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Study of Macro-Instabilities in Stirred Tanks Using a Velocity Decomposition Technique
0263±8762/00/$10.00+0.00 q Institution of Chemical Engineers Trans IChemE, Vol. 78, Part A, October 2000
STUDY OF MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE V. T. ROUSSINOVA, B. GRGIC and S. M. KRESTA Department of Chemical & Materials Engineering, University of Alberta, Alberta, Canada
I
n this paper a quantitative measure of errors introduced in the turbulent velocity RMS signal due to the presence of macroinstabilities (MI) in the velocity ®eld is presented. The velocity time series were measured for four commonly used impellers (PBT, A310, HE3 and RT) with a one component LDV. Two locations in the tank, the impeller stream I and upper corner U, were studied. Three aspects of the geometry were varied: impeller diameter (D T/2 and D T /4); number of baf¯es (two and four); and off-bottom clearance (C/D 1.0 and C/D 0.5). By resampling and smoothing the velocity records the RMS velocity due to MI, v MI , was determined. Further velocity decomposition recovered the high frequency component of the signal, v HF , due to the random ¯uctuations and the blade passages. Inclusion of the non-stationary, non-equilibrium MI component in the calculation of the RMS velocity can result in an overestimation of up to 50%. Analysis of the time series records shows that the MI is present in all con®gurations tested. In some cases (PBT and RT) the MI dominates the signal while for others (HE3 and A310) the amplitude of the signal is low and the MI is much less pronounced. The MI is very sensitive to geometry: for the impeller stream of the RT, increasing the number of baf¯es from two to four completely changes the velocity time series. The MI can have as dramatic an impact on experiments and analysis of the ¯ow as the trailing vortices observed behind impeller blades. An understanding of this phenomenon is important for accurate analysis of the velocity and turbulence ®elds, for measurement of blend time and solids distribution, and for improving understanding of mesomixing. Keywords: stirred vessel; velocity ®eld; macroinstability; velocity decomposition; time series analysis
INTRODUCTION Traditionally, studies concerned with the turbulent ¯ow pattern in a stirred vessel are based on the mean (VÅi ) and RMS ¯uctuating velocity (vÅ i2 ). The latter is used to determine the different turbulence characteristics such as turbulence intensity, turbulent kinetic energy and estimates of the rate of dissipation of turbulent kinetic energy. In spite of the fact that there are numerous conceptual dif®culties with examining turbulence characteristics in a stirred tank (Kresta 1 ), the region close to the impeller has been studied in some detail (Wu and Patterson 2 , Rutherford et al.3 , Stoots and Calabrese4 , Rao and Brodkey5 , Cutter6 ). Signi®cant progress has been made in the understanding of this region of the tank. The mean centred velocity signal close to the impeller contains three signi®cant components: the random turbulent ¯uctuations, which have frequencies higher than the blade passage frequency and up to the Kolmogorov wavenumber; the blade passages, or trailing vortices; and a much slower variation on the mean, which is here termed macroinstability. This macroinstability indicates the existence of eddy ®elds whose time and length scale considerably exceeds those associated with blade passages and smallscale turbulence. This observation is not a new one: the existence of large-scale low-frequency phenomena in stirred
tanks has been regularly reported in the literature since 1988 (Winardi et al.7 , Winardi and Nagase8 , Bruha et al.9 12 , Kresta and Wood13,14 , Chapple and Kresta15 , Myers et al.16 , Montes et al.17 ). Winardi et al.7 were among the ®rst to report that the instantaneous circulation pattern for a paddle impeller is different from the mean velocity ®eld as measured using LDV. Quite different ¯ow patterns and various combinations of patterns appear over time. In a second paper, Winardi and Nagase8 , used several ¯ow visualization techniques (visual observation, high speed VTR, and video recording), and LDV to examine the time varying ¯ow pattern of a marine propeller. They found that the ¯ow pattern was asymmetric with respect to the impeller shaft. They identi®ed three circulation patterns and measured the lifetime distribution of these patterns using 2D ¯ow visualization. They found that ¯ow pattern changes were random in order and that the lifetime of a given pattern could range from half a second to several minutes. Kresta and Wood13,14 observed variations in the bulk circulation pattern when they analysed the ¯ow ®eld of a PBT impeller using ¯ow visualization, LDV, and spectral analysis. A later paper by Chapple and Kresta15 considers the in¯uence of geometric parameters such as off bottom clearance, impeller diameter and number of baf¯es on ¯ow stability for two axial impellers: the Lightnin’ A310
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MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE 1041
impeller and the pitched blade turbine (PBT). Chapple and Kresta15 also showed that the geometric variables which are most signi®cant for stability depend on the type of impeller. Bruha et al.9 investigated 3, 4 and 6 bladed, 458 axial impellers. They studied the effect of impeller rotational speed and the different geometries in which macroinstabilities (MI) occur. Their investigations covered two impeller diameters (D T /3 and T/4), and three different off-bottom clearances (C/T 0.33, 0.4 and 0.5). They observed a vortex which appears as a welling up of the ¯uid surface, again on a much longer time and length scale than that of the turbulent eddies. To capture the shape and dimensions of the large vortex they used a video camera. A round probe above the ¯uid surface was used to measure the frequency of this `surface swelling’. They concluded that this frequency is linearly dependent on the impeller rotational speed. The effect of geometry was not systematically examined, although several variations were studied. More recent work from this group (Bruha et al.10 12 ) uses ¯ow visualization and a special mechanical device called a `tornadometer’. The tornadometer is placed below the surface in the vicinity of the impeller and de¯ects whenever macro-instabilities are strong enough to change the direction of the mean ¯ow and de¯ect the target. The authors were able to con®rm their earlier report that the frequency of the macro-instabilities is linearly related to the impeller rotational speed. They also observed that the low frequency macro-instabilities are accompanied by changes in the angle of the impeller discharge ¯ow and the appearance of an unstable secondary circulation loop. This is in agreement with Kresta and Wood13 . Montes et al.17 used LDV measurements, ¯ow visualization, spectral analysis, and wavelet transforms to analyse ¯ow macro-instabilities in a stirred tank equipped with a six bladed 458 PBT impeller (D T /3, C/T 0.35, four baf¯es). They con®rm that the occurrence of macroinstabilities is accompanied by the presence of a large vortex in the upper part of the vessel, and it is linearly coupled with the frequency of the impeller revolution. The RMS velocity underwent a sudden rise in ¯uctuating intensity at Re 600. This rise might correspond to the ®rst appearance of the macro-instability. For ReI > 600, the frequency spectrum also shows a distinct peak for low frequency oscillations. This implies that both the transitional and fully turbulent ¯ow regimes are subject to this type of instability. Hasal et al.18 implemented a new type of analysis of MI. The ¯ow ®eld in a stirred vessel was treated as a pseudostationary high-dimensional dynamical system. Analysis of the velocity time series was done with the proper orthogonal decomposition (POD) technique. The proposed method was successfully applied at both low and moderate Reynolds numbers. The authors showed that the MI has no coherent frequency for the geometry investigated (a six bladed 458 PBT impeller, D T/3, C 0.35T, four baf¯es). The frequency of the MI could not be determined in zones close to the vessel wall, or at high ReI . In chemical processing, the velocity ®eld is used to accomplish process objectives such as heat transfer, mass transfer and chemical reaction. Reports in the literature provide evidence of the effect of the MI on the local heat transfer coef®cient (Haam and Brodkey19 ), on intermittency Trans IChemE, Vol 78, Part A, October 2000
phenomena in the feedstream (Houcine et al.20 ) and on solids distribution at low solids concentrations (Bittorf 21 ). Haam and Brodkey19 measured the local heat transfer ¯ux and temperature of the vessel wall. Time variations in the local heat transfer coef®cient were observed. The authors hypothesized that the low frequency ¯uctuations in the signal were due to the precession of an axial vortical structure, which rotated between the surface and the impeller. Feedstream jet intermittency was studied by Houcine et al.20 using laser induced ¯uorescence (LIF). The feedstream jet was injected at the baf¯e in the upper part of the tank. The authors observed three states of the feedstream, which were characterized as no intermittency (the feed stream is always vertical), slight intermittency (chaotic movement of the feedstream jet axis in the plane of the laser sheet) and effective intermittency with a period of 1-2 s. This phenomenon can be linked to the existence of MI. Bittorf and Kresta22 provide another example of how MI can affect process results. At low solids concentrations, the MI is strong and the solids are distributed throughout the vessel. As the solids concentration is increased, the MI is damped out before it reaches the top of the vessel and a clear ¯uid layer appears. Bujalski et al.23 showed that this clear ¯uid layer can exhibit mixing times many times longer than the well mixed zone below the interface. Finally, large scale macroinstabilities can have considerable dynamic effects on the solid surfaces inside the vessel. When large scale eddies, sometimes as large as the vessel diameter, interact with components of the system (baf¯es, shaft, impeller or vessel wall) they cause dramatic force ¯uctuations, leading to (for example) abrupt changes in the torque on the shaft. In some cases, this can result in mechanical failure. In summary, the MI is an important component of the large-scale motions in a stirred tank. It has implications for many of the operations which are carried out in stirred tanks, and for the structural integrity of the vessel components. The MI produces a broad, low frequency band in the power spectrum. It does not necessarily show a coherent frequency, although this may be observed for some cases. The time scale of the MI is of the order of the tank turnover time. No frequencies are observed in the interval between the MI and the blade passage frequency, so these motions are clearly associated with the mean velocity as determined over a short time interval, rather than with the turbulent ¯uctuations. While these large scale phenomena will eventually decay to turbulent ¯uctuations (probably in the top third of the tank), they are a non-stationary, non-equilibrium component of the ¯ow which should not be included in the turbulent rms velocity. In this paper, a quantitative measure of the MI is provided for two regions of the stirred tank: the impeller region and the top corner of the tank. Four commonly used impellers (PBT, RT, HE3 and A310) are discussed. The experiments were performed using a one component LDV. For each impeller eight geometric con®gurations were used in order to investigate: (a) the impact of the tank and impeller geometry on the MI; (b) the portion of the measured RMS velocity which is due to the MI. This is analogous to isolating the effect of the blade
ROUSSINOVA et al.
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passages (Stoots and Calabrese4 , Wu and Patterson 2 , Yianneskis et al.24 , Rutherford et al.3 ) for the Rushton turbine. EXPERIMENTAL LDV Experimental Set Up The forward scattering, one dimensional laser Doppler velocimeter (LDV) described in Zhou and Kresta25 was used for this study. Two beams from an Argon Ion Laser with a beam separation of 0.01691 m were focused to form a measuring volume 168 mm in diameter with fringe spacing of 9.1 mm. The tank was moved horizontally using computer-controlled traverses with an accuracy of 6 0.5 mm and in the vertical direction using a manual traverse (accuracy of 6 1 mm). An Aerometrics Doppler signal analyser (DSA) was used to convert the analogue signal from the receiving optics into Doppler frequencies and instantaneous velocities. The DSA operates in the frequency domain burst detection mode. The analogue signal was sampled at a 2.5 MHz sampling frequency. The signal sampling frequency is different from the frequency of the velocity measurements, which is ®xed by the particle arrival rate, or seeding density. The particle arrival rate is affected by the velocity of the ¯uid, so the velocity data rate is much lower at the top of the tank than it is in the impeller stream. The DSA allows the user to set either the sample size (number of velocity measurements) or the sample time (length of record in seconds). Experimental results showed that both the sample size and the sampling time are critical to accurate velocity measurement. If the sample time is too short, the reproducibility is poor even with a sample size larger than 10000 data points. For all the measurements the sample size was set equal to 10000. The length of the record varies from 8 to 12 seconds. This is long enough to obtain a meaningful average of all components of the signal. Geometry of the Tank and Impellers The stirred vessel (STR) used for this work is shown in Figure 1. The tank diameter T 0.24 m is equal to the liquid height (H T ). To prevent air entrainment and surface vortexing, a lid was placed on the top of the tank and covered with 5 cm of water to form a seal. Either two or four vertical rectangular baf¯es of width W T /10 were spaced at equal intervals around the periphery of the tank ¯ush with the wall. A tank with four baf¯es is considered fully baf¯ed; the two-baf¯e con®guration is used to test the sensitivity of instabilities to the number of baf¯es. A large diameter (9 mm) was used in order to eliminate shaft run-out (or wobbling). The shaft was machined down to ®t into smaller hubs ID 12.75 mm. The experimental variables were combined in a factorial design (Box et al.26 ) to give 8 tank geometries for each impeller as de®ned in Table 1. This experimental design was followed, with minor variations, for all four impellers studied, with results reported in Tables 2(a)±(h). The three axial impellers (A310, HE3, PBT) were placed at dimensionless off bottom clearances of C/D 0.5 and 1. The impeller diameters for the HE3 and PBT were D T/2 and T/4; for the A310, the diameters were D 0.55T and 0.35T. The A310 was supplied by Lightnin’ Inc; the
Figure 1. Schematic of the measurement locations for the PBT and HE3: IÐimpeller stream; UÐupper corner.
