CFD simulation of pilot-scale bubble columns with internals: Influence of interfacial forces

Accepted Manuscript Title: CFD Simulation of Pilot-Scale Bubble Columns with Internals: Influence of Interfacial Forces Authors: Xiaoping Guan, Ning Yang PII: DOI: Reference:

S0263-8762(17)30431-8 http://dx.doi.org/10.1016/j.cherd.2017.08.019 CHERD 2794

To appear in: Received date: Revised date: Accepted date:

11-4-2017 7-8-2017 22-8-2017

Please cite this article as: Guan, Xiaoping, Yang, Ning, CFD Simulation of Pilot-Scale Bubble Columns with Internals: Influence of Interfacial Forces.Chemical Engineering Research and Design http://dx.doi.org/10.1016/j.cherd.2017.08.019 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

CFD Simulation of Pilot-Scale Bubble Columns with Internals: Influence of Interfacial Forces Xiaoping Guan1*, Ning Yang1* 1

State Key Laboratory of Multi-phase Complex System, Institute of Process Engineering, Chinese

Academy of Sciences, P.O. Box 353, Beijing 100190, PR China

To

whom correspondence should be addressed. E-mail: [email protected] (X. Guan), [email protected] (N. Yang)

Graphical abstract

Highlights 

Effects of interfacial forces in bubble columns with internals are examined.



Lateral forces are required in bubble columns with internals.



Presence of internals increases sensitivity of bubbly flow to lateral forces.



Internals decrease turbulent viscosity and hence enhance large-scale circulation.

Abstract In the present study, influence of interfacial forces, including drag force and lateral forces (lift force, turbulent dispersion force and wall force) on the hydrodynamics in pilot-scale bubble columns with internals is analyzed. The results indicate that the lateral forces may be optional for the hollow columns, but they are 1

required to accurately predict flow characteristics in the bubble columns with internals. Furthermore, it has been found that the bubbly flow behavior in the bubble columns with internals is more sensitive to the lateral forces in comparison with those without internals, and the complex geometry significantly alters the response of bubbly flow to the interfacial forces. In addition, despite the insignificant effect on gas holdup, the presence of internals gives rise to an enhancement of large-scale liquid circulation due to the remarkable decrease of turbulent viscosity. Key words: Bubble columns; Internals; CFD; Interfacial force; Churn-turbulent flow

Nomenclature CD

drag force coefficient

CD 0

single bubble drag force coefficient

CL

lift force coefficient

CTD

turbulent dispersion force coefficient

CW

wall force coefficient

CW 1 , CW 2 C

wall force related constant

constant in the turbulence model

dB

bubble diameter, m

dt

internals diameter, m

D

column diameter, m

Eo

Eotvos number

FD

drag force, N·m-3

FL

interface force, N·m-3

FTD turbulent dispersion force, N·m-3 FW

wall force, N·m-3

g

gravity acceleration, m ·s-2

k

turbulent kinetic energy, m2·s-2

n

internals number

nW

unit inward vector normal to the wall 2

ReB

bubble Reynolds number

P

pressure, pa

r

radial position, m

R

column radius, m

t

time, s

Tm

molecular stress, s

T Re

turbulent stress, s

u

velocity, m·s-1

UG

superficial gas velocity, m·s-1

VB

bubble volume, m3

y

distance from wall, m

z

axial height, m

Greek symbols 

phase fraction



turbulent kinetic energy dissipation rate, m2·s-3



percentage of CSA

m

molecular viscosity, kg·m-1·s-1

t

turbulent viscosity, kg·m-1·s-1



density, kg·m-3

 k , 

Schmidt number in the turbulence model



angular velocity, s-1

subscripts G

gas phase

i

phase

L

liquid phase

1. Introduction Heat removal is a crucial issue in the design and operation of commercial Fisher-Tropsch (F-T) synthesis slurry bubble column reactors. Dense heat-exchanging 3

