Chemical Engineering Science 62 (2007) 7205 – 7213 www.elsevier.com/locate/ces

Simulation of oscillatory baffled column: CFD and population balance K. Ekambara ∗,1 , M.T. Dhotre2 Institute of Chemical Technology, University of Mumbai, Matunga, Mumbai 19, India Received 19 April 2007; received in revised form 12 July 2007; accepted 13 August 2007 Available online 23 August 2007

Abstract In the present work, an attempt has been made to combine population balance and a CFD approach for simulating the flow in oscillatory baffled column (OBC). Three-dimensional Euler–Euler two-fluid simulations are carried out for the experimental data of Oliveira and Ni [2001. Gas hold-up and bubble diameter in a gassed oscillatory baffled column. Chemical Engineering Science 56, 6143–6148]. The experimental data include the average hold-up profile and bubble size distribution in the OBC. All the non-drag forces (turbulent dispersion force, lift force) and the drag force are incorporated in the model. The coalescence and breakage effects of the gas bubbles are modeled according to the coalescence by the random collision driven by turbulence and wake entrainment while for bubble breakage by the impact of turbulent eddies. Predicted liquid velocity and averaged gas hold-up are compared with the experimental data. The profile of the mean bubble diameter in the column and its variation with the superficial gas velocity is studied. Bubble size distribution obtained by the model is compared with the experimental data. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Two-fluid model; Population balance; CFD; Coalescence and break-up

1. Introduction Gas–liquid contacting is one of the most common operations used for mass transfer processes in chemical and biochemical applications. Gas-sparged stirred tanks, bubble columns (BCs) and airlift columns are the most commonly used devices for enhancing gas–liquid mass transfer. The oscillatory baffled column (OBC) can be regarded as a new generation of BC (Oliveira et al., 2003). Both columns are cylindrical and can be operated in a batch mode, but OBC uses equally spaced orifice baffles together with fluid oscillation, providing a significantly different flow pattern compared with the BC. Fluid oscillation in this kind of column is achieved by means of a piston or bellows at the base or by moving a set of baffles up and down the column. The flow passing through the baffles induces vortices, ∗ Corresponding author. Tel.: +1 780 264 7201; fax: +1 780 492 2881.

E-mail address: [email protected] (K. Ekambara). 1 Present address: Department of Chemical and Materials Engineering,

University of Alberta, Edmonton, Alta., Canada T6G 2G6. 2 Present address: Thermal-Hydraulics Laboratory, Nuclear Energy and Safety Department, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland. 0009-2509/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2007.08.048

which provide significant axial and radial mixing in the column. The intensity of mixing can be controlled by varying the oscillatory (amplitude and frequency) and geometrical (baffle spacing and baffle free area) conditions (Ni et al., 1995). The flow passing through the baffles induces vortices, which provide intensive radial motion within the column. It was found that the gas–liquid volumetric mass transfer coefficient in an OBC could be six times higher than that in a BC and 75% higher than that in a stirred tank fermenter involving a yeast culture (Ni et al., 1995). It is necessary to have a better knowledge of the local hydrodynamics to increase the predictability of the reactor design and to improve the efficiency of the process. The use of the computational fluid dynamics (CFD) should be able to improve this knowledge, by providing a complete description of the local hydrodynamics, if an adequate model is used. However, the complex interactions between the gas and the liquid phases cause many modeling problems still to be solved. Application of population balance models towards modeling bubbly flows has received unprecedented attention. With the continuous advancement of computer technologies, the use of CFD methodology and population balance models has been shown to expedite a more thorough understanding of different flow regimes and

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further enhance the description of the bubble characteristics in BC designs (Olmos et al., 2001). Importantly, population balance models have provided the way to incorporate fundamental mechanisms of bubble coalescence and bubble breakage in the numerical simulations. In this work, an attempt has been made to combine population balance with CFD for the case of gas–liquid flow in the OBC reactor. The flow hydrodynamics is simulated considering the bubble coalescence and break-up. Hydrodynamic variables such as liquid velocity, global and local gas hold-up are calculated and compared with experimental data. 2. Mathematical Modeling 2.1. Population balance model The type of flow encountered in OBCs is referred to as polydispersed multiphase flow, i.e., flow in which the dispersed phase covers a broad range of size groups. One of the attributes of polydispersed multiphase flow is that the different sizes of the dispersed phase interact with each other through the mechanisms of break-up and coalescence. To deal with this type of flow, a population balance equation must be formulated. Population balance is a well-established method in computing the size distribution of the dispersed phase and accounting for the breakage and coalescence effects in bubbly flows. A general form of the population balance equation is jni + ∇ · (uG ni ) = BB + BC − DB − DC , jt

