〈�� 〉 (-)

1

(a)

D= 0.019 m

Predicted gas holdup, 〈�� 〉 (-)

1

0.2

0.4

0.6

0.8

Experimental gas holdup, 〈�� 〉 (-)

1 0.8 0.6 0.4

26

0.2

Hibiki & Ishii (2000) ±20% error bands

0 0

0.2 0.4 0.6 0.8 Experimental gas holdup, 〈�� 〉 (-)

0.8 0.6 0.4 0.2

Abdulkadir et al. (2010) ±20% error bands 0

1

(c)

D= 0.102 m

(b)

D= 0.067 m

0

1

Predicted gas holdup, 〈�� 〉 (-)

Predicted gas holdup, 〈�� 〉 (-)

0.12 𝛼𝑡𝑟𝑎𝑛𝑠 = 0.59𝐵1.5(𝜌0.96 /𝜌𝑙) 𝑔 𝜎

0.2 0.4 0.6 Experimental gas holdup, 〈�� 〉0.8 (-)

(d)

D= 0.152 m

0.8 0.6 0.4 0.2

Schlegle et al. (2013) ±20% error bands

0 1

1

0

0.2

0.4

0.6

0.8

Experimental gas holdup, 〈�� 〉 (-)

1

Fig. 10. Performance of the proposed void fraction correlation for different column diameters.

1

(e)

D= 0.203 m

Predicted gas holdup, 〈�� 〉 (-)

Predicted gas holdup, 〈�� 〉 (-)

1 0.8 0.6 0.4 0.2

Schlegel et al. (2013) ±20% error bands

0 0

0.2

0.4

0.6

0.8

Experimental gas holdup, 〈�� 〉 (-)

0.8 0.6 0.4 0.2

Schlegel et al. (2013) ±20% error bands

0 1

27

(f)

D= 0.304 m

0

0.2 0.4 0.6 Experimental gas holdup, 〈�� 〉 (-)0.8

1

Predicted gas holdup, 〈�� 〉 (-)

1

(a)

P= 100 kPa

0.8 0.6 0.4 0.2

Schlegel et al. (2012) ±20% error bands

0 0

Predicted gas holdup, 〈�� 〉 (-)

1

0.2 0.4 0.6 Experimental gas holdup, 〈�� 〉 (-)0.8

(b)

P= 180 kPa

0.8 0.6 0.4 0.2

Schlegel et al. (2010) ±20% error bands

0 0

0.2 0.4 0.6 Experimental gas holdup, 〈�� 〉 (-)0.8

1

Predicted gas holdup, 〈�� 〉 (-)

1

1

(c)

P= 280 kPa 0.8 0.6 0.4 0.2

Schlegel et al. (2010) ±20% error bands

0 0

0.2 0.4 0.6 Experimental gas holdup, 〈�� 〉 (-)0.8

1

Fig. 11. Performance of the proposed void fraction correlation for different system pressures.

28

0.6 0.4 0.2

Rollbucsh et al. (2015) ±10% error bands

1

0.2

0.4

0.6

0.8

Experimental gas holdup, 〈�� 〉 (-)

μl = 7.9 mPa.s

0.6 0.4 Xing et al. (2013) Besagni et al. (2017)

1

0.2

0.4

0.6

0.8 0.6 0.4

Xing et al. (2013) ±10% error bands 0

0.2

0.4

0.6

0.8

Experimental gas holdup, 〈�� 〉 (-)

0.6 0.4 0.2

Xing et al. (2013) ±10% error bands 0

0.2

0.4

0.6

1

(f)

μl = 40- 185 mPa.s

0.8 0.6 0.4 Das et al. (1992) Esmaeili (2015) ±20% error bands

0.2

0

0.2

0.4 0.6 Experimental gas holdup, 〈�� 〉0.8 (-)

Fig. 12. Performance of the proposed void fraction correlation for different liquid viscosities.

29

0.8

Experimental gas holdup, 〈�� 〉 (-)

0 1

1

(d)

μl = 20.1 mPa.s

1

(e)

0

0.4 0.6 Experimental gas holdup, 〈�� 〉0.8 (-)

0.8

1

Experimental gas holdup, 〈�� 〉 (-)

0.2

0.2

0

0.8

μl = 39.6 mPa.s

Schlegel et al. (2012) Yang & Fan (2003) ±20% error bands

0.2

0

±20% error bands 0

0.4

1

0.8

0

0.6

0

(c)

0.2

(b)

μl = 1.0 mPa.s

0.8

1

Predicted gas holdup, 〈�� 〉 (-)

0

Predicted gas holdup, 〈�� 〉 (-)

Predicted gas holdup, 〈�� 〉 (-)