tubes are equipped in the reactors to maintain the desired reaction environment. Therefore, the effects of dense tube internals on the hydrodynamics in bubble columns have gradually received attention in recent years (Al Mesfer, 2013; Bernemann et al., 1991; Besagni and Inzoli, 2016; Chen et al., 1999; Forret et al., 2003; Guan et al., 2015; Guan et al., 2014b; Jhawar and Prakash, 2014; Kagumba and Al-Dahhan, 2015; Youssef and Al-Dahhan, 2009; Youssef et al., 2013a; Youssef et al., 2013b; Youssef et al., 2012; Zhang et al., 2011; Zhang et al., 2009). Chen et al. (1999) found that the presence of internals had insignificant on gas holdup and liquid velocity profiles, but substantially reduced turbulent stress and eddy diffusivities. Forret al. (2003) reported internals intensified liquid large-scale circulation and decreased liquid fluctuation velocity and radial dispersion. Zhang (2009) observed gas holdup and liquid velocity profiles became steep in the presence of internals. Youssef and Al-Dahhan (2009) and Youssef et al. (2012) measured bubble properties in bubble columns with internals, and the results indicated that dense internals increased gas holdup and interfacial area and reduced bubble chord length and bubble velocity. Furthermore, column size combined with internals drastically effected bubble motion direction near the wall region. The measured results by Guan et al. (2014b, 2015) indicated that the internals extended the gas distributor region, and the effect was intensified in large-scale bubble columns so that the well-developed region in a 0.8 m diameter bubble column equipped with internals vanished. Kagumba and Al-Dahhan (2015) studied the effects of internals diameter on bubble properties and found that smaller size internals gave higher specifific interfacial area and lower bubble velocity. Recently, Besagni and Inzoli (2016) investigated hodlup, flow regime transition and local flow properties in an annular gap configuration bubble column, and observed that the presence of internals stablized homogeneous regime and postponed regime transition. In comparison to experimental investigation, numerical studies on the effects of internals are relatively sparse (Guan et al., 2014a; Laborde-Boutet et al., 2010; Larachi et al., 2006; Zhang et al., 2011). Larachi et al. (2006) and Laborde-Boutet et al. (2010) simulated the effects of internals in the bubble columns. The meandering 4

gas twirls were replaced by smaller pockets in the size of inter-tube gaps in the presence of internals. Moreover, higher gas holdup was observed around the tube internals. In their model, drag force was assumed as the sole interfacial force and other lateral interfacial forces (such as lift force, wall force and turbulent dispersion force) were neglected. Zhang et al. (2011) developed a 1D porous media model to simulate hydrodynamics in bubble columns with internals, and concluded that the enhancement of large-scale circulation was relevant to the decrease of turbulent viscosity. In addition, Guan et al. (2014a) employed volume of fluid (VOF) method to simulate bubble dynamics in bubble columns with internals and the simulated results showed that the bubble shape and rise velocity were significantly altered in the presence of internals. Guédon et al. (2017) employed a bi-dispersed bubble model to model an annular gas bubble column, and found that relative amount of small bubbles was important and should be provided according to empirical correlations. Recently, Bhusare et al. (2017) have simulated a bubble column with internals by Openfoam, and the predictions showed good agreement with experimental data. However, an imposed zero-void wall boundary condition was employed to force bubbles to move away from wall. As discussed by Jakobsen et al. (2005), prescribing the wall void fraction leads to an overconstrained set of equations, and should be avoided. By contrast, numerous studies focused on the simulation of hydrodynamics in bubble columns without internals (Chen et al., 2005; Jakobsen et al., 2005; Jakobsen et al., 1997; Joshi, 2001; Sokolichin et al., 2004; van Baten and Krishna, 2004). Yet consensus on whether lateral forces is required to accurately predict gas-liquid flow has not been reached. Some studies (Laborde-Boutet et al., 2010; Laborde-Boutet et al., 2009; Larachi et al., 2006; Sokolichin et al., 2004) reported that the drag force is the most important interfacial force, and the lateral forces can be neglected to give good predictions, whereas other simulations (Ekambara et al., 2008; Jakobsen et al., 1997; Krepper et al., 2005; Tabib et al., 2008) indicated that the lateral forces, such as lift force and turbulent dispersion force, are of significance to quantitively simulate bubbly flows. However, in the bubble columns with internals, the effects of interfacial forces on the hydrodynamics have not been clarified, and whether the lateral forces 5

are indispensible to predict flow characteristics remains open question. This paper is intended to target this issue. First, the governing equations and interfacial models as well as numerical strategies are thoroughly presented. Second, the effects of interfacial forces, including the drag law and other lateral forces, on the simulated results are evaluated through the CFD simulation of a pilot-scale bubble column with internals of Zhang et al. (2009). Then the mechanism underlying different hydrodynamic behaviour close to column wall and tube internal wall without considering the lateral forces is revealed, and the simulated hydrodynamics in the presence of internals is elaborated. Lastly, the related conclusions are drawn from the foregoing analysis and discussion.

