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1113Gas–liquid flow in stirred reactors Trailing vortices and gas accumulation behind impeller blad


Chemical Engineering Science 54 (1999) 2305}2315

Gas}liquid #ow in stirred reactors: Trailing vortices and gas accumulation behind impeller blades
Vivek V. Ranade *, Vaibhav R. Deshpande
Reaction Engineering Group, Faculty of Chemical Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Swiss Center for Scientixc Computing, Via Cantonale, CH-6928 Manno, Switzerland

Abstract In a gas}liquid stirred reactor, gas tends to accumulate in low-pressure regions behind the impeller blades. Such gas accumulation signi"cantly alters impeller performance characteristics. We have computationally investigated gas}liquid #ow generated by a Rushton (disc) turbine. Rotating Rushton turbine generates trailing vortices behind the blades, which in#uence the gas accumulation in the impeller region. Characteristics of these trailing vortices were "rst investigated by considering a model problem of #ow over a single impeller blade. Predicted results were compared with the published experimental data. Circulation velocity and turbulent kinetic energy of the trailing vortices were found to scale with blade tip velocity. Several numerical experiments were carried out to understand interaction of gas bubbles and trailing vortices. Gas}liquid #ow in stirred vessel was then simulated by extending the computational snapshot approach of Ranade and Dometti (Chem. Engng Res. Des., 74, 476}484, 1996). The approach was able to capture the main features of gas}liquid #ow in stirred vessels. The detailed analysis of predicted results with reference to experimental data and the results obtained for #ow over a single impeller blade will be useful for extending the scope of computational #uid dynamics (CFD) based tools for engineering gas}liquid stirred reactors. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Trailing vortices; Gas}liquid #ow; Stirred vessels; CFD

1. Introduction Stirred reactors are widely used for gas dispersion in chemical and bio-chemical industry. Despite the widespread use of stirred reactors, #uid dynamics of these reactors, especially for multiphase systems, is poorly understood. It is essential to develop and apply new tools to enhance our understanding of #uid dynamics of stirred reactors. In this paper, we have computationally studied the behaviour of gas}liquid #ow near Rushton (disc) turbine blades. In a stirred reactor agitated by a Rushton turbine, the #ow approaching from above and below the disc interact with the rotating vertical #at blade to generate trailing vortices behind the blade. A schematic of trailing vortex is shown in Fig. 1. The #ow processes associated with the trailing vortex region are of interest, especially for

*Corresponding author; Present address: Chemical Engineering Division, National Chemical Laboratory, Pune 411008, India. Tel./fax: 00 91 203 93041; e-mail: vvr@dalton.ncl.res.in.

gas}liquid stirred reactors. When gas bubbles are introduced in such a vessel, they rise upward from the sparger (along with the liquid #ow) and interact with trailing vortices. These vortices trap gas bubbles by centrifugal action and enhance gas accumulation behind the impeller blades. Such gas accumulation (the so-called gas cavities) signi"cantly alters #ow around impeller region and therefore performance of gas}liquid stirred reactor. Apart from the local #ow characteristics near the impeller, two very important gross characteristics, namely the power dissipation and the pumping capacity of the impeller get a!ected. It is therefore essential to understand and develop a capability to predict the #ow and gas accumulation around impeller blades. This will lead to better computational tools for engineering gas}liquid stirred reactors. Most of the earlier attempts of modelling #uid dynamics of stirred reactors were based on steady-state analyses and were for single-phase #ows (reviewed by Ranade, 1995). Most of these investigators have treated the rotating impeller as a black box. This approach can not capture details of the #ow between the impeller blades

0009-2509/99/$ } see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 3 0 1 - 7

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Fig. 1. Schematic of trailing vortex behind an impeller blade (from Yianneskis and Whitelaw, 1993).

