Particle Image Velocimetry Experiments and Large Eddy Simulations

Article pubs.acs.org/IECR

Particle Image Velocimetry Experiments and Large Eddy Simulations of Merging Flow Characteristics in Dual Rushton Turbine Stirred Tanks Zhipeng Li, Mengting Hu, Yuyun Bao,* and Zhengming Gao State Key Laboratory of Chemical Resource Engineering, School of Chemical Engineering, Beijing University of Chemical Technology, Mailbox 230, Beijing 100029, PR China ABSTRACT: The merging flow characteristics in dual Rushton turbine stirred tanks were investigated using particle image velocimetry (PIV) experiments and large eddy simulation (LES) methods. The velocity and turbulent kinetic energy (TKE) were carefully measured with a high resolution PIV system. The regions with high TKE levels are affected by the movement of the trailing vortices generated behind the blades of the two turbines. The effects of the blade arrangements between the upper and lower turbines on the flow characteristics were discussed, but they are negligible for the phase-averaged flow fields. However, the phase-resolved data are totally different under various blade arrangements. The LES results of velocity, TKE, and trajectories of the trailing vortex cores were quantitatively compared with the PIV experiments and the laser Doppler velocimetry (LDV) data in the literature. Both the phase-averaged and phase-resolved LES results are in good agreement with the PIV experimental data and are better than the simulation results of the k−ε model. The good agreement between LES simulations and PIV experiments shows that the LES method has great potential for predicting complex flow fields in stirred tanks.

1. INTRODUCTION Stirred tanks with multiple impellers are frequently used for various engineering processes, such as fermentation, polymerization, and bioengineering, because a single impeller cannot provide sufficient momentum, heat, or mass transfer. The type, combination, and interaction of multiple impellers are the main factors to determine the flow characteristics and fulfill the mixing objectives in such stirred tanks. Gogate et al.1 reviewed the flow characteristics in multiple-impeller bioreactors under different conditions and made recommendations for the design and scale-up of bioreactors. Moucha et al.2 investigated the gas dispersion characteristics for 18 multiple-impeller configurations. Stirred tanks with multiple Rushton turbines are considered as fundamental models for investigations of power consumption,3,4 liquid circulation,5,6 flow pattern,7 and mixing characteristics.8 Nocentini et al.9 measured the characteristics of two gas−liquid systems, including mass transfer coefficient, gas holdup, and power consumption. Woziwodzki et al.10 measured the mixing characteristics of shear-thinning fluids in vessels agitated with three, four, and five Rushton turbines in the transitional regime. Nurtono et al.11 investigated the macroinstability characteristics in a dual Rushton turbine stirred tank and found three dominant instantaneous flow patterns: radial discharge, asymmetric radial discharge, and cross-pass radial discharge. Different combinations of these dominant flow patterns were also observed. Because of these complex instantaneous flow patterns, materials can be transferred throughout the whole tank and mixed well even under parallel flow pattern (a flow structure with four main circulations in a dual Rushton turbine stirred tank) with large impeller separation.12 Rutherford et al.13 systematically investigated the flow patterns in a dual Rushton turbine stirred tank by means of laser Doppler velocimetry (LDV). They found that the overall © 2012 American Chemical Society

flow structures were strongly affected by three parameters: the clearance of the lower impeller from the bottom of the tank, the separation between the two impellers, and the submergence of the upper impeller below the fluid surface. They reported that stable flow patterns, including parallel flow, merging flow (a flow pattern with two main circulations in a dual Rushton turbine stirred tank), and diverging flow (a flow pattern with three main circulations in a dual Rushton turbine stirred tank), were observed under different geometric parameters. They then investigated in detail the merging flow pattern for its typical characteristics with interaction between impellers. However, they only presented the velocity and turbulent kinetic energy (TKE) distributions using vectors and contours, respectively. Lee and Yianneskis14 obtained the transient mixing characteristics of merging flow in a dual Rushton turbine stirred tank using a liquid crystal thermographic technique and found that the mixing number is around 15. As for the effect of D/T on merging flow characteristic, Pan et al.15 found that if D/T is increased from 0.33 to 0.5, the separation of two Rushton turbines should be decreased from 0.38T to 0.27T to keep the merging flow pattern stable. This result is in agreement with the effect of D/T on the single-loop flow pattern.16 The hydrodynamic characteristics in multiple Rushton turbines stirred tanks have also been increasingly simulated by means of computational fluid dynamics (CFD) techniques, and most works mainly focused on single-phase flow field. Joshi et al.17 recently reviewed in detail various impeller modeling techniques and different turbulence models used to predict the single-phase turbulent flow characteristics in stirred tanks. Received: Revised: Accepted: Published: 2438

