Fractal Impeller Amol Kulkarni

Published: April 18, 2011

r 2011 American Chemical Society 7667 dx.doi.org/10.1021/ie200301y | Ind. Eng. Chem. Res. 2011, 50, 7667–7676

ARTICLE

pubs.acs.org/IECR

Fractal Impeller for Stirred Tank ReactorsAmol A. Kulkarni,* Neha Jha, Abhishek Singh, Sumit Bhatnagar, and Bhaskar D. Kulkarni

Chemical Engineering Division, National Chemical Laboratory, Pune—411 008, India

ABSTRACT: Stirred tank reactors are used for variety of applications at different scales of operation. The conventional impellerstend to develop regions having nonuniform energy dissipation rates in the stirred reactor. In this work, we propose a novel fractalimpeller, which helps in reducing such nonuniformities and help develop a uniform randomness throughout the reactor. Theimpeller geometry is discussed in detail. Experimental measurements of the power consumption, mixing time, suspension quality,and the ability for gas dispersion were carried out, and the performance is compared with the conventional impellers. The impeller isseen to have a low power number, and it can generate a uniform suspension of particles even at relatively lower impeller speeds andcan efficiently disperse gas into liquid to yield relatively higher gas hold-up values. The Fourier analysis of the power consumptiontime series data indicates that no specific prominent frequency events exist in the reactor, and the spectrum showed severalfrequency events to exist in the reactor with almost identical prominence.

1. INTRODUCTION

Stirred tank reactors (STR) form an integral component ofchemical, pharmaceutical, and the fermentation industry. Thesetypes of reactors are in operation for last several decades and anumber of investigators have analyzed them in detail to optimizethe designs based on the power consumption, mass and heattransfer, and the internal hydrodynamics. In the stirred reactors,the energy is supplied in the form of a kinetic energy by rotatingthe impeller at desired speed. STRs are largely used for (i) mixingor blending of two miscible liquids, (ii) generation of dispersionsfor gas�liquid and liquid�liquid reactions, (iii) keeping thesolid particles in suspension to facilitate the solid�fluid contactto achieve solid dissolution, (iv) crystallization, etc. The energyrequirement of these processes forms a significant part of thetotal energy and contributes toward major expenses. Thus, theefficiency of a stirred tank reactor mainly depends on the impellerdesign and its location in the stirred reactor. The variations in thedesign and operational protocols make the fluid mechanicsprevailing in a STR complex, and hence the design procedureshave largely been empirical. However, over the years detailedexperiments and theoretical analyses have contributed to someextent in making the operation of a STR more efficient.1 Ingeneral, the industrial inclination toward following the efficientdesign procedures or using efficient impellers has increased,while a large fraction of the industry still believes in stirring hardto achieve desired mixing with the same old generation impellers.

Typically (except for the highly viscous fluids), the systemoperates in turbulent regime. Usually, the distribution of energydissipation is considerably heterogeneous. Thus for instance, fora paddle mixer, 90% of the input energy is dissipated below theimpeller while the remaining 10% is dissipated above theimpeller.2 Also, for a pitched blade downflow turbine (PBTD),30% energy is dissipated in the impeller region, 57% below theimpeller and just 13% above the impeller.3 Usually, the impellerregion is the most active zone of the reactor and also a regionyielding high transient shear gradients. Thus, uniform spatialdistribution of energy is difficult to achieve in the conventionalSTRs and this implies that it is necessary to look for alternatives

that would make the entire reactor active in a hydrodynamicallysimilar manner. Also, for achieving uniform temperaturethroughout the reactor while operating it at lower impeller speedto avoid high shear zones (mainly for shear sensitive media), theconventional impellers may not be applicable.

In the present work, we propose a new impeller design thatoccupies less than 0.4% of the volume of the reactor, which issimilar to the conventional impeller system, but the design allowsit to spread over almost the entire vessel, yielding a structure withrelatively large voids. The objectives are (i) to achieve uniformitythroughout the stirred tank and (ii) to develop an innovative andefficient impeller that can yield better mixing and low shear atrelatively low power consumption. Both of these objectives canbe achieved using the principle of self-similarity, and hence afractal design will be more appropriate for an impeller. In thepresent work, we report on the comparative performance of suchfractal impellers (referred hereafter as FI) vis-a-vis the conven-tional ones. While several fractal geometries, configurations andresolutions can be used for such a concept, in this manuscript ouremphasis is on proving this concept and hence only one impellerdesign has been used for this study. In section 2 we havediscussed the experiments, details of the fractal impeller, andmeasurement techniques followed by observations and discus-sion in section 3.