HE3 was supplied by Chemineer. One radial impeller, the Rushton turbine (RT: D T/2 and T /3; disc diameter 2/3D; blade height D/5; blade width D/4; blade and disc thickness 0.002 m) was placed at off bottom clearances of C T /2 and T/3. Different off bottom clearances were used for the radial impeller due to the inherently different ¯ow pattern produced by this impeller. For the radial impeller, the off bottom clearance determines the symmetry or asymmetry of the two circulation loops. For the axial impeller, the dimensionless off-bottom clearance allows for the effect of C and D on changes in the ¯ow pattern14 . For the PBT, RT, and HE3, the tip speed (Vtip ) was the same for both impeller diameters, and for all three impellers. For the A310, the tip speeds were slightly different. In order to compare the different impellers directly, all reported velocities are normalized with Vtip . Since the ¯ow is in the fully turbulent regime, the velocity ®eld can be expected to scale with the characteristic velocity, or the tip speed.
Table 1. Geometrical con®gurations used in the experiments *. Run 1 3 3 4 5 6 7 8
D
C/D
Nf
T/2 T/2 T/2 T/2 T/4 T/4 T/4 T/4
1 0.5 1 0.5 1 0.5 1 0.5
2 2 4 4 2 2 4 4
* For the RT, D T/4 is replaced by D T/3, and C/D 0.5 by C T/3 for the A310, D 0.55T and 0.35T.
Trans IChemE, Vol 78, Part A, October 2000
MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE 1043 Table 2(a). Results for the PBT impeller stream.
Run 1 2 3 4 5 6 7 8
N (rpm)
VÅres Vtip
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
200 200 200 200 400 400 400 400
0.392 0.412 0.360 0.456 0.348 0.337 0.40 0.332
0.158 0.139 0.135 0.122 0.084 0.178 0.086 0.084
0.090 0.071 0.072 0054 0.031 0.079 0.026 0.032
0.121 0.109 0.107 0.103 0.077 0.158 0.080 0.074
56.82 46.74 51.13 33.07 11.85 48.85 5.60 11.62
31.01 26.69 25.32 17.93 9.23 12.76 6.85 11.82
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
34.9 16.03 25.68 14.74 4.71 0.76 0.56 0.52
40.94 33.55 20.34 21.33 15.40 25.69 21.43 13.39
Table 2(b). Results for the PBT upper corner stream.
Run 1 2 3 4 5 6 7 8
N (rpm) 200 200 200 200 400 400 400 400
VÅres Vtip 0.021 0.057 0.002 0.012 0.011 0.025 0.007 0.010
v res Vtip
v MI Vtip
v HF Vtip
0.083 0.065 0.085 0.070 0.054 0.033 0.033 0.034
0.053 0.033 0.042 0.033 0.025 0.018 0.018 0.021
0.059 0.049 0.071 0.058 0.047 0.026 0.027 0.030
Table 2(c). Results for the HE3 impeller stream.
Run 1 2 3 4 5 6 7 8
N (rpm)
VÅres Vtip
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
200 200 200 200 400 400 400 400
0.236 0.236 0.226 0.206 0.255 0.194 0.224 0.194
0.062 0.058 0.060 0.054 0.049 0.045 0.048 0.047
0.026 0.025 0.021 0.016 0.010 0.013 0.019 0.016
0.055 0.052 0.053 0.052 0.047 0.043 0.042 0.043
2.41 5.15 2.69 0.0 0.0 0.0 0.45 0.0
12.52 12.48 9.96 4.47 4.82 5.24 14.78 8.77
Table 2(d). Results for the HE3 upper corner.
Run 1 2 3 4 5 6 7 8
N (rpm)
VÅres Vtip
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
200 200 200 200 400 400 400 400
0.002 0.012 0.006 0.008 0.004 0.008 0.013 0.018
0.044 0.033 0.036 0.040 0.034 0.033 0.033 0.029
0.023 0.013 0.017 0.020 0.016 0.014 0.014 0.010
0.036 0.030 0.030 0.033 0.030 0.029 0.029 0.026
2.33 0.36 0.6 0.0 0.62 0.0 0.0 0.03
22.19 8.85 18.9 20.31 13.67 11.84 14.17 9.99
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Table 2(e). Results for the A310 impeller stream.
Run 1 2 3 4 5 6 7 8
N (rpm)
VÅres Vtip
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
150 150 150 150 400 400 400 400
0.199 0.227 0.275 0.259 0.281 0.221 0.269 0.161
0.088 0.138 0.079 0.086 0.060 0.065 0.057 0.096
0.051 0.060 0.032 0.036 0.018 0.025 0.018 0.032
0.068 0.118 0.070 0.076 0.055 0.058 0.054 0.085
33.54 37.49 13.02 15.51 0.0 4.38 0.04 11.55
29.11 16.9 11.85 14.48 9.37 11.72 5.7 12.81
Table 2( f ). Results for the A310 upper corner.
Run 1 2 3 4 5 6 7 8
N (rpm) 150 150 150 150 400 400 400 400
VÅres Vtip 0.013 0.003 0.028 0.030 0.011 0.003 0.041 0.044
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
0.047 0.045 0.052 0.056 0.037 0.032 0.039 0.041
0.024 0.018 0.030 0.033 0.018 0.019 0.018 0.023
0.039 0.040 0.042 0.042 0.031 0.026 0.034 0.031
5.41 0.6 14.33 9.23 2.01 1.08 4.23 4.27
23.18 13.33 24.33 32.92 20.86 18.96 16.15 30.27
Table 2(g). Results for the RT impeller stream.
Run 1 2 3 4 5 6 7 8
N (rpm)
VÅres Vtip
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
200 200 200 200 300 300 300 300
0.455 0.568 0.712 0.829 0.596 0.293 0.756 0.682
0.346 0.395 0.327 0.385 0.390 0.415 0.355 0.339
0.107 0.139 0.044 0.051 0.152 0.235 0.077 0.053
0.319 0.354 0.321 0.379 0.355 0.364 0.343 0.335
65.38 68.48 25.56 34.51 73.92 90.32 53.02 32.82
8.45 11.49 1.84 1.59 9.86 14.06 3.61 1.2
Table 2(h). Results for the RT upper corner.
Run 1 2 3 4 5 6 7 8
N (rpm)
VÅres Vtip
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
200 200 200 200 300 300 300 300
0.172 0.154 0.088 0.172 0.071 0.038 0.024 0.019
0.153 0.146 0.149 0.133 0.096 0.066 0.083 0.043
0.065 0.073 0.053 0.051 0.064 0.033 0.041 0.021
0.129 0.115 0.133 0.119 0.067 0.053 0.068 0.035
45.02 46.60 33.93 34.22 51.50 15.24 23.32 2.31
18.85 26.94 11.85 12.30 43.51 23.44 22.95 19.13
Trans IChemE, Vol 78, Part A, October 2000
MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE 1045 Two locations in the stirred tank were chosen for detailed investigation: the impeller stream and the top corner of the tank. These locations were chosen based on tuft visualization work reported by Chapple and Kresta15 . The impeller stream velocity signal can capture all frequencies present in the ¯ow. It contains the blade passages, the high frequency turbulent components, and the macroinstabilities of interest for this work. The second probe location is in the upper corner of the tank. This region is considered highly unstable, since the velocities are low and the ¯ow changes direction frequently from radially inward to radially outward. The exact measurement locations for the PBT and HE3 are shown in Figure 1. The radial component of velocity was measured in the impeller stream for the RT. In all other cases, the axial velocity was measured, since it gives the clearest indication of the ¯ow pattern and it is the primary velocity component for axial impellers. For the axial impellers, the impeller stream measurement point was located 3 mm below the impeller blade at a radial distance of 2r/D 0.8 for the PBT and HE3 and 2r/D 0.5 for the A310. For the RT, the impeller stream measurements were taken 3 mm away from the tip of the blade (r D/2 3 mm) and at the centreline of the impeller. For the upper corner, measurements were taken at a radial position 3 mm away from the edge of the baf¯e (r 0.9T 3 mm), and 5 mm down from the top surface. For each of the 32 geometric con®gurations and each measurement location, ®ve repeat measurements were done. The averaged mean and RMS velocities are reported in Tables 2a±h. DATA PROCESSING STEPS Time Series Analysis
Figure 2. Effect of the data processing steps on the signal. (a) raw signal, 10,000 points, (b) resampled signal, 5000 points, (c) smoothed signal, 5000 points.
A typical time series of the axial velocity signal, measured in the impeller stream for the PBT, is shown in Figure 2a. The velocity record shows the turbulent ¯uctuations, the ¯uctuations due to the individual blade passages, and the ¯uctuations due to the MI. In this time series, the high intensity turbulent ¯uctuations tend to obscure the longer time scale ¯uctuations. The objective of this work was to separate and analyse the macroinstability, or low frequency, component of the signal. Several possible methods are available for this signal processing task. The ®rst class of these methods operates in the frequency domain using various forms of notch ®ltering (Press et al.27 ). The autocorrelation function has been used in combination with a notch ®lter (Rao and Brodkey5 , Wu and Patterson 2 , Hockey and Nouri28 ) to characterize the pure turbulent component in the measured velocity. The main objective in all of these studies was to eliminate the blade passage frequency from the turbulent analysis. From the signal processing perspective, a notch ®lter can be used only if a single frequency is to be removed from the signal. Kresta and Wood13 applied this method to remove the combined effect of blade passages and a coherent low frequency from the integral of the autocorrelation function. In this work, the MI is not a ®xed single frequency (i.e. it is not necessarily coherent). In this case, if a notch ®lter is used, the width of the window selected and the sharpness of the ®lter function signi®cantly affect the results of the analysis. Similar dif®culties arise with low pass ®ltering. This topic is discussed in detail by Grgic29 .