and therefore can not be used for understanding gas accumulation behind impeller blades. Recently attempts have been made to simulate full time-varying #ow within and outside the impeller region with the combination of moving and deforming grids or sliding mesh (for example, Harris et al., 1996). These methods, however, require excessive computational resources. Application of this approach for the simulation of gas}liquid #ows has not been attempted yet. Ranade and Van den Akker (1994) have used a simpli"ed quasi-steady state (computational snapshot) approach to simulate gas}liquid #ow in stirred vessels. The approach showed promising results and indicated a possibility of simulating accumulation of gas behind the impeller blades for the "rst time. Ranade and Dometti (1996) have further developed the snapshot approach to make it applicable for impeller of any shape. Ranade (1997) has presented detailed evaluation of this approach for simulating single-phase #ow generated by a standard Rushton turbine. It was also successfully used by Ranade and Deshpande (1997) to simulate interaction between multiple Rushton impellers. In this work, we examine the possibility of using the modi"ed snapshot approach to simulate gas}liquid #ow in stirred reactors. The complex nature of the #ow "elds in stirred reactors and need for validating CFD computations have motivated a number of experimental studies. In particular, a large body of knowledge has been accumulated on the characteristics of #ow generated by a single Rushton turbine. Most of the studies, however, were restricted to measurements of mean and turbulence characteristics in the bulk region of the vessel. Some of the recent studies have provided quantitative information about #ow within and near impeller blades (Yianneskis and Whitelaw, 1993; Stoots and Calabrese, 1995; Schafer et al., 1997; Lee and Yianneskis, 1998). Analysis of these data indicates that the well-de"ned vortices (diameter of about half of the blade width) trail the blades of Rushton

turbines. Considering the complex and small scale #ow structures, it is not surprising that none of the CFD simulations of stirred vessels so far could capture these trailing vortices. The grid size requirements to resolve these small-scale #ow structures will be prohibitive while simulating #ow in the whole stirred vessel. Considering the importance of trailing vortices in the case of gas}liquid #ow in stirred reactors (Van't Riet and Smith, 1973, 1975), it will be useful to obtain detailed quantitative information about these vortices and their interaction with gas bubbles. No such information is available in the published literature. We therefore thought it to be desirable to formulate a model problem, which mimics the main features of trailing vortices, observed in stirred vessels. Recently, Rigby et al. (1997) have investigated bubble break-up and #ow around a ventilated gas cavity by considering a model #ow problem. They have, however, considered a two-dimensional, single-phase #ow around an assumed shape of the cavity and have not observed any trailing vortices. In this work, we have considered a model problem of #ow over a single impeller blade. Apart from enhancing our understanding and giving useful information for numerical simulation of vortices, such a study may lead to development of a useful sub-model which can be incorporated in overall simulation of gas}liquid stirred reactors. Simulations of model problem and of stirred reactor are discussed in Sections 2 and 3, respectively. These results will have signi"cant implications for extending the scope of CFDbased tools for engineering gas}liquid stirred reactors.

2. Vortices behind a single blade 2.1. Formulation of model problem A model problem of #ow of water over a single blade was considered. A Cartesian geometry (as shown in Fig. 2) was considered to avoid complications of radially outward #ow due to rotational e!ects. The considered solution domain (0.1;0.1;0.2 m) was signi"cantly larger than the blade dimensions (0.025;0.02;0.001 m). A face parallel to the blade was considered as an inlet of the solution domain where boundary conditions of velocity, turbulence intensity and volume fraction were speci"ed. To maintain the similarity with the rotating #ow in stirred vessel, inlet velocity was speci"ed as a linear function of distance (x) from the origin (this is indicated by the arrows shown at the top row of inlet cells in Fig. 2). The incoming liquid was assumed to have zero turbulence intensity. The face parallel to the inlet at the opposite end of the solution domain was de"ned as an outlet with standard outlet boundary conditions. All the remaining four faces of the solution domain parallel to the inlet #ow were speci"ed as zero gradient surfaces,