July 20, 2011 December 17, 2011 January 4, 2012 January 4, 2012 dx.doi.org/10.1021/ie201579t | Ind. Eng.Chem. Res. 2012, 51, 2438−2450

Industrial & Engineering Chemistry Research

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Micale et al.18 predicted the parallel, merging, and diverging turbulent flow fields in vessels stirred by dual Rushton turbine using three impeller modeling techniques: impeller boundary condition method (IBC), sliding grid method (SG), and inner outer iterative procedure (IO). They used the standard k−ε model with wall functions to model turbulence and found that the k−ε model overpredicted the angle between the impeller stream and the horizontal plane and seriously underestimated the distribution of turbulent kinetic energy for the merging and diverging flow patterns. They suggested that the dual Rushton turbine problem can be considered as an excellent benchmark to assess the capability of modeling approaches. However, they also pointed out that the experimental velocity and TKE results were read from the published vectors and contours of Rutherford et al.,13 respectively, so there might be errors involved in reading the data. Brucato et al.19 assessed the impeller modeling applicability of IO procedure using the dual Rushton turbine problems and the experimental results of Rutherford et al.13 Deshpande and Ranade20 verified the computational snapshot approach using the LDV data of Rutherford et al.13 and the simulated results of Micale et al.18 The dual Rushton turbine problem under merging flow pattern was used by Chiu et al.21,22 to validate their CFD modeling method before they conducted the flow and reactive mixing simulation of industrial ethoxylation reactors. Montante and Magelli23 investigated the homogenization process in a multiple Rushton turbines stirred vessel using a Reynolds averaged Navier−Stokes (RANS) based CFD method and found that the turbulent Schmidt number has great influence on the simulated mixing characteristics, and a value of 0.10 is recommended to obtain accurate predicted results. Because the RANS based turbulence models fail to provide good simulated results, especially the TKE distributions, for various complex flow fields in stirred tanks, the large eddy simulation (LES) method is increasingly becoming a useful tool in the simulation of the flow fields in stirred tanks. LES techniques, including nonuniform grid,24 discretization methods,25−31 and subgrid scale (SGS) models,16,32−35 have been widely used in stirred tank studies and applied in multiphase systems.36−40 In our previous work,41 the dual Rushton turbine problem was investigated by the LES method with two SGS models, and the results obtained by LES with a dynamic SGS model are in better agreement with experimental data from the literature. However, the LES results need further verification because quantitative phase-averaged, especially phase-resolved, experimental data are not available in the literature. Furthermore, because of the inherent complexity of the merging flow field, it has so far been proved difficult to predict the characteristics of this flow pattern, especially at the turbulence level, and there is no further published work dealing with this problem with the LES method. The objective of this work is to investigate the merging flow patterns in dual Rushton turbine stirred tanks using particle image velocimetry (PIV) experiments and LES approaches. Both phase-averaged and phase-resolved velocities, TKE, trailing vortices, and the corresponding transportations were quantitatively measured, and the effects of blade arrangements on the flow fields were also considered for the first time. The experimental PIV data are used to validate the simulated results obtained by the LES method and k−ε model. The trajectories of trailing vortices are also discussed.

2. PIV EXPERIMENT A schematic view of the experimental apparatus is shown in Figure 1. A flat bottom Perspex tank of inner diameter T = 0.48

Figure 1. Schematic diagram of experimental apparatus.