0.8

0

Predicted gas holdup, 〈�� 〉 (-)

1

(a)

μl = 0.3 mPa.s

Predicted gas holdup, 〈�� 〉 (-)

Predicted gas holdup, 〈�� 〉 (-)

1

1

The accuracy of the proposed gas holdup correlation is also assessed based on the statistical parameters such as mean relative error (𝑒𝑦) and the standard deviation (𝑠) between the experimental and predicted values of gas holdup. The mean relative error and the standard deviation are determined using Equations (26) and (27), respectively. 1 𝑒𝑦 = × 𝑁

𝑁

∑| 1

𝑁

∑ [|

1 𝑠= 𝑁―1

|

𝛼𝑔,𝑒𝑥𝑝 ― 𝛼𝑔,𝑝𝑟𝑒𝑑 𝛼𝑔,𝑒𝑥𝑝

,

(26)

| ]

𝛼𝑔,𝑒𝑥𝑝 ― 𝛼𝑔,𝑝𝑟𝑒𝑑 𝛼𝑔,𝑒𝑥𝑝

1

2

― 𝑒𝑦 ,

(27)

The comprehensive datasets listed in Table 3, are divided into three different ranges of gas holdup based on the associated two-phase flow patterns. The first range of gas holdup of 0 < 𝛼𝑔 ≤ 0.25 is based on bubbly flow regime, the second range is based on the churn turbulent/slug flow pattern that approximately occupies a gas holdup range of 0.25 < 𝛼𝑔 ≤ 0.75. Finally, the last division is related to the annular flow pattern that is in a range of 0.75 < 𝛼𝑔 ≤ 1.0. The quantitative performance of different existing correlations for gas holdup is represented in Table 9. Table 9: Statistical comparison of the performance of the proposed model against the existing correlations. Gas holdup range

0 < 𝛼𝑔 ≤ 0.25

0.25 < 𝛼𝑔 ≤ 0.75

0.75 < 𝛼𝑔 ≤ 1.0

(705 Data points)

(1025 Data points)

(149 Data points)

Correlations

𝑒𝑦

s

Present study

0.0003

0.0097

Hills (1976)

0.0004

Shipley (1984)

% of Data

% of Data

s

% of Data

𝑒𝑦

s

79

0.0001

0.0067

93

0.0004

0.0058

100

0.0127

40

0.0002

0.0070

64

0.0013

0.0184

54

0.0003

0.0097

65

0.0001

0.0058

90

0.0007

0.0093

100

Kataoka and Ishii (1987)

0.0003

0.109

49

0.0001

0.0071

91

0.0006

0.0078

100

Clark and Flemmer (1985)

0.0004

0.0138

48

0.0002

0.0056

67

0.0022

0.0267

0

Clark and Flemmer (1986)

0.0003

0.0105

63

0.0002

0.0076

55

0.0027

0.0330

0

Kawanishi et al. (1990)

0.0006

0.0164

15

0.0001

0.0050

70

0.0007

0.0096

98

Hibiki and Ishii (2003)

0.0003

0.0110

46

0.0001

0.0083

89

0.0006

0.0078

100

Ishii (1977)

0.001

0.0444

35

0.0003

0.0167

49

0.002

0.0421

64

𝑒𝑦 ≤ 20%

30

𝑒𝑦 ≤ 20%

𝑒𝑦

𝑒𝑦 ≤ 20%

For the first range of gas holdup, the proposed correlation predicts 79% of the data points within ±20% error bands. For this range, the developed model gives the lowest mean relative error (0.0097). For the second range of gas holdup, approximately representing churn turbulent/slug flow regime, the proposed correlation predicts 93% of data points within ±20% error bands and is comparable to the performance of Shipley (1984), and Kataoka and Ishii (1987). For 0.75 < 𝛼𝑔 ≤ 1.0, the suggested correlation in this study yields the highest accuracy by predicting 100% of data points within ±20% error bands. 4

Conclusion

In view of the practical importance of the drift-flux model for the analysis of two-phase flows, a comprehensive investigation of different flow conditions was conducted, the distribution parameter and drift velocity were studied, and a new approach for calculating the drift velocity and gas holdup was presented. The results can be summarized as follows: 1- Based on the experimental local flow parameters obtained by Shawkat et al. (2008) and Shen et al. (2010), the distribution parameter and drift velocity were calculated directly from their definitions, and the results were compared with those of several existing models proposed to estimate drift-flux parameters. The comparisons show that the models of Ishii and of Clark and Flemmer cannot predict the distribution parameter. Furthermore, Hibiki and Ishii’s correlation can only predict the correct values at low velocities, and overestimates when

〈𝑗𝑔〉 〈𝑗〉

is greater than 0.2.