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assuming that solution domain is su$ciently large to avoid any disturbance at the domain edges. Typical grid distribution and boundary conditions are shown in Fig. 2. Time-averaged Navier}Stokes equations were solved (along with a turbulence model) using the commercial CFD code, FLUENT (of Fluent Inc., USA). A two-#uid model was used to simulate gas}liquid #ow. Model equations and boundary conditions used in this work are summarised in Table 1. Additional details can

be found in Ranade and van den Akker (1994) or documentation of FLUENT. Simulation results are discussed below. 2.2. Results and discussion Typical predicted results for the single-phase case are shown in Fig. 3 in the form of vector and contour plots. It can be seen that the model problem leads to trailing vortices, which are qualitatively similar to those observed in stirred reactors. Strong circulating #ow can be seen in the planes parallel to the blade. Particle streaklines showed in this "gure clearly indicate the circulatory #ow extending from the blade. These vortices were found to retain their identity much longer than those observed in stirred vessels. In stirred vessels, radially outward #ow pushes trailing vortices away from the blade, where they eventually decay. In the considered model problem, vortices remain more or less behind the impeller blades and therefore retain their identity much longer. Before comparing predicted characteristics of the trailing vortices in the region immediately behind impeller blades with available data, an in#uence of various numerical parameters was examined. Simulations were carried out for di!erent number of computational cells

Fig. 2. Solution domain and grid used for model problem of #ow over a blade.

Table 1 Model equations and boundary conditions Mass and momentum balance equations: * * ( )# ( < )"0, *t I I *x I I IG H * *p * ( < )# ( < < )"! # g #F I *x I I G IG *x I I IG IH *t I I IG H G * *< *< 2 * *< IG# IH ! IK , # I I *x I I *x *x *x 3 *x H H G H K where F is interphase momentum exchange term: IG 3 C " < !< "(< !< ) G G G . F "! * % " G G 4d Balance equations listed here are before time averaging. For more details of time averaged two phase balance equations, the reader is referred to Ranade and van den Akker (1994) and FLUENT manual. Equations for the turbulence model (k! ), are listed below: * * * *k R ( k)# ( < k)" # (G! ), * *x * * *t * * *x * * *G *x H H I H * * * * R ( )# ( < )" # (C G!C ), * *x *k   * *t * * *x * * *G *x H H C H where G is turbulence generation rate and is turbulent viscosity: R *< *< *< C k H# G G G" " * I . R *x R *x *< G H G Standard values of k} model parameters were used in the present simulations. Wall functions were used to specify wall boundary conditions. At the inlet, three velocity components for both the phases, volume fractions and turbulent intensity and scale of turbulence were speci"ed. Standard outlet and zero gradient boundary conditions were used wherever appropriate. For stirred vessel simulations, upper surface of the dispersion was assumed to be #at and was modelled as a wall. Bubbles exiting from the vessel were simulated by specifying an appropriate sink at the top row of computational cells.

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Fig. 3. Predicted results for single-phase #ow over a blade (at 3 planes parallel to blade, distance between blade/plane"1 blade length). Inlet velocity, = "20;. Vector plot: maximum arrow corresponds to 0.33  m/s. Contours of vorticity: !59, !37, !25, !14, !2 1/s. Particle streak lines.

(63 000 and 154 560), discretisation schemes ("rst- and second-order upwind) and turbulence models (standard and renormalisation group version of k} model). Convergence problems were experienced while carrying out simulations with the renormalisation group theory-based version of k} model. A recent study of the wake characteristics at the rear of the blu! body (by Dally et al., 1995) indicated that the k} model performs adequately in comparison with the signi"cantly more expensive Reynolds stress model (by suitable tuning of constant C ). Therefore, all the subsequent computations were  carried out with standard k} model with second-order upwind scheme. Single-phase computations of #ow over an impeller blade were carried out for three inlet velocities (corresponding to 1, 1.5 and 2 m/s at the tip of the impeller blade). Pro"les of predicted circulation velocity and turbulent kinetic energy at a distance of about one impeller blade length behind the blade are shown in Fig. 4a and b. Circulation velocity and turbulent kinetic energy were made dimensionless using the inlet liquid velocity at the tip of the blade, ; and its square. The vertical  distance was made dimensionless by dividing it with half of the blade width (with origin located at the midway between the blade height). It can be seen that dimensionless pro"les for all the three inlet conditions are almost the same. Maximum circulation velocity is about 0.25 ; , which is in good agreement with the value reported  by Yianneskis and Whitelaw (1993). Maximum predicted turbulent kinetic energy behind the blade is about 0.08; which is rather less than that reported Yiannes kis and Whitelaw (1993). It must be noted that in the model problem considered here, inlet #uid was assumed

