CFD Simulations for Continuous Flow of Bubbles through Gas-Liquid Columns: Application of VOF Method.

Curtin University of Technology From the SelectedWorks of Abid Akhtar Mr March, 2007 CFD Simulations for Continuous Flow of Bubbles through Gas-Liquid Columns: Application of VOF Method. Abid Akhtar Available at: https://works.bepress.com/abid_akhtar/3/

Chemical Product and Process Modeling Volume 2, Issue 1 2007 Article 9 CFD Simulations for Continuous Flow of Bubbles through Gas-Liquid Columns: Application of VOF Method Abid Akhtar Vishnu Pareek Moses Tadé Curtin University of Technology, Perth, Australia, m.akhtar@curtin.edu.au Curtin University of Technology, Perth, Australia, v.pareek@curtin.edu.au Curtin University of Technology, Perth, Australia, m.o.tade@curtin.edu.au Copyright c 2007 The Berkeley Electronic Press. All rights reserved.

CFD Simulations for Continuous Flow of Bubbles through Gas-Liquid Columns: Application of VOF Method Abid Akhtar, Vishnu Pareek, and Moses Tadé Abstract Hydrodynamics study of a continuous bubble chain rising through liquid column has been performed for laboratory scale bubble column using the volume-of-fluid (VOF) approach. The effect of operating and design parameters on the bubble size distribution and rise trajectory has been investigated for air-water system. For the same distributor, simulation results have indicated the formation of small bubbles at low superficial gas velocity and relatively large bubbles at higher velocities. The increase in the hole-size of distributor has shown similar behaviour. Analysis of bubble trajectories for different superficial gas velocities and distributors has demonstrated an oscillatory behaviour exhibited by small bubbles formed at low superficial gas velocity. A reasonable agreement between the predicted values of gas hold-up with the experimental work has validated the present model. KEYWORDS: bubble column, VOF, gas distributor, CFD, superficial gas velocity The authors gratefully acknowledge the financial support provided by the Australian Research Council (ARC) to conduct this research (Grant Application ID: DP0451314). Abid also thanks the Western Australian Energy Research Alliance (WA-ERA) for providing a top-up scholarship. Authors also express their thanks to Prof Dudukovic and Prof Al-Dahhan for granting permission to Abid to perform experiments in their labs at Washington University at St Louis.

Akhtar et al.: VOF Simulations for Continuous Flow of Bubbles 1. Introduction Bubble columns are presently used for a wide range of applications in both chemical and biochemical industry due to their relative simple construction; favourable heat and mass transfer properties and low operating cost (Deckwer, 1992; Degaleesan et al., 2001; Akhtar et al., 2006). However, due to the complexity of the flow involved, these reactors are not yet fully understood. The knowledge of bubble dynamics is of considerable importance for the proper design and operation of bubble columns. Among bubble properties the bubble size distribution is an important design parameter. It has direct influence on gas holdup and interfacial area. Therefore in order to qualify and quantify the performance of bubble columns, it is necessary to understand thoroughly the effects of different parameters like superficial gas velocity and distributor on the bubble dynamics prevailing in these columns (Clift et al., 1978; Chen et al., 2005). Among available simulation approaches, the volume of fluid (VOF) (Hirt et al., 1981) is one of the most well known methods for volume tracking in which the motion of all phases is modelled by solving a single set of transport equations with appropriate jump boundary conditions at the interface (Delnoij et al., 1997; Krishna et al., 1999). The only drawback of VOF method is the so-called artificial (or numerical) coalescence of gas bubbles which occurs when their mutual distances is less than the size of the computational cell, which also makes this approach memory intensive for simulation of dispersed multiphase flows in large equipment (Ranade 2002). In the literature the motion of single bubble is relatively well understood and extensive experimental data on shape and terminal velocity is available (Clift et al., 1978). Using that experimental data, simulations for single bubble rising in stagnant fluid have been performed by many researchers mostly in two-dimension (Krishna et al., 1999; Essemiani et al., 2001; Liu, Zheng, Jia et al., 2005) and few in three-dimension (Olmos et al., 2001; Van Sint Annaland et al., 2005). The rise trajectories of bubbles, their size and shape, rise velocity; effect of fluid properties on bubble dynamics and gas hold-up has been largely analysed in these efforts. Although these studies are useful in predicting the dynamic behaviour of single bubble, but cannot accurately model the behaviour of a bubble swarm or continuous chain in bubble column. Another aspect is the bubble-bubble interaction and higher level of turbulence, which becomes more important with continuous chain of bubbles where the inter-phase forces models based on single bubble studies do not provide good results. Therefore, despite all of these significant efforts in predicting the characteristics of flow with single bubble rise, the path instability of chain of bubbles in an otherwise stagnant fluid is still one of least understood aspects of bubble column which requires CFD investigations. Published by The Berkeley Electronic Press, 2007 1