m was used, and four equally spaced baffles with width Wb = T/ 10 were arranged along the internal surface of the tank. The cylindrical tank was placed in a square tank to minimize optical refraction. Two standard Rushton turbines with diameter D = T/3 were used. The upper one, shown in Figure 1 in red, is denoted by RTU, and the lower one is denoted by RTL. The diameter and thickness of the disk were 0.118 and 0.004 m, respectively, and the length, height, and thickness of the blade were 0.04 m, 0.032 m, and 0.003 m, respectively. The offbottom clearance of RTL, the separation between RTL and RTU, and the submergence of RTU were equal to the turbine diameter D, so the liquid height of tap water H was equal to T. For the fully turbulent flow field in a stirred tank, our previous work showed that the Reynolds number had no influence on the normalized flow characteristics,16 so the rotational speed was chosen as N = 97 rpm, and the corresponding Reynolds number, defined as Re = ND2/ν, was 40 000, which was equal to that used by Rutherford et al.13 A commercial 2D PIV system (TSI Inc., USA) was used in this work, consisting of a dual Nd:YAG 532 nm pulsed laser (New Wave Research Solo, 200 mJ, 15 Hz), a frame-straddling CCD camera (PowerView Plus 11MP, 4008 × 2672 pixels), a synchronizer, and Insight 3G software. The plane in the middle of two successive baffles was chosen as the measurement plane, and the phase angle between the current (or measured) blade of RTL and the measurement plane was denoted by θ, as shown in Figure 1. A shaft encoder was used to obtain the phaseresolved measurements in this work, and the TTL signals triggered by the encoder once-per-revolution were used by the synchronizer to synchronize the blade angular position of RTL, image acquisition, and laser firing. The images at different blade angular positions were acquired by increasing the trigger delay time. Twelve phases were measured between the current blade and its trailing blade of RTL; i.e., the phase-resolved measurements were obtained every 5° behind the current blade of RTL, and then, the phase-averaged results were computed. To investigate the influence of the blade arrangements of the two Rushton turbines on the flow characteristics of the stirred tank, four different angles β = 0°, 15°, 30°, and 45° between the two current blades of RTU and RTL were used 2439

dx.doi.org/10.1021/ie201579t | Ind. Eng.Chem. Res. 2012, 51, 2438−2450


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The dynamic eddy viscosity approach introduced by Germano et al.47 provides a self-consistent way to compute the Smagorinsky coefficient from the information of resolved scales. The dynamic approach requires a second “test” filter (denoted by a tilde) to the large eddy governing equation. The subgrid scale stress in the test filter is as follows:

to measure the phase-resolved and phase-averaged flow fields and evaluate the corresponding characteristics. To resolve the interacted flow region of the two Rushton turbines, a rectangular area of about 0.26 m × 0.17 m was recorded by the high resolution camera. A cross-correlation algorithm was applied to the 32 × 32 pixels interrogation windows with 50% overlap using the Insight 3G software to obtain the instantaneous velocity distribution, so the vector resolution in this work is about 1 mm. The time difference between laser pulses was chosen as 150 μs in our experiments, which was carefully optimized to ensure that the maximum inplane and out-of-plane displacements of seeding particles were less than one-quarter of the sizes of interrogation windows and the thickness of light sheet.42 The statistical convergence on the mean velocity, fluctuating velocity, and TKE was verified; the difference between the fluctuating velocities calculated from 200 and 300 pairs of images was less than 2%. Thus, three hundred pairs of images were captured at each phase angle behind the blade. In a 2D PIV system, the tangential component of fluctuating velocity cannot be directly measured but usually can be estimated using a pseudoisotropic assumption, resulting in the following estimation equation of turbulent kinetic energy:

k = 0.75( u′ 2 + v′ 2)

Tij = uiu͠ j − ∼∼ ui uj Tij is associated with larger scales and similar to τij

∼ 1 δijTkk = − 2(CsΔ̃)2 |S | S͠ ij 3 The stresses Tij and τij are related as follows: Lij ≡ Tij − τ̃ij Tij −

Lij −

(3)

1 δijLkk = CS 2Mij 3

(10)

2∼ Mij = − 2Δ̃ |S | S͠ ij + 2Δ2 |S |͠Sij

Following the work of Lilly, calculated as

CS 2 =

3. LARGE EDDY SIMULATION The basic governing equations of LES for incompressible flows are the filtered continuity and Navier−Stokes equations:

∂ ui uj ∂τij ∂ ui ∂ 2 ui 1 ∂p ̅ + +ν − =− ∂t ∂xj∂xj ∂xj ρ ∂xi ∂xj

(9)

where

where u′ and v′ are fluctuating radial and axial velocities, respectively. Equation 1 was widely used in the literature and verified by many works.43−45

(2)

(8)

where Lij is the stress associated with the small resolved scales between the test and grid filters. The above equations can be combined into

(1)

∂ ui =0 ∂xi

(7)

48

(11)

the model coefficient is

LijMij MijMij

(12)

