Analysis of Turbulence Structure in the Stirred Tank with a Deep Hollow Blade Disc Turbine by Time-resolved PIV

FLUID FLOW AND TRANSPORT PHENOMENA Chinese Journal of Chemical Engineering, 18(4) 588ü599 (2010)

Analysis of Turbulence Structure in the Stirred Tank with a Deep Hollow Blade Disc Turbine by Time-resolved PIV* LIU Xinhong (ঞ໯‫)܀‬, BAO Yuyun (ͧုၩ), LI Zhipeng (ᴢᄝକ) and GAO Zhengming (‫غ‬ჾੜ)** School of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China Abstract The turbulence structure in the stirred tank with a deep hollow blade (semi-ellispe) disc turbine (HEDT) was investigated by using time-resolved particle image velocimetry (TRPIV) and traditional PIV. In the stirred tank, the turbulence generated by blade passage includes the periodic components and the random turbulent ones. Traditional PIV with angle-resolved measurement and TRPIV with wavelet analysis were both used to obtain the random turbulent kinetic energy as a comparison. The wavelet analysis method was successfully used in this work to separate the random turbulent kinetic energy. The distributions of the periodic kinetic energy and the random turbulent kinetic energy were obtained. In the impeller region, the averaged random turbulent kinetic energy was about 2.6 times of the averaged periodic one. The kinetic energies at different wavelet scales from a6 to d1 were also calculated and compared. TRPIV was used to record the sequence of instantaneous velocity in the impeller stream. The evolution of the impeller stream was observed clearly and the sequence of the vorticity field was also obtained for the identification of vortices. The slope of the energy spectrum was approximately 5/3 in high frequency representing the existence of inertial subrange and some isotropic properties in stirred tank. From the power spectral density (PSD), one peak existed evidently, which was located at f0 (blade passage frequency) generated by the blade passage. Keywords stirred tank, time-resolved particle image velocimetry, wavelet analysis, energy spectrum, power spectral density, turbulent kinetic energy

1

INTRODUCTION

Stirred vessels are widely used in chemical industries for a variety of mixing processes such as gas-liquid, liquid-liquid, and gas-liquid-solid systems. Turbulent flow, existing in most stirred tank, contains a series of eddies of a wide range of time and length scales and hence helps to promote macro- and micromixing. An obvious phenomenon in the stirred tank is that the blade motion can induce trailing vortices which are the source of the turbulence and have an effect on fluid mixing. The flow structures developed in stirred vessels are complex, which have been investigated by many researchers. Using a photographic technique, Sachs and Rushton [1], who are usually considered to be the earliest authors, examined the flow characteristics in a tank with a single Rushton turbine and found that strong radial discharge flow existed and the radial velocity component varied with the impeller blade angle. Van’t Riet and Smith [2, 3] observed the structure of the trailing vortex pair originating at the back of each blade, and recorded the locus of the center of trailing vortices. With the rapid development of laser, computational and photographic techniques, Laser-Doppler Anemometry (LDA) and Particle Image Velocimetry (PIV) have been used by a great number of researchers to investigate the flow structures developed in stirred vessels. Wu and Patterson [4] used LDA to ob-

tain mean velocities, turbulence correlation functions, turbulent intensities, one-dimensional energy spectra and turbulence macro- and micro-scales in a Rushton turbine stirred tank. Their results showed that about 60% of the energy input from the impeller was dissipated in the immediate vicinity of the impeller and jet flow regions. Lee and Yianneskis [5] determined the time and length scales of turbulence in a stirred tank using LDA. The PIV technique is more efficient than LDA since thousands of velocity vectors in a single plane can be obtained simultaneously. A wide range of turbulent scales exist in the turbulent regime in a stirred tank. Sharp and Adrian [6] studied the small-scale flow structures and turbulent kinetic energy dissipation around a Rushton turbine in a stirred tank with the PIV technique. A triple decomposition for velocity was employed to separate the periodic fluctuations generated by blade passage and the random turbulent fluctuations. Escudié and Liné [7] analyzed the transfer of kinetic energy, and estimated the kinetic energy dissipation from the energy balance using PIV data. Baldi and Yianneskis [8] obtained mean velocities, Reynolds stresses and turbulent kinetic energy dissipation from fluctuating velocity gradient measurements with 2-D and 3-D PIV in a Rushton turbine stirred tank. The time-resolved PIV (TRPIV) can not only give the instantaneous map of velocity field but also capture the temporal sequence of turbulence flow. Using TRPIV (time-resolved PIV), Huchet et al. [9]

Received 2010-03-23, accepted 2010-07-02. * Supported by the National Natural Science Foundation of China (20776008, 20821004, 20990224) and the National Basic Research Program of China (2007CB714300). ** To whom correspondence should be addressed. E-mail: [email protected]

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estimated local kinetic energy dissipation rate in the impeller stream of a Rushton turbine by applying the direct measurement without any simplification. In this work, a novel impeller, deep hollow blade (semi-ellipse) disc Turbine (HEDT) as shown in Fig. 1, was investigated with both TRPIV and the traditional PIV. HEDT consists of six semi-ellipse blades mounted on the disk, and is often used as the bottom impeller for multi-impeller stirred reactor to gain better performance of gas dispersion and solid suspension [10].

Figure 2 The geometry of the experimental tank Figure 1 (HEDT)

Deep hollow blade (semi-ellispe) disc turbine

In the stirred tank, the turbulence generated by blade passage includes the periodic component and the random turbulent one. Many researchers tried to identify the random turbulent fluctuations by using angle-resolved measurement [6, 7, 11, 12]. In this work, the wavelet analysis was used to separate the random turbulent kinetic energy from the periodic one. The fluctuating velocity was decomposed to several scales with discrete wavelet transform, and the kinetic energies at different scales were discussed. The time evolution of trailing vortices was recorded by using TRPIV instead of with angle-resolved PIV like most of the former researchers did [5,1315]. The energy spectrum and power spectral density (PSD) were evaluated to investigate the isotropic properties of turbulence of low frequencies generated by impeller blade passage. 2

EXPERIMENTAL

tem (TSI, USA) are used in the work. The traditional PIV system, consisting of a laser, a CCD camera, a synchronizer, an external trigger and a computer, was used to capture the flow with angle-resolved measurement. Experiments were carried out for only one sextant since the turbine is symmetric in 60° increments. Within the sextant, measurements were carried out at T 0°, T 10°, T 20°, T 30°, T 40° and T 50°, respectively. 200 pairs of image were captured at each angular blade position. The TRPIV system consisted of a laser, a CMOS camera, a frequency controller and a computer. The capture was carried out at the frequency of 640 Hz. 3000 instantaneous velocity fields were sampled. The water inside the cylindrical tank was seeded with neutrally buoyant glass beads with diameter of 812 ȝm. Cylindrical coordinates were used to designate the directions and the notation, r, t and z were identified to the radial, tangential and axial directions, respectively. The origin of the coordinates was at the centre of the vessel base. The measurement plane was illuminated by the light sheet set through one baffle (being transparent) as shown in Fig. 3.

All experiments were carried out in a cylindrical tank of diameter T 0.19 m stirred by a six-blade HEDT of diameter D 0.4T. The tank was filled with water of the height H T as shown in Fig. 2. To minimize the optical refraction, the cylindrical tank was placed inside a large rectangular tank made of perspex and the gap between them was filled with water. The four equal spaced baffles had a width of T/10. In the stirred tank, the Reynolds number was defined by Re

D2 N U

P

(1)

The rotational speed N 90 r·min1 was chosen for our experiments, corresponding to Re 8847. The TRPIV system and the traditional PIV sys-

Figure 3 The PIV measurement setup

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RESULTS AND DISCUSSION Comparison of TRPIV and PIV

TRPIV enables the instantaneous measurements of the velocity field with high frequency fluctuations, but the traditional PIV can’t. It is necessary to validate TRPIV results with traditional PIV. The whole instantaneous velocity fields of TRPIV are averaged and compared with the results of the traditional PIV as shown in Fig. 4, both normalized by the impeller tip velocity Utip. It can be seen that the averaged radial and axial velocities by TRPIV are in good agreement with that by the traditional PIV, which confirms that the TRPIV with the quantity of 3000 samples can produce correct velocity fields. 3.2

Sequence of impeller stream

The impeller stream generated by the blade includes one jet flow and two trailing vortices. Several researchers have investigated the trailing vortices induced by single and dual Rushton turbine systems using different techniques [2, 3, 5, 1315]. In this work, TRPIV was used to analyse the evolution of the impeller stream and trailing vortices. 3.2.1 Sequence of the instantaneous velocity field Figure 5 shows the TRPIV recorded sequence of the instantaneous superimposed velocity in the impeller stream in the same plain between two impeller blades. The velocity is normalized by Utip. In the map of t 0 s, one stream is shown far away from the impeller, where the high velocity area locates in the stream. When the blade passes the plane measured, the stream moves away from the blade, above and below which two trailing vortices formed. The velocity of the stream decreases with the blade passing away from the measured plane. The two vortices, travelling with the stream, becomes blurry and irregular in the maps after t 0.082 s.

(a) Radial velocity

3.2.2 Sequence of the instantaneous vorticity field In order to observe vortices clearly, the vortices are best identified by determining the distribution of vorticity in the impeller stream in this work, although a series of methods can be used to identify vortices [16]. The vorticity component in the r-z plain is shown as

Z

wU z wU r  wr wz

(2)

where Uz and Ur represent the instantaneous axial and radial velocities, respectively. Fig. 6 show the sequence of the vorticity field normalized by the rotational speed, N, with the corresponding instantaneous velocity superimposed. High positive and negative values of the vorticity shown in red and blue indicate the presence of the vortex cores, representing clockwise and counter-clockwise, respectively, above and below the stream. Besides the trailing vortices, many small vortex structures moving with the stream exist in the impeller stream, which facilitates the significant mixing in a stirred tank and also confirms the turbulence structures. The figures of the vortices change continually with the time evolution. The colour of Ȧ shows clearly that Ȧ decreases with the trailing vortices moving far away from the impeller. 3.3 Average velocity, vorticity, and turbulent kinetic energy Velocities in the impeller stream include both periodic fluctuations and random turbulent ones, and the former is believed generated by the impeller blade passage. The instantaneous velocity could be decomposed as

U i U i  ui (3) where U i is the instantaneous velocity, U i is the averaged velocity, and ui is the fluctuating velocity including both the periodic components and the random turbulent ones. In a stirred tank, the angle-resolved measurement with PIV was often carried out to iden-

(b) Axial velocity

Figure 4 Comparison of the averaged radial and axial velocities of TRPIV and traditional PIV at z/T 0.245 normalized by Utip at Re 8847 Ƶ TRPIV 640 Hz;ƽtraditional PIV

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Figure 5

Sequence of instantaneous velocity fields

Figure 6

Sequence of instantaneous vorticity fields

tify the random turbulent fluctuations [6, 7, 11, 12, 14, 17]. In the present work, the wavelet transform analysis will be employed to obtain sequences of periodic component and random turbulent one. Figure 7 shows the average velocity normalized by the impeller tip velocity, U tip , in the plain between two blades. The high velocity locates in the impeller stream region which is at the centre line of the impeller. The impeller stream jets from the blade near horizontally (the inclination angle almost 0°). As a com-

591

parison, the impeller stream produced by a Rushton turbine jets towards the vessel wall with a slight upward inclination of around 3°4° to the horizontal [5]. The distribution of the instantaneous vorticity has been shown and discussed in Section 3.2.2, and the average vorticity Z is shown on the map of averaged velocity in Fig. 8. Comparing Fig. 8 with Fig. 7, it is clear that the high average vorticity is not located in the impeller stream but above and below the stream where the two trailing vortices traverse, and the averaged

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cause some error because of the anisotropy characteristics of the turbulent fluctuation at low frequency. Fortunately, the error is acceptable for the baffled vessel and a reasonably accurate estimation of the energy distribution could be obtained [18]. Fig. 9 shows 2 . The the turbulent kinetic energy normalised by U tip high turbulent kinetic energy locates in the area where the trailing vortices pass through, which implies that vortices provide a source of turbulence. In this section, the turbulent kinetic energy involves the periodic turbulent one and the random turbulent one.

Figure 7

Average velocity normalised by U tip

Figure 9

3.4 Figure 8 Average vorticity normalised by N

velocity and vorticity distribute symmetrically with the centre line of blade. Since the fluctuating component can take positive or negative values, it is usually to characterize it via the root mean square or RMS u ,

uii

U i  U i

2

ui2

(4)

The turbulent kinetic energy k provides a mixing mechanism due to turbulent dispersion. In Cartesian coordinate, k may be determined from 3 velocity components as [5] 1 2 2 2 k (5) uiz  uir  uit 2 Since 2-D TRPIV was employed, the fluctuating component at tangential direction uit can only be determined from the local isotropic approximation. 2 1 2 2 uit (6) ui  uir 2 z Hence, 3 2 2 k (7) ui  uir 4 z The turbulent kinetic energy calculated by Eq. (7) will













2 Turbulent kinetic energy normalised by U tip

Time signals

The time signals of the axial and radial fluctuations ur and u z at r/T 0.220, z/T 0.330 are shown in Fig. 10, which provides both the periodic component and the random turbulent one. The fluctuations with large magnitude in long time interval is the periodic component generated by the impeller blade passage, and the fluctuations with small magnitude superimposed on the periodic component is generated by the turbulence. 3.5

Spectral analysis

Turbulence consists of a range of time and spatial scale eddies. It is necessary to compute the energy spectrum E(f) to describe a turbulent flow, especially to quantify the slope of the spectrum at high frequencies. The one-dimensional energy spectrum E(f) can be estimated from Fast Fourier Transform (FFT) of u 2 , which represents the energy contained between frequencies f and f  df (Hz) [5]: u2

f

³0

E ( f )df

(8)

where u is the fluctuating velocity. The one-dimensional wavenumber spectrum E(ț) can be converted from one-dimensional energy spectrum E(f) using an appropriate convective velocity Uc with Eq. (9):

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Figure 10

Figure 11

N

Time signals of ur and uz at r/T 0.220, z/T

One-dimensional energy spectrum of ur and uz at r/T 0.220, z/T=0.330

2Sf Uc

(9)

Kolmogorov’s spectrum law states that the 5/3 slope of the energy spectrum in high frequencies represents the inertial subrange existing and some isotropic properties [19]. AH 2 / 3N 5/ 3

E (N )

Ab N

where N is rotational speed of 1.5 s1, Ab is blade number, 6 for HEDT impeller and f0 equals to 9 Hz in this case. Figure 12 shows the power spectral density (PSD) in low frequency, which reveals that there are peaks at

(10)

Figure 11 shows the energy spectrums of the radial and the axial velocity plotted in log-log scales at r / T 0.220 , z / T 0.330 . A straight line of 5/3 slope is also drawn on the figure representing the energy distribution predicted by Kolmogorov’s spectrum law. It is clear that the slopes of the energy spectrum are approximately 5/3 in high frequency, representing the inertial subrange existing and some isotropic properties in stirred tank. The plots of the radial and axial velocity are similar, especially in high frequency. Peak values at 9 Hz in Fig. 11 imply the existence of the blade passage frequency. Hence, the blade passage frequency in this configuration could be calculated by f0

0.330

(11)

Figure 12 Power spectral density (PSD) of ur and uz at r/T 0.220, z/T 0.330

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9 Hz (f0) corresponding to the blade passage frequency. Since the HEDT impeller discharge flow radially, the peak for ur at 9 Hz (f0) is much more obvious than that for uz. The other peaks shown in the PSD for uz may correspond to large scale structures of the flow [20], which are called macro instabilities due to the geometry, wall and baffles [21]. 3.6

Wavelet analysis

The theory of wavelets, a new branch of mathematics developed in the last two decades, has been widely applied in a variety of engineering and scientific disciplines. Farge [22] firstly employed the wavelet analysis for turbulence research. Kitagawa and Nomura [23] illuminated the relation between the turbulent eddies and the wavelet. 3.6.1 Continuous wavelet transform (CWT) The continuous wavelet transform W f (s,W ) of a

finite energy signal f (t ) is defined as its scalar prod1 § t W · uct with the wavelet I ( x) i.e.: I¨ ¸ . In other s © s ¹ words, the wavelet transform represents the correlation of the signal f(t) and the wavelet s 1/ 2I > (t  W ) / s @ . The following equation gives both representations: 1 § t W ·  W f ( s,W ) ³ f (t ) I¨ ¸ dt , s  R , W  R (12) R s © s ¹ It is noted that there is a constraint on f(t): its Fourier transform, \ (Z ) , must satisfies

³R

\ (Z ) dZ  f Z

(13)

3.6.2 Discrete wavelet transform (DWT) As a result of the quantization of the parameters s and IJ with s s0j , W ks0jW 0 , we obtain the following equation, which corresponds to s 1/ 2I > t  W / s @ in CWT: 

j

I j , k (t ) s0 2 I  s0 j t  kW 0 , 2 , W0

In particular, we can choose s0

I j , k (t ) 2



j 2I

2

j

j, k  Z



tk ,

(14)

1 to obtain j, k  Z

(15)

Hence, we can obtain the Discrete Wavelet Transform W f  j, k

³R f (t )I j ,k (t )dt

(16)

In this work, the data of sampling frequency 640 Hz is employed to analysis with DWT, which can decompose the signal to several levels with db4 wavelet. The multi-resolution analysis is carried out using the db4 wavelet to seven levels of detail. Fig. 13 shows multi-scale decomposition of signal, where d is high

Figure 13

Multi-scale decomposition of signal

frequency band, and a is low frequency one. G and H represent high pass filter and low pass filter. The fluctuating velocity u is decomposed by discrete wavelet transform as below: a6  d1  d 2  d3  d 4  d5  d 6

u

(17)

Table 1 shows discrete wavelet transform detail frequencies according to wavelet scales in the present work. Table 1

Discrete wavelet transform detail frequency according to wavelet scale

J

Wavelet scale

Frequency range/Hz

1

d1

160320

2

d2

80160

3

d3

4080

4

d4

2040

5

d5

1020

6

d6

510

6

a6

<5

From band d1 to a6, the frequency range decreases. Band d6 will be focused since its range of 510 Hz covers the blade passage frequency, 9 Hz. The radial fluctuating velocity at r/T 0.245, z/T 0.330 with DWT was calculated and shown in Fig. 14. Band d6 can be identified as the periodic fluctuation generated by the blade passage. The sum of the other bands can be considered as the random turbulent fluctuations. The fluctuating velocity ui comprises of periodic component upi and random turbulent component uicc . ui upi  uicc (18) Hence, upi

d6 ,

u cc a6  d1  d 2  d3  d 4  d5

(19)

According to Eq. (7), periodic kinetic energy kp and random turbulent kinetic energy k cc could be expressed as 3 2 3 2 2 kp k cc (20) upr  up2z , uicc  uiccz 4 4 r Based on the energy conservation law, it is known that the sum of periodic kinetic energy kp and random turbulent kinetic energy k cc equal to turbulent kinetic energy k defined in Eq. (7): k kp  k cc (21)









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Figure 14

595

Discrete wavelet transform of radial fluctuating velocity at r/T 0.245, z/T 0.330

The periodic and the random turbulent sequences are calculated by Eq. (20) and shown in Fig. 15.

Figure 15 Radial periodic fluctuation and turbulent fluctuation at r/T 0.245, z/T 0.330

It is clear that only fluctuations with long period existing in periodic sequence are generated by the impeller blade passage, while the detail fluctuations are generated by the turbulence in turbulent sequence. Fig. 16 shows the power spectral density (PSD) of ur, random turbulent fluctuation and periodic fluctuation at r/T 0.245, z/T 0.330. Compared with PSD of ur, it is found that one peak at 9 Hz (f0) exists in the PSD of periodic fluctuation, but not in that of turbulent fluctuation. It verifies that we obtain the periodic fluctuation and random turbulent fluctuation successfully. In fact, the whole processing is similar to the digital filter which filters out some unwanted input spectral components by allowing certain frequencies to pass while attenuating other frequencies. For the periodic turbulent fluctuation, the frequencies from 5 to 10 Hz containing 9 Hz

Figure 16 Power spectral density (PSD) of ur, random turbulent fluctuation and periodic fluctuation at r/T 0.245, z/T 0.330

(f0) are allowed to pass, and the other frequencies are filtered out. For the random turbulent fluctuation, the case is reverse. 3.7 Periodic kinetic energy and random turbulent kinetic energy

In this work, traditional PIV with angle-resolved measurement and TRPIV with wavelet analysis are both used to obtain the random turbulent kinetic energy as a comparison. Fig. 17 shows axial distributions of random turbulent kinetic energy obtained by both wavelet analysis and angel-resolve methods at r/T 0.245. Good agreement between each other reveals that wavelet analysis can be employed to calculate the

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Figure 17 Axial distributions of random turbulent kinetic energy obtained by wavelet analysis and angel-resolve methods at r/T 0.245 Ƶ wavelet analysis; ƽangle-resolve measurement

random turbulent kinetic energy successfully. By comparing these two methods, it is found that in fact, they can both filter out unwanted frequency components, and reserve wanted frequency ones. Angle-resolved measurement, carried out between two blades, actually filter out the periodic component. For wavelet analysis, the frequencies from 5 to 10 Hz containing 9 Hz (f0) are filtered out, and the other frequencies are reversed. Angle-resolved measurement only filters out the periodic frequency, and the wavelet analysis filters out a range of 5 to 10 Hz including the periodic frequency, the results for both methods are almost same since no obvious peaks exist in the range of 5 to 10 Hz for the power spectral density (PSD) except 9 Hz. The periodic kinetic energy was also obtained by wavelet analysis. The axial distributions of random turbulent kinetic energy and periodic kinetic energy at r/T 0.245 were shown in Fig. 18. The random turbulent kinetic energy was larger than the periodic kinetic energy. In the impeller stream (0.30.39 or z/T<0.27), the periodic kinetic energy is very close to zero, implying that the flow in these areas is hardly affected by the blade passage.

Figure 18 Axial distributions of random turbulent kinetic energy and periodic kinetic energy at r/T 0.245 Ƶ periodic kinetic energy;ƽrandom turbulent kinetic energy

Figure 19 shows the periodic kinetic energy and the random turbulent kinetic energy normalised by 2 2 U tip . The large periodic kinetic energy ( kp / U tip ! 0.03 ) and the large random turbulent kinetic energy 2 ( k cc / U tip ! 0.06 ) distribute differently. By comparing Fig. 19 (a) with Fig. 8, it is found that the distribution of the large random turbulent kinetic energy is similar to that of the vorticity since the trailing vortices produces intense turbulent fluctuations. However, the large periodic kinetic energy distributes only near the blade where the large vorticity locates. It seems that the random turbulent kinetic energy mainly depends on the trailing vortices, and the periodic kinetic energy depends on both the trailing vortices and the blade passage. With the increasing of r/T, the periodic energy decreases evidently. In the impeller region (0.207
(a) Random turbulent kinetic energy

(b) Periodic kinetic energy Figure 19 Random turbulent kinetic energy and periodic kinetic energy distributions

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(a) a6

(b) d6

(c) d5

(d) d4

(e) d3

(f) d2

(g) d1

Figure 20

Fluctuation kinetic energy distributions at scales from a6 to d1

3.8 Kinetic energies at different scales

The turbulent mixing in the stirred tank is very complex, including macro-, meso- and micro-mixing, which depend on turbulent fluctuations of the different scales. The investigation on the distributions of kinetic energy at different scales is helpful for a better understanding of the mixing process and the optimal design of the stirred vessel. The fluctuating velocity was decomposed to several scales with discrete wavelet transform. The ki-

netic energies at different scales obtained by Eq. (7) were shown in Fig. 20 from 20 (a) to 20 (g), corresponding to the wavelet scale a6 to d1, respectively. It should be noticed that the range of the contour level for Figs. 20 (a) to 20 (c) is from 0 to 0.03, whereas that for Figs. 20 (d) to 20 (g) is from 0 to 0.003. The frequency range of a6 is 05 Hz, which belongs to the lowest frequency and corresponds to large scale structures of the flow in the stirred tank. The large kinetic 2 energy ( ka6 / U tip ! 0.022 ) at a6 exists in the region of 0.23
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Fig. 20 (b) with 20 (c), it can be found that the large kinetic energy at d6 locates close to the blade, which is similar as the distribution of that at d5 since both of the kinetic energy at d6 and d5 are mainly affected by the blade passage. From Figs. 20 (d) to 20 (g), it seems that the distributions of the kinetic energies at d4 to d1 are similar to each others, which is because the turbulence at d4 to d1 locates in the inertial subrange where the turbulence eddies at the scales represent the similar properties. In the stirred tank, three main factors, including the large scale flow structures, the blade passage and the turbulence intensity, control the kinetic energy distributions at scales corresponding to low frequency, middle frequency and high frequency fluctuations, respectively. With the frequency increasing, the kinetic energies decrease from a6 to d1. 4

CONCLUSIONS

The turbulence structure in a stirred vessel of 0.19 m diameter with a novel impeller named deep hollow blade (semi-ellipse) disc turbine (HEDT) has been investigated with both TRPIV and traditional PIV. The evolution of the impeller stream was observed clearly and the sequence of the vorticity field was also obtained for the identification of vortices. The instantaneous velocity and vorticity fields taken by TRPIV show the evolution of the impeller stream and vortices clearly. The average velocity field reveals that the angle of the impeller stream is almost 0°. Energy spectrum and power spectral density (PSD) are also calculated for analysis. The slopes of the energy spectrum are approximately 5/3 in high frequency, representing the existence of the inertial subrange and some isotropic properties in stirred tank. From PSD, one peak exists evidently, which is located at f0 (blade passage frequency) generated by the blade passage. The random turbulent fluctuation was calculated by a new proposed method based on wavelet analysis with TRPIV in this work instead of using angle-resolved PIV measurement like other researchers did. Low periodic frequency, like 9 Hz in this work, required small filtered out range of 5 to 10 Hz, which can produce less errors in the results compared with the higher periodic frequency case. The bandpass filter is recommended to control the range of frequency to be filtered out in the high periodic frequency case. The distribution of the random turbulent kinetic energy is similar as that of the vorticity since the intense turbulent fluctuations are mainly produced by the trailing vortices. The large periodic kinetic energy locates very close to the blade. In the impeller region, the averaged random turbulent kinetic energy is about 2.6 times of the averaged periodic kinetic energy. The kinetic energies at different wavelet scales from a6 to d1 were calculated, respectively. The distributions of the kinetic energies at different scales are not same since they are controlled by different factors including the large scale structures of the flow, the

blade passage and turbulence intensity, corresponding to low, middle and high frequency fluctuations, respectively. With the frequency increasing, the kinetic energies decrease from a6 to d1. NOMENCLATURE a D d E(f) E(ț) f f0 H K kp

kcc N Re T U

U Utip U

u up ucc ț ȝ ȡ Ȧ

low frequency band, m·s1 impeller diameter, m high frequency band, m·s1 one-dimensional frequency spectrum, m2·s1 one-dimensional wavenumber spectrum, m3·s1 frequency, Hz blade passage frequency, Hz liquid height in the tank, m turbulent kinetic energy, m2·s2 periodic turbulent kinetic energy, m2·s2 random turbulent kinetic energy, m2·s2 rotational speed, s1 Reynolds number tank diameter, m instantaneous velocity, m·s1 averaged velocity, m·s1 impeller tip velocity, m·s1 fluctuating velocity, m·s1 root mean square intensity, m·s1 periodic component, m·s1 true turbulent component, m·s1 wave number, m1 dynamic viscosity, Pa·s density, kg·m3 vorticity, s1

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9 10 11

12

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