Numerical simulations of gas–liquid mass transfer in bubble columns with a CFD–PBM coupled model

Chemical Engineering Science 62 (2007) 7107 – 7118 www.elsevier.com/locate/ces

Numerical simulations of gas–liquid mass transfer in bubble columns with a CFD–PBM coupled model Tiefeng Wang ∗ , Jinfu Wang Beijing Key Laboratory of Green Reaction Engineering and Technology, Department of Chemical Engineering, Tsinghua University, Beijing 100084, China Received 18 April 2007; received in revised form 9 August 2007; accepted 13 August 2007 Available online 22 August 2007

Abstract Gas–liquid mass transfer in a bubble column in both the homogeneous and heterogeneous flow regimes was studied by numerical simulations with a CFD–PBM (computation fluid dynamics–population balance model) coupled model and a gas–liquid mass transfer model. In the CFD–PBM coupled model, the gas–liquid interfacial area a is calculated from the gas holdup and bubble size distribution. In this work, multiple mechanisms for bubble coalescence, including coalescence due to turbulent eddies, different bubble rise velocities and bubble wake entrainment, and for bubble breakup due to eddy collision and instability of large bubbles were considered. Previous studies show that these considerations are crucial for proper predictions of both the homogenous and the heterogeneous flow regimes. Many parameters may affect the mass transfer coefficient, including the bubble size distribution, bubble slip velocity, turbulent energy dissipation rate and bubble coalescence and breakup. These complex factors were quantitatively counted in the CFD–PBM coupled model. For the mass transfer coefficient kl , two typical models were compared, namely the eddy cell model in which kl depends on the turbulent energy dissipation rate, and the slip penetration model in which kl depends on the bubble size and bubble slip velocity. Reasonable predictions of kl a were obtained with both models in a wide range of superficial gas velocity, with only a slight modification of the model constants. The simulation results show that CFD–PBM coupled model is an efficient method for predicting the hydrodynamics, bubble size distribution, interfacial area and gas–liquid mass transfer rate in a bubble column. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Gas–liquid mass transfer; Population balance model (PBM); Bubble size distribution; Bubble breakup and coalescence

1. Introduction Gas–liquid/slurry reactors have received much attention for their wide applications in chemical, petrochemical and environmental processes, especially the gas-to-liquid processes of Fischer–Tropsch synthesis, and methanol and dimethyl ether syntheses. For reactions occurring in a gas–liquid/slurry reactor, the gas–liquid mass transfer rate is an important factor in determining the production rate. In the design and scale-up of the gas–liquid/slurry reactor, gas–liquid mass transfer is usually described in terms of a volumetric mass transfer coefficient. However, gas–liquid mass transfer is very complex because it depends on the turbulence, interfacial phenomena and bubble behaviors. Early works focused on the measurement ∗ Corresponding author. Tel.: +86 10 627 97490; fax: +86 10 627 72051.

E-mail address: wangtf@flotu.org (T. Wang). 0009-2509/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2007.08.033

and correlation of the volumetric mass transfer coefficient. Recent works have more focused on the theoretical or predictive models to give a better understanding of the mass transfer mechanism (Garcia-Ochoa and Gomez, 2004). Most results on the gas–liquid mass transfer behavior available in the literature were limited to the volumetric mass transfer coefficient, kl a. However, kl a is a global parameter and insufficient for understanding the mass transfer mechanism. The separation of kl and a allows in identifying which one of kl and a controls the mass transfer rate (Bouaifi et al., 2001; Yang et al., 2001; Garcia-Ochoa and Gomez, 2004), and can provide an insight into the mass transfer mechanism. There are also efforts in separation of kl and a in the computational fluid dynamics (CFD) simulations. Darmana et al. (2005) used a three-dimensional discrete bubble model to investigate the complex behavior involving hydrodynamics, mass transfer and chemical reactions in a gas–liquid bubble column reactor. Talvy et al. (2007)

7108

T. Wang, J. Wang / Chemical Engineering Science 62 (2007) 7107 – 7118

studied the physical modeling and numerical simulation of mass transfer in two-phase bubbly flow in an airlift internal loop reactor. The specific interfacial area a can be calculated from the gas holdup and bubble size distribution. In a gas–liquid/slurry reactor, the bubble size distribution plays an important role in the hydrodynamics and mass transfer behaviors. It determines the bubble rising velocity and gas residence time, and governs the gas holdup and gas–liquid interfacial area (Shimizu et al., 2000). The population balance model (PBM) is an effective method for calculating the bubble size distribution in a gas–liquid flow, and has received an increasing attention in recent years (Millies and Mewes, 1999; Colella et al., 1999; Lehr and Mewes, 2001; Olmos et al., 2001; Lehr et al., 2002; Ramkrishna and Mahoney, 2002; Chen et al., 2005; Wang et al., 2005, 2006). The key issue in using the PBM is the selection of reasonable bubble coalescence and breakup models. Wang et al. (2005) compared four typical bubble coalescence and breakup modes, and found that the calculated bubble size distributions were quite different when different bubble coalescence and breakup models were used. The results show that it is very important to include multiple bubble breakup and coalescence mechanisms and choose reasonable models for them to get a reliable prediction of the bubble size distributions in a wide range of superficial gas velocity. To predict the mass transfer coefficient kl , many approaches have been developed to be more and more realistic toward capturing the interfacial mass transfer in last several decades (Kulkarni, 2007). Classical mass transfer models include the two-film model (Lewis and Whitman, 1924), penetration theory (Higbie, 1935), surface renewal model (Danckwerts, 1951), film penetration model (Toor and Marchello, 1958) and eddy cell model (Lamont and Scott, 1970). Some more complex models were also proposed in recent years, e.g., the surfacerenewal-stretch model (Jajuee et al., 2006), and model based on the vertical velocity gradient at the interface (Xu et al., 2006). More comprehensive summary of the mass transfer models is referred to the publications by Jajuee et al. (2006) and by Kulkarni (2007). The present work aims to study the gas–liquid mass transfer behavior in a bubble column by numerical simulations with a CFD–PBM coupled model. Great efforts have been devoted by several authors to simulate the gas–liquid mass transfer by CFD coupled with the PBM. Burris et al. (2002) developed a simple model that predicted the oxygen transfer in an airlift reactor based on discrete bubbles, but did not model the detailed hydrodynamics. Dhanasekharan et al. (2005) modeled the oxygen transfer in bioreactors using the PBM and CFD. Due to the simplified bubble breakup and coalescence models used in their work, the accuracy of predictions of the mass transfer coefficient was not satisfactory. Furthermore, the ability of this approach to predict the mass-transfer rate in the typical heterogeneous flow regime was not investigated. In the CFD–PBM coupled model of the present work, multiple mechanisms for bubble coalescence and breakup were considered, and the effects of the bubble size on the interphase forces were also taken into account. Previous work showed that this CFD–PBM

coupled model had a good ability to predict the hydrodynamics in both the homogeneous and heterogeneous regimes. In the framework of the CFD–PBM coupled model, the mass transfer model was included to calculate the mass transfer coefficient. Different mass transfer models were compared. The complex effects of the bubble size distribution and turbulent energy dissipation rate on the mass transfer coefficient were discussed in detail.