2. Mathematical model 2.1 Euler-Euler governing equations The numerical simulations are based on two-fluid Euler-Euler approach. The continuity equation for each phase is ( k k )    ( k k u k )  0 t

(1)

The momentum equation for each phase is ( k k uk )    ( k k uk u k )   k P    [ k (Tkm  TkRe )]  Fi ,k   k k g t

(2)

2.2 Interfacial forces The total interfacial forces Fi,k between the two phases are given by the drag force, lift force, turbulent dispersion force and wall force: Fi ,G  Fi , L  FD, L  FL, L  FTD, L  FW , L

(3)

The drag represents the resistance experienced by bubbles passing through the liquid, including skin friction and form drag. The interfacial momentum transfer between gas and liquid due to drag is given by:

C 3 FD, L   L  L D uG  u L (uG  u L ) 4 dB

(4)

where CD is the drag coefficient taking into account of hydrodynamic interactions among bubbles, and dB is the bubble diameter. 6

The lift force comes from the net lateral effect of pressure and shear stress on the bubble surface, and is expressed as:

FL, L  CLG L (uG  u L )    u L 

(5)

where CL is the lift force coefficient and its sign depends on the bubble diameter and bubble shape as observed by Tomiyama et al. (2002). In our simulation, the lift force coefficient is -0.02, as suggested by Tabib et al. (2008). The turbulent dispersion force is due to the liquid velocity turbulent fluctuation and is derived by Lopez de Bertodano (1992) based on the analogy with molecular thermal motion: FTD, L  CTD L k L

(6)

where CTD is the turbulent dispersion force coefficient and its value is 0.3, in the range of 0.1~0.5 as recommended by Lahey et al. (1993). The wall force is proposed by Antal et al. (1991) based on the potential flow theory and is used to drive bubbles away from the wall: u  uL  CW  G  L G nW dB 2

FW , L

(7)

where CW is the wall force coefficient, and nW is the unit inward vector normal to the wall. The wall force coefficient is given as:

  d CW  max  CW 1  CW 2 B , 0  y  

(8)

where y is the distance between the bubble and wall. The wall force constants CW1 and CW2 as suggested by Ekambara et al. (2008) are -0.01 and 0.05, respectively. 2.3 Turbulence modeling The dispersed RNG k-ε turbulence model, which is recommended by Laborde-Boutet et al. (2009) to simulate gas-liquid flows, is adopted to calculate turbulent eddy viscosity:

   m  Lt   L  L k L       L  Lu L kL      L  L kL    TLRe : u L   L  L L t    k

7

(9)

   m  Lt   L  L  L        L  Lu L L      L  L  L     C 1TLRe : u L  C 2 L  L L  t  k     L R , L

  C  L t L

(10)

kL2

(11)

L

The related constants in the turbulence model are illustrated in detail by Laborde-Boutet et al. (2009).

3. Numerical details 3.1 Geometry The pilot-scale bubble columns with and without internals of Zhang et al. (2009) are simulated in this study. The details of geometry and operating conditions are given in Table 1. 16 steel tubes (0.025 m in diameter and 5 m in height) were installed in a 0.476 m in diameter and 5 m in height pilot-scale bubble column. The steel tubes were distributed in equilateral triangular with pitch of 0.09 m to cover about 5% of column cross sectional area. The gas was introduced into the column through a perforated plate. The perforated plate gas distributor had triangular arranged holes with 5×10-3 m in diameter and 0.06 m in pitch, which resulted in total free area of 0.61%. Local gas holdup and liquid velocity were measured in the well-developed region (axial position of 2.2 m above the gas distributor) by conductivity probe and Pavlov tube, respectively. the measured data were employed to validate the CFD model and to illustrate the effects of interfacial forces in the pilot-scale bubble columns with internals. 3.2 Mesh Gambit 2.4.6 was employed to generate hexahedral meshes for the fluid domain. The 5×10-3 m in diameter circle holes on the distributor were simplified as square holes to reduce the mesh number and computation burden. Typical grid layout for bubble columns with internals is illustrated in Fig. 1. Two layers of dense mesh were imposed near the wall to capture minimum liquid velocity point near wall and to satisfy standard wall function criterion (11.5~30<= y+ <=200~400). The first layer 8

