- Email: [email protected]
Influence of Gas Flow Rate on the Structure of Trailing Vortices of a Rushton Turbine: PIV Measurements and CFD Simulations
0263–8762/01/$10.00+0.00 # Institution of Chemical Engineers Trans IChemE, Vol 79, Part A, November 2001
INFLUENCE OF GAS FLOW RATE ON THE STRUCTURE OF TRAILING VORTICES OF A RUSHTON TURBINE: PIV Measurements and CFD Simulations V. V. RANADE 1 , M. PERRARD2 , C. XUEREB2 , N. LE SAUZE 2 and J. BERTRAND2 1
Industrial Flow Modeling Group, National Chemical Laboratory, Pune, India. 2 INP ENSIGC, Toulouse, Cedex 4, France.
T
railing vortices behind rotating impeller blades play crucial role in determining gas accumulation behind them. The gas accumulation behind blades affects the pumping and power dissipation capacity of the impeller and thus signi cantly affects the performance of gas–liquid stirred reactors. Understanding uid dynamic characteristics of these trailing vortices and capability to computationally simulate these vortices is, therefore, essential for reliable design and scale-up of stirred reactors. In this paper, we have used particle image velocimetry (PIV) technique and CFD model based on computational snapshot approach for systematically studying in uence of gas ow rate on structure of trailing vortices behind blades of a Rushton turbine. PIV measurements were carried out in a standard, fully baf ed stirred vessel (H/T 1) with a at bottom. Vessel diameter was 0.4 m. A six bladed standard Rushton turbine was placed at one third of liquid height with a ring sparger. Four baf es of 1/10 T width were placed at equal spacing. Tap water was used as a medium in the vessel. Measurements were carried out at ve different gas ow rates to vary the dimensionless gas ow number in the range of 0.01 to 0.06. Both, angle resolved and angle averaged ow elds near the impeller blades were measured. The structure of trailing vortices in presence of gas was studied in detail. A Eulerian–Eulerian, two uid model was used to simulate dispersed gas–liquid ow in stirred vessel. A computational snapshot approach was used to simulate impeller rotation. The computational model was implemented using the commercial CFD code, FLUENT (of Fluent Inc., USA) with the help of user de ned subroutines. The computational model was used to simulate ow in stirred vessel operating under conditions used in the experiments. The results of this study will have important implications for extending the applicability of CFD models for simulating multiphase stirred reactors. Keywords: trailing vortices; PIV measurement; CFD simulations.
INTRODUCTION
have used a particle image velocimetry (PIV) technique and CFD model based on a computational snapshot approach for systematically studying in uence of gas ow rate on structure of trailing vortices behind blades of a Rushton turbine. In recent years, signi cant work has been carried out to characterize complex uid dynamics near rotating impeller blades1,2. Most of these studies used laser Doppler anemometer (LDA) to measure the angle resolved ow characteristics. LDA is essentially a single point technique. Instantaneous measurements of large-scale structures are, therefore, not possible with LDA. Recently attempts were made to use particle image velocimetry (PIV), which is a whole eld technique, to characterize instantaneous ow structures around rotating impeller blades3–5. Such PIV data will be very useful for validating detailed computational uid dynamic models of stirred vessels. This work was, therefore, undertaken with two fold objectives:
Stirred reactors, in which one or more impellers are used to generate desired ow and mixing within the reactor, are among the most widely used reactors in chemical and allied industries. These reactors are commonly used for carrying out gas–liquid processes. The rotating impeller generates extremely complex ow within the stirred vessel. Interaction of rotating blades with the surrounding blades generates trailing vortices behind the rotating blades. These trailing vortices play a crucial role in determining gas accumulation behind impeller blades. The gas accumulation affects the pumping and power dissipation capacity of the impeller and thus signi cantly affects the performance of gas–liquid stirred reactors. Understanding uid dynamic characteristics of these trailing vortices and capability to computationally simulate these vortices is, therefore, essential for reliable design and scale-up of stirred reactors. In this paper, we 957
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to extend the application of PIV to characterize gas–liquid ow structure near impeller blades to use this data to evaluate computational snapshot approach for simulating ow in stirred vessels. In this paper, we report detailed experimental data on gas–liquid ow eld generated by standard Rushton turbine. Both, angle resolved and angle averaged ow elds near the impeller blades were measured using PIV to study the structure of trailing vortices. The structure of trailing vortices in presence of gas was studied in detail. A Eulerian–Eulerian, two uid model was used to simulate dispersed gas–liquid ow in stirred vessel. A computational snapshot approach was used to simulate impeller rotation. The computational model was implemented using the commercial CFD code, FLUENT (of Fluent Inc., USA) with the help of user de ned subroutines. The computational model was used to simulate ow in stirred vessel operating under conditions used in the experiments. The experimental set-up and computational models are discussed in the following. Experimental results and comparison with the predicted results are presented in a later section. The results of this study will have important implications for extending the applicability of CFD models for simulating multiphase stirred reactors.
Figure 3. Typical predicted ow eld. Left side-vectors of gas phase, right side-vectors of liquid phase.
EXPERIMENTAL In PIV, small tracer particles, which follow the liquid ow, are used as seeding particles. A part of the ow domain is
Figure 4. Typical predicted ow eld. Red min. (0.1), blue min. (0.0). Left side-contours of dimensionless turbulent kinetic energy, right side-contours of gas hold-up.
Figure 1. Schematic diagram of experimental set-ups.
