Chemical Engineering Science 61 (2006) 2799–2806www.elsevier.com/locate/ces

Angle-resolved stereo-PIV measurements close to a down-pumpingpitched-blade turbine

F.R. Khana, C.D. Riellya, ∗, D.A.R. Brownb

aDepartment of Chemical Engineering, Loughborough University, Loughborough, Leics LE11 3TU, UKbThe Fluid Engineering Centre, BHR Group Ltd., Cranfield, MK43 0BN, UK

Received 27 June 2005; received in revised form 13 October 2005; accepted 16 October 2005Available online 6 January 2006

Abstract

The present work employs a stereoscopic-PIV technique to obtain angle-resolved fields of all three velocity components close to a T/3,45◦ down-pumping pitched-blade turbine operated at 300 rpm in a 0.29 m diameter vessel. The measurements were made at blade angles 7.5◦apart, with 300 measurements taken at each blade position, in order to calculate angle-resolved mean velocity fields and turbulence quantities.Turbulent kinetic energy (k) distributions were obtained using (i) a pseudo-isotropic approximation, from two velocity components and (ii) afull calculation from all three velocity components. The two calculation methods for k yielded similar results, indicating that data from 2-DPIV measurements yield reasonable estimates of the turbulence kinetic energy. The tangential velocity components at the impeller dischargefrom PIV were in good agreement with data from LDA analysis. A kinetic energy balance across the impeller was performed (i) rigorouslyand (ii) using approximations which neglected second- and higher-order velocity cross-correlations. Both analyses show that around 44% ofthe total power consumed by the impeller is dissipated in the impeller region. The average rate of dissipation of kinetic energy is about 40times higher in the impeller region than the volume-average dissipation rate in the whole vessel.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Particle image velocimetry; Stirred vessels; Pitched blade turbine; Turbulent kinetic energy; Kinetic energy balance

1. Introduction

Stirred vessels are used in the process industries for a varietyof applications, such as liquid blending, chemical reaction, gasdispersion in liquids and solid suspensions in liquids. Turbu-lence and mixing in stirred reactors are key issues to understandthe above operations. Extensive research work has been there-fore carried out and has been widely reported in the literaturein the last four decades. Various experimental techniques suchas hot-wire anemometry, laser-doppler anemometry (LDA) andmore recently particle image velocimetry (PIV) have been em-ployed to study complex flow fields in stirred vessels (Mavros,2001). The majority of work reported in literature is generallyone or two-dimensional and represents time-averaged flow pat-terns (e.g. Stoots and Calabrese, 1995; Sharp and Adrian, 2001;

∗ Corresponding author. Tel.: +01509 222504; fax: +01509 223923.E-mail address: C.D.Rielly@lboro.ac.uk (C.D. Rielly).

0009-2509/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.10.067

Schäfer et al., 1998). However, it is well-established that thetrailing vortex structures result in fully three-dimensional pe-riodic flow variations close to an impeller (e.g. Schäfer et al.,1998).

The turbulence properties such as the Reynolds stresses, theturbulent kinetic energy and the dissipation rate require instan-taneous and coincident velocity field information in the axial,radial and tangential directions. In the absence of all three ve-locity components, isotropic assumptions are often applied toestimate turbulence quantities in the vessel. The applicabilityof these assumptions used to obtain the turbulence kinetic en-ergy is investigated here. Developments in experimental meth-ods allow simultaneous measurement of all the three velocitycomponents. For example, Derksen et al. (1999) made three-dimensional LDA measurements in the vicinity of a Rushtondisc turbine (RDT) and used this information to study Reynoldsstress distributions. They demonstrated that the trailing vorticesinduced by a Rushton turbine were associated with a strongdegree of anisotropy in this region. Similarly, Hill et al. (2000)

2800 F.R. Khan et al. / Chemical Engineering Science 61 (2006) 2799–2806

used three-dimensional PIV in a vessel stirred by a RDT andconcluded that in the blade region, the radial and tangentialvelocities were of approximately equal magnitude. Moreover,they pointed out that detailed stereoscopic measurements arerequired to understand the complex flows close to an impeller.