(1)

where uG is the gas velocity, ni represents the number density of size group i and terms on the right-hand side BB , BC , DB and DC are, respectively, the ‘birth’ and ‘death’ due to breakup and coalescence of bubbles. The bubble number density ni is related to the gas volume fraction εG by εG fi = ni Vi ,

(2)

where fi represents the volume fraction of bubbles of group i and Vi is the corresponding volume of a bubble of group i. 2.1.1. Bubble break-up model The break-up of bubbles in turbulent dispersions employs the model developed by Luo and Svendsen (1996). Binary break-up of the bubbles is assumed and the model is based on the theories of isotropic turbulence. For binary breakage, a dimensionless variable describing the sizes of daughter drops or bubbles (the breakage volume fraction) can be defined as fBV

d3 d3 Vi = i3 = 3 i 3 , = V d di + dj

(3)

where di and dj are diameters (corresponding to Vi and Vj ) of the daughter bubbles in the binary breakage of a parent bubble with diameter d (corresponding to volume V ). The value interval of the breakage volume fraction is between 0 and 1.

The break-up rate of bubbles of volume Vj into volume sizes of Vi (=Vf BV ) can be obtained as  1/3  1 (1 + )2 (Vj : Vi )  =C 2 11/3 (1 − εG )nj dj min  ⎞ ⎛ 12cf  ⎠ d, × exp ⎝− 5/3 2/3 L  dj 11/3

(4)

where  = /dj is the size ratio between an eddy and a particle in the inertial sub-range and consequently min =min /dj , C and  are determined, respectively, from fundamental consideration of drops or bubbles breakage in turbulent dispersion systems to be 0.923 and 2.0 in Luo and Svendsen (1996) and cf is the increase coefficient of surface area given by 2/3

cf = [fBV + (1 − fBV )2/3 − 1].

(5)

The birth rate of group i bubbles due to break-up of larger bubbles is BB =

N

(Vj : Vi )nj .

(6)

j =i+1

The death rate of group i bubbles due to break-up into smaller bubbles is DB = i ni

with i =

N

ki .

(7)

i=1

2.1.2. Bubble coalescence model According to Prince and Blanch (1990), the coalescence of two bubbles occurs in three steps. The first step where the bubbles collide and trap a layer of liquid between them, a second step where this liquid layer drains until it reaches a critical thickness and a last step during which this liquid film disappears and the bubbles coalesce. The collisions between bubbles may be caused by three mechanisms: turbulence ( Tij ), laminar B shear ( LS ij ) and buoyancy ( ij ). Collision is also induced by the difference in rise velocities of bubbles with different sizes. The calculations showed that laminar shear collisions are negligible because of the low superficial gas velocities considered in this investigation. The coalescence rate considering turbulent collision can be expressed as 

tij

= ( Tij + B , (8) ) exp − ij ij where ij is the contact time for two bubbles given by (dij /2)2/3 /1/3 and tij is the time required for two bubbles to coalesce having diameter di and dj estimated to be {(dij /2)3 L /16}1/2 ln(h0 / hj ). The equivalent diameter dij is calculated as suggested by Chesters and Hoffman (1982): dij = (2/di + 2/dj )−1 . The parameters h0 and hj represent the film thickness when collision begins and critical film thickness at which rupture occurs, respectively. The turbulent collision

K. Ekambara, M.T. Dhotre / Chemical Engineering Science 62 (2007) 7205 – 7213

rate Tij for two bubbles of diameter di and dj is given by Tij =

[di + dj ]2 (u2ti + u2tj )1/2 , 4

becomes larger: (9)

where the turbulent velocity ut in the inertial sub-range of isotropic turbulence (Rotta, 1974) is ut = 1.41/3 d 1/3 .

(10)

The buoyancy contribution to collision frequency is modeled as B ij = FCB Sij |uij − uri |, where FCB is a calibration factor and 2.14 uri = + 0.505gd i . C di

(11)

(12)

The birth rate of group i due to coalescence of group k and group l bubbles is 1

i,kl ni nj . 2 N

BC =

N

(13)

k=1 l=1

The death rate of group i due to coalescence with other bubbles is DC =

N

ij ni nj .