An alternative approach uses short time averages in the time domain to smooth the signal and isolate the nonstationary component of the velocity. The premise of data smoothing is that one is measuring a variable that is both slowly varying and is also corrupted by random noise. Replacement of each data point with an appropriate local average of the surrounding points will discard the noise (in this case, the random turbulent component) without signi®cantly biasing the mean. Smoothing is a form of low pass ®ltering, which is applied in the time domain, not the frequency domain. In this case, the adjustable parameters are the length of the smoothing window, and the number of repetitions of the smoothing process. Press et al.27 discuss the correct application of smoothing techniques in some detail. Other forms of signal analysis have been applied to velocity signals from stirred tanks with varying degrees of success. The correlation dimension (Brandstater and Swinney30 ) is dominated by the blade passage frequency and is unable to distinguish between noise and the MI (Richardson31 ). Wavelets can be used to separate various frequency ranges17 , but the adjustable parameters in this technique are both more numerous and less easily interpreted than those used in the smoothing technique. The end result is equivalent to application of a low pass ®lter. Proper orthogonal decomposition18 allows reconstruction of the MI after several stages of signal processing. While the
Trans IChemE, Vol 78, Part A, October 2000
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Table 3. Number of blade passages used for smoothing window. Impeller type D impeller diameter Blade passages
PBT D 4
T/4 4
HE3 D 2
T/2 4
D 4
T/4 6
authors show accurate decomposition of the signal, the method is signi®cantly more complex than that selected for this work. In the Fourier domain burst detection mode of LDV signal processing, velocity data is recorded every time a particle traverses the measurement volume. The resulting LDV time series is represented by Sn , where n 1 . . . N. The corresponding arrival times are tn , and the intervals between particle arrival times (D tn tn 1 tn ) are approximately Poisson distributed. The unevenly spaced data precludes the use of the FFT algorithm, and makes the computation of the power spectrum non-trivial. Sampling based on particle arrival times also allows the possibility of velocity biasing (McLaughlin and Tiederman32 ). This is a bias due to the tendency to collect more data when the velocity is high than when it is low. On the other hand, the random sampling times can be used to eliminate errors in the spectrum due to a biasing of high frequencies onto lower ones. In this study the main objective was to isolate the component of the signal due to MI, so calculation of the power spectrum was of secondary importance. Corrections for possible velocity biasing were implemented through signal resampling. The raw LDV signal was interpolated using zero-order interpolation, also known as `sample and hold’ in the LDV literature. In sample and hold interpolation, the value Sn measured at time tn is retained until the next particle is observed at tn 1 , at which point the record jumps to the new value Sn 1 . When the signal is resampled to simplify the signal analysis and to remove any velocity biasing, a ®xed sampling rate is set with equal time intervals tres . The new time series Qres is constructed from the zero-order interpolated signal. For the new time m tres , m 1 . . . M, one ®nds a data point Sn such that tn # m tres < tn 1 and the velocity at m tres , Q m tres is set equal to S tn . In this way a new data set is produced Q tm , m 1 . . . M de®ned at M regularly spaced new times tm . A resampling window equal to the mean data rate was applied. An interpolation density d M/N is further de®ned, which must be less than or equal to 1. This value was used to adjust the width of the sampling window between two measurement locations. In the impeller stream, the turbulence intensities are the highest, and the data rate is the highest, while in the vessel corners lower velocities give a lower data density. Mean sampling frequencies varied from several hundred hertz (600 Hz) up to 2 kHz. Once the raw signal is resampled it is smoothed using a moving average. This is equivalent to applying a low-pass ®lter at the time scale of the averaging window. The averaging time, or the smoothing window, was based on a number of blade passages, as given in Table 3. The ®rst round of smoothing proved insuf®cient for isolation of the MI: higher frequency ¯uctuations were still apparent in the signal. These could have been removed in one step by
A310 D 4
T/2
0.35T
D
6
2
2
RT 0.55T
D 2
2
D 6
T/3
D
3
4
T/2 4
using a longer averaging window, but at this point the smoothed signal cuts off the peaks in the MI ¯uctuations. In order to retain the peaks while removing the high frequencies, a second round of smoothing was used, typically with the same averaging window as the ®rst smoothing. The procedure was considered complete when the smoothed signal overlapped the slow variation in the initial velocity signal, and no observable higher ¯uctuations remained. The effect of each step of data processing on the signal is shown in Figures 2(b and c). The shape of the smoothed signal is virtually identical to that of the raw signal in terms of the slow variation on the mean. Some slow variations on the mean were observed for all geometries and impellers examined. Velocity Decomposition and Error De®nition The concept of velocity decomposition is useful to determine how much turbulence a given impeller can generate, and the relative contribution of the regular blade passages and large scale MI ¯uctuations to the total ¯uctuations. The instantaneous mean centred velocity, v res can be de®ned as follows: v res
v BP
v MI
v rand
1
The subscripts BP and MI denote the blade passage and low frequency components, and rand denotes the pure random or turbulent component. The v res is the mean-centred RMS velocity component after resampling. With selection of a suitable window length relative to the blade passage frequency, data smoothing ®lters out the random (turbulent) and periodic (due to the blade passages) velocity components. Then the low frequency RMS velocity Ðv MI , can be calculated from the smoothed velocity pro®le as: s n 1X v MI v2 2 n i 1 MIi
As a result of the resampling procedure the original signal was transformed to an equally spaced sequence, with a change of mean and RMS velocities of less than 1%. The smoothed signal represents the variations of the original signal due to the MI. In order to quantify the distortion of the RMS velocity due to the MI, the value of the smoothed MI velocity was subtracted from the original resampled velocity. The RMS of the difference, v HF quanti®es the portion of the total RMS which is due to the random turbulent component and the blade passages. v uX n u 2 u v v MIi t 1 res i v HF 3 n Trans IChemE, Vol 78, Part A, October 2000
MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE 1047
It should be carefully noted that v res is not a simple sum of v MI and v HF . The percent error due to the long time scale MI can be de®ned as: v res v HF error ´ 100% 4 v HF This value was used to compare the various con®gurations. Another quantity which can provide information about the intensity of the MI relative to the total ¯uctuating velocity was de®ned as: v MI Intensity 5 v res It is useful to de®ne a relationship between the error and the intensity de®ned at equation (4) and equation (5) in terms of the relative size of the signals. This provides a reference comparison between the two quantities. If the resampled RMS velocity, v res is randomly correlated with v MI , then the ratio x can be de®ned as follows
x
n X
v
2 resi
v
2 MIi
1
i
X n
i
v res v MI
2
6
1
Rewriting equations (4) and (5) in terms of the ratio x, the error and intensity can be de®ned as r 1 Intensity 7 1 x r 1 x % error 1 8 x In Figure 3, equations (7) and (8) are plotted as a function of x. The error is always substantially less than the intensity. Expanding the scale of the graph and considering x between 1 and 2 (the region of our experimental results) shows an error of 30% for an intensity of up to 65%. A large MI component is needed to produce a signi®cant impact on the raw RMS velocity calculation.
Figure 3. Comparison between error and intensity of MI as a function of its size x.
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RESULTS Analysis of the results was done on several levels. Selected time series are presented to show the overall characteristics of the MI and the effect of tank geometry on the signal. These qualitative observations are then put into the context of the larger data set through analysis of the absolute size of the errors, the percent of time the MI exceeds 5% of the tip speed, and through factorial analysis of the geometric effects. The percent of time the MI exceeds 5% of the tip speed correlates well with the intensity, de®ned in the previous section: only the time percent is discussed in detail here. A combination of measures is needed to describe the MI characteristics: both the error and the time percent must be signi®cant in order to have a MI which truly dominates the signal.
Time Series Characteristics Before quantitative analysis of the errors due to the MI, some general observations can be made based on the resampled time series. This provides an overview of the dominant characteristics of the velocity signal, and the effect of impeller and tank geometry on these characteristics. It is already well known that the ¯ow pattern for the four impellers of interest varies signi®cantly. Differences in the time series between the two measurement locations are also expected. The ®rst observation is the extent to which the MI persists throughout the tank. In Figures 4a and b the times series for the PBT (D T/2, C/D 0.5 and Nf 4), at the impeller and close to the tank wall are shown. The slow variations due to the MI are evident in both locations. In this case, the MI appears as a regular frequency, so it is possible to verify that it retains its character throughout the tank. It must be emphasized that this is not always the case. The MI coherence strongly depends on the impeller and tank geometry. As expected, the turbulent (random) component is larger in the impeller stream than at the wall. The second observation relates to the effect of baf¯es on the MI. Figures 5a and b show the effect of reducing the number of baf¯es from 4 to 2 for the RT. Although the effect of baf¯ing was consistent for all impellers (less baf¯es leads to more MI), in most cases the effect is small. For the RT, the effect is dramatic. In the four baf¯e con®gurations, the dominant characteristic of the time series is the BPF (Figure 5a). Changing to two baf¯es leads to a signi®cant change in the observed time series (Figure 5b): the dominant effect in this geometry is the slow variation due to MI. The amplitude of the velocity ¯uctuations for the other two impellers, HE3 and A310, (Figures 6a and b) is smaller than that observed for the RT and the PBT. The time scale of the MI is much longer for these purely axial impellers and the MI can be very dif®cult to discern in the raw signal. It could be argued that the amplitude of variation is not suf®cient to warrant consideration of a separate MI component. For the scale of tank examined in this work, a time series record at least 10 s long is needed to ensure the capture of one full cycle of MI for these impellers.
ROUSSINOVA et al.
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Figure 4. PBT time series: (a) in the impeller stream and (b) close to the baf¯es showing the decay of turbulence and persistence of MI (D T/2, C/D 0.5, four baf¯es).
Quantitative Measures The results of the quantitative evaluation are presented in Tables 2a to h (more extensive data is given in Grigic29 ), and are discussed from three perspectives. First, the percent of time the MI exceeds 5% of VTIP (time percentage) is considered. This gives a measure of the amplitude and impact of the MI. Second, the error in RMS velocity for all impellers and geometric variations is examined. To simplify the error analysis, a threshold value of 12% is used as a reference point. Third, a factorial analysis is used to identify the dominant geometric variables affecting the MI. Top corner of the tank In the top corner of the tank the size of the MI, as measured by the time percentage, is small (<15%) with the exception of the large diameter PBT cases and all but one case for the RT. This is partly to be expected, since the mean velocity in the top of the tank is very small for all axial impellers, and larger for the RT; however, the size of the MI for the large diameter PBT is out of proportion to the mean velocity, which is almost zero. The errors calculated for the upper corner are shown in Figure 7, with numerical results in Tables 2b, d, f and h. The errors vary from 10% to 45%, and no clear trends emerge from the factorial analysis based on geometric variables. All but two of the 32 cases exceed the threshold value of
Figure 5. Time series for the RT (D T/3, C/D effect of baf¯ing (a) four baf¯es and (b) two baf¯es.
0.5) showing the
12%. In this region of the tank the mean velocity changes direction frequently, implying the dominance of the MI. This trend was previously observed for the PBT and HE3 impellers by Myers et al.16 , and by Chapple and Kresta15 . In the upper third of the tank, the turbulent ¯uctuations are small and the MI is a substantial part of the ¯uctuating velocity signal. The prevalence of a large impact of MI on the calculated RMS velocity requires a reconsideration of how the velocity ¯uctuations in the top of the tank are characterized. The one dimensional, long time record LDV measurements provide repeatable mean and RMS velocities, showing that in some sense the ¯ow is stationary in time. Time averaging, however, obscures the presence of the MI. The one-dimensional energy spectrum provides information about the average distribution of the energy in the turbulent cascade and the range of the scales of motion. If the energy cascade is in equilibrium, all the energy that enters the turbulent motion at large scales is dissipated at the same rate at the smallest scales of motion, where viscous dissipation is most effective. In a non-equilibrium cascade, energy is transferred and dissipates at all scales. When the mean velocity gradients are constantly changing, as they are in the presence of a dominant MI, the supply of energy to the turbulent cascade will change continually, and the equilibrium assumption is likely to introduce signi®cant errors in the analysis. In the present results, there is a large gap between the broad banded MI and the range of dissipative scales, indicating that the MI participates in Trans IChemE, Vol 78, Part A, October 2000
MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE 1049 numerical simulation and LES, large eddy simulation) are based on direct computation of the Navier-Stokes equations without time averaging, at least on the larger scales of motion. If these modelling approaches are not used, appropriate models for the ¯ow in the top of the tank must be selected. If the ¯ow is transitional or non-equilibrium, rather than a fully turbulent equilibrium cascade, established models of turbulence such as the k « model are not likely to perform well. Fluid viscosity may play an important role, complicating both accurate simulations and scale up. Impeller stream Figure 8 is a composite of the errors for all impeller stream experiments, with the threshold level of 12% error. From this, it is clear that the large diameter PBT and A310 runs show the most signi®cant impact of MI. This result agrees well with the time percentage measure for the PBT, A310, and HE3. For the RT, even though the error is small, the time percentage is very high in all cases. Factorial analysis of the major geometric variables con®rms the observations taken from the time series characteristics. Impeller stream: PBT The time percentage of signi®cant MI in the PBT impeller discharge stream is very large, and for the D T /2 impeller, it is always larger than 30%. The error due to MI ranges from 18±31% for the large impeller. A factorial analysis (Figure 9a) con®rms that the impeller diameter has the largest impact on the size of the MI for the PBT. 4 and (b) A310 Figure 6. (a) HE3 time series, D T/2, C/D 0.5, Nf time series, D 0.55T, C/D 0.5, Nf 4. Both have a much smaller amplitude than the PBT and RT.
an unsteady mean ¯ow, rather than entering directly into the turbulent cascade. The dominance of the MI in the top of the tank also raises questions about models to be used in CFD (computational ¯uid dynamic) simulations. Available algorithms which address the time varying nature of eddies (DNS, direct
Impeller stream: RT The RT impeller generates radially outward ¯ow emerging from the impeller tip. This high-speed radial jet entrains the surrounding ¯uid and slows down as it approaches the tank wall. Near the tank wall this stream divides and circulates through the rest of the tank, forming a pair of circulation loops. The vertical circulation for the RT decays as an annular wall jet when the tank is fully baf¯ed, while it decays as four separate three-dimensional jets for the axial impellers22 . Since the RT has an inherently different
50
PBT
45
A310 HE3
40
RT 35
30
25
20
15
10
5
0 1
2
3
4
5
6
7
Run number
Figure 7. Errors in RMS due to MI for the upper corner, U.
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PBT A310
30
HE3 RT 25
20
15
10
5
0 1
2
3
4
5
6
7
8
Run number
Figure 8. Errors in RMS velocity due to MI for the impeller stream, I.
¯ow pattern from the axial impellers, the experimental design was also different, emphasizing the symmetry of the circulation loops for the RT, vs. dimensionless off bottom clearance for the axial impellers.
The two baf¯e con®guration for the RT shows a strong MI, with time percentages ranging from 65±90%, while the four baf¯e con®guration shows time percentages ranging from 25±53%. Because the turbulence intensity is
Figure 9. Normal probability plots of the effects of the geometric variables and their interactions on the ¯ow in the impeller stream for the: (a) PBT (b) RT (c) HE3 and (d) A310.
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MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE 1051
very high for the RT, the error due to the MI is relatively small (close to 10% for all four two baf¯e con®gurations, and close to 2% for the four baf¯e con®gurations). The MI for the RT is affected most strongly by the number of baf¯es, Nf , as shown in Figure 9b. Four baf¯e con®gurations are more stable because the decay of the strong vertical circulation is directed and controlled due to the additional imposed drag. Unlike the PBT, the RT shows greater instabilities when a small (D T/3) impeller is used. This relationship between impeller diameter and instabilities was also observed for RTs in liquid and liquidair systems by McFarlane and Nienow33 . It is hypothesized that when a small impeller is used, the longer distance to impingement allows the radially discharged jet to develop instabilities which propagate into the bulk of the tank. Impeller stream: HE3 and A310 Although the HE3 and A310 ¯ow ®elds are fairly similar to that of the PBT, the MI for these two impellers are characterized by small amplitudes and very slow variations (longer than the 10 second measuring time). The time percentage never exceeds 5% for the HE3, and for the A310 most of the con®gurations show a signi®cant MI less than 15% of the time. Only one run exceeds the error threshold for the HE3. For the A310, the error reaches 12% in six of the eight con®gurations. The factorial analysis shown in Figures 9(c) and (d) suggests an interaction between the impeller diameter and the number of baf¯es for the HE3, but is inconclusive for the A310. This presents a picture somewhat different to that suggested by the composite error plot in Figure 7, and is accepted as the more accurate outcome. In any case, the MI for these impellers are much less violent than those observed for the large diameter PBT and the underbaf¯ed RT. Their timescale is so long and their amplitude so small that the ¯ow may be close to a stationary equilibrium condition. The size of the error is due more to the small size of the turbulent ¯uctuations than due to the intensity of the MI, as evidenced by the time percentage measure. CONCLUSIONS The importance of recognizing and accounting for macroinstabilities in the analysis of turbulence arises from two assumptions made for fully developed turbulent ¯ow. The ®rst is equilibrium of the turbulent energy cascade, wherein all of the energy which enters the turbulent motion at large scales is dissipated at the same rate at the smallest (viscous) scales of motion. In a non-equilibrium cascade, energy may dissipate at all scales. When the mean velocity gradients are constantly changing, as they are in the presence of a dominant MI, the supply of energy to the turbulent cascade will change continually, and the equilibrium assumption is not likely to hold. The second assumption of fully developed turbulence is stationary ¯ow, as measured on a time scale long relative to the turbulent ¯uctuations, but short relative to the process of interest (e.g. mixing). In the present measurements, there is a large gap between the broad banded low frequency of the MI (of the order of the tank turnover time) and the range of dissipative scales (bounded on the low end by the blade passage frequency). This indicates that the MI participates in an unsteady Trans IChemE, Vol 78, Part A, October 2000
mean ¯ow, rather than entering directly into the turbulent cascade. While the existence of macro-instabilities in the stirred tank ¯ow ®eld has been regularly reported since 1988, this work provides information on the geometric conditions which amplify these instabilities, and a measure of both the size of the MI and the impact of MI on RMS velocity measurements for a range of impellers and tank geometries. The conclusions can be summarized as follows: 1. Macroinstabilities in the stirred tank ¯ow ®eld must be taken into account for analysis of the velocity ¯uctuations. This is always the case at the top of the tank, and is important in the impeller stream for two classes of the geometries studied (large diameter PBT, D T /2; and underbaf¯ed RT, 2 baf¯es). This most directly affects (a) the experimental characterization of turbulence, and (b) selection of appropriate turbulence models for CFD simulations. 2. The amplitude of the MI is considered signi®cant when it exceeds 5% of the impeller tip speed. In cases where the presence of a MI in the impeller stream is evident from inspection of a 10 second time series, the amplitude is signi®cant at least 35% of the time. 3. MI are both low amplitude and slowly varying in the impeller stream of the HE3, the fully baf¯ed RT, and the D T/4 PBT. In these cases, errors resulting from an assumption of steady ¯ow are small. In the case of the A310, the size of v MI relative to v HF leads to a measurable change in RMS velocity on removal of the MI component, but other measures of the MI (time series characteristics, percent of time MI amplitude is signi®cant, and factorial analysis) suggest that it can be neglected. 4. If the MI are not properly accounted for, errors can reach 30% in the impeller stream (for a D T /2 PBT) and 40% at the top of the tank. The importance of the MI for process results will depend on the time scale of the process. In the case of slow reactions, the response is slow, so the time averaged result will contain all necessary details. If, however, the reaction (or process) has a time scale shorter than the lifetime of the largest eddies, then the process will be affected by the MI and some knowledge of these scales will be required. Feed jet intermittency, as studied by Houcine et al.20 , provides a good example of how meso-mixing can be affected by macro-instabilities. NOMENCLATURE C d D H M N N Nf r Re1 Qres Sn tn tn D tn T
impeller off bottom clearance, m interpolation density, d M/N impeller diameter, m liquid height, m number of points in the resampled signal number of points in the raw LDV signal impeller rotational speed, rpm or s 1 number of baf¯es radial position, m impeller Reynolds number ND2 /n resampled LDV sequence raw LDV Sequence particle arrival time interval between particle arrival times tank diameter, m
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1052 VÅi vi v BP v HF
v MI v MI v res v res v rand Vtip W
mean centered velocity, m s 1 instantaneous velocity, m s 1 instantaneous velocity due to the BPF, m s 1 RMS velocity which is due to the combined effect of random turbulent component and the blade passages, v HF v n uX u 2 v v t res MI i i 1 . n
instantaneous velocity due to MI (m s 1 ) s n 1X RMS velocity due to the MI, v MI v 2MI . ni 1
instantaneous velocity in the resampled signal, m s 1 RMS velocity of the resampled signal. instantaneous velocity due to the pure random or turbulent component, m s 1 tip speed of the impeller (pND), m s 1 blade width, m
Abbreviations A310 three bladed aerofoil impeller by Lightnin’ BPF blade passage frequency CFD computational ¯uid dynamics DNS direct numerical simulations DSA Doppler signal analyser impeller stream I FFT fast Fourier transform HE3 three bladed high ef®ciency turbine by Chemineer MI macroinstability LDV laser Doppler velocimeter LES large eddy simulation LIF laser induced ¯uorescence PBT four bladed, 458 pitched blade turbine PIV particle image velocimetery POD proper orthogonal decomposition RT 6 bladed Rushton turbine upper corner U
14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
24. 25. 26.
REFERENCES 1. Kresta, S. M., 1998, Turbulence in stirred tanks: anisotropic, approximate, and applied, Can J Chem Eng, 76: 563±576. 2. Wu, H. and Patterson, G. K., 1988, Laser-Doppler measurements of turbulent ¯ow parameters in a stirred mixer, Chem Eng Sci, 44: 2207±2221. 3. Rutherford, K., Lee, K. C., Mahmoudi, S. M. S. and Yianneskis, M., 1996, Hydrodynamic characteristics of dual Rushton impeller stirred vessel, AIChE J, 42: 332±346. 4. Stoots, C. M. and Calabrese, R. V., 1989, Flow in the impeller region of a stirred tank, Chem Eng Prog, 43. 5. Rao, M. A. and Brodkey, R. S., 1972, Continuous ¯ow stirred tank turbulence parameters in the impeller stream, Chem Eng Sci, 27: 137±156. 6. Cutter, L., 1966, Flow and turbulence in a stirred tank, AIChE J, 12: 35±45. 7. Winardi, S., Nakao, S. and Nagase, Y., 1988, Pattern recognition in ¯ow visualization around a paddle impeller, J Chem Eng Japan, 21: 503±508. 8. Winardi, S. and Nagase, Y., 1991, Unstable phenomenon of ¯ow in mixing vessel with a marine propeller, J Chem Eng Japan, 24: 243±249. 9. Bruha, O., Fort, I. and Smolka, P., 1993, Large scale unsteady phenomenon in a mixing vessel, Acta Polytechnica, Czech Tech Univ Prague, 27: 33. 10. Bruha, O., Fort, I. and Smolka, P., 1994, Flow transition phenomenon in an axially agitated system, Proc Eighth Europ Conf Mixing, IChemE Symp series No. 136, Cambridge, UK, pp 121±128. 11. Bruha, O., Fort, I. and Smolka, P., 1995, Phenomenon of turbulent macroinstabilities in agitated systems, Collect Czech Chem Commun, 60: 85±94. 12. Bruha, O., Fort, I., Smolka, P. and Jahoda, M., 1996, Experimental study of turbulent macroinstabilities in an agitated system with axial high-speed impeller and with radial baf¯es, Collect Czech Chem Commun, 61: 856±867. 13. Kresta, S. M. and Wood, P. E., 1993a, The ¯ow ®eld produced
27. 28. 29. 30. 31. 32. 33.
by a pitched blade turbine: Characterization of the turbulence and estimation of the dissipation rate, Chem Eng Sci, 48: 1761±1774. Kresta, S. M. and Wood, P. E., 1993b, The mean ¯ow ®eld produced by a 458 pitched blade turbine: changes in the circulation pattern due to off-bottom clearance, Can J Chem Eng, 71: 42±53. Chapple, D. and Kresta, S. M., 1994, The effect of geometry on the stability of ¯ow patterns in a stirred tank, Chem Eng Sci, 49(21): 3651±3660. Myers, K. J., Ward, R. W. and Bakker, A., 1997, A digital particle image velocimetry investigation of ¯ow ®eld instabilities of axial ¯ow impellers, J Fluids Eng, 119: 623±632. Montes, J. L., Boisson, H. C., Fort, I. and Jahoda, M., 1997, Velocity ®eld macro-instabilities in an axial agitated mixing vessel, Chem Eng J, 67: 139-145. Hasal, P., Montes, J. L., Boisson, H. C. and Fort, I., 2000, Macroinstabilities of velocity ®eld in stirred vessel: detection and analysis, Chem Eng Sci, 55: 391±401. Haam, S., Brodky, R. S. and Fasano, B., 1992, Local heat transfer in a mixing vessel using the heat ¯ux sensors, Ind Eng Chem Res, 31: 1384±1391. Houcine, I., Plasari, E., David, R. and Villermaux, J., 1999, Feedstream jet intermittency phenomena in continuous stirred tank reactor, Chem Eng J, 72: 19±30. Bittorf, K., 2000, The application of wall jets in stirred tanks with solids distribution, PhD Thesis, (University of Alberta, Edmonton, Canada). Bittorf, K. and Kresta, S., 1999, The prediction of the solids cloud height within a stirred tank, Mixing XVII, Banff, Canada, August. Bujalski, W., Takenaka, K., Paglianti, A., Takahashi, K., Poalini, S., Nienow, A. W. and Etchells, A. W., 1999, Suspension and liquid homogenization in high solids concentration stirred chemical reactors, Trans IChemE, Part A, 77: 241±247. Yianneskis, M., Popiolek, Z. and Whitelaw, J. H., 1987, An experimental study of the steady and unsteady ¯ow characteristics of stirred reactors, J Fluid Mech, 175: 537±555. Zhou, G. and Kresta, S. M., 1996, Impact of the tank geometry on the maximum turbulence energy dissipation rate for impellers, AIChE J, 42: 2476±2490. Box, G. P., Hunter, W. G. and Hunter, J. S., 1978, Statistics for Experimenters, (John Wiley and Sons, Toronto). Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T., 1994, Numerical Recipes, The Art of Scienti®c Computing, (Cambridge University Press, New York). Hockey, R. M. and Nouri, J. M., 1996, Turbulent ¯ow in a baf¯ed vessel stirred by a 60 degrees pitched blade impeller, Chem Eng Sci, 51: 4405±4421. Grigic, B., 1998, In¯uence of the impeller and tank geometry on low frequency phenomena and ¯ow stability, MSc Thesis, (University of Alberta, Canada). Brandstater, H. L. and Swinney, H. L., 1987, Strange attractor in weakly turbulent Couette Taylor ¯ow, Phys Rev A, 35: 2207±2220. Richardson, S., 1992, Chaotic Analysis of Two Impellers, (Internal Report, University of Alberta). McLaughlin, D. K. and Tiederman Jr, W. G., 1973, Biasing correction for individual realizations of Laser Anemometer measurements in turbulent ¯ow, Physics of Fluids, 16: 2082±2088. McFarlane, K. and Nienow, A. W., 1995, Studies of High Solidity Ratio Hydrofoil Impellers for Aerated Bioreactors, Biotechnol Prog, 11: 601±607.
ACKNOWLEDGEMENTS The authors wish to thank both NSERC and Lightnin for ®nancial support of this work.
ADDRESS Correspondence concerning this paper should be addressed to Professor S. Kresta, Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G, 2G6, Canada, (E-mail: [email protected]). This manuscript was communicated via our International Editor for the USA, Professor Richard Calabrese. It was received 21 January 2000 and accepted for publication after revision 28 July 2000.
Trans IChemE, Vol 78, Part A, October 2000
STUDY OF MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE V. T. ROUSSINOVA, B. GRGIC and S. M. KRESTA Department of Chemical & Materials Engineering, University of Alberta, Alberta, Canada
I
n this paper a quantitative measure of errors introduced in the turbulent velocity RMS signal due to the presence of macroinstabilities (MI) in the velocity ®eld is presented. The velocity time series were measured for four commonly used impellers (PBT, A310, HE3 and RT) with a one component LDV. Two locations in the tank, the impeller stream I and upper corner U, were studied. Three aspects of the geometry were varied: impeller diameter (D T/2 and D T /4); number of baf¯es (two and four); and off-bottom clearance (C/D 1.0 and C/D 0.5). By resampling and smoothing the velocity records the RMS velocity due to MI, v MI , was determined. Further velocity decomposition recovered the high frequency component of the signal, v HF , due to the random ¯uctuations and the blade passages. Inclusion of the non-stationary, non-equilibrium MI component in the calculation of the RMS velocity can result in an overestimation of up to 50%. Analysis of the time series records shows that the MI is present in all con®gurations tested. In some cases (PBT and RT) the MI dominates the signal while for others (HE3 and A310) the amplitude of the signal is low and the MI is much less pronounced. The MI is very sensitive to geometry: for the impeller stream of the RT, increasing the number of baf¯es from two to four completely changes the velocity time series. The MI can have as dramatic an impact on experiments and analysis of the ¯ow as the trailing vortices observed behind impeller blades. An understanding of this phenomenon is important for accurate analysis of the velocity and turbulence ®elds, for measurement of blend time and solids distribution, and for improving understanding of mesomixing. Keywords: stirred vessel; velocity ®eld; macroinstability; velocity decomposition; time series analysis
INTRODUCTION Traditionally, studies concerned with the turbulent ¯ow pattern in a stirred vessel are based on the mean (VÅi ) and RMS ¯uctuating velocity (vÅ i2 ). The latter is used to determine the different turbulence characteristics such as turbulence intensity, turbulent kinetic energy and estimates of the rate of dissipation of turbulent kinetic energy. In spite of the fact that there are numerous conceptual dif®culties with examining turbulence characteristics in a stirred tank (Kresta 1 ), the region close to the impeller has been studied in some detail (Wu and Patterson 2 , Rutherford et al.3 , Stoots and Calabrese4 , Rao and Brodkey5 , Cutter6 ). Signi®cant progress has been made in the understanding of this region of the tank. The mean centred velocity signal close to the impeller contains three signi®cant components: the random turbulent ¯uctuations, which have frequencies higher than the blade passage frequency and up to the Kolmogorov wavenumber; the blade passages, or trailing vortices; and a much slower variation on the mean, which is here termed macroinstability. This macroinstability indicates the existence of eddy ®elds whose time and length scale considerably exceeds those associated with blade passages and smallscale turbulence. This observation is not a new one: the existence of large-scale low-frequency phenomena in stirred
tanks has been regularly reported in the literature since 1988 (Winardi et al.7 , Winardi and Nagase8 , Bruha et al.9 12 , Kresta and Wood13,14 , Chapple and Kresta15 , Myers et al.16 , Montes et al.17 ). Winardi et al.7 were among the ®rst to report that the instantaneous circulation pattern for a paddle impeller is different from the mean velocity ®eld as measured using LDV. Quite different ¯ow patterns and various combinations of patterns appear over time. In a second paper, Winardi and Nagase8 , used several ¯ow visualization techniques (visual observation, high speed VTR, and video recording), and LDV to examine the time varying ¯ow pattern of a marine propeller. They found that the ¯ow pattern was asymmetric with respect to the impeller shaft. They identi®ed three circulation patterns and measured the lifetime distribution of these patterns using 2D ¯ow visualization. They found that ¯ow pattern changes were random in order and that the lifetime of a given pattern could range from half a second to several minutes. Kresta and Wood13,14 observed variations in the bulk circulation pattern when they analysed the ¯ow ®eld of a PBT impeller using ¯ow visualization, LDV, and spectral analysis. A later paper by Chapple and Kresta15 considers the in¯uence of geometric parameters such as off bottom clearance, impeller diameter and number of baf¯es on ¯ow stability for two axial impellers: the Lightnin’ A310
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MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE 1041
impeller and the pitched blade turbine (PBT). Chapple and Kresta15 also showed that the geometric variables which are most signi®cant for stability depend on the type of impeller. Bruha et al.9 investigated 3, 4 and 6 bladed, 458 axial impellers. They studied the effect of impeller rotational speed and the different geometries in which macroinstabilities (MI) occur. Their investigations covered two impeller diameters (D T /3 and T/4), and three different off-bottom clearances (C/T 0.33, 0.4 and 0.5). They observed a vortex which appears as a welling up of the ¯uid surface, again on a much longer time and length scale than that of the turbulent eddies. To capture the shape and dimensions of the large vortex they used a video camera. A round probe above the ¯uid surface was used to measure the frequency of this `surface swelling’. They concluded that this frequency is linearly dependent on the impeller rotational speed. The effect of geometry was not systematically examined, although several variations were studied. More recent work from this group (Bruha et al.10 12 ) uses ¯ow visualization and a special mechanical device called a `tornadometer’. The tornadometer is placed below the surface in the vicinity of the impeller and de¯ects whenever macro-instabilities are strong enough to change the direction of the mean ¯ow and de¯ect the target. The authors were able to con®rm their earlier report that the frequency of the macro-instabilities is linearly related to the impeller rotational speed. They also observed that the low frequency macro-instabilities are accompanied by changes in the angle of the impeller discharge ¯ow and the appearance of an unstable secondary circulation loop. This is in agreement with Kresta and Wood13 . Montes et al.17 used LDV measurements, ¯ow visualization, spectral analysis, and wavelet transforms to analyse ¯ow macro-instabilities in a stirred tank equipped with a six bladed 458 PBT impeller (D T /3, C/T 0.35, four baf¯es). They con®rm that the occurrence of macroinstabilities is accompanied by the presence of a large vortex in the upper part of the vessel, and it is linearly coupled with the frequency of the impeller revolution. The RMS velocity underwent a sudden rise in ¯uctuating intensity at Re 600. This rise might correspond to the ®rst appearance of the macro-instability. For ReI > 600, the frequency spectrum also shows a distinct peak for low frequency oscillations. This implies that both the transitional and fully turbulent ¯ow regimes are subject to this type of instability. Hasal et al.18 implemented a new type of analysis of MI. The ¯ow ®eld in a stirred vessel was treated as a pseudostationary high-dimensional dynamical system. Analysis of the velocity time series was done with the proper orthogonal decomposition (POD) technique. The proposed method was successfully applied at both low and moderate Reynolds numbers. The authors showed that the MI has no coherent frequency for the geometry investigated (a six bladed 458 PBT impeller, D T/3, C 0.35T, four baf¯es). The frequency of the MI could not be determined in zones close to the vessel wall, or at high ReI . In chemical processing, the velocity ®eld is used to accomplish process objectives such as heat transfer, mass transfer and chemical reaction. Reports in the literature provide evidence of the effect of the MI on the local heat transfer coef®cient (Haam and Brodkey19 ), on intermittency Trans IChemE, Vol 78, Part A, October 2000
phenomena in the feedstream (Houcine et al.20 ) and on solids distribution at low solids concentrations (Bittorf 21 ). Haam and Brodkey19 measured the local heat transfer ¯ux and temperature of the vessel wall. Time variations in the local heat transfer coef®cient were observed. The authors hypothesized that the low frequency ¯uctuations in the signal were due to the precession of an axial vortical structure, which rotated between the surface and the impeller. Feedstream jet intermittency was studied by Houcine et al.20 using laser induced ¯uorescence (LIF). The feedstream jet was injected at the baf¯e in the upper part of the tank. The authors observed three states of the feedstream, which were characterized as no intermittency (the feed stream is always vertical), slight intermittency (chaotic movement of the feedstream jet axis in the plane of the laser sheet) and effective intermittency with a period of 1-2 s. This phenomenon can be linked to the existence of MI. Bittorf and Kresta22 provide another example of how MI can affect process results. At low solids concentrations, the MI is strong and the solids are distributed throughout the vessel. As the solids concentration is increased, the MI is damped out before it reaches the top of the vessel and a clear ¯uid layer appears. Bujalski et al.23 showed that this clear ¯uid layer can exhibit mixing times many times longer than the well mixed zone below the interface. Finally, large scale macroinstabilities can have considerable dynamic effects on the solid surfaces inside the vessel. When large scale eddies, sometimes as large as the vessel diameter, interact with components of the system (baf¯es, shaft, impeller or vessel wall) they cause dramatic force ¯uctuations, leading to (for example) abrupt changes in the torque on the shaft. In some cases, this can result in mechanical failure. In summary, the MI is an important component of the large-scale motions in a stirred tank. It has implications for many of the operations which are carried out in stirred tanks, and for the structural integrity of the vessel components. The MI produces a broad, low frequency band in the power spectrum. It does not necessarily show a coherent frequency, although this may be observed for some cases. The time scale of the MI is of the order of the tank turnover time. No frequencies are observed in the interval between the MI and the blade passage frequency, so these motions are clearly associated with the mean velocity as determined over a short time interval, rather than with the turbulent ¯uctuations. While these large scale phenomena will eventually decay to turbulent ¯uctuations (probably in the top third of the tank), they are a non-stationary, non-equilibrium component of the ¯ow which should not be included in the turbulent rms velocity. In this paper, a quantitative measure of the MI is provided for two regions of the stirred tank: the impeller region and the top corner of the tank. Four commonly used impellers (PBT, RT, HE3 and A310) are discussed. The experiments were performed using a one component LDV. For each impeller eight geometric con®gurations were used in order to investigate: (a) the impact of the tank and impeller geometry on the MI; (b) the portion of the measured RMS velocity which is due to the MI. This is analogous to isolating the effect of the blade
ROUSSINOVA et al.
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passages (Stoots and Calabrese4 , Wu and Patterson 2 , Yianneskis et al.24 , Rutherford et al.3 ) for the Rushton turbine. EXPERIMENTAL LDV Experimental Set Up The forward scattering, one dimensional laser Doppler velocimeter (LDV) described in Zhou and Kresta25 was used for this study. Two beams from an Argon Ion Laser with a beam separation of 0.01691 m were focused to form a measuring volume 168 mm in diameter with fringe spacing of 9.1 mm. The tank was moved horizontally using computer-controlled traverses with an accuracy of 6 0.5 mm and in the vertical direction using a manual traverse (accuracy of 6 1 mm). An Aerometrics Doppler signal analyser (DSA) was used to convert the analogue signal from the receiving optics into Doppler frequencies and instantaneous velocities. The DSA operates in the frequency domain burst detection mode. The analogue signal was sampled at a 2.5 MHz sampling frequency. The signal sampling frequency is different from the frequency of the velocity measurements, which is ®xed by the particle arrival rate, or seeding density. The particle arrival rate is affected by the velocity of the ¯uid, so the velocity data rate is much lower at the top of the tank than it is in the impeller stream. The DSA allows the user to set either the sample size (number of velocity measurements) or the sample time (length of record in seconds). Experimental results showed that both the sample size and the sampling time are critical to accurate velocity measurement. If the sample time is too short, the reproducibility is poor even with a sample size larger than 10000 data points. For all the measurements the sample size was set equal to 10000. The length of the record varies from 8 to 12 seconds. This is long enough to obtain a meaningful average of all components of the signal. Geometry of the Tank and Impellers The stirred vessel (STR) used for this work is shown in Figure 1. The tank diameter T 0.24 m is equal to the liquid height (H T ). To prevent air entrainment and surface vortexing, a lid was placed on the top of the tank and covered with 5 cm of water to form a seal. Either two or four vertical rectangular baf¯es of width W T /10 were spaced at equal intervals around the periphery of the tank ¯ush with the wall. A tank with four baf¯es is considered fully baf¯ed; the two-baf¯e con®guration is used to test the sensitivity of instabilities to the number of baf¯es. A large diameter (9 mm) was used in order to eliminate shaft run-out (or wobbling). The shaft was machined down to ®t into smaller hubs ID 12.75 mm. The experimental variables were combined in a factorial design (Box et al.26 ) to give 8 tank geometries for each impeller as de®ned in Table 1. This experimental design was followed, with minor variations, for all four impellers studied, with results reported in Tables 2(a)±(h). The three axial impellers (A310, HE3, PBT) were placed at dimensionless off bottom clearances of C/D 0.5 and 1. The impeller diameters for the HE3 and PBT were D T/2 and T/4; for the A310, the diameters were D 0.55T and 0.35T. The A310 was supplied by Lightnin’ Inc; the
Figure 1. Schematic of the measurement locations for the PBT and HE3: IÐimpeller stream; UÐupper corner.