2. EXPERIMENTAL SECTION

2.1. Fractal Impeller (FI). Conceptually, the self-similarity inthe geometry of an impeller at different scales can be expected toreplicate in the self-similar distribution of energy to achieveuniformity in the flow properties in a STR. It is known that formixing at small scale, generation of local chaotic advection bydifferent mechanisms including the mechanical movements

Received: February 12, 2011Accepted: April 18, 2011Revised: April 7, 2011

7668 dx.doi.org/10.1021/ie200301y |Ind. Eng. Chem. Res. 2011, 50, 7667–7676

Industrial & Engineering Chemistry Research ARTICLE

helps to achieve better mixing.4 Here we attempt to generatesuch chaotic advection by using a novel impeller which has self-similar geometrical features at different scales. The schematic ofthe impeller, the fabricated unit, and the setup are shown inFigure 1. The impeller has four main branches, each of whichfurther gets split in three sub-branches. On each of such sub-branch we have four blades. Of which 2 blades are horizontal andthe remaining two are vertical. Importantly, in the entire design,the orientation of the blades is kept such that none of the bladesactually sweep any liquid with them but simply fragment the fluidas they pass through it. An additional sub-branch at the bottom ofthe impeller helps to generate the necessary flow in the regionclose to the tank bottom. Also, for a given impeller rotationspeed, the angular distances covered by the blades vary and yieldvariation in the local blade passage velocity. However with theconfined nature of the entire system, such variations do not makesignificant effects on the flow uniformity. Since the design ofimpeller is expected to distribute energy in a more uniformmanner throughout the tank, it was thought desirable to char-acterize the performance of this concept. More details on thefractals are given in Appendix 1.2.2. Experimental Setup. The experiments were carried out

in an acrylic stirred tank (T = H = 0.3 m) with a single impellersystem. The vessel was fitted with four baffles (width,W =T/10).The impeller shaft was connected to a DC motor via a shaftmounted torque transducer. Experiments were carried out with

three different impellers: 6 blade-disk turbine, 6 blade-PBTD,and the FI. For DT and PBTD, the impeller diameter was D =T/3 = 100 mm, and the off-bottom clearance (C) was equal toT/3. The FI was supported from the bottom bymaking a countergroove on the shaft (Figure 1D) and for the FI, DFI = T/1.58. Asteel ball (6 mm diameter) was sandwiched between the bottomof the shaft and the center of the vessel bottom. The volume ofthe liquid inside the reactor was 21.2 L for all experiments. Thesolidity of all the impellers was maintained in a close range.2.3. Measurements. The experiments were carried out to

compare the performance of the FI with the conventionalimpellers, vis DT and PBTD. To facilitate such a comparison,the power consumption by the impeller, the mixing character-istics, and the efficacy of suspending the particles andmaking gas�liquid dispersion were considered as the measurable parameters.2.3.1. Power Consumption. Power draw can be measured

using various methods.5 In our experiments we used a rotarytorque transducer (CTime Sync, UK). These transducers arenoncontact optical devices, which function using the displace-ment principle causing a variation of volume of light. Dependingupon the extent of torsion experienced by the impeller shaftduring itsmotion, a proportional volume of light is generated by alow power demand solid state laser. This volume of light iscaptured by the optical components attached to the transducertorsion shaft, and the value helps us to know the torqueexperienced for a given impeller rotation speed. An in-built shaft

Figure 1. Schematic of the fractal impeller: (A) front view, (B) top view, (C) photograph of the fractal impeller, (D) photograph of the stirred tank withfractal impeller.



encoder helps to monitor the impeller rotation speed. Signalprocessing is done within the transducer, and the transducer canbe fixed either by base flange or in-line, between suitablecouplings. The torque data were acquired online on a PC usinga data acquisition system and were later subjected to Fourieranalysis to identify the possible dominant frequencies that wouldaffect the flow and which may be characteristics to the impeller.The FI impeller structure was given a support at the bottom. Itwas seated on a steel ball and was seen to have a very smoothmotion without offering any significant friction due to thecontact between the impeller bottom and the steel ball, and thusthe measured torque was entirely due to the friction experiencedby the impeller.2.3.2. Mixing Time. The mixing time was measured by giving a

tracer (of 0.3% of the total reactor volume) in the form ofconcentrated salt solution (1 M, NaCl in the form of pulse of) atthe liquid surface. The tracer concentration wasmeasured in timeusing the conductivity probe (connected to a standard conduc-tivity electrode with cell constant of 1.0 along with a digitalconductivity meter) fixed at a given location in the tank. Themixing time is considered as the time at which the measuredconcentration of the tracer reaches to within 95�98% of the finalconcentration. The transient variation in the concentration wasused for the estimation of θmix. In general, under turbulent flowconditions, θmix is inversely proportional to the impeller speed,and the product N 3 θmix known as dimensionless mixing time isused as a performance parameter.2.3.3. Solid�Liquid Suspension. The FI was also used for

checking its ability to suspend solid particles. Two different typesof particles were used: (i) resin particles (Fs = 1080 kg/m3) of theparticle size in the range of 350�500 μm and (ii) glass beadparticles (Fs = 2500 kg/m

3) of diameter 250 μm ((6 μm) in tapwater (FW≈ 1000 kg/m3). For the case of resin particles the localparticle concentration at different distances from the bottom ofthe tank was measured, and for the suspension of glass particles,cloud height was measured. A SS316 straight tube (4.5 mm o.d.and 3 mm i.d.) was used to collect the resin particles locally, andtheir mass was measured to estimate the local solid mass fraction.No external suction was used to capture the particles as thatwould affect the local flow.2.3.4. Gas�Liquid Dispersion. To study the performance of

the FI for dispersion of gas into liquid, experiments were carriedout by sparging compressed air in the stirred tank using a ringsparger located at the bottom of the reactor. The sparger had 16holes of 1 mm diameter spaced at equal distance. The superficialgas velocity was monitored and controlled using precalibratedrotameter. The power consumption during the stirring at differ-ent impeller rotation speeds and over a range of superficialvelocities was measured as mentioned before. The fractional gashold-up was estimated from the difference in the height ofdispersed liquid and clear liquid. The bubble size was estimatedfrom the images obtained from a high speed camera (Red lake).The observations from these experiments are discussed insection 3.4.

3. RESULTS AND DISCUSSIONS

3.1. Power Consumption. The actual power consumption(P) by the impeller was estimated using the measured torquedata (τ) at different impeller rotation speeds as P = 2πτN, whereN is the impeller rotation speed (per second). Subsequentlythe volumetric power draw (P/V) and the power consumption

per unit mass PW (W/kg) were calculated. The impeller powernumber NP was estimated as NP = P/(FLN

3D5), where FL is thebulk fluid density (estimated by taking into account the dispersedphase properties). Typical variation in PW with increasingimpeller Reynolds number (Re= ND2F/μ) showed power lawrelations (Figure 2A). Since the energy dissipation per unit massor the energy draw scales as N3D2 in the turbulent regime, theplot of PW vs N3D2 showed positive relationship for all the threeimpellers. Interestingly, while the linear relation exists for the DTand PBTD in the turbulent regime, for the FI, a linear variationwas noticed for the entire range of impeller rotation. While thevalues of the intercept for the linear straight line for DT andPBTD were very close (Figure 2B), the slope of the relation forDT was higher than that for PBTD. For the FI, the intercept aswell as the slope were very small as compared to the other twoimpellers. This indicates that the overall energy draw with the FIis much lesser than that for the other two impellers. In otherwords, at identical Re, PW for the FI is lower than the conven-tional impellers. Typical plot of power versus speed (not shownhere) also shows a change in the slope. The plots showed an earlytransition for the FI than the conventional impellers. However,with the varying distance of blades and branches, it may needsome other way to define the Re to characterize the conventionallaminar and turbulent regimes. For the present calculations, wehave used the actual lateral distance between the farthest bladesas the impeller diameter (D). On estimating the power numberfor these cases (Figure 2C), the NP value for a single DTindependent of Re was 6.014 (which is close to the value knownfor standard DT: 6), while for the PBTD of the same dimensionsand standard geometry, NP independent of Re was 1.84. How-ever, for identical rotation speed of the FI, the Re was muchhigher due to larger diameter, and the value ofNP independent ofRewas 0.38, which is much lower than theNP for DT and PBTD.Thus, beyond the critical Re, PW,FI is lower by many times that ofthe conventional impellers. One of the reasons for such lowerpower consumption is the lack of significant energy dissipationgradients in the tank as they exist for the conventional impellersand where the energy dissipation occurs due to the cascadingprocess originating from large eddies. The presence of uniformstructure helps to maintain similarity in the flow throughout thusmaking the energy distribution uniform. Moreover, since thereare no fluid sweeping blades, the formation of large eddies isavoided, and the impeller basically cuts the fluid continuously.This practically eliminates the typical energy dissipation andshear zones observed in conventional impellers. Also, the ab-sence of any wakes behind the blades helps further reduce thedrag and hence the energy consumption. Reduction in thecontact area of the impeller also helps to decrease the extent ofform and skin drag. The design yields stream lines that wouldfollow the flow separation over the blades and interaction withother streamlines in the compartments formed due to self-similarfeature. This specifically reduces the value of form drag to a greatextent, and also the possibility of any wake formation behind theblade is almost zero. However, continuous passage of blades inthe same plane helps develop local circulation zones restricted tothe blade dimensions thereby creating several similar localcirculating zones that interact with each others. More detailedwork on flow visualization of the zones and the interactingcirculation zones is in progress. Different design alternativeswith varied blade angles, etc. may yield better flow, and moreefforts on understanding the effect of design of the FI on theperformance are under investigation. To further characterize the



performance of FI, we studied the liquid�solid suspension,gas�liquid dispersion, and the mixing time.3.2. Suspension. Usually, the flow pattern from an axial flow

impeller is conducive to easier suspension than that of by a radialflow impeller, while the mixed flow impellers show an inter-mediate performance. Suspension of solids in liquid in a stirredtank reactor has been studied over many decades, and certainguidelines on the selection of suitable impeller is known.6�8

Typically weak recirculation induced loops occur just below theimpeller and also at the junction of the tank base and the wall. Forthe case of the impeller operating close to the tank base, theefficiency of energy transfer from impeller to particles is max-imum. The particulate mass trapped in the stagnant zone belowthe impeller is, therefore, easily driven to the corners with enoughvelocity to get suspended. If the off-bottom clearance of theimpeller is increased, then the stagnant zone below the impelleralso increases and more particles get accumulated there. In suchcases, higher impeller rotation speed would be needed to lift theparticles from the bottom and then get completely suspended atfurther greater impeller rotation speeds.The flow generated by the FI is largely a tangential flow as all

the blades simply cut the fluid in different planes therebyavoiding any possibility of sweeping or pushing the fluid in itspath. Thus, the flow separation over the blades is a prominent

phenomena, and the fluid interacting with different rotatingzones mix with each other. This results in a strong tangentialflow at the bottom of the impeller, and thus helps to lift theparticles while pushing them toward the wall; however, oncethese particles are lifted, they are trapped in the rotating structurewhich keeps the particle floating between different zones. Also,the velocity gradients in the vicinity of the blade were seen to helpget the particles lifted in the direction perpendicular to themotion of the blade. For different suspension densities of theresin particles (as mentioned in section 2), the value of PW wasseen to increase with increasing impeller Re, (Figure 3A) and atall suspension densities, the following relation was seen to hold:

PW ¼ C1ε1:34Re1:5 ð1Þ

where the value of C1 is 2.63� 10�7 and would strongly dependon the physical properties of the suspended particles (volume,density, shape, etc.). Re was estimated using the fluid density andviscosity at different solid mass fraction ε (%). With increasing ε(%), Re continued to decrease, and the corresponding variationin the estimated NP values is shown in Figure 3B. Interestingly,on comparing the data with similar solid concentration for astirred tank of 1 m diameter with PBTD (filled symbols)(Sardeshpande et al.9), the FI showed much lower power

Figure 2. (A) Variation in the power consumed per unit mass (PW) with impeller Re for different impellers under consideration, (B) PW vs N3D2,(C) NP vs log (Re).



numbers even with solid particles. Importantly, the Re range forwhich the complete suspension was achieved using a FI was atleast less than 50% than that of PBTD. Here, by completesuspension, we refer to the situation where the particles aresuspended in the entire liquid phase and there remains only anegligible fraction at the bottom of the tank. However completesuspension does not mean uniform spatial distribution of parti-cles. In addition to PW, uniformity in the solid concentration inthe suspension would help to quantify the performance of thisimpeller. To understand the level of suspension (particle cloud)in the liquid, the local concentration of solid particles at variouslevels from the bottom of the tank was measured at differentimpeller rotation speeds. The variation in the local particle massfraction is shown in Figure 4. For all solid loadings, for less than70 rpm, most of the particles were close to the bottom and werefar from being lifted. At higher impeller rotation speed, the localsolid mass fraction was well suspended with a standard deviationof (3%. Thus, for all ε values, an impeller rotation speed of100 rpm was sufficient to keep all the particles in suspendedcondition. On achieving complete suspension, for ε = 5%, thelocal solid concentration decreased slightly from bottom to topof the stirred tank, while for ε = 7%, it was slightly higher towardthe bottom as well as at top of the tank. In general, the observationsindicated that once the solid particles are lifted from the bottom,increasing energy input to the reactor by increasing impeller speedprimarily helps in dispersing the particles, achieving a less nonuni-form suspension. It would be interesting to track the particlemotion throughout the tank, and such experiments are in progress.The relatively large volume of the FI results in a better effect in

keeping the particles suspended. Importantly, the presence ofmultiple blades in the section close to the bottom develops astrong tangential flow, which helps the particles to experience liftin the direction perpendicular to themotion of the blades. On theother hand, the localized vortex generated due to the motion ofthe blades perpendicular to the bottom helped lift the particles inthe center of the stirred tank. This vortex was seen to have aperiodic behavior and details will be studied by measuring the

local velocity field. Further, unlike the circulation cells that getdeveloped in the tank with conventional impellers (DT andPBTD), in the presence of multiple blades and self-similarbehavior at different levels of the geometry of the FI, no suchcirculation cycles were visible. As a result, the particles lifted fromthe bottom remain mostly floating between different branches ofthe impeller thereby reducing the extent of nonuniformity in thesuspension quality. At any given tank cross section, at differentimpeller rotation speeds, the variation in the particle mass waslargely due to the local regions generated by the interaction offlow domains from different blades in different branches. Withincreasing particle loading, the extent of uniformity in thesuspended particle concentration also increased. In the case ofa PBTD, two different circulation loops, one below and anotherabove the impeller are established. Unlike this, in the FI, highersolidity practically breaks such loops and develops several net-works of zones which interact with each other. This also helps tokeep the particles suspended while being transferred from onezone to another self-similar zone. More detailed experiments ontracking of particles between zones would help to quantify theresidence time of particles in different zones and therebycharacterize the energy distribution in the stirred reactor withsuch an impeller. Such details are being investigated and will bereported separately.In another set of experiments, the performance of FI for

suspending solid glass particles was studied. The Pw variation fordifferent glass particle suspension densities is shown inFigure 5A. While the nature of plots is similar to that of Figure 3,the value of Pw at similar impeller rotation speed is almost twicethat of the suspension of particles with density 1069 kg/m3. Thisextent of difference is almost equivalent to the settling velocity ofthese particles in water, which is proportional to their densities.Onmeasuring the cloud height for suspension of glass particles, itwas seen that the extent of lifting of the particles in the bulkincreased with increasing suspension density. This implies thatthe increase in power consumption at higher suspension den-sities was indeed utilized in suspending particles. Visual

Figure 3. (A) PW for fractal impeller for different solid loadings in the tank. The values of Re at different solid concentrations are estimated based on theslurry density and viscosity. (B) Variation in Np with impeller Re for different solid loadings. Open symbols belong to FI while the filled symbolscorrespond to PBTD.9 For the FI, the experiments were limited for lower impeller speed as the particles were completed suspended and there was noneed for experiments at higher Re.



observations showed that at very low suspension density, theparticles always remained in the lower half of the impeller. Thiswas largely because the particles were seen to remain entrappedin the smaller mixing zones formed by the blades of fractalimpeller. This observation was also seen for higher suspensiondensity, but at lower impeller rotation speed (<100 rpm) theparticles were seen to get aligned in a peculiar manner at the tankbottom, in two lines each connecting the diagonally oppositebaffles. On increasing impeller speed, they eventually get sus-pended. This indicates that at lower N, the confluence of radialflow, tangential flow, and the presence of baffles makes theparticles to assemble in a specific manner. At higher suspensiondensity (5%), the particles were seen to get easily suspended,which is not very common largely because of the variation in thebulk property which helped the particles get suspended easily.Equation 1 was seen as valid even for the suspension of glassparticles with the value of C1 = 5.5 � 10�7.The performance of suspending identical glass particles in a

stirred tank using FI andPBTD(in a large tank having identicalT/Hratio10) is shown in Figure 5B. The observations for three different

solid concentrations can be summarized as follows: (i) at 1% solidconcentration, PBTD (filled symbols) performs much better thanFI (open symbols) in suspending particles even at very low impellerspeed; (ii) at 3% and 5% solid concentration, the power required forlifting of particles with PBTD is relatively lower than that of FI.Withglass particles, the terminal velocity being higher, while achievingcomplete suspension was possible at lower Pw, achieving uniformsuspension needed relativelymuchhigher power. (iii)With 5% solidfraction, the FI is efficient in suspending particles at higherconcentrations; (iv) the trend in the efficiently suspending theparticles at different solid concentrations for PBTD and for FI areexactly opposite. The values of PW for PBTD were estimated fromthe values of Re, D reported in Sardeshpande et al.10 and NP forPBTD in turbulent region. These observations indicate that thisnovel impeller design is useful in efficiently suspending particles athigher solid loadings, which is not very easy with the conventionalimpellers. Importantly, the solid concentration for glass particlesalong the height (measured in similar manner as for low densityparticles) (not shown here) when the dimensionless cloud height is1 was very much uniform with a standard deviation of 6%.

Figure 4. Variation in the solid phase mass fraction at different impeller rotation speeds and at different solid loadings from bottom to the top ofthe tank.



3.3. Mixing Time. Before we bring out the experimentalobservations, it is necessary to highlight a few important issuesthat relate impeller diameter and mixing time θmix. In general, fora given tank diameter θmix � 1/D2. Thus, variation in theimpeller diameter modifies the flow pattern (relative magnitudesof convective and turbulent motions and relative magnitudes ofaxial and radial mean components) and hence also the mixingefficacy. This also means that for the conventional impellersystems, impellers with larger D and lower NP are morebeneficial.11 However, it is necessary to avoid a too large impeller

diameter that can inhibit the secondary flow. Also, since gen-erating a relatively larger radial component of the mean velocityresults into large energy dissipation at the wall and thus lowersthe mixing efficiency, it is always preferred to avoid such asituation. Further, a larger impeller diameter demands highertorque and hence higher capital cost. Hence, the selection shouldbe made on the basis of capital and operating costs. While theseobservations are valid for the conventional impellers, it does notnecessarily apply for the FI. Hence experiments were carried outto understand the characteristic mixing time for a FI.The mixing time was measured as described in the section 2.

The conductivity signal was smoothened to eliminate thespurious effects due to data acquisition noise, and the smooth-ened signal was analyzed to measure the mixing time. The mixingtime at identical N for PBTD was 2 to 3.5 times higher than theFI. This particular situation can again be explained on the basis ofthe existence of fractal structure which develops self-similar flowstructures in the entire vessel and hence a uniform randomness.As a result, the tracer gets continuously distributed in severalmixing zones existing in the reactor due to the fractal structure ofthe impeller, and it helps achieve better mixing. However acomparison of the θmix variation as a function of the PW showsthat both impellers have similar performance (Figure 6). Experi-ments were also carried out to measure the mixing of a tracerliquid in the bulk viscous liquid (50% glycerol solution). In thiscase, the mixing time was 40% higher than that of water, and thisobservation was consistent over the entire range of impellerrotation speed for FI. Mixing time was alsomeasured for differentsolid loadings for the suspensions and the dimensionless mixingtime showed a positive dependence on the solid loading. Thisobservation is consistent with an increase in the solid loading; thebulk viscosity and density increase thereby leading to enhancedviscous forces, higher drag, and hence a longer mixing time.The fact that the presence of number of blades and mixing

zones would create a uniform randomness was verified by takingthe fast Fourier transforms of the acquired time series of thetorque data. The resulting power spectra for one such experimentwithN = 100 rpm is shown in Figure 7. It can be clearly seen that

Figure 6. Mixing characteristics of fractal impeller (θmix vs Pw).Theopen symbols indicate the dimensionless mixing time and the closedsymbols indicate the actual mixing time (s).

Figure 5. (A) PW for FI for different solid loadings of glass particles inthe tank. The values of Re at different solid concentrations are estimatedon the basis of the slurry density and viscosity. (B) Dimensionless cloudheight vs power consumption per unit mass of the contents for the caseof FI and PBTD11 for glass particles of identical size. Open symbolscorrespond to FI, while closed symbols are for PBTD.



unlike the literature information on variety of conventionalimpellers, where the impeller rotation frequency, blade passagefrequency are prominently seen in the power spectra, in the caseof a FI no specific dominance was seen. The power distributionover a range of frequencies showed similar features and thussupport the notion that with the help of such a self-similarstructure for mixing of fluids, one can attain a uniform random-ness in the flow at different scales, and no specific instabilities(associated with certain frequency)12 exist that are usuallyconsidered to promote spatial mixing. Thus, the scaling effectscan be minimized by achieving local mixing effects, and theprinciple of self-similarity can be maintained to achieve similarperformance even in the scale-up of such systems.More work on understanding the effect of the extent and

distribution of void volumes and the interzonal mixing on theperformance of the impeller designs is in progress. The concepthas been shown to achieve uniformmixing even at lower impellerspeeds thereby reducing the shear rates throughout and can givea fresh look to the STR design used for the synthesis ofbiomaterials and specialty chemicals.Figure 7. Fourier spectrum of the transient torque data forN = 100 rpm

Figure 8. Performance of FI for a gas�liquid dispersion: (A) power consumption vs VG, (B) RPD vs impeller rotation speed, and (C) RPD vs flownumber.



3.4. Gas�Liquid Dispersion. On measuring the power con-sumption per unit volume of the reactor, it was seen that thevalue of PW decreased continuously with increasing superficialgas velocity. For the case of conventional impellers, the gasloading reduces the power consumption due to the formation ofcavities behind the impeller blades which modified the pumpingaction of the impeller. For the case of fractal impeller as well, theextent of reduction in PWG was higher for higher impellerrotation speed as well as at higher gas flow rate (Figure 8A).The relative power demand (RPD)1,13 estimated as PWG/PW at agiven VG was seen to go through a maximum (Figure 8B). Theoverall values of RPD decreased with increasing VG. The plot ofRPD vs flow number (NQ = Q/ND3) clearly indicated a locusalong which the RPD goes through a maximum value (Figure 8C).Also, the RPD corresponding to the point of inflection decreasedwith increasing gas velocity. Values of RPD were seen to get wellcorrelated (not shown here) with Fl�0.2Fr�0.25, with the pro-portionality constant as a function of the dimensionless group(N 3VG/g), where g is the acceleration due to gravity. The value ofproportionality constant deviated from the 0.18 as reported forother standard impellers. One of the reasons for such a deviation isthe spatial distribution of impeller blades, which will not supportthe use of a single value of impeller diameter. Thus, for example,the Froude number (Fr = N2D/g) which depends on the tipvelocity (ND) of the impeller: the variation in the tip distance fromblade to blade will now allow a fixed value of the proportionalityconstant in estimating the RPD based on Fl and Fr. Interestingly, inthe FI, none of the blades actually sweep the fluid with it, and hencethe possibility of formation of cavities is almost negligible. Howeverfor the case of gas�liquid dispersion, a thin gas layer was seen to staybelow the horizontally flat blades yielding a compressible boundarylayer thatwould help reduce the drag.This further helps in reductionin the actual power consumption.On comparing the variation in theFr vs Fl for a FI with that of a standard Rushton turbine, it is quiteevident that for a FI, the flow is still in the regime where cavities areunder formation and the possibility of flooding may occur at highergas velocities. Such a comparison may look unrealistic as thebehavior of FI and that of a Rushton turbine are verymuch different.On estimating the fractional gas hold-up at different impeller

Re, the hold-up was seen to go through a maximum. However at

all impeller rotation speeds, an increase in gas velocity resulted inan increase in hold-up (Figure 9). The measured average bubblesize decreased with increasing power consumption, which sub-sequently yielded higher effective interfacial area. Importantly, itwas seen that the uniform dissipation of energy throughout thereactor helped yield very narrow bubble-size distribution. Theeffectiveness of the reduction in bubble size and increased hold-up on the overall mass transfer coefficient is being studiedseparately. The purpose of this work was to demonstrate thefeasibility of a new impeller design and not to bring outcorrelations that would help estimate different hydrodynamicproperties. Since the FI geometry that we have reported here isjust one such design/configuration of blades and energy dis-tribution, a detailed analysis on the quantitative dependence ofthe hydrodynamic properties on the fractal structure (design ofimpeller, blade configuration, nature of fractal structure, etc.) isunder progress.

4. CONCLUSIONS

A new type of impeller with self-similar features is proposedand demonstrated. The performance of the FI was comparedwith the conventional impellers (DT and PBTD) on the basis ofmeasured power consumption, mixing characteristics, and theefficacy in suspending the particles. The power number of FI wasrelatively lower than the other standard impellers. The self-similar structure leading to reduced drag in the absence of anypossibility of wake formation behind the impeller blades helps togenerate a uniform randomness throughout the stirred tank.Importantly, at identical N, although the Re for FI would behigher than that of a PBTD or DT, in reality the flow is laminar.For suspensions, while the low density particles were seen to getcompletely suspended even at very low impeller rotation speed,the suspension of high density particles required only twice theamount of power for identical solid loading and impeller Re. TheFI when used for gas�liquid dispersion showed that relativepower demand continues to decrease with increasing impellerrotation speed as well as the superficial gas velocity. The bubblesize distribution was very much narrow throughout the reactorsupporting the hypothesis of possible uniformity in spatial energy

Figure 9. Hydrodynamics of a stirred reactor with FI for gas�liquid system. (A) Variation in the fractional gas hold-up at different impeller rotationspeeds and at different superficial gas velocities, (B) Variation in the average bubble size with power consumption at different superficial gas velocities.



dissipation. Different design alternatives with varied blade angles,etc. may yield better flow but at relatively higher power con-sumption. More details on the effect of design of FI on theperformance for different applications are under investigation.

’APPENDIX 1

Fractals: Since their discovery by Mandelbrot,12 fractals haveexperienced considerable success inquantifying the complex structureexhibited by many scientific and engineering problems. Fractals aredisordered systems described in terms of noninteger dimensions, andthe disorder is their intrinsic property. The fractals are characterizedby the property of geometrical self-similarity and independency at anyscale. The fractal character of an object/region/domain can bequantified by a parameter called the fractal dimension, D, whichquantifies the fractal scaling relationship between the patterns,observed at different magnifications. Fractal dimension can be bestcalculated by the box counting method, which means drawing a boxof size R and counting the mass inside (M). On plotting ln(M) vsln(R),we get a value ofDmwhich is the fractal dimension.A vast bodyof literature on fractals, their features, mathematical analysis, themethods of characterization, and finally their applications in variousfields may be referred for details.14�18 A detailed account of thefractals in flow and turbulence can be seen in Sreenivasan.19

’AUTHOR INFORMATION

Corresponding Author*Tel.: þ91-20-25902153. Fax: þ91-20-25902621. E-mail:[email protected].

’ACKNOWLEDGMENT

Among the authors, A.S. (Indian Institute of Technology,Kharagpur) and S.B. (University of Minnesota) thank theirrespective Chemical Engineering Departments for encouragingthem to do their summer training at the National ChemicalLaboratory, Pune. N.J. (NIT, Surathkal) thanks the SummerResearch Fellowship from the JNSCR, Bangalore.

’NOTATIONSC = off bottom clearance (m)D = impeller diameter (m)Fl = flow number (Q/ND3)Fr = Froud number (N2D/g)g = acceleration due to gravity (m/s2)N = impeller rotation speed (per second)NP = impeller power number (-)P = power consumption (w)PW = power consumption per unit mass (W/kg)PWG = power consumption per unit mass with gas sparging

(W/kg)Q = gas flow rate (m3/s)Re = impeller Reynolds number (ND2F/μ)T = tank diameter (m)ε = mass fraction of solid phase (�)FL = bulk fluid density (kg/m3)τ = measured torque (N.m)

SubscriptsDT = disk turbinePBTD = pitched blade impeller downwardFI = fractal impeller

’REFERENCES

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(13) Paglianti, A.; Fujasova, M.; Montante, G. A simple model forpower consumption in gassed and boiling stirred vessels. AIChE J. 2008,54 (3), 646.

(14) Mandelbrot, B. B. The fractal Geometry of Nature; Freeman:New York, 1977.

(15) Geake, J.; Landini, G. Individual differences in the perception offractal curves. Fractals 1997, 5, 129–43.

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Fractal Impeller Amol Kulkarni

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