Volumetric 3-component velocimetry (V3V) measurements of the turbulent flow in stirred tank reactors

Transcription

1 Volumetric 3-component velocimetry (V3V) measurements of the turbulent flow in stirred tank reactors David Hill 1, Daniel Troolin 2, Geoffrey Walters 3, Wing Lai 4, Kendra Sharp 5 1: Department of Civil and Env. Engr., Penn State University, University Park, PA, USA, dfh4@psu.edu 2: TSI, Incorporated, 500 Cardigan Rd., Shoreview, MN, USA dtroolin@tsi.com 3: Department of Civil and Env. Engr., Penn State University, University Park, PA, USA, gsw122@psu.edu 4: TSI, Incorporated, 500 Cardigan Rd., Shoreview, MN, USA wlai@tsi.com 5: Department of Mech. and Ind. Engr., Penn State University, University Park, PA, USA, kvs10@psu.edu Abstract: Volumetric 3-component velocimetry (V3V) measurements have been made of the flow field near a Rushton turbine in a stirred tank reactor. This particular flow field is highly unsteady and threedimensional, and is characterized by a strong radial jet, large tank-scale ring vortices, and small-scale blade tip vortices. V3V uses a single camera head with three apertures and particle tracking methods to obtain approximately 15,000 vectors in a cubic volume 10 cm on a side. The data are then interpolated onto a regular grid, allowing for the ensemble averaging of individual realizations and the calculation of gradientbased quantities such as vorticity. The volumetric velocity data from the present experiments offer the most comprehensive view to date of this flow field. The mean data clearly show the structure of the tip vortices and the Gaussian nature of the radial jet. The turbulent statistics reveal the production of turbulence in the high-shear region near the impeller. 1. Introduction Particle imaging methods have developed rapidly over the past two decades (Adrian 2005). Much of this development has been facilitated by steadily increasing experimental capabilities, in the forms of increased data storage, larger CCD (charge coupled device) arrays, and faster camera frame rates and laser repetition rates. In addition to being assisted by technological capacity, this development has been motivated by the scientific goal of increasing the dimensionality and resolution of the obtained data. The ultimate goal is to obtain instantaneous volumetric measurements of the complete (three components) velocity vector with very high temporal and spatial resolutions. These data will allow for the precise evaluation of velocity and acceleration fields and of gradient-based variables such as vorticity and dissipation. Hinsch (1995; Fig. 1 of that paper) summarizes how this trajectory from standard laser Doppler anemometry (LDA) to multi-component LDA to standard particle image velocimetry (PIV), and so on, has steadily increased the dimensionality of the available data space. Classic PIV (Adrian 1991), both in its original film and modern digital formats, uses a single camera to obtain two components of fluid velocity on a planar surface, with low to moderate temporal resolution. Stereoscopic methods (Arroyo and Greated 1991; see Prasad 2000 for a review) use two cameras with lateral and / or angular offset between them to obtain three components of fluid velocity on a planar surface. Generally, the uncertainty in the out-of-plane velocity component is higher than for the in-plane components. Soloff et al. (1997) describe a generalized distortion compensation procedure for stereoscopic PIV that uses calibration images of an in-situ target to map the image planes of the two cameras to the object plane. This method absorbs all sources of distortion between the field of view and the cameras, allowing for the straightforward registration of multiple cameras and the recombination of individual (one from each camera) two-dimensional displacement fields into a single three-dimensional displacement field