The present work studies three-dimensional flows close to apitched-blade impeller (PBT) and how the flow field varies withthe angle of blade rotation. It does so using stereoscopic PIV,in which two cameras view the same object plane illuminatedby a laser sheet. The presence of the impeller, shaft and baffleslimit the field of view which is accessible to both camerasand the laser sheet. Furthermore, the curvature of the vesseldistorts images taken close to the wall, even when the vesselis surrounded by an optical enclosure. All these factors resultin a restricted optical access to many parts of the vessel andtherefore only the regions close to the impeller were analysedhere.

Stereoscopic particle imaging systems have broadly devel-oped into two configurations (Hill et al., 2000): (i) lateral, alsoknown as translational offset, and (ii) angular, also known asrotational offset. In the lateral offset configuration, the imageplanes, lens planes, and object plane are all parallel. This con-figuration allows a limited overlap area in the flow field thatis viewed by both cameras. In the angular offset configuration,the two camera axes are no longer parallel to the object planebut are rotated such that they intersect the object plane at thesystem axis. Both the cameras are mounted in the Scheimpflugconfiguration, in which the object plane, the lens plane and theimage plane are set collinear, bringing all the points in the il-lumination plane into focus in the image plane. Thus, angularoffset provides an increased common field of view with greateraccuracy of the out-of-plane component, compared to the lat-eral offset configuration (Prasad, 2000).

Lawson and Wu (1997a) found up to 40% greater out-of-plane accuracy with the angular-based stereo-PIV, as comparedto an equivalent translational system. Generally the offset half-angle can be varied from 15◦ to 45◦. Different researchersused various offset half-angles; e.g. Hill et al. (2000) used 20◦,Willert (1997) used 35◦, while Lawson and Wu (1997b) studiedrange of offset half-angles up to 45◦. In order to balance thein-plane and out-of-plane errors, an offset half-angle of 45◦ isdesirable, however, larger angles cause greater non-uniformityin magnification and reduce the overlapped regions viewed byboth cameras. Lawson and Wu (1997b) have shown that in or-der to balance the non-uniform magnification with the in-planeand out-of-plane errors, the optimum performance is obtainedby using offset half-angles of between 20◦ and 30◦. In thepresent work, the angular offset configuration is used with anoffset half-angle of 30◦. The detailed experimental set up isdescribed in the next section.

2. Experimental set up

A 290 mm Perspex vessel was used with four baffles mountedon the walls at 90◦ intervals. The baffles were 3 mm thickand 29 mm (T/10) wide. A 45◦ PBT (T/3) with four down-pumping blades was used for the mixing. The PBT impeller

InsightSynchroniser

Cylindrical LensSpherical Lens

PIV CAM10-30

Encoder

Laser enteringfrom the tank base

New WaveNd:Yag laser

Fig. 1. Stereo-PIV set up.

was located at a distance of T/3 from the base. The experi-mental set up is shown in Fig. 1 and more details of the im-peller geometry can be obtained from Khan et al. (2004). ThePIV system used here consisted of two 1k ×1k TSI PIV 10–30CCD cameras with a 50 mJ Nd:Yag pulsed laser. The flow wasseeded using 10 �m hollow spherical glass particles, with al-most neutral buoyancy. The Stokes particle relaxation time was6 �s, while the corresponding Kolmogorov time scale was ofthe order 3 ms. Thus, particles can be expected to follow theturbulent motion. The laser sheet thickness was around 1 mm.Both cameras were mounted in the Scheimpflug configuration,in which all the points in the illumination plane can be broughtinto focus in the image plane. As shown in Fig. 1, the cylindri-cal section mixing tank was placed inside a pentagonal Perspexbox, in which the two front faces formed an angle of 120◦.Both cameras were placed perpendicular to these two faces,such that the axis of both the lenses met at an angle of 60◦ onthe object plane, forming offset half-angle of 30◦.

To reduce the obstruction to the cameras by a following bladeentering into the field of view, the PBT impeller was constructedfrom 3 mm thick Perspex.

For camera calibration, images of a calibration target are re-quired. The target was mounted on a micrometer for fine con-trol, whilst traversing the out-of-plane depth (Z-direction). Thetarget contained calibration markers (dots), for which the truepositions (horizontal X and vertical Y in plane co-ordinates)were known by the calibration software (PIV Calib). The targetwas traversed to acquire calibration images at 5 different planesspaced at 0.25 mm covering the full thickness of the light sheet(1 mm).

The impeller was rotated at 300 rpm (Re = 47 500) and 300image pairs were taken at each of 12 fixed blade positions be-tween 0◦ and 90◦, at measurement planes 7.5◦ apart, to yieldthe angle-resolved velocity field statistics (averaging took placeover much less than 1◦ of rotation). A blade angle of 0◦ corre-sponds to the vertical centre line of the blade being in the lasersheet measurement plane (refer to Khan et al. (2004) for de-tails). A vector resolution of 1.28 mm × 1.28 mm was obtained(i.e., the size of the 32 × 32 pixel PIV interrogation windows)and the field of view covered an area of 40 × 40 mm. The

F.R. Khan et al. / Chemical Engineering Science 61 (2006) 2799–2806 2801

Taylor microscale is estimated to be about 4 mm in the presentcase and thus the chosen vector resolution is sufficient to re-solve turbulent structures of this size.

Sharp and Adrian (2001) assessed the convergence flow ofthe statistics from various numbers of measured vector fieldsand reported that the difference in the calculation of mean andrms velocities between 50 and 100 frames was less than 3%.Statistical convergence has also been checked in the presentwork and it was found that the difference in the rms valuescalculated from 200 to 300 frames is about 1.7%. Thus rmsvelocities calculated from the 300 frames should be statisticallyacceptable.

3. Results and discussions

3.1. Angle-averaged mean-velocity fields

Figs. 2(a) and (b) shows the angle-averaged mean velocityfields for blade angles of 0◦–30◦, respectively, and are sampleframes from animated sequences. The vectors represent radial

0.39

0.36

0.33

0.30

0.270.185 0.150 0.115 0.08

r/T

z/T

Wm/s

1.000.800.600.400.200.00-0.20-0.40-0.60-0.80-1.00

0.5 m/s

(a)

Wm/s

1.000.800.600.400.200.00-0.20-0.40-0.60-0.80-1.00

0.5 m/s

0.39

0.36

0.33

0.30

0.27

0.185 0.150 0.115 0.08

r/T

z/T

(b)

Fig. 2. Angle resolved mean velocity fields for the PBT at blade angles of:(a) 0◦ and (b) 30◦.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.00 0.05 0.10 0.15 0.20 0.25r/ T

u'/V

tip ,

v' /

Vtip

.

3D PIV, u'3D PIV, v'2D PIV, u'2D PIV, v'

Fig. 3. Normalised axial and radial rms velocities from 2-D/3-D PIV at 0◦blade angle (z/T = 0.39).

0.05

0.07

0.09

0.11

0.13

0.15

0.17

0.19

0.21

0.23

0.25

0.00 0.05 0.10 0.15 0.20 0.25r/ T

u' /

Vtip

, v' /

Vtip

3D PIV, u'

3D PIV, v'2D PIV. u'2D PIV, v'

Fig. 4. Normalised axial and radial rms velocities from 2-D and 3-D PIV at0◦ blade angle (z/T = 0.27).

and axial velocity components, whilst the background contourcolours show the distribution of the tangential velocity. Fig. 2shows that the tangential velocity is highest in the trailing vortexregion, reaching up to 67% of the tip velocity (Vtip = 1.5 m/s).From the animations prepared from the angle-resolved meanflow fields, the vortex moves axially downward and fol-lows the same pattern as recorded in 2-D PIV measurements(Khan et al., 2004).

3.2. Root mean square velocity distributions

The angle-resolved radial (u′) and axial (v′) rms velocitieswere compared between the 2-D and 3-D PIV data. Figs. 3and 4 shows the variation of axial and radial rms velocities(normalised by the tip speed of 1.5 m/s) for two different ax-ial locations obtained from 2-D and 3-D PIV and at 0◦ ofblade rotation. The 2-D rms velocity data used here are fromthe multi-block velocity fields reported by Khan et al. (2004).The impeller centre is located at z/T = 0.33, so the location

2802 F.R. Khan et al. / Chemical Engineering Science 61 (2006) 2799–2806

-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0 0.05 0.1 0.15 0.2

r/T

w/V

tip

PIV, z / T=0.29

LDA, z / T=0.29

PIV, z / T=0.38

LDA, z / T=0.38

Fig. 5. Comparison of tangential velocity from 3-D PIV and LDA.

z/T =0.39 is a plane just above the impeller and z/T =0.27 isjust below the impeller. The comparisons show higher axial andradial rms velocities at z/T =0.27 than at z/T =0.39; the flowis more turbulent in the discharge stream (z/T = 0.27) than inthe suction stream (z/T = 0.39). For z/T = 0.27 (Fig. 4), therms velocities exhibit a maximum close to the blade tip (aboutr/T = 0.15), because of the presence of the trailing vortex atthe bottom edge of the PBT. This is the region where the vortexleaves the blade and is more turbulent. At both locations, thedistributions of rms velocities were in fairly good agreementbetween the 2-D and 3-D PIV data; the in-plane velocity mea-surements in the present 3-D analysis were satisfactorily mea-sured and systematic differences from the 2-D measurementswere not evident.

The 2-D PIV data used here were obtained using a multi-block approach as explained in Khan et al. (2004), whichshowed discontinuities of up to 10% between adjacent blocks.Some of the scatter in the 2-D data shown in Fig. 3 may be dueto these discontinuity errors.

3.3. Tangential velocity comparisons

To check the validity of the out-of-plane measurements (thetangential velocity component) a comparison was performedwith time-averaged LDA measurements obtained at BHRGroup. Both velocity data sets were taken from an identicaltank (T = 290 mm) and at the same speed of impeller rota-tion (300 rpm); other geometric variables such as the impellerclearance (C =T/3) and the number and width of baffles weresimilar. Fig. 5 shows a comparison of the time-averaged LDAdata, with the 3-D PIV tangential velocities, which have beenaveraged here over all blade angles. The two data sets for thetangential velocity distributions are in reasonable agreement;both show a similar minimum velocity at about r/T = 0.15,which is where the main discharge stream carrying the trailingvortices leaves the blade. The slight difference in the locationof this minimum point may be due to minor differences in the

impeller dimensions, such as the blade thickness (3 mm forthe Perspex blades used in the present work). The tangentialvelocities at the suction or inlet side of the PBT (z/T = 0.29)are much higher than on the discharge side (z/T = 0.38). Theflow exhibits a much higher degree of swirl in the discharge,compared to the inlet stream to the impeller. This is an im-portant fact to know for a kinetic energy balance across theimpeller, which is discussed later.

3.4. Turbulent kinetic energy

A rigorous calculation of the turbulent kinetic energy, k, re-quires knowledge of all three rms-velocity components as in

k = 1

2

(u′2 + v′2 + w′2

). (1)

Use of the angle-resolved rms velocities avoids broadening ofthe turbulence kinetic energy due to the periodicity of this flow.

With 2-D PIV data, there is no knowledge of the third rms-velocity component (usually tangential) and k can only be es-timated using a pseudo-isotropic assumption:

k = 3

4

(u′2 + v′2

). (2)

Here it was possible to check out the validity of this assumption,using both 2-D and 3-D PIV measurements.

Fig. 6 shows a comparison of turbulent kinetic energy esti-mated from all the three velocity components using Eq. (1) andusing the pseudo-isotropic assumption, Eq. (2). In both cases,the kinetic energy is normalised by the square of the tip veloc-ity and both are at a blade angle of 0◦.

Comparing these contour plots it is clear that the turbulencekinetic energy distribution and magnitudes are very similar inthe two cases. Slight differences exist in the centre of vortex,where the rigorously calculated turbulence kinetic energy esti-mated from the 3-D data is slightly higher than that estimatedby Eq. (2), which indicates that the contribution from the tan-gential velocity in the vortex is more than that predicted by thepseudo-isotropic assumption. In this paper, Eq. (2) has beendescribed as using a pseudo-isotropic assumption and it shouldnot be inferred that the turbulence is truly isotropic. All that isclaimed here is that the tangential rms component is reasonablywell estimated as the average of the axial and radial rms ve-locities; the 3-D LDA measurements by Derksen et al. (1999)and the data presented in Figs. 3 and 4 show that close to theimpeller the turbulence can be strongly anisotropic.

Zhou and Kresta (1996) carried out a similar comparison,using all three rms components from their 3-D LDA data forRDT, pitched blade turbine and A310 impellers and they alsogave an estimate based on only one velocity component (as32v′

i2, where i can be r , z, or �). They obtained a conclusion

similar to the above and found that using only the axial rmscomponent for axial flow impellers or only the radial rms com-ponent for the disk turbine, yielded reasonable estimates of k.

F.R. Khan et al. / Chemical Engineering Science 61 (2006) 2799–2806 2803

0.39

0.36

0.33

0.30

0.27

0.175 0.14 0.105 0.07

r/T

z/T

0 0.005 0.01 0.015 0.02 0.03 0.04 0.05 0.06

k

Vtip2

(a)

0 0.005 0.01 0.015 0.02 0.03 0.04 0.05 0.060.39

0.36

0.33

0.30

0.270.175 0.14 0.105 0.07

r/ T

z/T

k

Vtip2

(b)

Fig. 6. Normalised angle-resolved turbulent kinetic energy from: (a) two rmsvelocity components and (b) all three rms velocity components.

3.5. Kinetic energy balance

In stirred vessels, the power input to rotate the impeller isused to supply kinetic energy to the fluid. The largest vorticesare produced in the wake of the impeller. These trailing vor-tices transfer their kinetic energy to the smaller eddies througha process known as the energy cascade (Tennekes and Lumley,1972). Eventually the energy is transferred to very small ed-dies, where viscous forces dominate and dissipation of the tur-bulence kinetic energy to heat takes place. The local dissipationrate is very important to understand mixing and dispersion ofone phase into another. In order to estimate the energy dissipa-tion in various zones (e.g. the impeller region, impeller streamand in the bulk), a kinetic energy balance can be performed. Inthe impeller region, the total energy flows into and out of theimpeller can be compared with the power input via the impeller

itself. A significant portion of the kinetic energy is dissipatedbetween the blades and in the impeller stream (Jaworski andFort, 1991). A kinetic energy balance may be applied to under-stand what percentage of total energy supplied is dissipated indifferent regions.

Cutter (1966) was the first to obtain estimates of the energydissipation rate by carrying out an energy balance across sec-tions of the flow generated by a RDT. He derived Eq. (3) be-low based on a number of simplifying assumptions, such as theturbulence was approximately isotropic, viscous stresses werenegligible and the flow contained negligible axial mean veloc-ity components:

2�r

∫ ∞

−∞� dz = d

dr

[r2�

∫ ∞

0(K2

t v + 2wu′w′) dz

], (3)

where K2t = v2 + u2 + w2 + u′2 + v′2 + w′2, u, v and w are

radial, axial and tangential mean velocities, respectively, u′, v′and w′ are instantaneous fluctuating velocity components.

Cutter (1966) used streak photography to measure local fluidvelocities and was able to justify his assumption that the ax-ial mean velocities were negligible compared to the radial andtangential components. He concluded that around 20% of thetotal kinetic energy is dissipated in the impeller region, 50% inthe impeller stream and 30% in the bulk. Although Gunkel andWeber (1975) used a similar approach to Cutter (1966), theyfound that most of energy was dissipated in the bulk of the tankand not in the impeller region or impeller discharge. The mainreason for this disagreement may be the difference in the D/T

ratio. Cutter (1966) used an impeller of D = 0.3 T whereasGunkel and Weber (1975) used an impeller diameter of 0.5 T.Furthermore, from 2-D PIV measurements of the flow field inRDT stirred flows (Khan, 2005) it has been found that the axialvelocity component is not small enough to be neglected. Theflows generated by the PBT have strong axial components andso Cutter’s Eq. (3) is not useful for performing a kinetic energybalance. A further criticism of Eq. (3) is that Cutter neglectedperiodic fluctuations in his decomposition of the instantaneousvelocity components and hence omitted some kinetic energytransfer terms between mean flow, turbulence and periodic mo-tion (Escudié and Liné, 2003). Hence, a more direct approach,in which a kinetic energy balance is performed using over amacroscopic region surrounding the impeller is sought, and isdescribed below.

Fig. 7 shows the control volume used for the kinetic energybalance carried out in the current work, based on the 3-D PIVdata for the pitched blade turbine. The kinetic energy flow inthe radial direction through surface BC of Fig. 7 is given by

KEr = 2��R

∫ z2

z1

Ku dz, (4)

where K is the kinetic energy obtained from the instantaneousvelocities:

K = 1

2((u + u′)2 + (v + v′)2 + (w + w′)2) (5)

and the overbar represents an angle-resolved mean velocity.Escudié and Liné (2003) discussed a triple decomposition

2804 F.R. Khan et al. / Chemical Engineering Science 61 (2006) 2799–2806

Axial IN

Axial

OUT

Radial

Flow

AB

C D

u

v

v

19.1

4.2

4.1

2.3

z1

z2z

ru

v

v

Fig. 7. Control volume over a PBT (all dimensions are in mm).

technique to split each instantaneous velocity component intoa mean velocity (U ), a periodic fluctuation due to the organ-ised blade passages (u) and a turbulent fluctuation (u′). Theythen derived kinetic energy balances for mean flow, organisedflow and turbulent flow. In principle, although Eq. (4) is amacroscopic balance, the decomposition is no different fromthat proposed by Escudié and Liné (2003), with u = U + u.In terms of the angle-resolved and fluctuating velocities, andfollowing averaging, Eq. (4) yields the angle-resolved kineticenergy given by

KEr = ��R

∫ z2

z1

(u3 + 3uu′2 + u′3 + uv2 + v′2u + 2vu′v′

+ v′2u′ + uw2 + w′2u + 2w · u′w′ + w′2u′) dz. (6)

Similarly for the axial flow of kinetic energy (AB and CD),

KEz = ��∫ r2

r1

(v3 + 3vv′2 + v′3 + u2v + u′2v + 2u · u′v′

+ v′u′2 + vw2 + w′2v + 2w · v′w′ + w′2v′)r dr . (7)

For the angle-resolved flows of kinetic energy in the radial andthe axial directions Eqs. (6) and (7) can be used. However, theseequations require calculation of second- and higher-order ve-locity cross-correlations, which are only readily available from3-D measurements, such as stereoscopic PIV. Typically, with2-D PIV or LDA measurements, the cross-correlation terms aresimply neglected (see for example, Zhou and Kresta (1996))and the kinetic energy flows from the radial and axial surfacesof the control volume are estimated from Eqs. (8) and (9), re-spectively:

KEr = ��r

∫ z2

z1

u(u2 + v2 + w2 + 3u′2 + v′2 + w′2) dz, (8)

KEz = ��∫ r2

r1

v(u2 + v2 + w2 + u′2 + 3v′2 + w′2)r dr . (9)

Wu and Patterson (1989) carried out a kinetic energy balanceacross a RDT impeller, using data obtained from LDA. Theflow of KE across the radial and axial planes was computedfrom Eqs. (8) and (9), respectively. Thus, after neglecting thecross-correlations terms, they found that about 30% of the totalkinetic energy was dissipated in the impeller region. The 3-D PIV from the current study will now be applied to check

the effect of neglecting these terms on the kinetic energy flowestimations.

Work by Jaworski and Fort (1991) considered potential en-ergy, kinetic energy and internal energy (expressed by staticpressure work) to analyse the kinetic energy dissipation distri-bution in a tank agitated by pitched blade turbines (six-bladed)of various sizes. A three-hole pitot tube was employed to mea-sure velocities and pressure (the pressure obtained from thepitot tube is the total of the static and dynamic pressures). Intheir analysis they considered only axial velocities and pressureprofiles and radial flow was neglected. However, a flow balancein the current PIV study indicates that the radial flows cannotbe neglected. Jaworski and Fort (1991) reported that 54% ofthe total input energy is dissipated in the region below the im-peller, 32% in the impeller region and 14% in the remainingvolume of the tank.

Zhou and Kresta (1996) carried out a kinetic energy bal-ance for three different impellers (RDT, PBT and Fluid foil im-peller A310) from the velocities measured by one-dimensionalLDA. They used the approximate Eqs. (8) and (9), as proposedby Wu and Patterson (1989), to estimate the energy dissipa-tion in the impeller region. For the RDT, they noticed that15% energy was dissipated in the impeller region, while forPBT it was 52%. They concluded that unlike in the RDT, thelocal dissipation for the PBT is a maximum in the impellerregion.

In the present work, the consistency of the 3-D PIV experi-mental data was first checked by performing a material balanceon the control volume shown in Fig. 7. For each blade position,all net flows were calculated as angle-resolved averages from300 sets of measurements across each section; they were thenaveraged over 360◦ of blade rotation to yield the time aver-age. The mass balance for the 360◦ average fields was in goodagreement between in and outflows, with only a 9% error asshown below:

total flow IN = 3860 cm3/s,

total flow OUT = 3540 cm3/s.

The power input to the impeller was 1.75 W at 300 rpm, mea-sured directly using strain gauges and an FM telemetry system;the power number was 1.52, averaged over 10 measurements.Fig. 8 shows the flows of kinetic energy into and out of the

F.R. Khan et al. / Chemical Engineering Science 61 (2006) 2799–2806 2805

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 20 40 60 80Blade Angle (°)

Ene

rgy

flow

(W

)

KE OUT (CD)-1

KE OUT (CD)-2

KE IN (AB)-1

KE IN (AB)-2

Net KE IN (BC)-1

Net KE IN (BC)-2

Fig. 8. Angle resolved flow of kinetic energy.

control volume for different blade angles. The effects ofneglecting the second- and higher-order velocity cross-correlations on the flows of kinetic energy are also shown onFig. 8: the dotted lines (denoted by −2 in the legend) repre-sents the kinetic energy flows estimated from the full sets ofEqs. (6) and (7), whereas the solid lines (denoted by −1 inthe legend) represent energy flows estimated after neglectingcorrelation terms via Eqs. (8) and (9).

The flows across BC contain both positive and negative radialcomponents and so the net kinetic energy flows are shown forthis surface in Fig. 8: Both methods of calculation are in goodagreement. The variations of axial kinetic energy flows throughAB and CD are almost the same using both methods. Thus,neglecting the second and higher correlation terms in Eqs. (6)and (7) does not significantly affect the mean + organised +turbulent kinetic energy flow estimations. It can be seen thatthe radial net flow of kinetic energy through BC is very small,compared to the axial flows. Thus, the kinetic energy mainlyenters the control volume through the upper surface with avertical downward velocity and flows out of the control volumethrough the lower surface with a vertical downward velocity.The kinetic energy flow leaving the lower edge of the bladedecreases to minimum at around a blade angle 20◦, which iswhere the vortex starts leaving the impeller region. The strongcirculations in the vortex, result in part of the axial flow re-entering into impeller region, reducing the net kinetic energyflow out of the blade. The kinetic energy flows in the axialdirections vary as the angle of blade rotation varies, whereasthe energy flows in the radial directions are not much affectedby the blade rotation.

Fig. 9 shows the total flows of kinetic energy in and out ofthe whole control volume. Averaging over all blade angles, thenet flow of KE entering into control volume was 0.41 W whilstaround 1.39 W was exiting from the control volume.

Net kinetic energy flow across the control volume = 0.98 W,

power input to the impeller = 1.75 W.

Thus power consumed within the control volume

surrounding the impeller = 0.77 W.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 20 40 60 80Blade angle (°)

Ene

rgy

flow

(W

)

Total KE OUT

Total KE IN

Average = 1.39 W

Average = 0.41 W

Fig. 9. Total flows of kinetic energy in and out.

By these calculations around 44% of the total power consumedby the impeller is dissipated in the impeller region. These resultsare similar to Zhou and Kresta (1996) findings for a pitchedblade turbine: 52% of the power was dissipated in the impellerregion. The small discrepancy however may be due to the dif-ferent size of control volume used by them and larger impellerdiameter (D = T/2) and larger impeller clearance (C = T/2)used than the present work (D = T/3 and C = T/3).

The volume occupied by the control volume is Vcontrol =0.0002089 m3, compared to a total vessel volume of 0.0192 m3.Thus, although tank volume average dissipation is 0.091 W/kg,the average dissipation of energy in the control volume is

�control = Pconsumed in control volume

�Vcontrol= 3.58 W/kg.

Thus, the average rate of dissipation of kinetic energy is about40 times higher in the impeller region than the average dis-sipation rate in the vessel. It should be noted that there willbe gradients of the dissipation rate within the impeller sweptvolume and so the maximum value of � may be significantlygreater than 40 times the mean.

4. Conclusions

Using stereoscopic PIV measurements, angle-resolved flow-fields close to a PBT impeller have been investigated at mea-surement planes 7.5◦ apart. Close to the impeller in the trailingvortices, the tangential velocity can be up to 67% of the tip ve-locity. A comparison of the angle-resolved mean velocities andthe radial and axial rms velocities shows fairly good agreementwith previously obtained measurements from 2-D PIV. Simi-larly, tangential velocity obtained from three-dimensional LDAcompared very favourably with the present data. Thus, both thein-plane and out-of-plane mean velocity fields obtained withthe stereoscopic system have been validated against previouswork.

The turbulent kinetic energies, calculated using thetwo-dimensional instantaneous velocity fields and a

2806 F.R. Khan et al. / Chemical Engineering Science 61 (2006) 2799–2806

pseudo-isotropic assumption, were compared with a more rig-orous method employing the full three-dimensional data. Thepseudo-isotropic method was found to be a reasonable assump-tion for the majority of the field in view. However, in the trailingvortices, the tangential rms velocities were higher than thosepredicted from the pseudo-isotropic assumption and hence k

was slightly underpredicted. Khan et al. (2004) showed thatfurther from the impeller, the flow is approximately isotropicand therefore the estimate of k using only 2-D velocity com-ponents is likely to be accurate throughout the tank.

A turbulent kinetic energy flow balance was carried outaround the PBT impeller. It was shown that inclusion of the fullset of velocity cross-correlations makes very little contributionto the flows of kinetic energy through a control volume. Bothsets of calculations showed around 44% of the total power con-sumed by the impeller is dissipated in the impeller region. Theaverage rate of dissipation of kinetic energy is 40 times higherin the impeller region than the average dissipation rate in thevessel.

Notation

C impeller offbottom clearanceD impeller diameterk turbulent kinetic energy, m2/s2

K kinetic energy, m2/s2

T tank diameteru, v, w instantaneous radial, axial and tangential

velocity, respectivelyu′, v′, w′ fluctuating radial, axial and tangential velocity,

respectivelyu, v, w angle-resolved radial, axial and tangential ve-

locity, respectivelyVtip impeller tip velocity, m/s

Acknowledgements

FRK and CDR gratefully acknowledge financial supportfrom Syngenta plc and thank Dr. Frans Muller and Prof. AlanHall of Syngenta plc for a number of useful discussions andhelpful suggestions.

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