Figure 2. Solution domain and boundary conditions.
illuminated using a laser sheet. The motion of particles in such illuminated section is recorded with a digital camera. Two subsequent ow images with short time delay dt are used to compute instantaneous velocity eld. The value of dt depends on the studied velocity range. Different delays have been tested between 20 and 900 ms. The best results, according to signal to noise ratio and velocity eld continuity are obtained with dt 100 ms. Some experiments have been performed with shorter dt ms in the region of high velocities. They give the same results as those obtained with dt 100 ms so that the same dt has been used for the whole studied region. The use of a camera to record subsequent images allows to x their chronology, which is necessary and suf cient to determine the velocity direction. Each of these images are divided into small ‘interrogation areas’ (IA). Each IA in the rst image is then correlated with the corresponding IA in the subsequent image (cross correlation was carried out using FFT). The resulting correlation eld exhibits a distinct peak corresponding to the displacement of particles in the IA. The instantaneous velocity is determined Trans IChemE, Vol 79, Part A, November 2001
PIV MEASUREMENTS AND CFD SIMULATIONS
Figure 5. Contours of gas hold-up at impeller centre plane (anticlockwise rotation). 10 uniform contours between 0 and 0.1; Red 0.1, Blue 0.00.
by dividing the measured displacement by the exposure time delay between two images. Such velocity values computed for all IAs are assigned to the center point of each of these IA. PIV measurements were carried out in a standard, fully baf ed stirred vessel (H/T l) with at bottom. Vessel diameter was 0.4 m. Schematic of the experimental set-up is shown in Figure 1. A six bladed standard Rushton turbine of diameter 0.13 m was placed at one third of liquid height (C T/3). The values of impeller blade width and blade length were 0.026 m and 0.032 m respectively. Four baf es of 1/10 T width were placed at equal spacing. Tap water was used as a medium in the vessel. Gas was introduced in the vessels through a ring sparger of diameter 0.1 m. Sparger has 30 holes of 1 mm diameter. The ow was seeded with rhodamine particles (of DANTEC), which re ect the laser sheet with a wavelength
Figure 6. PIV data (vector plots and contours of turbulent kinetic energy) at different angles (dimensionless gas ow number, Flg
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0.01).
RANADE et al.
greater than 575 nm. This wavelength is longer than the wavelength of light re ected by the bubbles. A lter, which captures light with a wavelength greater than 550 nm, is used so that the camera only notices the light re ected by the rhodamine. A 15 Hz Continuum make double pulsed Nd:YAG laser with a beam expanding lens was used to create a light sheet of 1 mm thickness. A CCD camera (Kodak) was placed at right angles to the light sheet. This camera was used to record ow images with 1000 1000 pixels resolution. A digital encoder was mounted on the shaft, which allows synchronization of laser pulse and camera. The laser and the camera was synchronized with the encoder signal in such a way that measurements can be carried out at a speci ed angle from the impeller blade. The software program used to calculate the instantaneous velocity elds determines velocity vectors over the whole image including in the gas phase. In order to detect the presence of gas phase, a set threshold value of signal to noise ratio was used based on the preliminary experiments6. Impeller rotational speed was set to 160 rpm, which corresponds to impeller Reynolds number of 5 104 and tip velocity of 1.09 m s 1. After preliminary experiments, detailed measurements were carried out for two values of dimensionless gas ow numbers: 0.01 (dispersion regime) and 0.03 (loaded regime). PIV measurements were carried out in both, r-z plane as well as r-y planes. Preliminary experiments were carried out to select appropriate parameters of the PIV measurement. Based on these results, exposure time delay between images was set to 100 ms for the r-z planes and 50 ms for the r-y planes. Experimental measurements were carried out for 6 vertical and 8 horizontal planes. The captured images were divided into interrogation areas (IA) of 16 16 pixels (corresponds to area of 1 mm2) for the r-z planes and 32 32 pixels (corresponds to area of 4 mm2) for the r-y planes. The PIV technique determines a velocity, which is spatially averaged over each of such IAs. The data was processed using commercial software, V2IP of Lotoriel/Coria, France. For obtaining average ow eld, ensemble averaging was carried out using 600 images. Use of a larger number of images did not change the average ow eld signi cantly. Instantaneous and ensemble averaged velocity eld and turbulence kinetic energy elds are reported in Section 4 along with the simulated results. COMPUTATIONAL MODEL Various approaches have been used for modelling ow in stirred vessels. Ranade7 recently carried out a detailed evaluation of these approaches. Based on this study, we have used a computational snapshot approach in this paper. In the computational snapshot approach, impeller blades are considered as xed at one particular position (similar to taking a snapshot of rotating impeller). The ow generated by an impeller of any shape is governed mainly by pressure and centrifugal forces generated because of the impeller rotation and the corresponding rotating ows. The shape of the impeller blades controls the direction and the characteristics of impeller discharge stream via generated pressure and centrifugal forces. The blade rotation causes suction of uid at the backside of blades and equivalent ejection of uid from the front side of the blades. This phenomenon of ejection and suction needs to be modelled correctly to
simulate impeller rotation in a steady framework proposed in a computational snapshot approach. Recently, Ranade et al.8 discussed the development of snapshot approach in detail. In this work, snapshot approach was extended for the simulation of gas–liquid ows. The framework for such an extension was discussed by Ranade et al.8 and is not repeated here. In the present work, a two- uid model was used to simulate gas–liquid ow in stirred vessel. The model equations and boundary conditions are listed below. Mass balance equations: @ ra @t k k
@ r a V @xj k k ki
0
1
Momentum balance equations: @ ra V @t k k ki
@ r a V V @xj k k ki kj @r ak rk ak gi @xi @ @Vki a m @xj k k @xj
Fki @Vkj @xj
2 @ @Vkm a m 3 @xi k k @xm
2
where Fki is interphase momentum exchange term: 3aL aG CD V2i
F2i
V1i V2i 4dB
V1i
3
The 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 Akker9 and the FLUENT manual. Turbulence was modelled using a standard k-e turbulence model. Governing equations for turbulent kinetic energy, k, and turbulent energy dissipation rates e are listed below: @ r a k @t L L
@ r a V k @xj L L Li
m @k @ a t @xj L sk @xj @ r a e @t L L
aL G
4
rL e
@ r a V e @xj L L Li
m @e @ aL t se @xj @xj
aL
e C G k 1
C 2 rL e
5
where G is turbulence generation rate and mt is turbulent viscosity which are given by: G
mt
@Vj @xi
@Vi @Vi @xj @Vj
;
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mt
rL C u k 2 e
6
Standard values of k-model parameters were used in the present simulations C1 1:44; C2 1:92; CD 0:09; sk 1:0; se 1:3 . In addition to these parameters, it is necessary to use appropriate values of drag coef cient (appearing in Equation 3). The turbulence levels in stirred Trans IChemE, Vol 79, Part A, November 2001
PIV MEASUREMENTS AND CFD SIMULATIONS
tanks are usually much higher than those found around rising bubbles in isolation. Such increased turbulence levels are expected to increase the value of drag coef cient. However, it should be noted that in stirred vessels, swarm of bubbles exist. In such a bubble swarm, motions of surrounding bubbles/wakes tend to reduce the value of drag coef cient. Because of these two opposing effects and because of lack of systematic information, usually values of drag coef cients obtained from single bubble rise velocity are used for simulations of gas–liquid ows in stirred vessels. In this work, we have used a following simpli cation of interphase drag force proposed by Schwarz and Turner10: F2i
5
104 aL aG V2i
V1i
7
Wall functions were used to specify wall boundary conditions. Gas was introduced at the sparger by de ning an appropriate source of gas at the sparger cells. The top surface of the dispersion was assumed to be at and was modelled as a wall. The top boundary condition through which gas bubbles escape the solution domain require special treatment. Recently, Ranade7 discusses different formulations of such top boundary conditions. Following his recommendations, in this work, the bubbles exiting from the vessel were simulated by specifying an appropriate sink at the top row of computational cells. In the present work a snapshot approach for gas–liquid ows was implemented using a commercial CFD code, FLUENT (of Fluent Inc., USA). User de ned subroutines were used for this purpose. Half of the vessel was considered as a solution domain. Preliminary numerical experiments carried out by Ranade et al.5,8 indicated that it is necessary to use more than 100 cells covering the impeller blades. The solution domain and details of nite volume grid used in the present work are shown in Figure 2. The QUICK discretization scheme with SUPERBEE limiter function was used for integrating all the equations (Fluent User Guide, 1997). Simulations were carried out for three values of dimensionless gas ow rates, 0.01, 0.02 and 0.03. Experimental and computational results are discussed in the following section. RESULTS AND DISCUSSION Numerical simulations carried out with the computational snapshot approach show the well-known ow patterns generated by the Rushton turbine. The snapshot approach was shown to be able to capture the details of ow around rotating impeller blades including the trailing vortices5,8. Simulated results for the gas–liquid ows are discussed here. Predicted gas–liquid ow elds for the typical r-z planes are shown in Figure 3 (for dimensionless gas ow number of 0.01) and Figure 4. The simulations indicate signi cant upward inclination of the radial jet issuing from the impeller in the presence of gas, which is in agreement with the published experimental evidence. It can be seen that even at such a low gas ow rate, simulations indicate that gas bubbles are not dispersed in the lower circulation loop (left side of Figure 3). Signi cant upward inclination in the presence of gas is also obvious from the contours of turbulent kinetic energy shown in Figure 4. Contours of gas hold-up con rm that the impeller is unable to re-circulate Trans IChemE, Vol 79, Part A, November 2001
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gas bubbles in the lower loop. The contours of predicted gas hold-up at the impeller centre plane are shown in Figure 5 (impeller rotation direction is counter clockwise). It can be seen that snapshot simulations of the gas–liquid ow clearly show the presence of gas accumulation at the locations of trailing vortices behind the impeller blades. The gas hold-up just behind the blade is orders of magnitude larger than the average gas hold-up. Such gas accumulation signi cantly modi es ow around impeller blades. In order to study characteristics of such gas accumulation and its interaction with the trailing vortices in detail, an experimental program with PIV was carried out. PIV data obtained at three different r-z planes behind the leading impeller blade is shown in Figure 6. As mentioned in the experimental section, if the signal to noise ratio is below a certain threshold value (1.5), the value of the velocity vector is set to zero. Since gas cavities do not contain any tracer particles, the zero velocity eld is expected to indicate the presence of cavities. Vector plots shown in Figure 6 clearly indicate the accumulation of gas and formation of cavities behind impeller blades. The turbulent kinetic energy is determined with only two components in each planes and are respectively calculated by equations (8) and (9) for r-z and r-y planes: k
3 V 4
2
U
2
8
3 V2 W2 9 4 Gas–liquid values appears to be much higher than the values obtained for single phase ows (maximum of 0.5 U 2tip compared to 0.1 U 2tip obtained for only liquid ows). It must, however, be noted that a zone of such high turbulence kinetic energy lies at or near the interface of the gas cavities. The accuracy of PIV technique to make meaningful measurements very near to the gas–liquid interface has not yet been clearly established. If the small region near the interface of gas cavities is neglected, the measured turbulence kinetic energy values are almost the same as those obtained for only liquid ows. PIV measurements obtained for the impeller centre plane are shown in Figure 7. This data also indicate the formation of gas cavities in the region of trailing vortices. The locations of gas accumulation and trailing vortices are shown schematically in Figure 8. As shown in this gure, gas accumulation starts in the lowpressure region generated by trailing vortices. Accumulation of gas leads to rapid coalescence and formation of gas cavity clinging to the impeller blade. The lower (below the disk) gas cavity, however, ows upward due to the buoyancy force and eventually merges with the upper gas cavity. It will be instructive to examine whether the computational model captures such phenomenon. The predicted contours of gas hold-up at different r-z planes near the impeller region are shown in Figure 9. It can be seen that just behind the leading blade, gas accumulates in the core of two trailing vortices. In the present computational model, coalescence was not modeled and hence the model was incapable of simulating gas cavity formation. However, even in absence of a coalescence model, computations could capture signi cant gas accumulation in the region of trailing vortices. As one moves away from the leading blade, the lower region of gas accumulation shifts
k
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Figure.7. PIV data (vector plots and contours of turbulent kinetic energy) at 2z/Bw
Figure 8. Schematic of trailing vortices and gas cavities behind the blade.