2 Kahler and Kompenhas (2000) describe a variant on standard stereoscopic PIV called dual-plane stereoscopic PIV. In this technique, stereoscopic PIV fields are recorded, instantaneously, on two parallel planes (offset in the lateral, out-of-plane direction). Possession of velocity data on these two planes allows for the evaluation of derivatives in the out-of-plane direction. Stereoscopic PIV requires a level of optical access that can be difficult to obtain in some experimental configurations. Wernet (2004) and Willert et al. (2006) both describe a hybrid method that uses PIV to obtain in-plane velocity components and Doppler global velocimetry (DGV) to obtain the out-of-plane velocity component. While this hybrid methodology may be different from stereoscopic PIV, the end result of three-dimensional velocity on a planar surface is the same. A quasi-instantaneous volumetric method is obtained by using stereoscopic PIV with a laser sheet that is rapidly scanned in the out-of-plane direction. Hori and Sakakibara (2004) applied this method to the case of a round turbulent jet. In their method, the two cameras were on opposite sides of the light sheet, both oriented to receive forward-scattered light. The vertical light sheet was controlled via a rotating mirror, allowing it to be scanned in an angular fashion throughout the measurement volume. While this method does not yield an instantaneous snapshot of the flow in the measurement volume, the authors argued that since the scanning time (0.22 s) was less than the Kolmogorov time scale (estimated at 0.45 s), their method was adequate to resolve the smallest eddies in the flow. Burgmann et al. (2008) discuss a scanning stereoscopic PIV system that, rather than mechanically scanning a single laser sheet with a mirror, uses a series of 10 independent laser heads, all pre-aligned at different out-of-plane locations. By firing these lasers in sequence, a finite volume is rapidly illuminated and imaged. The maximum volumetric imaging rate was reported to be in excess of 90 Hz. One method capable of providing truly instantaneous volumetric data is holographic PIV (HPIV) (e.g. Meng et al. 2004). HPIV involves illuminating a flow field seeded with particles, similar to standard PIV. An interference pattern, generated by the interference of a reference wave with the light scattered by the seed particles, is then recorded and interrogated to obtain three dimensional displacements over a volume. Traditional HPIV uses photographic films as the recording medium. Given the time required to process films, this severely limits the number of realizations that can be obtained. Digital HPIV uses digital cameras as the recording medium. However, the present inferior spatial resolution of a CCD array compared to photographic film limits the number of velocity vectors that can be obtained. A second method of obtaining instantaneous volumetric data is tomographic PIV (TomoPIV) (Elsinga et al. 2006, Schroder et al. 2008). TomoPIV uses diffuse laser light to illuminate seed particles over a volumetric region of interest. Scattered light from the seed particles is collected by an array (3-4) of cameras oriented at different angles. The apertures on the cameras are set to ensure good focus over the entire depth of the illuminated volume. Three dimensional particle distributions are reconstructed from the individual camera images and the distributions from the two camera exposures are cross-correlated to obtain the three dimensional particle displacement field. Finally, three-dimensional particle tracking velocimetry (3dPTV) (Maas et al. 1993, Virant and Dracos 1997) is similar to TomoPIV, but uses lower seeding densities, allowing individual particles to be tracked during the time interval between successive camera images. The present paper describes volumetric three component velocimetry (V3V) measurements of the flow in a cylindrical tank stirred by a Rushton turbine. This work is based upon 3dPTV, but allows for higher seeding densities (and generates more velocity vectors) than previous 3dPTV - 2 -

3 studies. The Rushton turbine flow was chosen for two primary reasons. First, flows in stirred tanks are extremely important to a wide range of engineering disciplines. Second, the flow is extremely complex and highly three-dimensional, making it a suitable test case for the volumetric measurements. Rushton turbines have been widely studied and used since the 1950s. They are high-shear devices that commonly find application to gas dispersion processes. The literature on their powerconsumption and flow-field characteristics is quite extensive and only the most relevant studies are discussed here. The flow field in a tank stirred by a Rushton turbine is characterized by a strong radial jet which sets up large ring vortices, one in each half (in the vertical sense) of the tank. Additionally, in the absence of baffles, a strong circumferential flow will be established. Finally, and as discussed extensively (e.g. Van t Riet and Smith 1975, van der Molen and van Maanen 1978, Wu and Patterson 1989), each blade produces a pair of counter-rotating tip vortices to promote mixing on a much smaller scale than the ring vortices. Sharp and Adrian (2001) carried out a thorough two-dimensional PIV study of the flow in the very near field of the impeller. Their phase-locked data in the radial azimuthal plane provided insight into the dynamics of the tip vortices (location, strength, etc.) as well as highly resolved maps of the mean and turbulent flow fields. Hill et al. (2000) carried out stereoscopic PIV measurements, also in the near-field of the impeller. They interpolated their results onto a curved surface (the angular axial plane) spanning two adjacent blade tips, providing a clearer picture of the three-dimensional structure of the tip vortices. This work was extended by Yoon et al. (2005) in an effort to understand the scaling of the flow between differently sized tanks. 2. Experimental Methods and Procedures Experimental Setup The present experiments were conducted in an unbaffled cylindrical mixing tank (Fig. 1) with an inner diameter (D) of 44.3 cm. The tank was filled with water to a depth equal to D and a lid was used to prevent free surface displacement (Froude number effects) and to help ensure vertical symmetry in the mixing tank. The cylindrical tank was then placed in a square glass tank of larger dimension and the space between the two tanks was filled with water. This was done to greatly reduce light refraction from the curved surface of the cylindrical tank. Figure 1: Plan (a) and elevation (b) views of the experimental tank. Mixing in the tank was provided by the rotation of a six-blade Rushton turbine (Fig. 1), located at - 3 -