mesh attached on the wall was 3×10-3 m and 2×10-3 m in height for the column wall and the tube internal wall, respectively. The grid independence studies indicate that the mesh number listed in Table 2 (835K cells for the bubble column without internals and 945K cells for the case with internals) is suitable taking the accuracy and computing resources into account. For example, for the cases with internals, the grid sizes for coarse mesh, medium mesh and dense mesh are 473K, 945K and 1.4M, respectively. Fig.2 shows the predicted overall gas holdup, local gas holdup and axial liquid velocity profiles at axial position of 2.2 m for different mesh sizes. The overall gas holdup discrepancy between coarse mesh and medium mesh is up to 21.8%, whereas it is only 4.2% between medium mesh and dense mesh. The local gas holdup profiles of medium mesh and dense mesh are almost overlapped, while evident gap appears between coarse mesh and medium mesh. Identical trend emerges for the axial liquid velocity profile. Therefore, medium mesh was employed for the CFD simulation. Table 1 Bubble column geometrical and simulation parameters. Configuration

Without internals

With internals

Column diameter (m)

0.476

0.476

Aspect ratio

10

10

Internals number

0

18

Internals diameter (m)

—

0.025

Triangular pitch (m)

—

0.09

Superficial gas velocity (m·s-1)

0.12, 0.31

0.12

Cell number

835K

945K

9

Fig. 1. Typical grid layout for bubble columns with internals.

Fig.2 Grid sensitivity analysis: (a) overall gas holdup; (b) local gas holdup profile; (c) axial liquid velocity profile 3.3 Numerical settings The related boundary conditions are set as follows: a uniform velocity inlet boundary condition was assigned at the inlet according to the operating superficial gas 10

velocity, and constant pressure outlet was set for the exit. It should be mentioned that the turbulent quantities at the inlet were defined according to liquid turbulent kinematic viscosity and the average mixing length as suggested by Guédon et al. (2017). For the column wall and the equipped tube wall, no-slip and free slip boundary conditions were set for liquid and gas phase, respectively. The pressure-velocity coupling was handled with phase-coupled SIMPLE method, and the first order upwind and first order implicit schemes were used for spatial and transient discretization, respectively. The convergence criteria was the absolute residuals for all transport equations below 10-3. To prevent numerical divergence, the time step was 0.0005 s for the first 5 s, 0.001 s for the following 10s and 0.002 s until the end. This time step method combined with maximal number iterations per time step of 20 guaranteed convergence for the simulated cases. The flow was simulated for 180 s and time averaging of flow properties was initiated when the quasi-steady state was reached (the physical time was about 60s). Therefore, the simulated data was time averaged over 120 s.

4. Results and discussion 4.1 Influence of drag force Drag law is vital to predict the hydrodynamics in bubble columns (Tabib et al., 2008; Xiao et al., 2013b; Zhang et al., 2006). The expressions of the considered drag laws are given in Table 2. For the drag models of Schiller and Naumann (1935) and Tomiyama (1998), bubble size of 5×10-3 m was suggested in the literature (Masood and Delgado, 2014; Xiao et al., 2013), and it was also set as 5×10-3 m in the simulations; furthermore, the following default correction term in the ANSYS fluent 15.0 is used to consider the swarm effect.

CD  CD 0 (1  G )

(12)

The drag model of Xiao et al. (2013) was deduced from the heterogeneous structure based on the dual-bubble size (DBS) distribution and energy-minimization multi-scale (EMMS) concept. Its advantages over other drag models are that the bubble size and swarm effect correction are no longer explicitly specified, since these 11

effects have already been considered in the original stability-constrained DBS model [39, 40]. Table 2 Drag laws investigated. Author

Model

Schiller and Naumann

CD 0

[36] Tomiyama [37]

Xiao et al. [38]

24(1  0.15ReB0.687 ) / ReB ReB  1000  ReB  1000 0.44

  16 48  8  Eo   CD 0  max  min  (1  0.15ReB0.687 ), ,   Re Re  B B  3  Eo  4   

CD

2 431.14  6729.02U G  35092.2U G U G  0.101m / s  d B 122.49  553.94U  741.24U 2 U G  0.101m / s G G 