to be non-turbulent (zero turbulent intensity). In the case of a stirred vessel, #uid approaching the blade is turbulent. This may be one of the major reasons for the under-prediction of turbulence behind the blade. The circulation velocity, kinetic energy and turbulent energy dissipation rates were found to scale with inlet velocity (Table 1). It can be concluded that considered model problem of #ow over a single blade mimics major characteristics of the trailing vortices observed in stirred reactors. It was therefore used to understand interaction of gas bubbles and trailing vortices. Gas (density"1 kg/m, bubble diameter"2.E!3 m) was introduced in the solution domain in two ways. In the "rst mode (mode 1), gas}liquid stream with constant gas volume fraction was assumed to enter the solution domain at the inlet (shown in Fig. 2). The inlet liquid velocity of the gas and liquid phase was assumed to be same. Gas volume fraction at the inlet was speci"ed in the range of 0.01}0.03. These simulations were carried out to understand interaction of gas bubbles and centrifugal "eld of trailing vortices in absence of gravity force term. In the second mode (mode 2), inlet boundary conditions of the single-phase simulations were used without any change. The gas was introduced at the sparger cells with user speci"ed mass source per cell using the user de"ned subroutines of FLUENT. In this case, the #ow is not symmetric in vertical direction. Gravity terms were activated for these simulations. Standard inter-phase coupling terms were used in all the simulations. In#uence of inlet velocity at "xed inlet gas volume fraction and in#uence of inlet gas volume fraction for a "xed inlet velocity was examined. Typical predicted results for the gas}liquid #ow over the blade (gas introduced from the inlet with a "xed volume fraction of 0.02) are shown in Fig. 5. Vorticity contours and vector plots behind the blade are quite similar to those obtained for single-phase case (refer Fig. 3) with somewhat lower values. The shown iso-surface (clinging to the back of the blade) of gas volume fraction of 0.15 clearly indicates the accumulation of bubbles in the low pressure region (and in trailing vortices) behind the impeller blade. Predicted volume fraction of gas within these vortices is much higher than the average gas volume fraction. Predicted results indicate that circulation velocity and turbulence intensity is lower in presence of gas, but still scale with inlet velocity. Predicted pro"les of gas volume fraction (normalised by inlet gas volume fraction) within the trailing vortices are shown in Fig. 6a and b. For a "xed inlet velocity, normalised accumulation of gas in the vortices is constant (Fig. 6a). The maximum volume fraction predicted is about three times that of inlet gas volume fraction. It can be seen that as the inlet velocity increases (that is, circulation velocity increases), more and more gas accumulates within the vortices (Fig. 6b). The predicted turbulent energy dissipation rates decrease as gas accumulation behind the blade increases (Table 2).

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Fig. 4. Predicted pro"les for single-phase #ow over a blade. Across a vertical line passing through the center of the vortices and in a plane parallel to the blade, at one blade length away from the blade. (a) Dimension circulation velocity, ; /; . (b) Dimensionless turbulent kinetic energy, k/; . A  

In the second mode, gas was introduced from a gas sparger located below the impeller blade (Fig. 2). Body force terms due to gravity were considered in these simulations. For these cases also, gas tends to accumulate within the trailing vortices, leading to much higher gas hold-up behind impeller blade than the average gas hold-up. However, contrary to earlier simulations (with gas introduced at the inlet, without activating body force terms), because of the asymmetric gas #ow, two symmetric vortices were not observed. The predicted volume integrated energy dissipation decreases signi"cantly in

presence of gas. These results are not presented here for the sake of brevity. In any case, it will be much more fruitful to use advanced #ow visualisation tools to process and interpret these simulated results than to show a few black and white "gures. The simulated results and a video based on numerical animation of these results are kept on a web site (http://www.cscs.ch/& jfavre/Projects/ Reactor.html) for the interested reader. In addition to these, interaction of gas bubbles with trailing vortices was also simulated in a Lagrangian framework (for gas phase). These results will be useful to understand

