On the main flow features and instabilities in an unbaffled vessel agitated with an eccentrically located impeller

Chemical Engineering Science 63 (2008) 4494 -- 4505

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On the main flow features and instabilities in an unbaffled vessel agitated with an eccentrically located impeller Chiara Galletti ∗ , Elisabetta Brunazzi Department of Chemical Engineering, Industrial Chemistry and Materials Science, University of Pisa, via Diotisalvi 2, I-56126 Pisa, Italy

A R T I C L E

I N F O

Article history: Received 17 April 2008 Received in revised form 12 June 2008 Accepted 14 June 2008 Available online 19 June 2008 Keywords: Hydrodynamics Mixing Turbulence Eccentric agitation Unbaffled vessel Macro-instability

A B S T R A C T

The hydrodynamics of an unbaffled vessel stirred by an eccentrically located Rushton turbine is investigated with both Laser Doppler Anemometry and flow visualisation techniques. The flow field is shown to be characterised by a strong circumferential motion which develops itself around two main vortices, one above and one below the impeller, both inclined with respect to the vertical plane. Such vortices are not steady but move periodically very slowly in comparison to the impeller rotational timescale. Accordingly, two low frequency components, whose values are linearly dependent on the impeller rotational speed, are identified across the vessel. The energetic contribution to the turbulent kinetic energy of such flow instabilities is significant so that they should be taken into account when evaluating micro-mixing information from turbulence quantities. Besides, an additional low frequency component is observed and related to vortex shedding phenomena from the flow-shaft interaction which occur in eccentric agitation operation. The flow discharged from the impeller is also measured and discussed. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Mixing is one of the most common unit operations in the process industry and it is typically achieved in stirred tanks. These are usually equipped with baffles in order to promote small scale mixing by breaking the primary vortex. Indeed most of the research on stirred vessels focuses on baffled configuration (Tatterson, 1991; Harnby et al., 1997; Paul et al., 2004). However, there are applications where unbaffled vessels are preferred, as for instance in food and pharmaceutical industry where cleaning is a major issue (Assirelli et al., 2008) or as in crystallisation because baffles can damage growing particles or as in laminar regime where baffles can cause the formation of dead regions. Most of the available literature on unbaffled vessels concerns this latter application. Among others, Lamberto et al. (1996) and Alvarez et al. (2002) pointed out that the unbaffled stirred vessels were not optimised with large waste of power, because of the presence of large poor mixing regions with a typical toroidal shape, located both above and below the impeller plane. Therefore, two different strategies have been suggested in order to improve mixing in unbaffled vessel in the laminar regime: the use of variable speed protocols (Lamberto et al., 1996) and the eccentric position of the shaft (Alvarez et al., 2002). The second strategy performs better as it enhances the axial circulation

∗

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0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.06.007

in the tank destroying the toroidal segregated regions and the separation plane between the upper and lower part of the vessel. This was noted even for impellers without blades (see the disc impeller of Alvarez et al., 2002) and this finding was exploited lately by Sánchez Cervantes et al. (2006) who suggested a disc impeller in eccentric position for the culture of suspended mammalian cells; in this manner mixing conditions are ensured with low mechanical stresses, so that damage to the cells is limited (Aloi and Cherry, 1996). Recently a combination of two eccentric impellers was studied for single-phase (Ascanio and Tanguy, 2005) and gas–liquid (Cabaret et al., 2008) systems with both Newtonian and non-Newtonian fluids. Only a few works regard eccentric agitation for turbulent flows, even though this operation mode is also suggested in the industrial practice (e.g. paint and food processes, Karcz et al., 2005) in alternative to baffles to break the primary vortex. Eccentric agitation for turbulent flows was pioneered by Kramers et al. (1953), Nishikawa et al. (1979) and Medek and Fort (1985). Kramers et al. carried out a comparative study on the rate of mixing and power consumption of three kinds of agitators, a three-blade marine impeller and twoimpeller type agitators. Nishikawa et al. evaluated the effect of eccentricity on mixing time of a paddle impeller, whereas Medek and Fort determined energetic efficiencies for different eccentricity values and number of baffles. More recently Karcz et al. (2005) investigated an unbaffled tank stirred by a D = T/3 propeller placed at different eccentric positions from e/T = 0 to 0.285. The authors carried out comprehensive computer-aided mixing time measurements, showing the strong