To study the influence of drag model, the following two situations were simulated: drag force as the sole interfacial force and the other lateral forces considered besides the drag force. 4.1.1 Without lateral forces Fig. 3 shows the influence of drag model on the predicted gas holdup contour without considering lateral forces in the bubble columns with and without internals. The drag model of Tomiyama gives the highest gas holdup, while the drag model of Schiller and Naumann predicts the lowest value. This is related to Cd/dB values predicted by different drag models: 88(1-αG) m-1 for the drag law of Schiller and Naumann, 67 m-1 for the drag law of Xiao et al., and 245(1-αG) m-1 for the drag law of Tomiyama. Furthermore, for the bubble columns without internals, lower void fraction is observed near the column wall than in the column core, whereas in the cases with internals, it is observed that higher gas holdup around the internal tube wall and lower near the column wall. These observations are in accord with the simulated results of Larachi et al. (2006) and Laborde-Boutet et al. (2010). However, there is a remarkable discrepancy between the predictions and the experimental observations, since the experimental results in the gas-liquid pipe flows in the presence of rod bundles (Yang et al., 2013; Hosokawa et al., 2014; Arai et al., 2012) showed that close 12

to the tube wall, gas holdup tended to be lower rather than to be higher. It is worth noting that the gas holdup shows different behavior close to the tube wall and column wall, and the underlying physical mechanism will be illustrated in the discussion section.

Fig. 3. Influence of drag model on gas holdup contour without considering lateral forces at z=2.2 m under UG=0.12 m·s-1, (a) (d): Schiller and Naumann; (b) (e): Xiao et al.; (c) (f): Tomiyama. The influence of drag model on axial liquid velocity contour without considering lateral forces is displayed in Fig. 4. For the bubble columns without internals, all the drag model predicts the familiar core-annulus structure (up-flow in the core and down-flow near the wall). The drag law of Schiller and Naumann gives the most intensified large-scale circulation, and lowest circulation intensity is predicted with the drag model of Tomiyama. This may be related to the different gas holdup steepness in the column radial direction as shown in Fig. 3. In the bubble columns with internals, the core-annulus structure is still predicted, but the axial liquid velocity in the core is remarkably reduced in comparison with the hollow columns. This is 13

inconsistent with the experimental observation of an enhancement of large-scale liquid circulation (Al Mesfer, 2013; Bernemann et al., 1991; Chen et al., 1999; Jhawar and Prakash, 2014; Zhang et al., 2009). In addition, higher axial liquid velocity is predicted near the internal tubes, which is a result of higher gas holdup around the internals in Fig. 3. This trend has also been reported by Larachi et al. (2006) and Laborde-Boutet et al. (2010).

Fig. 4. Influence of drag model on axial liquid velocity contour without considering lateral forces at z=2.2 m under UG=0.12 m·s-1, (a) (d): Schiller Naumann; (b) (e): Xiao et al.; (c) (f): Tomiyama. Fig. 5 compares simulated results without considering lateral forces and experimental data of Zhang et al. (2009). In the hollow columns, the simulated gas holdup profiles are flatter than the measured data for all drag models. The drag law of Schiller and Naumann gives best agreement with measured gas holdup and liquid velocity, and the drag model of Tomiyama shows poorest agreement with experimental data. In the bubble columns with internals, all drag models predict much flatter holdup profile and drastically underestimate axial liquid velocity in the column 14

core in comparison with measured results. In addition, the two humps in the gas holdup and liquid velocity profiles corresponds to the higher gas holdup and axial liquid velocity near the tube internals as shown in Fig. 3-4. From Fig. 5, proper drag model without considering the lateral forces is capable of predicting hydrodynamics in the hollow columns, and this is in consistency with the conclusion in the literature (Sokolichin et al., 2004). Yet considering the drag as the sole interfacial force cannot guarantee good prediction for the bubble columns with internals. This different behavior in the bubble columns with and without internals indicates that column configurations may determine the applicability of CFD models.