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the impeller blades as S "! A = , (1) +I I I @A @A where is density and is volume fraction of phase k. I I A is area of the surface of the computational cell which @A is adjacent to the impeller blade and = is normal vector @A of the rotational velocity of the blade averaged for that computational cell surface, respectively. The corresponding sink terms for the other variables,
, for the computational cells adjacent to the back side of the impeller blades can be de"ned as S "!S
, (2) (I +I A where
is the value of
prevailing over the computaA tional cell. Now consider all the computational cells adjacent to the front side of the impeller blades. For any such cell, blade rotation will cause movement of blade into the computational cell. Such an inward movement will displace #uid from the computational cell. Since in the snapshot approach, the blade is de"ned as a stationary solid, a mass source needs to be de"ned for the cells adjacent to the front surface of the blade to simulate the displacement of #uid caused by the blade rotation. These mass sources were de"ned by Eq. (1) with positive sign to the right-hand side. The inward movement of impeller blade, however, will not add any corresponding source for other variables like momentum, kinetic energy. The stirred reactor system investigated here consists of a standard, fully ba%ed cylindrical vessel of diameter 0.3 m. Standard disc turbine dimensions were considered (impeller diameter, D"?/3, blade length"D/4, blade width"D/5, disc diameter"0.72D, ba%e width "0.1 T). Gas was introduced in the vessel from a sparger located below the impeller (sparger diameter"0.78D, height from the bottom"0.675D). Cylindrical coordinate system was used with origin located at the centre of the vessel bottom. Half of the vessel was considered as a solution domain as shown in Fig. 7. Upper surface of the vessel was considered as a wall. Fluid properties were set as those of water and air (gas density was speci"ed as 1 kg/m). The bubble diameter was set to 2.E-3 m in all the simulations. The snapshot approach was mapped onto commercial CFD code FLUENT using user de"ned subroutines. Mass sources for gas phase were speci"ed at sparger cells to simulate gas introduction at sparger. Appropriate mass sinks were speci"ed for the top-most live cells to simulate gas exit from the liquid pool. Simulations were carried out at two grid sizes (r; ;z: 17;32;32 and 35;63;80). Typical grid distribution is shown in Fig. 7. 3.2. Results and discussion Single-phase simulations were carried out "rst to establish the reference case. Details of single-phase results

Fig. 5. Predicted results for gas-liquid #ow over a blade (without body force terms). Inlet liquid velocity, = "20;; Inlet gas volume frac tion"0.02. Vector plot: maximum arrow corresponds to 0.28 m/s. Contours of vorticity: !39, !30, !21, !12, !2 (outer contour) 1/s. Iso-surface of gas volume fraction"0.15.

formation of a standing bubble behind the blade if coupled with appropriate bubble break-up and coalescence model.

3. Gas+liquid 6ow in stirred reactor 3.1. Formulation We have extended the snapshot approach (Ranade and Dometti, 1996; Ranade, 1997) developed for single-phase #ows to simulate gas}liquid #ow in a stirred vessel agitated with a standard Rushton turbine. This approach is based on the assumption that if the pressure and centrifugal forces around the blade are modelled correctly, it may be possible to simulate #ow in stirred vessel using a steady framework. Impeller rotation generates high pressure at the front side and low pressure at the backside of the blade. In the snapshot approach, suitable mass sources (and sinks) are speci"ed at the front and backside of the blades (to simulate suction and ejection of #uid occurring at the back and front sides of impeller blade). The knowledge of impeller geometry and rotational speed are su$cient to specify these sources as discussed below. Consider "rst all the computational cells adjacent to the backside of the impeller blades. For any such cell, if the blade moves away from the adjacent surface, #uid from the computational cell will move out to "ll the gap generated between the computational cell surface and the back side of impeller blade. This outgoing #uid will carry momentum and all the other properties of #uid in that computational cell. Thus the rotating impeller blades can be simulated by de"ning the mass sink for phase k, S , +I for all the computational cells adjacent to the backside of