C. Galletti, E. Brunazzi / Chemical Engineering Science 63 (2008) 4494 -- 4505

decrease of mixing time with increasing the impeller eccentricity. They also provide shear rate, friction and heat transfer coefficients at the vessel walls. Recently, their investigation was extended to different types of eccentrically located impellers (e.g. radial, axial and mixed flow) (Karcz and Cudak, 2006; Cudak and Karcz, 2006, 2008). Hall et al. (2005a) investigated with particle image velocimetry (PIV) the best configuration of small tanks (high throughput experimentation reactors, HTE) stirred by an up-pumping pitched blade turbine with blades inclined by 45◦ . In particular the authors considered both baffled and unbaffled configurations with centric agitation, as well as unbaffled eccentric agitation. Mean flow field, rms and turbulent kinetic energy distribution were provided. The authors showed that eccentricity improves the axial motion which becomes comparable to baffled configuration. In addition the eddy dissipation rate was evaluated across the vessel through the Smagorinsky subgrid scale (SGS) model. Planar induced fluorescence (PLIF) was also used in the same work to evaluate mixing times which indicated eccentric agitation to perform better than baffled agitation. In a later work of the same group the eccentric agitation was discussed also for gas–liquid application of HTE reactors (Hall et al., 2005b). Recently, Montante et al. (2006) used both PIV and computational fluid dynamics (CFD) to analyse the flow field in a vessel stirred by a Rushton turbine placed eccentrically and for Re = 41, 000. The authors showed that the flow field was characterised by two vortices, departing from the impeller towards the top and the bottom of the vessel, and inclined with respect to the vertical plane. The authors addressed also numerical issues and suggested the use of transient calculations (unsteady RANS) for the CFD prediction of eccentric agitation operation. Actually steady RANS approaches can give satisfactory results for unbaffled vessel with centric agitation if coupled with direct turbulence model (Reynolds stress model), but steady RANS predictions for eccentric agitation are always poor. As far as eccentric agitation is concerned, modelling difficulties were also pointed out by Rivera et al. (2004). Laser Doppler Anemometry (LDA) was used by Wang et al. (2006) to determine mean flow and turbulent characteristics (turbulent kinetic energy and Reynolds stresses) induced by a semi-ellipse impeller placed in eccentric position in a rectangular vessel. In the present work two experimental techniques, LDA and flow visualisation, are combined in order to determine the main flow features in an unbaffled vessel agitated by a Rushton turbine in eccentric position and in turbulent regime. An analysis is also performed on the various instabilities which were found to characterise the flow. Indeed flow instabilities in stirred vessels have received much attention in the last decade and their implication on meso- and macromixing have been investigated (see among others Roussinova et al., 2003; Galletti et al., 2003; Ducci and Yianneskis, 2007; Paglianti et al., 2006). However, to our knowledge no indication of instabilities in unbaffled vessel with eccentric agitation is reported in the literature.

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T e

H

C

D

B

y

x A

2. Experimental apparatus Measurements were performed in a cylindrical vessel made of Perspex with inner diameter T = 290 mm. The vessel was filled with distilled water up to a height of H = T and was covered with a lid (equipped with plugs of different sizes) in order to avoid air entrainment. The agitation was provided by a standard Rushton turbine with D = T/3. The impeller blade thickness to diameter ratio was tb /D = 0. 01. The impeller was positioned in an off-axis location, at a distance of e = 60 mm, i.e., e/T = 0. 21, from the vessel axis. The impeller off-bottom clearance was C = T/3. The stirred vessel is depicted Fig. 1. The impeller was driven by a 0.3 kW power motor and the agitation speed could be varied by means of a speed controller. Measurements were taken with impeller rotational speeds of N=200, 250 and

Fig. 1. Scheme of: (a) agitation system; (b) reference coordinate system (the impeller rotates clockwise when seen from above).

300 rpm, corresponding to impeller Reynolds number Re = 36, 300, 45,300 and 55,500, respectively. 3. Measuring system Measurements were performed with a single-component LDA operating in back scatter mode. A pair of blue beams ( = 488 nm) was separated from a 3 W Argon-ion Spectra Physics laser through a Bragg cell which provided also frequency shifting. The scattered light is transmitted by means of optical fibres to photo-multipliers and then processed by a Dantec Burst Spectrum Analyser. The

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measurement volume was an ellipsoid with a diameter of 0.141 mm and a length of 3.7 mm. The working fluid was seeded with silver coated hollow glass spheres with a mean diameter of 10 m and a density of 1400 kg/m3 . The acquisition time was 1000 s. The number of samples ranged from 9000 to 220,000 with average sampling rates of 60 Hz. The vessel was encased by a square trough in order to minimise refraction effects of laser beams at the tank curvature. The vessel and the trough were positioned on a traversing system which allowed moving along the three spatial directions with an accuracy of 0.05 mm in the x- and y-directions and of 0.1 mm in the z-direction. A bespoke computer program was used in order to determine the exact location of the measurements volume based on the calculation of refraction of laser beams when crossing different mediums. Measurements regarded velocity components along the x- and ydirections and were performed at different x- and y-values and for different levels, both below and above the impeller mid-plane, i.e., from z/T = 0. 1 to 0.7. Measurements were taken from x/T = −0. 42 to 0.42 along the x-axis, whereas from y/T = −0. 34 to 0.42 along the y-axis. Some of the measurements were also performed near the impeller blades and in additional locations to capture specific flow features. Frequency analysis was applied to the velocity recordings. Frequency analysis was based on fast Fourier transforms (FFTs) which were performed through subroutines available in the Matlab software package. Since FFTs require samples distributed evenly in time, a resampling of velocity–time data was necessary. Data were resampled with the “nearest neighbour” technique using the same number of samples of the velocity recording, thus the resampling frequency was equal to the mean data rate. Flow visualisation experiments were also performed at N = 200 to 400 rpm by allowing some air to be entrained into the flow field through plugs in the lid, in order to trace the flow with bubbles. A commercial digital video camera capturing at 25 frames per second was used. 4. Results 4.1. Flow field 4.1.1. Flow visualisation results Fig. 2 shows a typical flow visualisation image taken from view A (Fig. 1b). Two main vortices occur, one above and one below the impeller plane. These vortices will be denoted as upper vortex (UV) and lower vortex (LV), respectively. The former vortex shows an axis deviating from the y-axis, whereas the second vortex axis is almost coincident with the y-axis if seen from above. This is confirmed by Fig. 3a which shows the stirred vessel from view B (Fig. 1b). The vortex above the impeller, appears stronger than that below the impeller; however, this has to be verified as the position of the upper vortex leads to a larger entrainment of bubbles used for flow visualisation. The position of the upper vortex axis inverts itself when inverting the direction of the impeller rotation (see Fig. 3b). Conversely, the location of the lower vortex below the impeller does not change substantially when inverting the direction of rotation of the impeller. Importantly, flow visualisation experiments showed that the positions of both vortices are not steady but move slightly. These movements were observable with the naked eye, thus proving that they occur with a timescale much larger than the impeller rotational timescale. This evidence will be discussed in “Flow instabilities” section. The presence of two main vortical structures is in agreement with the findings of Montante et al. (2006) who investigated an unbaffled