The geometric parameters used for the LES model in this work were the same as those used for the configuration of blade angular difference β = 0° described in the section on PIV experiment. The total computational grid consisted of approximately 2 000 000 unstructured, nonuniformly distributed, hexahedral cells. The averaged cell size in the impeller discharge region was about 2 mm, and the averaged size in the rest of the tank was about 5 mm. Both cell sizes were selected after an extensive grid refinement study, and a similar grid distribution has been validated in our previous work.16 Because the Reynolds number was relatively large in the present work, wall functions, which require that the centroid of the walladjacent cell falls within the logarithmic region of the boundary layer, were utilized to calculate the wall shear stresses. The numerical simulations were conducted using the commercial software package FLUENT 6.2, which is based on the finite volume method of unstructured grids. The linear equations were solved using an algebraic multigrid procedure. The impeller movement was modeled using the sliding mesh model for unstructured grids. The center differencing scheme was used for the spatial discretization of momentum equations and the second-order accurate implicit scheme for time advancement. The coupling between the continuity and momentum equations was achieved using the pressure implicit with splitting of operators (PISO) algorithm. To provide good initial values for the transient LES simulation, a steady state k−ε calculation was utilized. The condition of no-slip velocity was applied to all solid wall boundaries; the free surface of the tank was considered as a symmetry boundary. For each impeller revolution, 216 time steps were used; 20 iterations per time step were required to ensure that every normalized residual dropped below the

where the overbars denote the resolvable scales obtained from grid filtering. The effect of subgrid scales appears through the subgrid scale stress tensor τij = uiuj − ui uj (4) which must be modeled. The subgrid scale stresses represent the interaction between the resolved larger scales and the unresolved smaller scales of flow. In the eddy viscosity model, the unknown subgrid scale stress tensor τij and the filtered rate of strain tensor S̅ij are related by

1 δijτkk = − 2νt Sij (5) 3 In the first subgrid scale model suggested by Smagorinsky,46 the eddy viscosity is modeled as τij −

νt = (CSΔ)2 |S ̅ | (6) where CS is the Smagorinsky constant, Δ is the filter width, and |S̅| is the modulus of strain rate for the resolved scales. However, no universal Smagorinsky constant can be adopted for different flow fields, and it might require case-by-case adjustment, which is the most serious shortcoming of this model. 2440



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specified convergence tolerance (10−4). The velocities and impeller torques were also monitored during the calculation. LES results could reach quasi-steady state after about 20 revolutions, and flow data were collected and statistically processed in the subsequent 40 revolutions. Because gathering data for time statistics inside a sliding zone is not meaningful in the commercial code FLUENT,49 a user-defined subroutine was programmed for the LES postprocessing in the present work. In the subroutine, the instantaneous velocity fields in absolute Cartesian coordinates were collected at specified time steps and statistically processed at the end of the simulation, and then, the results were exported. In our LES simulation, the calculated TKE included contributions from both resolved and subgrid scales, but the SGS contribution is less than 1% of the total TKE at all phase angles, consistent with the results reported by Eggels,25 Yeoh et al.,28 and Hartmann et al.34 The simulation was performed on a 36-node cluster with dualsocket six-core Intel Xeon processors, 24 GB memory, and a Linux operating system in each node. The RANS based realizable k−ε model50 was also adopted using the same grid model as that used for the LES. Twelve phase-resolved data were obtained between two successive blades, and then, the phase-averaged results were calculated. The second-order upwind scheme was used for the momentum equations, and the SIMPLE algorithm for the coupling between the continuity and momentum equations.

Figure 2. Phase-averaged plot of velocity and TKE field obtained by PIV.