2- The comparison between the estimations using the existing drift velocity correlations and the collected experimental data showed that none of the selected correlations can estimate the drift velocity. 3- Furthermore, the drift-flux parameters in small- and large-diameter columns were compared, and a significant difference was observed. The formation of cap bubbles and the occurrence of liquid recirculation can affect the flow characteristics in large-diameter columns. 4- New correlations for the drift velocity and gas holdup were derived. As the two-phase flow characteristics could be influenced by the bubble size, two different correlations were selected, for small and large bubble diameters. The derived correlations yield practically

31

reasonable predictions for an extensive range of column diameters, system pressures, and fluid properties. 5- As the majority of datasets utilized in this study are based on Newtonian liquids, with the exception of some data of Das et al., it is recommended to verify the accuracy of the model against more diversified fluid systems with non-Newtonian liquids. According to the overall performance of the proposed correlation, it is recommended for utilization to predict void fractions in the ranges listed in Table 3. Acknowledgment We gratefully acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC).

32

NOMENCLATURE AF

Acceleration factor (-)

C0

Distribution parameter (-)

C1

Drift velocity (m/s)

C1,B

Ishii’s Drift velocity (m/s)

C1,P

Kataoka & Ishii’s Drift velocity (m/s)

𝐶𝐷

Drag force coefficient of a bubble in the swarm (-)

𝐶𝐷∞

Drag force coefficient of an isolated bubble (-)

𝐶𝛼

Void fraction covariance (-)

𝐶𝜇

Constant in k-ε model (-)

𝐶𝜀1, 𝐶𝜀2

Constants in k-ε model (-)

𝑑𝑏

Bubble diameter (m)

𝐷𝐻

Hydraulic diameter of the flow channel (m)

𝐷𝑠𝑚

Sauter mean diameter (m)

𝑒𝑦

Mean relative error (-)

𝐹𝑘𝑚

Total interfacial forces (N/m3)

𝐹𝐷

Drag force (N/m3)

𝐹𝐿

Lift force (N/m3)

𝐹𝑉𝑀

Virtual mass force (N/m3)

𝐹𝑊𝐿

Wall lubrication force (N/m3)

g

Gravity (m/s2)

G

Generation of turbulent kinetic energy (J/m3.s)

𝑗

Mixture volumetric flux (m/s)

𝑗𝑔

Superficial gas velocity (m/s)

𝑗𝑙

Superficial liquid velocity (m/s)

k

Turbulent kinetic energy (m2/s2)

N

Number of data points (-)

P

Pressure (Pa)

𝑃𝐷

Correction factor of drag coefficient (-)

Re

Reynolds number (-)

33

s

Standard deviation (-)

SF

Scale correction factor (-)

u

Velocity (m/s)

𝑢𝑙

Local liquid velocity (m/s)

𝑢𝑔

Local gas velocity (m/s)

𝑢∞

Terminal velocity (m/s)

𝑈𝑡𝑟𝑎𝑛𝑠

Transition velocity (m/s)

𝑉𝑠

Slip velocity (m/s)

Greek letters α

Phase void fraction (-)

α𝑡𝑟𝑎𝑛𝑠

Transition void fraction (-)

𝜇𝑇

Turbulent viscosity (Pa.s)

𝜇𝑒𝑓𝑓

Effective viscosity (Pa.s)

𝜎𝑘

Prandtle number for turbulent kinetic energy (-)

𝜎𝜀

Prandtle number for turbulent energy dissipation rate (-)

𝜏𝑘

Shear stress of phase k (Pa)

µ

Molecular viscosity (Pa.s)

αg

Local gas holdup (-)

αg,exp

Experimental gas holdup (-)

αg,pred

Predicted gas holdup (-)

ε

Turbulent energy dissipation rate (m2/s3)

𝜐𝑔𝑗

Drift velocity of gas phase (m/s)

𝜐𝑔

Gas phase velocity (m/s)

𝜐𝑙

Liquid phase velocity (m/s)

𝜐𝑟

Relative velocity between phases (m/s)

ν

Kinematic viscosity (m2/s2)

ρ

Density (kg/m3)

σ

Surface tension (N/m)

Subscripts g

Gas phase

34

k

Phase index

l

Liquid phase

p

Primary phase in Schiller and Naumann model

q

Secondary phase in Schiller and Naumann model

tp

Two phase

35

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Highlights 

The existing correlations for 𝐶0 and 𝐶1 were re-evaluated against the published experimental data.



The values of the distribution parameter and drift velocity in small- and large-diameter columns were compared.



Semi-empirical correlations for the drift velocity and gas holdup were developed.



The proposed correlations were verified against a wide range of experimental data.

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Declaration of interests

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

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

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