Fig. 5. Influence of drag model on the predictions without considering the lateral forces under UG=0.12 m·s-1: (a) gas holdup, without internals; (b) axial liquid velocity, without internals; (c) gas holdup, with internals; (d) axial liquid velocity, with internals. For traditional drag models, such as Schiller and Naumann, or Tomiyama used in the present work, bubble size strongly impacts slip velocity and the simulated results 15

(Tabib et al., 2008; Xu et al., 2013), and how to select an appropriate bubble size for various superficial gas velocities or sparger designs is purely empirical. However, the drag law of Xiao et al. circumvents the issues through establishing relationship between Cd/dB and operating conditions with EMMS approach. If the operated superficial gas velocity in the hollow bubble column is increased to 0.31 m/s, the predictions with drag models of Schiller and Naumann (bubble diameter of 5×10-3 m) and Xiao et al. are depicted in Fig. 6. As is seen, the drag model of Schiller and Naumann substantially over-predicts gas holdup and slightly overestimates axial liquid velocity, while the drag model of Xiao et al. gives much better predictions. Therefore, drag model of Xiao et al. shows advantages over other drag models due to its wide flexibility with respect to operation conditions.

Fig. 6. Influence of drag model on the predictions without considering the lateral forces under UG=0.31 m·s-1: (a) gas holdup, without internals; (b) axial liquid velocity, without internals. 4.1.2 With lateral forces Fig. 7-8 displays the influence of drag model on the predicted gas holdup and axial liquid velocity contours when considering lateral forces. In the bubble columns without internals, all drag models predict gas holdup enrichment in the column core and give the large-scale core-annulus structure, which is similar to the estimations without considering lateral forces in Fig. 3-4. Moreover, the circulation intensity difference among three drag models is reduced when considering lateral forces. In the bubble columns with internals, however, considering lateral forces or not 16

gives rise to different results. In Fig. 7, lower gas holdup is observed around the internals, whereas it shows opposite trends without lateral forces in Fig. 3. Moreover, in Fig. 8, the higher axial liquid velocity around the internals observed in Fig. 4 disappears, and the internals’ enhancement of large-scale liquid circulation is surprisingly reproduced. Therefore, considering lateral forces in the bubble columns with internals is of more significance than in the hollow columns.

Fig. 7. Influence of drag model on gas holdup contour when considering lateral forces at z=2.2 m under UG=0.12 m·s-1, (a) (d): Schiller Naumann; (b) (e): Xiao et al.; (c) (f): Tomiyama.

17

Fig. 8. Influence of drag model on axial liquid velocity contour contour when considering lateral forces at z=2.2 m under UG=0.12 m·s-1, (a) (d): Schiller Naumann; (b) (e): Xiao et al.; (c) (f): Tomiyama. Fig. 9 shows comparison between simulated results and the experimental data when considering lateral forces. In the hollow columns, the gas holdup profile turns slightly steeper than that without lateral forces in Fig. 5. Predicted axial liquid velocity perfectly fits with measured data with all three drag models in Fig. 9(b). it indicates that considering lateral forces decreases the sensitivity of axial liquid velocity to the drag model in the hollow columns. In the bubble columns with internals, two valleys are clearly observed at the gas holdup and axial liquid velocity profiles in Fig. 9, which corresponds to the lower gas holdup and axial liquid velocity near the tube internals as shown in Fig. 7-8. For gas holdup, the drag model of Schiller and Naumann seems to give good estimation near the wall, and the drag model of Xiao et al. predicts well in the core region. As with axial liquid velocity, the predictions with the drag model of Xiao et al. amazingly match with experiments, while the drag model of Schiller and Naumann substantially 18

overestimates the results. Hence, the drag model of Xiao et al. is superior to other drag models in the bubble columns with internals when considering lateral forces. In the following, the drag model of Xiao et al. will be employed to study the sensitivity of lift force, turbulent dispersion force and wall force with UG=0.12 m·s-1.

Fig. 9. Influence of drag model on the predicted results with UG=0.12 m·s-1 when considering the lateral forces : (a) gas holdup, without internals; (b) axial liquid velocity, without internals; (c) gas holdup, with internals; (d) axial liquid velocity, with internals. 4.2 Influence of lift force To study the influence of lift force, simulations were performed with or without lift force, and the other interfacial forces were included. As shown in Fig. 10, the gas holdup and axial liquid velocity profiles tend to be flatter without the lift force. Furthermore, the axial liquid velocity is more sensitive to the lift force in the bubble column with internals, since when the lift force is included, the centerline axial liquid velocity increases 138% and 20.5% for the cases with and without internals, respectively. In addition, comparison with experimental data demonstrates that lift force may be optional in the hollow columns, but it cannot be ignored in the bubble 19

columns with internals to accurately predict hydrodynamics.

Fig. 10. Influence of lift force coefficient on the predicted results with UG=0.12 m·s-1: (a) gas holdup, without internals; (b) axial liquid velocity, without internals; (c) gas holdup, with internals; (d) axial liquid velocity, with internals. 4.3 Influence of turbulent dispersion force In order to explore the influence of turbulent dispersion force, cases with and without turbulent dispersion force were simulated. From Fig. 11, the gas holdup tends to increase when considering turbulent dispersion force. The gas holdup becomes flatter and lowers the liquid velocity in the bubble columns without internals. Nonetheless, for the cases with internals, turbulent dispersion force enhances large-scale liquid circulation, and this may come from the increased gas holdup in the center region. Thereby, the complex geometry drastically alters the flow sensitivity to the turbulent dispersion force.

20

Fig. 11. Influence of turbulent dispersion force coefficient on the predicted results with UG=0.12 m·s-1: (a) gas holdup, without internals; (b) axial liquid velocity, without internals; (c) gas holdup, with internals; (d) axial liquid velocity, with internals. 4.4 Influence of wall force Fig. 12 shows the predictions with or without the wall force. The wall force has effect only near the wall, as indicated in Eq. (8). In the hollow columns, the wall force decreases the gas holdup near the column wall, and the liquid velocity profile becomes steeper in Fig. 12(b). For the bubble columns with internals, the wall force also reduces the gas holdup close to the tube internal wall, as indicated by the two valleys observed on the gas holdup profile. Comparison between simulated results shows that the impact of the wall force is magnified in the bubble columns with internals. The wall force is limited to 5dB from the solid wall in Eq. (8). Thereby, the percentage of area impacted by wall force on the cross sectional area of the bubble column is:



[ D 2  ( D  5d B )2 ]  n[(dt  5d B ) 2  dt2 ] 10d B ( D  ndt )  25d B2 (n  1)  D2 D2 21

(13)

where dt is tube diameter, D is column diameter, and n is tube number. In the current simulation, φ is 10.2% for bubble columns without internals, whereas it increases to 25.3% for the cases with internals. Therefore, the bubble columns with internals are much more sensitive to the wall force.

Fig. 12. Influence of wall force on the predicted results with UG=0.12 m·s-1: (a) gas holdup, without internals; (b) axial liquid velocity, without internals; (c) gas holdup, with internals; (d) axial liquid velocity, with internals. Fig. 13 shows the contour plots of the predicted results without the wall force. Higher gas holdup and liquid velocity are observed around tube internals, similar to Fig. 3-4 without lateral forces. This implies that the decreased gas holdup close to the wall in Fig. 7 with lateral forces comes from the wall force rather than the lift force or the turbulent dispersion force. The conclusion can be also drawn from Fig. 10-12, since the two valleys on gas holdup profiles still can been observed without lift or turbulent dispersion force, while the absence of wall force causes the vanish of the valleys. Consequently, the wall force is more essential to predict the flow characteristics in the bubble columns with internals.

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Fig. 13. contour plots of the predicted contours without wall force at z=2.2 m with UG=0.12 m·s-1: (a) (b) gas holdup; (c) (d) axial liquid velocity. 4.5 Column wall and tube wall As mentioned above, without considering the lateral forces, gas holdup tends to increase close to the tube wall and to decrease near the column wall. In other words, gas bubbles seem to accumulate near the tube wall, and to escape from the column wall. The instantaneous liquid velocity vector without considering the lateral forces with UG=0.12 m·s-1 is depicted in Fig. 14. It is clearly indicated that constrained by the no-slip boundary of the wall, the instantaneous swirling flow can be observed close to the wall. Similar to the cyclone separator, the instantaneous centrifugal field exerts buoyancy force on the bubble and drives the bubble to the field center: Fb  (  L  G )VB 2 r