tips and cannot be extended to as farther than r/T > 0.25. These regions of high TKE are recommended for the injection of reactive material or catalyst. Owing to the interaction between the two discharge streams, the regions with k/Vtip2 = 0.03−0.08 have been enlarged over those of the single-loop flow pattern.16 This enlargement will be helpful to the mixing in circulation region. To illustrate the dynamic flow field, the phase-resolved velocity and TKE distributions at four different phase angles 0°, 15°, 30°, and 45° are shown in Figure 3. The formation and subsequent evolution of the trailing vortices and impeller discharge streams are resolved in detail by the high resolution PIV system and provide more elaborate descriptions of the flow hydrodynamics than the LDV results of Rutherford et al.13 Behind the upper and lower turbines, two pairs of counterrotating vortices are generated and then move toward each other in a zigzag pattern. During the lifetime of the trailing vortices, the two pairs of vortices are transported at a merging tendency. However, the trailing vortices originated from RTU and RTL cannot move far enough to interact directly with each other, and the interactions only take place between the two inclined impeller discharge streams at all phase angles. The development of the trailing vortices is also closely related with the distributions of the high TKE regions. Accompanied by the movement of the trailing vortices, mechanical energy is transported radially and axially away from the blade tips to the whole tank. From Figure 3, at the phase angle of 15°, two regions with high TKE levels can be clearly identified: the one on the left is newly generated by the current blade, and the one on the right is associated with the leading blade, 60° before the current one. In other words, the region with k/Vtip2 > 0.10 can last about 75° behind the leading blade. The contributions from the pattern of TKE transportation and the strong interaction between impeller streams can lead to 20% lower mixing time in merging flow than in parallel or diverging flow, as reported by Rutherford et al.13

4. RESULTS AND DISCUSSION In the following discussion of PIV experimental and CFD simulated results, the radial, axial, and tangential components of mean velocity are represented by U, V, and W, respectively, and the corresponding fluctuating components are represented by u′, v′, and w′, respectively. Both mean and fluctuating velocities are normalized with the impeller tip velocity Vtip, and TKE is normalized with Vtip2. The origin of the coordinates is located at the center of the bottom of the stirred tank, and the axial and radial coordinates are normalized with the diameter of the tank T and are expressed by z/T and r/T, respectively. 4.1. Flow Field Measured by PIV. 4.1.1. Distributions of Velocity and TKE. Figure 2 shows the phase-averaged velocity (represented by vectors) and TKE (represented by contours) distributions in the configuration of blade angular difference β = 0°, and the regions with normalized TKE lower than 0.02 are not presented for the sake of conciseness. Two impeller streams mainly discharged from the lower part of the side edge of RTU and the upper part of the side edge of RTL begin to merge at about r/T = 0.28 and then form two large circulations in the domain of the tank, as shown in Figure 2. In the region between these two streams, secondary flow phenomena can also be identified. The impeller streams, the main circulations, and the secondary flows are almost perfectly symmetrical around the line of z/T = 0.5, so the difference between the boundary conditions of the top free surface and the bottom tank wall does not have significant influence on the flow characteristics in the region between the two Rushton turbines. If we analyze only the flow pattern of RTU, we can find that the flow pattern is very similar to the typical singleloop flow field for the case of a single RT, except that there are two small vortices for the single-loop flow,16 but only one for the merging flow of this work. The distribution of phase-averaged TKE is closely associated with the impeller discharge streams. The high TKE regions with k/Vtip2 greater than 0.08 are mainly located near the blade 2441



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Figure 3. Phase-resolved plots of velocity and TKE fields obtained by PIV at θ = 0°, 15°, 30°, and 45°.

4.1.2. Effect of Blade Arrangements on Flow Field. When the angle β between the current blades of RTU and RTL deviates from 0°, the two pairs of trailing vortices are generated in different tangential directions. The corresponding phaseresolved flow fields, which were obtained at fixed phase angles, i.e., the angle between the current blade of RTL and the measurement plane, are obviously different from those obtained at β = 0°. On the other hand, because of the short lifetime of the trailing vortices, interaction occurs only between two inclined impeller streams, as mentioned in Section 4.1.1. Therefore, it is interesting to notice that, for example, the phase-resolved result at θ = 0° and β = 15° is almost the same as a combination of the lower part of Figure 3a and the upper

part of Figure 3b. The results at other phase angles and blade angles also have similar phenomena and are not presented here for conciseness. From the viewpoint of design and optimization, phaseaveraged flow fields are more commonly used than phaseresolved results in process engineering, so the phase-averaged flow characteristics at different blade angles were quantitatively compared and partly presented in Figure 4. The phase-averaged radial and axial velocity distributions at different radial positions are only slightly different near the peak of the profiles, so the blade angle almost has no effect on the phase-averaged velocity distribution for the merging flow pattern. As for the TKE distributions, at r = 0.18T, the maximum TKE value at β = 0° is 2442



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Figure 4. Phase-averaged velocity and TKE profiles obtained by PIV at r = 0.18T and r = 0.25T for β = 0°, 15°, 30°, and 45°.

a little smaller than those at β = 15°, 30°, and 45°, but this difference decreases with increasing distance from the blade tip, as shown in Figure 4f for r = 0.25T. Therefore, it is not necessary to adopt the arrangements with different blade angles for the merging flow, especially for applications in industrial reactors. Multiple blade angles may increase the complexity of the mechanical design and manufacture of shaft.