4 the vertical mid-point of the tank. The turbine followed the traditional specification of having a diameter (blade tip to blade tip; D T ) equal to D/3. In a slight variation from the usual configuration, the impeller shaft passed through the impeller disk and extended to the bottom of the tank, where it was supported by a bearing. This was done in an effort to minimize lateral oscillations of the shaft at high rotational rates. The impeller was controlled by a mixer (Lightnin, Inc., L5U10F) and was painted a flat black in order to reduce reflections of laser light. In the present experiments, data were obtained at impeller rotational speeds of 42, 295, and 379 rpm. For impeller flows, the Reynolds number is generally defined as 2 NDT Re =, ν where N is the rotational rate in revolutions sec -1 and ν is the fluid s kinematic viscosity. Therefore, the present data sets correspond to Re values of approximately 15,000, 107,000, and 137,000. Data Acquisition The flow field was illuminated (Fig. 2) from the side with a dual-head Nd:Yag laser emitting light with a wavelength of 532 nm, a pulse width of 12 ns, and energy of 120 mj pulse -1. The two beams were combined to travel on co-linear paths. A light cone was generated with two -25 mm cylindrical lenses mounted at 90 to each other. These cylindrical lenses spread the beam in the horizontal and vertical directions to illuminate a measurement volume approximately 120 mm height 120 mm width 100 mm depth. Beam blocks were placed at the edge of the rectangular water tank in order to clip the laser cone to illuminate only the volume depth being measured and reduce background noise. The flow was seeded with 100 µm polycrystalline particles. Figure 2: Front elevation view (as seen by camera) of experimental tank, showing location of laserilluminated region. The V3V camera probe (TSI, Inc.), shown in Fig. 3, consists of three apertures, each containing 4 million pixels ( ). The images are 12 bits, with a pixel size of 7.4 microns. 50 mm camera lenses were used with a fixed aperture of f16. The imaged volumetric field of view was approximately 10 cm on a side. The V3V camera frames and laser pulses were triggered by a TSI synchronizer with 1 ns resolution. The pulses from each laser were timed to straddle neighboring camera frames in order to produce images suitable for 3D particle tracking. The time between frame-straddled laser pulses ( t) was 1500 µs, 250 µs, and 200 µs, for the speeds of 42, 295, and 379 rpm, respectively. The synchronizer was externally triggered by a shaft encoding trigger mounted on the shaft of the turbine axis. This allowed velocity captures to be taken consistently at the same phase in the shaft cycle. The images were streamed to a HYPER2-4 -

5 HyperStreaming computer, and subsequently analyzed. Figure 3: Elevation (side) view of experimental tank, showing location and optical path of the V3V camera head. Data Processing The 3D particle tracking algorithm is based upon the work of Ohmi and Li (2000) and is done in three primary steps. The first is determining the 2D particle locations in each image, then determining the 3D particle locations in space, then tracking the particles in the volume. 1. 2D Particle Identification. One V3V capture consists of 6 separate images, one from each of the three apertures at instance A, and one from each of the three apertures at instance B. The first step in the tracking involves identifying the 2D particle locations in each of the individual images. This is done by setting two parameters. The first is a baseline intensity threshold. Any valid particle must have a peak intensity above this threshold. This quickly reduces the search area to include valid particles and eliminate background noise. The second parameter is a local ratio, in which the particle peak intensity must be larger than the local background by this ratio. Finally a Gaussian intensity profile is fitted to the particle image, the peak of which represents the center of the particle. 2. 3D Particle Identification. Images from each of the apertures are effectively combined in order to determine the 3D location of each particle. Each triplet represents a single particle in the flow, as seen in Fig. 4. The centroid of the triplet represents the x and y location and the size of the triangle represents the z location. The correspondence of a 2D particle in one image to the same particle in the other two images is achieved through a volumetric spatial calibration. The spatial calibration is done by traversing a single plane target through the measurement region, and capturing images at regular intervals. Dots on the calibration target are regularly spaced at 5 mm in the horizontal and vertical directions, and the target is traverse by 5 mm increments in the depth direction. The calibration dot locations from each image are combined to define a signature graph, in which a defining triplet size is determined for each plane