Chemical Product and Process Modeling, Vol. 2 [2007], Iss. 1, Art. 9 In this innovative work, a numerical study has been conducted to study the behaviour of a continuous chain of bubbles under different operating conditions. For the first time computational intensive 3D volume-of-fluid simulations have been performed with comparatively larger laboratory scale bubble column (D = 20 cm, H = 1 m). The VOF approach implemented in the commercial code for CFD FLUENT has been used to investigate the characteristics of associated bubble rising through a. Simulations findings were compared with the Four Point Optical Probe experimental work performed at CREL, USA and a good agreement was obtained 2. Governing Equations and Numerical Scheme FLUENT 6.1 has used to simulate the motion of a continuous chain of bubbles rising through stagnant liquid. In the VOF model of FLUENT, the movement of the gas liquid interface is tracked based on the distribution of α G, the volume fraction of gas in a computational cell, where α G = 0 in the liquid phase and α G = 1 in the gas phase. Therefore, the gas liquid interface exists in the cell where α G lies between 0 and 1. The geometric reconstruction scheme that is based on the piece linear interface calculation (PLIC) method is applied to reconstruct the bubble-free surface. The surface tension is included by defining a constant value (0.072 N/m), while turbulence is modelled with k-ε turbulence model. 2.1. Modeling Equations. The VOF approach used to simulate bubble motion is based on the Navier-Stokes equations, which are given for the mixture phase: 2.1.1. The Continuity Equation t r ( ρ) +.( ρ v) = 0 (1) 2.1.2. The Momentum Equation. A single momentum equation, which is solved throughout the domain and shared by all the phases, is given by: http://www.bepress.com/cppm/vol2/iss1/9 2

Akhtar et al.: VOF Simulations for Continuous Flow of Bubbles t r r r r rt r r ( ρv) +.( ρ v v) = - p +. µ ( v + v + ρg + F (2) 2.1.3. The Volume Fraction Equation. The tracking of the interface between the gas and liquid is accomplished by the solution of a continuity equation for the volume fraction of gas, which is: t r ( α ) + v. α = 0 G G (3) The volume fraction equation is not solved for the liquid; the liquid volume fraction is computed based on the following constraint: α + α = 1 (4) G L Where α G and α L is the volume fraction of gas and liquid phase respectively. 2.1.4. Surface Tension The surface tension model in FLUENT is the continuum surface force (CSF) model proposed by Brackbill et al. (FLUENT 6.1 Manual). With this model, the addition of surface tension to the VOF calculation results in a source term in the momentum equation 2.2. Turbulence Modeling. The turbulence in the continuous phase has been modelled using a modified k-ε turbulence model available in FLUENT 6.1, which is widely used turbulence model to simulate turbulence eddies. This model accounts for the transport not only of the turbulence velocity scale but also of the length scale. It employs a transport equation for the length scale that allows the length scale distribution to be determined even in complex flow situations like in bubble column. It is the simplest model that promise success for flows for which the length scale cannot be prescribed empirically in an easy way. Published by The Berkeley Electronic Press, 2007 3