4.1.3. Trajectories of Measured Trailing Vortex Cores. Trailing vortices behind each blade of the impeller have significant influence on the mixing characteristics of stirred tanks, and they are also closely related to the transportation of kinetic energy, as mentioned in Section 4.1.2. Escudie and Line51 derived a simplified procedure based on Jeong and Hussain’s definition52 to identify a vortex and its core using 2D PIV measurements, and the results are in good agreement with 2443



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those calculated from 3D PIV data.53 Therefore, this approach was applied to determine the trailing vortex cores in the present work. The trajectories of the upper and lower vortices of RTU and RTL measured at β = 0° are shown in Figure 5. The

with the results of Rutherford et al.13 It can be concluded that the movement of the trailing vortices scales well for geometrically similar stirred tanks under fully turbulent conditions. The other two vortices move a little farther in the radial direction, as shown in Figure 5. However, the four trailing vortices travel shorter distances in the radial direction in the merging flow pattern than in the double-loop flow pattern.53 A similar phenomenon was also reported by Liu et al.54 for the merging flow pattern under the configuration of D = 0.4T and C2 = 0.31T. These phenomena may be attributed to the partially axial flow characteristics and the impeller stream inclination of the merging flow pattern. 4.2. Flow Characteristics Simulated Using LES Method. 4.2.1. Phase-Averaged Flow Field. To preliminarily validate the LES results, the pressure and shear stress acting on the turbines were integrated, and the power number of the dual Rushton turbine under merging flow was calculated. Our LES simulated power number was found to be 8.6, which is in good agreement with the experimental value of 8.4. The corresponding power number under parallel flow is 10.13,18 A sharp power drop occurs during the transition from parallel flow to merging flow, and this sharp power drop has been accurately predicted by Montante and Magelli23 using a RANS based CFD method. Figure 6 shows the phase-averaged plots of velocity and TKE fields simulated by LES with the dynamic Smagorinsky-Lilly (DSL) SGS model and by the realizable k−ε model. A comparison of Figures 6a and 2 shows that both the shapes of the two impeller discharge streams and the secondary flows between the two turbines are well predicted by LES with the DSL model, but the angle between the stream and the horizontal plane is slightly underestimated by LES. This underestimation will be discussed in the following quantitative comparison. According to the results of the realizable k−ε model, as shown in Figure 6b, the impeller streams are partly discharged from the lower edge of RTU and the upper edge of RTL, so the secondary flow regions are severely compressed in

Figure 5. Trajectories of the trailing vortex cores of RTU and RTL obtained by PIV. (The upper and lower vortices of Escudie et al.53 were generated under classical double-loop flow pattern.)

trajectory measured by Rutherford et al.13 was generated by the lower Rushton turbine RTL and identified from the locations where the axial velocity was zero. The trajectories of the trailing vortices under the classical double-loop flow pattern measured by Escudie et al.53 are also presented. Figure 5 was plotted on the r−θ plane as viewed from the top of the tank, with the turbines rotating in the clockwise direction. The trajectories of the upper vortex of RTU and the lower vortex of RTL coincide well with each other, in good agreement

Figure 6. Phase-averaged plots of velocity and TKE fields simulated by LES and realizable k−ε model. 2444



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Figure 7. Phase-averaged radial velocity profiles at different radial positions (LDV from Rutherford et al.13 and SG from Micale et al.18).

the radial direction, and the distribution of TKE is physically incorrect in shape and seriously underpredicted in value. Figures 7 and 8 compare the simulated axial profiles of phaseaveraged radial velocity and TKE, respectively, with the corresponding LDV and PIV data at different radial positions. The LDV data were taken from Rutherford et al.,13 and the SG (sliding grid) modeling results with the standard k−ε model were taken from Micale et al.18 Because the results obtained by the realizable k−ε model in the present work do not provide better estimates than the SG data, and similar CFD results were also obtained by Chiu et al.22 using the k−ε model with the sliding grid impeller modeling technique, only the SG data of Micale et al.18 are given here for conciseness. The profiles in Figure 7 clearly show that the LES method with the DSL model can successfully predict the velocity characteristics of the merging flow pattern, especially near the tip of the blade at r = 0.18T. Since LES slightly underpredicts the convergence of the two impeller discharge streams, the simulated velocity peaks are slightly higher than the PIV results with increasing radial distance from the center, as shown in Figure 7(b,c,d). Similar phenomena can also be observed, especially as shown in Figure 8c, for the distribution of TKE. However, obvious underestimations of phase-averaged TKE appear in the region near the blade, as shown in Figure 8a. The treatment of solid wall boundary condition is a key issue in