6 Figure 4: Sample V3V triplets. Figure 5 shows a schematic of the V3V aperture arrangement. The drawing on the left shows each aperture looking into the page and to the right. The cube represents the measurement volume. For purposes of simplification, only one particle is shown in the measurement volume, represented by a small sphere within the cube. The right side of Fig. 5 displays the 2D representations of the view from each camera aperture. Green represents the ray extending from the left aperture through the particle, red represents the ray extending from the right aperture through the particle, and blue represents the ray extending from the top aperture through the particle. Consider the 2D particle as seen by the top image (labeled Top Image in Fig. 5). The blue ray appears as a single dot; however, in the left and right images, the blue ray appears as a line. The algorithm searches along this defining ray in the left and right images simulaneously for possible 2D particle matches. The search is done in two steps, a coarse search, then a fine search. The fine search requires the particle position to match within 0.5 pixels in all three images. If a triplet is not found that falls within the 0.5 pixel criteria, the 2D particles are not used. This process is repeated for all particles in the field. Figure 5: Schematic of V3V aperture arrangement and the 2d representations of a single particle and the optical rays from all three cameras. 3. 3D Particle Tracking (relaxation method). Once the volumetric particle locations are determined in frame A and frame B, the particles are divided into subgroups called clusters according to their spatial locations. Clusters here can be thought of as similar to interrogation regions in PIV. Clusters in B are larger in volume than corresponding clusters in A because particles may move out of the cluster area. Within a cluster, each pair of - 6 -

7 corresponding particles is assigned a number representing match probabilities. For example, P(m,n) is the match probability between particle m in frame A and particle n in frame B. Initially each particle pair has the same probability, 1/N, where N is the number of possible pairs between A and B for each cluster. The probability computation is based on the assumption that neighboring particles move similarly. These probabilities are then iteratively recomputed for all particles in the cluster, until they converge. For particle m in frame A, the maximum match probability P(m,n) is found among P(m,1), P(m,2), If this maximum probability is greater than a given threshold, then (m,n) is considered a matched pair. As shown in Fig. 6, the probability is high (case shown on the left) when, if the displacement from particle m to particle n is applied to the other particles in the cluster, a matching particle is nearby. The probability is low (case shown on the right) when the displacement of other particles in the cluster results in no nearby particle matches. Figure 6: Illustration of potential (high probability and low probability) particle matches between successive camera frames. After the 3D particle tracking step, the vectors lie on a randomly spaced grid, according to particle locations. In order to computer quantities such as vorticity, it is useful to have vectors on a rectangular grid. This was done through regular Gaussian-weighted interpolation. For the experiments carried out here, the average number of 2D particles identified per image was approximately 60,000 which resulted in approximately 30,000 valid 3D particles. Of these particles, the average number of independent particle-tracked vectors was approximately 15,000. These randomly spaced vectors were interpolated onto a rectangular grid with 4 mm vector spacing which resulted in an average of approximately 23,000 vectors per field. For each run, 500 phaselocked realizations were obtained, and the velocity fields were ensemble-averaged to yield mean and turbulent flow fields. 3. Experimental Results and Discussion Figure 7 gives results for the mean, non-dimensionalized, angular vorticity (ω θ *) for the 295 rpm dataset. Note that the impeller diameter and the blade tip speed have been used as characteristic length and velocity scales in normalizing these and all subsequent results. An asterisk symbol is used to denote scaled variables. Next, note that the impeller has been included in order to help establish the relative position of the volumetric data region. The heavy gray line indicates the axis of the impeller shaft and the angular velocity vector of the impeller is aligned with the negative y axis. Isosurfaces of angular vorticity are shown, clearly revealing the structure of the counterrotating tip vortices. In both the above and below perspective views, two pairs of tip vortices are present, as shed from adjacent impeller blades. Vertical slices of angular vorticity are also - 7 -