0.23 (dimensionless gas ow number, Flg
0.01).
upwards and eventually merges with the upper region as observed in the experimental data. Thus it can be said that the computational snapshot approach can capture the essential features of gas accumulation behind the impeller blades. If a suitable coalescence model is combined with the present computational model, formation of gas cavities may be simulated. After establishing that the computational model captures key features of the gas–liquid ows, it can be used to examine the in uence of gas ow rate on generated ow in more detail. In uence of gas ow rate on predicted radial velocity and turbulent kinetic energy pro le at the impeller centre plane is shown in Figures 10 and 11, respectively. It can be seen that as gas ow rate increases, both, the predicted radial mean velocity as well as turbulent kinetic energy decrease. It must be noted that as the gas ow rate increases, the impeller stream bends upwards due to the
Figure 9. Computational simulation of accumulation of gas behind impeller blades. Blue
0, Red
0.1.
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PIV MEASUREMENTS AND CFD SIMULATIONS
Figure 10. In uence of gas ow rate on radial velocity at impeller centre plane.
Figure 11. In uence of gas ow rate on turbulent kinetic energy at impeller centre plane.
Figure 12. In uence of gas ow rate on pumping capacity.
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buoyancy force exerted by the gas bubbles. Such upward bending results in a signi cant decrease in the predicted values of radial mean velocity and turbulent kinetic energy at the impeller centre plane (that is, the impeller center plane is no longer the centre plane for the impeller stream). Further experiments are in progress to evaluate these predictions. The predicted angular variation of impeller pumping capacity is shown in Figure 12 for different gas ow rates. It can be seen that for gas–liquid ows, gas accumulated behind the leading blade and signi cantly reduced the pumping capacity of the impeller in that region. As one moves away from the leading blade, the pumping capacity slowly approaches that of only liquid ow. For a dimensionless gas ow rate of 0.03, overall pumping capacity is only 60% of that predicted for the only liquid case. Further work on error analysis and tuning of PIV technique is in progress, which is expected to provide more accurate experimental data to quantitatively (especially in the impeller region) evaluate predictions of the computational model. An attempt is also being made to include a coalescence and break-up model within the CFD framework to simulate formation of gas cavities behind the blades. These results will be published separately.
dt k N NQ r R Sf T U U Ut V V Vcell Vtip W W z
delay between two images, s turbulent kinetic energy, m2 s 2 impeller rotational speed, 1 s 1 pumping number radial co-ordinate direction, measured from the centre of the impeller, m impeller radius, m source of f tank radius, m axial velocity, m s 1 axial Root Mean Square velocity, m s 1 impeller tip velocity, m s 1 radial velocity m s 1 radial Root Mean Square Velocity m s 1 volume of computational cell, m3 impeller tip velocity, m s 1 tangential velocity, m s 1 tangential Root Mean Square velocity m s 1 axial co-ordinate direction, measured from the disk of the impeller (m)
Greek symbols f general variable G effective transport coef cient m2 s y tangential co-ordinate direction r density kg m 2
1
ACKNOWLEDGEMENT
CONCLUSIONS A computational snapshot approach was extended for simulating gas–liquid ows in a baf ed stirred vessel. A particle image velocimetry (PIV) technique was used to characterize gas–liquid ows in stirred vessels. Angle resolved PIV measurements (velocity elds and contours of kinetic energy) clearly show gas accumulation behind impeller blades. The locations of gas cavities detected in the present study coincide with the location of trailing vortices observed in only liquid ows. The regions of gas accumulation were found to retain their coherent structure up to about 30 behind the leading blade. PIV measurements seemed to over-predict the turbulent kinetic energy at the interface of the gas cavity and dispersion. Further work is necessary to identify limits and error bounds of the PIV technique for characterizing gas–liquid ows with high gas hold-up (as in impeller region). The computational model appears to capture most of the key features of the gas–liquid ows in stirred vessels, including the accumulation of gas behind impeller blades. Simulated results indicate signi cant upward bending of the impeller stream in presence of gas. The computational model was also found to capture the overall trends in pumping capacity and power dissipation of the impeller correctly. Further re nements of experimental technique to improve the accuracy of the experimental data and a more quantitative evaluation of the predicted results will allow application of the computational model presented in this work for design and optimization of gas–liquid stirred reactors. NOMENCLATURE Bw FlG
height of the impeller blade, m dimesionless gas ow number
Authors wish to acknowledge the Indo-French Center for Promotion of Advanced Research (IFCPAR) for providing nancial support for the work presented in this paper.
ADDRESS Correspondence regarding this paper should be addressed to C. Xuereb, INP ENSIGC, 18, Chemin de la Loge, 310 78 Toulouse, Cedex 4, France. E-mail: [email protected]
REFERENCES 1. Schafer, M., Hofken, M. and Durst, F., 1997, Detailed LDV measurements for visualization of the ow eld within a stirred tank reactor equipped with a Rushton turbine, Trans IChemE, Chem Eng Res Des, 75: 729–736. 2. Lee, K. C. and Yianneskis, M., 1998, Turbulence properties of the impeller stream of a Rushton turbine, AIChE J, 44: 13. 3. Sharp, K. V., Kim, K. C. and Adrian, R. J., 1998, Int Symp on Applications of Laser Techniques to Fluid Mechanics. Lisbon, 233. 4. Deen, N. G. and Hjertager, B. H., 1999, Multiphase Particle Image Velocimetry Measurements in an Aerated Tank, AIChE Meeting Dallas, November. 5. Ranade, V. V., Perrard, M., Xuereb, C., Le Sauze, N. and Bertrand, J., 2001, Trailing vortices of Rushton turbine: PIV measurements and CFD simulations with snapshot approach, Trans IChemE, Chem Eng Res Des. 6. Perrard, M., Le Sauze, N., Xuereb, C. and Bertrand, J., 2000, In: Proc 10th European Conference on Mixing, July, Delft, The Netherlands. 7. Ranade, V. V., 2001, Computational Flow Modeling for Chemical Reactor Engineering, Academic Press (accepted for publication). 8. Ranade, V. V., Tayalia, Y. and Krishnan, H., 2001, CFD predictions of ow near impeller blades in baf ed stirred vessels, Chem Eng Commun (accepted for publication). 9. Ranade, V. V and Van den Akker, H. E. A., 1994, A computational snapshot of gas–liquid ow in baf ed stirred reactors, Chem Eng Sci, 49: 5175–5192. 10. Schwarz, M. P. and Turner, W. J., 1988, Applicability of the standard k-e model to gas stirred baths, Appl Math Modelling, 12: 273–279.