applying the LES technique, and there are two possible approaches: either resolving the near-wall dynamics directly or modeling them. The use of a sufficiently fine grid to directly resolve the wall boundary layer is not practical in the present stirred tank with Reynolds number of about 40 000. Therefore, in our LES calculations the wall function method was used for all solid walls to model the near-wall characteristics. Sagaut55 pointed out that most wall models can present satisfactory mean flow results with appropriate grids, but the turbulence intensities are usually unphysical or spurious. The LES simulated profiles of velocity and TKE near the tip of the blade in our work are similar in that the simulated velocity results are in better agreement with experimental data than the simulated TKE results are. At different radial positions, the radial velocity profiles from PIV are in good agreement with those from LDV; i.e., for geometrically similar stirred tanks, the normalized flow fields scale well under fully turbulent conditions. As for the TKE profiles from the literature, since the LDV data were read from the published contour figures and specifically no data could be distinguished and read at k/Vtip2 lower than 0.027,18 large differences may occur in the regions of r/T < 0.3 and r/T > 0.7, as shown in Figure 8a,b. As a consequence of the excessive convergence of the two streams predicted by the k−ε model, the radial velocities are 2445



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Figure 8. Phase-averaged TKE profiles at different radial positions (LDV from Rutherford et al.13 and SG from Micale et al.18).

corresponding PIV experimental results at θ = 15°, as shown in Figures 9b and 3b. These differences lead to, for example, some overestimations of radial velocities at r = 0.25T and θ = 45°, as illustrated in Figure 10b, and obvious underestimations of TKE at r = 0.18T and θ = 45°, as shown in Figure 10d, although the simulated plots of velocity and TKE are quite similar to the corresponding experimental plots in shape. As reviewed by Joshi et al.,17 because few researchers have quantitatively compared and validated the phase-resolved flow characteristics in stirred tanks using the LES methods, more attention should be paid to these areas in the future. The disagreement between the LES and PIV data may be caused by two reasons. First, the use of interface to connect the moving and the static zones leads to some oscillations near the interface position, as shown in Figure 9 at about r = 0.32T, but this method has been the only suitable way to simulate the instantaneous flow fields of stirred tanks in the commercial code FLUENT until now. Second, the grid refinement in the present work is mainly focused on the inner rotating region for the sake of computational efficiency, and this focus may have some effect on the convergence of the two impeller discharge streams. 4.2.3. Trajectories of Trailing Vortex Cores Simulated by LES. To quantitatively identify the movements of the trailing

severely underpredicted at r = 0.18T and completely merged at r = 0.25T. Furthermore, the underpredictions of TKE with the k−ε model have been extensively reported by many other researchers.17 A possible explanation for the underpredictions might be the underlying isotropic assumption and the strong swirling flow characteristics of the merging flow field. 4.2.2. Phase-Resolved Flow Characteristics. The phaseresolved plots of the velocity and TKE fields simulated by LES with the DSL SGS model, as shown in Figure 9, are in good agreement with the PIV experimental results presented in Figure 3. The results compared include the generation and movement of the trailing vortices, the distribution and transportation of TKE, and the dynamic interactions of two impeller discharge streams. The quantitative comparisons between PIV and LES at θ = 15° and 45°, as shown in Figure 10, further confirm the accuracy of the LES results. The corresponding predictions of the k−ε model do not agree with the PIV results and are not presented here for conciseness. Some differences still exist between the PIV and LES data. For example, the high TKE regions generated by the two current blades are smaller in the LES results than in the PIV results, and the LES predicted regions with high TKE levels produced by the two leading blades are moving slower in the axial direction but faster in the radial direction than the 2446



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Figure 9. Phase-resolved plots of velocity and TKE fields simulated by LES at θ = 0°, 15°, 30°, and 45°.

merging flow pattern do not extend as far as those under the double-loop flow pattern, further confirming that the LES method is suitable for the complex flows in stirred tanks. However, the disagreement of the simulated trajectories with the PIV ones is strongly related to the distributions of the phase-resolved TKE levels and need further investigations in the future.

vortices, the simulated trajectories of the trailing vortex cores of RTL and the corresponding PIV measurements are compared in Figure 11. Figure 11 shows the successful prediction of the trajectories of the trailing vortex cores, except that from about 40° to 60° after the blade, the predicted vortex cores move a little farther in the radial direction than the PIV ones measured in this work and the ones from Rutherford et al.13 The simulated trajectories of the upper and the lower vortex of RTU are almost the same as those of the lower and the upper vortex of RTL, respectively, and are not presented here for conciseness. The method used to determine the simulated cores of trailing vortices is the same as that mentioned in Section 4.1.3. In comparison with the results obtained by Escudie et al.,53 our LES calculated trajectories under the

5. CONCLUSIONS Both phase-averaged and phase-resolved velocities and turbulent kinetic energy (TKE) distributions of the merging flow pattern in a dual Rushton turbine stirred tank were measured by means of a commercial PIV system with high resolution. Two pairs of trailing vortices are generated behind 2447



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Figure 10. Phase-resolved velocity and TKE profiles simulated by LES at different radial positions for θ = 15° and 45°.

the blades of the upper and the lower turbine and then transported toward the central part of the tank. However, the interaction and convergence only occur between the two impeller discharge streams. The regions with high TKE levels are closely related with the movement of the trailing vortices and are transferred from the blade tip to the bulk of the tank. This complex flow field can be considered as a benchmark for CFD modeling and calculation. The phase-averaged velocity profiles at different radial positions measured by PIV in the present work are in good agreement with the laser Doppler velocimetry (LDV) results from the literature. However, the literature TKE data were read from published contours and should be further verified in the future. The angle between the respective current blades of the upper and lower turbines has no significant influence on the phase-averaged flow fields, although the phase-resolved flow patterns at various angle values are totally different. The predicted flow characteristics obtained using the LES method with the dynamic subgrid scale model, including the phase-averaged and phase-resolved velocities and TKE distributions, and the trajectories of the trailing vortex cores, are in good agreement with the PIV experimental results and the literature LDV data. However, the RANS based k−ε model fails to provide reasonable simulation results for the merging

Figure 11. Trajectories comparison of trailing vortex cores of RTL. (The upper and lower vortices of Escudie et al.53 were generated under classical double-loop flow pattern.)

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flow pattern. The LES method has great potential in predicting the complex flow in stirred tanks, although the causes of some discrepancies need to be further investigated in the future.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

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ACKNOWLEDGMENTS The authors would like to thank Professor G. M. Evans (Department of Chemical Engineering, University of Newcastle, Australia) for the helpful discussion on the PIV experiments. The financial support from the National Natural Science Foundation of China (Nos. 20776008, 20821004, 20990224) and the National Basic Research Program of China (973 Program, No. 2007CB714300) is gratefully acknowledged.

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NOMENCLATURE C2 = separation of two Rushton turbines (m) Cs = Smagorinsky constant (-) D = diameter of Rushton turbine (m) H = height of liquid in tank (m) k = turbulent kinetic energy (m2 s−2) N = rotational speed (s−1) p = pressure (Pa) r = radial coordinate (m) Re = Reynolds number (-) Sij = strain-rate tensor (s−1) t = time (s) T = diameter of stirred tank (m) ui = instantaneous velocity in ith direction (m s−1) u′, v′, w′ = fluctuating radial, axial, and tangential velocity, respectively (m s−1) U, V, W = mean radial, axial, and tangential velocity, respectively (m s−1) Vtip = tip velocity of Rushton turbine (m s−1) Wb = width of baffle (m) xi = spatial component in ith direction (m) z = axial coordinate (m)

Greek letters

Δ = filter width (m) δij = Kronecker’s delta (-) ε = dissipation rate of turbulent kinetic energy (m2 s−3) θ = angle between current blade and measurement plane (rad) β = angle between current blades of upper and lower turbines (rad) ν = kinematic viscosity (m2 s−1) νt = eddy viscosity (m2 s−1) τij = SGS stress tensor in grid filter (m2 s−2)

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REFERENCES

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