8 included, showing the rapid radial decay of the fluid vorticity. Figure 7: Isosurfaces and vertical slices of mean, normalized angular vorticity (ω θ *) for the 295 rpm case. left and right figures give above and below views of the impeller and the volumetric data region. The Figure 8: Isosurface of mean, normalized fluid speed for the 295 rpm case. The isosurfaces of mean vorticity are also shown to indicate the locations of the tip vortices. Vectors of the mean velocity field are provided. The left and right figures give above and below views of the impeller and the volumetric data region. Similar perspective views of the mean velocity field are given in Fig. 8. The vorticity isosurfaces from Fig. 7 are retained and one new isosurface of normalized fluid speed is presented. From this surface, the Gaussian profile of the radial jet becomes quite evident. Velocity vectors are also shown and indicate the strong circumferential flow established in the non-baffled experimental tank. Note that only every third vector in each of the three Cartesian directions is shown for the sake of visual clarity

9 Figure 9 presents results for some of the turbulent flow characteristics. Both the turbulent kinetic energy and the radial axial Reynolds stress term, given by ' ' < u ru y > *, are shown. Note that the turbulent kinetic energy is given by 1 ' ' * ' ' * ' ' * tke* = ( < urur > + < uθ uθ > + < u yu y > ). 2 As with the mean flow data, flow variables have been nondimensionalized by using the impeller diameter and the blade tip velocity as characteristic length and velocity scales. Select slices through the data volume and isosurfaces have been used to bring out the key characteristics of the distributions. For the turbulent kinetic energy, values outside of the radial jet are extremely small and the peak values are found on the vertical center-plane, not in the tip vortex cores. In fact, each tip vortex pair merges into a single concentrated region of high turbulent kinetic energy. For the radial - axial Reynolds stress, the distribution makes clear physical sense. For a classic Gaussian jet (which describes the radial jet in this flow), positive stresses are found above the axis, and negative stresses are found below the axis. In the present flow, the tip vortices modulate this distribution, creating local peaks at several radial locations. Note that this figure picks up a third tip vortex pair, located at approximately z* = 1.1 on the x* = 0 plane. Figure 9: Computed turbulent statistics for the 295 rpm case. The left figure shows slices and one isosurface of turbulent kinetic energy. The right figure shows slices and two isosurfaces of the ensemble averaged radial axial turbulent stress. The ability to manipulate these three-dimensional data, by interactively rotating the impeller data volume system, making animations of moving slices, etc., allows for the most detailed study to date of the complex flows in mixing tanks. Previous studies (Sharp and Adrian 2001, Hill et al. 2000) have been limited to two-dimensional or three-dimensional velocity measurements on a single plane. These previous studies nevertheless provide a useful comparison with the present data. As an example, adapted data from Sharp and Adrian (2001) are presented in Fig. 10. Their experiments were in a mixing tank 1/3 the size of the present tank and their experimental Reynolds number was approximately The various sub-figures correspond to the number of degrees since blade passage. The generation of the tip vortex pair and its trajectory (increasing radial location) are clearly evident. In this way, Fig. 10 essentially represents two-dimensional slices from the type of data shown in Fig. 7. The advantage of the experiments of Sharp and Adrian (2001) is the much greater spatial resolution of the in-plane velocity field. The reason for this is - 9 -

10 that the computational resources are being expended on a much narrower task; namely two components of velocity on a single plane. The advantage of the present experiments is their ability to capture the full three-dimensional structure of not just the tip vortex pair from the closest blade, but also that of the preceding blade. Figure 10: Two-dimensional PIV data on the radial - axial plane, as obtained by Sharp and Adrian (2001). Only 11% of the vectors are shown, for the sake of visual clarity. Color contours denote angular vorticity. The ensemble averaged fields correspond to: (a) - 10 deg, (b) - 20 deg, (c) - 30 deg, (d) 40 deg since passage of 4. Concluding Remarks the impeller blade, which is indicated by the rectangular masked region. The present measurements are the first instantaneous volumetric measurements of the threedimensional flow inside a stirred tank reactor. The V3V method greatly simplifies the process of acquiring this dense dataset by integrating three fixed camera apertures into a single head. In this way, acquisition of data requires only a simple in-situ calibration, and no complex alignment of cameras or Scheimpflug lens mounts, as is commonly the case in stereoscopic measurements. The V3V is able to return approximately 15,000 velocity vectors in a cubic volume roughly 10 cm on a side. The irregularly spaced vectors are easily re-interpolated onto a regular grid to facilitate ensemble averaging and the computation of gradient-based quantities. The data obtained in the present experiments greatly increase the understanding of the complex flow in stirred tanks. Previous single-camera PIV and stereoscopic PIV studies have provided glimpses of the flow features such as the tip vortices and the radial jet. With the present method, a full understanding of these features is now available. Isosurfaces of quantities such as velocity, vorticity and turbulent kinetic energy help to visualize the structure of the flow and to identify regions of high mixing potential