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Fig. 6. Predicted pro"les of gas volume fraction (with body force terms). Across a vertical line passing through the center of the vortices and in a plane parallel to the blade, at one blade length away from the blade. Gas volume fraction is made dimensionless by inlet gas volume fraction. (a) Inlet liquid velocity, = "20;. (b) Inlet gas volume fraction, "0.02.  

are recently published by Ranade (1997) and are therefore not included here. For the grids used in this work, predicted results did not show trailing vortices behind the blades. The computational model was still used to simulate gas}liquid #ow in stirred vessel to examine di!erences and similarities of predicted results with those obtained for #ow over a blade. At a "xed impeller tip speed, gas}liquid #ow was simulated for three gas #ow rates. In addition, in#uence of impeller tip speed was examined at a "xed gas #ow rate. Unfortunately, very limited experimental data

is available to make systematic comparison of predicted results with measured data. Even in those limited cases where data is available, a complete set of data, that is gas and liquid phase velocities along with gas volume fractions, is not available for the same vessel geometry. In absence of possibility of making systematic comparison with experimental data, we restrict the scope to qualitative discussion of the predicted results. Predicted velocity "elds of liquid and gas phase for typical r}z planes are shown in Fig. 8. The vector plot

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Table 2 Macroscopic results for #ow over a blade Flow Liquid Liquid Liquid Gas}liquid Gas}liquid Gas}liquid Gas}liquid Gas}liquid ; (m/s)  1.0 1.5 2.0 1.0 1.5 2.0 1.0 1.0  * * * 0.02 0.02 0.02 0.03 0.04 ; /;  0.24 0.30 0.27 0.198 0.2 0.2 0.195 0.194 k /;  0.08 0.08 0.08 0.068 0.067 0.067 0.068 0.068  /; (m\)   * * * 0.0212 0.0216 0.0219 0.0318 0.0421  * * * 0.12 0.156 0.181 0.18 0.22

0.64 0.61 0.61 0.58 0.55 0.53 0.55 0.515

Fig. 7. Solution domain and grid for stirred vessel.

Fig. 8. Predicted velocity "eld at typical r}z planes. ; "2 m/s,  Q /ND"0.01. Left side: gas phase; Right side: liquid phase. Max% imum vector corresponds to 1 m/s.

for liquid phase shows the well-known #ow pattern generated by Rushton turbines. Gas velocity in the region above the impeller is much smaller than that in the near wall region. Predicted contours of turbulent energy dissipation rates and gas volume fractions at the same r}z planes are shown in Fig. 9. As expected, most of the turbulent energy gets dissipated in the impeller stream region. Predicted relative distribution of dissipation rates within the vessel agrees quite well with the experimental data for the single-phase case (Ranade, 1997). Gas volume fraction distribution indicates that impeller pumping is not su$cient to circulate gas bubbles in the lower part of the vessel. Gas bubbles rise from the sparger and are transported radially outward by the impeller stream. Gas volume fraction in the region immediately above the

impeller is higher than that in the near wall region because of the downward #ow of liquid in that region. Further accumulation of gas in that region may eventually change the usual #ow pattern generated by disc turbine, by generating upward #ow above the impeller. Typical results obtained for the impeller centre plane are shown in Fig. 10. Vector plots at this plane clearly show the high-velocity jets issuing from the impeller blades. Predicted contours of gas volume fraction for the impeller centre plane show the accumulation of gas around impeller blades. In general, the average gas volume fraction of the impeller swept volume is about "ve times that of the whole vessel. Predicted in#uence of gas #ow rate on gross characteristics, power and pumping numbers is also of interest. Pumping and power numbers were

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Fig. 11. Predicted in#uence of gas #ow rate on power and pumping numbers.

Fig. 9. Predicted results at typical r}z planes. ; "2 m/s,  Q /ND"0.01 Left side: gas volume fraction (11 uniform contours % between 0 to 0.03); Right side: turbulent energy dissipation rates (11 uniform contours between 0 to 0.8 m2/s3)

Fig. 10. Predicted distribution of gas volume fraction at impeller center plane. ; "2 m/s, Q /ND"0.01 (impeller is moving anti-clockwise).  %

calculated from the simulated results as > U L R; d dz 2 * \ U  , N " / ND





(3)

2 d< * N" 4 , . ND



(4)

where B is blade width, D is impeller diameter, N is U impeller speed, R is impeller radius and ; is radial velocity. The calculated values of pumping and power

number with and without gas #ow (Fig. 11) show the well-known trends. It can be seen that computational model has captured essential features of the gas}liquid #ow in the bulk region of the stirred vessel. It will be instructive to examine details of predicted results near the impeller blades. Predicted #ow characteristics near the impeller blades are shown in Fig. 12 in the form of contours of gas volume fraction, vorticity and turbulent energy dissipation rates. The snapshot approach correctly simulates generation of high-pressure region at the front surface of the blade and low-pressure region at the back surface of the blade. It can be seen that gas accumulates more in the low-pressure region behind the blade. When the #ow around impeller is dominated by upward gas #ow than the radial pumping of the impeller, gas tends to accumulate on both sides of the blade. This accumulated gas signi"cantly reduces pumping number and power dissipated within the impeller region (as shown in Fig. 11). The contours of energy dissipation and vorticity indicate the complex #ow "eld within the impeller blades. Both, vorticity and dissipation seem to concentrate in the region behind the blade. These results indicate the usefulness of snapshot approach to simulate gas}liquid #ow in stirred reactors. However, as indicated by the results obtained for the model problem of #ow over a blade, the simulation of trailing vortices is important to correctly estimate the gas accumulation behind the blades. The lack of vortices behind impeller blades leads to under-prediction of accumulation of gas behind the blade. Apart from the number of grids needed to capture the trailing vortices, it must be pointed out that, the formulation of snapshot approach relies on approximating blade rotation by specifying mass sources at the front and backsides. Speci"cation of sources may not be su$cient to simulate vorticity generation and transport around the blade. Further work in this regard is underway to enhance the snapshot approach.

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this model problem were found to be similar to those observed in stirred reactors. Gross characteristics of these vortices like maximum circulation velocity and maximum turbulent kinetic energy were found to scale with tip speed. Modi"ed computational snapshot approach was used to simulate gas}liquid #ow in stirred vessels. This approach was found to be adequate for simulating #ow characteristics in the bulk region of the vessels. Predicted #ow characteristics near the impeller blade, however, were found to be considerably di!erent compared to those predicted for the case of #ow over a blade. The detailed analysis of predicted results with reference to experimental data and the results obtained for #ow over a single impeller blade will be useful for extending the scope of CFD based tools for engineering gas}liquid stirred reactors.

Notation B U C ,C ,C   I C " d D k N . N / ? ; AK ;  blade width parameters of k} model drag coe$cient bubble diameter impeller diameter turbulent kinetic energy power number pumping number vessel diameter maximum circulation velocity in the vortex tip speed

Greek letters volume fraction turbulent energy dissipation rate viscosity density parameters of k} model

, I C Subscripts
Fig. 12. Details of #ow near impeller blades. ; "2 m/s,  Q /ND"0.01 (impeller is moving from left to right). (a) Gas volume % fraction (7 uniform contours between 0 to 0.03); (b) Energy dissipation rates (7 uniform contours between 0 to 2 m/s); (c) Vorticity (z-direction, 7 uniform contours between !75 to 75 s\).

1,2 G i, j k ? m t

of region 1, 2 of gas phase of direction i or j of phase k of liquid phase maximum turbulent

4. Conclusions Trailing vortices and their interaction with gas bubbles were investigated by considering a model problem of #ow over a single impeller blade. Simulated vortices of

References
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