Trans IChemE, Vol 79, Part A, November 2001
INFLUENCE OF GAS FLOW RATE ON THE STRUCTURE OF TRAILING VORTICES OF A RUSHTON TURBINE: PIV Measurements and CFD Simulations V. V. RANADE 1 , M. PERRARD2 , C. XUEREB2 , N. LE SAUZE 2 and J. BERTRAND2 1
Industrial Flow Modeling Group, National Chemical Laboratory, Pune, India. 2 INP ENSIGC, Toulouse, Cedex 4, France.
T
railing vortices behind rotating impeller blades play crucial role in determining gas accumulation behind them. The gas accumulation behind blades affects the pumping and power dissipation capacity of the impeller and thus signi cantly affects the performance of gas–liquid stirred reactors. Understanding uid dynamic characteristics of these trailing vortices and capability to computationally simulate these vortices is, therefore, essential for reliable design and scale-up of stirred reactors. In this paper, we have used particle image velocimetry (PIV) technique and CFD model based on computational snapshot approach for systematically studying in uence of gas ow rate on structure of trailing vortices behind blades of a Rushton turbine. PIV measurements were carried out in a standard, fully baf ed stirred vessel (H/T 1) with a at bottom. Vessel diameter was 0.4 m. A six bladed standard Rushton turbine was placed at one third of liquid height with a ring sparger. Four baf es of 1/10 T width were placed at equal spacing. Tap water was used as a medium in the vessel. Measurements were carried out at ve different gas ow rates to vary the dimensionless gas ow number in the range of 0.01 to 0.06. Both, angle resolved and angle averaged ow elds near the impeller blades were measured. The structure of trailing vortices in presence of gas was studied in detail. A Eulerian–Eulerian, two uid model was used to simulate dispersed gas–liquid ow in stirred vessel. A computational snapshot approach was used to simulate impeller rotation. The computational model was implemented using the commercial CFD code, FLUENT (of Fluent Inc., USA) with the help of user de ned subroutines. The computational model was used to simulate ow in stirred vessel operating under conditions used in the experiments. The results of this study will have important implications for extending the applicability of CFD models for simulating multiphase stirred reactors. Keywords: trailing vortices; PIV measurement; CFD simulations.
INTRODUCTION
have used a particle image velocimetry (PIV) technique and CFD model based on a computational snapshot approach for systematically studying in uence of gas ow rate on structure of trailing vortices behind blades of a Rushton turbine. In recent years, signi cant work has been carried out to characterize complex uid dynamics near rotating impeller blades1,2. Most of these studies used laser Doppler anemometer (LDA) to measure the angle resolved ow characteristics. LDA is essentially a single point technique. Instantaneous measurements of large-scale structures are, therefore, not possible with LDA. Recently attempts were made to use particle image velocimetry (PIV), which is a whole eld technique, to characterize instantaneous ow structures around rotating impeller blades3–5. Such PIV data will be very useful for validating detailed computational uid dynamic models of stirred vessels. This work was, therefore, undertaken with two fold objectives:
Stirred reactors, in which one or more impellers are used to generate desired ow and mixing within the reactor, are among the most widely used reactors in chemical and allied industries. These reactors are commonly used for carrying out gas–liquid processes. The rotating impeller generates extremely complex ow within the stirred vessel. Interaction of rotating blades with the surrounding blades generates trailing vortices behind the rotating blades. These trailing vortices play a crucial role in determining gas accumulation behind impeller blades. The gas accumulation affects the pumping and power dissipation capacity of the impeller and thus signi cantly affects the performance of gas–liquid stirred reactors. Understanding uid dynamic characteristics of these trailing vortices and capability to computationally simulate these vortices is, therefore, essential for reliable design and scale-up of stirred reactors. In this paper, we 957
RANADE et al.
958
to extend the application of PIV to characterize gas–liquid ow structure near impeller blades to use this data to evaluate computational snapshot approach for simulating ow in stirred vessels. In this paper, we report detailed experimental data on gas–liquid ow eld generated by standard Rushton turbine. Both, angle resolved and angle averaged ow elds near the impeller blades were measured using PIV to study the structure of trailing vortices. The structure of trailing vortices in presence of gas was studied in detail. A Eulerian–Eulerian, two uid model was used to simulate dispersed gas–liquid ow in stirred vessel. A computational snapshot approach was used to simulate impeller rotation. The computational model was implemented using the commercial CFD code, FLUENT (of Fluent Inc., USA) with the help of user de ned subroutines. The computational model was used to simulate ow in stirred vessel operating under conditions used in the experiments. The experimental set-up and computational models are discussed in the following. Experimental results and comparison with the predicted results are presented in a later section. The results of this study will have important implications for extending the applicability of CFD models for simulating multiphase stirred reactors.
Figure 3. Typical predicted ow eld. Left side-vectors of gas phase, right side-vectors of liquid phase.
EXPERIMENTAL In PIV, small tracer particles, which follow the liquid ow, are used as seeding particles. A part of the ow domain is
Figure 4. Typical predicted ow eld. Red min. (0.1), blue min. (0.0). Left side-contours of dimensionless turbulent kinetic energy, right side-contours of gas hold-up.
Figure 1. Schematic diagram of experimental set-ups.