Chemical Product and Process Modeling, Vol. 2 [2007], Iss. 1, Art. 9 2.3. Differencing Schemes. In order to minimize numerical diffusion, the first order up-wind differencing scheme is applied for the solution of momentum equation. Higher order scheme (3 rd order QUICK) has given similar results. The pressure-implicit with splitting of operators (PISO) pressure velocity-coupling scheme, part of the SIMPLE family of algorithms, is used for the pressure velocity-coupling scheme, which is recommended for usual transient calculations. Using PISO allows for a rapid rate of convergence without any significant loss of accuracy. Pressure is discretized with a PRESTO scheme. Other schemes (linear or second-order schemes) lead to strong divergence or to slow convergence (Body force weighted scheme). Segregated algorithms converge poorly unless partial equilibrium of pressure gradient and body forces is taken into account. FLUENT provides an optional implicit body force treatment that can account for this effect, making the solution more robust. The volume fraction equation for gas (Equation.3) was solved using an explicit time-marching scheme and the maximum allowed Courant number was set to 0.25. Under relaxation factor used for pressure and momentum were 0.6 and 0.4 respectively. For turbulence parameters, intensity and hydraulic diameter specification was used. A typical value of time step 5 10-4 s was used throughout the simulations. The solution was converged in less than 50 iteration at each time step. A simulation time of 5 and 2.5 s is used for each run of two and three-dimensional simulations respectively. 2.4. Physical Properties The properties of air and water were used in the transport equations when the computational cell is in the liquid or the gas phase, respectively. At interface between the gas and liquid phases, the mixture properties of the gas and liquid phases based on the volume fraction weighted average were used. The density and viscosity in each cell at interface were computed by the application of following equations: ρ = αg ρg + (1- αg) ρ L (5) µ = α G µ G + (1- α G) µ L (6) Where ρ G, ρ L, µ G and µ L is density and viscosity of gas and liquid phase respectively, while α G is the volume fraction of gas. http://www.bepress.com/cppm/vol2/iss1/9 4

Akhtar et al.: VOF Simulations for Continuous Flow of Bubbles 2.5. Interface Tracking. To overcome the problem of numerical diffusion which most standard differencing schemes suffer, the geometric reconstruction scheme, which is the most accurate scheme in FLUENT was used (Fluent 6.1 Manual). It assumes that the interface between two fluids has a linear slope within each cell, and uses this linear shape for calculation of the advection of fluid through the cell faces. The first step in this reconstruction scheme is calculating the position of the linear interface relative to the centre of each partially filled cell, based on information about the volume fraction and its derivatives in the cell. The second step is calculating the adverting amount of fluid through each face using the computed linear interface representation and information about the normal and tangential velocity distribution on the face. The third step is to calculate the volume fraction in each cell using the balance of fluxes calculated during the previous step (Rider et al., 1998). Pressure outlet 100 cm Velocity Inlet (a) (b) Figure 1. 3D Domain of the Bubble column (a) Grid configuration with boundary conditions, (b) Experimental Setup. Published by The Berkeley Electronic Press, 2007 5

Chemical Product and Process Modeling, Vol. 2 [2007], Iss. 1, Art. 9 2.6. Domain Description. Simulations were performed both for two and three-dimensional bubble column with different size distributors. Figure 1 (a) displays 3D computational domain of column with boundary conditions while experimental setup used for model validation at CREL, USA is shown in Figure 1 (b). Table 1 summaries different geometrical configurations used in the present work. Results with gas distributor size ranging from 2-10 cm are reported in this paper. Simulations were run for different mesh sizes and reported number of cells is for grid independent solution. Velocity inlet and pressure outlet boundary conditions were applied at inlet and outlet respectively. At the walls, no-slip boundary condition was imposed. The column was modelled as an open system, so the pressure in the gas space above the initial liquid column was equal to the ambient pressure (101.325 kpa). Table 1. Various Configurations of Distributors Used for Simulation Hole size D (cm) Ug (cm/s) 2 D 3 D No of mesh nodes Hole size D (cm) 10 1, 5, 10 9900 10 Ug (cm/s) 1, 5, 10 No of mesh nodes 33440 5 0.25, 1, 5, 10 10020 5 1, 5, 10 2 0.01, 1, 5 12050 2 1, 5, 10 90984 97512 In order to reduce computational time for these memory intensive simulation studies, all simulations were performed on high-speed Xeon dual processor computers (having four 2.66GHz processors). Parallel processing of FLUENT was performed which allowed four simulations to be run simultaneously. Simulations were performed till a fully developed flow field was ensured by examining overall mass balance and time history of relevant flow variables. Depending on the geometrical configuration of the model each simulation required between 1 5 days of CPU time. http://www.bepress.com/cppm/vol2/iss1/9 6

Akhtar et al.: VOF Simulations for Continuous Flow of Bubbles 3. Results and Discussions Simulations were performed with VOF approach to study the effect of superficial gas velocity and single hole gas distributor on the flow characteristics and bubble dynamics of continuous bubble chain. First of all the effect of superficial gas velocity on bubble dynamics and gas hold-up was studied for same size gas distributor, then the effect of gas distributor was examined at constant superficial gas velocity. All simulations were carried out both in two and three-dimensions. As mentioned before, most of previous work has been done either with single bubble rise or otherwise with two-dimensional domain of bubble column. In present study, 2D simulations have been included to elaborate the significance of 3D work. A close agreement of bubble rise velocity and volume average gas holdup with experimental work performed at CREL, USA as well as with correlations from literature (Joshi et al., 1979; Hikita et al., 1980; Deckwer, 1992) has validated the present model. 3.1. Effect of Superficial Gas Velocity on Continuous Bubble Chain. In order to study the effect of superficial gas velocity on bubble size distribution for continuous bubble flow, simulations were performed with single distributor (hole size = 10 cm) and different superficial gas velocities ranging from 1-10 cm/s. Figure 2 and 3 displays snapshots of volume fraction of air for two and three-dimensional simulations respectively. It is clear that bubble size is a function of superficial gas velocity. It increases with the increase in superficial gas velocity, which is in accordance with the experimental observations of Clift (1978) and other researchers (Chen et al., 2004). Small bubbles produced at low superficial gas velocity (for example with 1 cm/s refer to animation 1) had low bubble rise velocity, which was measured using the rise position of nose of initial bubble produced from the orifice (Figure 2). It was also noted that the coalescence times for leading bubble with 1.0, 5.0 and 10 cm/s superficial gas velocity were 4.5, 3.5 and 2.75 s respectively. Leading bubble produced from orifice had bigger diameter when compared to the trailing bubbles which might be due to less effects of wall and other surface forces on the trailing bubbles. The trailing bubbles were observed to move in a rectilinear manner at low superficial gas velocities (1.0 cm/s) while at higher superficial gas velocities these bubbles exhibit a slightly zigzag or oscillatory behaviour. This can be explained by considering the behaviour of a single rising bubble. A rising bubble pushes the liquid in front of it while the liquid behind it is sucked by the bubble wake which lies directly behind the bubble; therefore, it is due to the greater degree of turbulence and the formation wake behind the bubbles which is responsible for the oscillatory Published by The Berkeley Electronic Press, 2007 7

Chemical Product and Process Modeling, Vol. 2 [2007], Iss. 1, Art. 9 behaviour of the trailing bubbles behind large rising bubbles, which is in accordance with experimental observations of Olmos et al. (2001) and Chen et al. (2004). This behaviour is more clear in the animation files at Ug = 10 cm/s as given in the supplement material (refer to Animation No. 1: d = 10 cm, Ug = 1 cm/s and Animation No. 2: d = 10 cm, Ug = 10 cm/s). These animations show the formation of small spherical bubbles at low velocities and large ellipsoidal bubbles at higher velocities. The obtained bubble shapes are in agreement with previous experimental observations (Clift et al., 1978). An oscillatory or zigzagging behaviour was exhibited by small spherical 0.25s 0.5s 0.75s 1.0s 1.25s 1.5s 1.75s 2.0s 2.25s 2.5s 2.75s 3.0s 3.25s 3.5s 3.75s 4.0s 4.25s 4.5s 4.75s 5.0s (a) 0.25s 0.5s 0.75s 1.0s 1.25s 1.5s 1.75s 2.0s 2.25s 2.5s 2.75s 3.0s 3.25s 3.5s 3.75s 4.0s 4.25s 4.5s 4.75s 5.0s (b) 0.25s 0.5s 0.75s 1.0s 1.25s 1.5s 1.75s 2.0s 2.25s 2.5s 2.75s 3.0s 3.25s 3.5s 3.75s 4.0s 4.25s 4.5s 4.75s 5.0s (c) Figure 2. Two-dimensional effect of gas superficial velocity on bubble size distribution. (Column Width = 20 cm, H = 100 cm, Hole size = 10 cm ). (a) Ug = 1 cm/s. (b) Ug = 5 cm/s. (c) Ug = 10 cm/s. bubbles which are produced at low superficial gas velocities. The observed behaviour is consistent with experimental results reported by Liu et al. (2005). This zigzagging behaviour could be attributed due to the ability of small bubbles to travel toward the wall in case of air-water two-phase bubbly flow as mentioned by Tomiyama et al. (1993). It is also well evident from Animation 1 that nearly same size bubbles were produced at comparatively low superficial gas velocities. http://www.bepress.com/cppm/vol2/iss1/9 8

Akhtar et al.: VOF Simulations for Continuous Flow of Bubbles In case of three-dimensional simulations, the effect of superficial gas velocity similar to that observed in two-dimensional simulations (Figure 3). For example, a similar qualitative increase in bubble size and zigzag trajectory of trailing bubbles was observed with the increase in gas velocity. 0.25s 0.5s 0.75s 1.0s 1.25s 1.5s 1.75s 2.0s 2.25s 2.5s (a) 0.25s 0.5s 0.75s 1.0s 1.25s 1.5s 1.75s 2.0s 2.25s 2.5s (b) Figure 3. 0.25s 0.5s 0.75s 1.0s 1.25s 1.5s 1.75s 2.0s 2.25s 2.5s (c) Three-dimensional effect of gas superficial velocity on bubble size distribution. (Column Width = 20 cm, H = 100 cm, Hole size = 10 cm). (a) Ug = 1 cm/s. (b) Ug = 5 cm/s. (c) Ug = 10 cm/s. However, under the same conditions, quantitative comparisons of bubble position have shown some interesting trends in bubble rise velocity in case of 3D simulations when compared with those shown under 2D simulations. For example, at 5 cm/s superficial velocity, the first bubble ejected in 2D simulation had taken 3.5 s to reach to the top of the column (coalescence) while in case of 3D simulations, it took only 2.5 s. In order to validate the qualitative results in Figures 2 3, bubble rise velocities were calculated from a linear regression of the z-coordinates of the nose of the bubble during its upward motion (a similar approach to that used by Krishna et al. (2000)). Calculated values of bubble rise velocities are plotted Published by The Berkeley Electronic Press, 2007 9

Chemical Product and Process Modeling, Vol. 2 [2007], Iss. 1, Art. 9 against superficial gas velocity and are shown in Figure 4. This quantitative increase in bubble rise velocity with increase in superficial gas velocity is consistent with the qualitative behaviour. The bubble rise velocity was varied from 34 45 cm/s in 3D and from 24.5 27 cm/s in 2D simulations. Calculated rise velocities in 2D simulations were ~ 30% lower than those in 3D simulations which is in accordance with Krishna et al. (2000) results obtained with 2D circular cap bubble and 3D spherical cap single bubble rise. This discrepancy with 2D simulations can be explained on the basis of the 3D wake exhibited by the bubbles (Olmos 2001), which cannot be accurately modeled with 2D simulations. Therefore for a precise prediction of bubble rise velocity and other hydrodynamic characteristics of bubble, 3D simulation is better choice. 50 10 cm Distributor (2D) 10 cm Distributor (3D) Deckwer Correlation 45 40 Ub (cm/s) 35 30 25 20 1 2 3 4 5 6 7 8 9 10 Ug (cm/s) Figure 4. Quantitative comparison of the bubble rise velocities with constant diameter gas distributor (D = 20 cm, H = 100 cm, d = 10 cm) In order to provide further verification of CFD simulations, a comparison of bubble rise velocity was made with experimental correlation from Deckwer (1992). Dotted line plotted with Deckwer equation in Figure 4 was very close to the calculated rise velocities with 3D simulations particularly at lower superficial gas velocities. A slight deviation at higher superficial gas velocities could be attributed to the use of single 10 cm distributor instead of actual gas distributors. The results with 2D simulations were significantly different from Deckwer s correlation which indicates the significance of 3D simulations for bubble rise velocity prediction. http://www.bepress.com/cppm/vol2/iss1/9 10

Akhtar et al.: VOF Simulations for Continuous Flow of Bubbles 3.2. Effect of Gas Distributor (Hole size) on Bubble Size Distribution. Effect of distributor hole size on bubble size distribution and hydrodynamics was studied at constant superficial gas velocity (1.0 cm/s). Three distributors with hole size ranging from 2-10 cm (Table 1) were selected for two and three-dimensional simulations. Figure 5 shows two-dimensional effect of gas distributor on bubble size distribution. Although these results appear similar if just looked at bubble sizes, but one can observe decrease in rise velocity with decrease in hole-size which for example, becomes quite apparent by comparing bubble rise trajectories for 5 and 2 cm size distributor. Therefore, it can be inferred that the bubble size has similar dependency on hole size as superficial gas velocity. Small bubbles usually formed from small size distributor and have low rise velocities. A steady state rise of these bubbles without much deflection from centre position can also be observed. Three-dimensional simulations shown in Figure 6 show a similar behaviour. Relatively bigger size bubbles were formed with 10 cm hole when compared with 5 and 2 cm and their rise time to the same height of column was less (refer to supplementary material, Animation No. 3: d = 2 cm, Ug = 1 cm/s). Comparison of Figure 5 with Figure 6 shows that at relatively low superficial bubbles exhibited rectilinear trajectory. Results obtained are overall in well agreement with the earlier work reported by Cartland et al. (2004) for identical problem by using algebraic slip mixture model. In that effort inlet velocity conditions were applied to 80 % of the base of the column in contrast to the 50 % in present work, still the rectilinear trajectory of bubbles is comparable for the range of superficial gas velocity considered in our simulations. Published by The Berkeley Electronic Press, 2007 11

Chemical Product and Process Modeling, Vol. 2 [2007], Iss. 1, Art. 9 0.25s 0.5s 0.75s 1.0s 1.25s 1.5s 1.75s 2.0s 2.25s 2.5s 2.75s 3.0s 3.25s 3.5s 3.75s 4.0s 4.25s 4.5s 4.75s 5.0s (a) 0.25s 0.5s 0.75s 1.0s 1.25s 1.5s 1.75s 2.0s 2.25s 2.5s 2.75s 3.0s 3.25s 3.5s 3.75s 4.0s 4.25s 4.5s 4.75s 5.0s (b) 0.25s 0.5s 0.75s 1.0s 1.25s 1.5s 1.75s 2.0s 2.25s 2.5s 2.75s 3.0s 3.25s 3.5s 3.75s 4.0s 4.25s 4.5s 4.75s 5.0s (c) Figure 5. Two-dimensional effect of gas distributor on bubble size distribution. (Column Width = 20 cm, H = 100 cm, Ug = 1 cm/s ) (a) 10 cm hole. (b) 5 cm hole. (c) 2 cm hole. http://www.bepress.com/cppm/vol2/iss1/9 12

Akhtar et al.: VOF Simulations for Continuous Flow of Bubbles 0.25s 0.5s 0.75s 1.0s 1.25s 1.5s 1.75s 2.0s 2.25s 2.5s (a) 0.25s 0.5s 0.75s 1.0s 1.25s 1.5s 1.75s 2.0s 2.25s 2.5s (b) 0.25s 0.5s 0.75s 1.0s 1.25s 1.5s 1.75s 2.0s 2.25s 2.5s (c) Figure 6. 3D effect of gas distributor on bubble size distribution. (Column Width = 20 cm, H = 100 cm, Ug = 1 cm ) (a) 10 cm hole. (b) 5 cm hole. (c) 2 cm hole. Figure 7 plots calculated bubble rise velocities as a function of hole-size at constant superficial gas velocity (1.0 cm/s). For the investigated range of holesizes (2-10 cm), bubble rise velocities varied from 13-23 cm/s and 15-34 cm/s for two and three-dimensional simulations respectively. Interestingly, as in bubble rise velocity curves (Figure 5), there was an offset between 2 and 3D results (10 30 %), it can still be observed that this difference is not prominent with small size distributor at low superficial gas velocity. When compared with Deckwer s equation, both 2D and 3D simulations failed to agree at smaller distributor diameters. However, this apparent disagreement can be attributed to the use of single hole-size in these simulations that resulted in very low perforations compared to those used in Deckwer s work. Even so, 2D simulations, in general gave a very poor comparison with Deckwer s equation throughout the entire domain. In contrast, 3D simulations gave reasonable quantitative approximation of bubble rise velocities, particularly those with bigger distributor hole-sizes. Published by The Berkeley Electronic Press, 2007 13

Chemical Product and Process Modeling, Vol. 2 [2007], Iss. 1, Art. 9 35 Ug = 1 cm/s (2D) Deckwer Correlation Ug = 1 cm/s (3D) 30 Bubble rise velocity (cm/s) 25 20 15 10 2 3 4 5 6 7 8 9 10 Gas distributor hole size (cm) Figure 7. Quantitative comparison of the bubble rise velocities with constant superficial gas velocity (D = 20 cm, H = 100 cm, Ug = 1.0 cm/s) 3.3. Effect of Grid Size. The effect of varying the mesh size on the accuracy of the solution was investigated for 2 D column. Three sets of grids with size ranging from 3 6 mm and 10 cm gas distributor were used for grid dependency study. Figure 8 shows the effect of mesh size on the calculated bubble rise velocity as a function of time. Figure 8. Effect of grid size on bubble rise velocity (Column Width = 20 cm, Height = 100 cm, hole size = 10cm, Time = 2.5 s, Ug = 5 cm/s). http://www.bepress.com/cppm/vol2/iss1/9 14

Akhtar et al.: VOF Simulations for Continuous Flow of Bubbles The intermediate grid size (5mm) is shown to yield solutions of sufficient accuracy (with respect to the trad-off between computational cost and the features being studied), and is chosen for the results present in this paper. 4. Model Validation In order to achieve quantitative validation of simulation results, volume-averaged gas hold-ups for various distributors were plotted as a function of superficial gas velocity as shown in Figure 9 which shows that the gas hold-up increased with superficial gas velocity and this is in accordance with the data available in the literature. Predicted values of volume-averaged gas hold-up for 3 D simulations have been compared with the experimental work of Joshi et al.(1979) and Hikita et al.(1980) (d = 1.1 cm; single nozzle). A close agreement between simulations and experimental work is obvious from Figure 9. It is also noticeable from these curves that there is slight increase in gas hold-up value with decrease in hole-size, which can be attributed due to the formation of small bubbles. Interestingly, this dependency of gas hold-up on superficial gas velocity was less at relatively low and high superficial gas velocities, which is also in accordance with the Andou and coworkers (1996) experiments. 0.25 0.2 Joshi & Coworkers (1979) Hikita & Coworkers (1980) 3D VOF Model (10 cm Distributor) 3D VOF Model (5 cm Distributor) 3D VOF Model (2 cm Distributor) Gas Hold-up (Fraction) 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 9 10 Gas Superficial Velocity ( cm/s) Figure 9. Comparison o f volume average gas hold-up for 3-D simulations with literature data. Further validation of the present VOF model was achieved by performing a series of experiments with 10 cm (ID) and 80 cm high column at Chemical Published by The Berkeley Electronic Press, 2007 15

Chemical Product and Process Modeling, Vol. 2 [2007], Iss. 1, Art. 9 Reaction Engineering Laboratory (CREL), Washington University in St. Louis, USA. A single hole distributor (5 mm) was utilized for this work. Figure 10 compared overall gas hold-up results obtained from experiments and simulation. A close agreement between experiments and 3D simulations is evident which has validated the significance of VOF simulations with continuous chain of bubbles. 30 Experimal Results 3 D Simulation Results 25 Overall Gas Holdup (%) 20 15 10 5 0 0 2 4 6 8 10 12 14 16 18 20 Ug (cm/s) Figure 10. Experimental Validation of VOF simulations (Hole diameter = 5 mm, Column Diameter = 10 cm, Column Height = 80cm, Static Liquid Height = 59 cm) 5. Conclusions In this paper, an analysis of the applicability of the VOF model for simulation of continuous bubble flow for gas-liquid system has been carried out. The effect of superficial gas velocity and distributor hole-size on the bubble dynamics is studied both for two and three-dimensional bubble columns. It has been observed that bubble rise velocity and bubble size increase with increase in gas superficial velocity. Increase in bubble size with distributor hole has also been demonstrated by simulation results. The bubble rise velocity, bubble shape, typical rise trajectories and gas hold-up has been compared with the experimental results and a close agreement with 3D simulations is demostrated. In conclusion, the http://www.bepress.com/cppm/vol2/iss1/9 16

Akhtar et al.: VOF Simulations for Continuous Flow of Bubbles efficiency of 3D VOF method to compute the shape and the motion of continuous bubble chain with different superficial gas velocity and distributors has been established. Nomenclature Ug = Gas superficial velocity [ms -1 ] v r = Velocity vector [ms -1 ] G = Acceleration due to gravity [ms -2 ] F r = External body force [N] T = Time [s] P = Static pressure [Pa] D = Column diameter [m] D = Hole diameter [m] Greek Symbols α G α L ρ L ρ G µ L µ G = Volume fraction of the gas phase in the computational cell = Volume fraction of the liquid phase in the computational cell = Liquid density [kgm -3 ] = Gas density [kgm -3 ] = Liquid viscosity [kgm -1 s -1 ] = Gas viscosity [kgm -1 s -1 ] σ = Surface tension [N/m] REFERNCES: Akhtar, M. A., M. O. Tade and V. K. Pareek; "Two-Fluid Eulerian Simulation of Bubble Column Reactors with Distributors," Journal of Chemical and Engineering of Japan, Vol. 39, No. 8, 831-841 (2006) Andou, S., K. Yamagiwa and A. Ohkawa; "Effect of Gas Sparger Type on Operational Characteristics of a Bubble Column under Mechanical Foam Control," Journal of Chemical Technology and Biotechnology, Vol. 66, No., 65-71 (1996) Bertola, F., G. Baldi, D. Marchisio and M. Vanni; "Momentum transfer in a swarm of bubbles: estimates from fluid-dynamic simulations," Chemical Engineering Science, Vol. 59, No. 22-23, 5209-5215 (2004) Published by The Berkeley Electronic Press, 2007 17

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