(14)

where ω is the angular velocity and VB is bubble volume. For the column wall, the centrifugal field center is the column center, and hence the bubble tends to escape from the column wall. However, for the wall of tube internals, the tube center is the centrifugal field center and leads to higher gas holdup closer to the tube outer wall. At the same time, the instantaneous horizontal component of the drag force may block 23

bubble movement in the centrifugal field, similar to the vertical gravity field. Therefore, without considering the lateral forces, the instantaneous secondary tangential flow field is the driving force for the large-scale flow. However, when considering the lateral forces (much larger than the buoyancy caused by the instantaneous centrifugal field), the large-scale circulation is mainly determined by the balance between the lateral forces (Krepper et al., 2005; Zhang et al., 2011): the wall force drives the bubble away from the wall (including column wall and tube wall), the negative lift force coefficient causes bubble moving to the higher liquid velocity region, and the turbulent dispersion force tends to reduce phase holdup spatial non-uniformity.

Fig. 14. Instantaneous liquid velocity vector with the drag model of Xiao et al. without considering lateral forces at z=2.2 m with UG=0.12 m·s-1 (colored by horizontal velocity magnitude): (a) without internals; (b) with internals. 4.6 Effect of internals As discussed in the previous section, some consensuses have been achieved in the effects of internals: the overall gas holdup is slightly increased, the large-scale liquid circulation is strengthened, and the turbulence properties is weakened. As to the gas holdup profile, Zhang et al. (2009), Youssef and Al-Dahhan (2009), Youssef et al. (2012) appeared to become steeper, and Chen et al. (1999) and Al Mesfer (2013) suggested insignificant change of slope. 24

Fig. 15 shows the effect of internals on the simulated results with the drag model of Xiao et al. when considering the lateral forces. The gas holdup slightly decreases in the center region (except the two valleys), and steep gradient is observed near the wall. For the liquid velocity, the internals strengthen the large-scale circulation, and this conforms to the measured data

(Al Mesfer, 2013; Bernemann et al., 1991; Forret et

al., 2003; Zhang et al., 2009). So the question is, why the internals slightly change gas holdup profiles, whereas large-scale circulation is remarkably enhanced as shown in Fig. 15? The underlying mechanism is that the turbulent viscosity is substantially decreased by the internals as shown in Fig. 16. The 1D model in the fully developed region [46] is:

  G  G   L  L  g 

du  dP 1 d   r L  Lm  Lt  L   0  dz r dr  dr 

(15)

The presence of internals shows insignificant effects on the first and second terms on the LHS, as observed in Fig. 15. Hence, for the third term on the LHS, the remarkable decrease of turbulent viscosity owing to the presence of internals in Fig. 16, tends to steepen the axial liquid velocity profile.

Fig. 15. Influence of internals on the predicted results with the drag model of Xiao et al. when considering lateral forces.

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Fig. 16. The predicted liquid turbulent viscosity with the drag model of Xiao et al. when considering lateral forces at z=2.2 m with UG=0.12 m·s-1: (a) without internals; (b) with internals.

5. Concluding remarks The effects of drag law and lateral forces (lift force, turbulent dispersion force and wall force) have been studied in the pilot-scale bubble columns with internals. The results showed that the drag model of Xiao al. is superior to that of Schiller and Naumann, and Tomiyama, since it is based on heterogeneous bubble size structure in bubble columns and has wide flexibility with respect to operation conditions. Furthermore, the lateral forces may be optional for the hollow columns, but they are required to accurately predict flow characteristics in the presence of internals. The lift force tends to steepen both gas holdup and liquid velocity profile. The turbulent dispersion force reduces non-uniformity of gas holdup and liquid velocity in the bubble columns without internals, whereas it tends to enhance large-scale circulation for the cases with internals. The wall force drives the bubble to escape from the wall and leads to lower gas holdup near the tube internals. Without wall force, unphysical higher gas holdup is observed around the internals. Moreover, the hydrodynamics in the bubble columns with internals is more sensitive to lateral forces in comparison with those without internals. In addition, due to the remarkable decrease of turbulent 26

viscosity, the equipped internals give rise to an enhancement of large-scale circulation in spite of insignificant change of gas holdup.

Acknowledgement The authors wish to acknowledge the long term support from the National Natural Science Foundation of China (Grant No. 91434121 and 91634203), the Ministry of Science and Technology of China (National Key R&D Plan, Grant No. 2017YFB0602500), and the Research Center for Mesoscience at Institute of Process Engineering, Chinese Academy of Sciences (Grant No. COM2016A004).

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