11 Acknowledgements Funding for this work came, in part, from the American Chemical Society (ACS PRF# AC9). The authors thank Corbitt Kerr and Scott Davison for their assistance in preparing several of the figures. References Adrian, R.J. (1991), Particle imaging techniques for experimental fluid mechanics, Annual Review of Fluid Mechanics, 23: Adrian, R.J., (2005), Twenty years of particle image velocimetry, Experiments in Fluids, 39: Arroyo, M.P., Greated, C.A., (1991), Stereoscopic particle image velocimetry, Measurement Science Technology, 2: Burgmann, S., Dannemann, J., Schroder, W., (2008), Time-resolved and volumetric PIV measurements of a transitional separation bubble on an SD7003 airfoil, Experiments in Fluids, 44: Elsinga, G.E., Scarano, F., Wieneke, B., van Oudheusden, B.W., (2006), Tomographic particle image velocimetry, Experiments in Fluids, 41: Hill, D.F., Sharp, K.V., Adrian, R.J., (2000), Stereoscopic particle image velocimetry measurements around a Rushton turbine, Experiments in Fluids, 29: Hinsch, K.D., (1995), Three-dimensional particle velocimetry, Measurement Science Technology, 6: Hori, T., Sakakibara, J., (2004), High-speed scanning stereoscopic PIV for 3D vorticity measurements in fluids, Measurement Science Technology, 15: Kahler, C.J., Kompenhans, C., (2000), Fundamentals of multiple plane stereo particle image velocimetry, Experiments in Fluids, 29:S70-S77. Maas, H.G., Gruen, A., Papantoniou, D., (1993), Particle tracking velocimetry in threedimensional flows, Experiments in Fluids, 5: Meng, H., Pan, G., Pu, Y., Woodward, S., (2004), Holographic particle image velocimetry: from film to digital recording, Measurement Science Technology, 15: Ohmi, K., Li, H.-Y., (2000), Particle tracking velocimetry with new algorithms, Measurement Science Technology, 11: Schroder, A., Geisler, R., Elsinga, G., Scarano, F., Dierksheide, U., (2008), Investigation of a turbulent spot and a tripped turbulent boundary layer flow using time-resolved tomographic PIV, Experiments in Fluids, 44: Sharp, K.V., Adrian, R.J., (2001), PIV study of small-scale flow structure around a Rushton turbine, AICHE Journal, 47: Soloff, S.M., Adrian, R.J., Liu, Z.-C., (1997), Distortion compensation for generalized stereoscopic particle image velocimetry, Measurement Science Technology, 8: van der Molen, K., van Maanen, H., (1978), Laser Doppler measurements of the turbulent flow in stirred vessels to establish scaling rules, Chemical Engineering Science, 33: Van t Riet, K., Smith, J., (1975), The trailing vortex system produced by Rushton turbine agitators, Chemical Engineering Science, 30: Virant, M., Dracos, T., (1997), 3D PTV and its application on Lagrangian motion, Measurement Science Technology, 8: Wernet, M.P., (2004), Planar particle imaging Doppler velocimetry: a hybrid PIV/DGV technique for three-component velocity measurements, Measurement Science Technology, 15: Willert, C., Hassa, C., Stockhausen, G., Jarius, M., Voges, M., Klinner, J., 2006, Combined PIV

12 and DGV applied to a pressurized gas turbine combustion facility, Measurement Science Technology, 17: Wu, H., Patterson, G.K., (1989), Laser Doppler measurements of turbulent flow parameters in a stirred mixed, Chemical Engineering Science, 44: Yoon, H.S., Hill, D.F., Balachandar, S., Adrian, R.J., Ha, M.Y., (2005), Reynolds number scaling of flow in a Rushton turbine stirred tank. Part I mean flow, circular jet, and tip vortex scaling, Chemical Engineering Science, 60: