Using positron emission particle tracking (PEPT) to study the turbulent flow in a baffled vessel agitated by a Rushton turbine: Improving data treatment and validation

chemical engineering research and design 8 9 ( 2 0 1 1 ) 1947–1960

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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Using positron emission particle tracking (PEPT) to study the turbulent flow in a baffled vessel agitated by a Rushton turbine: Improving data treatment and validation Fabio Chiti a , Serafim Bakalis a , Waldemar Bujalski a , Mostafa Barigou a , Archie Eaglesham b , Alvin W. Nienow a,∗ a

School of Chemical Engineering, College of Physical Sciences and Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK b Huntsman Polyurethanes, Everslaan 45, B-3078 Everberg, Brussels, Belgium

a b s t r a c t Positron emission particle tracking (PEPT) is a relatively new technique allowing the quantitative study of flow phenomena in three dimensions in opaque systems that cannot be studied by optical methods such as particle image velocimetry (PIV) or laser Doppler anemometry (LDA). Here, velocity measurements made using PEPT in two sizes of baffled vessel (∼0.20 m and ∼0.29 m diameter) and two different viscosity fluids agitated by a Rushton turbine are compared for the first time directly in depth with some studies reported in the literature made by LDA for the turbulent regime in the equivalent geometry. Initially, the paper considers how the Lagrangian data obtained by PEPT can be converted into Eulerian in order to make the comparison most effective. It also considers ways of data treatment that improve the accuracy of both the raw PEPT data and the velocities determined from it. It is shown that excellent agreement is found between the PEPT and literature results, especially for the smaller vessel, except for the radial velocity just off the tip of the blade in the plane of the disc of the Rushton turbine. This difference is attributed to the very rapid changes in both magnitude and direction that occurs in that region and also to the different way of ensemble averaging in the two techniques. In addition, the results for the absolute velocities normalised by the impeller tip velocity for all the rectangular cross-section toroidal cells in each size of vessel and each fluid and a range of agitator speeds are compared in the form of frequency histograms. In this analysis, the velocities for each run are obtained from PEPT based on tracking a particle for 30 min and the mean and mode of the velocities each decrease slightly with decreasing scale and Reynolds number. The possible reasons for this variation in the mode and the mean are discussed. Overall, it is concluded that for the radial flow Rushton turbine the PEPT technique can be used to obtain accurate velocity data throughout the entire complex three-dimensional turbulent flow field in an agitated, baffled vessel except very close to the impeller in the radial discharge stream. © 2011 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: PEPT; Baffled stirred tank; Rushton turbine; Turbulent flow; Mean flow field; Scale

1.

Introduction

Laser Doppler anemometry (LDA) and particle image velocimetry (PIV) are currently the techniques of choice for studying the velocities in stirred vessels to give a wide range of information, particularly in turbulent flow. These techniques use laser light to track seeding particles, at a point or within a light sheet, to follow their very rapid fluctuations

∗

to give detailed Eulerian information on the statistics of the fluid motion. However, in cases where the fluid is not completely transparent, these laser-based techniques are ineffective. For opaque systems, Lagrangian techniques have been developed based on recording the passage of a flow follower. The frequency response of Lagrangian techniques is inferior to that of Eulerian and in general until recently they have only been used to study flow through the impeller region,

Corresponding author. E-mail address: [email protected] (A.W. Nienow). Received 14 July 2010; Received in revised form 19 January 2011; Accepted 21 January 2011 0263-8762/$ – see front matter © 2011 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2011.01.015

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chemical engineering research and design 8 9 ( 2 0 1 1 ) 1947–1960

Nomenclature C D dli dp dT fopt H l* N Po Re T St t t1/2 u v vtip r, th, z x, y, z

impeller clearance (m) diameter of Rushton turbine blades (m) distance between two consecutive recorded tracer locations (m) particle diameter (m) time resolution (s) fraction of events used in tracer location height of fluid in the tank (m) dimension of a cell (m) impeller rotation speed (rpm) power number Reynolds number diameter of the tank (m) Stokes number ( p / f ) time (s) tracer half-time life (s) fluid velocity (m/s) fluid speed (m/s) impeller tip speed (m/s) cylindrical coordinates (radial, azimuthal and axial respectively) Cartesian coordinates

Greek letters  fluid viscosity (Pa s) s particle density (kg/m3 ) particle response time (s) p f fluid response time (s)

with radio or magnetic followers and aerial detection being used. As a result, only very limited statistical information (visits per unit of time) and circulation times around the aerial were obtained and a full description of the flow in the whole system has still not been achieved in this way. In the last few years, more promising Lagrangian techniques to study opaque systems have been developed based on the detection of a tracer that emits gamma rays. Computer-automated radioactive particle tracking (CARPT) (Khopkar et al., 2005) and positron emission particle tracking (PEPT) (Hawkesworth et al., 1986; Stellema et al., 1998) are the main two techniques. In the work reported here, positron emission particle tracking (PEPT) has been used. PEPT enables the quantitative study of flow phenomena in three dimensions in opaque systems from multi-phase, low viscosity to high viscosity non-Newtonian pastes (Barigou et al., 2009). It is based on the detection of a tracer that can be followed everywhere throughout the vessels as it emits backto-back gamma rays and thus gives Lagrangian data for the whole space of interest. The data obtained can also be converted into Eulerian information. The technique has already been used extensively for studying particulate systems such as solids mixing in vee- (Kuo et al., 2005) and rotating drum mixers (Parker et al., 1997; Wong, 2006) and fluidised beds (Parker et al., 1997; Wong, 2006). The work published on particulate systems reports that PEPT can track particles at velocities of 2 m/s with an accuracy of 0.5 mm and a frequency of 50 Hz (Barigou, 2004). However, such capabilities are related to dry systems. When the tracer is immersed in a liquid, the gamma rays emitted are absorbed by the media surrounding the particle, which potentially lowers its effectiveness. An earlier study

in liquid systems validated the use of PEPT for laminar flows in a pipe, where an error of less than 10% for velocities of 0.5 m/s was reported (Bakalis et al., 2004). The use of PEPT for studying stirred vessels was initiated in 1999 (Fangary et al., 1999) and there were other early applications (Fangary et al., 2002; Fishwick et al., 2003; Fishwick et al., 2005). However, in these early papers, its use was not validated against other well-established techniques. More recently Pianko-Oprych et al. (2009) have validated it using a pitched blade turbine under turbulent flow conditions but the comparison with the literature proved difficult, partly because of the wide range of configurations reported there. In addition, it was felt to be valuable to assess the use of the PEPT technique by comparing results with well-validated data based on PIV or LDA from the most extensively studied geometry in the literature, the Rushton turbine. In the early studies on stirred vessels, the generic developments of Parker et al. (2002) have been assumed to be directly applicable. Hence, this work also aimed to optimise the way Lagrangian data are treated to produce more useful information than can be obtained from cruder Lagrangian techniques involving magnetic or radioactive flow followers as reported in detail elsewhere (Chiti, 2008).

2.

Materials and methods

2.1.

Equipment

All the experiments were carried out in geometrically similar systems at two different scales. The geometry is shown in Fig. 1 and consisted of a flat-based fully baffled cylindrical vessel made of Perspex, filled with fluid to a height, H, equal to the diameter, T. The four metallic baffles were T/10 in width and about T/100 in thickness, vertical and close to the wall. The stirrer used was a six blade Rushton turbine with diameter, D, equal to T/3, positioned at a clearance C = T/4 from the bottom of the tank. Each blade of the turbine had a length and a height equal to D/4 and D/5 respectively. The impeller was driven by a shaft of diameter equal to 0.09D using a 3 kW motor with variable speed control. Of the two sizes used, the larger had a diameter, T, of 287 mm (T29 ) and the smaller, 204 mm (T20 ). The size of the tanks was limited by the size of the PEPT camera, i.e. they should be small enough to fit within the camera but big enough to ensure a satisfactory resolution of the experimental measurements, which depends on the size of the tracers and relative to that of the vessel. For T29 , only one rotational speed, 300 rpm, was used. However, for the smaller scale, speeds from 100 to 600 rpm were used. The Reynolds number for each experiment is shown in Table 1. For T29 , an aqueous salt solution was used whilst for T20 , both a salt and a sucrose solution were employed. The salt solutions matched the density of the tracer used for the PEPT experiments, which varied slightly. Neutral buoyancy was obtained in each case by slightly varying the salt concentration. For the experiments in T20 , the use of two different solutions allowed the effect of viscosity to be investigated by also using a sucrose solution of 50 g/100 ml of water whilst still allowing the densities to be well matched. For the salt solution, the viscosity was assumed equal to that of water whilst for the sucrose solution, it was measured using a Rheomat 30 using the double-gap concentric cylinder head. This gave a viscosity 3.19 mPa s at 20 ◦ C in good agreement with value of 3.25 mPas in the literature (Norrish, 1967).

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Fig. 1 – Diagram of the experimental rig.

2.2.

Positron emission particle tracking (PEPT)

PEPT was used to obtain particle paths. Tracers used in this study were 250 and 600 ␮m diameter ion-exchange resin particles, doped with 18 F ions coated with paint to prevent leaching, of density ∼1150 kg/m3 . The salt solution was finely adjusted to minimise particle rise or fall. The ability of a particle to follow the motion of a fluid with a rapidly changing velocity is characterised by the use of Stokes number, St =  p / f , where  p is the particle response time and  f is the fluid response time (to an external disturbance, in this case the impeller rotation). For Stokes numbers close to zero, the particle and fluid velocities behave identically (equilibrium flow); typically, equilibrium flow conditions can be assumed (Schetz, 1996) for St < 0.1. For these experiments, taking the worst case of 600 rpm, and assuming that the flow is Stokesian,  p (=s dp 2 /18) has a value of approximately 0.02 s. If  f = D/ND, i.e. the impeller diameter divided by the impeller tip speed, it gives a value of approximately 0.03 s, giving a Stokes number of the order of 0.6. This value of the Stokes number will occur at the tip of the impeller where velocities are highest and  f is at its lowest. Thus, for the greater proportion of the vessel, St was much lower. These values of St, therefore, suggest that the neutrally buoyant tracer particle would follow the fluid motion. One would thus expect the tracer to follow the mean flow especially as it is neutrally buoyant. Slightly larger tracers (2.36 mm) were assumed to closely follow the liquid when Table 1 – List of experiments done in T29 and in T20 . Experiments done T29 , salt solution RT300 T20 , salt solution RT600 RT480 RT200 RT100 T20 , sugar solution RT600 RT480 RT300 RT200

N [rpm]

Re

vtip [m/s]

300

1.61 × 105

1.58

600 480 200 100

1.38 × 105 1.10 × 105 4.60 × 104 2.30 × 104

1.98 1.58 0.66 0.33

600 480 300 200

4.25 × 104 3.40 × 104 2.12 × 104 1.42 × 104

1.98 1.58 0.99 0.66

the densities of the fluid and particle were closely matched in earlier Lagrangian studies in stirred vessels (Rammohan et al., 2001). Pre-processed PEPT data consists of two dimensional locations of gamma rays detected by two cameras with the very fast resolution time of 12 ns. A previously developed location algorithm (Parker et al., 1993) was used to transform raw data to particle paths. In it, the data recorded by the PEPT cameras are divided into subsets (slices) of sequential events and for each slice, the co-ordinates of the tracer are estimated using an optimisation routine. An important element in it assesses events which lie furthest from a point and discards them as corrupted, and a new point is recalculated using only the events left. This reconstruction process is repeated until only a fraction, fopt , of the all the events listed in the subset is kept. The following subset of data is then processed. The final data has the form of a list of temporal and spatial positions, t, x, y & z as well as an error in the position location (in mm) defined as the diameter of the confidence cloud where the particle has been located. The values of # of events/slice and fopt are selected for each experiment and they depend on the particular geometry, velocities and also on activity of the tracer which decays with time. Since the events are continuously recorded whilst the tracer is moving, the choice of the above parameters plays an important role in the quality of the particle paths obtained. It has also been shown that for a particle moving on a circular path at constant velocity, increasing the size of the event subset, up to a certain limit, results in more accurate reconstruction of the position of the particle (Parker et al., 1993). However, on extending these findings to larger sets, the accuracy decreases again. This finding suggests that, for a given speed, an optimum value of the number of events per slice exists at which the best possible location is achieved. Furthermore, they also showed that with increasing velocity, the optimum value of the number of events per slice steadily decreases. Similarly, the fopt parameter had an optimum value for a given set of events. In fact, a small fopt should cut off all the corrupted events, leaving only the good ones. However, if the # of events/slice is small, e.g., 80, then a small fopt , e.g., 0.05, would lead to the use of very few (in this example, only 4 events) for the triangulation process.

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Fig. 2 – Coefficient of variation of the location error for different values of fopt and # of events/slice. Defining a general rule for the value that should be ascribed to these parameters is challenging, as they depend on the radioactivity of the tracer, its absolute velocity and fluctuation characteristics, including the rate of change of direction of the mean velocity of the tracer. The fully three dimensional turbulent flow with its very wide velocity range found in stirred tanks is therefore among the most challenging cases to analyse. The analysis is made more difficult as the mass of water surrounding the tracer absorbs a portion of the ␥rays, decreasing the signal to noise ratio. Furthermore the absolute velocity of the tracer and its angular and linear acceleration are high in the vicinity of the impeller. Experience can help in the first choice of these parameters, but only a trial and error approach leads to the finding of the optimum. In order to quantify the effect of the processing parameters on the raw data, the same experiment was processed with different values of # of events/slice from 100 to 300 and fopt from 10% to 30%. The relative standard deviation of the location error has been calculated for every combination of # of events/slice and fopt , during 20 min of data recording.

3.

Results

3.1. Choice of initial raw data processing parameters on accuracy of particle location A series of experiments were performed to evaluate the applicability of using PEPT to study turbulent flows. The first objective was to identify the effect that the PEPT data processing parameters had on the accuracy of the obtained particle location and subsequent paths. In Fig. 2, one can see the effect of the parameters # of events/slice and fopt on the coefficient of variation of the location error. Clearly, there is an optimum combination of processing parameters that minimise the location error and this is highlighted by the darker region on the plot (values between 0.34 and 0.36). Moving away from these sets of processing parameters, the error location increases and data location becomes less accurate. The minimum relative standard deviation is, in fact, obtained for values of # of

Fig. 3 – Total number of location recorded and integrated over the height of the liquid collapsed onto an x–y plane.

events/slice and fopt equal to 120 and 17 respectively, which also represents the middle of the darker region mentioned above. The optimum values of # of events/slice and fopt obtained in this way are typical of those used for processing all the data in this work.

3.2. The impact of tracer position on probability of detection Once the optimum # of events/slice and fopt parameters have been selected for each experiment, further pre-processing as set out below is used to obtain flow field and other fluid dynamic parameters of interest. The contour plot in Fig. 3 is an example of typical experimental locations recorded by PEPT during a 2 h experiment, collapsed onto a single horizontal (x–y) plane. For a neutral density particle tracking the fluid in a well mixed closed system, such as the one used here, one would expect that the probability of the tracer being in any one position within the vessel would be uniform, i.e. the number of locations should be the same everywhere. However, the figure demonstrates that closer to the camera, a smaller number of locations are detected when compared to the ones that are further away. Consider a tracer at the two different positions in the vessel marked in Fig. 3 as a and b, both on the same plane normal to the cameras and that it is emitting ␥-rays equally in every directions. The number of ␥-ray pairs detected by the cameras is proportional to the solid angle defined by the relative position of the tracer to the cameras and the dimensions of the cameras themselves. From Fig. 3, it can be seen that the detection angle (˛) for the tracer at position a close to detector 1 is larger than the angle (ˇ) when it is at position b. It is this geometric effect that results in areas close to detector 1 with lower number of ␥ rays being detected and hence fewer locations. The pre-processing allowed for the geometric effect by azimuthally averaging the data. Finally, more locations were obtained in the right part than the left part. This difference probably occurred because the tank was not precisely centralised even though it was positioned by means of static radioactive tracers fixed to its extremities.

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coordinates: x˙ = −ωR sin(ωt + rndt ) + rndx y˙ = ωR cos(ωt + rndt ) + rndy

Fig. 4 – Cartesian and cylindrical coordinates vs. time for artificial data.

3.3. Estimating Lagrangian velocities using PEPT location data Velocities were estimated using particle paths according to an earlier method which proposed that the initial location data be transformed so that velocities can be estimated as the slope of a straight line fitted to the data (Bakalis et al., 2004). In stirred tanks, the fluid flows in a predominantly cylindrical motion around the vessel axis. Many researchers have therefore reported data in cylindrical coordinates assuming axial symmetry or by azimuthally averaging variables. Notionally, the transformation of Cartesian velocities to a cylindrical reference frame might be expected to give the same result as using cylindrical coordinates to estimate velocities in a cylindrical reference frame. However, numerically, due to the swirling nature of the flow in stirred vessels, the latter transformation is significantly better. This may be illustrated by applying both approaches to the set of artificial data shown in Fig. 4, representing flow in a spiral. In Fig. 5, the results of this comparison are shown; Fig. 5a is for the two cylindrical velocity components obtained by converting the Cartesian locations in Fig. 4a, whilst on the right hand side are reported the same velocities calculated by ‘linear fitting’ of the cylindrical coordinates in Fig. 5b. In both cases, the velocities are compared with the theoretical cylindrical velocities, which are found by converting the Cartesian velocities, x˙ and y˙ in Eq. (1) into velocities, uth and ur in cylindrical

(1)

In order to ensure that the artificial date are representative of those obtained from experiments, a random noise (up to 10% of the value) was added (rndx , rndy ) as indicated in Eq. (1). In Fig. 5a, the method used underestimates the tangential velocity, uth , for higher speeds and is not accurate for the radial component, ur , whilst in Fig. 5b, the method gives very close agreement to the theoretical velocities based on Eq. (1) for both components. Though this comparison is based on artificial data, the tendency of the tracer to move circumferentially around the vessel axis with curved trajectories makes the latter approach more effective for stirred vessels and this method was therefore used to treat all the experimental PEPT data obtained in this work.

3.4. Decaying activity effect on tracking time resolution Once the preferred algorithms to estimate local Lagrangian velocities were established, they were applied to data from experiments. The first experiment consisted of a long run in the large vessel with a constant speed of 300 rpm. Over time, the tracer activity decays according to the following equations: A = A0 exp(−t)

=

ln(2) t1/2

(2)

where A is the activity which is a function of time, t, the initial activity, A0 , and the tracer half-time life, t1/2 (=110 min for 18 F). Thus, at the beginning of an experiment, the high activity leads to a large number of events per second being recorded by the cameras, and under these conditions, the parameters used to process the data were 200 events/slice with fopt = 0.15. Using these parameters, Fig. 6 shows the mean and the standard deviation of the time resolution, dT, by averaging 500 consecutive time steps between pairs of locations as a function of time up to 1 h as the activity decayed. The average dT at the beginning is about 40 ms whilst it is 58 ms towards the end, which corresponds to an average of 5.0k events/s and 3.45k events/s (69% of the starting value) recorded by the cameras respectively. This fall in events recorded corresponds with the loss of activity of the tracer (according to Eq. (2) after 1 h, the tracer has 68.5% of its original activity remaining). On the other hand,

Fig. 5 – Cylindrical velocities (real) from Fig. 4 vs. equivalent (theoretical) velocities from Eq. (1): (a) real ur and uth calculated from Fig. 4a and (b) real ur and uth calculated from Fig. 4b.

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Fig. 6 – Time resolution vs. experiment time. the relative standard deviation, defined as the ratio between the data standard deviation and its mean, was found to be constant during the experiment leading to the conclusion that the scatter of dT did not change with time though its mean value did. As a result of this experiment, it was decided to break the data into different batches, adapting the parameters # of events/slice and fopt to the frequency of the raw data collection.

3.5. Azimuthally averaged variables and velocities contour plots The first set of data analysed with all the routines discussed was undertaken in T29 at 300 rpm with the 600 ␮m tracer for 5 h until the number of events detected by the camera was considered the minimum acceptable (approximately 5k events/s). Once all the positions and the velocities were calculated in cylindrical coordinates, they were averaged on an imposed grid. As stated above, due to the axial symmetry of stirred vessels, azimuthal averages are most commonly used in the literature. Here, it was first assumed that over a sufficient period of time, the tracer would spend an equal time in each part of the entire tank. Then, a vertical 2D grid with variable distances in the radial direction and constant in the vertical was established to give equal volume toroidal cells throughout the vessel (Fig. 7a). Once the grid is built, there are different approaches that can be used in order to define a mean value from all the locations falling within a toroidal cell: (a) average on the number of location recorded in a cell, (b) average on the passes from the cell, and (c) average based on weighting the time spent in the cell.

Fig. 7b shows an example of a cell containing eight locations associated with three passes. From the example, since the locations belonging to pass 1 are sparser than the ones associated with pass 2 and pass 3, it is reasonable to assume that the velocity of pass 1 is higher than that of pass 2, which in turn is higher than pass 3. It is clear from this example that it would be wrong to obtain the average velocity for the cell based on the total number of location in the cell, because pass 3 would swamp the effect of the other two. On the other hand, the averaging of passes is computationally very demanding since it involves, firstly, identifying the different passes, secondly, finding an average velocity per pass and, finally, averaging the pass velocities. To overcome the complexity of this approach, a simpler method was used. This method involved a time weighted average, where the single velocity associated with each location, being multiplied by the half interval time between the previous and the subsequent location is added to all the other time–velocity products in the same cell. Finally, this total time–velocity summation is divided by the total time spent in the cell. This method can be summarised as follows:

 v tj t − ti−1 j j v¯ =  tj = i+1 j

tj

2

(3)

where j is the jth location and i indicates the index of that particular cell. This method also takes into account the effect of the radioactive decay of the tracer as, for two different passes having the same velocities but different tracer activities, the time spent in a cell will be the same, but in one case there are more locations than the other. By using Eq. (3) these two passes result in the same velocity in each case, as it should be. Since the total time spent by the tracer in each cell was recorded, it was also possible to calculate the value of occupancy, defined as time spent in a cell divided by the total time of the experiment, as well as record the total number of times the tracer visited each cell. Finally, the average velocities were made dimensionless by dividing by the velocity of the impeller tip (as is usually done for treating such data) and the height and radial coordinates of the tracer position using the height and the radius of the vessel respectively. Fig. 8 shows the dimensionless velocity components for the larger vessel (radial, tangential and axial respectively) for T29 at 300 rpm with a grid of 25 and 50 cells in the radial and axial direction respectively.

Fig. 7 – (a) Azimuthal grid scheme used to spatially average data and (b) locations and passes from a cell.

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Fig. 8 – Dimensionless velocity contour plots (a) ur /vtip ; (b) uth /vtip and (c) uz /vtip (T29 , 300 rpm, vtip = 1.58 m/s, tracer 600 ␮m).

3.6.

Radial velocity profile in the impeller discharge

3.6.1.

Using the 600 m tracer

Contour plots are useful in visualising the velocity flow fields and in a qualitative analysis of the results within the entire geometry. However, for a quantitative comparison with data reported in literature, a single velocity component profile as a function of position is really required. From the literature, a suitable profile is the radial discharge velocity, which is available at various radial positions. The present values using the 600 ␮m tracer are shown in Fig. 9 (as RT300La) at r/R equal to 0.33, 0.5 and 0.66 compared with similar data reported in the literature (Wu and Patterson, 1989; Dyster et al., 1993). At locations further away from the impeller the velocities obtained using PEPT are in very good agreement with the literature (within 3%). Very close to the impeller at r/R = ∼0.33, it can be seen that there is a major differences with the velocities at the level of the disc obtained from the PEPT measure-

Fig. 10 – Axial velocity profiles at r/R = 0.33, 0.5 and 0.66 compared with some LDA data from the literature (PEPT data, T29 , N = 300 rpm, vtip = 1.58 m/s). ments being less than 50% of those in the literature based on LDA. This difference probably is due to the fact that in the region of the impeller, the fluid accelerates rapidly to the highest velocities and also changes direction by 90◦ , from vertically up or down to radial. The high velocity and acceleration leads to a smaller number of locations and difficulty in estimating velocities accurately. With increasing distance away from the impeller in the radial discharge stream, the agreement improves and at r/R = 0.66, it is good. This improvement is because with lower velocities and no further rapid change in direction, the number of locations is sufficient to obtain accurate velocities that accord with the literature. Fig. 9 – Impeller discharge radial velocity profiles at r/R = 0.33, 0.5 and 0.66 compared with some LDA data from the literature (PEPT data, T29 , N = 300 rpm, vtip = 1.58 m/s).

3.6.2.

Using the 250 m tracer

In order to establish whether the poor agreement with the literature very close to the impeller was due to the relatively

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Fig. 11 – (a) Axial locations frequency distribution and (b) contour number of passes per cell (PEPT data, T29 , N = 300 rpm, vtip = 1.58 m/s). large size of the tracer, another long run at 300 rpm in T29 was conducted with a tracer of 250 ␮m, which lowers  p . In addition, this data set would help assess the reproducibility of the experiments. The results are also given in Fig. 9 as RT300Lb. It can be seen that despite the use of a smaller tracer, the results are essentially the same. This finding leads to the conclusion that the low calculated velocity in the impeller region is not due to the inability of the larger tracer to follow the liquid. Furthermore, the results in Fig. 10 show the good reproducibility of the technique and the routines which process the data recorded during the experiments. As a further comparison, Fig. 10 shows the axial velocity component for the two PEPT experiments along with the laser based in previous results (Wu and Patterson, 1989). As in Fig. 9, the zero value of the dimensionless velocity is positioned to correspond with the appropriate r/R value. In this case, the agreement with the literature is generally better even close to the impeller, especially picking up the change of direction from above to below the impeller plane. It is also interesting to note that at r/R = 0.66, the axial velocities are close to zero so that in this plane, the flow is mainly tangential and radial.

3.7. Improving the results by using selective interpolation In the previous section, it was shown that close to the impeller where the velocities and acceleration are at their maximum, there are a smaller number of locations recorded by the PEPT cameras than in the rest of the vessel where they are lower. This difference happens because the cameras have a constant acquisition time resolution, so that the locations are more distant from each other when the particle moves faster. Thus, if the particle moves at 1 m/s and the time between two locations is 30–40 ms, the positions will be 30–40 mm apart. Fig. 11a shows the normalised frequency distribution of the tracer locations over the vertical axes for 40 time intervals. From the well-mixed assumption proposed above, the average frequency for the vessel for 40 intervals should be around 2.5% per interval. In fact, in the proximity of the impeller, the value decrease to ∼2% and rises to almost 3% in the regions of the upper and lower loop where the velocities are slightly lower.

Fig. 12 – Selective interpolation working scheme. Fig. 11b is a contour plot of the number of passes from each cell. Since the radial discharge stream divides approximately equally into the upper and lower circulation loop, it is reasonable to expect that the passes in the discharge stream should be of the order of twice those in the two loops. Fig. 11b shows that, on the contrary, the number of passes in this region is in fact the lowest. This low value is probably due to the high velocity, acceleration and change of direction in that region so that the particle moves between the cells without its passage being recorded. In order to overcome this problem, a selective linear interpolation was applied to the data in the cases where the distance between locations was found to be more distant than a certain value. This value was chosen to be the cell dimension used for the azimuthal averaging process. Hence, if two locations have a relative distance larger than the cell dimension, a number of locations are interpolated in such way that the single relative distances between the new locations are smaller than the cell dimension. When instead, the particle moves slowly and the positions are close to each other, there is no need for interpolation. The concept and problem are shown schematically in Fig. 12. In this figure, l* is the dimension of the cell whilst dli is the distance between two consecutive locations. When dli is greater than l*, the missing locations are interpolated and with them, their time as well. The outcome of the selective interpolation on the frequency distribution and passes is shown in Fig. 13. Comparing the modified results in Fig. 13 to the previous ones in Fig. 11, it is obvious that the interpolation has a big influence on the region around the impeller. The number of passes in the radial discharge stream is higher than everywhere else and much higher than before the interpolation. Although the new locations are interpolated, this improvement in the number of passes in the area close to the impeller suggests that recalculating the velocities by processing the new set of data, with the interpolated locations, should have an impact on the velocities calculated from the PEPT technique. In Fig. 13, the contour plots of the velocity components from the interpolated data are presented. These plots can be compared with those without interpolation in Fig. 8. It can be seen that the radial discharge velocities are obviously higher.

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3.8. Effect of scale of tank and viscosity in the turbulent regime

Fig. 13 – The impact of selective interpolation: (a) axial locations frequency distribution and (b) contour number of passes per cell (interpolated PEPT data, T29 , N = 300 rpm, vtip = 1.58 m/s).

This increase is confirmed more quantitatively in Fig. 14 where the radial velocity profiles at two radial positions are shown which can be compared with those in Fig. 9. The largest change is in the radial component at the level of the disc close to the impeller where the maximum velocity is now 0.43vtip compared to 0.36vtip , i.e. an increase of 17% resulting in an error of 30% in this part of the vessel. To better quantify this improvement, two radial velocity profiles, at r/R equal to 0.33 and 0.66, for the original and interpolated data are shown in Fig. 15. It can be seen how the inclusion of interpolated points increases the ‘measured’ radial velocities close to the impeller though still lower than measured by LDA. Further away, the agreement is now excellent. Except where noted, all of the later data presented in this paper is based on using this selective interpolation technique.

As shown earlier, the number of locations recorded by the detector varies with the position of the tracer with respect to the cameras. In fact, since the particles emit ␥-rays in every direction, the probability that a ␥-ray will hit a detector increases as the distance of the particle from it decreases. Thus, by going to a smaller diameter, geometrically similar vessel and moving the two cameras as close as possible to the vessel, more ␥-rays will be detected. Further to that one would expect that provided that the frequency of acquisition remains the same a larger number of data points would be obtained. Therefore, it can be speculated that flow followers in smaller sized equipment are tracked more effectively enabling the flow to be more accurately analysed than in larger ones. To assess this idea, experiments in the smaller geometrically similar vessel, T20 , were made in the same salt solution as in T29 . In addition, to improve the ability of the tracer to follow the liquid by reducing the particle response time,  p , the same experiments were also undertaken using a media with higher viscosity, again matching the liquid and tracer density. Table 1 lists all the experiments carried out in T29 and T20 . From Table 1, it can be observed that the Re values used were all essentially in the turbulent region (>2 × 104 ) and approximately overlapped at the two scales. Similar tip speeds were also used in each fluid and scale. However in the turbulent regime, it is postulated, and experimentally has been shown many times that for any point in the impeller discharge stream (e.g., see Costes and Couderc, 1988; Dyster et al., 1993) that the velocity is directly proportional to the tip velocity. Hence, the measured velocity at any point for all the experiments should be comparable when made dimensionless by dividing by the impeller tip speed, DN. Since in the smaller geometry, it was expected that a larger number of locations should be detected per unit of time, resulting in a higher number of passes, the first comparison between this scale and the larger one was to compare the passes at different times using a geometrically similar meshing grid. Fig. 16 shows the contour plots of the passes for T29

Fig. 14 – Contour plots after selective interpolation: (a) ur /vtip ; (b) uth /vtip and (c) uz /vtip (PEPT interpolated data, T29 , N = 300 rpm, vtip = 1.58 m/s).

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Fig. 15 – Comparison of radial velocity profiles for original and interpolated data. (PEPT, T29 , N = 300 rpm, vtip = 1.58 m/s, tracer a = 600 ␮m and b = 250 ␮m). and T20 as a function of time in the salt solution, i.e. after 5, 15, 30 and 60 min at the same tip velocity in both. Since this comparison was to assess if the size of the equipment had an effect on the acquisition of the data, non-

interpolated data were used to avoid selective interpolation influencing the comparison. For T20 , the number of passes increases faster compared to T29 so that after 30 min, the number of passes recorded in the former was almost the same as

Fig. 16 – Passes after 5, 15, 30 and 60 min in the salt solution: (a) T29 at 300 rpm and (b) T20 at 480 rpm; in both case vtip = 1.58 m/s.

chemical engineering research and design 8 9 ( 2 0 1 1 ) 1947–1960

Fig. 17 – Comparison of radial velocity profiles for large and small scales (PEPT, L = T29 , a = 600 ␮m tracer, b = 250 ␮m tracer, S = T20 , w = salt solution). in 60 min in T29 . This result leads to the conclusion that in a small tank, the higher number of events per time recorded allows a quicker description of the system and the experiments can be run for a shorter time in order to achieve the same level of information. In addition, the colour in the plots in Fig. 16 is almost identical everywhere except very close to walls and where the impeller prevents its passage. This evenness of colour shows that in a relatively short time the tracer visits the whole vessel with a similar frequency and is not significantly held in any recirculation loop such as that found behind a baffle. This observation also validates the earlier assumption in the paper that in a well-mixed vessel, the tracer would spend an equal time in each part of the entire tank. Similar confirmation has been established for a vessel agitated by a pitched blade turbine under turbulent conditions (Pianko-Oprych et al., 2009). In yield stress fluids where caverns form, the assumption only holds for the cavern region (Adams et al., 2008). Having shown how the size of the equipment influenced the data acquisition process, the velocities found in T20 were compared with those in the literature. Hence the radial dimensionless velocity profiles for all the runs in the salt solution are again compared in Fig. 17. It can be seen that the results are very similar for the two sizes. Close to the impeller, r/R = 0.25 and 0.4, for T20 , the profile is more narrow around the impeller plane and the maximum dimensionless radial velocities are higher, especially at the lowest speed. However, most importantly, the maximum dimensionless velocities close to the impeller are still underestimated with respect to the ones reported in the literature. Before continuing further this comparison of the radial velocity profiles from the present PEPT experiments and literature data from LDA measurements, it is useful to consider the different ways the raw data from each are averaged. With LDA, data are collected at a single, relatively small point defined by the crossing position and angle and size of the laser light beams. The Eulerian data obtained is then temporally averaged for each radius for which a measurement is made to give an ensemble-average radial component with respect to the impeller rotation. The PEPT technique leads to Lagrangian data being collected throughout the vessel, and after manipulation of the raw data as explained above, those data that

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can be ascribed to the appropriate rectangular cross-section toroidal cell with respect to the radius and height of interest can then be ensemble-averaged to give the radial component of velocity for that cell. Thus, with PEPT, data are averaged with respect to the impeller rotation and the baffles. Clearly, there are significant differences in the way averaging is conducted in the two approaches and perhaps even more importantly, where the velocity is changing rapidly with position, the substantially larger area over which PEPT data are averaged compared to LDA makes obtaining precise agreement very unlikely. This difference has been shown to be a particular problem with 45◦ pitched blade turbines (Pianko-Oprych et al., 2009). Given the difference in the way LDA velocity data are ensemble-averaged compared to PEPT, it is also interesting to consider all the velocities measured by the tracer over the period of the experiment, which are subsequently averaged. Such dimensionless velocity data for the radial, circumferential and axial components are shown in Fig. 18 along the axis of the vessel for the cells at the radial plane, r/R = 0.3. The first point to be noted is the very wide scatter in the velocities in the cells. For example, Fig. 18a shows that for the radial discharge at the impeller plane, z/H = 0.25, it is possible for the velocity away from the impeller to reach values close to the impeller tip velocity (ur /vtip = ∼1) whilst at the same plane, the velocity can also be towards the impeller (ur /vtip < 0). When all these values in Fig. 18a in the region of the impeller are averaged as described above, the data shown in Fig. 17 are obtained. The tangential component, over all the height of the vessel, shows on average positive values, i.e. the same direction as the impeller rotation, but significant negative values are also recorded meaning that are times when the fluid also moves in the opposite direction to the impeller. In the impeller discharge stream, the maximum tangential values are generally higher than the radial as has also been reported when using LDA (Koutsakos and Nienow, 1990). Finally, the axial velocity component shows on average negative (downward) velocities above the impeller plane and positive (upward) velocities below it as expected as the liquid is sucked into the impeller region impeller from the top and the bottom to be, subsequently, radially discharged. However, in the upper part of the vessel, the liquid occasionally reverses its flow direction and at the impeller discharge significantly higher velocities up to ∼0.5vtip in either vertical direction are detected. Even though the difference in the ways the raw data are averaged in order to obtain the results in the discharge stream which are being compared was recognised, it was decided to see if increasing the liquid viscosity, which might enhance the ability of the flow follower to track the flow accurately by reducing  p , would change the results obtained. Since the viscosity increase still left the experiments in T20 mainly in the turbulent regime, it was felt that if the dimensionless velocities recorded were the same as those previously obtained with salt solutions, it would show that the deviations from the literature were not due to the inability of the tracer to follow the motion of the liquid. Essentially, the results in the radial discharge stream were the same (data not shown) for the salt and sugar solutions in T20 . However, recognising that the PEPT and LDA measurements might inherently be different, it was decided that it would be interesting to compare the PEPT results for the salt and sugar solutions by determining the mean absolute dimensionless velocities, |v|/vtip (the vector sum of ur /vtip , uth /vtip and uz /vtip ) at the same vtip for each of the fluids for the whole

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Fig. 18 – Velocity components at a constant radial position r/R = 0.33, T20 , salt solution, N = 480 rpm. matrix of cells in the vessel for which ensemble-averaged values were obtained. The ability to carry out such data treatment relatively easily for the whole vessel is unique to PEPT. These velocities for each fluid at 480 rpm were then plotted as a frequency distribution as shown in Fig. 19 and in this way, all the velocities determined would be included in the comparison. As can be seen, visually the two distributions appear identical within experimental error. Though the Reynolds number is different for the two experiments, Re = 1.1 × 105 and 3.4 × 104 for the salt and sucrose solution respectively, the flow is still turbulent so that with this method of normalising, equality of results would be expected within the accuracy of the experimental technique. To make further comparisons therefore, it was decided to determine the mean, mode, standard deviation and skewness

Fig. 19 – Normalised absolute velocity distribution for the all the cells in T20 (N = 480 rpm, w = salt solution, s = sucrose solution).

of the distributions for all the experiments listed in Table 1. The calculated values are reported in Table 2. Since the flow is turbulent or very nearly so, as stated above, it would be expected that all velocity parameters when normalised by vtip should be the same. Clearly here that does not seem to apply to the either the mean, (|v|/vtip )mean or the mode. The reason for this is unclear and is worthy of some detailed discussion which will be related to the mean but equally applicable to the mode. Firstly, it might be a measure of the reproducibility of the data and that the mean of all the mean velocities, (|v|/vtip )mean in Table 2 is 0.20 ± 0.06 for all conditions. However, the repeat identical runs in T29 do not support such a hypothesis. On the other hand, it might be a real effect due to the different vessel sizes. The (|v|/vtip )mean values in T20 (average 0.191) in all cases are less than those in T29 (average 0.233). Previous work (Bujalski et al., 1987) has shown that, even very precise geometric similarity, with increasing scale, the dimensionless group, the power number, Po, for the Rushton turbine in the turbulent regime increases with scale, i.e. Po ∝ T0.065 . Previous to that study, it had been assumed that under these conditions, Po should be scale independent. Though the exponent on T is very small, it predicts what has been found by many workers, namely that Po at the laboratory scale is less than the value of 5 quoted for large vessels. However, an explanation for this scale dependency has never been published. The variation in (|v|/vtip )mean might also be linked to the fall in Re from 16.1 × 104 to 1.42 × 104 , i.e. (|v|/vtip )mean ∝ ∼Re0.09 The precise Re at which the flow changes from completely turbulent throughout the whole tank to one in which, in some extremities, transitional flow occurs is not established. The tank flow normally defined as turbulent at the value of Re above which Po becomes constant, ∼2 × 104; and for a Rushton turbine, it slowly falls for Re < 2 × 104 . It is conceivable that at Re values around this value, in the extremities of the vessel, the flow is no longer turbulent. In that case, when |v|/vtip values from all parts of the tank are involved in obtaining the mean,

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Table 2 – Mean, mode, STD and skewness for the normalised absolute velocity distributions for all the experimental conditions used. Experiment

N [rpm]

Re

vtip [m/s]

Mode

Mean

STD

Skewness

RT300a T29 RT300b T29

300 300

1.61 × 105 1.61 × 105

1.58 1.58

0.1767 0.1667

0.2333 0.2347

0.1308 0.1342

1.612 1.666

RT600w T20 RT480w T20 RT200w T20 RT100w T20

600 480 200 100

1.38 × 105 1.10 × 105 4.60 × 105 2.30 × 104

1.98 1.58 0.66 0.33

0.1464 0.1368 0.1266 0.1377

0.2006 0.1929 0.1949 0.2089

0.1341 0.1266 0.1300 0.1354

2.072 2.109 2.141 1.940

RT600s T20 RT480s T20 RT300s T20 RT200s T20

600 480 300 200

4.25 × 104 3.40 × 104 2.12 × 104 1.42 × 104

1.98 1.58 0.99 0.66

0.1264 0.1352 0.1371 0.1253

0.1945 0.1864 0.1850 0.1647

0.1260 0.1173 0.1157 0.1289

2.029 1.960 1.846 2.104

the assumption of constant (|v|/vtip )mean for all tank sizes and Re values may not apply. Another possible explanation for why normalising with vtip within Re range has not pulled all the data together here might be the length of time (30 min) over which the raw data that lead to the calculated parameters are collected in PEPT. When LDA or PIV are used, though much raw data are utilised in order to obtain the statistical information on the Eulerian velocities in the flow field, the rate of collection is so high, that the only a few seconds of measurement are required in each case for the data at a point or in a plane respectively. An F-test of the hypothesis that (|v|/vtip )mean ∝ ∼Re0.09 rather than (|v|/vtip )mean = 0.20 for all Re in this study shows that the more complex relationship is statistically significant at 2% level. Overall, a clear explanation for the variation of these normalised global parameter which can probably not be measured realistically by any other technique and which to the authors’ knowledge has never been measured before cannot be given.

4.

Conclusions

For the first time, velocity measurements made using PEPT in two sizes of baffled vessel (∼0.20 m and ∼0.29 m diameter) and two different viscosity fluids agitated by a Rushton turbine are compared directly in depth with some studies reported in the literature made by LDA for the turbulent regime in the equivalent geometry. Initially, the basic parameters in the program track used to produce the raw PEPT data have been shown to affect the accuracy of the final locations generated by the system. In particular, the parameters depend on the rate of data collection which in turn depend on the velocities in the system under study and the level of activity of the irradiated tracking particle which decays with time. Hence, it is recommended that a study to assess the optimum parameters for each experimental set up is undertaken, with, for long runs, the parameters being changed with time. Once the locations are obtained from the PEPT system, Lagrangian velocities are calculated by differentiating position (defined in Cartesian coordinates) versus time. However, most velocity data in the literature are Eulerian, presented in cylindrical coordinates. It is shown that since the flow in general has a strong rotational component, it is better to convert the original Cartesian coordinates into cylindrical coordinates before determining velocities in cylindrical coordinates rather than determine Cartesian velocities first and then converting. Methods of azimuthally averaging these

velocities in rectangular cross sectional torroids are also discussed and optimised. Having optimised the raw data treatment, it was shown that the size of the neutral density tracer (250 ␮m and the 600 ␮m) used in this study had no significant influence on the results, since both of these sizes are small enough to follow the mean liquid velocity closely. Also, comparing the radial velocities obtained with PEPT in the impeller discharge stream to well established published data, good agreement was obtained except very close to the impeller at the plane of the disc. Here, PEPT results seriously underestimated velocities (about 50% lower than literature values). This problem is associated with the low data capture rate because of the very high velocity and the rapid changes in both magnitude and direction in this region. It is also associated with the different ensemble averaging and spatial resolution between PEPT and LDA. Nevertheless, this problem was less significant in the smaller vessel because of the enhanced data capture rate. The latter is related to the higher probability of both of the back-to-back ␥rays impacting the detector plates in the smaller rig. However, to improve the accuracy of the velocity data in the discharge stream, selective interpolation of the raw data has been proposed (and justified), leading to an increase in the calculated velocities, which significantly brings them closer to the literature values. It also ensures that the occupancy is realistic for this location compared to elsewhere in the vessel. Finally, the absolute velocities normalised by the impeller tip velocity for all the toroidal cells in each size of vessel and each fluid and a range of agitator speeds are reported for the first time and are presented in the form of frequency histograms, where the velocities for each run were based on tracking a particle for 30 min. Surprisingly, it is found that though the flow is essentially always just turbulent, the mean and mode of the normalised velocities each decrease with decreasing scale. The possibility that this fall is an experimental artefact is shown to be unlikely and other possible reasons are discussed. It is concluded that this calibration study shows that the PEPT technique can be used to obtain accurate velocity data throughout the entire complex three-dimensional turbulent flow field in a mechanically agitated, baffled vessel. Subsequent papers from this study will present further Lagrangian data for the whole vessel that cannot be obtained by previous Lagrangian methods or by Eulerian techniques.

Acknowledgements The authors would like to acknowledge the support of Huntsman Polyurethanes (Europe) for this work and especially for

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financial support for Mr. (now Dr.) Fabio Chiti; also the Engineering and Physical Sciences Research Council (UK) through Grant GR/S70517/01 for funding the experimental PEPT studies. We would also like to thank Professor D. J. Parker and Dr. X. Fan of the School of Physics and Astronomy of the University of Birmingham for the assistance with the production of radioactive tracers and help with the experimental program of work.

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