2. Mathematical modeling Fig. 1 shows the algorithm for simulating the gas–liquid mass transfer in the framework of the CFD–PBM coupled model developed in our previous work (Wang et al., 2006). The flow field, gas holdup and turbulent energy dissipation rate were calculated by CFD, which were then used to solve the PBM. The bubble size distribution calculated from the PBM was used to identify the flow regime, and calculate the interphase forces and turbulence modification for closing the two-fluid model. In this work, multiple mechanisms for bubble coalescence, including coalescence due to turbulent eddies, different bubble rise velocities and bubble wake entrainment, and for bubble breakup due to eddy collision and instability of large bubbles were considered, as shown in Fig. 2. The volumetric mass transfer coefficient kl a was commonly used to describe the interfacial mass transfer rate, and the influence of the concentration distribution near the gas–liquid interface was taken into account in the mass transfer model. The averaged transport equations for species concentration are required only when species concentration distribution is of interest. In such transport equations, kl a is also the key parameter for calculating the source terms resulted from interfacial mass transfer. In this work, the mass transfer coefficient kl and the interfacial area a were modeled separately to better understand

CFD Two-fluid model

Gas holdup Flow field Dissipation rate

a

Regime identification Gas-liquid interaction Turbulence modification

kl a Bubble breakup and coalescence

kl

Bubble size distribuiton

PBM Population balance model

Fig. 1. Simulation of gas–liquid mass transfer in the framework of the CFD–PBM coupled model.

T. Wang, J. Wang / Chemical Engineering Science 62 (2007) 7107 – 7118

7109

Fig. 2. Bubble breakup and coalescence due to different mechanism.

Table 1 Governing equations of the two-fluid model Models

Equations

Mass conservation

∇ · (u)i = 0, i = g, l

Momentum conservation

∇ · (uu)i = −i ∇P  + ∇ · (eff (∇u + ∇uT ))i + Fi,j + ()i g, i = g, l

k–ε turbulence model for the liquid phase

k Equation

∇ · (l l kl ul ) = ∇ · (l (lam,l + (t,l + tb )/k )∇kl ) + l (Gk,l − l εl )

ε Equation

∇ · (l l εl ul ) = ∇ · (l (lam,l + (t,l + tb )/ε )∇εl ) + l kεll (Cε1 Gk,l − Cε2 l εl )

Generation rate and eddy viscosity

Gk,l = eff,l ∇ul · (∇ul + (∇ul )T ) − 23 ∇ · ul (eff,l ∇ · ul + l kl ), t.l = C (l kl2 /εl )

Turbulence modification

eff,l = lam,l + t,l + tb , tb = Cb l g dbs |ug − ul | kl,t = kl + kl,g , εl,t = εl + εl,g , kl,g = 21 g Cvm u2slip , εl,g = g guslip

t,g = t,l g /l

Turbulent viscosity of the gas phase Drag force

FD =

M  i=1

Di kb,large fi g l 3C 4dbi (ug − ul )|ug − ul |, kb,large = max(1.0, 50.0g fb,large )

0.687 ), 8 Eo/(Eo + 4)] CDi = max[24Re−1 i (1 + 0.15Rei 3

Virtue mass force Interphase forces

Transverse lift force

D (ug − ul ), CV M = 0.25 FV M = g l CV M Dt ⎧ ⎨ min(0.288 tanh(0.121Rei ), f (Eoi )), Eoi < 3.4 M  jul FL = − fi CLi g l (ug − ul ) jr , CLi = f (Eoi ), 3.4 < Eoi < 5.3 ⎩ i=1 −0.29, Eoi > 5.3 



f (Eoi ) = 0.00925Eoi3 − 0.0995Eoi2 + 1.088 j

Turbulent dispersion force

FT D = −CT D g l kl jr , CT D = 0.7

Wall lubrication force

FW = −

M  i=1

1 2 fi CW i g dbi [(R

the gas–liquid mass transfer. The gas–liquid interfacial area a was calculated from the gas holdup and bubble size distribution. Two widely used mass transfer models, namely the slip penetration model (Higbie, 1935) and the eddy cell model (Lamont (and Scott, 1970), were compared. Many parameters may affect the mass transfer coefficient, including the bubble size distribution, bubble slip velocity, turbulent energy dissipation rate of the liquid phase and bubble coalescence and breakup rates. These complex factors account for the different mass transfer behaviors in the heterogeneous regime reported by Krishna and van Baten (2003). All these influencing parameters were quantitatively included in the CFD–PBM coupled model.

− r)−2 − (R + r)−2 ]l (ug − ul )2 , CW i = 0.1

2.1. CFD–PBM coupled model The equations that govern the hydrodynamics of a two-phase flow were extensively discussed in the literature. The main difficulty in the two-fluid model is the modeling of interphase forces and turbulence. The time-averaged two-fluid model for steady-state flows used in this work was listed in Table 1. The formulation of turbulence is based on the two-time-constants model proposed by de Bertodano et al. (1994), in which the total turbulent energy dissipation rate includes two parts: one is the shear induced and the other is bubble induced. To couple the PBM with the two-fluid model, the population balance equation is expressed in terms of fi , which is the

7110

T. Wang, J. Wang / Chemical Engineering Science 62 (2007) 7107 – 7118

volume fraction of bubble group i in the gas holdup g (Wang et al., 2006). The final equation is j (g fi ) + ∇ · (g ub fi ) jt j k