HE3 was supplied by Chemineer. One radial impeller, the Rushton turbine (RT: D T/2 and T /3; disc diameter 2/3D; blade height D/5; blade width D/4; blade and disc thickness 0.002 m) was placed at off bottom clearances of C T /2 and T/3. Different off bottom clearances were used for the radial impeller due to the inherently different ¯ow pattern produced by this impeller. For the radial impeller, the off bottom clearance determines the symmetry or asymmetry of the two circulation loops. For the axial impeller, the dimensionless off-bottom clearance allows for the effect of C and D on changes in the ¯ow pattern14 . For the PBT, RT, and HE3, the tip speed (Vtip ) was the same for both impeller diameters, and for all three impellers. For the A310, the tip speeds were slightly different. In order to compare the different impellers directly, all reported velocities are normalized with Vtip . Since the ¯ow is in the fully turbulent regime, the velocity ®eld can be expected to scale with the characteristic velocity, or the tip speed.
Table 1. Geometrical con®gurations used in the experiments *. Run 1 3 3 4 5 6 7 8
D
C/D
Nf
T/2 T/2 T/2 T/2 T/4 T/4 T/4 T/4
1 0.5 1 0.5 1 0.5 1 0.5
2 2 4 4 2 2 4 4
* For the RT, D T/4 is replaced by D T/3, and C/D 0.5 by C T/3 for the A310, D 0.55T and 0.35T.
Trans IChemE, Vol 78, Part A, October 2000
MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE 1043 Table 2(a). Results for the PBT impeller stream.
Run 1 2 3 4 5 6 7 8
N (rpm)
VÅres Vtip
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
200 200 200 200 400 400 400 400
0.392 0.412 0.360 0.456 0.348 0.337 0.40 0.332
0.158 0.139 0.135 0.122 0.084 0.178 0.086 0.084
0.090 0.071 0.072 0054 0.031 0.079 0.026 0.032
0.121 0.109 0.107 0.103 0.077 0.158 0.080 0.074
56.82 46.74 51.13 33.07 11.85 48.85 5.60 11.62
31.01 26.69 25.32 17.93 9.23 12.76 6.85 11.82
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
34.9 16.03 25.68 14.74 4.71 0.76 0.56 0.52
40.94 33.55 20.34 21.33 15.40 25.69 21.43 13.39
Table 2(b). Results for the PBT upper corner stream.
Run 1 2 3 4 5 6 7 8
N (rpm) 200 200 200 200 400 400 400 400
VÅres Vtip 0.021 0.057 0.002 0.012 0.011 0.025 0.007 0.010
v res Vtip
v MI Vtip
v HF Vtip
0.083 0.065 0.085 0.070 0.054 0.033 0.033 0.034
0.053 0.033 0.042 0.033 0.025 0.018 0.018 0.021
0.059 0.049 0.071 0.058 0.047 0.026 0.027 0.030
Table 2(c). Results for the HE3 impeller stream.
Run 1 2 3 4 5 6 7 8
N (rpm)
VÅres Vtip
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
200 200 200 200 400 400 400 400
0.236 0.236 0.226 0.206 0.255 0.194 0.224 0.194
0.062 0.058 0.060 0.054 0.049 0.045 0.048 0.047
0.026 0.025 0.021 0.016 0.010 0.013 0.019 0.016
0.055 0.052 0.053 0.052 0.047 0.043 0.042 0.043
2.41 5.15 2.69 0.0 0.0 0.0 0.45 0.0
12.52 12.48 9.96 4.47 4.82 5.24 14.78 8.77
Table 2(d). Results for the HE3 upper corner.
Run 1 2 3 4 5 6 7 8
N (rpm)
VÅres Vtip
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
200 200 200 200 400 400 400 400
0.002 0.012 0.006 0.008 0.004 0.008 0.013 0.018
0.044 0.033 0.036 0.040 0.034 0.033 0.033 0.029
0.023 0.013 0.017 0.020 0.016 0.014 0.014 0.010
0.036 0.030 0.030 0.033 0.030 0.029 0.029 0.026
2.33 0.36 0.6 0.0 0.62 0.0 0.0 0.03
22.19 8.85 18.9 20.31 13.67 11.84 14.17 9.99
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ROUSSINOVA et al.
1044
Table 2(e). Results for the A310 impeller stream.
Run 1 2 3 4 5 6 7 8
N (rpm)
VÅres Vtip
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
150 150 150 150 400 400 400 400
0.199 0.227 0.275 0.259 0.281 0.221 0.269 0.161
0.088 0.138 0.079 0.086 0.060 0.065 0.057 0.096
0.051 0.060 0.032 0.036 0.018 0.025 0.018 0.032
0.068 0.118 0.070 0.076 0.055 0.058 0.054 0.085
33.54 37.49 13.02 15.51 0.0 4.38 0.04 11.55
29.11 16.9 11.85 14.48 9.37 11.72 5.7 12.81
Table 2( f ). Results for the A310 upper corner.
Run 1 2 3 4 5 6 7 8
N (rpm) 150 150 150 150 400 400 400 400
VÅres Vtip 0.013 0.003 0.028 0.030 0.011 0.003 0.041 0.044
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
0.047 0.045 0.052 0.056 0.037 0.032 0.039 0.041
0.024 0.018 0.030 0.033 0.018 0.019 0.018 0.023
0.039 0.040 0.042 0.042 0.031 0.026 0.034 0.031
5.41 0.6 14.33 9.23 2.01 1.08 4.23 4.27
23.18 13.33 24.33 32.92 20.86 18.96 16.15 30.27
Table 2(g). Results for the RT impeller stream.
Run 1 2 3 4 5 6 7 8
N (rpm)
VÅres Vtip
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
200 200 200 200 300 300 300 300
0.455 0.568 0.712 0.829 0.596 0.293 0.756 0.682
0.346 0.395 0.327 0.385 0.390 0.415 0.355 0.339
0.107 0.139 0.044 0.051 0.152 0.235 0.077 0.053
0.319 0.354 0.321 0.379 0.355 0.364 0.343 0.335
65.38 68.48 25.56 34.51 73.92 90.32 53.02 32.82
8.45 11.49 1.84 1.59 9.86 14.06 3.61 1.2
Table 2(h). Results for the RT upper corner.
Run 1 2 3 4 5 6 7 8
N (rpm)
VÅres Vtip
v res Vtip
v MI Vtip
v HF Vtip
% of time v MI exceeds 0.05Vtip
% error vres v HF v HF
200 200 200 200 300 300 300 300
0.172 0.154 0.088 0.172 0.071 0.038 0.024 0.019
0.153 0.146 0.149 0.133 0.096 0.066 0.083 0.043
0.065 0.073 0.053 0.051 0.064 0.033 0.041 0.021
0.129 0.115 0.133 0.119 0.067 0.053 0.068 0.035
45.02 46.60 33.93 34.22 51.50 15.24 23.32 2.31
18.85 26.94 11.85 12.30 43.51 23.44 22.95 19.13
Trans IChemE, Vol 78, Part A, October 2000
MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE 1045 Two locations in the stirred tank were chosen for detailed investigation: the impeller stream and the top corner of the tank. These locations were chosen based on tuft visualization work reported by Chapple and Kresta15 . The impeller stream velocity signal can capture all frequencies present in the ¯ow. It contains the blade passages, the high frequency turbulent components, and the macroinstabilities of interest for this work. The second probe location is in the upper corner of the tank. This region is considered highly unstable, since the velocities are low and the ¯ow changes direction frequently from radially inward to radially outward. The exact measurement locations for the PBT and HE3 are shown in Figure 1. The radial component of velocity was measured in the impeller stream for the RT. In all other cases, the axial velocity was measured, since it gives the clearest indication of the ¯ow pattern and it is the primary velocity component for axial impellers. For the axial impellers, the impeller stream measurement point was located 3 mm below the impeller blade at a radial distance of 2r/D 0.8 for the PBT and HE3 and 2r/D 0.5 for the A310. For the RT, the impeller stream measurements were taken 3 mm away from the tip of the blade (r D/2 3 mm) and at the centreline of the impeller. For the upper corner, measurements were taken at a radial position 3 mm away from the edge of the baf¯e (r 0.9T 3 mm), and 5 mm down from the top surface. For each of the 32 geometric con®gurations and each measurement location, ®ve repeat measurements were done. The averaged mean and RMS velocities are reported in Tables 2a±h. DATA PROCESSING STEPS Time Series Analysis
Figure 2. Effect of the data processing steps on the signal. (a) raw signal, 10,000 points, (b) resampled signal, 5000 points, (c) smoothed signal, 5000 points.
A typical time series of the axial velocity signal, measured in the impeller stream for the PBT, is shown in Figure 2a. The velocity record shows the turbulent ¯uctuations, the ¯uctuations due to the individual blade passages, and the ¯uctuations due to the MI. In this time series, the high intensity turbulent ¯uctuations tend to obscure the longer time scale ¯uctuations. The objective of this work was to separate and analyse the macroinstability, or low frequency, component of the signal. Several possible methods are available for this signal processing task. The ®rst class of these methods operates in the frequency domain using various forms of notch ®ltering (Press et al.27 ). The autocorrelation function has been used in combination with a notch ®lter (Rao and Brodkey5 , Wu and Patterson 2 , Hockey and Nouri28 ) to characterize the pure turbulent component in the measured velocity. The main objective in all of these studies was to eliminate the blade passage frequency from the turbulent analysis. From the signal processing perspective, a notch ®lter can be used only if a single frequency is to be removed from the signal. Kresta and Wood13 applied this method to remove the combined effect of blade passages and a coherent low frequency from the integral of the autocorrelation function. In this work, the MI is not a ®xed single frequency (i.e. it is not necessarily coherent). In this case, if a notch ®lter is used, the width of the window selected and the sharpness of the ®lter function signi®cantly affect the results of the analysis. Similar dif®culties arise with low pass ®ltering. This topic is discussed in detail by Grgic29 .
An alternative approach uses short time averages in the time domain to smooth the signal and isolate the nonstationary component of the velocity. The premise of data smoothing is that one is measuring a variable that is both slowly varying and is also corrupted by random noise. Replacement of each data point with an appropriate local average of the surrounding points will discard the noise (in this case, the random turbulent component) without signi®cantly biasing the mean. Smoothing is a form of low pass ®ltering, which is applied in the time domain, not the frequency domain. In this case, the adjustable parameters are the length of the smoothing window, and the number of repetitions of the smoothing process. Press et al.27 discuss the correct application of smoothing techniques in some detail. Other forms of signal analysis have been applied to velocity signals from stirred tanks with varying degrees of success. The correlation dimension (Brandstater and Swinney30 ) is dominated by the blade passage frequency and is unable to distinguish between noise and the MI (Richardson31 ). Wavelets can be used to separate various frequency ranges17 , but the adjustable parameters in this technique are both more numerous and less easily interpreted than those used in the smoothing technique. The end result is equivalent to application of a low pass ®lter. Proper orthogonal decomposition18 allows reconstruction of the MI after several stages of signal processing. While the
Trans IChemE, Vol 78, Part A, October 2000
ROUSSINOVA et al.
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Table 3. Number of blade passages used for smoothing window. Impeller type D impeller diameter Blade passages
PBT D 4
T/4 4
HE3 D 2
T/2 4
D 4
T/4 6
authors show accurate decomposition of the signal, the method is signi®cantly more complex than that selected for this work. In the Fourier domain burst detection mode of LDV signal processing, velocity data is recorded every time a particle traverses the measurement volume. The resulting LDV time series is represented by Sn , where n 1 . . . N. The corresponding arrival times are tn , and the intervals between particle arrival times (D tn tn 1 tn ) are approximately Poisson distributed. The unevenly spaced data precludes the use of the FFT algorithm, and makes the computation of the power spectrum non-trivial. Sampling based on particle arrival times also allows the possibility of velocity biasing (McLaughlin and Tiederman32 ). This is a bias due to the tendency to collect more data when the velocity is high than when it is low. On the other hand, the random sampling times can be used to eliminate errors in the spectrum due to a biasing of high frequencies onto lower ones. In this study the main objective was to isolate the component of the signal due to MI, so calculation of the power spectrum was of secondary importance. Corrections for possible velocity biasing were implemented through signal resampling. The raw LDV signal was interpolated using zero-order interpolation, also known as `sample and hold’ in the LDV literature. In sample and hold interpolation, the value Sn measured at time tn is retained until the next particle is observed at tn 1 , at which point the record jumps to the new value Sn 1 . When the signal is resampled to simplify the signal analysis and to remove any velocity biasing, a ®xed sampling rate is set with equal time intervals tres . The new time series Qres is constructed from the zero-order interpolated signal. For the new time m tres , m 1 . . . M, one ®nds a data point Sn such that tn # m tres < tn 1 and the velocity at m tres , Q m tres is set equal to S tn . In this way a new data set is produced Q tm , m 1 . . . M de®ned at M regularly spaced new times tm . A resampling window equal to the mean data rate was applied. An interpolation density d M/N is further de®ned, which must be less than or equal to 1. This value was used to adjust the width of the sampling window between two measurement locations. In the impeller stream, the turbulence intensities are the highest, and the data rate is the highest, while in the vessel corners lower velocities give a lower data density. Mean sampling frequencies varied from several hundred hertz (600 Hz) up to 2 kHz. Once the raw signal is resampled it is smoothed using a moving average. This is equivalent to applying a low-pass ®lter at the time scale of the averaging window. The averaging time, or the smoothing window, was based on a number of blade passages, as given in Table 3. The ®rst round of smoothing proved insuf®cient for isolation of the MI: higher frequency ¯uctuations were still apparent in the signal. These could have been removed in one step by
A310 D 4
T/2
0.35T
D
6
2
2
RT 0.55T
D 2
2
D 6
T/3
D
3
4
T/2 4
using a longer averaging window, but at this point the smoothed signal cuts off the peaks in the MI ¯uctuations. In order to retain the peaks while removing the high frequencies, a second round of smoothing was used, typically with the same averaging window as the ®rst smoothing. The procedure was considered complete when the smoothed signal overlapped the slow variation in the initial velocity signal, and no observable higher ¯uctuations remained. The effect of each step of data processing on the signal is shown in Figures 2(b and c). The shape of the smoothed signal is virtually identical to that of the raw signal in terms of the slow variation on the mean. Some slow variations on the mean were observed for all geometries and impellers examined. Velocity Decomposition and Error De®nition The concept of velocity decomposition is useful to determine how much turbulence a given impeller can generate, and the relative contribution of the regular blade passages and large scale MI ¯uctuations to the total ¯uctuations. The instantaneous mean centred velocity, v res can be de®ned as follows: v res
v BP
v MI
v rand
1
The subscripts BP and MI denote the blade passage and low frequency components, and rand denotes the pure random or turbulent component. The v res is the mean-centred RMS velocity component after resampling. With selection of a suitable window length relative to the blade passage frequency, data smoothing ®lters out the random (turbulent) and periodic (due to the blade passages) velocity components. Then the low frequency RMS velocity Ðv MI , can be calculated from the smoothed velocity pro®le as: s n 1X v MI v2 2 n i 1 MIi
As a result of the resampling procedure the original signal was transformed to an equally spaced sequence, with a change of mean and RMS velocities of less than 1%. The smoothed signal represents the variations of the original signal due to the MI. In order to quantify the distortion of the RMS velocity due to the MI, the value of the smoothed MI velocity was subtracted from the original resampled velocity. The RMS of the difference, v HF quanti®es the portion of the total RMS which is due to the random turbulent component and the blade passages. v uX n u 2 u v v MIi t 1 res i v HF 3 n Trans IChemE, Vol 78, Part A, October 2000
MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE 1047
It should be carefully noted that v res is not a simple sum of v MI and v HF . The percent error due to the long time scale MI can be de®ned as: v res v HF error ´ 100% 4 v HF This value was used to compare the various con®gurations. Another quantity which can provide information about the intensity of the MI relative to the total ¯uctuating velocity was de®ned as: v MI Intensity 5 v res It is useful to de®ne a relationship between the error and the intensity de®ned at equation (4) and equation (5) in terms of the relative size of the signals. This provides a reference comparison between the two quantities. If the resampled RMS velocity, v res is randomly correlated with v MI , then the ratio x can be de®ned as follows
x
n X
v
2 resi
v
2 MIi
1
i
X n
i
v res v MI
2
6
1
Rewriting equations (4) and (5) in terms of the ratio x, the error and intensity can be de®ned as r 1 Intensity 7 1 x r 1 x % error 1 8 x In Figure 3, equations (7) and (8) are plotted as a function of x. The error is always substantially less than the intensity. Expanding the scale of the graph and considering x between 1 and 2 (the region of our experimental results) shows an error of 30% for an intensity of up to 65%. A large MI component is needed to produce a signi®cant impact on the raw RMS velocity calculation.
Figure 3. Comparison between error and intensity of MI as a function of its size x.
Trans IChemE, Vol 78, Part A, October 2000
RESULTS Analysis of the results was done on several levels. Selected time series are presented to show the overall characteristics of the MI and the effect of tank geometry on the signal. These qualitative observations are then put into the context of the larger data set through analysis of the absolute size of the errors, the percent of time the MI exceeds 5% of the tip speed, and through factorial analysis of the geometric effects. The percent of time the MI exceeds 5% of the tip speed correlates well with the intensity, de®ned in the previous section: only the time percent is discussed in detail here. A combination of measures is needed to describe the MI characteristics: both the error and the time percent must be signi®cant in order to have a MI which truly dominates the signal.
Time Series Characteristics Before quantitative analysis of the errors due to the MI, some general observations can be made based on the resampled time series. This provides an overview of the dominant characteristics of the velocity signal, and the effect of impeller and tank geometry on these characteristics. It is already well known that the ¯ow pattern for the four impellers of interest varies signi®cantly. Differences in the time series between the two measurement locations are also expected. The ®rst observation is the extent to which the MI persists throughout the tank. In Figures 4a and b the times series for the PBT (D T/2, C/D 0.5 and Nf 4), at the impeller and close to the tank wall are shown. The slow variations due to the MI are evident in both locations. In this case, the MI appears as a regular frequency, so it is possible to verify that it retains its character throughout the tank. It must be emphasized that this is not always the case. The MI coherence strongly depends on the impeller and tank geometry. As expected, the turbulent (random) component is larger in the impeller stream than at the wall. The second observation relates to the effect of baf¯es on the MI. Figures 5a and b show the effect of reducing the number of baf¯es from 4 to 2 for the RT. Although the effect of baf¯ing was consistent for all impellers (less baf¯es leads to more MI), in most cases the effect is small. For the RT, the effect is dramatic. In the four baf¯e con®gurations, the dominant characteristic of the time series is the BPF (Figure 5a). Changing to two baf¯es leads to a signi®cant change in the observed time series (Figure 5b): the dominant effect in this geometry is the slow variation due to MI. The amplitude of the velocity ¯uctuations for the other two impellers, HE3 and A310, (Figures 6a and b) is smaller than that observed for the RT and the PBT. The time scale of the MI is much longer for these purely axial impellers and the MI can be very dif®cult to discern in the raw signal. It could be argued that the amplitude of variation is not suf®cient to warrant consideration of a separate MI component. For the scale of tank examined in this work, a time series record at least 10 s long is needed to ensure the capture of one full cycle of MI for these impellers.
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Figure 4. PBT time series: (a) in the impeller stream and (b) close to the baf¯es showing the decay of turbulence and persistence of MI (D T/2, C/D 0.5, four baf¯es).
Quantitative Measures The results of the quantitative evaluation are presented in Tables 2a to h (more extensive data is given in Grigic29 ), and are discussed from three perspectives. First, the percent of time the MI exceeds 5% of VTIP (time percentage) is considered. This gives a measure of the amplitude and impact of the MI. Second, the error in RMS velocity for all impellers and geometric variations is examined. To simplify the error analysis, a threshold value of 12% is used as a reference point. Third, a factorial analysis is used to identify the dominant geometric variables affecting the MI. Top corner of the tank In the top corner of the tank the size of the MI, as measured by the time percentage, is small (<15%) with the exception of the large diameter PBT cases and all but one case for the RT. This is partly to be expected, since the mean velocity in the top of the tank is very small for all axial impellers, and larger for the RT; however, the size of the MI for the large diameter PBT is out of proportion to the mean velocity, which is almost zero. The errors calculated for the upper corner are shown in Figure 7, with numerical results in Tables 2b, d, f and h. The errors vary from 10% to 45%, and no clear trends emerge from the factorial analysis based on geometric variables. All but two of the 32 cases exceed the threshold value of
Figure 5. Time series for the RT (D T/3, C/D effect of baf¯ing (a) four baf¯es and (b) two baf¯es.
0.5) showing the
12%. In this region of the tank the mean velocity changes direction frequently, implying the dominance of the MI. This trend was previously observed for the PBT and HE3 impellers by Myers et al.16 , and by Chapple and Kresta15 . In the upper third of the tank, the turbulent ¯uctuations are small and the MI is a substantial part of the ¯uctuating velocity signal. The prevalence of a large impact of MI on the calculated RMS velocity requires a reconsideration of how the velocity ¯uctuations in the top of the tank are characterized. The one dimensional, long time record LDV measurements provide repeatable mean and RMS velocities, showing that in some sense the ¯ow is stationary in time. Time averaging, however, obscures the presence of the MI. The one-dimensional energy spectrum provides information about the average distribution of the energy in the turbulent cascade and the range of the scales of motion. If the energy cascade is in equilibrium, all the energy that enters the turbulent motion at large scales is dissipated at the same rate at the smallest scales of motion, where viscous dissipation is most effective. In a non-equilibrium cascade, energy is transferred and dissipates at all scales. When the mean velocity gradients are constantly changing, as they are in the presence of a dominant MI, the supply of energy to the turbulent cascade will change continually, and the equilibrium assumption is likely to introduce signi®cant errors in the analysis. In the present results, there is a large gap between the broad banded MI and the range of dissipative scales, indicating that the MI participates in Trans IChemE, Vol 78, Part A, October 2000
MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE 1049 numerical simulation and LES, large eddy simulation) are based on direct computation of the Navier-Stokes equations without time averaging, at least on the larger scales of motion. If these modelling approaches are not used, appropriate models for the ¯ow in the top of the tank must be selected. If the ¯ow is transitional or non-equilibrium, rather than a fully turbulent equilibrium cascade, established models of turbulence such as the k « model are not likely to perform well. Fluid viscosity may play an important role, complicating both accurate simulations and scale up. Impeller stream Figure 8 is a composite of the errors for all impeller stream experiments, with the threshold level of 12% error. From this, it is clear that the large diameter PBT and A310 runs show the most signi®cant impact of MI. This result agrees well with the time percentage measure for the PBT, A310, and HE3. For the RT, even though the error is small, the time percentage is very high in all cases. Factorial analysis of the major geometric variables con®rms the observations taken from the time series characteristics. Impeller stream: PBT The time percentage of signi®cant MI in the PBT impeller discharge stream is very large, and for the D T /2 impeller, it is always larger than 30%. The error due to MI ranges from 18±31% for the large impeller. A factorial analysis (Figure 9a) con®rms that the impeller diameter has the largest impact on the size of the MI for the PBT. 4 and (b) A310 Figure 6. (a) HE3 time series, D T/2, C/D 0.5, Nf time series, D 0.55T, C/D 0.5, Nf 4. Both have a much smaller amplitude than the PBT and RT.
an unsteady mean ¯ow, rather than entering directly into the turbulent cascade. The dominance of the MI in the top of the tank also raises questions about models to be used in CFD (computational ¯uid dynamic) simulations. Available algorithms which address the time varying nature of eddies (DNS, direct
Impeller stream: RT The RT impeller generates radially outward ¯ow emerging from the impeller tip. This high-speed radial jet entrains the surrounding ¯uid and slows down as it approaches the tank wall. Near the tank wall this stream divides and circulates through the rest of the tank, forming a pair of circulation loops. The vertical circulation for the RT decays as an annular wall jet when the tank is fully baf¯ed, while it decays as four separate three-dimensional jets for the axial impellers22 . Since the RT has an inherently different
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Figure 7. Errors in RMS due to MI for the upper corner, U.
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Figure 8. Errors in RMS velocity due to MI for the impeller stream, I.
¯ow pattern from the axial impellers, the experimental design was also different, emphasizing the symmetry of the circulation loops for the RT, vs. dimensionless off bottom clearance for the axial impellers.
The two baf¯e con®guration for the RT shows a strong MI, with time percentages ranging from 65±90%, while the four baf¯e con®guration shows time percentages ranging from 25±53%. Because the turbulence intensity is
Figure 9. Normal probability plots of the effects of the geometric variables and their interactions on the ¯ow in the impeller stream for the: (a) PBT (b) RT (c) HE3 and (d) A310.
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MACRO-INSTABILITIES IN STIRRED TANKS USING A VELOCITY DECOMPOSITION TECHNIQUE 1051
very high for the RT, the error due to the MI is relatively small (close to 10% for all four two baf¯e con®gurations, and close to 2% for the four baf¯e con®gurations). The MI for the RT is affected most strongly by the number of baf¯es, Nf , as shown in Figure 9b. Four baf¯e con®gurations are more stable because the decay of the strong vertical circulation is directed and controlled due to the additional imposed drag. Unlike the PBT, the RT shows greater instabilities when a small (D T/3) impeller is used. This relationship between impeller diameter and instabilities was also observed for RTs in liquid and liquidair systems by McFarlane and Nienow33 . It is hypothesized that when a small impeller is used, the longer distance to impingement allows the radially discharged jet to develop instabilities which propagate into the bulk of the tank. Impeller stream: HE3 and A310 Although the HE3 and A310 ¯ow ®elds are fairly similar to that of the PBT, the MI for these two impellers are characterized by small amplitudes and very slow variations (longer than the 10 second measuring time). The time percentage never exceeds 5% for the HE3, and for the A310 most of the con®gurations show a signi®cant MI less than 15% of the time. Only one run exceeds the error threshold for the HE3. For the A310, the error reaches 12% in six of the eight con®gurations. The factorial analysis shown in Figures 9(c) and (d) suggests an interaction between the impeller diameter and the number of baf¯es for the HE3, but is inconclusive for the A310. This presents a picture somewhat different to that suggested by the composite error plot in Figure 7, and is accepted as the more accurate outcome. In any case, the MI for these impellers are much less violent than those observed for the large diameter PBT and the underbaf¯ed RT. Their timescale is so long and their amplitude so small that the ¯ow may be close to a stationary equilibrium condition. The size of the error is due more to the small size of the turbulent ¯uctuations than due to the intensity of the MI, as evidenced by the time percentage measure. CONCLUSIONS The importance of recognizing and accounting for macroinstabilities in the analysis of turbulence arises from two assumptions made for fully developed turbulent ¯ow. The ®rst is equilibrium of the turbulent energy cascade, wherein all of the energy which enters the turbulent motion at large scales is dissipated at the same rate at the smallest (viscous) scales of motion. In a non-equilibrium cascade, energy may dissipate at all scales. When the mean velocity gradients are constantly changing, as they are in the presence of a dominant MI, the supply of energy to the turbulent cascade will change continually, and the equilibrium assumption is not likely to hold. The second assumption of fully developed turbulence is stationary ¯ow, as measured on a time scale long relative to the turbulent ¯uctuations, but short relative to the process of interest (e.g. mixing). In the present measurements, there is a large gap between the broad banded low frequency of the MI (of the order of the tank turnover time) and the range of dissipative scales (bounded on the low end by the blade passage frequency). This indicates that the MI participates in an unsteady Trans IChemE, Vol 78, Part A, October 2000
mean ¯ow, rather than entering directly into the turbulent cascade. While the existence of macro-instabilities in the stirred tank ¯ow ®eld has been regularly reported since 1988, this work provides information on the geometric conditions which amplify these instabilities, and a measure of both the size of the MI and the impact of MI on RMS velocity measurements for a range of impellers and tank geometries. The conclusions can be summarized as follows: 1. Macroinstabilities in the stirred tank ¯ow ®eld must be taken into account for analysis of the velocity ¯uctuations. This is always the case at the top of the tank, and is important in the impeller stream for two classes of the geometries studied (large diameter PBT, D T /2; and underbaf¯ed RT, 2 baf¯es). This most directly affects (a) the experimental characterization of turbulence, and (b) selection of appropriate turbulence models for CFD simulations. 2. The amplitude of the MI is considered signi®cant when it exceeds 5% of the impeller tip speed. In cases where the presence of a MI in the impeller stream is evident from inspection of a 10 second time series, the amplitude is signi®cant at least 35% of the time. 3. MI are both low amplitude and slowly varying in the impeller stream of the HE3, the fully baf¯ed RT, and the D T/4 PBT. In these cases, errors resulting from an assumption of steady ¯ow are small. In the case of the A310, the size of v MI relative to v HF leads to a measurable change in RMS velocity on removal of the MI component, but other measures of the MI (time series characteristics, percent of time MI amplitude is signi®cant, and factorial analysis) suggest that it can be neglected. 4. If the MI are not properly accounted for, errors can reach 30% in the impeller stream (for a D T /2 PBT) and 40% at the top of the tank. The importance of the MI for process results will depend on the time scale of the process. In the case of slow reactions, the response is slow, so the time averaged result will contain all necessary details. If, however, the reaction (or process) has a time scale shorter than the lifetime of the largest eddies, then the process will be affected by the MI and some knowledge of these scales will be required. Feed jet intermittency, as studied by Houcine et al.20 , provides a good example of how meso-mixing can be affected by macro-instabilities. NOMENCLATURE C d D H M N N Nf r Re1 Qres Sn tn tn D tn T
impeller off bottom clearance, m interpolation density, d M/N impeller diameter, m liquid height, m number of points in the resampled signal number of points in the raw LDV signal impeller rotational speed, rpm or s 1 number of baf¯es radial position, m impeller Reynolds number ND2 /n resampled LDV sequence raw LDV Sequence particle arrival time interval between particle arrival times tank diameter, m
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1052 VÅi vi v BP v HF
v MI v MI v res v res v rand Vtip W
mean centered velocity, m s 1 instantaneous velocity, m s 1 instantaneous velocity due to the BPF, m s 1 RMS velocity which is due to the combined effect of random turbulent component and the blade passages, v HF v n uX u 2 v v t res MI i i 1 . n
instantaneous velocity due to MI (m s 1 ) s n 1X RMS velocity due to the MI, v MI v 2MI . ni 1
instantaneous velocity in the resampled signal, m s 1 RMS velocity of the resampled signal. instantaneous velocity due to the pure random or turbulent component, m s 1 tip speed of the impeller (pND), m s 1 blade width, m
Abbreviations A310 three bladed aerofoil impeller by Lightnin’ BPF blade passage frequency CFD computational ¯uid dynamics DNS direct numerical simulations DSA Doppler signal analyser impeller stream I FFT fast Fourier transform HE3 three bladed high ef®ciency turbine by Chemineer MI macroinstability LDV laser Doppler velocimeter LES large eddy simulation LIF laser induced ¯uorescence PBT four bladed, 458 pitched blade turbine PIV particle image velocimetery POD proper orthogonal decomposition RT 6 bladed Rushton turbine upper corner U
14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
24. 25. 26.
REFERENCES 1. Kresta, S. M., 1998, Turbulence in stirred tanks: anisotropic, approximate, and applied, Can J Chem Eng, 76: 563±576. 2. Wu, H. and Patterson, G. K., 1988, Laser-Doppler measurements of turbulent ¯ow parameters in a stirred mixer, Chem Eng Sci, 44: 2207±2221. 3. Rutherford, K., Lee, K. C., Mahmoudi, S. M. S. and Yianneskis, M., 1996, Hydrodynamic characteristics of dual Rushton impeller stirred vessel, AIChE J, 42: 332±346. 4. Stoots, C. M. and Calabrese, R. V., 1989, Flow in the impeller region of a stirred tank, Chem Eng Prog, 43. 5. Rao, M. A. and Brodkey, R. S., 1972, Continuous ¯ow stirred tank turbulence parameters in the impeller stream, Chem Eng Sci, 27: 137±156. 6. Cutter, L., 1966, Flow and turbulence in a stirred tank, AIChE J, 12: 35±45. 7. Winardi, S., Nakao, S. and Nagase, Y., 1988, Pattern recognition in ¯ow visualization around a paddle impeller, J Chem Eng Japan, 21: 503±508. 8. Winardi, S. and Nagase, Y., 1991, Unstable phenomenon of ¯ow in mixing vessel with a marine propeller, J Chem Eng Japan, 24: 243±249. 9. Bruha, O., Fort, I. and Smolka, P., 1993, Large scale unsteady phenomenon in a mixing vessel, Acta Polytechnica, Czech Tech Univ Prague, 27: 33. 10. Bruha, O., Fort, I. and Smolka, P., 1994, Flow transition phenomenon in an axially agitated system, Proc Eighth Europ Conf Mixing, IChemE Symp series No. 136, Cambridge, UK, pp 121±128. 11. Bruha, O., Fort, I. and Smolka, P., 1995, Phenomenon of turbulent macroinstabilities in agitated systems, Collect Czech Chem Commun, 60: 85±94. 12. Bruha, O., Fort, I., Smolka, P. and Jahoda, M., 1996, Experimental study of turbulent macroinstabilities in an agitated system with axial high-speed impeller and with radial baf¯es, Collect Czech Chem Commun, 61: 856±867. 13. Kresta, S. M. and Wood, P. E., 1993a, The ¯ow ®eld produced
27. 28. 29. 30. 31. 32. 33.
by a pitched blade turbine: Characterization of the turbulence and estimation of the dissipation rate, Chem Eng Sci, 48: 1761±1774. Kresta, S. M. and Wood, P. E., 1993b, The mean ¯ow ®eld produced by a 458 pitched blade turbine: changes in the circulation pattern due to off-bottom clearance, Can J Chem Eng, 71: 42±53. Chapple, D. and Kresta, S. M., 1994, The effect of geometry on the stability of ¯ow patterns in a stirred tank, Chem Eng Sci, 49(21): 3651±3660. Myers, K. J., Ward, R. W. and Bakker, A., 1997, A digital particle image velocimetry investigation of ¯ow ®eld instabilities of axial ¯ow impellers, J Fluids Eng, 119: 623±632. Montes, J. L., Boisson, H. C., Fort, I. and Jahoda, M., 1997, Velocity ®eld macro-instabilities in an axial agitated mixing vessel, Chem Eng J, 67: 139-145. Hasal, P., Montes, J. L., Boisson, H. C. and Fort, I., 2000, Macroinstabilities of velocity ®eld in stirred vessel: detection and analysis, Chem Eng Sci, 55: 391±401. Haam, S., Brodky, R. S. and Fasano, B., 1992, Local heat transfer in a mixing vessel using the heat ¯ux sensors, Ind Eng Chem Res, 31: 1384±1391. Houcine, I., Plasari, E., David, R. and Villermaux, J., 1999, Feedstream jet intermittency phenomena in continuous stirred tank reactor, Chem Eng J, 72: 19±30. Bittorf, K., 2000, The application of wall jets in stirred tanks with solids distribution, PhD Thesis, (University of Alberta, Edmonton, Canada). Bittorf, K. and Kresta, S., 1999, The prediction of the solids cloud height within a stirred tank, Mixing XVII, Banff, Canada, August. Bujalski, W., Takenaka, K., Paglianti, A., Takahashi, K., Poalini, S., Nienow, A. W. and Etchells, A. W., 1999, Suspension and liquid homogenization in high solids concentration stirred chemical reactors, Trans IChemE, Part A, 77: 241±247. Yianneskis, M., Popiolek, Z. and Whitelaw, J. H., 1987, An experimental study of the steady and unsteady ¯ow characteristics of stirred reactors, J Fluid Mech, 175: 537±555. Zhou, G. and Kresta, S. M., 1996, Impact of the tank geometry on the maximum turbulence energy dissipation rate for impellers, AIChE J, 42: 2476±2490. Box, G. P., Hunter, W. G. and Hunter, J. S., 1978, Statistics for Experimenters, (John Wiley and Sons, Toronto). Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T., 1994, Numerical Recipes, The Art of Scienti®c Computing, (Cambridge University Press, New York). Hockey, R. M. and Nouri, J. M., 1996, Turbulent ¯ow in a baf¯ed vessel stirred by a 60 degrees pitched blade impeller, Chem Eng Sci, 51: 4405±4421. Grigic, B., 1998, In¯uence of the impeller and tank geometry on low frequency phenomena and ¯ow stability, MSc Thesis, (University of Alberta, Canada). Brandstater, H. L. and Swinney, H. L., 1987, Strange attractor in weakly turbulent Couette Taylor ¯ow, Phys Rev A, 35: 2207±2220. Richardson, S., 1992, Chaotic Analysis of Two Impellers, (Internal Report, University of Alberta). McLaughlin, D. K. and Tiederman Jr, W. G., 1973, Biasing correction for individual realizations of Laser Anemometer measurements in turbulent ¯ow, Physics of Fluids, 16: 2082±2088. McFarlane, K. and Nienow, A. W., 1995, Studies of High Solidity Ratio Hydrofoil Impellers for Aerated Bioreactors, Biotechnol Prog, 11: 601±607.
ACKNOWLEDGEMENTS The authors wish to thank both NSERC and Lightnin for ®nancial support of this work.
ADDRESS Correspondence concerning this paper should be addressed to Professor S. Kresta, Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G, 2G6, Canada, (E-mail: [email protected]). This manuscript was communicated via our International Editor for the USA, Professor Richard Calabrese. It was received 21 January 2000 and accepted for publication after revision 28 July 2000.
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