7207

(14)

j =1

2.2. Flow equations The numerical simulations presented are based on the twofluid model Eulerian–Eulerian approach. Turbulence is taken into consideration for the continuous phase. The dispersed gas phase is modeled as laminar but influences the turbulence in the continuous phase by a bubble-induced turbulence model. The k–  turbulence model is used to model the turbulence phenomena in the continuous phase of the gas–liquid flow. The details about the governing equations can be found in Yeoh and Tu (2004, 2005). The model proposed by Sato et al. (1981) has been used to take account of the turbulence induced by the movement of the bubbles. All forces (the drag force, the lift force and the turbulence dispersion forces) except the virtual mass force have been used. According to Hunt (1987), the contribution of the virtual mass force becomes negligible for column diameters greater than 0.15 m. Thakre and Joshi (1999), Deen et al. (2001) and Dhotre et al. (2007) have also pointed out the negligible effect of the virtual mass force. A brief description of each interfacial force component is presented below. The origin of the drag force is due to the resistance experienced by a body moving in the liquid. Viscous stress creates skin drag and pressure distribution around the moving body creates form drag. The later mechanism is due to inertia and becomes significant as the particle Reynolds number

Re =

dS |ur | .

L

(15)

In Eq. (15), ur = uG − uL is the slip velocity, dS is the disperse phase Sauter mean bubble diameter and L is the liquid kinetic viscosity. The interphase momentum transfer between gas and liquid due to drag force is given by 1 3 FD = CD εG L |uG − uL |(uG − uL ), 4 dS

(16)

where CD is the drag coefficient taking into account the character of the flow around the bubble. Hydrodynamic interaction of the bubble with other particles also influences the drag coefficient. This phenomenon can be taken into account (Ishii and Zuber, 1979): 24 (17) (1 + 0.15Re0.687 ). Re The lift force arises from the interaction between bubble and shear stress in liquid. The lift force in terms of the slip velocity and the curl of the liquid phase velocity can be described as

CD =

FL = CL εG L (uG − uL ) × ∇ × uL .

(18)

The sign of this force depends on the orientation of slip velocity with respect to the gravity vector. The constant CL has been modeled using Tomiyama et al. (2002) relationship. The origin of the wall lubrication force is due to the fact that liquid flow rate between bubble and the wall is lower than between the bubble and the outer flow. This results in a hydrodynamic pressure difference driving bubble away from the wall. This force density is approximated as (ur − (ur · nw )nw ) FW L = − εG L dS

 dS × max Cw1 + Cw2 , 0 , yw

(19)

where nw is the outward unit vector perpendicular to the wall and yw is the distance from the wall to the bubble. The wall lubrication constants Cw1 and Cw2 as suggested by Antal et al. (1991) are −0.01 and 0.05, respectively. The turbulent dispersion force, derived by Lopez de Bertodano (1992), is based on the analogy with molecular movement. It approximates a turbulent diffusion of the bubbles by the liquid eddies. It is formulated as FT D = −CT D L k∇εL ,

(20)

where k is the turbulent kinetic energy per unit of mass. The turbulent dispersion coefficient of CT D = 0.5 was found to give a good result which is in the recommended range of 0.1–1.0. By definition, the interfacial area aij for bubbly flows can be determined through the relationship aij =

6εG , dS

(21)

where dS is the Sauter mean bubble diameter. The local Sauter mean bubble diameter based on the calculated values of the

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scalar fraction fi and the discrete bubble sizes di can be deduced from 1 . f i i /di

dS = 

(22)

From the drag and the non-drag forces above, it is evident that the interfacial area aij as well as the Sauter mean bubble diameter in Eq. (21) are essential parameters that link the interaction between the liquid and the gas (bubbly) phases. In most two-phase flow studies, the common approach of prescribing constant bubble sizes through the Sauter mean bubble diameter is still prevalent. Such an approach does not allow dynamic representation of the changes in the interfacial structure. 3. Numerical solution The simulations were carried out for three-dimensional transient flow in an OBC using the commercial software ANSYSCFX-10.0. Water was considered as the continuous phase, and air was considered as the dispersed phase. The multiple size group (MUSIG) model used in this study combines the population balance method with the break-up (Luo and Svendsen, 1996) and coalescence (Prince and Blanch, 1990) models in order to predict the bubble size distribution of the dispersed phase and it uses the Eulerian–Eulerian two-fluid model. A standard two-phase flow calculation, with equation for continuity, momentum and turbulence for a continuous and a dispersed phase, can be extended to include mass fraction of bubbles within several size ranges using the MUSIG model. The size range of the bubbles is split into several groups with, for example, bands of equal diameter or equal volume. Equations are solved for the mass fraction in each band. The MUSIG model takes into account non-uniform bubble size distribution in a gas–liquid mixture. The MUSIG model has been extensively used for different systems (Lo, 1996; Olmos et al., 2001; Yeoh and Tu, 2004, 2005; Chen et al., 2005). These size fractions provide a more accurate measure of the interfacial area density and therefore allow better calculation of the heat and mass transfer taking place between the continuous and dispersed phases. In this present study, bubbles ranging from 1 to 20 mm diameter are equally divided into 10 classes (see Table 1). The fate of the discrete bubble sizes so prescribed was tracked using the population balance model. Instead of considering 11 different complete phases, it was assumed that each bubble class travels at the same mean algebraic velocity to reduce the computational time. Therefore, it results in 10 continuity equations for the gas phase coupled with a single continuity equation for the liquid phase. Sensitivity of the number of size groups needed to describe a meaningful distribution was examined by dividing

the bubble diameters equally into 10, 15 and 20 size groups. The results revealed that no appreciable difference was found for the predicted maximum Sauter mean bubble diameter using the 10, 15 or 20 bubble size groups. In view of computational resources and times, it was therefore concluded that the subdivision of the bubble sizes into 10 size groups was sufficient and all subsequent computational results are based on the discretization of the bubble sizes into 10 groups. Solution to the two sets of governing equations for the balances of mass and momentum of each phase was sought. The conservation equations were discretized using the control volume technique. The velocity–pressure linkage was handled through the SIMPLE procedure. The hybrid-upwind discretization scheme was used for the convective terms. Non-uniform grid of 142,330 cells were used and is shown in Fig. 1. The time step of 0.001 s was used. 3.1. Boundary conditions Specifying the spatial and temporal periodicity is the trivial task in applying boundary condition for OBC. We have adopted the procedure suggested by Jian and Ni (2005). The spatial periodicity is achieved by forcing the flow at the inlet and the outlet to be identical, including grids. For liquid, the temporal periodicity is implemented by discretizing the mass flow rate of mL into many known small time intervals over an oscillation cycle. The flow rate for mL is given as mL = L

 4

 D 2 2 f x 0 cos(2 f t).

(23)

For the air, inlet velocity is specified calculated using the aeration rate. At each time interval, the mass flow rates at both the inlet and the outlet are calculated corresponding to the time as well as the phase positions of the oscillation and are then forced to be the same. No slip boundary conditions were used at all the walls. 3.2. Experiments Oliveira and Ni (2001) carried out the experiments in the OBC, which was a 50 mm internal diameter and 1.5 m tall Perspex column. The column was stationed vertically onto a metal table and was filled with 2.5 l of tap water. A set of 14 orifices baffles was used with the baffles being made of 2 mm thick polyethylene plate. The baffles were equally spaced at 75 mm apart (1.5 times the tube internal diameter). The orifice diameter is 24 mm, giving a 23% free cross-sectional area. The liquid oscillation was achieved using stainless steel bellows mounted at the base of the OBC and driven by an inverter connected to

Table 1 Diameter of each bubble class tracked in the simulation Class index Bubble diameter, di (mm)

1

2

3

4

5

6

7

8

9

10

1.0

3.0

5.0

7.0

9.0

11.0

13.0

15.0

17.0

19.0

K. Ekambara, M.T. Dhotre / Chemical Engineering Science 62 (2007) 7205 – 7213

7209

Fig. 1. Grid configuration used in the numerical simulations.

an electrical motor. Flow visualization was carried out using a CCD camera with a resolution of 1000 × 1000 TV lines and a maximum shutter speed of 1/10,000 s. More details about the experimental set-up is given in Oliveira and Ni (2001). 4. Results and discussion The CFD simulations are carried out using a combination of two-fluid k–  model and population balance approach with bubble size distribution (1–20 mm) for the experimental conditions of Oliveira and Ni (2001). The predicted average gas hold-up and mean bubble diameter has been compared with the experimental data. Figs. 2 and 3 show the flow field colored with local gas hold-up for three time instances for two oscillatory Reynolds numbers (Re0 = 2513.3 and 7539.8). The oscillatory motion in the baffled column produces substantial modifications in bubble trajectories. This induces a complex liquid–bubble mixing pattern which is highly affected by the Re0 . At the low oscillatory Reynolds number, the flow is symmetric with clear and visible eddies or vortices within the baffled cell (Fig. 2). With the increase of Re0 , the flow becomes highly chaotic and spatially asymmetric (Fig. 3). However, at long times, both flow configurations behave in identical way giving long time averaged profile. In order to compare quantitatively, we have compared the prediction for both the Re0 in the following section. First, the averaged fractional gas hold-up, Sauter mean bubble diameter, followed by bubble size distribution predictions, are compared with the experimental data. 4.1. Average fractional gas hold-up The experimental data for averaged fractional gas hold-up is available in the presence of both baffles and oscillation.

In order to compare the results with the measured data, the simulations were carried out for five oscillatory Reynolds numbers and three aeration rates. The details of the runs are given in Table 2. Fig. 4 shows the comparison of the predicted averaged fractional gas hold-up with the experimental data. The predictions and the measurements of the gas hold-up as a function of the oscillatory Reynolds number reveal two patterns as observed in the experiments: at low oscillatory Reynolds numbers the changes in the hold-up are very small, but at high oscillatory Reynolds numbers gas hold-up (εG ) increases with the increased oscillation intensities. For the latter case, the oscillatory motion causes increased breakage, resulting in more of small and tiny bubbles from the experimental observations, hence longer residence times and higher hold-ups in the column (Oliveira and Ni, 2001). Further, it can be seen that at a given oscillatory Reynolds number, the average gas hold-up again increases with the increase of the aeration rate. At low Re0 , predictions are in agreement with the experimental data; however, for higher Re0 , some discrepancies can be observed. Unfortunately, no experimental data for local gas hold-up was were available. 4.2. Sauter mean bubble diameter Fig. 5 illustrates the comparison of the predicted Sauter mean bubble diameter and the experimental data of the mean bubble diameter. A good agreement between the experimental data and the predictions of the Sauter mean bubble diameter can be observed. For low oscillatory Reynolds numbers, oscillation has little effect on the mean diameter, and the aeration rate is the dominating factor. As with the increase of aeration rate, hold-up becomes higher, while dissipation rate is lower, which results in larger bubbles because of the higher coalescence rate and lower break-up rate (Fig. 5).

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Fig. 2. Computational snapshots of velocity flow field and gas hold-up distribution for Re0 = 2513.3: (a) 2 s, (b) 10.5 s, (c) 15.5 s (10 uniform contours in 0.0–0.1 and maximum velocity of 1.14 m/s).

Fig. 3. Computational snapshots of velocity flow field and gas hold-up distribution for Re0 = 7539.8: (a) 2 s, (b) 10.5 s, (c) 15.5 s (10 uniform contours in 0.0–0.2 and maximum velocity of 2.63 m/s).

K. Ekambara, M.T. Dhotre / Chemical Engineering Science 62 (2007) 7205 – 7213 Table 2 Simulations

1 Aeration rate (vvm)

Oscillation frequency (Hz)

0.05 0.1 0.2

1–3 1–3 1–3

Re0

1 2 3

583–12,600 583–12,600 583–12,542

0.8 BUBBLE CLASS p. d. f

Run no.

7211

0.06

0.6

0.4

AVG FRACTIONAL HOLD UP

0.2 0.05 0 0.04

0

0.03

5 10 15 BUBBLE MEAN DIAMETER (mm)

20

Fig. 6. Bubble classes hold-up based on probability distribution. Baffle, oscillation (x0 = 8 mm), aeration rate = 0.05 vvm: Re0 = 2513.3 (f = 1 Hz): —experiment, —CFD; Re0 = 5026.5 (f = 2 Hz): —experiment, —CFD; Re0 = 7539.8 (f = 3 Hz): ◦—experiment, —CFD.

0.02 0.01 0 0

2000

4000

6000

8000

10000

12000

14000

Re0 Fig. 4. Comparison of the prediction of averaged hold-up with the experimental data. Experiment: aeration rate: ◦—0.05 vvm; —0.1 vvm; —0.2 vvm; —CFD.

14

MEAN BUBBLE SIZE (mm)

12 10 8

attributed to the strong velocity gradient at the higher oscillatory Reynolds number, which causes further break-up of the bubbles into smaller sizes. In these cases, the effect of superficial gas velocity is hindered and the oscillation becomes the dominating factor. Fig. 6 shows the comparison between the experimental data and the prediction of bubble class hold-up based on probability function in OBC. Though not quantitatively captured, the trend is well predicted. The increase in the oscillatory Reynolds number via the oscillation frequency gave narrower BSD and reduced the mean bubble size. This is due to the fact that the increase in the oscillation velocity causes an increase in the power density to the system, subsequently resulting in smaller bubbles (Oliveira and Ni, 2001).

6

5. Conclusions

4 2 0 0

2000

4000 Re0 (-)

6000

8000

Fig. 5. Comparison of the predicted mean bubble size with experimental data. Experiment: aeration rate: ◦—0.05 vvm; —0.1 vvm; —0.2 vvm; —CFD.

For high oscillatory Reynolds number, a Sauter mean bubble diameter is almost identical. The mean bubble size tends to reduce with increase of Re0 (as shown in Fig. 5). This can be

CFD simulation coupled with the population balance approach is successfully shown to handle the gas–liquid flows in OBC reactor. The multiple size group (MUSIG) model is used for gas–liquid flows in order to determine the temporal and spatial geometrical changes of the gas bubbles. Population balance combined with coalescence and break-up models was taken into consideration. A detailed comparison has been presented between the CFD predictions and the experimental data reported by Oliveira and Ni (2001). The results were compared with the averaged fractional gas hold-up and Sauter mean bubble diameter. The model predictions show good agreement with the experimental data. The approach presented in this communication shows promising outcome for design of the OBC.

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yw z

Notation

aij BB BC cf CD CL CT D Cw1 , Cw2 d d, di , dj dij dS D DB DC fBV fi FCB FD FL FT D FV M FW L g h0 , hj

k ni nj nw p r Re Re0 Si t tij uG ur ut Vi Vj

interfacial area, 1/m the ‘birth’ due to break-up of bubbles, defined in Eq. (6), 1/m3 s the ‘birth’ due to coalescence of bubbles, defined in Eq. (13), 1/m3 s the incr6 coefficient of surface area, defined in Eq. (5) drag coefficient, dimensionless lift coefficient, dimensionless the turbulent dispersion coefficient, dimensionless the wall lubrication constants, dimensionless bubble diameter, m diameters (corresponding to Vi and Vj ) of the daughter bubbles, m the equivalent diameter =(2/di + 2/dj )−1 , m Sauter mean bubble diameter, m column diameter, m the ‘death’ due to break-up of bubbles, defined in Eq. (7), 1/m3 s the ‘death’ due to coalescence of bubbles, defined in Eq. (14), 1/m3 s breakage volume fraction, dimensionless volume fraction of bubbles of group i, dimensionless calibration factor, dimensionless drag force, defined in Eq. (16),N lift force, defined in Eq. (18), N turbulent dispersion force, defined in Eq. (20), N virtual mass force, N wall lubrication force, defined in Eq. (19), N gravitational acceleration, m/s2 parameters representing the film thickness when collision begins and critical film thickness at which rupture occurs turbulent kinetic energy, m2 /s2 the number density of size group i, m3 the number density of size group j, m3 the outward unit vector perpendicular to the wall pressure, Pa radial coordinate, m Reynolds number, defined in Eq. (15), dimensionless oscillatory Reynolds number, dimensionless source term due to coalescence and break-up, kg/m3 s time, s time required for two bubbles having diameter di and dj to coalesce, s gas velocity, m/s slip velocity, m/s turbulent velocity, defined in Eq. (10), m/s volume of a bubble of group i, m3 break-up rate of bubbles of volume, m3

the distance from the wall to the bubble, m axial coordinate, m

Greek letters   εG εL  B ij LS ij Tij  

L  C ij ij



measured constant in Eq. (4), dimensionless turbulent eddy dissipation rate, m2 /s3 gas volume fraction, dimensionless liquid volume fraction, dimensionless the size ratio between an eddy and a particle in the inertial sub-range (=/dj ) buoyancy collision rate, defined in Eq. (11), 1/m3 s laminar shear collision rate, 1/m3 s turbulence collision rate, defined in Eq. (9), 1/m3 s size of an eddy, m viscosity, Pa s liquid kinetic viscosity, m2 /s density, kg/m3 density of the continuous phase, kg/m3 the contact time for two bubbles, s the pressure–strain correlation parameter defined in Eq. (8), 1/m3 s break-up rate, 1/m3 s

Subscripts G L w

gas liquid wall

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