Figure 2. Solution domain and boundary conditions.
illuminated using a laser sheet. The motion of particles in such illuminated section is recorded with a digital camera. Two subsequent ow images with short time delay dt are used to compute instantaneous velocity eld. The value of dt depends on the studied velocity range. Different delays have been tested between 20 and 900 ms. The best results, according to signal to noise ratio and velocity eld continuity are obtained with dt 100 ms. Some experiments have been performed with shorter dt ms in the region of high velocities. They give the same results as those obtained with dt 100 ms so that the same dt has been used for the whole studied region. The use of a camera to record subsequent images allows to x their chronology, which is necessary and suf cient to determine the velocity direction. Each of these images are divided into small ‘interrogation areas’ (IA). Each IA in the rst image is then correlated with the corresponding IA in the subsequent image (cross correlation was carried out using FFT). The resulting correlation eld exhibits a distinct peak corresponding to the displacement of particles in the IA. The instantaneous velocity is determined Trans IChemE, Vol 79, Part A, November 2001
PIV MEASUREMENTS AND CFD SIMULATIONS
Figure 5. Contours of gas hold-up at impeller centre plane (anticlockwise rotation). 10 uniform contours between 0 and 0.1; Red 0.1, Blue 0.00.
by dividing the measured displacement by the exposure time delay between two images. Such velocity values computed for all IAs are assigned to the center point of each of these IA. PIV measurements were carried out in a standard, fully baf ed stirred vessel (H/T l) with at bottom. Vessel diameter was 0.4 m. Schematic of the experimental set-up is shown in Figure 1. A six bladed standard Rushton turbine of diameter 0.13 m was placed at one third of liquid height (C T/3). The values of impeller blade width and blade length were 0.026 m and 0.032 m respectively. Four baf es of 1/10 T width were placed at equal spacing. Tap water was used as a medium in the vessel. Gas was introduced in the vessels through a ring sparger of diameter 0.1 m. Sparger has 30 holes of 1 mm diameter. The ow was seeded with rhodamine particles (of DANTEC), which re ect the laser sheet with a wavelength
Figure 6. PIV data (vector plots and contours of turbulent kinetic energy) at different angles (dimensionless gas ow number, Flg
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0.01).
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greater than 575 nm. This wavelength is longer than the wavelength of light re ected by the bubbles. A lter, which captures light with a wavelength greater than 550 nm, is used so that the camera only notices the light re ected by the rhodamine. A 15 Hz Continuum make double pulsed Nd:YAG laser with a beam expanding lens was used to create a light sheet of 1 mm thickness. A CCD camera (Kodak) was placed at right angles to the light sheet. This camera was used to record ow images with 1000 1000 pixels resolution. A digital encoder was mounted on the shaft, which allows synchronization of laser pulse and camera. The laser and the camera was synchronized with the encoder signal in such a way that measurements can be carried out at a speci ed angle from the impeller blade. The software program used to calculate the instantaneous velocity elds determines velocity vectors over the whole image including in the gas phase. In order to detect the presence of gas phase, a set threshold value of signal to noise ratio was used based on the preliminary experiments6. Impeller rotational speed was set to 160 rpm, which corresponds to impeller Reynolds number of 5 104 and tip velocity of 1.09 m s 1. After preliminary experiments, detailed measurements were carried out for two values of dimensionless gas ow numbers: 0.01 (dispersion regime) and 0.03 (loaded regime). PIV measurements were carried out in both, r-z plane as well as r-y planes. Preliminary experiments were carried out to select appropriate parameters of the PIV measurement. Based on these results, exposure time delay between images was set to 100 ms for the r-z planes and 50 ms for the r-y planes. Experimental measurements were carried out for 6 vertical and 8 horizontal planes. The captured images were divided into interrogation areas (IA) of 16 16 pixels (corresponds to area of 1 mm2) for the r-z planes and 32 32 pixels (corresponds to area of 4 mm2) for the r-y planes. The PIV technique determines a velocity, which is spatially averaged over each of such IAs. The data was processed using commercial software, V2IP of Lotoriel/Coria, France. For obtaining average ow eld, ensemble averaging was carried out using 600 images. Use of a larger number of images did not change the average ow eld signi cantly. Instantaneous and ensemble averaged velocity eld and turbulence kinetic energy elds are reported in Section 4 along with the simulated results. COMPUTATIONAL MODEL Various approaches have been used for modelling ow in stirred vessels. Ranade7 recently carried out a detailed evaluation of these approaches. Based on this study, we have used a computational snapshot approach in this paper. In the computational snapshot approach, impeller blades are considered as xed at one particular position (similar to taking a snapshot of rotating impeller). The ow generated by an impeller of any shape is governed mainly by pressure and centrifugal forces generated because of the impeller rotation and the corresponding rotating ows. The shape of the impeller blades controls the direction and the characteristics of impeller discharge stream via generated pressure and centrifugal forces. The blade rotation causes suction of uid at the backside of blades and equivalent ejection of uid from the front side of the blades. This phenomenon of ejection and suction needs to be modelled correctly to
simulate impeller rotation in a steady framework proposed in a computational snapshot approach. Recently, Ranade et al.8 discussed the development of snapshot approach in detail. In this work, snapshot approach was extended for the simulation of gas–liquid ows. The framework for such an extension was discussed by Ranade et al.8 and is not repeated here. In the present work, a two- uid model was used to simulate gas–liquid ow in stirred vessel. The model equations and boundary conditions are listed below. Mass balance equations: @ ra @t k k
@ r a V @xj k k ki
0
1
Momentum balance equations: @ ra V @t k k ki
@ r a V V @xj k k ki kj @r ak rk ak gi @xi @ @Vki a m @xj k k @xj
Fki @Vkj @xj
2 @ @Vkm a m 3 @xi k k @xm
2
where Fki is interphase momentum exchange term: 3aL aG CD V2i
F2i
V1i V2i 4dB
V1i
3
The 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 Akker9 and the FLUENT manual. Turbulence was modelled using a standard k-e turbulence model. Governing equations for turbulent kinetic energy, k, and turbulent energy dissipation rates e are listed below: @ r a k @t L L
@ r a V k @xj L L Li
m @k @ a t @xj L sk @xj @ r a e @t L L
aL G
4
rL e
@ r a V e @xj L L Li
m @e @ aL t se @xj @xj
aL
e C G k 1
C 2 rL e
5
where G is turbulence generation rate and mt is turbulent viscosity which are given by: G
mt
@Vj @xi
@Vi @Vi @xj @Vj
;
960
mt
rL C u k 2 e
6
Standard values of k-model parameters were used in the present simulations C1 1:44; C2 1:92; CD 0:09; sk 1:0; se 1:3 . In addition to these parameters, it is necessary to use appropriate values of drag coef cient (appearing in Equation 3). The turbulence levels in stirred Trans IChemE, Vol 79, Part A, November 2001
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tanks are usually much higher than those found around rising bubbles in isolation. Such increased turbulence levels are expected to increase the value of drag coef cient. However, it should be noted that in stirred vessels, swarm of bubbles exist. In such a bubble swarm, motions of surrounding bubbles/wakes tend to reduce the value of drag coef cient. Because of these two opposing effects and because of lack of systematic information, usually values of drag coef cients obtained from single bubble rise velocity are used for simulations of gas–liquid ows in stirred vessels. In this work, we have used a following simpli cation of interphase drag force proposed by Schwarz and Turner10: F2i
5
104 aL aG V2i
V1i
7
Wall functions were used to specify wall boundary conditions. Gas was introduced at the sparger by de ning an appropriate source of gas at the sparger cells. The top surface of the dispersion was assumed to be at and was modelled as a wall. The top boundary condition through which gas bubbles escape the solution domain require special treatment. Recently, Ranade7 discusses different formulations of such top boundary conditions. Following his recommendations, in this work, the bubbles exiting from the vessel were simulated by specifying an appropriate sink at the top row of computational cells. In the present work a snapshot approach for gas–liquid ows was implemented using a commercial CFD code, FLUENT (of Fluent Inc., USA). User de ned subroutines were used for this purpose. Half of the vessel was considered as a solution domain. Preliminary numerical experiments carried out by Ranade et al.5,8 indicated that it is necessary to use more than 100 cells covering the impeller blades. The solution domain and details of nite volume grid used in the present work are shown in Figure 2. The QUICK discretization scheme with SUPERBEE limiter function was used for integrating all the equations (Fluent User Guide, 1997). Simulations were carried out for three values of dimensionless gas ow rates, 0.01, 0.02 and 0.03. Experimental and computational results are discussed in the following section. RESULTS AND DISCUSSION Numerical simulations carried out with the computational snapshot approach show the well-known ow patterns generated by the Rushton turbine. The snapshot approach was shown to be able to capture the details of ow around rotating impeller blades including the trailing vortices5,8. Simulated results for the gas–liquid ows are discussed here. Predicted gas–liquid ow elds for the typical r-z planes are shown in Figure 3 (for dimensionless gas ow number of 0.01) and Figure 4. The simulations indicate signi cant upward inclination of the radial jet issuing from the impeller in the presence of gas, which is in agreement with the published experimental evidence. It can be seen that even at such a low gas ow rate, simulations indicate that gas bubbles are not dispersed in the lower circulation loop (left side of Figure 3). Signi cant upward inclination in the presence of gas is also obvious from the contours of turbulent kinetic energy shown in Figure 4. Contours of gas hold-up con rm that the impeller is unable to re-circulate Trans IChemE, Vol 79, Part A, November 2001
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gas bubbles in the lower loop. The contours of predicted gas hold-up at the impeller centre plane are shown in Figure 5 (impeller rotation direction is counter clockwise). It can be seen that snapshot simulations of the gas–liquid ow clearly show the presence of gas accumulation at the locations of trailing vortices behind the impeller blades. The gas hold-up just behind the blade is orders of magnitude larger than the average gas hold-up. Such gas accumulation signi cantly modi es ow around impeller blades. In order to study characteristics of such gas accumulation and its interaction with the trailing vortices in detail, an experimental program with PIV was carried out. PIV data obtained at three different r-z planes behind the leading impeller blade is shown in Figure 6. As mentioned in the experimental section, if the signal to noise ratio is below a certain threshold value (1.5), the value of the velocity vector is set to zero. Since gas cavities do not contain any tracer particles, the zero velocity eld is expected to indicate the presence of cavities. Vector plots shown in Figure 6 clearly indicate the accumulation of gas and formation of cavities behind impeller blades. The turbulent kinetic energy is determined with only two components in each planes and are respectively calculated by equations (8) and (9) for r-z and r-y planes: k
3 V 4
2
U
2
8
3 V2 W2 9 4 Gas–liquid values appears to be much higher than the values obtained for single phase ows (maximum of 0.5 U 2tip compared to 0.1 U 2tip obtained for only liquid ows). It must, however, be noted that a zone of such high turbulence kinetic energy lies at or near the interface of the gas cavities. The accuracy of PIV technique to make meaningful measurements very near to the gas–liquid interface has not yet been clearly established. If the small region near the interface of gas cavities is neglected, the measured turbulence kinetic energy values are almost the same as those obtained for only liquid ows. PIV measurements obtained for the impeller centre plane are shown in Figure 7. This data also indicate the formation of gas cavities in the region of trailing vortices. The locations of gas accumulation and trailing vortices are shown schematically in Figure 8. As shown in this gure, gas accumulation starts in the lowpressure region generated by trailing vortices. Accumulation of gas leads to rapid coalescence and formation of gas cavity clinging to the impeller blade. The lower (below the disk) gas cavity, however, ows upward due to the buoyancy force and eventually merges with the upper gas cavity. It will be instructive to examine whether the computational model captures such phenomenon. The predicted contours of gas hold-up at different r-z planes near the impeller region are shown in Figure 9. It can be seen that just behind the leading blade, gas accumulates in the core of two trailing vortices. In the present computational model, coalescence was not modeled and hence the model was incapable of simulating gas cavity formation. However, even in absence of a coalescence model, computations could capture signi cant gas accumulation in the region of trailing vortices. As one moves away from the leading blade, the lower region of gas accumulation shifts
k
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Figure.7. PIV data (vector plots and contours of turbulent kinetic energy) at 2z/Bw
Figure 8. Schematic of trailing vortices and gas cavities behind the blade.
0.23 (dimensionless gas ow number, Flg
0.01).
upwards and eventually merges with the upper region as observed in the experimental data. Thus it can be said that the computational snapshot approach can capture the essential features of gas accumulation behind the impeller blades. If a suitable coalescence model is combined with the present computational model, formation of gas cavities may be simulated. After establishing that the computational model captures key features of the gas–liquid ows, it can be used to examine the in uence of gas ow rate on generated ow in more detail. In uence of gas ow rate on predicted radial velocity and turbulent kinetic energy pro le at the impeller centre plane is shown in Figures 10 and 11, respectively. It can be seen that as gas ow rate increases, both, the predicted radial mean velocity as well as turbulent kinetic energy decrease. It must be noted that as the gas ow rate increases, the impeller stream bends upwards due to the
Figure 9. Computational simulation of accumulation of gas behind impeller blades. Blue
0, Red
0.1.
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Figure 10. In uence of gas ow rate on radial velocity at impeller centre plane.
Figure 11. In uence of gas ow rate on turbulent kinetic energy at impeller centre plane.
Figure 12. In uence of gas ow rate on pumping capacity.
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buoyancy force exerted by the gas bubbles. Such upward bending results in a signi cant decrease in the predicted values of radial mean velocity and turbulent kinetic energy at the impeller centre plane (that is, the impeller center plane is no longer the centre plane for the impeller stream). Further experiments are in progress to evaluate these predictions. The predicted angular variation of impeller pumping capacity is shown in Figure 12 for different gas ow rates. It can be seen that for gas–liquid ows, gas accumulated behind the leading blade and signi cantly reduced the pumping capacity of the impeller in that region. As one moves away from the leading blade, the pumping capacity slowly approaches that of only liquid ow. For a dimensionless gas ow rate of 0.03, overall pumping capacity is only 60% of that predicted for the only liquid case. Further work on error analysis and tuning of PIV technique is in progress, which is expected to provide more accurate experimental data to quantitatively (especially in the impeller region) evaluate predictions of the computational model. An attempt is also being made to include a coalescence and break-up model within the CFD framework to simulate formation of gas cavities behind the blades. These results will be published separately.
dt k N NQ r R Sf T U U Ut V V Vcell Vtip W W z
delay between two images, s turbulent kinetic energy, m2 s 2 impeller rotational speed, 1 s 1 pumping number radial co-ordinate direction, measured from the centre of the impeller, m impeller radius, m source of f tank radius, m axial velocity, m s 1 axial Root Mean Square velocity, m s 1 impeller tip velocity, m s 1 radial velocity m s 1 radial Root Mean Square Velocity m s 1 volume of computational cell, m3 impeller tip velocity, m s 1 tangential velocity, m s 1 tangential Root Mean Square velocity m s 1 axial co-ordinate direction, measured from the disk of the impeller (m)
Greek symbols f general variable G effective transport coef cient m2 s y tangential co-ordinate direction r density kg m 2
1
ACKNOWLEDGEMENT
CONCLUSIONS A computational snapshot approach was extended for simulating gas–liquid ows in a baf ed stirred vessel. A particle image velocimetry (PIV) technique was used to characterize gas–liquid ows in stirred vessels. Angle resolved PIV measurements (velocity elds and contours of kinetic energy) clearly show gas accumulation behind impeller blades. The locations of gas cavities detected in the present study coincide with the location of trailing vortices observed in only liquid ows. The regions of gas accumulation were found to retain their coherent structure up to about 30 behind the leading blade. PIV measurements seemed to over-predict the turbulent kinetic energy at the interface of the gas cavity and dispersion. Further work is necessary to identify limits and error bounds of the PIV technique for characterizing gas–liquid ows with high gas hold-up (as in impeller region). The computational model appears to capture most of the key features of the gas–liquid ows in stirred vessels, including the accumulation of gas behind impeller blades. Simulated results indicate signi cant upward bending of the impeller stream in presence of gas. The computational model was also found to capture the overall trends in pumping capacity and power dissipation of the impeller correctly. Further re nements of experimental technique to improve the accuracy of the experimental data and a more quantitative evaluation of the predicted results will allow application of the computational model presented in this work for design and optimization of gas–liquid stirred reactors. NOMENCLATURE Bw FlG
height of the impeller blade, m dimesionless gas ow number
Authors wish to acknowledge the Indo-French Center for Promotion of Advanced Research (IFCPAR) for providing nancial support for the work presented in this paper.
ADDRESS Correspondence regarding this paper should be addressed to C. Xuereb, INP ENSIGC, 18, Chemin de la Loge, 310 78 Toulouse, Cedex 4, France. E-mail: [email protected]
REFERENCES 1. Schafer, M., Hofken, M. and Durst, F., 1997, Detailed LDV measurements for visualization of the ow eld within a stirred tank reactor equipped with a Rushton turbine, Trans IChemE, Chem Eng Res Des, 75: 729–736. 2. Lee, K. C. and Yianneskis, M., 1998, Turbulence properties of the impeller stream of a Rushton turbine, AIChE J, 44: 13. 3. Sharp, K. V., Kim, K. C. and Adrian, R. J., 1998, Int Symp on Applications of Laser Techniques to Fluid Mechanics. Lisbon, 233. 4. Deen, N. G. and Hjertager, B. H., 1999, Multiphase Particle Image Velocimetry Measurements in an Aerated Tank, AIChE Meeting Dallas, November. 5. Ranade, V. V., Perrard, M., Xuereb, C., Le Sauze, N. and Bertrand, J., 2001, Trailing vortices of Rushton turbine: PIV measurements and CFD simulations with snapshot approach, Trans IChemE, Chem Eng Res Des. 6. Perrard, M., Le Sauze, N., Xuereb, C. and Bertrand, J., 2000, In: Proc 10th European Conference on Mixing, July, Delft, The Netherlands. 7. Ranade, V. V., 2001, Computational Flow Modeling for Chemical Reactor Engineering, Academic Press (accepted for publication). 8. Ranade, V. V., Tayalia, Y. and Krishnan, H., 2001, CFD predictions of ow near impeller blades in baf ed stirred vessels, Chem Eng Commun (accepted for publication). 9. Ranade, V. V and Van den Akker, H. E. A., 1994, A computational snapshot of gas–liquid ow in baf ed stirred reactors, Chem Eng Sci, 49: 5175–5192. 10. Schwarz, M. P. and Turner, W. J., 1988, Applicability of the standard k-e model to gas stirred baths, Appl Math Modelling, 12: 273–279.
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