Fig. 2. Flow visualisation of the two vortices occurring with the eccentric impeller configuration. Impeller rotational speed N = 300 rpm. View from A of Fig. 1b.

vessel agitated by a D = T/3 Rushton turbine placed eccentrically at C =T/2 and e=T/4 with both experimental (PIV) and numerical (CFD) techniques. 4.1.2. Velocity measurements results It is not possible to identify radial and tangential velocity components as usually made for stirred vessels, because of the eccentric configuration of the impeller. Therefore a Cartesian system, referring to x- and y-components of velocity as illustrated in Fig. 1, will be used. Even though in standard vessel configurations (centric agitation with baffles) the Reynolds number values of the present work are considered fully turbulent conditions, the flow regime in the unbaffled vessel with eccentric agitation has to be verified preliminarily. Fig. 4a and b show profiles at z/T = 0. 4 along the x-axis of the ycomponent of the mean and rms velocity, respectively, for different impeller rotational speeds from N = 200 to 300 rpm. It can be noticed that measurements overlap when normalised with respect to the impeller blade tip velocity (Vtip = DN), indicating the fully turbulent characteristics of the flow. Fig. 5 shows the y-component of the mean velocity as a function of the x-coordinate for different axial levels and for y/T = 0. Velocity values have been normalised with respect to the impeller blade tip velocity. The impeller rotational speed was N = 300 rpm. It can be observed that above the impeller and for all axial levels, Vy tends to −0. 3Vtip for large negative x-coordinates, whereas tends to 0. 35Vtip for large positive x-coordinates. An inversion of velocity direction occurs for x/T = 0–0. 1, indicating an inversion of the flow field. This is explained by the strong circumferential motion which characterises the flow field. Besides, it can be noticed from Fig. 5 that the inversion position moves closer to the vessel axis with decreasing the axial coordinate. This is in agreement with flow visualisation results indicating that the upper vortex, located above the impeller is inclined with respect to the vertical plane. Since the x-coordinate of the vortex centre may be evaluated approximately as the position at which the y-component of the velocity in the x-axis is zero, the upper vortex inclination with respect

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Fig. 3. Flow visualisation of the two vortices occurring with the eccentric configuration of the impeller rotating: (a) clockwise (b) counter clockwise when seen from above. Impeller rotational speed N = 300 rpm. View from B of Fig. 1b.

Fig. 4. Component along y of the (a) mean velocity and (b) root-mean-square velocity as a function of x-coordinate for different impeller rotational speeds: y/T = 0. 4, z/T = 0. 4.

to the vertical plane can be simply determined as   yVy=0 UV = arctan z

Fig. 5. Component along y of the mean velocity as a function of x-coordinate for different axial levels z. Impeller rotational speed N = 300 rpm. y/T = 0.

(1)

The inclination of the upper vortex resulted to be roughly 15◦ just above the impeller, and 8◦ near the top of the vessel. This is in agreement with the visual observation of the flow (see Fig. 2). The inclination of the lower vortex was estimated to be roughly 20◦ . The upper vortex observed by Montante et al. (2006) in eccentric configuration was inclined by 30◦ with respect to the vertical plane, whereas the lower vortex of 10◦ . Discrepancies with the present investigation are due to the different configuration of the two vessels: in the aforementioned work, the larger C/T (i.e., C = T/2) and e/T (i.e., e = T/4) values lead to a larger inclination of the upper vortex. Below the impeller (z/T = 0. 2) the y-component of the velocity along the x-axis reaches the absolute value of 0. 35Vtip for both large positive and negative x-coordinates. As mentioned previously, flow visualisation experiments indicated that both the upper and the lower vortex are not steady but move slightly and periodically. This was evident also from the analysis of row LDA velocity data, i.e., time series. For instance Fig. 6a

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Fig. 6. (a) Time series and (b) histogram of y-component of velocity acquired at z/T = 0. 1, x/T = 0, y/T = −0. 09. N = 300 rpm.

shows the instantaneous velocity as a function of time acquired at z/T = 0. 1, y/T = −0. 09 and x/T = 0, which is below the impeller blade. For clarity of representation a time interval of 10 s has been extracted from the 1000 s LDA acquisition. It can be observed that the instantaneous velocity oscillates cyclically. This is evidenced by the solid line which was obtained from a moving-window average technique over 20 consecutive samples. In such a location the shape of velocity distribution is bimodal (see Fig. 6b). The mean velocity is very low, 0.02 m/s but of course it is not representative of what happens. Considering roughly that two phenomena are occurring, one characterised by positive y-velocity component and one by negative y-component, the bimodal histogram of Fig. 6b may be divided into two unimodal distributions, whose mean values are 0. 35Vtip and −0. 32Vtip , respectively. The bimodal distribution may be imputed to the presence of the movement of the vortices axes: the vortices axes move slightly, so that the measurement volume is periodically located at the RHS (seeing a negative y-velocity component) and at the LHS of them (seeing a positive y-velocity component). The absolute values of the mean velocity when positive velocities are discerned from negative ones, are in agreement with tangential velocity of about 0.3–0. 35Vtip , which characterises the strong circumferential flow. It is worth noting that the above observation on bimodal distribution was found for many of the velocity acquisitions. These velocity data have to be handled with care, therefore not only the first two moments (mean and rms) of the velocity data have to be checked, but also the skewness and the kurtosis. Measurements of the velocity component discharged from the impeller have been made in four positions equally spaced by 90◦ each, and will be referred as N, W, S and E (see Fig. 7). The measurements were taken at  = 1. 5 mm from the impeller blade tip at different axial levels. Fig. 8a compares the axial profiles of discharged velocity from the impeller blade tip in the N and S positions, corresponding to x/T = 0 and y/T = (−e + D/2 + )/T and y/T = (−e − D/2 − )/T, respectively. For clarity of representation the y-component of the velocity is taken positive when directed outwards from the impeller. In addition the z-coordinate has been reported to the impeller mid-plane, so that z = (z − C)/(Hb /2). In this manner z = 1 and −1 correspond to the upper and lower edges of the impelled blade tip, respectively.

y

N x

W

E

S

discharged velocity vortical motion resulting velocity

Fig. 7. Sketch for the superimposition of impeller discharged and circumferential flows.

It can be noticed that profiles vary with changing the location, indicating that the discharged flow is affected by the non-axial symmetry of the flow in eccentric impeller configurations. Both profiles show a velocity peak which is located below the impeller mid-plane. This is logical as the strong vortical motion induced by the upper vortex interferes with the discharged flow, which is deflected towards the bottom of the vessel. This is also proved by the positive velocity values which are found also for z < − 1. It can be also noticed that in the S position the discharged velocity profile differs from the typical “bell” profile for a Rushton turbine, being S-shaped, with negative velocity values for z > 0. 4. This observation may be imputed to the strong discharged flowwall interactions which lead to a flow inversion. Indeed in the S location the distance from the vessel walls is minimum, of just 0. 11T. In S location the y-component of the velocity was found to have a maximum of 0. 81Vtip , which is rather large for a Rushton impeller. When the same Rushton impeller (same tb /D and D/T) is mounted in centric configuration, the maximum velocity is 0. 83Vtip and 0. 45Vtip in baffled and unbaffled vessels, respectively (Galletti et al., 2004; Vinci, 2004).

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Fig. 8. Discharged velocity profiles taken at 1.5 mm from the impeller blade tip and for: (a) N and S positions; (b) W and E positions.

The larger value found for the eccentric configuration with respect to the centric configuration in unbaffled vessels may be easily explained by considering that the strong vortical motion induced by the upper vortex overlaps to the discharged flow from the impeller (see sketch of Fig. 7). In this manner, the discharged component of the velocity in the S location is augmented by nearly the radial velocity component of the vertical motion. Conversely, in N position the interaction with the radial component of the vortical structures is minimum, as the distance from the vortex axis is smaller (Fig. 7). In this case the radial velocity of the vortex dumps the discharged flow velocity, explaining the lower velocity peak, of about 0. 56Vtip . Fig. 8b shows axial profiles of the discharged velocity in the W and E positions, corresponding to y/T = −e/T and x/T = (D/2 + )/T and y/T = (−D/2 − )/T, respectively. In these cases the x-velocity component is measured and for clarity of representation the positive values are taken with velocity directed outwards from the impeller. It can be noticed that in the W position, velocities are much larger than in E position. In particular a peak velocity of 0. 65Vtip is observed in W position, whereas it is of just 0. 27Vtip in E positions. Again this observation is explainable with the interaction with the vortical motion (see sketch of Fig. 7); in E position, the nearly tangential component of the main vortical structure has to be subtracted to the impeller discharged velocity. The discharged flow is less affected by the vortical structure in the W location as this location is downstream the impeller–vortex interaction. The variation of discharged velocity profiles with varying the angular position with respect to the impeller is peculiar of eccentric configurations, as in normal stirred vessels configuration the ensemble-averaged axial profiles of radial velocity components out of the blade tip are the same across the 360◦ . Consequently a flow number cannot be defined in a conventional manner. The pumping capacity of the impeller, QP , can be estimated by integrating the discharged velocity profile over the blade height. Then the flow number is defined as NQ =

QP N · D3

(2)

When based on the S profiles, the pumping number would be 0.75; however, for the N profiles would be 0.72. Such a small difference is imputed to the fact the profiles show also a different shape, the one with the larger peak value being also the narrower. The pumping

numbers calculated from the W and E profiles were of 0.93 and 0.23, respectively. Such huge variation is well explainable with the interaction with the vortical motion (see sketch of Fig. 7), which augments velocities in the W location but reduces strongly those in E location leading to a very small pumping number. It is worth observing that unbaffled vessels with centric agitation are characterised by a much lower pumping number than baffled vessels. Brunazzi et al. (2003) determined with LDA a pumping number NQ = 0. 25 in an unbaffled vessel stirred by a D = T/3 Rushton turbine with a tb /D = 0. 035, whereas NQ = 0. 61 in baffled configuration. For the present tb /D Rushton turbine, a flow pumping number of NQ = 0. 41 is reported in unbaffled configurations (Vinci, 2004), thus much lower than in baffled vessel (NQ = 0. 79). 4.2. Flow instabilities Velocity recordings were analysed in order to obtain the main frequency components. This was motivated by the visual observation of periodic phenomena as that of Fig. 6a, whose time series shows well-visible periodic oscillations of the instantaneous velocity. Such oscillations occur with a timescale much lower than the impeller blade passage timescale. The presence of periodic oscillations of the flow has been largely observed in baffled stirred vessels with centred shaft, for different types of impellers and such oscillations are usually referred to macro-instabilities, MIs. These flow instabilities may be present in a vessel at all times and for most configurations (precessional MIs of Nikiforaki et al., 2003; Galletti et al., 2005; Ducci and Yianneskis, 2007), and constantly or intermittently for some vessel/impeller combinations (impeller stream MIs, see Roussinova et al., 2003, and instabilities due to clearance changes, CIs, Galletti et al., 2003). However, there is lack of data on flow instabilities in unbaffled configurations and especially for eccentric impellers. In the present work, three characteristic frequencies, which were much lower than the impeller blade passage frequency (i.e., fBP = 6N), were identified across the vessel. Runs at different impeller rotational speeds indicated that these frequencies are scalable with the rotational speed, i.e., f  =f/N =const. Such a linear dependence on N is in agreement with previous works on flow instabilities in stirred vessels with centric shaft configuration (see for instance Nikiforaki et al., 2003).

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As far as stirred vessels with eccentric impeller are concerned, some data on flow instabilities can be found in the comprehensive work of Paglianti et al. (2006), who used a pressure transducer to determine oscillations in the flow. The authors investigated a baffled vessel with two different position (centric and eccentric) of a Rushton turbine and for the specific case found negligible influence of the impeller position on the MI frequency. The authors suggested a novel correlation for the evaluation of MI frequency based on the flow number. However, as shown in section “Velocity measurements results” for eccentric unbaffled agitation the impeller flow number cannot be defined in a conventional manner. In the present work three characteristic frequencies were identified: • • •

f  = 0. 105 ± 0. 007, f  = 0. 155 ± 0. 01, f  = 0. 94 ± 0. 06.

These frequencies were observed in different locations across the vessel. The frequency spectrum illustrated in Fig. 9a was obtained from the analysis of velocity data taken above the impeller (x/T = 0, y/T = −0. 034, z/T = 0. 5). It can be observed the presence of a clear peak with f  = 0. 155. The frequency spectrum of Fig. 9b was obtained from the analysis of velocity data taken below the impeller (x/T = 0, y/T = −0. 12, z/T = 0. 15) and indicated two main frequency components, i.e., f  = 0. 105 and 0. 155. Specifically, the f  = 0. 105 and 0.155 frequencies may be related to the periodic movements of the lower and upper vortices' axis, respectively, which are visible from flow visualisation experiments. For instance, approximately 8 and 14 oscillations of the lower and upper vortices, respectively, could be identified within 17 s of flow visualisation recording taken at N = 300 rpm. These oscillations correspond to frequencies of 0.47 and 0.82 Hz, thus f  =0. 094 and 0.164, respectively. These values are in fair agreement with results from LDA signal analysis. The movements of the upper and lower vortices are shown in the images of Fig. 10, which were captured from a flow visualisation recording at N = 400 rpm. It is chosen to shown N = 400 rpm experiments as in this case less air was let to be entrained by the flow, resulting in more clear images than those of N = 300 rpm (Figs. 2 and 3). In the first image of Fig. 10a (at the time t = 0 s), the upper vortex departs from above the impeller and it is inclined so that it touches the free liquid surface for slightly positive y. In the second image (after 0.52 s), the vortex shows a S-shape, touching the free liquid level for nearly zero y. Finally in the third image (at t = 1. 04 s) the vortex looks like that in the first image. Such periodic movement of the vortex corresponds to a frequency f = 1/1. 04 Hz, leading to a non-dimensional frequency f  = 0. 144 ± 0. 006, which is in good agreement with the non-dimensional frequency of 0.155 observed from the frequency analysis of the LDA signals. The errors are due to the frame rate (25 fps) of the video camera. Similarly, the three frames of Fig. 10b were captured from a flow visualisation recording at N=400 rpm in order to illustrate the movement of the lower vortex. A time t = 0 s was assigned to the first of the three frames, as made also for Fig. 10a. In this case the vortex is S-shaped at t = 0 s, departing from the inner edge of the impeller blades towards the bottom of the vessel. After 0.72 s the shape of the vortex is different and it is originated from about the middle of the impeller blade. Finally in the third image, the vortex shows features similar to those of the first image. The resulting frequency is f = 1/1. 44 Hz leading to a f  = 0. 104 ± 0. 003, in agreement with frequency analysis of LDA signals.

Fig. 9. Frequency spectrum of velocity recordings taken in: (a) z/T = 0. 5, x/T = 0, y/T = −0. 034; (b) z/T = 0. 15, x/T = 0, y/T = −0. 12; (c) z/T = 0. 5, x/T = 0. 31, y/T = 0. N = 300 rpm.

The analysis of locations at which the frequency components were found indicates that the f  = 0. 155 frequency was observable across the entire vessel; it was more pronounced above the impeller, but also in some location below of the impeller. This indicates that the upper vortex affects the motion across most of the vessel. Conversely, the f  = 0. 105 frequency was found mostly below the impeller, indicating that the influence of the lower vortex on the motion is limited to the lower part of the tank.

C. Galletti, E. Brunazzi / Chemical Engineering Science 63 (2008) 4494 -- 4505

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Fig. 10. Frames taken from flow visualisation experiments with sketches at N = 400 rpm for: (a) upper vortex; (b) lower vortex. Eccentricity e/T = 0. 21.View from A of Fig. 1b.

As far as the third frequency is concerned, f  = 0. 94, such a frequency was observed at different axial levels and for positive xvalues, especially from x/T = 0. 2 to 0.31. The frequency spectrum of Fig. 9c was obtained at x/T = 0. 31, y/T = 0 and z/T = 0. 5. The f  = 0. 94 frequency was found alone or together with the f  = 0. 155 frequency (i.e., upper vortex frequency). The f  = 0. 94 frequency may be explained by considering that the locations at which it was observed are located downstream of the vortical structure—shaft interaction, which lead to vortex shedding phenomena. The frequency of the shedding vortex can be expressed with the Strohual number which is defined as St = fSV

Dshaft Dshaft L = fSV = fSV VSV 0. 32Vtip 0. 32ND

(3)

L is the dimension of the object behind which the shedding vortex is originated, thus it is the diameter of the shaft, whereas VSV is the velocity of the flow impacting the object. Such velocity can be roughly estimated to be 0. 32Vtip , according to Fig. 5. Consequently the characteristic frequency is  = fSV = St 0. 32D fSV N Dshaft

with St = 0. 2, which is in agreement with the present observation of f  = 0. 94. In order to assess the effect of the flow instabilities identified above on the flow field and turbulence characteristics, an analysis of their energetic content was performed. Indeed it is well known that flow instabilities can broaden the real turbulence levels, as the blade passage does in the vicinity of the impeller region. If such broadening is not taken into account, there could be an erroneous estimation of micro-mixing length and temporal scales. Besides flow instabilities superimpose an additional fluid motion to the mean flow pattern, hence they may improve significantly both meso- and macro-mixing. The estimation of the energetic contribution of each kind of flow instabilities in different locations of the vessel is required to evaluate the amount of broadening of real turbulence levels, as well as to indicate whether flow instabilities can be exploited for mixing optimisation. The frequency spectra illustrated above are one-dimensional. The integral of the spectrum over the entire range of frequencies constitutes the variance of the signal:  ∞ v2  = amplitude(f ) df (5) 0

(4)

A Strohual number St = 0. 2 has been reported in literature (Roshko, 1993) for the flow past a circular cylinder and for the Reynolds  = 0. 97 number Reshaft = 11, 000. The above expression leads to fSV

ETOT = 12 v2 

(6)

where v is the fluctuating velocity v = v − V

(7)

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Fig. 11. Energetic content of the 3 characteristic frequencies for (a) y/T = 0 and different axial levels; (b) x/T = 0 and different axial levels.

and ETOT is the total kinetic energy in the coordinate direction of measurement. The kinetic energetic contribution of each instability to the total energetic content can be evaluated as half of the area underneath the corresponding frequency peak. In the procedure, preliminary frequency spectra were smoothed to a frequency resolution of f = 0. 01. This operation was made because the FFT analysis uses a periodogram, so that the calculated spectra may show fake frequency peaks at nearby frequencies. Spectral smoothing is a simple moving-average calculation that improves the stability of the spectral density function reducing the variance of the spectral estimates and making dominant trend easier to be identified. An alternative method of estimating the energetic contribution of each flow instabilities would be to filter the relevant peaks in the frequency domain and then reconstruct the signal in the time domain through the inverse of the Fourier transform. The energy associated with the flow instabilities may be determined from the comparison between the original signal and the reconstructed one, from which flow instability contributions have been removed. However, as mentioned by Galletti et al. (2005), the two methods give similar results in the characterisation of flow instabilities, with differences of a few percents, so that the former method has been followed in the present work. Subsequently, the energetic contributions of instabilities associated with the lower vortex (ELV ), upper vortex (EUV ), and shedding

vortices (ESV ), from shaft, were evaluated as areas underneath the curves of the f  = 0. 105, 0.155 and 0.94 peaks, respectively. In a few locations, near the impeller, the contribution of the blade passage, i.e., f  = 6, was also considered. Since not only the amplitude but also the extent of the different peaks varied between the frequency spectra, the extent of the peaks for the calculation of the integrals was selected individually. As the energetic contribution of the above peaks is removed from the signal, the energy contained in the remainder of the signal represents the turbulent energy (ETUR ). ETUR = ETOT − ELV − EUV − ESV − (EBP )

(8)

Fig. 11a and b show the energetic contribution to the turbulent kinetic energy content of the three main characteristic frequencies in different locations in the vertical planes at y/T = 0 and x/T = 0, respectively. Fig. 11a indicates that the f  = 0. 155 frequency is present across a large part of the vessel, mostly for slightly positive or near zero x/T. This is in agreement with the spatial location of the upper vortex (see Fig. 3a). Values of the energetic content related to the upper vortex were up to EUV /ETUR = 52% just above the impeller; this means that fluctuations due to the upper vortex may broaden real turbulence levels of up to about 23%. As far as the lower vortex is concerned, its influence is negligible almost everywhere in the y/T = 0 plane. This may be explained by considering that the vortex develops itself for negative y/T (see Fig. 2). Conversely, the characteristic frequency

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associated with vortex shedding from the shaft is well visible for positive x/T (from x/T = 0. 2 to 0.42) and for all levels above the impeller. Very large values of energetic content are associated with the shedding vortex with ESV /ETUR up to 81% (i.e., broadening of turbulence levels of 35%). Similar values were also confirmed by additional measurements in this region but for negative y/T. In the x/T = 0 vertical plane (Fig. 11b) the frequency associated with the upper vortex are present across a large part of the vessel, mainly near y/T = 0. Below the impeller and for slightly negative y/T, the frequency related to the lower vortex, i.e., f  = 0. 105, is also well visible with energetic content up to ELV /ETUR = 35%. Such locations are also affected by the upper vortex so that both vortices contribute to the broadening of real turbulence levels. For instance EUV /ETUR = 50% and ELV /ETUR = 35% were evaluated at x/T = 0, y/T = −0. 12 and z/T = 0. 1. This means that the energetic content of the two vortices is (EUV + ELV )/ETUR = 85%, thus real turbulence levels are broadened of about 36%. The bimodal distribution of Fig. 6b (taken at z/T = 0. 1, y/T = −0. 09 and x/T = 0) was obtained in these locations where two main frequency peaks were observed. A visual observation of the time series (Fig. 6a) indicates about eight periodic oscillations within 10 s of acquisition, resulting in a frequency f = 0. 8 Hz. This corresponds to f/N =0. 16 which is in fair agreement with the frequency related to the upper vortex. This is explainable by considering that the upper vortex peak is the dominant one, because its energetic contribution is larger than that of the lower vortex. Actually in this location the superimposition of the two main frequency components identified by the LDA signal analysis occur. Vortex shedding frequency is not visible. This makes sense by considering that in the clockwise impeller rotation, the flow shed by the shaft firstly encounters y/T = 0 and positive x locations and subsequently x/T = 0 and positive y locations, in which shedding vortices have lost most of their energy. Actually for positive x/T and y/T vortex shedding frequencies were observed with an energetic content lower than 80% of ETUR . For instance at x/T = 0. 21 and y/T = 0. 1 in the axial level z/T = 0. 5, the energetic content of the shedding vortex was 48% ETUR ; in such a location the upper vortex was also well visible with an energetic content of approximately 10% ETUR . 5. Discussion and conclusions An experimental investigation of an unbaffled stirred vessel with eccentric impeller configuration is presented. LDA is combined to flow visualisation in order to determine the main flow features. Two main vortical structures, one above and one below the impeller have been observed. The former vortex departs the impeller towards the top of the vessel and it is inclined at 15◦ with respect to the vertical plane near the impeller and 8◦ far away. Such vortex leads to a strong circumferential flow around it, which dominates all vessel motion. The lower vortex originates from the impeller blades and it is directed towards the bottom of the vessel, near the axis. Both vortices are not steady but move slightly, inducing periodic oscillations in the flow field. Indeed in many locations LDA data showed bimodal distributions which have to be handled with care. The periodic oscillations are like flow instabilities which have been extensively reported in baffled stirred vessels with centric impeller configurations. The beneficial effects of flow instabilities on the meso- and macro-mixing has been acknowledged in the literature. Recently Ducci and Yianneskis (2006) showed that a reduction of mixing time by around 30% could be achieved when inserting the reactants near the MI vortex axis in a baffled vessel stirred by a RT. However flow instabilities lead to a broadening of turbulence levels, which has to be evaluated to determine accurately micro-mixing information (Nikiforaki et al., 2003).

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Two characteristic frequency components, related with the periodic oscillations of the two vortices, were observed in different positions across the vessel. The frequencies were scalable with the impeller rotational speed, i.e., f/N = const. A f/N = 0. 105 frequency component, related to the slow movement of the lower vortex, was observed in a few locations below the impeller, with an energetic contribution to the turbulent kinetic energy up to 35%. A f/N = 0. 155 frequency component associated to the movement of the upper vortex was found in many locations across the vessel, both below and above of the impeller, indicating that the upper vortex is stronger and affects the flow in great part of the vessel. Its energetic contribution was as high as 52% of the turbulent kinetic energy. It was shown that the contribution of both vortices to oscillations of the flow may broad the real turbulence levels of up to 36%, so that this has to be considered when determining micro-mixing information. The origin of the oscillations of the two vortices needs further investigation. Recently Ducci and Yianneskis (2007) showed that precessional macro-instabilities (with f/N = 0. 02) in baffled vessel with centric agitation are driven by two whirlpool-type vortices, extending from the impeller to the top/bottom of the vessel, and moving precessionally about the vessel axis. In eccentric configuration, it can be argued that the lack of axial symmetry leads to an inclination of the vortices which becomes also the two main axes (one above and one below the impeller) of the circumferential flow. The processional motion of the vortices or something like may be still present in some extent inducing oscillations superimposed to the mean flow field. However the frequencies determined in the present work (f  = 0. 105 and 0.155) are much larger than the precessional MIs frequencies (f  = 0. 02), with values more similar to frequencies typical of impeller stream MIs (Roussinova et al., 2003; Paglianti et al., 2006). It is also congruent to think that the eccentric position of the shaft improves the importance of the impeller stream–wall interactions. The strong circumferential flow and the eccentric position of the shaft generate vortex shedding phenomena from the flow-shaft interaction. This is a new evidence in stirred vessel as it does not occur in conventional shaft configuration. A frequency component of f  = 0. 94 was measured in some locations downstream of the flowshaft interaction and explained through the Strohual number typical of vortex shedding from circular cylinders. Finally, the eccentric impeller configuration prevents from defining properly a pumping number, as the impeller discharged velocity profiles change with varying the angular position. The work has shown that a combined use of different experimental investigation techniques, LDA and flow visualisation, allows capturing the main flow features. LDA is a local technique but provides a great number of samples per unit time, so that frequency analysis can be carried out to determine main frequency components. Nevertheless, the use of high speed PIV would allow to gain a global picture of the flow fields with also information on frequency component of the flow.

Notation C

D Dshaft e E

impeller off-bottom clearance, defined as the distance between the vessel base and the middle of the impeller blades, m impeller diameter, m shaft diameter, m eccentricity, i.e., distance of the shaft from the vessel axis, m energy, m2 s−2

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EBP ELV ESV EUV ETOT ETUR f f fBP H Hb L N NQ Qp Re Reshaft t tb T v v V Vtip x y z z

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energy associated with cyclic variation due to the blade passage, m2 s−2 energy associated with the lower vortex, m2 s−2 energy associated with the vortex shedding, m2 s−2 energy associated with the upper vortex, m2 s−2 total kinetic energy along the direction of measurement, m2 s−2 turbulent energy, m2 s−2 frequency, Hz non-dimensional flow instability frequency, f  = f/N blade passage frequency, Hz liquid level, m impeller blade height, m dimension of the objects originating the shedding vortex, m impeller rotational speed, s−1 impeller flow number, dimensionless impeller pumping capacity, m3 s−1 impeller Reynolds number, Re =  · N · D2 · −1 shaft Reynolds number, Reshaft =  · VSV · Dshaft · −1 time, s impeller blade thickness, m tank inner diameter, m instantaneous velocity, m s−1 fluctuating velocity, m s−1 mean velocity, m s−1 impeller blade tip velocity Vtip = ND, m s−1 spatial coordinate, m spatial coordinate, m axial coordinate measured from the vessel bottom, m axial coordinate measured from the impeller centre and adimensionalised with respect to the blade height, dimensionless

Greek letters

UV  f    

inclination of the upper vortex, degrees distance, m frequency resolution, Hz laser beam wavelength, m dynamic viscosity, kg m−1 s−1 3.1415 density, kg m−3

Subscripts BP LV SV UV x y

blade passage lower vortex shedding vortex upper vortex component along the x-axis component along the y-axis

Symbols 

temporal mean

Abbreviations BP CFD CI FFT LDA MI PIV rms RT

blade passage computational fluid dynamics instabilities from impeller clearance variations fast Fourier transform Laser Doppler Anemometer precessional macro-instability particle image velocimetry root-mean-square Rushton turbine

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