=



j, k

1 1 − j,k 2



vi−1 (vj + vk )vi+1 × i,j k cj,k g fj g fk vi /vj /vk −  g fi

M 

ci,j g fk /vk +

k=1

M 

ni,k bk g fk vi /vk − bi g fi

k=i

(1) To solve Eq. (1), the bubble coalescence rate c(v, v  ), bubble breakup rate b(v) and daughter bubble size distribution (v, v  ) are needed. The total coalescence rate is calculated as the sum of the rates due to turbulent eddies ct , different bubble rise velocities cu and bubble wake entrainment cw . The total bubble breakup rate is calculated as the sum of the rates due to eddy collision bt and instability of large bubbles bl . In most works reported in the literature, only the mechanism due to turbulent eddies was considered for bubble coalescence. This simplification is reasonable only at low-to-medium superficial gas velocities since coalescence due to turbulent eddies is the main mechanism in such conditions. However, at a high superficial gas velocity, bubble coalescence due to wake entrainment becomes significant for the formation of large bubbles. Furthermore, bubble coalescence due to different bubble rise velocity cannot be ignored when the bubble rise velocity is sensitive to the bubble size. The models for bubble coalescence and breakup are listed in Table 2. The details of the CFD–PBM coupled model are referred to our previous work (Wang et al., 2006). Note that the assumption of spherical bubbles was used in this work. In a gas–liquid system, even a single gas bubble has complex shapes depending on its size. Only very small bubbles are spherical, bubbles of several millimeters are ellipsoidal and bubbles of larger size are sphere-cap. In addition, the bubbles in a bubble column are much more complex, due to the bubble–bubble interaction and bubble–turbulence interaction. Accurate treatment of the bubble shape at high superficial gas velocities is too difficult and needs further studies and more experimental data. 2.2. Mass transfer model When the PBM is used to calculate the bubble size distribution, the bubbles are divided into a number of groups according to their sizes. Denote the number of bubbles with volume between vi and vi+1 as Ni , then the interfacial area a is  2 a= db,i Ni . (2) i

Great efforts have been devoted to understand the mechanics of gas–liquid mass transfer, and several models were proposed. One of the mostly used models is Higbie’s penetration model (Higbie, 1935). According to this theory, the liquid-phase mass transfer coefficient for a bubble with a mobile surface is given by Dl , (3) kl = 2 e where e is the exposure time. Higbie (1935) recommended that the exposure time e for gas bubbles could be estimated as db /uslip , where db is the bubble diameter and uslip is the bubble slip velocity. This results in the slip penetration model: Dl uslip , (4) kl = 2F dbe where F is a model constant. Combination of Eqs. (2) and (4) leads to  3/2 1/2 kl a = 2F ( Dl )1/2 db,i uslip,i Ni . (5) i

Danckwerts (1951) made a refinement of the penetration model. He assumed that the average surface renewal rate resulted from exposure to eddies with variable contact time, and obtained the surface renewal model:

(6) kl = Dl s, where s is the fractional rate of surface-element replacement. Lamont and Scott (1970) assumed that the small scales of turbulent motion, which extend from the smallest viscous motions to the inertial ones, affected the mass transfer. They calculated s through the Kolmogorov theory of isotropic turbulence, and obtained the eddy cell model as follows: 1/2

kl = KD l

 ε 1/4 l



,

(7)

where K is a model parameter. Combination of Eqs. (7) and (2) gives another equation for calculating the volumetric mass transfer coefficient:   1/2 εl 1/4 2 kl a = K Dl db,i Ni . (8)  i

Both the slip penetration model and the eddy cell model were widely used in the literature. However, these two models seem to be contradictory because the eddy cell model predicts a dependence of kl on the turbulent energy dissipation rate εl , while the slip penetration model predicts a dependence of kl on the bubble diameter db . Some experimental results support the slip penetration model (Alves et al., 2006), while some others support the eddy cell model (Vasconcelos et al., 2003; Linek et al., 2004). In this work, both the slip penetration model and the eddy cell model were tested for a detailed comparison of them.

T. Wang, J. Wang / Chemical Engineering Science 62 (2007) 7107 – 7118

7111

Table 2 Models of bubble breakup and coalescence Items Bubble breakup due to turbulent eddies

Equations Breakup rate Daughter bubble size distribution Complement equations

0.5 bt (d) = 0 b(fv |d)df −1

v (fv , d) = 2b(fv |d) 01 b(fv |d) dfv b(fv |d) = 0.923(1 − d )nε 1/3

db

min

Pb (fv |d, )( + d)2 −11/3 d , Pb (fv |d, ) =

Pe (e( )) = (1/e( )) exp(−e( )/e( )), e( ) =

∞ 0

Pb (fv |d, e( ), )Pe (e( )) de( )

3 1 2 min((21/3 − 1), e( )/( d 2 )), 12  c u¯ , cf,max = −1 (fv,max − fv,min ) , fv,max − fv,min   and fv,min

fv,min = ( 3 /(6e( )d))3 , Pb (fv |d, e( ), ) =

0

< fv < fv,max

else

m) bl (d) = b∗ (d − dc2 )m /((d − dc2 )m + dc2 (fv , d) = 2(0.5)

Bubble breakup due to instability of large bubbles

Breakup rate Daughter bubble size distribution

Coalescence rate due to turbulent eddies: c t = t P t

Collision rate

Coalescence efficiency

Pt (di , dj ) = exp(−(0.75(1 + 2ij )(1 + 3ij ))1/2 (g /l + )−1 (1 + ij )−3 W eij ), ij = di /dj

Coalescence rate due to different rise velocity: cu = u P u

Collision rate Coalescence efficiency

u (di , dj ) = 41 g,max (g,max − g )−1 2ε1/3 (di + dj )2 (di2/3 + dj2/3 )1/2

Coalescence due to bubble wake: c w = w P w

Collision rate

w (di , dj ) = 12.0di2 u¯ slip,i , u¯ slip,i = 0.71 gd i

 = (dj − dc /2)6 /((dj − dc /2)6 + (dc /2)6 ) for dj  dc /2;  = 0 for dj < dc /2 with dc = 4 /(g ) 1/2 Pw (di , dj ) = exp(−0.46l ε 1/3 −1/2 (di dj /(di + dj ))5/6 )

√

t (di , dj ) = 41 g,max (g,max − g )−1 ij 2ε1/3 (di + dj )2 (di2/3 + dj2/3 )1/2 

1/3 m /(l m m 2 2 ij = lbt,ij bt,ij + hb,ij ), lbt,ij = lbt,i + lbt,j ,lbt = 0.89db , hb,ij = (Ni + Nj )

Coalescence efficiency

1/2

√

Pu (di , dj ) = 0.5



3. Simulation details A bubble column of 0.19 m diameter and 2.5 m height was simulated to compare with the measured hydrodynamic parameters reported by Degaleesan et al. (2001). The simulations were two-dimensional. Uniform inlet boundaries were used for both the gas and the liquid phases. The boundary conditions for the inlet were as follows (Wang et al., 2006): ug,in = Ug /¯g,in , kl,in = 0.004u2l,in ,

ul,in = Ul,D /(1 − ¯ g,in ),

g,in = ¯ g,in ,

3/4 3/2 εi,in = C kin /(0.07D).

Fully developed flow conditions were used at the top and symmetrical conditions were applied at the center axis for all the variables. The wall function was used for the liquid phase and the non-slip condition for the gas phase. The bubble size was divided into 30 sections using the geometry method (vi+1 = vi r) with the smallest bubble volume v1 = 1.0 × 10−10 m3 and factor r = 1.7. The grid used was 10 × 0.76 cm + 10 × 0.19 cm in the radial direction, and 10 × 3 cm + 20 × 10.5 cm in the axial direction. The CFD–PBM coupled model was implemented in CFX4.4. The PBM was added by defining 30 scalar equations for the population balance equations. The source terms resulted from

bubble breakup and coalescence were added by modifying the user Fortran routine USRSRC. The effects of the bubble size distribution on the drag force and other interphase forces were included in the user Fortran routines USRIPT and USRBF, respectively. The independence of the solution on the grid and group division was tested in our previous work (Wang et al., 2006).

4. Results and discussion 4.1. Gas holdup The hydrodynamics of the bubble columns are characterized by different flow regimes, namely the homogeneous and heterogeneous regimes, as shown in Fig. 3. The variation in the average gas holdup is different in different regimes (Degaleesan et al., 2001; Shaikh and Al-Dahhan, 2005). In the homogeneous regime, the gas holdup increases almost linearly with increasing superficial gas velocity. This increase in the gas holdup becomes less pronounced in the heterogeneous regime. The reason is that in the heterogeneous regime, large bubbles resulted from coalescence have notable bubble-wake attraction effect (Hibiki and Ishii, 2000; Wang et al., 2006).

7112

T. Wang, J. Wang / Chemical Engineering Science 62 (2007) 7107 – 7118

Fig. 3. Flow regimes in a bubble column (from Zhang et al., 1997).

0.60

0.4

HoR

Ug, m/s 0.02 0.08 0.12 0.16

0.45

HeR

αg

αg

0.3

data of Degaleesan CFD-PBM model total gas holdup small bubbles large bubbles

0.2

0.30

0.15

0.1

0.00 0.0 0.00

0.0 0.04

0.08 Ug, m/s

0.12

0.16

Fig. 4. Comparison between the predicted and measured gas holdup in a bubble column (air–water system, HoR: homogenous regime; HeR: heterogeneous regime).

Industrial slurry reactors are usually operated at high superficial gas velocity and solid concentration for high reactor productivity (Hyndman et al., 1997; Krishna and Sie, 2000). Thus, reliable predictions of the gas holdup in a wide rage of superficial gas velocity are crucial for simulation of the gas–liquid mass transfer. Reasonable predictions of the gas holdup for both the homogeneous and the heterogeneous regimes were obtained by the CFD–PBM coupled model as shown in Fig. 4. In the CFD–PBM coupled model, both the effects of the bubble size distribution on the drag force and the wake effect of large bubbles are considered. The need for including both small and large bubbles in the heterogeneous regime was discussed in detail by Krishna et al. (2000) and by Wang et al. (2006). Their results show that CFD models that do not take into account the effects of the bubble size distribution only give good

0.2

0.4

0.6

0.8

1.0

r/R Fig. 5. Simulated radial profiles of the gas holdup in a bubble column (symbols: data of Sanyal et al., 1999; lines and symbols: predictions by the CFD–PBM model; air–water system, D = 0.19 m).

predictions of the average gas holdup in the homogeneous regime, but greatly over-predict the gas holdup in the heterogeneous regime. The value of dc that distinguishes small and larger bubbles was estimated by the correlation of dc = 4(/g)1/2 (Ishii and Zuber, 1979). For the air–water system, this critical value is about 10 mm. Fig. 5 shows the simulated radial profiles of the average gas holdup. At low superficial gas velocities, the flow is in the homogeneous regime and the radial profile of the gas holdup is relatively uniform. With an increase in the superficial gas velocity, the flow enters the heterogeneous regime, and the radial non-uniformity of the gas holdup becomes pronounced. This variation is in accordance with the experimental results reported by other authors (Hills, 1974; Ohnuki and Akimoto, 2000; Kemoun et al., 2001; Shaikh and Al-Dahhan, 2005).

T. Wang, J. Wang / Chemical Engineering Science 62 (2007) 7107 – 7118

Ug=2cm/s r/R=0

Ug=8cm/s r/R=0

Ug=16cm/s r/R=0

300

Ug=2cm/s r/R=0.44

Ug=8cm/s r/R=0.44

Ug=16cm/s r/R=0.44

200

7113

0 a, 1/m

PDF(volume) of db, 1/mm

100

150 0

Ug=8cm/s r/R=0.95

Ug=2cm/s r/R=0.95

Ug=16cm/s r/R=0.95

CFD-PBM model total interfacial area small bubbles large bubbles

100

Measured data Grund (1988)

200 0 0

20 40 db, mm

0

20 40 db, mm

0

20 40 db, mm

Fig. 6. Bubble size distributions at different radial positions and superficial gas velocities.

0 0.00

0.03

0.06

0.09 Ug, m/s

0.12

0.15

0.18

Fig. 7. Comparison between the predicted and measured interfacial area (air–water system).

4.2. Bubble size distribution The bubble size distribution has significant effects on gas–liquid mass transfer, because it determines the interfacial area per volume of the gas phase. The bubble size distribution is determined by bubble coalescence and breakup. In a given system, bubble coalescence and breakup rates are mainly affected by the local gas holdup and turbulent energy dissipation rate. Due to the non-uniform radial profiles of the gas holdup and dissipation rate, especially in the heterogeneous regime, the bubble size distribution varies not only with the superficial gas velocity, but also with the radial position. Fig. 6 shows the bubble size distribution at different superficial gas velocities and radial positions. At low superficial gas velocities, the bubble size distributions are very similar at different radial positions. With an increase in the superficial gas velocity, the difference in the bubble size distributions at different radial positions becomes pronounced, especially after the flow enters the heterogeneous regime. In the heterogeneous regime, the volume fraction of large bubbles in the central region remarkably increases and the bubble size distribution has an obvious tail. Fig. 6 also shows that in the wall region the variation of the bubble size distribution with the superficial gas velocity is not so remarkable as in the central region, indicating that the bubbles in the central region play an important role in determining the gas phase structure and the flow regime transition. Measured bubble size distributions are available only at low superficial gas velocities. The validations of the calculated bubble size distributions at low superficial gas velocities were reported in our previous work (Wang et al., 2006). At high superficial gas velocities, quantitative experimental data of the bubble size distribution are unavailable. However, the simulation results are qualitatively in agreement with the experimental results. The bimodal bubble size distribution is consistent with the results of dynamic gas disengagement experiments at high superficial gas velocities (Krishna et al., 2000). The simulation results that large bubbles are mainly present in the central

region are in agreement with the observed phase structure in a bubble column reported by Chen et al. (1994). 4.3. Gas–liquid interfacial area The CFD–PBM coupled model combines the advantages of CFD to calculate the entire flow field and of the PBM to calculate the local bubble size distribution. The gas–liquid interfacial area is calculated from the gas holdup and bubble size distribution. Fig. 7 shows the comparison between the predicted and measured gas–liquid interfacial area in a wide range of superficial gas velocity. The CFD–PBM coupled model successfully predicts that the interfacial area increases with the superficial gas velocity, but with a smaller increasing rate at a higher superficial gas velocity. With increasing superficial gas velocity, the increase in the gas holdup is less pronounced at high superficial gas velocity, and the average bubble size becomes larger due to intense bubble coalescence. Both account for the slower increase in the interfacial area at high superficial gas velocities. Note that the deviation of the predictions is relatively larger at higher superficial gas velocities. The reason may be two fold: one is that the reliable measurement of the gas–liquid interfacial area is much difficult at high superficial gas velocities because of the severe deformation and complex movement of bubbles in such conditions; the other is that modeling of the bubble coalescence and breakup at high superficial gas velocities still need further improvement and more experimental validation. The contributions of small and large bubbles to the interfacial area are also shown in Fig. 7. With increasing superficial gas velocity, both the contributions of small and large bubbles to the interfacial area increase. The contribution of small bubbles is much larger than that of large bubbles. The measurement of interfacial areas contributed by small and large bubbles is difficult and only the total interfacial area data were reported by Grund (1988). Fig. 8 shows the radial profiles of the gas–liquid interfacial area. At low superficial gas velocities, the interfacial

7114

T. Wang, J. Wang / Chemical Engineering Science 62 (2007) 7107 – 7118

400

300

a, 1/m

Ug, m/s 0.02 0.08 0.16

200

100

0 0.0

0.2

0.4

0.6

0.8

1.0

r/R Fig. 8. Radial profile of the interfacial area at different superficial gas velocities (air–water system).

0.15

kla, 1/s

0.10

0.05 Data of Tekasa Penatration model F=1 Penatration model F=0.8 Eddy cell model K=0.27

0.00 0.00

0.04

0.08 Ug, m/s

0.12

0.16

Fig. 9. Comparison of the predicted and measured volumetric mass transfer coefficients (air–water system).

area is relatively uniform, which is evident because both the gas holdup and the bubble size distribution are uniform in such operating conditions. With an increase in the superficial gas velocity, the radial non-uniformity of the gas–liquid interfacial area increases, but not so pronounced as that of the gas holdup. The reason is that at high superficial gas velocities, most large bubbles resulted from coalescence are in the central region, as shown in Fig. 7. Compared with small bubbles, large bubbles have smaller specific interfacial area. 4.4. Volumetric mass transfer coefficient The volumetric mass transfer coefficient data in a bubble column measured by Terasaka et al. (1998) were used to validate the numerical simulations. Fig. 9 shows a comparison between

the predicted and measured volumetric mass transfer coefficients in a bubble column in a wide range of superficial gas velocity. The variation in the volumetric mass transfer coefficient was very similar to that of the gas holdup. This is consistent with the results in the literature. Many other authors also found that the superficial gas velocity had insignificant influence on kl a/g , which was almost constant between 0.4 and 0.5 (Letzel et al., 1999; Jordan and Schumpe, 2001; Vandu et al., 2004). Both the slip penetration model and the eddy cell model were tested. The results showed that the CFD–PBM coupled model gave good predictions of the gas–liquid volumetric mass transfer coefficient with both the eddy cell and slip penetration models, with only a slight modification of the model constants. With the slip penetration model, simulation results using a theoretical model constant are higher than the experimental data. Good predictions of the volumetric mass transfer coefficient are obtained with a scaling factor of 0.8, indicating that the mass transfer rate of the bubble swarms is smaller than that of a single bubble. The difference between the mass transfer behaviors of a single bubble and bubble swarm was studied by Koynov et al. (2005) by numerical simulations. They found that in bubble swarms, bubbles no longer traveled by themselves, but rather in liquid perturbed by the wakes of neighboring bubbles. In addition, the concentration of gas dissolved in the liquid around the bubble in a swarm no longer depended only on the mass transfer from the bubble itself, but also on the mass transfer from the other bubbles in the swarm. These two factors resulted in a decrease in the mass-transfer coefficient of the bubble swarm compared with a single bubble. With the eddy cell model, good predictions of the volumetric mass transfer coefficient are obtained with the model constant K = 0.27. Different values of K were used in the literature, e.g. 0.4 (Lamont and Scott, 1970), 1.13 (Kawase et al., 1987), 0.301 (Kawase et al., 1992) and 0.523 (Linek et al., 2004). One reason for this inconsistence is the difficulty in accurate determination of the turbulent energy dissipation rate (Sheng et al., 2000). Another reason is that the eddy cell model also did not take into account the difference between mass transfer behaviors of a single bubble and bubble swarm. Fig. 10 shows the radial profiles of the volumetric mass transfer coefficient. At low superficial gas velocities, the volumetric mass transfer coefficient is relatively uniform, which is a result of uniform radial profiles of both the gas holdup and the bubble size distribution in such operating conditions. With an increase in the superficial gas velocity, the radial profile of the volumetric mass transfer coefficient becomes non-uniform, but not so pronounced as that of the gas holdup.

4.5. Mass transfer coefficient Eqs. (4) and (7) predict the local mass transfer coefficient. The global mass transfer coefficient is defined as the ratio of the global volumetric mass transfer coefficient to interfacial area. The results from the volumetric mass transfer coefficient with both the slip penetration model and the eddy cell model are compared in Fig. 11. The relationships between the mass

T. Wang, J. Wang / Chemical Engineering Science 62 (2007) 7107 – 7118

0.20 Ug, m/s 0.02 0.08 0.16

kla, 1/m

0.15

0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0

r/R Fig. 10. Radial profile of the volumetric mass transfer coefficient at different superficial gas velocities (air–water system).

kl, 10-4m/s

4.5

3.0

1.5 0.00

0.05

0.10 Ug, m/s

0.15

0.005 0.006 db, mm

0.007

0.5

0.7

0.9

1.1

l1/4, m1/2/s3/4

kl, 10-4m/s

4.5

3.0

1.5 0.004

7.0

7.5

8.0

8.5

9.0

(db/uslip)1/2

Fig. 11. Variation of the mass transfer coefficient with the superficial gas velocity Ug , turbulent energy dissipation rate εl , bubble diameter db and (db /uslip )1/2 (hollow symbols: slip penetration model; solid symbols: eddy cell model).

transfer coefficient and the superficial gas velocity Ug , turbulent energy dissipation rate εl , average bubble diameter db and (uslip /db )1/2 are analyzed. In general, the mass transfer coefficient increases with the superficial gas velocity and average bubble size. The variation in the mass transfer coefficient with the bubble size is different from the results of Alves et al. (2006) for a single bubble, where the mass transfer coefficient was found to decrease with an increase in the bubble size. The predicted mass transfer coefficient in the bubble column is lower than that of a single bubble, in accordance with the simulation results of Koynov et al. (2005). The slip penetration model and the eddy cell model are usually considered to be contradictory because the eddy cell model

7115

predicts a dependence of kl on the turbulent energy dissipation rate εl , while the penetration model predicts a dependence of kl on the bubble diameter db . Experimental evidence tends to support the penetration model in low turbulence conditions and supports the eddy cell model in high turbulence conditions (Vasconcelos et al., 2003; Linek et al., 2004; Alves et al., 2006). However, the results of this work showed that in a bubble column, the predictions of the slip penetration model and the eddy cell model are consistent in high turbulence conditions. The global mass transfer coefficient calculated from the volumetric mass transfer coefficient with the slip penetration model also has a dependence on the εl . This is not contradictory with Eq. (5) where εl is not involved, because the equilibrium bubble size depends on the gas holdup and the turbulent energy dissipation rate; thus the influence of εl is implicitly involved. Inconsistency of the two mass transfer models exists in low turbulence range. Most experimental evidence supports the slip penetration model in low turbulent conditions (Vasconcelos et al., 2003; Alves et al., 2006). Vasconcelos et al. (2003) found that kl agreed with the slip penetration model in a bubble column with clean water within the superficial gas range of 0.28–2.2 cm/s. Alves et al. (2006) measured the gas–liquid mass transfer coefficient in a system where a single bubble was kept stationary by a low turbulence downward liquid flow, and compared the data with predictions by the slip penetration and eddy cell models. They found that turbulence did not affect mass transfer in the range of low turbulence intensities and the slip penetration model was valid for calculating the gas–liquid mass transfer coefficient of clean bubbles in systems with εl up to 0.04 m2 /s3 . They also pointed that the level of turbulent energy dissipation required for the eddy cell model to replace the slip penetration model needed future research. Above discussion shows that the slip penetration model can be used in a wider range of operating conditions. Its validity in low turbulence intensity has been verified by several authors. The validity of the slip penetration model in high turbulence intensity is verified by the results in this work that showed a good agreement between the predictions of the slip penetration model with the measured data, and also with the eddy cell model which has been validated in high turbulence intensities. In the slip penetration model, the effects of the turbulent energy dissipation rate are indirectly taken into account through the slip velocity and bubble diameter. For example, when the dissipation rate increases while keeping other operating conditions unchanged, the bubble size will decrease due to bubble breakup, and result in an increase in the mass transfer coefficient. It should be pointed out that the present mass transfer models are not strictly predictive. In fact, most of the models for multi-phase flow are not predictive in a strict sense. Even the commonly used two-fluid models have the problems of closing the interphase forces and turbulence. The formulations of the interphase forces include parameters that have a complex dependence on the multiphase flow behaviors and some of them need to be treated as adjustable parameters. In this work, the unknown model parameters (K in the eddy cell model and F in the slip penetration model) are fitted to match experimental

7116

T. Wang, J. Wang / Chemical Engineering Science 62 (2007) 7107 – 7118

data. Such treatment attributes to the different mass transfer behavior of a bubble swarm from that of a single bubble (Koynov et al., 2005). However, a prior quantitative formulation of the difference between the bubble swarms in turbulent bubbling column and a single bubble still requires more detailed experimental and simulation studies on bubble scale.

CW d dc dc2 dbH

5. Conclusions Gas–liquid mass transfer in a bubble column in both the homogeneous and the heterogeneous flow regimes was studied by numerical simulations with a CFD–PBM coupled model. This coupled model successfully predicted the hydrodynamics of a bubble column in a wide range of superficial gas velocities The mass transfer coefficient kl and the interfacial area a were modeled separately. The interfacial area a was calculated from the gas holdup and bubble size distribution. The mass transfer coefficient kl was calculated with the slip penetration model and the eddy cell model. The average gas holdup, volumetric mass transfer coefficient and interfacial area increase with an increase in the superficial gas velocity. In addition, the radial profiles of these parameters become more non-uniform with increasing superficial gas velocity. With both the slip penetration model and the eddy cell model, reasonable predictions of kl a are obtained in a wide range of superficial gas velocity, with only a slight modification of the model constants. The mass transfer coefficient of the bubble swarm is smaller than a single bubble because of the difference of the flow field and concentration distribution around the bubble. For a bubble column, the predictions of the slip penetration model and eddy cell model are consistent in high turbulence range. While in low turbulence range, there is a marked deviation between these two models. The slip penetration model can be used in a wider range of operating conditions, only if the effects of the gas holdup and turbulent energy dissipation rate on the bubble size and bubble slip velocity are correctly counted. The simulation results show that the CFD–PBM coupled model is an efficient method for predicting the hydrodynamics, bubble size distribution, interfacial area and gas–liquid mass transfer rate in a bubble column.

Dl e( ) e( ) Eo Eo fb,large fv fi FL FT D FV M FW hb,ij kl

lbt lbt,ij ni,k

Ni (t) P (di , dj ) Pe (e( ))

Notation a b(v), b(d) b(fv |d)

c(vi , vj ), c(di , dj ) CL CT D CV M

gas–liquid interfacial area, m−1 bubble breakup rate, s−1 breakup rate of a bubble with size d breaking with breakup fraction fv , s−1 bubble coalescence kernel function, m3 s−1 lift force coefficient, dimensionless turbulent dispersion force coefficient, dimensionless virtual mass force coefficient, dimensionless

Pb (fv |d, )

Pb (fv |d, e( ), )

Re uslip v

wall lubrication force coefficient, dimensionless diameter of the mother bubble, m critical size of bubbles having wake effect for bubble coalescence, m critical size of bubbles with breakup due to instability, m maximum horizontal dimension of the bubble, m diffusion coefficient, m2 s−1 kinetic energy of an eddy of size , J mean kinetic energy of an eddy of size , J Eötvös number, g(l − g )db2 /, dimensionless modified Eötvös number, g(l − 2 /, dimensionless g )dbH fraction of large bubbles, dimensionless breakup fraction defined by v1 /v, dimensionless volume fraction of bubble group i in the gas holdup, dimensionless transverse lift force, N m−3 turbulent dispersion force, N m−3 virtual mass force, N m−3 wall lubrication force, N m−3 mean distance between bubbles of size di and dj , m gas–liquid mass transfer coefficient, m s−1 ; kinetic energy of the liquid phase, m2 s−2 bubble turbulent path, m mean relative turbulent path of bubbles of size di and dj , m transfer coefficient between bubble groups due to bubble breakup, dimensionless number of bubbles with volume between vi and vi+1 , m−3 bubble coalescence efficiency, dimensionless energy-distribution density function for eddy of size , J −1 breakup probability for a bubble of size d breaking with breakup fraction fv when hit by an eddy of size , dimensionless breakup probability for a bubble of size d breaking with breakup fractionfv when hit by an eddy of size and kinetic energy e( ), dimensionless bubble Reynolds number, dimensionless bubble slip velocity, m s−1 bubble volume, m3

T. Wang, J. Wang / Chemical Engineering Science 62 (2007) 7107 – 7118

Greek letters  phase holdup, dimensionless g,max possible maximum gas holdup, set to 0.8 in this work (v, v  ) daughter bubble size distribution function, dimensionless ε turbulent energy dissipation rate, m2 s−3 i,j k transfer coefficient between bubble groups due to bubble coalescence, dimensionless  model parameter, dimensionless eddy size, m  viscosity, Pa s  dynamic viscosity, m2 s−1 ij relative size defined as di /dj , dimensionless

(di , dj ) specific collision frequency between bubbles of size  density, kg m−3  surface tension, N m−1

e exposure time in the penetration model, s Acknowledgments The authors gratefully acknowledge the financial supports by the National Natural Science Foundation of China (No. 20606021), Foundation for the Author of National Excellent Doctoral Dissertation of PR China (No. 200757), and National 973 Project of China (No. 2007CB714302).

References Alves, S.S., Vasconcelos, J.M.T., Orvalho, S.P., 2006. Mass transfer to clean bubbles at low turbulent energy dissipation. Chemical Engineering Science 61, 1334–1337. Bouaifi, M., Hebrard, G., Bastoul, D., Roustan, M., 2001. A comparative study of gas holdup, bubble size, interfacial area and mass transfer coefficients in stirred gas–liquid reactors and bubble columns. Chemical Engineering Processing 40, 97–111. Burris, V.L., McGinnis, D.F., Little, J.C., 2002. Predicting oxygen transfer and water flow rate in airlift aerators. Water Research 36, 4605–4615. Chen, R.C., Reese, J., Fan, L.S., 1994. Flow structure in a three-dimensional bubble column and three-phase fluidized bed. A.I.Ch.E. Journal 40, 1093–1104. Chen, P., Sanyal, J., Dudukovic, M.P., 2005. Numerical simulation of bubble columns flows: effect of different breakup and coalescence closures. Chemical Engineering Science 60, 1085–1101. Colella, D., Vinci, D., Bagatin, R., Masi, M., Bakr, E.A., 1999. A study on coalescence and breakage mechanisms in three different bubble columns. Chemical Engineering Science 54, 4767–4777. Danckwerts, P.V., 1951. Significance of liquid–film coefficients in gas absorption. Industrial and Engineering Chemistry 43, 1460–1467. Darmana, D., Deen, N.G., Kuipers, J.A.M., 2005. Detailed modeling of hydrodynamics, mass transfer and chemical reactions in a bubble column using a discrete bubble model. Chemical Engineering Science 60, 3383–3404.

7117

de Bertodano, L.M., Lahey, R.T., Jones, O.C., 1994. Development of a k–ε model for bubbly two phase flow. Journal of Fluids Engineering, Transactions of the ASME 116, 128–134. Degaleesan, S., Dudukovic, M., Pan, Y., 2001. Experimental study of gas-induced liquid structures in bubble columns. A.I.Ch.E. Journal 47, 1913–1931. Dhanasekharan, K.M., Sanyal, J., Jain, A., Haidari, A., 2005. A generalized approach to model oxygen transfer in bioreactors using population balances and computational fluid dynamics. Chemical Engineering Science 60, 213–218. Garcia-Ochoa, F., Gomez, E., 2004. Theoretical prediction of gas–liquid mass transfer coefficient, specific area and holdup in sparged stirred tanks. Chemical Engineering Science 59, 2489–2501. Grund, G., 1988. Hydrodynamische Parameter und Stoffaustauscheigenschaften in Blasensäulen mit organischen medien. Ph.D. Thesis, Universität Oldenburg, Germany. Hibiki, T., Ishii, M., 2000. Two-group interfacial area transport equations at bubbly-to-slug transition. Nuclear Engineering and Design 202, 39–76. Higbie, R., 1935. The rate of absorption of a pure gas into a still liquid during short periods of exposure. Transactions of the American Institute of Chemical Engineers 31, 365–389. Hills, J.H., 1974. Radial non-uniformity of velocity and voidage in a bubble column. Transactions of the Institution of Chemical Engineers 52, 1–9. Hyndman, C.L., Larachi, F., Guy, C., 1997. Understanding gas-phase hydrodynamics in bubble columns: a convective model based on kinetic theory. Chemical Engineering Science 52, 63–77. Ishii, M., Zuber, N., 1979. Drag coefficient and relative velocity in bubbly, droplet or particulate flows. A.I.Ch.E.. Journal 25, 843–855. Jajuee, B., Margaritis, A., Karamanev, D., Bergougnou, M.A., 2006. Application of surface-renewal-stretch model for interface mass transfer. Chemical Engineering Science 61, 3917–3929. Jordan, U., Schumpe, A., 2001. The gas density effect on mass transfer in bubble columns with organic liquids. Chemical Engineering Science 56, 6267–6272. Kawase, Y., Halard, B., Moo-Young, M., 1987. Theoretical prediction of volumetric mass transfer coefficients in bubble columns for Newtonian and non-Newtonian fluids. Chemical Engineering Science 42, 1609–1617. Kawase, Y., Halard, B., Moo-Young, M., 1992. Liquid-phase mass transfer coefficients in bioreactors. Biotechnology and Bioengineering 39, 1133–1140. Kemoun, A., Ong, B.C., Gupta, P., Al-Dahhan, M.H., Dudukovic, M.P., 2001. Gas holdup in bubble columns at elevated pressure via computed tomography. International Journal of Multiphase Flow 27, 929–946. Koynov, A., Khinast, J.G., Tryggvason, G., 2005. Mass transfer and chemical reactions in bubble swarms with dynamic interfaces. A.I.Ch.E. Journal 51, 2786–2800. Krishna, R., van Baten, J.M., 2003. Mass transfer in bubble columns. Catalysis Today 79–80, 67–75. Krishna, R., Sie, S.T., 2000. Design and scale-up of the Fischer–Tropsch bubble column slurry reactor. Fuel Processing Technology 64, 73–105. Krishna, R., van Baten, J.M., Urseanu, M.I., 2000. Three-phase Eulerian simulations of bubble column reactors operating in the churn-turbulent regime: a scale up strategy. Chemical Engineering Science 55, 3275–3286. Kulkarni, A.A., 2007. Mass transfer in bubble column reactors: effect of bubble size distribution. Industrial and Engineering Chemistry Research 46, 2205–2211. Lamont, J.C., Scott, D.S., 1970. An eddy cell model of mass transfer into surface of a turbulent liquid. A.I.Ch.E. Journal 16, 513–519. Lehr, F., Mewes, D., 2001. A transport equation for the interfacial area density applied to bubble columns. Chemical Engineering Science 56, 1159–1166. Lehr, F., Millies, M., Mewes, D., 2002. Bubble-size distributions and flow fields in bubble columns. A.I.Ch.E. Journal 48, 2426–2443. Letzel, H.M., Schouten, J.C., Krishna, R., van den Bleek, C.M., 1999. Gas holdup and mass transfer in bubble column reactors operated at elevated pressure. Chemical Engineering Science 54, 2237–2246. Lewis, W.K., Whitman, W.G., 1924. Principles of gas absorption. Industrial and Engineering Chemistry 16, 1215–1220. Linek, V., Kordaˇc, M., Fujasová, M., Moucha, T., 2004. Gas–liquid mass transfer coefficient in stirred tanks interpreted through models of idealized

7118

T. Wang, J. Wang / Chemical Engineering Science 62 (2007) 7107 – 7118

eddy structure of turbulence in the bubble vicinity. Chemical Engineering and Processing 43, 1511–1517. Millies, M., Mewes, D., 1999. Interfacial area density in bubbly flow. Chemical Engineering and Processing 38, 307–319. Ohnuki, A., Akimoto, H., 2000. Experimental study on transition of flow pattern and phase distribution in upward air–water two-phase flow along a large vertical pipe. International Journal of Multiphase Flow 26, 367–386. Olmos, E., Gentric, C., Vial, C., Wild, G., Midoux, N., 2001. Numerical simulation of multiphase flow in bubble column reactors: influence of bubble coalescence and breakup. Chemical Engineering Science 56, 6359–6365. Ramkrishna, D., Mahoney, A.W., 2002. Population balance modeling. Promise for the future. Chemical Engineering Science 57, 595–606. Sanyal, J., Vasquez, S., Roy, S., et al., 1999. Numerical simulation of gas–liquid dynamics in cylindrical bubble column reactors. Chemical Engineering Science 54, 5071–5083. Shaikh, A., Al-Dahhan, M., 2005. Characterization of the hydrodynamic flow regime in bubble columns via computed tomography. Flow Measurement and Instrument 16, 91–98. Sheng, J., Meng, H., Fox, R.O., 2000. A large eddy PIV method for turbulence dissipation rate estimation. Chemical Engineering Science 55, 4423–4434. Shimizu, K., Takada, S., Minekawa, K., Kawase, Y., 2000. Phenomenological model for bubble column reactors: prediction of gas hold-ups and volumetric mass transfer coefficients. Chemical Engineering Journal 78, 21–28.

Talvy, S., Cockx, A., Line, A., 2007. Modeling of oxygen mass transfer in a gas–liquid airlift reactor. A.I.Ch.E. Journal 53, 316–326. Terasaka, K., Hullmann, D., Schumpe, A., 1998. Mass transfer in bubble columns studied with an oxygen optode. Chemical Engineering Science 53, 3181–3184. Toor, H.L., Marchello, J.M., 1958. Film-penetration model for mass and heat transfer. A.I.Ch.E. Journal 4, 97–101. Vandu, C.O., Koop, K., Krishna, R., 2004. Volumetric mass transfer coefficient in a slurry bubble column operating in the heterogeneous flow regime. Chemical Engineering Science 59, 5417–5423. Vasconcelos, J.M.T., Rodrigues, J.M.L., Orvalho, S.C.P., Alves, S.S., Mendes, R.L., Reis, A., 2003. Effect of contaminants on mass transfer coefficients in bubble column and airlift contactors. Chemical Engineering Science 58, 1431–1440. Wang, T.F., Wang, J.F., Jin, Y., 2005. Population balance model for gas–liquid flows: influence of bubble coalescence and breakup models. Industrial and Engineering Chemistry Research 44, 7540–7549. Wang, T.F., Wang, J.F., Jin, Y., 2006. A CFD–PBM coupled model for gas–liquid flows. A.I.Ch.E. Journal 52, 125–140. Xu, Z.F., Khoo, B.C., Carpenter, K., 2006. Mass transfer across the turbulent gas–water interface. A.I.Ch.E. Journal 52, 3363–3374. Yang, W.G., Wang, J.F., Wang, T.F., Jin, Y., 2001. Experimental study on gas–liquid interfacial area and mass transfer coefficient in three-phase circulating fluidized beds. Chemical Engineering Journal 84, 485–490. Zhang, J.P., Grace, J.R., Epstein, N., Lim, K.S., 1997. Flow regime identification in gas–liquid flow and three-phase fluidized beds. Chemical Engineering Science 52, 3979–3992.